Layering as Optimization Decomposition: A Mathematical Theory of Network Architectures

Transcription

1 INVITED PAPER Layering a Optimization Decompoition: A Mathematical Theory of Network Architecture There are variou way that network functionalitie can be allocated to different layer and to different network element, ome being more deirable than other. The intellectual goal of the reearch urveyed by thi article i to provide a theoretical foundation for thee architectural deciion in networking. By Mung Chiang, Member IEEE, Steven H. Low, Senior Member IEEE, A. Robert Calderbank, Fellow IEEE, and John C. Doyle ABSTRACT Network protocol in layered architecture have hitorically been obtained on an ad hoc bai, and many of the recent cro-layer deign are alo conducted through piecemeal approache. Network protocol tack may intead be holitically analyzed and ytematically deigned a ditributed olution to ome global optimization problem. Thi paper preent a urvey of the recent effort toward a ytematic undertanding of Blayering[ a Boptimization decompoition,[ where the overall communication network i modeled by a generalized network utility maximization problem, each layer correpond to a decompoed ubproblem, and the interface among layer are quantified a function of the optimization variable coordinating the ubproblem. There can be many alternative decompoition, leading to a choice of different Manucript received July 24, 2005; revied September 6, The work at Princeton Univerity and Caltech that are ummarized in thi paper were upported by the National Science Foundation (NSF) Grant ANI , EIA , CNS , CNS , CCF , CCF , CNS , CNS , CNS , CCF , and CNS , by the Air Force Office of Scientific Reearch (AFOSR) Grant F and FA , by the ARO Grant DAAD , by the Defene Advanced Reearch Project Agency (DARPA) Grant HR and CBMANET program, and by the Cico Grant GH M. Chiang i with the Electrical Engineering Department, Princeton Univerity, Princeton, NJ USA ( chiangm@princeton.edu). S. H. Low i with the Computer Science and Electrical Engineering Department, California Intitute of Technology, Paadena, CA USA ( low@caltech.edu). A. R. Calderbank i with the Electrical Engineering and Mathematic Department, Princeton Univerity, Princeton, NJ USA ( calderbk@math.princeton.edu). J. C. Doyle i with the Control and Dynamic Sytem, California Intitute of Technology, Paadena, CA USA ( doyle@cd.caltech.edu). Digital Object Identifier: /JPROC /$25.00 Ó2007 IEEE layering architecture. Thi paper urvey the current tatu of horizontal decompoition into ditributed computation, and vertical decompoition into functional module uch a congetion control, routing, cheduling, random acce, power control, and channel coding. Key meage and method ariing from many recent work are ummarized, and open iue dicued. Through cae tudie, it i illutrated how BLayering a Optimization Decompoition[ provide a common language to think about modularization in the face of complex, networked interaction, a unifying, top-down approach to deign protocol tack, and a mathematical theory of network architecture. KEYWORDS Ad hoc network; channel coding; computer network; congetion control; cro-layer deign; ditributed algorithm; feedback control; game theory; Internet; Lagrange duality; medium acce control (MAC); network utility maximization (NUM); optimization; power control; revere-engineering; routing; cheduling; tochatic network; tranmiion control protocol (TCP)/Internet protocol (IP); wirele communication I. INTRODUCTION A. Overview 1) Structure of the Layered Protocol Stack: Network architecture determine functionality allocation: Bwho doe what[ and Bhow to connect them,[ rather than jut reource Vol. 95, No. 1, January 2007 Proceeding of the IEEE 255

2 allocation. It i often more influential, harder to change, and le undertood than any pecific reource allocation cheme. Functionality allocation can happen, for example, between the network management ytem and network element, between end-uer and intermediate router, and between ource control and in-network control uch a routing and phyical reource haring. The tudy of network architecture involve the exploration and comparion of alternative in functionality allocation. Thi paper preent a et of conceptual framework and mathematical language for a foundation of network architecture. Architecture have been quantified in field uch a information theory, control theory, and computation theory. For example, the ource-channel eparation principle i a fundamental reult on architecture in information theory. The choice of architectural deciion are even more complicated in networking. For example, the functionality of rate allocation among competing uer may be implemented through variou combination of the following control: end-to-end congetion control, local cheduling, per-hop adaptive reource allocation, and routing baed on end-to-end or per-hop action. However, we do not yet have a mature theoretical foundation of network architecture. Layered architecture form one of the mot fundamental tructure of network deign. They adopt a modularized and often ditributed approach to network coordination. Each module, called layer, control a ubet of the deciion variable, and oberve a ubet of contant parameter and the variable from other layer. Each layer in the protocol tack hide the complexity of the layer below and provide a ervice to the layer above. Intuitively, layered architecture enable a calable, evolvable, and implementable network deign, while introducing limitation to efficiency and fairne and potential rik to manageability of the network. There i clearly more than one way to Bdivide and conquer[ the network deign problem. From a dataplane performance point of view, ome layering cheme may be more efficient or fairer than other. Examining thee choice of modularized deign of network, we would like to tackle the quetion of Bhow to[ and Bhow not to[ layer. While the general principle of layering i widely recognized a one of the key reaon for the enormou ucce of data network, there i little quantitative undertanding to guide a ytematic, rather than an ad hoc, proce of deigning layered protocol tack for wired and wirele network. One poible perpective to undertand layering i to integrate the variou protocol layer into a ingle theory, by regarding them a carrying out an aynchronou ditributed computation over the network to implicitly olve a global optimization problem modeling the network. Different layer iterate on different ubet of the deciion variable uing local information to achieve individual optimality. Taken together, thee local algorithm attempt to achieve a global objective. Such a deign proce can be quantitatively undertood through the mathematical language of decompoition theory for contrained optimization [104]. Thi framework of BLayering a Optimization Decompoition[ expoe the interconnection between protocol layer a different way to modularize and ditribute a centralized computation. Even though the deign of a complex ytem will alway be broken down into impler module, thi theory will allow u to ytematically carry out thi layering proce and explicitly tradeoff deign objective. The core idea in BLayering a Optimization Decompoition[ are a follow. Different vertical decompoition of an optimization problem, in the form of a generalized network utility maximization (NUM), are mapped to different layering cheme in a communication network. Each decompoed ubproblem in a given decompoition correpond to a layer, and certain function of primal or Lagrange dual variable (coordinating the ubproblem) correpond to the interface among the layer. Horizontal decompoition can be further carried out within one functionality module into ditributed computation and control over geographically diparate network element. Since different decompoition lead to alternative layering architecture, we can alo tackle the quetion of Bhow and how not to layer[ by invetigating the pro and con of decompoition method. Furthermore, by comparing the objective function value under variou form of optimal decompoition and uboptimal decompoition, we can eek Beparation theorem[ among layer: condition under which layering incur no lo of optimality. Robutne of thee eparation theorem can be further characterized by enitivity analyi in optimization theory: how much will the difference in the objective value (between different layering cheme) fluctuate a contant parameter in the generalized NUM formulation are perturbed. There are two intellectually freh cornertone behind BLayering a Optimization Decompoition.[ The firt i Bnetwork a an optimizer.[ The idea of viewing protocol a a ditributed olution (to ome global optimization problem in the form of the baic NUM) ha been uccefully teted in the trial for tranmiion control protocol (TCP) [56]. The key innovation from thi line of work (e.g., [64], [72], [73], [87], [89], [90], [96], [116], [125], and [161]) i to view the TCP/IP network a an optimization olver, and each variant of congetion control protocol a a ditributed algorithm olving a pecified baic NUM with a particular utility function. The exact hape of the utility function can be revere-engineered from the given protocol. In the baic NUM, the objective i to the um of ource utilitie a function of rate, the contraint are linear flow contraint, and optimization variable are ource rate. Other recent reult alo how how to revere-engineer border gateway protocol (BGP) a a olution to the table path problem [44], and contention-baed medium acce control (MAC) protocol 256 Proceeding of the IEEE Vol.95,No.1,January2007

3 a a game-theoretic elfih utility maximization [76], [78]. Starting from a given protocol originally deigned baed on engineering heuritic, revere-engineering dicover the underlying mathematical problem being olved by the protocol. Forward-engineering baed on the inight obtained from revere-engineering then ytematically improve the protocol. The econd key concept i Blayering a decompoition.[ A will be dicued in Section I-A2, generalized NUM problem can be formulated to repreent a network deign problem involving more degree of freedom than jut the ource rate. Thee generalized NUM problem put the end-uer utilitie in the Bdriver eat[ for network deign. For example, benefit of innovation in the phyical layer, uch a better modulation and coding cheme, are now characterized by the enhancement to application rather than jut the drop in bit-error rate (BER), which the uer do not directly oberve. Note that an optimal olution to a generalized NUM formulation automatically etablihe the benchmark for all layering cheme. The problem itelf doe not have any predetermined layering architecture. Indeed, layering i a human engineering effort. The overarching quetion then become how to attain an optimal olution to a generalized NUM in a modularized and ditributed way. Vertical decompoition acro functional module and horizontal decompoition acro geographically diparate network element can be conducted ytematically through the theory of decompoition for nonlinear optimization. Implicit meage paing (where the meage have phyical meaning and may need to be meaured anyway) or explicit meage paing quantifie the information haring and deciion coupling required for a particular decompoition. There are many way to decompoe a given problem, each of which correpond to a different layering architecture. Even a different repreentation of the ame NUM problem may lead to different decompoability tructure even though the optimal olution remain the ame. Thee decompoition have different characteritic in efficiency, robutne, aymmetry of information and control, and tradeoff between computation and communication. Some are Bbetter[ than other depending on the criteria et by network uer and operator. A ytematic exploration in the pace of alternative decompoition i poible, where each particular decompoition lead to a ytematically deigned protocol tack. Given the layer, croing layer i tempting. A evidenced by the large and ever growing number of paper on cro-layer deign over the lat few year, we expect that there will be no hortage of cro-layer idea baed on piecemeal approache. The growth of the Bknowledge tree[ on cro-layer deign ha been exponential. However, any piecemeal deign jointly over multiple layer doe not bring a more tructured thinking proce than the ad hoc deign of jut one layer. What eem to be lacking i a level ground for fair comparion among the variety of cro-layer deign, a unified view on how and how not to layer, and fundamental limit on the impact of layer-croing on network performance and robutne metric. BLayering a Optimization Decompoition[ provide a candidate for uch a unified framework. It advocate a firt-principled way to deign protocol tack. It attempt at hrinking the Bknowledge tree[ on cro-layer deign rather than growing it. It i important to note that BLayering a Optimization Decompoition[ i not the ame a the generic phrae of Bcro-layer optimization.[ What i unique about thi framework i that it view the network a the optimizer itelf, put the end-uer application need a the optimization objective, etablihe the globally optimal performance benchmark, and offer a common et of methodologie to deign modularized and ditributed olution that may attain the benchmark. There have been many recent reearch activitie along the above line by reearch group around the world. Many of thee activitie were inpired by the eminal work by Kelly et al. in 1998 [64], which initiated a freh approach of optimization-baed modeling and decompoition-baed olution to implify our undertanding of the complex interaction of network congetion control. Since then, thi approach ha been ubtantially extended in many way, and now form a promiing direction toward a mathematical theory of network architecture. Thi paper 1 provide a ummary of the key reult, meage, and methodologie in thi area over the lat 8 year. Mot of the urveyed work focu on reource allocation functionalitie and performance metric. The limitation of uch focu will alo be dicued in Section V. 2) NUM: Before preenting an overview of NUM in thi ection, we emphaize the primary ue of NUM in the framework of BLayering a Optimization Decompoition[ a a modeling tool, to capture end-uer objective (the objective function), variou type of contraint (the contraint et), deign freedom (the et of optimization variable), and tochatic dynamic (reflected in the objective function and contraint et). Undertanding architecture (through decompoition theory), rather than computing an optimum of a NUM problem, i the main goal of our tudy. The Baic NUM problem i the following formulation [64], known a Monotropic Programming and tudied ince the 1960 [117]. TCP variant have recently been revere-engineered to how that they are implicitly olving thi problem, where the ource rate vector x 0ithe 1 Variou abridged verion of thi urvey have been preented in 2006 at the Conference of Information Science and Sytem, IEEE Information Theory Workhop, and IEEE MILCOM. Two other horter, related tutorial can be found in [85] and [105]. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 257

4 only et of optimization variable, and the routing matrix R and link capacity vector c are both contant U ðx Þ ubject to Rx c: (1) Utility function U areoftenaumedtobemooth,increaing, concave, and dependent on local rate only, although recent invetigation have removed ome of thee aumption for application where they are invalid. Many of the paper on BLayering a Optimization Decompoition[ are pecial cae of the following generic problem [18], one of the poible formulation of a Generalized NUM for the entire protocol tack: U ðx ; P e; Þþ V j ðw j Þ j ubject to Rx cðw; P e Þ; x 2C 1 ðp e Þ; x 2C 2 ðfþ or 2 /ðwþ; R 2R; F 2F; w 2W: (2) Here, x denote the rate for ource and w j denote the phyical layer reource at network element j. The utility function U and V j may be any nonlinear, monotonic function. R i the routing matrix, and c are the logical link capacitie a function of both phyical layer reource w and the deired decoding error probabilitie P e. For example, the iue of ignal interference and power control can be captured in thi functional dependency. The rate may alo be contrained by the interplay between channel decoding reliability and other hop-by-hop error control mechanim like Automatic Repeat Requet (ARQ). Thi contraint et i denoted a C 1 ðp e Þ. The iue of ratereliability tradeoff and coding i captured in thi contraint. The rate are further contrained by the medium acce ucce probability, repreented by the contraint et C 2 ðfþ, where F i the contention matrix, or, more generally, the chedulability contraint et /. The iue of MAC (either random acce or cheduling) i captured in thi contraint. The et of poible phyical layer reource allocation cheme, of poible cheduling or contentionbaed medium acce cheme, and of ingle-path or multipath routing cheme are repreented by W, F, and R, repectively. The optimization variable are x, w, P e, R, and F. Holding ome of the variable a contant and pecifying ome of thee functional dependencie and contraint et will then lead to a pecial cla of thi generalized NUM formulation. Utility function and contraint et can be even more general than thoe in problem (2), poibly at the expene of loing pecific problem tructure that may help with finding ditributed olution. A determinitic fluid model i ued in the above formulation. Stochatic network dynamic change the NUM formulation in term of both the objective function and the contraint et. A will be dicued in Section V-D, tochatic NUM i an active reearch area. Whether modeled through a baic, general, or tochatic NUM, there are three eparate tep in the deign proce of BLayering a Optimization Decompoition:[ Firt formulate a pecific NUM problem, then devie a modularized and ditributed olution following a particular decompoition, and finally explore the pace of alternative decompoition that provide a choice of layered protocol tack. The following quetion naturally arie: How to pick utility function, and how to guarantee quality-of-ervice (QoS) to uer? Firt of all, in revere-engineering, utility function are implicitly determined by the given protocol already, and are to be dicovered rather than deigned. In forwardengineering, utility function can be picked baed on any combination of the following four conideration: Firt, a in the firt paper [122] that advocated the ue of utility a a metric in networking, elaticity of application traffic can be repreented through utility function. Second, utility can be defined by human pychological and behavioral model uch a mean opinion core in voice application. Third, utility function provide a metric to define optimality of reource allocation efficiency. Fourth, different hape of utility function lead to optimal reource allocation that atify well etablihed definition of fairne (e.g., a r of -fair utilitie parameterized by 0: UðxÞ ¼ð1 Þ 1 x 1 [96] can be proved to be an -fair reource allocation). In general, depending on who i intereted in the outcome of network deign, there are two type of objective function: um of utility function by end uer, which can be function of rate, reliability, delay, jitter, power level, etc., and a network-wide cot function by network operator, which can be function of congetion level, energy efficiency, network lifetime, collective etimation error, etc. Utility function can be coupled acro the uer, and may not have an additive tructure (e.g., network lifetime). Maximizing a weighted um of all utility function i only one of the poible formulation. An alternative i multiobjective optimization to characterize the Paretooptimal tradeoff between the uer objective and the operator objective. Another et of formulation, which i not covered in thi urvey, i game-theoretic between uer and operator, or among uer or operator themelve. While utility model lead to objective function, the contraint et of a NUM formulation incorporate the following two type of contraint. Firt i the collection of 258 Proceeding of the IEEE Vol.95,No.1,January2007

5 Table 1 Summary of 10 Key Meage phyical, technological, and economic retriction in the communication infratructure. Second i the et of peruer, hard, inelatic QoS contraint that cannot be violated at the equilibrium. Thi i in contrat to the utility objective function, which may repreent elatic QoS demand of the uer. Given a generalized NUM formulation, we do not wih to olve it through centralized computation. Intead, we would like to modularize the olution method through decompoition theory. Each decompoed ubproblem control only a ubet of variable (poibly a calar variable), and oberve only a ubet of contant parameter and value of other ubproblem variable. Thee correpond, repectively, to the limited control and obervation that each layer ha. The baic idea of decompoition i to divide the original large optimization problem into maller ubproblem, which are then coordinated by a mater problem by mean of ignaling. Mot of the exiting decompoition technique can be claified into primal decompoition and dual decompoition method. The former i baed on decompoing the original primal problem, wherea the latter i baed on decompoing the Lagrange dual of the problem. Primal decompoition method have the interpretation that the mater problem directly give each ubproblem an amount of reource that it can ue; the role of the mater problem i then to properly allocate the exiting reource. In dual decompoition method, the mater problem et the price for the reource to each ubproblem which ha to decide the amount of reource to be ued depending on the price; the role of the mater problemithentoobtainthebetpricingtrategy. Mot paper in the vat, recent literature on NUM ue a tandard dual-decompoition-baed ditributed algorithm. Contrary to the apparent impreion that uch a decompoition i the only poibility, there are in fact many alternative to olve a given NUM problem in different but all ditributed manner [104], including multilevel and partial decompoition. Each of the alternative provide a poibly different network architecture with different engineering implication. Coupling for generalized NUM can happen not only in contraint, but alo in the objective function, where the utility of ource, U ðx ; fx i g i2iðþ Þ, depend on both it local rate x and the rate of a et of other ource with indice in et IðÞ. If U i an increaing function of fx i g i2iðþ, thi coupling model cooperation, for example, in a clutered ytem, otherwie it model competition, uch a power control in wirele network or pectrum management in digital ubcriber loop (DSL). Such coupling in the objective function can be decoupled through Bconitency price[ [130]. 3) Key Meage and Methodologie: The ummary lit of key meage in Table 1 illutrate the conceptual implicity in thi rigorou and unifying framework, which i more important than any pecific cro-layer deign derived from thi framework. In Table 2, the ummary lit of main method developed in many recent publication aim at popularizing thee analytical technique o that future reearch can invoke them readily. Each method will be ummarized in a tand-alone paragraph at the end of the aociated development or explanation. Section II and III cover the revere- and forwardengineering apect for both horizontal and vertical decompoition, a outlined in Table 3. After preenting the main point of horizontal and vertical decompoition, we turn to a more general dicuion on decompoition method in Section IV. At thi point, curiou reader may tart to raie quetion, for example, on the iue involving tochatic network dynamic, the difficultie aociated with nonconvex optimization formulation, the coverage of accurate model, the comparion metric for decompoition alternative, the engineering implication of aymptotic convergence, and the jutification of performance optimization in the firt place. Some of thee quetion have recently been anwered, while other remain underexplored. Indeed, there are many challenging open problem and intereting new direction in thi emerging reearch area, and they will be outlined in Section V. In concluding thi opening ection, we highlight that, more than jut an enemble of pecific cro-layer deign for exiting protocol tack, BLayering a Optimization Decompoition[ i a mentality that view network a Vol. 95, No. 1, January 2007 Proceeding of the IEEE 259

6 Table 2 Summary of 20 Main Method Surveyed optimizer, a common language that allow reearcher to quantitatively compare alternative network architecture, and a uite of methodologie that facilitate a ytematic deign approach for modularized and ditributed network architecture. Notation: Unle otherwie pecified, vector are denoted in boldface mall letter, e.g., x with x i a it ith component; matrice are denoted by boldface capital letter, e.g., H, W, R; and et of vector or matrice are denoted by cript letter, e.g., W n, W m, R n, R m. Inequalitie between two vector denote component-wie inequalitie. We will ue the term Buer,[ Bource,[ Beion,[ and Bconnection[ interchangeably. Due to the wide coverage of material in thi urvey paper, notational conflict occaionally arie. Conitency i maintained within any ection, and main notation i ummarized in the table of notation for each ection: Table 4 9. B. From Theory to Practice 1) Optimization: Linear programming ha found important application in communication network for everal decade. In particular, network flow problem, i.e., minimizing linear cot ubject to linear flow conervation and capacity contraint, include important pecial cae uch a the hortet path routing and maximum flow problem. Recently, there have been many reearch activitie that utilize the power of recent development in nonlinear convex optimization to tackle a much wider cope of problem in the analyi and deign of communication ytem. Thee reearch activitie are driven by both new demand in the tudy of communication and networking, and new tool emerging from optimization theory. In particular, a major breakthrough in optimization over the lat two decade ha been the development of powerful theoretical tool, a well a highly efficient computational algorithm like the interior-point method, for nonlinear convex optimization, i.e., minimizing a convex function (or maximizing a concave function a i often een in thi paper) ubject to upper bound inequality contraint on other convex function and affine equality contraint minimize f 0 ðxþ ubject to f i ðxþ 0; i ¼ 1; 2;...; m Ax ¼ a (3) where the variable are x 2 R n. The contant parameter are A 2 R ln and a 2 R l. The objective function f 0 to be minimized and the m contraint function f i are convex function. Since the early 1990, it ha been recognized that the waterhed between efficiently olvable optimization Table 3 Organization of Section II and III 260 Proceeding of the IEEE Vol.95,No.1,January2007

7 problem and intractable one i convexity. It i well known that for a convex optimization problem, a local minimum i alo a global minimum. The Lagrange duality theory i alo well-developed for convex optimization. For example, the duality gap i zero under contraint qualification condition, uch a Slater condition [9] that require the exitence of a trictly feaible olution to nonlinear inequality contraint. When put in an appropriate form with the right data tructure, a convex optimization problem can alo be efficiently olved numerically, uch a the primal-dual interior-point method, which ha wort-cae polynomial-time complexity for a large cla of function and cale gracefully with problem ize in practice. Special cae of convex optimization include convex quadratic programming, econd-order cone programming, and emidefinite programming [9], a well a eemingly nonconvex optimization problem that can be readily tranformed into convex problem, uch a geometric programming [19]. The lat decade ha witneed the appreciation-application cycle for convex optimization, where more application are developed a more people tart to appreciate the capabilitie of convex optimization in modeling, analyzing, and deigning communication ytem. When tackling the much more difficult nonconvex optimization problem, there are ome claical approache, which have been enhanced by new one in recent year. The phrae Boptimization of communication ytem[ in fact carrie three different meaning. In the mot traight-forward way, an analyi or deign problem in a communication ytem may be formulated a minimizing a cot, or maximizing a utility function, or determining feaibility over a et of variable confined within a contraint et. Decompoition, robutne, and fairne, in addition to optimality of the olution, can then be tudied on top of the optimization model. In a more ubtle and recent approach, emphaized in Section II, a given network protocol may be interpreted a a ditributed algorithm olving an implicitly defined global optimization problem. In yet another approach, the underlying theory of a network control method or a communication trategy may be generalized uing nonlinear optimization technique, thu extending the cope of applicability of the theory. In addition to optimization theory and ditributed algorithm theory, the reult urveyed here alo naturally borrow tool from feedback control theory, tochatic network theory, game theory, and general market equilibrium theory. They are alo connected with other branche of mathematic, uch a algebraic geometry and differential topology. 2) Practice: Indutry adoption of BLayering a Optimization Decompoition[ ha already tarted. For example, inight from revere-engineering TCP have led to an improved verion of TCP in the FAST Project (Fat AQM Scalable TCP) [56], [57], [146], [147]. Putting end-uer application utilitie a the objective function ha led to a new way to leverage innovation in the phyical and link layer beyond the tandard metric uch a BER, e.g., in the BFAST Copper[ Project (here FAST tand for frequency, amplitude, pace, time) for an order-of-magnitude boot to rate in fiber/dsl broadband acce ytem [38]. FAST TCP [37] i a joint project between computer cience, control and dynamic ytem, electrical engineering, and phyic department at Caltech and UCLA, and involve partner at variou national laboratorie around the world. It integrate theory, algorithm, implementation, and experiment o that they inform and influence each other intimately. It goal i to undertand the current TCP congetion control, deign new algorithm, implement and tet them in real high-peed global network. Through revere-engineering, a will be dicued in Section II-A2, the NUM and duality model allow u to undertand the limitation of the current TCP and deign new algorithm. Until about ix year ago, the tate of the art in TCP reearch had been imulation-baed uing implitic cenario, with often a ingle bottleneck link and a ingle cla of algorithm. We now have a theory that can predict the equilibrium behavior of a TCP-like algorithm in any arbitrary network topology. Moreover, we can prove, and deign, their tability propertie in the preence of feedback delay for large cale network. A explained in detail in Section II-A5, the inight from thi erie of theoretical work have been implemented in a oftware prototype FAST TCP and it ha been ued to break world record in data tranfer in the lat few year. FAST Copper [38] i a joint project at Princeton Univerity, Stanford Univerity, and Fraer Reearch Intitute, aiming at providing at leat an order-of-magnitude increae in DSL broadband acce peed, through a joint optimization of frequency, amplitude, time, and pace dimenion to overcome the attenuation and crotalk bottleneck in today DSL ytem. One of the key idea i to treat the DSL network a a multiple-input multipleoutput (MIMO) ytem rather than a point-to-point channel, thu leveraging the opportunitie of multiuer cooperation and mitigating the current bottleneck due to multiuer competition. Another key idea i to leverage burtine of traffic over broadband acce network under QoS contraint. The overarching reearch challenge i to undertand how to engineer the functionality allocation acro module and network element. BLayering a Optimization Decompoition[ provide a framework for thee deign iue in the interference environment of fiber/dsl broadband acce. Clean-late deign of the entire protocol tack i another venue of application of BLayering a Optimization Decompoition.[ For example, Internet 0 [54] i a project initiated at the Center for Bit and Atom at MIT and jointly purued by an indutrial conortium. It goal i to develop theory, algorithm, protocol, and implementation to connect a large number of mall device. 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8 Table 4 Summary of Main Notation for Section II-A opportunity of clean-late protocol tack deign for wirele ad hoc network i the control-baed MANET program by DARPA. Eventually, BLayering a Optimization Decompoition[ may even be ued to develop computeraided deign tool for protocol deign and implementation. There are alo other potential point of interaction between the theoretical foundation urveyed in thi paper and indutry practice, e.g., ditributed joint rate and power control through decompoition theory by cellular network infratructure vendor [47], and Bviibility[ acro layer enabled by ervice provider [70]. II. HORIZONTAL DECOMPOSITION It i well-known that phyical layer algorithm try to olve the data tranmiion problem formulated by Shannon: maximizing data rate ubject to the contraint of aymptotically vanihing error probability. Widely ued network protocol, uch a TCP, BGP, and IEEE DCF, were intead deigned baed primarily on engineering intuition and ad hoc heuritic. Recent progre ha put many protocol in layer 2 4 (of the tandard even-layer reference model) on a mathematical foundation a well. The congetion control functionality of TCP ha been revere-engineered to be implicitly olving the baic NUM problem [87], [88], [125]. While heterogeneou congetion control protocol do not olve an underlying NUM problem, their equilibrium and dynamic propertie can till be analyzed through a vector field repreentation and the Poincare Hopf index theorem [134], which together how that bounded heterogeneity implie global uniquene and local tability of network equilibrium. Interior gateway protocol of IP routing i known to olve variant of the hortet path problem, and the policy-baed routing protocol in BGP ha recently been modeled a the olution to the table path problem [44]. Scheduling-baed MAC protocol are known to olve variant of maximum weight matching problem [12], [14], [80], [121], [149], [150], [165] or 262 Proceeding of the IEEE Vol.95,No.1,January2007

9 graph-coloring problem [114] (and the reference therein), and random acce (contention-baed MAC) protocol have recently been revereengineered a a noncooperative utility maximization game [76], [78]. In Section II-A and II-B, the revere- and forwardengineering reult for TCP congetion control and random acce MAC are ummarized, repectively. A. TCP Congetion Control 1) Congetion Control Protocol: Congetion control i a ditributed mechanim to hare link capacitie among competing uer. In thi ection, a network i modeled a a et L of link (carce reource) with finite capacitie c ¼ðc l ; l 2 LÞ. They are hared by a et N of ource indexed by. Each ource ue a et LðÞ L of link. Let SðlÞ ¼f 2 Njl 2 LðÞg be the et of ource uing link l. The et flðþg define an L N routing matrix 2 R l ¼ 1; if l 2 LðÞ, i.e.,ourceue link l 0; otherwie. U ðx Þ a a function of it rate x. Conider the baic NUM propoed in [64] U ðx Þ ubject to Rx c (4) and it Lagrangian dual problem [90] minimize L0 DðLÞ :¼ max x 0! U ðx Þ x R l l þ l l c l l : (5) We now preent a general model of congetion control algorithm and how that they can be interpreted a ditributed algorithm to olve NUM (4) and it dual (5). Let y l ðtþ ¼ P R lx ðtþ be the aggregate ource rate at link l and let q ðtþ ¼ P l R l l ðtþ be the end-to-end price for ource. Invectornotation,wehave Aociated with each ource i it tranmiion rate x ðtþ at time t, in packet/econd. Aociated with each link l i a calar congetion meaure l ðtþ 0attimet. Wewillcall l ðtþ the link (congetion) price. A congetion control algorithm conit of two component: a ource algorithm that dynamically adjut it rate x ðtþ in repone to price l ðtþ in it path, and a link algorithm that update, implicitly or explicitly, it price l ðtþ and end it back, implicitly or explicitly, to ource that ue link l. On the current Internet, the ource algorithm i carried out by TCP, and the link algorithm i carried out by (active) queue management (AQM) cheme uch a DropTail or RED [43]. Different protocol ue different metric to meaure congetion, e.g., TCP Reno [55], [127] and it variant ue lo probability a the congetion meaure, and TCP Vega [10] and FAST [56], [147] ue queueing delay a the congetion meaure [89]. Both are implicitly updated at the link and implicitly fed back to ource through end-to-end lo or delay, repectively. Mathematical model for congetion control tarted [25] immediately after the development of the protocol, and many of the reult ince 1999 follow the approach advocated in [64] and focu on average model of the congetion avoidance phae in TCP. In thi ection, we how that a large cla of congetion control algorithm can be interpreted a ditributed algorithm to olve a global optimization problem. Specifically, we aociate with each ource a utility function 2 We abue notation to ue L and N to denote both et and their cardinalitie. and yðtþ ¼RxðtÞ qðtþ ¼R T LðtÞ: Here, xðtþ ¼ðx ðtþ; 2 NÞ and qðtþ ¼ðq ðtþ; 2 NÞ are in R N þ,andyðtþ ¼ðy lðtþ; l 2 LÞ and LðtÞ ¼ð l ðtþ; l 2 LÞ are in R L þ. In each period, the ource rate x ðtþ and link price l ðtþ are updated baed on local information. Source can oberve it own rate x ðtþ and the end-to-end price q ðtþ of it path, but not the vector LðtÞ, nor other component of xðtþ or qðtþ. Similarly, link l can oberve jut local price l ðtþ and flow rate y l ðtþ. The ource rate x ðtþ are updated according to x ðt þ 1Þ ¼F ðx ðtþ; q ðtþþ (6) for ome nonnegative function F. The link congetion meaure l ðtþ i adjuted in each period baed only on l ðtþ and y l ðtþ, and poibly ome internal (vector) variable v l ðtþ, uch a the queue length at link l. Thi can be modeled by ome function ðg l ; H l Þ:foralll l ðt þ 1Þ ¼ G l ðy l ðtþ; l ðtþ; v l ðtþþ (7) v l ðt þ 1Þ ¼ H l ðy l ðtþ; l ðtþ; v l ðtþþ (8) Vol. 95, No. 1, January 2007 Proceeding of the IEEE 263

10 where G l are nonnegative o that l ðtþ 0. Here, F model TCP algorithm (e.g., Reno or Vega) and ðg l ; H l Þ model AQM (e.g., RED, REM). We will often refer to AQM by G l, without explicit reference to the internal variable v l ðtþ or it adaptation H l. We now preent ome example. TCP Reno/RED: The congetion control algorithm in the large majority of current TCP implementation i (an enhanced verion of ) TCP Reno, firt propoed in [55]. A ource maintain a parameter called window ize that determine the number of packet it can tranmit in a round-trip time (RTT), the time from ending a packet to receiving it acknowledgment from the detination. Thi implie that the ource rate i approximately equal to the ratio of window ize to RTT, in packet per econd. The baic idea of (the congetion avoidance phae of ) TCP Reno i for a ource to increae it window by one packet in each RTT and halve it window when there i a packet lo. Thi can be modeled by (ee, e.g., [72], [87]) the ource algorithm F ðt þ 1Þ :¼ F ðx ðtþ; q ðtþþ to accommodate the equilibrium queue length o that Vega ource can converge to the unique equilibrium. In thi cae, there i no packet lo in equilibrium. Unlike TCP Reno, Vega ue queueing delay a congetion meaure l ðtþ ¼b l ðtþ=c l, where b l ðtþ i the queue length at time t. The update rule G l ðy l ðtþ; l ðtþþ i given by (dividing both ide of (10) by c l ) l ðt þ 1Þ ¼ l ðtþþ y þ lðtþ 1 : (13) c l Hence, AQM for Vega doe not involve any internal variable. The update rule F ðx ðtþ; q ðtþþ forourceratei given by x ðt þ 1Þ ¼x ðtþþ 1 T 2ðtÞ 1 ð d x ðtþq ðtþþ (14) F ðt þ 1Þ ¼ x ðtþþ 1 T 2 2 þ 3 q ðtþx 2 ðtþ (9) where T i the RTT of ource, i.e.,thetimeittakefor to end a packet and receive it acknowledgement from the detination. Here we aume T i a contant even though in reality it value depend on the congetion level and i generally time-varying. The quadratic term reflect the property that, if rate double, the multiplicative decreae occur at twice the frequency with twice the amplitude. The AQM mechanim of RED [43] maintain two internal variable, the intantaneou queue length b l ðtþ and average queue length r l ðtþ. They are updated according to b l ðt þ 1Þ ¼½b l ðtþþy l ðtþ c l Š þ (10) r l ðt þ 1Þ ¼ð1! l Þr l ðtþþ! l b l ðtþ (11) where! l 2ð0; 1Þ. Then, (the Bgentle[ verion of) RED mark a packet with a probability l ðtþ that i a piecewie linear, increaing the function of r l ðtþ with contant 1, 2, M l, b l,andb l 8 0; r l ðtþ b l >< 1 ðr l ðtþ b l Þ; b l r l ðtþ b l l ðtþ ¼ 2 r l ðtþ b l þ Ml ; b l r l ðtþ 2b l >: 1; r l ðtþ 2b l. (12) Equation (10) (12) define the model ðg; HÞ for RED. TCP Vega/DropTail: A duality model of Vega ha been developed and validated in [89]; ee alo [96]. We conider the ituation where the buffer ize i large enough where i a parameter of Vega, d i the round-trip propagation delay of ource, and1ðzþ ¼1ifz 9 0, 1 if z G 0, and 0 if z ¼ 0. Here T ðtþ ¼d þ q ðtþ i the RTT at time t. FAST/DropTail: The FAST algorithm i developed in [56], [57], and [147]. Let d denote the round-trip propagation delay of ource. Let l ðtþ denote the queueing delay at link l at time t. Letq ðtþ ¼ P l R l l ðtþ be the round-trip queueing delay, or in vector notation, qðtþ ¼R T LðtÞ. Each ource adapt it window W ðtþ periodically according to d W ðtþ W ðt þ 1Þ ¼ d þ q ðtþ þ þð1 ÞW ðtþ (15) where 2ð0; 1Š and 9 0 i a protocol parameter. A key departure from the model decribed above and thoe in the literature i that, here, we aume that a ource end rate cannot exceed the throughput it receive. Thi i jutified becaue of elf-clocking: within one RTT after a congetion window i increaed, packet tranmiion will be clocked at the ame rate a the throughput the flow receive. A conequence of thi aumption i that the link queueing delay vector LðtÞ i determined implicitly by the intantaneou window ize in a tatic manner: given W ðtþ ¼W for all,thelinkqueueingdelay l ðtþ ¼ l 0foralll are given by R l W d þ q ðtþ where again q ðtþ ¼ P l R l l ðtþ. ¼ c l ; if l ðtþ 9 0 c l ; if l ðtþ ¼0 (16) 264 Proceeding of the IEEE Vol.95,No.1,January2007

11 Hence, FAST i defined by the dicrete-time model (15), (16) of window evolution. The ending rate i then defined a x ðtþ :¼ W ðtþ=ðd ðtþþq ðtþþ. 2) Revere-Engineering: Congetion Control a Ditributed Solution of Baic NUM: Under mild aumption on ðf; G; HÞ, it can be hown uing Kakutani fixed point theorem that equilibrium ðx; LÞ of (6) (8) exit and i unique [96], [134]. The fixed point of (6) define an implicit relation between equilibrium rate x and end-toend congetion meaure q x ¼ F ðx ; q Þ: Aume F i continuouly differentiable =@q 6¼ 0 in the open et A :¼ fðx ; q Þjx 9 0; q 9 0g. Then, by the implicit function theorem, there exit a unique continuouly differentiable function f from fx 9 0g to fq 9 0g uch that q ¼ f ðx Þ 9 0: (17) To extend the mapping between x and q to the cloure of A, define f ð0þ ¼inffq 0jF ð0; q Þ¼0g: (18) If ðx ; 0Þ i an equilibrium point F ðx ; 0Þ ¼x,thendefine f ðx Þ¼0: (19) Define the utility function of each ource a Z U ðx Þ¼ f ðx Þdx ; x 0 (20) which i unique up to a contant. Being an integral, U i a continuou function. Since f ðx Þ¼q 0forallx, U i nondecreaing. We aume that f i a nonincreaing functionvthemoreeverethe congetion, the maller the rate. Thi implie that U i concave. If f i trictly decreaing, then U i trictly concave ince U 00ðx Þ G 0. An increaing utility function model a greedy ource (a larger rate yield a higher utility) and it concavity model diminihing marginal return. We aume the following condition: C1: For all 2 S and l 2 L, F and G l are nonnegative function. F are continuouly differentiable =@q 6¼ 0infðx ; q Þjx 9 0; q 9 0g; moreover, f in (17) are trictly decreaing. C2: R ha full row rank. C3: If l ¼ G l ð y l ; l ; v l Þ and v l ¼ H l ð y l ; l ; v l Þ, then y l c l, with equality if l 9 0. Condition C1 guarantee that ðxðtþ; LðtÞÞ 0 and ðx ; L Þ0, and that utility function U exit and are trictly concave. C2 guarantee uniquene of equilibrium price vector L. C3 guarantee the primal feaibility and complementary lackne of ðx ; L Þ. We can regard congetion control algorithm (6) (8) a ditributed algorithm to olve the NUM (4) and it dual (5) [87]. Theorem 1: Suppoe aumption C1 and C2 hold. Then (6) (8) ha a unique equilibrium ðx ; L Þ.Moreover,it olve the primal problem (4) and the dual problem (5) with utility function given by (20) if and only if C3 hold. Hence, the variou TCP/AQM protocol can be modeled a different ditributed olution ðf; G; HÞ to olve (4) and it dual (5), with different utility function U. Theorem 1 characterize a large cla of protocol ðf; G; HÞ that admit uch an interpretation. Thi interpretation i the conequence of end-to-end control: it hold a long a the end-to-end congetion meaure to which the TCP algorithm react i the um of the contituent link congetion meaure, and that the link price are independent of ource (thi would not be true in the heterogeneou protocol cae a in Section II-A4). Note that the definition of utility function U depend only on TCP algorithm F.TheroleofAQMðG; HÞ i to enure that the complementary lackne condition (condition C3) of problem (6) (8) i atified. The complementary lackne ha a imple interpretation: AQM hould match input rate to capacity to utilization at every bottleneck link. Any AQM that tabilize queue poee thi property and generate a Lagrange multiplier vector L that olve the dual problem. The utility function of everal propoed TCP algorithm turn out to belong to a imple cla of function defined in [96] that i parameterized by a calar parameter 0 U ðx Þ¼ w log x ; ¼ 1 w ð1 Þ 1 x 1 ; 6¼ 1 where weight w 9 0. In particular, it ha been hown that TCP Vega, FAST, and Scalable TCP correpond to ¼ 1, HTCP to ¼ 1:2, TCP Reno to ¼ 2, and maxmin fairne to ¼1. Maximizing -fair utility lead to Vol. 95, No. 1, January 2007 Proceeding of the IEEE 265

12 optimizer that atify the definition of -fair reource allocation in the economic literature. Method 1: Revere-Engineering Cooperative Protocol a a Ditributed Algorithm Solving a Global Optimization Problem. The potential and rik of network come from the interconnection of local algorithm. Often, intereting and counter-intuitive behavior arie in uch a etting where uer interact through multiple hared link in intricate and urpriing way. Revere-engineering of TCP/AQM ha alo led to a deeper undertanding of throughput and fairne behavior in large cale TCP network. For example, there i a general belief that one can deign ytem to be efficient or fair, but not both. Many paper in the networking, wirele, and economic literature provide concrete example in upport of thi intuition. The work in [132] prove an exact condition under which thi conjecture i true for general TCP network uing the duality model of TCP/AQM. Thi condition allow u to produce the firt counter-example and trivially explain all the upporting example found in the literature. Surpriingly, in ome counter-example, a fairer throughput allocation i alway more efficient. It implie for example that maxmin fair allocation can achieve higher aggregate throughput on certain network. Intuitively, we might expect that the aggregate throughput will alway rie a long a ome link increae their capacitie and no link decreae their. Thi turn out not to be the cae, and [132] characterize exactly the condition under which thi i true in general TCP network. Not only can the aggregate throughput be reduced when ome link increae it capacity, more trikingly, it can alo be reduced even when all link increae their capacitie by the ame amount. Moreover, thi hold for all fair bandwidth allocation. Thi paradoxical reult eem le urpriing in retropect: according to the duality model of TCP/AQM, raiing link capacitie alway increae the aggregate utility, but mathematically there i no a priori reaon that it hould alo increae the aggregate throughput. If all link increae their capacitie proportionally, however, the aggregate throughput will indeed increae, for -fair utility function. 3) Stability of Ditributed Solution: Theorem 1 characterize the equilibrium tructure of congetion control algorithm (6) (8). We now dicu it tability. We aume condition C1 and C2 in thi ection o that there i a unique equilibrium ðx ; L Þ.Inthiection,analgorithm i aid to be locally aymptotically table if it converge to the unique equilibrium tarting from a neighborhood of the equilibrium, and globally aymptotically table if it converge tarting from any initial tate. Global aymptotic tability in the preence of feedback delay i deirable but generally hard to prove. Mot paper in the literature analyze global aymptotic tability in the abence of feedback delay, or local tability in the preence of feedback delay. Proof technique that have been ued for global aymptotic tability in the abence of feedback delay include Lyapunov tability theorem, gradient decent method, paivity technique, and ingular perturbation theory. In the following, we ummarize ome repreentative algorithm and illutrate how thee method are ued to prove their tability in the abence of feedback delay. For analyi with delay, ee, e.g., [102], [103], [140], and [141] for local tability of linearized ytem and [90], [106], [107], and [115] for global tability; ee alo urvey in [63] [91], and [125] for further reference. In particular, unlike the Nyquit argument, [107] handle nonlinearity and delay with Lyapunov functional. Conider the algorithm (uing a continuou-time model) of [64] _x ¼ x ðtþ U 0 ð x ðtþþ q ðtþ (21) l ðtþ ¼g l ðy l ðtþþ (22) where 9 0 i a contant of gain parameter. Thi i called a primal-driven algorithm, which mean that there i dynamic only in the ource control law but not the link control law. To motivate (21) and (22), note that q ðtþ i the unit price for bandwidth that ource face end-toend. The marginal utility U 0ðx ðtþþ can be interpreted a ource it willingne to pay when it tranmit at rate x ðtþ. Then, according to (21), ource increae it rate (demand for bandwidth) if the end-to-end bandwidth price i le than it willingne to pay, and decreae it otherwie. Since g l i an increaing function, the price increae a the aggregate demand for bandwidth at link l i large. To prove that (21) and (22) are globally aymptotically table, conider the function VðxÞ :¼ U ðx Þ l Z y l Uing (21) and (22), it i eay to check that 0 g l ðzþdz: (23) _V :¼ d dt V ð xðtþ 9 0; for all xðtþ 6¼ x Þ ¼ ¼ 0; if xðtþ ¼x where x i the unique r of the trictly concave function VðxÞ. HenceVðxÞ i a Lyapunov function for the dynamical ytem (21), (22), certifying it global aymptotic tability. The function VðxÞ in (23) can be interpreted a the penalty-function verion of the NUM (4). Hence the algorithm in (21) and (22) can alo be thought of a a gradient acent algorithm to olve the approximate NUM. 266 Proceeding of the IEEE Vol.95,No.1,January2007

13 Method 2: Lyapunov Function Contruction to Show Stability. A dual-driven algorithm i propoed in [90]. where U 0 1 l ðt þ 1Þ ¼½ l ðtþþ 0 ðy l ðtþ c l ÞŠ (24) x ðtþ ¼U 0 1 ðq ðtþþ (25) i the invere of U 0. The algorithm i derived a the gradient projection algorithm to olve the dual (5) of NUM. The ource algorithm (25) i called the demand function in economic: the larger the end-to-end bandwidth price q ðtþ, the maller the demand x ðtþ. Thelink algorithm (24) i the law of upply and demand (for variable demand and fixed upply in thi cae): if demand y l ðtþ exceed upply, increae the price l ðtþ; otherwie, decreae it. By howing that the gradient rdðlþ of the dual objective function in (5) i Lipchitz, it i proved in [90] that, provided the tepize 0 i mall enough, xðtþ converge to the unique primal optimal olution of NUM and LðtÞ converge to it unique dual olution. The idea i to how that the dual objective function DðLðtÞÞ trictly decreae in each tep t. Hence, one can regard DðLÞ a a Lyapunov function in dicrete time. 3 The ame idea i extended in [90] to prove global aymptotic tability in an aynchronou environment where the delay between ource and link can be ubtantial, divere, and timevarying, ource and link can communicate at different time and with different frequencie, and information can be outdated or out of order. Method 3: Proving Convergence of Dual Decent Algorithm Through Decent Lemma. Several variation of the primal and dual-driven algorithm above can all maintain local tability in the preence of feedback delay [102], [103], [140], [141]. They are complementary in the ene that the primal-driven algorithm ha dynamic only at the ource, allow arbitrary utility function and therefore arbitrary fairne, but typically ha low link utilization, wherea the dualdriven algorithm ha dynamic only at the link, achieve full link utilization, but require a pecific cla of utility function (fairne) to maintain local tability in the preence of arbitrary feedback delay. The next algorithm ha dynamic at both. It allow arbitrary utility function, achieve arbitrarily cloe to full link utilization, and can maintain local tability in the preence of feedback delay. Algorithm that have dynamic at both link and ource are called primal-dual-driven algorithm. The algorithm of [71] extend the primal-driven algorithm (21), (22) to a 3 Indeed, for a continuou-time verion of (24) and (25), it i trivial to how that DðLÞ i a Lyapunov function. primal-dual-driven algorithm and the algorithm of [103] extend the dual-driven algorithm (24), (25) to a primaldual-driven algorithm. The paper [103] focue on local tability in the preence of feedback delay. We now ummarize the proof technique in [71] for global tability in the abence of feedback delay. The algorithm of [71] ue a ource algorithm that i imilar to (21) _x i ðtþ ¼w i U 0 i 1 R li l ðtþ: (26) ðx i ðtþþ It link algorithm adaptive virtual queue (AVQ) maintain an internal variable at each link called the virtual capacity ~c l that i dynamically updated _~c l l =@~c l ðc l y l ðtþþ; if ~c l 0 0; if ~c l ¼ 0andy l ðtþ 9 c l l (27) where 9 0iagainparameterandg l i a link Bmarking[ function that map aggregate flow rate y l ðtþ and virtual capacity ~c l into a price l ðtþ ¼g l ðy l ðtþ; c l ðtþþ: (28) Uing ingular perturbation theory, it i proved in [71] that, under (26) (28), xðtþ converge exponentially to the unique olution of the baic NUM, provided i mall enough. Furthermore, ðtþ then converge to the optimum of the dual problem. The idea i to eparately conider the tability of two approximating ubytem that are at different time cale when i mall. The boundary-layer ytem approximate the ource dynamic and aume that the virtual capacity ~c l are contant at the fat time cale _x ¼ w U 0 1 ð x R l g l ðyðtþ; ~c l Þ: (29) ðtþþ The reduced ytem approximate the link dynamic and aume the ource rate x are the unique r of (23) l _~c l ¼ c l y l (30) where y l ¼ P l R lx are contant and x are the unique r of VðxÞ defined in (23). Now we already know from above that the boundary-layer ytem (29) i aymptotically table. In [71], it i further hown that it i Vol. 95, No. 1, January 2007 Proceeding of the IEEE 267

14 exponentially table uniformly in ~c, and that the reduced ytem (30) i exponentially table provided the trajectory remain in a compact et. Singular perturbation theory then implie that the original ytem (26) (28) i globally exponentially table provided i mall enough (and the initial tate ðxð0þ; Lð0ÞÞ i in a compact et). Method 4: Proving Stability by Singular Perturbation Theory. A different approach i ued in [148] to prove global aymptotic tability for primal-dual-driven algorithm baed on paivity technique. A ytem, decribed by it tate zðtþ, inputuðtþ and output vðtþ, i called paive if there are poitive emidefinite function VðxÞ 0 and WðxÞ 0uchthat _V ðxðtþ Þ WðxðtÞÞþu T ðtþvðtþ: hence the forward ytem from LðtÞ L to _yðtþ i paive. For the revere ytem, conider the torage function V 2 ðy y Þ¼ l Z y l y l g l ðzþ g l ðz Þdz: V 2 i poitive emidefinite ince it Heian i a poitive emidefinite matrix. Moreover _V 2 ¼ ðlðtþ L Þ T _y and, hence, the revere ytem i paive. Then, VðxÞ :¼ V 1 ðxþþv 2 ðxþ can be ued a a Lyapunov function for the feedback ytem, becaue VðxÞ i called a torage function. The paivity theorem tate that the feedback interconnection of two paive ytem i globally aymptotically table and _V ¼ q ðtþ U 0 ð x 2 ðtþþ G 0; except for xðtþx : VðxÞ :¼ V 1 ðxþþv 2 ðxþ i a Lyapunov function for the feedback ytem, provided one of the torage function V 1, V 2 of the individual ytemarepoitivedefiniteandradiallyunbounded.conider the following variant of the primal-driven algorithm (21), (22): _x ðtþ ¼ U 0 ð x ðtþþ q ðtþ (31) l ðtþ ¼g l ðy l ðtþþ: (32) To how that it i the feedback interconnection of two paive ytem, the trick i to conider the forward ytem from LðtÞ L to _yðtþ, and the backward ytem from _yðtþ to LðtÞ L. FromLðtÞ L to _yðtþ, the torage function i V 1 ðxþ ¼ x q U ðx Þ: Then V 1 ðxþ i a poitive definite function ince it Heian i a poitive definite matrix for all x. Moreover, it can be hown, uing qðtþ ¼R T LðtÞ, that _V 1 ðxþ ¼ q ðtþ U 0 ð x 2 ðtþþ þ LðtÞ L ð Þ T _y Thi implie global aymptotic tability. The ame argument prove the global aymptotic tability of the dual-driven algorithm (24), (25) [148]. Moreover, ince the primal ource algorithm from L L to y y and the dual link algorithm from y y to L L are both paive, the paivity theorem aert the global aymptotic tability of their feedback interconnection, i.e., that of the following primal-dual-driven algorithm: _x ¼ U 0 ð x ðtþþ q ðtþ _ l ¼ l ðy l ðtþ c l where ðhþ þ z ¼ 0, if z ¼ 0 and h G 0, and ¼ h, otherwie. The global aymptotic tability of the AVQ algorithm (26) and (27) i imilarly proved in [148]. Method 5: Proving Stability by Paivity Argument. 4) Heterogeneou Congetion Control Protocol: Akeyaumption in the current model (6) (8) i that the link price l ðtþ depend only on link but not ource, i.e., the ource are homogeneou in that, even though they may control their rate uing different algorithm F,theyall adapt to the ame type of congetion ignal, e.g., all react to lo probabilitie, a in TCP Reno, or all to queueing delay, a in TCP Vega or FAST. When ource with heterogeneou protocol that react to different congetion ignal hare the ame network, the current convex Þ þ l þ x 268 Proceeding of the IEEE Vol.95,No.1,January2007

15 optimization and duality framework i no longer applicable. Thi i modeled in [133] and [134] by introducing price mapping function m l that map link price l to Beffective price[ een by ource. However, one can no longer interpret congetion control a a ditributed olution of the baic NUM when there are heterogeneou protocol. In thi ection, we ummarize the main reult of [134] on the equilibrium tructure of heterogeneou protocol. Dynamic propertie have alo recently been characterized. Suppoe there are J different protocol indexed by upercript j, andn j ource uing protocol j, indexedby ð j; Þ, where j ¼ 1;...; J and ¼ 1;...; N j. The total number of ource i N :¼ P j Nj. The L N j routing matrix R j for type j ourceidefinedbyr j l ¼ 1ifource ð j; Þ ue link l, and 0, otherwie. The overall routing matrix i denoted by R ¼½R 1 R 2 R J Š: Every link l ha an Bintrinic price[ l.atypej ource react to the Beffective price[ m j l ð lþ in it path, where m j l i a price mapping function, which can depend on both the link and the protocol type. By pecifying function m j l,we can let the link feed back different congetion ignal to ource uing different protocol, for example, Reno with packet loe and Vega with queueing delay. Let m j ðlþ ¼ðm j l ðl lþ; l ¼ 1;...LÞ and mðlþ ¼ðm j ðl l Þ; j ¼ 1;...JÞ. The aggregate price for ource ð j; Þ i defined a q j ¼ l R j l mj l ð lþ: (33) Let q j ¼ðq j ; ¼ 1;...; Nj Þ and q ¼ðq j ; j ¼ 1...; JÞ be vector of aggregate price. Then q j ¼ðR j Þ T m j ðlþ and q ¼ R T mðlþ. Letx j be a vector with the rate x j of ource ð j; Þ a it th entry, and x be the vector of x j h i T: x ¼ ðx 1 Þ T ; ðx 2 Þ T ;...; ðx J Þ T Source ðj; Þ ha a utility function U jðxj Þ that i trictly concave increaing in it rate x j.letu¼ðuj ; ¼ 1;...; N j ; j ¼ 1;...; JÞ. We call ðc; m; R; UÞ a network with heterogeneou congetion control protocol. A network i in equilibrium, or the link price L and ource rate x are in equilibrium, when each ource ð j; Þ it net benefit (utility minu bandwidth cot), and the demand for and upply of bandwidth at each bottleneck link are balanced. Formally, a network equilibrium i defined a follow. Given any price L, we aume that the ource rate x j are uniquely determined by x j h q j ¼ U j 0 1 i q j þ: Thi implie that the ource rate x j uniquely olve max z0 ½U jðzþ zqj Š. A uual, we ue xj ðq j Þ¼ðx j ðqj Þ; ¼ 1;...; N j Þ and xðqþ ¼ðx j ðq j Þ; j ¼ 1;...; JÞ to denote the vector-valued function compoed of x j. Since q ¼ R T mðlþ, weoftenabuenotationandwritex j ðþ, x j ðlþ, xðlþ. Define the aggregate ource rateyðlþ ¼ ð y l ðlþ; l ¼ 1;...; LÞ at link l by y j ðlþ ¼R j x j ðlþ; yðlþ ¼RxðLÞ: (34) In equilibrium, the aggregate rate at each link i no more than the link capacity, and they are equal if the link price i trictly poitive. Formally, we call L an equilibrium price, a network equilibrium, or jut an equilibrium if it atifie [from (33) and (34)] diagð l ÞðyðLÞ cþ ¼ 0; yðlþ c; L 0: (35) The theory in Section II-A2 correpond to J ¼ 1. When there are J 9 1 type of price, it break down becaue there cannot be more than one Lagrange multiplier at each link. In general, an equilibrium no longer aggregate utility, nor i it unique. It i proved in [134] that, under mild aumption, an equilibrium alway exit. There can be network ðr; c; m; UÞ that have uncountably many equilibria, but except for a et of meaure zero, all network have finitely many equilibria. Moreover, the Poincare Hopf index theorem implie that the number of equilibria i necearily odd. Specifically, uppoe the following aumption hold: C4: Price mapping function m j l are continuouly differentiable in their domain and trictly increaing with m j lð0þ ¼0. C5: For any 9 0, there exit a number max uch that if l 9 max for link l, then x j iðlþ G for all ðj; iþ with R j li ¼ 1: C6: Every link l ha a ingle-link flow ðj; iþ with ðu j iþ 0 ðc l Þ 9 0. Aumption C6 can be relaxed; ee [124]. We call an equilibrium L locally unique 6¼ 0atL. We call anetworkðc; m; R; UÞ regular if all equilibrium point are locally unique. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 269

16 Theorem 2: 1) There exit an equilibrium price L for any network ðc; m; R; UÞ. 2) Moreover, the et of link capacitie c for which not all equilibrium point are locally unique (i.e., the network i not regular) ha Lebegue meaure zero in R L þ. 3) A regular network ha a finite and odd number of equilibrium point. Depite the lack of an underlying NUM, heterogeneou protocol are till Pareto efficient for general network. Moreover, the lo of optimality can be bounded in term of the lope of the price mapping function m j l. Specifically, uppoe we ue the optimal objective value U of the following NUM a a meaure of optimality for heterogenou protocol: j U j ubject to Rx c: (36) Let UðL Þ :¼ P P j Uj ðxj ðl ÞÞ be the utility achieved by any equilibrium L of the heterogeneou protocol. Then it can be hown that, for any equilibrium L UðL Þ U min _mj l ðlþ max _m j l ðlþ where _m j l denote the derivative of mj l, and the minimization and maximization are over all type j, all link l ued by all type j flow, and all price L. For common AQM cheme uch a RED with (piecewie) linear m j l, the bound reduce to a imple expreion in term of their lope. For a homogeneou congetion control protocol, the utility function determine how bandwidth i hared among all the flow. For heterogeneou protocol, how i bandwidth hared among thee protocol (interprotocol fairne), and how i it hared among flow within each protocol (intraprotocol fairne)? It i hown in [133] (and a generalization of reult there) that any deired degree of fairne among the different protocol i achievable by appropriate linear caling of utility function. Within each protocol, the flow would hare the bandwidth among themelve a if they were in a ingle-protocol network according to their own utility function, except that the link capacitie are reduced by the amount conumed by the other protocol. In other word, intraprotocol fairne i unaffected by the preence of other protocol. Theorem 2 guarantee local unique equilibrium point for almot all network under mild condition. If the degree of heterogeneity, a meaured by the lope _m j l of the price mapping function m j l,imall,thenglobaluniquenei x j guaranteed: if _m j l do not differ much acro ource type at each link, or they do not differ much along link in every ource path, the equilibrium i globally unique. Moreover, under thi condition, global uniquene i equivalent to local tability. Specifically, conider the dual-driven algorithm (in continuou-time) x j _ l ¼ ðy l ðtþ c l Þ q j ðtþ ðtþ ¼U0 1 where the effective price q j ðtþ are defined by (33) (compare with (24) and (25) in the homogeneou cae). The linearized ytem with a mall perturbation L around an equilibrium point L i, in vector form _ ðl ÞL: (37) The equilibrium L i called locally table if all the eigenvalue Þ are in the left-half plane. Given the price mapping function m j l, we ay their degree of heterogeneity i mall if they atify any one of the following condition: 1) For each l ¼ 1;...; L, j ¼ 1;...; J h i _m j l ðl Þ2 a l ; 2L 1 al for ome a l 9 0 for any equilibrium L : (38) 2) For all j ¼ 1;...; J, l ¼ 1;...; L h i _m j l ðl Þ2 a j ; 2L 1 a j for ome a j 9 0 for any equilibrium L : (39) Theorem 3: For almot all network ðc; m; R; UÞ: 1) Suppoe their degree of heterogeneity i mall, then the equilibrium i globally unique. Moreover, it i locally table. 2) Converely, if all equilibrium point are locally table, it i alo globally unique. Aymptotically when L!1, both condition (38) and (39) converge to a ingle point. Condition (38) reduce to _m j l ¼ a l which eentially ay that all protocol are the ame ðj ¼ 1Þ. Condition (39) reduce to _m j l ¼ a j,whichi the cae where price mapping function m j l are linear and link independent. Variou pecial cae are hown to have a globally unique equilibrium in [134]. 270 Proceeding of the IEEE Vol.95,No.1,January2007

17 Method 6: Proving Equilibrium Propertie Through Vector Field Repreentation and Poincare Hopf Index Theorem. Recall that ince a network of homogeneou protocol olve the baic NUM, it alway ha a unique equilibrium pointalongatheroutingmatrixr ha full row rank. The equilibrium ource rate x doe not depend on link parameter, uch a buffer ize, a long a the AQM guarantee complementary lackne condition for the baic NUM. Moreover, x doe not depend on the flow arrival pattern. Thee propertie no longer hold in the heterogeneou cae. We now preent a imulation uing Network Simulator 2 (n2) that how that x can depend on the flow arrival pattern becaue of the exitence of multiple equilibria. The topology of thi network i hown in Fig. 1. All link run the RED algorithm. Link 1 and 3 are each configured with 9.1 pkt/m capacity (equivalent to 111 Mb/), 30 m one-way propagation delay and a buffer of 1500 packet. The RED parameter i et to be ðb; b; 1 Þ¼ð300; 1500; 10 4 Þ. Link 2 ha a capacity of 13.8 pkt per m (166 Mb/) with 30 m one-way propagation delay and buffer ize of 1500 packet. RED parameter i et to (0, 1500, 0.1). There are 8 Reno flow on path 3 utilizing all the three link, with one-way propagation delay of 90 m. There are two FAST flow on each of path 1 and 2. Both of them have one-way propagation delay of 60 m. All FAST flow ue a common parameter value ¼ 50 packet. Two et of imulation have been carried out with different tarting time for Reno and FAST flow. One et of flow (Reno or FAST) tart at time zero, and the other et tart at the 100th econd. Fig. 2 how the ample throughput trajectorie of one of FAST flow and one of Reno flow. The large difference in the rate allocation of FAST and Reno between thee two cenario reult from that the network reache two different equilibrium point, depending on which type of flow tart firt. The model introduced in [133] and [134] i critical in deepening our undertanding of uch complex behavior, and providing deign guideline to manage it in practice. Indeed, a ditributed algorithm i propoed in [135] that can Fig. 1. Scenario with multiple equilibria with heterogeneou congetion control protocol. Fig. 2. Sample throughput trajectorie of FAST and Reno. (a) FAST tart firt; (b) Reno tart firt. teer a heterogeneou network to the unique equilibrium point that aggregate utility. The baic idea i imple: Beide regulating their rate according to their congetion ignal, ource alo adapt a parameter in a low time cale baed on a common congetion ignal. Thi allow a ource to chooe a particular congetion ignal in a fat time cale (and therefore maintain benefit aociated with it) while aymptotically reaching the optimal equilibrium. The theoretical foundation and empirical upport of the algorithm are provided in [135]. 5) Forward-Engineering: FAST: The congetion control algorithm in the current TCP, which we refer to a Reno, wa developed in 1988 [55] and ha gone through everal enhancement ince. It ha performed remarkably well and i generally believed to have prevented evere congetion a the Internet caled up by ix order of magnitude in ize, peed, load, and connectivity. It i alo well-known, however, that a bandwidth-delay product continue to grow, TCP Reno will eventually become a performance bottleneck itelf. Even though, hitorically, TCP Reno wa deigned, implemented, and deployed without any conideration of NUM, and it equilibrium, fairne, and dynamic propertie were undertood only a an afterthought, it indeed olve a NUM implicitly a howninsectionii-a2. Several new algorithm have been propoed in the lat few year to addre the problem of Reno, including TCP Wetwood, HSTCP [42], FAST TCP [56], [57], STCP [67], BIC TCP [155], HTCP [123], MaxNet [153], [154], CP [62], and RCP [33], etc. (ee [147] for other reference). Some of thee deign were explicitly guided by the emerging theory urveyed in thi paper, which ha become indipenable to the ytematic deign of new congetion control algorithm. It provide a framework to undertand iue, clarify idea and ugget direction, leading to more undertandable and better performing implementation. The congetion control mechanim of FAST TCP i eparated into four component, a hown in Fig. 3. Thee four component are functionally independent o that they can be deigned eparately and upgraded aynchronouly. The data control component determine which packet to Vol. 95, No. 1, January 2007 Proceeding of the IEEE 271

18 Table 5 Summary of Main Notation for Section II-B Fig. 3. Schematic of FAST TCP. tranmit, window control determine how many packet to tranmit, and burtine control determine when to tranmit thee packet. Thee deciion are made baed on information provided by the etimation component. More pecifically, the etimation component compute two piece of feedback information for each data packet entva multibit queueing delay and an one-bit lo-orno-lo indicationvwhich are ued by the other three component. Data control elect the next packet to end from three pool of candidate: new packet, packet that are deemed lot (negatively acknowledged), and tranmitted packet that are not yet acknowledged. Window control regulate packet tranmiion at the RTT time cale, while burtine control moothe out the tranmiion of packet at a maller time cale. The theory urveyed in thi paper form the foundation of the window control algorithm. FAST periodically update the congetion window baed on the average RTT and average queueing delay provided by the etimation component, according to (15) in Section II-A1. The equilibrium value of window W and delay L of the network defined by (15) and (16) are obtained from the unique olution to the utility maximization problem over x ubject to w log x Rx c and it Lagrangian dual problem over L minimize c l l l w log l R l l : Thi implie that the equilibrium rate x i -weighted proportionally fair. In equilibrium, ource maintain packet in the buffer along it path. Hence, the total amount of buffering in the network mut be at leat P packet in order to reach the equilibrium. FAST TCP i proved in [145] to be locally aymptotically table for general network if all flow have the ame feedback delay, no matter how large the delay i. It i proved in [26] to be globally aymptotically table in the preence of heterogeneou feedback delay at a ingle link. We have implemented the inight from thi erie of theoretical work in a oftware prototype FAST TCP [56], [147] and have been working with our collaborator to tet it in variou network around the world [57]. Phyicit have been uing FAST TCP to break variou world record in data tranfer in the lat few year. Fig. 4 how it performance in everal experiment conducted during over a high-peed tran- Atlantic network, over a home DSL, and over an emulated loy link. B. MAC 1) Revere-Engineering: MAC a Noncooperative Game: If contention among tranmiion on the ame link in wired network, or acro different link in wirele network, are not appropriately controlled, a large number of colliion may occur, reulting in wate of reource uch a bandwidth and energy, a well a lo of ytem efficiency and fairne. There are two major type of MAC: chedulingbaed contention-free mode and random-acce-baed contention-prone mode. The firt i often hown to olve certain maximum weight matching or graph coloring problem. The econd ha been extenively tudied through the perpective of queuing-theoretic performance evaluation, but wa only recently revere-engineered to recover the underlying utility maximization tructure [75], [78]. 272 Proceeding of the IEEE Vol.95,No.1,January2007

19 In TCP revere-engineering conidered in the lat ection, the utility function of each ource depend only on it data rate that can be directly controlled by the ource itelf. TCP/AQM can be modeled a a ditributed algorithm that olve the baic NUM problem and it Lagrange dual problem. In contrat, in the exponential-backoff (EB) MAC protocol, the utility of each link directly depend not only on it own tranmiion (e.g., peritence probability) but alo tranmiion of other link due to colliion. We how that the EB protocol can be revere-engineered through a noncooperative game in which each link trie to, uing a tochatic ubgradient formed by local information, it own utility function in the form of expected net reward for ucceful tranmiion. While the exitence of the Nah equilibrium can be proved, neither convergence nor ocial welfare optimality i guaranteed. We then provide ufficient condition on uer denity and backoff aggreivene that guarantee uniquene and tability of the Nah equilibrium (i.e., convergence of the tandard bet repone trategy). Conider an ad hoc network repreented by a directed graph GðV; EÞ, e.g., a in Fig. 5, where V i the et of node and E i the et of logical link. We define L out ðnþ a a et of outgoing link from node n, L in ðnþ a a et of incoming link to node n, t l athetranmitternodeoflinkl, andr l a the receiver node of link l. We alo define Nto I ðlþ a the et of node whoe tranmiion caue interference to the Fig. 4. Performance of FAST TCP. (a) At 1 Gb/, FAST TCP utilized 95% of a tran-atlantic network bandwidth while maintaining a fairly contant throughput. Linux TCP on average ued 19% of the available bandwidth, while producing a throughput that fluctuate from 100 to 400 Mb/. (b) At an 512-Kb/ DSL uplink, data tranfer uing FAST TCP increaed the latency from 10 m to around 50 m, while Linux and Window TCP increaed it to a high a 600 m, an order of magnitude larger. (c) Over an emulated loy link, FAST TCP achieved cloe to optimal data rate while other (lo-baed) TCP variant collaped when lo rate exceeded 5%. Figure from unpublihed work by B. Wydrowki, S. Hegde, and C. Jin. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 273

20 receiver of link l, excluding the tranmitter node of link l (i.e., t l ), and L I fromðnþ a the et of link whoe tranmiion uffer interference from the tranmiion of node n, excluding outgoing link from node n (i.e., l 2 L out ðnþ). Hence, if the tranmitter of link l and a node in et Nto I ðlþ tranmit data imultaneouly, the tranmiion of link l fail. If node n and the tranmitter of link l in et L I from ðnþ tranmit data imultaneouly, the tranmiion of link l alo fail. Random-acce protocol in uch wirele network uually conit of two phae: contention avoidance and contention reolution. We focu only on the econd phae here. The EB protocol i a prototypical contention reolution protocol. For example, in the IEEE DCF (Ditributed Coordination Function) implementation, the EB protocol i window-baed: each link l maintain it contention window ize W l, current window ize CW l, and minimum and maximum window ize Wl min and Wl max. After each tranmiion, contention window ize and current window ize are updated. If tranmiion i ucceful, the contention window ize i reduced to the minimum window ize (i.e., W l ¼ Wl min ), otherwie it i doubled until reaching the maximum window ize Wl max (i.e., W l ¼ minf2w l ; Wl max g). Then, the current window ize CW l i choen to be a number between ð0; W l Þ uniformly at random. It decreae in every time lot, and when it become zero, the link tranmit data. Since the window ize i doubled after each tranmiion failure, the random acce protocol in DCF i called the binary exponential backoff (BEB) protocol, which i a pecial cae of EB protocol. We tudy the window-baed EB MAC protocol through a peritence probabilitic model, an approach analogou to the ource rate model for the window-baed TCP congetion control protocol in Section II-A2. Here each link l tranmit data with a probability p l,whichwe refer to a the peritence probability of link l. After each tranmiion attempt, if the tranmiion i ucceful without colliion, then link l et it peritence probability to be it maximum value p max l.otherwie,it multiplicatively reduce it peritence probability by a factor l ð0 G l G 1Þ until reaching it minimum value p min l. Thi peritence probability model i a memoryle one that approximate the average behavior of EB protocol. Since in the window-baed EB protocol the current window ize CW l of link l i randomly elected between ð0; W l Þ,whenitwindowizei W l,wemaythinkthat link l tranmit data in a time lot with an attempt probability 1=W l, which correpond to the peritence probability p l in our model for the average behavior of the EB protocol. In the window-baed protocol, after every tranmiion ucce, the attempt probability i et to be it maximum value (i.e., 1=Wl min ), which correpond to p max l in our model, and after every tranmiion failure, the attempt probability i et to be a fraction of it current value until it reache it minimum value, which correpond to reducing the peritence probability by a factor of ¼ 0:5 in BEB (and in general 2ð0; 1Þ in EB) until reaching the minimum peritence probability p min l. The update algorithm for the peritence probability decribed above can be written a p l ðt þ 1Þ ¼max p min l ; p max l þ l p l ðtþ1f Tl ðtþ¼1 1 Tl ðtþ¼1 f g1f Cl ðtþ¼1 g1f Cl ðtþ¼0g g þ p l ðtþ1f Tl ðtþ¼0g (40) where p l ðtþ i a peritence probability of link l at time lot t, 1 a i an indicator function of event a, andt l ðtþ and C l ðtþ are the event that link l tranmit data at time lot t and that there i a colliion to link l tranmiion given that link l tranmit data at time lot t, repectively. In the ret of thi ection, we will examine the cae when p min l ¼ 0. Given pðtþ, wehave ProbfT l ðtþ ¼1jpðtÞg ¼ p l ðtþ and ProbfC l ðtþ ¼1jpðtÞg ¼ 1 Y ð1 p n ðtþþ: n2l I to ðlþ Fig. 5. Logical topology graph of a network illutrating contention. Since the update of the peritence probabilitie for the next time lot depend only on the current peritence probabilitie, we will conider the update conditioning on the current peritence probabilitie. Note that p l ðtþ i a random proce whoe tranition depend on event T l ðtþ and C l ðtþ. We firt tudy it expected trajectory and will return to (40) later in thi ection. Slightly abuing the 274 Proceeding of the IEEE Vol.95,No.1,January2007

21 notation, we till ue p l ðtþ to denote the expected peritence probability. From (40), we have p l ðt þ 1Þ ¼p max l E 1f Tl ðtþ¼1g1f Cl ðtþ¼0gjpðtþ þ l E p l ðtþ1f Tl ðtþ¼1g1f Cl ðtþ¼1gjpðtþ þ E p l ðtþ1f Tl ðtþ¼0gjpðtþ ¼ p max l p l ðtþ Y ð1 p n ðtþþ n2l I to ðlþ 0 1 þ l p l ðtþp l ðtþ@ 1 Y ð1 p n ðtþþa n2l I to ðlþ þ p l ðtþð1 p l ðtþþ (41) where Efajbg i the expected value of a given b and 1 denote the indicator function of probabilitic event. We now revere-engineer the update algorithm in (41) a a game, in which each link l update it trategy, i.e., it peritence probability p l, to it utility U l baed on trategie of the other link, i.e., p l ¼ ðp 1 ; ; p l 1 ; p lþ1 ; ; p jej Þ. Formally, the game i G EB MAC ¼½E; l2e A l ; fu l g l2e Š, where E i a et of player, i.e., link, A l ¼fp l j0 p l p max l g i an action et of player l, and U l i a utility function of player l to be determined through revere-engineering. Theorem 4: The utility function i the following expected net reward (expected reward minu expected cot) that the link can obtain from it tranmiion: U l ðpþ ¼Rðp l ÞSðpÞ Cðp l ÞFðpÞ; 8l (42) where SðpÞ ¼p l Qn2L I ðlþð1 p nþ i the probability of to tranmiion ucce, FðpÞ ¼p l ð1 Q n2l I ðlþð1 p nþþ i to the probability of tranmiion failure, and Rðp l Þ¼ def p l ðð1=2þp max l ð1=3þp l Þ can be interpreted a the reward for tranmiion ucce, Cðp l Þ¼ def ð1=3þð1 l Þp 2 l can be interpreted a the cot for tranmiion failure. Furthermore, there exit a Nah equilibrium in the EB-MAC Game G EB MAC ¼½E; l2e A l ; fu l g l2e Š characterized by the following ytem of equation: p l p max Q l n2l ¼ I to ðlþ 1 p n 1 l 1 Q n2l I to ðlþ 1 ; 8l: (43) p n Note that the expreion of SðpÞ and FðpÞ come directly from the definition of ucce and failure probabilitie, while the expreion of Rðp l Þ and Cðp l Þ (thu exact form of U l ) are in fact derived in the proof by revere-engineering the EB protocol decription. In the EB protocol, there i no explicit meage paing among link, and the link cannot obtain the exact information to evaluate the gradient of it utility function. Intead of uing the exact gradient of it utility function a in (41), each link attempt to approximate it uing (40). It can be hown [76], [78] that the EB protocol decribed by (40) i a tochatic ubgradient algorithm to utility (42). Method 7: Revere-Engineer a Noncooperative Protocol a agame. The next tep i to invetigate uniquene of the Nah equilibrium together with the convergence of a natural trategy for the game: the bet repone trategy, commonly ued to tudy tability of the Nah equilibrium. In bet repone, each link update it peritence probabilityforthenexttimelotuchthatit it utility baed on the peritence probabilitie of the other link in the current time lot p l ðt þ 1Þ ¼ arg max 0 p l p max l U l p l ; p l ðtþ : (44) Hence, p l ðt þ 1Þ i the bet repone of link l given p l ðtþ. The connection between the bet repone trategy and tochatic ubgradient update trategy ha been quantified for the EB MAC Game [78]. Let K ¼ max l fjl I toðlþjg, which capture the amount of potential contention among link. We have the following theorem that relate three key quantitie: amount of potential contention K, backoff multiplier (peed of backoff), and p max that correpond to the minimum contention window ize (minimum amount of backoff). Theorem 5: If p max K=4ð1 p max Þ G 1, then 1) the Nah equilibrium i unique; 2) tarting from any initial point, the iteration defined by bet repone converge to the unique equilibrium. There are everal intereting engineering implication from the above theorem. For example, it provide guidance on chooing parameter in the EB protocol, and quantifie the intuition that with a large enough (i.e., link do not decreae the probabilitie uddenly) and a mall enough p max (i.e., link backoff aggreively enough), uniquene and tability can be enured. The higher the amount of contention (i.e., a larger value of K), the maller p max need to be. The key idea in the proof i to how the updating rule from pðtþ to pðt þ 1Þ i a contraction mapping by verifying the infinity norm of the Jacobian of the update dynamic in thegameilethanone. Method 8: Verifying Contraction Mapping by Bounding the Jacobian Norm. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 275

22 Revere-engineering for the vertical interaction between TCP Reno congetion control and DCF random acce ha alo been carried out [168]. A will be dicued in Section V, eion level tochatic effect need to be incorporated in the above revere-engineering model to include the arrival tatitic of finite-duration eion. Then MAC protocol can be analyzed and deigned through a union of tochatic tability reult in traditional queuing model and optimality reult in the utility maximization model. 2) Forward-Engineering: Utility-Optimal MAC Protocol: The Nah equilibrium attained by exiting EB MAC protocol may not be ocially optimal. Thi motivate forward-engineering where adequate feedback i generated to align elfih utility maximization by each logical link to the ocial welfare in term of total network utility. By impoing different utility function, different type of ervice and different efficiency-fairne tradeoff can be proviioned. Two uite of protocol are poible: cheduling-baed and random-acce-baed. We again focu on the econd in thi ubection on forward-engineering. Contention among link can be modeled by uing a contention graph firt propoed in [98]. An example i hown in Fig. 6, which i obtained from Fig. 5 auming that if the ditance between the receiver of one link and the tranmitter of the other link i le than 2d, there i interference between thoe two link. Each vertex in the contention graph correpond to a link in the network topology graph. If two link tranmiion interfere with each other, the vertice correponding to them in the contention graph are connected with an edge. Only one link at a time among link in the ame maximal clique in the contention graph can tranmit data without colliion. Thi contraint can be viualized by uing a bipartite graph, a in Fig. 7, where one partition of vertice correpond to link in the network (i.e., node in the contention graph) and the other correpond to maximal clique in the contention graph. An edge i etablihed in the bipartite graph if a node in the contention graph belong to a maximal clique. Hence, only network link repreented by the node in the bipartite graph that are covered by a matching can tranmit data imultaneouly without colliion. In [16], [39], and [98], a fluid approximation approach i ued where each maximum clique i defined a a reource Fig. 6. Contention graph derived from the logical topology graph. Fig. 7. Bipartite graph between maximal clique and link in the contention graph. with a finite capacity that i hared by the link belonging to the clique. Capacity of a clique i defined a the maximum value of the um of time fraction uch that each link in the clique can tranmit data without colliion. Conequently, a generalized NUM problem ha been formulated a follow, with capacity contraint C CLi at each maximal clique CL i : ubject to U l ðx l Þ l l2lðcl i Þ x l c l C CLi 8i: (45) Thi problem formulation eentially take the ame tructure a the baic NUM (4) for TCP congetion control, and can be olved following the ame dualdecompoition algorithm. We refer to thi a the determinitic approximation approach. An alternative approach i to explicitly model colliion probabilitie, a hown in [61] for log utility and in [76] and for general concave utility. Conider a random-accebaed MAC protocol in which each node n adjut it own peritence probability and alo the peritence probability of each of it outgoing link. Since peritent tranmiion deciion are made ditributively at each node, we need a hift from graph model baed on logical link to graph model that incorporate node a well. Let P n be the tranmiion probability of node n, andp l be that of link l. The appropriate generalized NUM thu formulated i a follow, with variable fx l g, fp n g, fp l g: U l ðx l Þ l Y ubject to x l ¼ c l p l l2l out ðnþ k2n I to ðlþ ð1 P k Þ; p l ¼ P n ; 8n 8l 0 P n 1; 8n 0 p l 1; 8l: (46) 276 Proceeding of the IEEE Vol.95,No.1,January2007

23 Without lo of generality, we can replace the equality in the firt contraint with an inequality. Thi i becaue uch an inequality will alway be achieved with an equality at optimality. The next tep of problem tranformation i to take the log of both ide of the firt contraint in problem (46) and a log change of variable and contant: x 0 l ¼ log x l, U 0 l ðx0 l Þ¼U lðe x0 l Þ,andc 0 l ¼ log c l.thireformulation turn the problem into ubject to Ul 0 x 0 l l2l c 0 l þlog p lþ l2l out ðnþ p l ¼ P n ; k2n I to ðlþ logð1 P k Þ x 0 l 0; 8n 0 P n 1; 8n 0p l 1; 8l: (47) Note that problem (47) i now eparable but till may not be a convex optimization problem, ince the objective Ul 0ðx0 lþ may not be a trictly concave function, even though U l ðx l Þ i a trictly concave function. However, the following imple ufficient condition guarantee it 2 U l ðx l 2 lðx l Þ l x l which tate that the curvature (degree of concavity) of the utility function need to be not jut nonpoitive but boundedawayfromzerobyamucha ð@u l ðx l Þ=x l Þ, i.e., the application repreented by thi utility function mut be elatic enough. Method 9: Log Change of Variable for Decoupling, and Computing Minimum Curvature Needed for Concavity After thechangeofvariable. Following dual decompoition and the ubgradient 4 method, the NUM problem (46) for random acce MAC protocol deign can be olved by the following algorithm. Algorithm 1: Utility Optimal Random Acce Algorithm Each node n contruct it local interference graph to obtain et L out ðnþ, L in ðnþ, L I from ðnþ,andni to ðlþ, 8l 2 L outðnþ. Each node n et t ¼ 0, l ð1þ ¼1, 8l 2 L out ðnþ, P n ð1þ ¼ jl out ðnþj=ðjl out ðnþj þ jl I from ðnþjþ,andp lð1þ ¼1=ðjL out ðnþjþ jl I from ðnþjþ, 8l 2 L outðnþ. 4 A ubgradient of a (poibly nondifferentiable) function f : R n! R at point x i a vector g uch that fðyþ fðxþþg T ðy xþ, 8y. 8l For each node n, do 1) Set t t þ 1. 2) Inform l ðtþ to all node in N I to ðlþ, 8l 2 L outðnþ and P n ðtþ to t l, 8l 2 L I from ðnþ. 3) Set k n ðtþ ¼ P l2l out ðnþ lðtþþ P k2l I from ðnþ kðtþ and ðtþ ¼1=t. 4) Solve the following problem to obtain P n ðt þ 1Þ, and x 0 l ðt þ 1Þ, p lðt þ 1Þ,and l ðt þ 1Þ, 8l 2 L out ðnþ: 8 P l2loutðnþ lðtþ >< Pl2LoutðnÞ P ; if k nðtþ 6¼ 0 P n ðt þ 1Þ ¼ lðtþþ k2l IfromðnÞ kðtþ >: jl out ðnþj jl out ðnþjþ j L I from ðnþj; if k nðtþ ¼0 8 l ðtþ >< Pl2LoutðnÞ P ; if k nðtþ 6¼ 0 p l ðt þ 1Þ ¼ lðtþþ k2l IfromðnÞ kðtþ >: 1 jl out ðnþjþ j L I from ðnþj; if k nðtþ ¼0 x 0 lðt þ 1Þ ¼ arg max Ul 0 x 0 l l ðtþx 0 l and x 0 l min x 0 x 0 max l " l ðt þ 1Þ ¼ l ðtþ ðtþ c 0 l þ log p lðtþ 13 þ log 1 P k ðtþ x 0 l ðtþa5: k2nto I ðlþ 5) Set it peritence probability P n ¼ P n ðtþ and the conditional peritence probability of each of it outgoing link q l ¼ p l ðtþ=p n ðtþ. 6) Decide if it will tranmit data with a probability P n, in which cae it chooe to tranmit on one of it outgoing link with a probability q l, 8l 2 L out ðnþ. while (1). Note that the above algorithm i conducted at each node n to calculate P n,andp l, l,andx 0 l for it outgoing link l (i.e., 8l 2 L out ðnþ). Hence, it i conducted at the tranmitter node of each link. If we aume that two node within interference range can communicate with each other (i.e., if node within ditance 2d in Fig. 5 can etablih a communication link), in the above algorithm each node require information from node within twohop ditance from it. To calculate P n and p l for it outgoing link l (i.e., 8l 2 L out ðnþ), node n need m from the tranmitter node t m of link m that i interfered from the tranmiion of node n (i.e., from t m, 8m 2 L I fromðnþ). Note that t m i within two-hop from node n. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 277

24 Fig. 8. Comparion of network utilitie in a numerical example. Fig. 9. Comparion of rate-fairne tradeoff in a numerical example. Alternatively, if l and x 0 l for each link l are calculated at it receiver node r l intead of it tranmitter node t l,a modified verion of Algorithm 1 can be devied in which each node require information only within one-hop ditance [76]. Theorem 6: Algorithm 1 converge to a globally optimal olution of (46) for ufficiently concave utility function. We now how a numerical example of the deired tradeoff between efficiency and fairne that can be achieved by appropriately adjuting the parameter of utility function. In thi experiment, the utility function for each link l, U l ðx l Þ i in the following tandard form of concave utility parameterized by, hifted uch that U l ðx min l Þ¼0andU l ðx max l U l ðx l Þ¼ Þ¼1 xð1 Þ l x maxð1 Þ l x minð1 Þ l x minð1 Þ l We et x min l ¼ 0:5 andx max l ¼ 5, 8l, varying the value of from 1 to 2 with a tep ize 0.1. We compare the performance of Algorithm 1 and it one-hop meage paing variant (modified Algorithm 1, not hown here) with determinitic fluid approximation and the BEB protocol in IEEE tandard. 5 In Fig. 8, we compare the network utility achieved by each protocol. We how the tradeoff curve of rate and fairne for each protocol in Fig. 9. Here, the fairne index i ðð P l x2 l Þ=jLj P l x2 l Þ. For each protocol hown in the graph, the area to the left and below of the tradeoff curve i the achievable region (i.e., every (rate, fairne) : point in thi region can be obtained), and the area to the right and above of the tradeoff curve i the infeaible region (i.e., it i impoible to have any combination of (rate, fairne) repreented by point in thi region). It i impoible to operate in the infeaible region and inferior to operate in the interior of the achievable region. Operating on the boundary of the achievable region, i.e., the Pareto optimal tradeoff curve, i the bet. Point on the Pareto optimal tradeoff curve are not comparable: which point i better depend on the deired tradeoff between efficiency and fairne. Since the BEB protocol i a tatic protocol, it alway provide the ame efficiency (rate) and fairne regardle of the choice of utility function. Hence, we cannot flexibly control the efficiency-fairne tradeoff in the BEB protocol. Algorithm 1 and it variant achieve higher network utilitie and wider dynamic range of rate-fairne tradeoff. Further dicuion on ditributed, uboptimal cheduling algorithm for different interference model can be found in Section III-D and V-A. III. VERTICAL DECOMPOSITION In thi ection, we turn to vertical decompoition acro the protocol tack. Following i a nonexhautive lit of ome of the recent publication uing BLayering a Optimization Decompoition[ for vertical decompoition. 6 Almot all of the following paper tart with ome generalized NUM formulation, and ue either dual decompoition or primal penalty function approach to modularize and ditribute the olution algorithm, followed by proof of optimality, tability, and fairne. The individual module in the 5 The performance of the BEB protocol highly depend on the choice of maximum and minimum window ize, Wl max and Wl min. It turn out that for the network in Fig. 5, the average-performance parameter are: Wl max ¼ 20 and Wl min ¼ Note that there are many more publication on joint deign acro thee layer that did not ue NUM modeling or decompoition theory. In addition, we apologize in advance for miing any reference we hould have included and would appreciate any information about uch citation. 278 Proceeding of the IEEE Vol.95,No.1,January2007

25 holitic olution range from adaptive routing and ditributed matching to information-theoretic ource coding and video ignal proceing, coupled through implicit or explicit meage paing of function of appropriate Blayering price[: variable that coordinate the layer. joint congetion control and adaptive coding or power control [18], [21], [75]; joint congetion and contention control [16], [61], [74], [143], [168] [170]; joint congetion control and cheduling [1], [12], [35], [92], [129]; joint routing and power control [58], [100], [158]; joint congetion control, routing, and cheduling [17], [35], [36], [82], [83], [99], [129]; joint routing, cheduling, and power control [27], [156]; joint routing, reource allocation, and ource coding [53], [167]; TCP/IP interaction [50], [49], [112], [144] and HTTP/TCP interaction [13]; joint congetion control and routing [46], [49], [60], [65], [69], [101]; network lifetime maximization [97]. Four cae tudie and the aociated illutrative numerical example are ummarized here, each picked mainly to convey everal key meage. We preent more detail on the firt two cae, which mainly illutrate the application to the analyi and deign apect, repectively. In all cae tudie, we firt formulate generalized NUM problem to capture the interaction acro uch functional module. Thee can be decompoed into ubproblem, each of which i olved by a layer, and the interface between the layer repreented by ome function of the optimization variable. Then in Section IV-C, we will how that thee cae tudie have only panned a ubet of alternative layering architecture. Table 6 Summary of Main Notation for Section III Vol. 95, No. 1, January 2007 Proceeding of the IEEE 279

26 There are obviouly many more cae tudie in the rapidly developing reearch literature in thi area. Many of thee are not covered in thi urvey, in part becaue of pace limitation, and in part becaue we hope to highlight the concept of the top-down approach to deign layered architecture from firt principle, rather than any et of pecific cro-layer cheme or their performance enhancement. Cae tudie are urveyed here only to illutrate the conceptual implicity in the tructured thinking of Blayering a decompoition,[ a implicity that we hope will not be buried under the rich detail in all thee recent publication. Even for thee elected illutrative example, there are many related work by variou reearch group. Our preentation i inevitably omewhat biaed toward relying on the material from publication by ourelve and coauthor. A. Cae Study 1: Jointly Optimal Congetion Control and Routing The word Brouting[ carrie different meaning in different part of the reearch literature. It can refer to dynamic or tatic routing, ingle-path or multipath routing, ditance-vector or link-tate-baed routing, inter-domain or intra-domain routing, fully ditributed routing or centralized-computation-aided routing, and other type of routing in wirele ad hoc network, optical network, and the Internet. Several notion of routing will be ued in the model in thi ection. 1) TCP/IP Interaction: Suppoe that there are K acyclic path from ource to it detination, repreented by a L K 0 1 matrix H,where H lj ¼ 1; if path j of ource ue link l 0; otherwie. Let H be the et of all column of H that repreent all the available path to. DefinetheL K matrix H a H ¼½H 1... H N Š path routing, and allow w j 2½0; 1Š for multipath routing. Collect the vector w, ¼ 1;...; N, intoak N blockdiagonal matrix W. LetW n be the et of all uch matrice correponding to ingle path routing, defined a WjW¼diagðw 1 ;...; w N Þ2f0; 1g KN ; 1 T w ¼ 1; 8: : Define the correponding et W m for multipath routing a WjW¼diagðw 1 ;...; w N Þ2½0; 1Š KN ; 1 T w ¼ 1; 8: : (48) A mentioned above, H define the et of acyclic path available to each ource, and W define how the ource load balance acro thee path. Their product define an L N routing matrix R ¼ HW that pecifie the fraction of flow at each link l. The et of all ingle-path routing matrice i R n ¼fRjR ¼ HW; W 2W n g (49) and the et of all multipath routing matrice i R m ¼fRjR ¼ HW; W 2W m g: (50) The difference between ingle-path routing and multipath routingitheintegercontraintonw and R. Ainglepath routing matrix in R n i an 0-1 matrix: R l ¼ 1; if link l i in the path of ource 0; otherwie. A multipath routing matrix in R m i one whoe entrie are in the range [0, 1] where K :¼ P K. H define the phyical topology of the network. Let w be a K 1 vector where the jth entry repreent the fraction of i flow on it jth path uch that w j 0 8j and 1 T w ¼ 1 where 1 i a vector of an appropriate dimenion with the value 1 in every entry. We require w j 2f0; 1g for ingle 9 0; if link l i in a path of ource R l ¼ 0; otherwie. The path of ource i denoted by r ¼½R 1... R L Š T,the th column of the routing matrix R. Wenowmodelthe interaction of congetion control at the tranport layer and hortet-path routing at the network layer. We firt conider the ituation where TCP-AQM operate at a fater time cale than routing update. We 280 Proceeding of the IEEE Vol.95,No.1,January2007

27 aume for now a ingle path i elected for each ourcedetination pair that minimize the um of the link cot in the path, for ome appropriate definition of link cot. In particular, traffic i not plit acro multiple path from the ource to the detination even if they are available. We focu on the time cale of the route change, and aume TCP-AQM i table and converge intantly to equilibrium after a route change. A explained in the lat ection, we interpret the equilibria of variou TCP and AQM algorithm a olution of NUM and it dual. Specifically, let RðtÞ 2R n be the (ingle-path) routing in period t. Let the equilibrium rate xðtþ ¼xðRðtÞÞ and price LðtÞ ¼LðRðtÞÞ generated by TCP-AQM in period t, repectively, be the optimal primal and dual olution, i.e., xðtþ¼ arg max x0 LðtÞ¼ arg min L0 þ l U ðx Þ ubject to RðtÞx c (51) max x 0! U ðx Þ x R l ðtþ l c l l : (52) The link cot ued in routing deciion in period t are the congetion price l ðtþ. Each ource compute it new route r ðt þ 1Þ 2H individually that minimize the total cot on it path r ðt þ 1Þ ¼arg min r 2H l l l ðtþr l : (53) We ay that ðr ; x ; L Þ i an equilibrium of TCP/IP if it i a fixed point of (51) (53), i.e., tarting from routing R and aociated ðx ; L Þ, the above iteration yield ðr ; x ; L Þ in the ubequent period. We now characterize the condition under which TCP/ IP a modeled by (51) (53) ha an equilibrium. Conider the following generalized NUM: R2Rn x0 U ðx Þ ubject to Rxc and it Lagrange dual problem minimize L0 max x 0 U ðx Þ x min r 2H l R l l! þ l (54) c l l (55) where r i the th column of R with r l ¼ R l.while(51) utility over ource rate only, problem (54) utility over both rate and route. While (51) i a convex optimization problem without duality gap, problem (54) i nonconvex becaue the variable R i dicrete, and generally ha a duality gap. 7 The intereting feature of the dual problem (55) i that the maximization over R take the form of minimum-cot routing with congetion price L generated by TCP-AQM a link cot. Thi ugget that TCP/IP might turn out to be a ditributed algorithm that attempt to utility, with a proper choice of link cot. Thi i indeed true, provided that an equilibrium of TCP/IP actually exit. Theorem 7: An equilibrium ðr ; x ; L Þ of TCP/IP exit if and only if there i no duality gap between (54) and (55). In thi cae, the equilibrium ðr ; x ; L Þ i a olution of (54) and (55). Method 10: Analyzing a Given Cro-Layer Interaction Through Generalized NUM. Hence, one can regard the layering of TCP and IP a a decompoition of the NUM problem over ource rate and route into a ditributed and decentralized algorithm, carried out on two different time cale, in the ene that an equilibrium of the TCP/IP iteration (51) (53), if it exit, olve (54) and (55). However, an equilibrium may not exit. Even if it doe, it may not be table [144]. The duality gap can be interpreted a a meaure of Bcot for not plitting.[ To elaborate, conider the Lagrangian LðR; x; LÞ ¼! U ðx Þ x R l l þ l l c l l : The primal (54) and dual (55) can then be expreed, repectively, a V np ¼ max min LðR; x; LÞ R2R n ; x0 L0 V nd ¼ min L0 max R2R n ; x0 LðR; x; LÞ: 7 The nonlinear contraint Rx c can be converted into a linear contraint (ee proof of Theorem 8 in [144]), o the integer contraint on R i the real ource of difficulty. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 281

28 If we allow ource to ditribute their traffic among multiple path available to them, then the correponding problem for multipath routing are V mp ¼ max min LðR; x; LÞ R2R m ; x0 L0 V md ¼ min max LðR; x; LÞ: (56) L0 R2R m ; x0 Since R n R m, V np V mp. The next reult clarifie the relation among thee four problem. Theorem 8: V p V d ¼ V mp ¼ V md. According to Theorem 7, TCP/IP ha an equilibrium exactly when there i no duality gap in the ingle-path utility maximization, i.e., when V np ¼ V nd.theorem8then ay that in thi cae, there i no penalty in not plitting the traffic, i.e., ingle-path routing perform a well a multipath routing, V np ¼ V mp. Multipath routing achieve a trictly higher utility V mp preciely when TCP/IP ha no equilibrium, in which cae the TCP/IP iteration (51) (53) cannot converge, let alone olve the ingle-path utility maximization problem (54) or (55). In thi cae the problem (54) and it dual (55) do not characterize TCP/IP, but their gap meaure the lo in utility in retricting routing to ingle-path and i of independent interet. Even though hortet-path routing i polynomial, the ingle-path utility maximization i NP-hard. Theorem 9: The primal problem (54) i NP-hard. Theorem 9 i proved [144] by reducing all intance of the integer partition problem to ome intance of the primal problem (54). Theorem 8, however, implie that the ubcla of the utility maximization problem with no duality gap are in P, ince they are equivalent to multipath problem which are convex optimization problem and hence polynomial-time olvable. Informally, the hard problem are thoe with nonzero duality gap. Theorem 7 ugget uing pure price LðtÞ generated by TCP-AQM a link cot, becaue in thi cae, an equilibrium of TCP/IP, when it exit, aggregate utility over both rate and route. It i hown in [144], however, that uch an equilibrium can be untable, and hence not attainable by TCP/IP. Routing can be tabilized by including a trictly poitive traffic-inenitive component in the link cot, in addition to congetion price. Stabilization, however, reduce the achievable utility. There thu eem to be an inevitable tradeoff between achievable utility and routing tability, when link cot are fixed. If the link capacitie are optimally proviioned, however, pure tatic routing, which i necearily table, i enough to utility. Moreover, it i optimal even within the cla of multipath routing: again, there i no penalty in not plitting traffic acro multiple path. Indeed, pure dynamic routing that ue only congetion price a link cot wa abandoned in APARNet preciely becaue of routing intability [3]. In practice, a weighted um of congetion price and a traffic inenitive component i often ued a link cot in hortet-path routing, i.e., (53) i replaced by r ðt þ 1Þ ¼arg min r 2H l ða l ðtþþb l Þr l (57) for ome poitive contant l.wewillinterpret l a the propagation delay over link l. The parameter a; b determine the reponivene of routing to network traffic: the larger the ratio of a=b, the more reponive routing i. The reult ummarized above correpond to pure dynamic routing b ¼ 0 which i never ued in practical network. When b 9 0, however, it can be hown that for any delayinenitive utility function U ðx Þ, there exit a network with ource uing thi utility function where TCP/IP equilibrium exit but doe not olve the joint utility maximization problem (54) and it dual (55). It turn out that when b 9 0, TCP/IP equilibrium, if it exit, a cla of delay-enitive utility function and their dual [112]. Specifically, Theorem 7 and 8 generalize directly to the cae with a 9 0andb 9 0 when utility function U ðx Þ in (51), (52), (53) are replaced by where U ðx ; Þ :¼ V ðx Þ b a x (58) :¼ l R l l i the end-to-end propagation delay, and V ðx Þ i a trictly concave increaing and continuouly differentiable function. Thi i an example of general delay-enitive utility function U ðx ; Þ where the utility of ource depend not only on it throughput x, but alo on it end-to-end propagation delay. Note that i determined by routing. The particular cla of utility function in (58) ha two ditinct component: V ðx Þ which i trictly increaing in throughput x and ðb=aþx which i trictly decreaing in delay. The weight a; b in the link cot in the routing deciion tranlate directly into a weight in the utility function that determine how enitive utility i to delay. In [112], ome counter-intuitive propertie are alo proved for any cla of delay-enitive utility function optimized by TCP/IP with a; b 9 0, a well a a ufficient condition for global tability of routing update for general network. 282 Proceeding of the IEEE Vol.95,No.1,January2007

29 In [50], three alternative time-cale eparation are further conidered for the joint congetion control and hortet-path routing dynamic baed on congetion price. Analytical characterization and imulation experiment demontrate how the tep ize of the congetion-control algorithm affect the tability of the ytem model, and how the time cale of each control loop and homogeneity of link capacitie affect ytem tability and optimality. In particular, the tringent condition on capacity configuration for TCP/IP interaction to remain table ugget that congetion price, on it own, would be a poor Blayering price[ for TCP and (dynamic routing-baed) IP. In a different routing model capturing today operational practice by ervice provider, [49] conider the following interaction between congetion control and traffic engineering. For a given routing configuration, the utilization of link l i u l ¼ y l =c l,wherey l ¼ P R lx.to penalize routing configuration that conget the link, candidate routing olution are evaluated baed on an increaing, convex cot function fðu l Þ that increae teeply a u l approache 1. The following optimization problem over R,forafixedxand c, capture the traffic-engineering practice: minimize f l R l x =c l!: (59) Thi optimization problem avoid olution that operate near the capacity of the link and conequently tolerate temporary traffic burt. The reulting routing configuration can, therefore, be conidered robut. It i proved [49] that, for certain clae of cot function f, the interaction between end-uer congetion control and the above traffic engineering (at the ame time cale) converge for ufficiently concave utilitie (i.e., ufficiently elatic traffic): ð@ 2 U ðx Þ=@x 2 ðx Þ. 2) Joint Congetion Control and Traffic Engineering: Reearcher have alo carried out forward-engineering of joint congetion control and traffic engineering over multiple path. Variou deign have been preented baed on omewhat different NUM formulation and decompoition method: MATE in [34], TeCP in [59], ditributed adaptive traffic engineering (DATE) in [49], Overlay TCP in [46], and other [60], [69], [84], [101], [142]. For example, in the DATE algorithm [49], edge and core router work together to balance load, limit the incoming traffic rate, and route around failure. The core router compute price baed on local information and feed it back to the edge router which adjut the end-to-end throughput on path. Uing the decompoition approach of Bconitency pricing[ (preented in Section IV-B), an algorithm i developed to update both congetion price and conitency price at core router and feedback to edge router for multipath load plitting. It i hown to be tochatically table (more dicuion in Section V-D) and converge to the joint and global optimum of the following NUM over both R and x: U ðx Þ l f! R l x =c l ubject to Rx c; x 0: (60) Note that the objective function above favor a olution that provide both high aggregate utility to end-uer and a low overall network congetion to the network operator, in order to atify the need for both performance (reflected through the utility function) and robutne (reflected through the cot function). Other related work have tudied different NUM formulation, e.g., without the linear capacity contraint or without the link congetion penalty term in the objective function in problem (60), uing different ditributed olution approache. Thi i one of the cae where alternative decompoition naturally arie and lead to different implementation implication. More dicuion on alternative vertical decompoition will appear in Section IV-C. B. Cae Study 2: Jointly Optimal Congetion Control and Phyical Reource Allocation Adaptive reource allocation per link, uch a power control and error correction coding conidered in thi ection, produce intriguing interaction with end-to-end congetion control. 1) Power Control: Firt conider a wirele multihop network with an etablihed logical topology repreented by R or equivalently fsðlþg, 8l, where ome node are ource of tranmiion and ome node act a relay node. Reviiting the baic NUM (4), we oberve that in an interference-limited wirele network, data rate attainable on wirele link are not fixed number c a in (4), and intead can be written a global and nonlinear function of the tranmit power vector P and channel condition c l ðpþ ¼ 1 T log ð 1 þ KSIR lðpþþ; 8l: Here contant T i the ymbol period, which will be aumed to be one unit without lo of generality, and contant K ¼ð 1 = logð 2 BERÞÞ, where 1 and 2 are contant depending on the modulation and BER i the required bit-error rate. The ignal-to-interference ratio for link l i defined a SIR l ðpþ ¼P l G ll =ð P k6¼l P kg lk þ n l Þ for a given et of path loe G lk (from the tranmitter on Vol. 95, No. 1, January 2007 Proceeding of the IEEE 283

30 logical link k to the receiver on logical link l) andagiven et of noie n l (for the receiver on logical link l). The G lk factor incorporate propagation lo, preading gain, and other normalization contant. Notice that G ll i the path gain on link l (from the tranmitter on logical link l to the intended receiver on the ame logical link). With reaonable preading gain, G ll i much larger than G lk, k 6¼ l, and auming that not too many cloe-by node tranmit at the ame time, KSIRimuchlargerthan1.In thi high-sir regime, c l can be approximated a logðksir l ðpþþ. With the above aumption, we have pecified the following generalized NUM with Belatic[ link capacitie: ubject to U ðx Þ x c l ðpþ; 2SðlÞ 8l x; P 0 (61) where the optimization variable are both ource rate x and tranmit power P. The key difference from the tandard utility maximization (4) i that each link capacity c l i now a function of the new optimization variable: the tranmit power P. The deign pace i enlarged from x to both x and P, which are clearly coupled in (61). Linear flow contraint on x become nonlinear contraint on ðx; PÞ. In practice, problem (61) i alo contrained by the maximum and minimum tranmit power allowed at each tranmitter on link l : P l;min P l P l;max, 8l. The major challenge are the two global dependencie in (61). Source rate x and link capacitie c are globally coupled acro the network, a reflected in the range of ummation f 2 SðlÞg in the contraint in (61). Each link capacity c l ðpþ, in term of the attainable data rate under a given power vector, i a global function of all the interfering power. We preent the following ditributive algorithm and later prove that it converge to the global optimum of (61). To make the algorithm and it analyi concrete, we focu on delay-baed price and TCP Vega window update (a reflected in item 1 and 2 in the algorithm, repectively), and the correponding logarithmic utility maximization over ðx; PÞ,where i a contant parameter in TCP Vega (not a the -fairne parameter here) ubject to log x x c l ðpþ; 2SðlÞ 8l x; P 0: (62) Algorithm 2: Joint Congetion Control and Power Control Algorithm During each time lot t, the following four update are carried out imultaneouly, until convergence: 1) At each intermediate node, a weighted queuing delay l i implicitly updated, where 1 9 0ia contant þ l ðt þ 1Þ ¼ l ðtþþ 4 x ðtþ c l ðtþa5 : (63) c l ðtþ 2SðlÞ 2) At each ource, total delay D i meaured and ued to update the TCP window ize w. Conequently, the ource rate x i updated 8 w ðtþþd 1 ðtþ ; if w ðtþ d w ðtþ D ðtþ G >< w ðt þ 1Þ ¼ w ðtþ 1 D ðtþ ; if w ðtþ d w ðtþ D ðtþ 9 >: w ðtþ; ele. x ðt þ 1Þ ¼ w ðt þ 1Þ : (64) D ðtþ 3) Each tranmitter j calculate a meage m j ðtþ 2 R þ 8 baed on locally meaurable quantitie, and pae the meage to all other tranmitter by a flooding protocol m j ðtþ ¼ jðtþsir j ðtþ P j ðtþg jj : 4) Each tranmitter update it power baed on locally meaurable quantitie and the received meage, where i a contant P l ðt þ 1Þ¼P l ðtþþ 2 l ðtþ P l ðtþ 2 G lj m j ðtþ: (65) With the minimum and maximum tranmit power contraint ðp l;min ; P l;max Þ on each tranmitter, the updated power i projected onto the interval ½P l;min ; P l;max Š. 8 Note that here m j doe not denote price-mapping function a in Section II-A4. j6¼l 284 Proceeding of the IEEE Vol.95,No.1,January2007

31 Item 2 i imply the TCP Vega window update [10]. Item 1 i a modified verion of queuing delay price update [89] (and the original update [10] i an approximation of item1).item3and4decribeanewpowercontroluing meage paing. Taking in the current value of j ðtþsir j ðtþ=p j ðtþg jj a the meage from other tranmitter indexed by j, the tranmitter on link l adjut it power level in the next time lot in two way: firt increae power directly proportional to the current price and inverely proportional to the current power level, then decreae power by a weighted um of the meage from all other tranmitter, where the weight are the path loe G lj. 9 Intuitively, if the local queuing delay i high, tranmit power hould increae, with a more moderate increae when the current power level i already high. If queuing delay on other link are high, tranmit power hould decreae in order to reduce interference on thoe link. To compute m j, the value of queuing delay j,ignalinterference-ratio SIR j, and received power level P j G jj can be directly meaured by node j locally. Thi algorithm only ue the reulting meage m j but not the individual value of j,sir j, P j and G jj. Each meage i a real number to be explicitly paed. To conduct the power update, G lj factor are aumed to be etimated through training equence. It i important to note that there i no need to change theexitingtcpcongetioncontrolandqueuemanagement algorithm. All that i needed to achieve the joint and global optimum of (62) i to utilize the value of weighted queuing delay in deigning power control algorithm in the phyical layer. The tability and optimality of thi layering price can be tated through the following. Theorem 10: For mall enough contant 1 and 2, Algorithm 2 (63), (64), (65) converge to the global optimum of the joint congetion control and power control problem (62). The key tep of thi vertical decompoition, which ue congetion price a the layering price, are again through dual decompoition. We firt aociate a Lagrange multiplier l for each of the contraint P 2SðlÞ x c l ðpþ. Uing the KKT optimality condition [4], [9], olving problem (62) [or (61)] i equivalent to atifying the complementary lackne condition and finding the tationary point of the Lagrangian. Complementary lackne condition tate that at optimality, the product of the dual variable and the aociated primal contraint mut be zero. Thi condition i atified ince the equilibrium queuing delay mut be zero if the total equilibrium ingre rate at a router i trictly maller than the egre link capacity. We alo need to find the tationary point of the Lagrangian: L ytem ðx; P; LÞ ¼ P U ðx Þ P l l P 2SðlÞ x þ P l lc l ðpþ. Bylinearityof the differentiation operator, thi can be decompoed into two eparate maximization problem x0 L congetion ðx;lþ¼ P0 L power ðp;lþ ¼ l U ðx Þ l c l ðpþ: l x l2lðþ The firt maximization i already implicitly olved by the congetion control mechanim for different U (e.g., TCP Vega for U ðx Þ¼ log x ). But we till need to olve the econd maximization, uing the Lagrange multiplier L a the hadow price to allocate exactly the right power to each tranmitter, thu increaing the link data rate and reducing congetion at the network bottleneck. Although the data rate on each wirele link i a global function of all the tranmit power, ditributed olution i till feaible through ditributed gradient method with the help of meage paing. Iue ariing in practical implementation, uch a aynchronou update and reduced meage paing, and their impact on convergence and optimality, are dicued in [18]. Method 11: Dual Decompoition for Jointly Optimal Cro- Layer Deign. The logical topology and route for four multihop connection are hown in Fig. 10 for a numerical example. The path loe G ij are determined by the relative phyical ditance d ij, which we vary in different experiment, by G ij ¼ d 4 ij.thetargetberi10 3 on each logical link. Tranmit power, a regulated by the propoed ditributed power control, and ource rate, a regulated through TCP Vega window update, are hown in Fig. 11. The initial condition of the graph are baed on the equilibriumtateoftcpvegawithfixedpowerlevelof 2.5 mw. With power control, the tranmit power P ditributively adapt to induce a Bmart[ capacity c and queuing delay L configuration in the network, which in turn lead to increae in end-to-end throughput a indicated by the rie in all the allowed ource rate. Notice that ome link capacitie actually decreae while the 9 Thi facilitate a graceful reduction of meage paing cope ince meage from far-away neighbor are weighted much le. Fig. 10. Logical topology and connection for a numerical example of joint congetion control power control. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 285

32 Fig. 11. Typical numerical example of joint TCP Vega congetion control and power control. The top left graph how the primal variable lp. The lower left graph how the dual variable L. The lower right graph how the primal variable x, i.e., the end-to-end throughput. In order of their y-axi value after convergence, the curve in the top left, top right, and bottom left graph are indexed by the third, firt, econd, and fourth link in Fig. 10. The curve in the bottom right graph are indexed by flow 1, 4, 3, 2. capacitie on the bottleneck link rie to the total network utility. Thi i achieved through a ditributive adaptation of power, which lower the power level that caue the mot interference on the link that are becoming a bottleneck in the dynamic demand-upply matching proce. Confirming our intuition, uch a Bmart[ allocation of power tend to reduce the pread of queuing delay, thu preventing any link from becoming a bottleneck. Queuing delay on the four link do not become the ame though, due to the aymmetry in traffic load on the link and different weight in the logarithmic utility objective function. 2) Adaptive Coding: In the econd half of thi ection, we dicu the interaction of per-hop adaptive channel coding with end-to-end congetion control. At the end hot, the utility for each uer depend on both tranmiion rate and ignal quality, with an intrinic tradeoff between the two. At the ame time, each link may alo provide a Bfatter[ (or Bthinner[) tranmiion Bpipe[ by allowing a higher (or lower) decoding error probability. In the baic NUM, the convexity and decompoability propertie of the optimization problem readily lead to a ditributed algorithm that converge to the globally optimal rate allocation. The generalized NUM problem for joint rate-reliability proviioning turn out to be noneparable and nonconvex. We review a price-baed ditributed algorithm, and it convergence to the globally optimal rate-reliability tradeoff under readily verifiable ufficient condition on link coding block length and uer utility curvature. In contrat to tandard price-baed rate control algorithm for the baic NUM, in which each link provide the ame congetion price to each of it uer and each uer provide it willingne to pay for rate allocation to the network, in the joint rate-reliability algorithm each link provide a poibly different congetion price to each of it uer and each uer alo provide it willingne to pay for it own reliability to the network. On ome communication link, phyical layer adaptive channel coding (i.e., error correction coding) can change the information Bpipe[ ize and decoding error probabilitie, e.g., through adaptive channel coding in DSL broadband acce network or adaptive diveritymultiplexing control in MIMO wirele ytem. Then each link capacity i a function of the ignal quality (i.e., decoding reliability) attained on that link. A higher throughput can be obtained on a link at the expene of lower decoding reliability, which in turn lower the end-toend ignal quality for ource travering the link and reduce uer utilitie. Thi lead to an intrinic tradeoff between rate and reliability. Thi tradeoff alo provide an additional degree of freedom for improving each uer utility a well a ytem efficiency. For example, if we allow lower decoding reliability, thu higher information capacity, on the more congeted link, and higher decoding reliability, thu lower information capacity, on the le congeted link, we may improve the end-to-end rate and reliability performance of each uer. Clearly, rate-reliability tradeoff i globally coupled acro the link and uer. In the cae where the rate-reliability tradeoff i controlled through the code rate of each ource on each link, there are two poible policie: integrated dynamic reliability policy and differentiated dynamic reliability policy. In integrated policy, a link provide the ame error probability (i.e., the ame code rate) to each of the ource travering it. Since a link provide the ame code rate to each of it ource, it mut provide the lowet code rate that atifie the requirement of the ource with the highet reliability. Thi motivate a more general approach called differentiated policy to fully exploit the rate-reliability tradeoff when there exit multicla ource (i.e., ource with different reliability requirement) in the network. Under the differentiated dynamic reliability policy, a link can provide a different error probability (i.e., a different code rate) to each of the ource uing thi link. We aume that each ource ha a utility function U ðx ; Þ,wherex i an information data rate and i reliability of ource. We aume that the utility function i a continuou, increaing, and trictly concave function of x and. Each ource ha a minimum reliability requirement min. The reliability of ource i defined a ¼ 1 p where p i the end-to-end error probability of ource. Each link l ha it maximum tranmiion capacity c max l. 286 Proceeding of the IEEE Vol.95,No.1,January2007

33 After link l receive the data of ource from the uptream link, it firt decode it to extract the information data of the ource and encode it again with it own code rate, r l;, where the code rate i defined by the ratio of the information data rate x at the input of the encoder to the tranmiion data rate t l; at the output of the encoder. Thi allow a link to adjut the tranmiion rate and the error probability of the ource, ince the tranmiion rate of ource at link l can be defined a t l; ¼ x r l; and the error probability of ource at link l can be defined a a function of r l; by p l; ¼ E l ðr l; Þ which i aumed to be an increaing function of r l;.rarely are there analytic formula for E l ðr l; Þ, and we will ue variou upper bound on thi function. The end-to-end error probability for each ource i p ¼ 1 Y ð1 p l; Þ¼1 Y 1 E l ðr l; Þ : l2lðþ l2lðþ Auming that the error probability of each link i mall (i.e., p l; 1), we can approximate the end-to-end error probability of ource a to each of the ource travering it, the aociated generalized NUM become the following problem with variable x, R, r: ubject to U ðx ; Þ 1 E l ðr l; Þ; 2SðlÞ x l2lðþ 8 Cl max ; r l; 8l min 1; 8 0 r l; 1; 8l; 2 SðlÞ: (66) There are two main difficultie in ditributively and globally olving the above problem. The firt one i the convexity of E l ðr l; Þ. If random coding baed on binary coded ignal i ued, a tandard upper bound on the error probability i p l G MðR 0 r l Þ where M i the block length and R 0 i the cutoff rate. In thi cae, E l ðr l Þ¼ð1=2Þ2 MðR 0 r l Þ i a convex function for given M and R 0. A more general approach for dicrete memoryle channel model i to ue the random code enemble error exponent that upper bound the decoding error probability p p l; ¼ E l ðr l; Þ: l2lðþ l2lðþ Hence, the reliability of ource can be expreed a 1 E l ðr l; Þ: l2lðþ Since each link l ha a maximum tranmiion capacity Cl max, the um of tranmiion rate of ource that are travering each link cannot exceed Cl max t l; ¼ 2SðlÞ 2SðlÞ x Cl max ; 8l: r l; For (the more general) differentiated dynamic reliability policy, in which a link may provide a different code rate p l expð ME r ðr l ÞÞ where M i the codeword block length and E r ðr l Þ i the random coding exponent function, which i defined a where E r ðr l Þ¼ max max 01 Q ½ E oð; QÞ r l Š " E o ð; QÞ ¼ log J 1 # K 1 1þ Q k P 1=ð1þÞ jk j¼0 k¼0 K i the ize of input alphabet, J i the ize of output alphabet, Q k i the probability that input letter k i choen, and P jk i the probability that output letter j i received given that input letter k i tranmitted. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 287

34 In general, E l ðr l Þ¼expð ME r ðr l ÞÞ may not be convex (even though it i known that E r ðr l Þ i a convex function). However, the following lemma provide a ufficient condition for it convexity. Lemma 1: If the abolute value of the firt derivative of E r ðr l Þ iboundedawayfrom0andabolutevalueof the econd derivative of E r ðr l Þ i upper bounded, then for a large enough codeword block length M, E l ðr l Þ i a convex function. Method 12: Computing Condition Under Which a General Contraint Set i Convex. The econd difficulty i the global coupling of contraint P 2SðlÞ ðx =r l; ÞCl max. Thi problem i tackled by firt introducing auxiliary variable c l;, which can be interpreted a the allocated tranmiion capacity to ource at link l ubject to U ðx ; Þ 1 E l ðr l; Þ; l2lðþ x c l; ; 8l; 2 SðlÞ r l; c l; Cl max ; 8l 2SðlÞ min 1; r l; 1; 8l; 2 SðlÞ 0 c l; C max l ; 8l; 2 SðlÞ: (67) Note that effectively a new Bcheduling layer[ ha been introduced into the problem: cheduling of flow by deciding bandwidth haring on each link fc l; g. Method 13: Introducing a New Layer to Decouple a Generalized NUM. A log change of variable x 0 ¼ log x can be ued to decouple the above problem for horizontal decompoition. Define a modified utility function U 0ðx0 ; Þ¼ U ðe x0 ; Þ, which need to be concave in order for the tranformed problem to remain a convex optimization problem, imilar to the curvature condition on utility function in Section II-B2. Define g ðx ; U ðx ; 2 x 2 2 U ðx ; Þ h ðx @2 U ðx ; 2 U ðx ; 2 U ðx ; ðx ; q ðx ; U ðx ; 2 : Lemma 2: If g ðx ; Þ G 0, h ðx ; Þ G 0, and q ðx ; Þ G 0, then U 0 ðx0 ; Þ i a concave function of x 0 and. Now the joint rate-reliability problem (66) can be olved ditributively through dual decompoition. Algorithm 3: Differentiated Dynamic Reliability Policy Algorithm In each iteration t, by olving (68) over ðx 0 ; Þ, each ource determine it information data rate and requeted reliability (i.e., x 0 ðtþ or equivalently, x ðtþ ¼e x0 ðtþ, and ðtþ) that it net utility baed on the price in the current iteration. Furthermore, by price update (69), the ource adjut it offered price per unit reliability for the next iteration. Source problem and reliability price update at ource : Source problem U x 0 ; ðtþx 0 ðtþ ubject to min 1 (68) where ðtþ ¼ P l2lðþ l;ðtþ i the end-to-end congetion price at iteration t. Price update (where tep ize can be et to ðtþ ¼ 0 =t for ome 0 9 0) 10 ðt þ 1Þ ¼½ ðtþ ðtþð ðtþ ðtþþš þ (69) where ðtþ ¼1 P l2lðþ E lðr l; ðtþþ i the end-toend reliability at iteration t. 10 Diminihing tepize, e.g., ðtþ ¼ 0 =t, can guarantee convergence when the primal optimization problem objective function i concave but not trictly concave in all the variable, wherea a contant tepize cannot. 288 Proceeding of the IEEE Vol.95,No.1,January2007

35 Concurrently in each iteration t, by olving problem (70) over ðc l; ; r l; Þ, 8 2 SðlÞ, each link l determine the allocated tranmiion capacity c l; ðtþ and the code rate r l; ðtþ of each of the ource uing the link, o a to the Bnet revenue[ of the network baed on the price in the current iteration. In addition, by price update (71), the link adjut it congetion price per unit rate for ource during the next iteration. Link problem and congetion price update at link l: Link problem: ubject to l; ðtþðlog c l; þ log r l; Þ 2SðlÞ ðtþe l ðr l; Þ c l; Cl max 2SðlÞ 0 c l; C max l ; 2 SðlÞ 0 r l; 1; 2 SðlÞ: (70) primal olution of the joint rate-reliability problem, i.e., x ¼ðe x0 Þ 8, R, c,andr are the globally optimal primal olution of problem (66). We now preent numerical example for the propoed algorithm by conidering a imple network, hown in Fig. 12, with a linear topology coniting of four link and eight uer. Utility function for uer i U ðx ; Þ in the following -fair form, hifted uch that U ðx min ; min Þ¼0 and U ðx max ; max Þ¼1 (where x min, min, x max, max are contant), and with utility on rate and utility on reliability ummed up with a given weight between rate and reliability utilitie U ðx ; Þ¼ x 1 x maxð1 Þ x minð1 Þ x minð1 Þ þð1 Þ ð1 Þ maxð1 Þ minð1 Þ minð1 Þ : Price update (where tep ize can be et to ðtþ ¼ 0 =t for ome 0 9 0) l; ðtþ1þ¼ l; ðtþ ðtþ log c l; ðtþþlog r l; ðtþ x 0 ðtþ þ ¼ l; ðtþ ðtþ log c l; ðtþþlog r l; ðtþ log x ðtþ þ 2 SðlÞ: (71) In the above algorithm, to olve problem (68), ource need to know ðtþ, the um of congetion price l; ðtþ of link that are along it path LðÞ. Thi can be obtained by the notification from the link, e.g., through acknowledgment packet. To carry out price update (69), the ource need to know the um of error probabilitie of the link that are along it path (i.e., it own reliability that i provided by the network, ðtþ). Thi can be obtained by the notification either from the link that determine the code rate for the ource [by olving problem (70)] or from the detination that can meaure it end-to-end reliability. To olve the link problem (70), each link l need to know ðtþ from each of ource uing thi link l. Thicanbe obtained by the notification from thee ource. To carry out price update (71), the link need to know the information data rate of each of the ource that are uing it (i.e., x ðtþ). Thi can be meaured by the link itelf. Different weight are given to the eight uer a follow: ¼ 0:5 v; if i an odd number 0:5 þ v; if i an even number (72) and vary v from 0 to 0.5 in tep ize of The decoding error probability on each link l i aumed to be of the following form, with contant M: p l ¼ 1 2 exp ð Mð1 r lþþ: We trace the globally optimal tradeoff curve between rate and reliability uing differentiated and integrated dynamic reliability policie, and compare the network utility achieved by the following three cheme: Static reliability: each link provide a fixed error probability Only rate control i performed to the network utility. Integrated dynamic reliability: each link provide the ame adjutable error probability to all it uer. Method 14: End Uer Generated Pricing for Ditributed Update of Metric in Uer Utility Function. Theorem 11: For ufficiently concave utilitie and ufficiently trong code, and diminihing tepize, the dual variable LðtÞ and MðtÞ converge to the optimal dual olution L and M and the correponding primal variable x 0, R, c, and r are the globally optimal Fig. 12. Network topology and flow route for a numerical example of rate-reliability tradeoff. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 289

36 concave utilitie, problem (73) i a convex optimization afteralogchangeofvariablefp l ; P k g. It olution can now be ditributively carried out uing either the penalty function approach or the dual-decompoition-baed Lagrangian relaxation approach. Both have tandard convergence propertie but now producing different implication to the time cale of TCP/MAC interaction (e.g., [74], [143], [169]), a hown in the ret of thi ection. Firt the penalty function approach i purued. We firt define h l ðp; x 0 Þ¼logð P 2SðlÞ ex0 Þ c 0 l log p l P k2n I to ðlþ logð1 P m2l out ðkþ p mþ and w n ðpþ ¼ P m2l out ðnþ p m 1. Then, problem (73) can be rewritten a Fig. 13. Comparion of data rate and reliability tradeoff in a numerical example by each policy for Uer 2, when are changed according to (72). Differentiated dynamic reliability: each link provide a poibly different error probability to each of it uer. Fig. 13 how the globally optimal tradeoff curve between rate and reliability for a particular uer, under the three policie of tatic reliability, integrated dynamic reliability, and differentiated dynamic reliability, repectively. The differentiated cheme how a much larger dynamic range of tradeoff than both the integrated and tatic cheme. The gain in total network utility through joint rate and reliability control i hown in Fig. 14. U 0 x 0 ubject to h l ðp; x 0 Þ0; 8l w n ðpþ 0; 8n x 0 min x 0 max x0 ; 8 0 p l 1; 8l: (74) Intead of olving problem (74) directly, we apply the penalty function method and conider the following problem: ubject to Vðp; x 0 Þ x 0 min x 0 max x0 ; 8 0 p l 1; 8l (75) C. Cae Study 3: Jointly Optimal Congetion and Contention Control Following the notation in Section II-B, for joint end-toend rate allocation and per-hop MAC, the generalized NUM problem for random-acce-baed MAC and TCP can be formulated a the following optimization over ðx; P; pþ: ubject to U ðx Þ Y x c l p l ð1 P k Þ; 2SðlÞ k2nto I ðlþ p l ¼ P n ; 8n l2l out ðnþ x min x x max ; 8 0 P n 1; 8n 8l 0 p l 1; 8l: (73) Similar to the dicuion on MAC forward-engineering and jointly optimal rate reliability control, for ufficiently Fig. 14. Comparion of the achieved network utility attained in a numerical example by the differentiated dynamic policy, the integrated dynamic policy, and the tatic policy, when are changed according to (72). 290 Proceeding of the IEEE Vol.95,No.1,January2007

37 where Vðp; x 0 Þ¼ P U0 ðx0 Þ P l maxf0; h lðp; x 0 Þg P n maxf0; w nðpþg and i a poitive contant. Since the objective function of problem (75) i concave, problem (75) i convex optimization with imple, decoupled contraint, which can be olved by uing a ubgradient projection algorithm. We can eaily how that and where x 0 Þ ¼ l p x 0 0 P k2l I from ðt lþ! k 1 P m2l out ðt l Þ p tl m x 0 e 0 l2lðþ l P k2sðlþ ex0 k 8 >< l ¼ 0; if P Q e x0 n cl p l 1 P n2sðlþ k2nto >: I ðlþ 1; otherwie n ¼ 0; if P m2l out ðnþ p m 1 1; otherwie. m2l out ðkþ p m! (76) (77) Then, an iterative ubgradient projection algorithm, with iteration indexed by t, that olve problem (75) i obtained a follow. Algorithm 4: Joint End-to-End Congetion Control and Local Contention Control Algorithm On each logical link, tranmiion i decided to take place with peritence probability and concurrently at each ource, the end-to-end rate i adjuted " x 0 ðtþ1þ¼ x0 # x 0 max x0 Þ ; 0 where ½aŠ b c ¼ maxfminfa; bg; cg. p¼pðtþ;x0 ¼x 0 ðtþ x 0 min (79) The joint control algorithm (78) and (79) can be implemented a follow. Each link l (or it tranmiion node t l ) update it peritence probability p l ðtþ uing (78), and concurrently, each ource update it data rate x ðtþ uing (79). To calculate the ubgradient in (76), each link need information only from link k, k 2 L I from ðt lþ, i.e., from link whoe tranmiion are interfered from the tranmiion of link l, and thoe link are in the neighborhood of link l. To calculate the ubgradient in (77), each ource need information only from link l, l 2 LðÞ, i.e.,fromlink on it routing path. Hence, to perform the algorithm, each ource and link need only local information through limited meage paing and the algorithm can be implemented in a ditributed way. In particular, note that n i calculated at the tranmitter node of each link to update the peritence probability of that link, and doe not need to be paed among the node. There i no need to explicitly pa around the value of peritence probabilitie, ince P their effect are included in f l g.quantitieucha m2l out ðt l Þ p m and P k2sðlþ expðx0 kþ can be meaured locally by each node and each link. To implement a dual-decompoition-baed algorithm intead, we can decompoe problem (73) into two problem, uing a tandard technique of dual decompoition alo ued in [16] and [143] ubject to U ðx Þ x y l ; 2SðlÞ x min 8l x x max ; 8 (80) where y l i the average data rate of link l, and " p l ðt þ 1Þ ¼ # 1 x0 Þ ; l p¼pðtþ;x0 ¼x 0 ðtþ 0 (78) ubject to ^UðpÞ p m 1; 8n m2l out ðnþ 0 p l 1; 8l (81) Vol. 95, No. 1, January 2007 Proceeding of the IEEE 291

38 where 8 ^UðpÞ¼max < U ðx Þ x y l ðpþ; 8l; : 2SðlÞ 0 Y y l ðpþ¼c l p g: 1 p m k2nto I ðlþ m2l out ðkþ 1 A; 8l For a given y, problem (80) can be olved by dual decompoition and ditributed ubgradient method jut a before. We now olve problem (81). To thi end, we firt add a penalty function to the objective function of the problem a ^VðpÞ ubject to 0 p l 1; 8l (82) where ^VðpÞ ¼ ^UðpÞ maxf0; P n ð1 P m2l out ðnþ p mþg and i a poitive contant. Since problem (82) i a convex problem with imple contraint, we can olve it by uing a ubgradient projection algorithm a " p l ðt þ 1Þ ¼ # 1 ^VðpÞ ; 8l l p¼pðtþ ^VðpÞ=@p l i a ubgradient of ^VðpÞ with repect to p l. It can be readily verified ^VðpÞ=@p l i obtained l ¼ l ðtþc l x min 0 1 p m k2nto I ðlþ m2l out ðkþ n2l I from ðt lþ Y k2n I to ðnþ;k6¼t l x x max ; 8 n ðtþc np n 1 m2l out ðkþ p m 1 A )( : ) 0 1 A tl (84) and L ðtþ i the optimal dual olution to dual problem of (80) with y ¼ yðpðtþþ. Thi dual-decompoition-baed algorithm can alo be implementedinaditributedway.ineachtimelot,each link determine it peritence probability by olving (83) with the help of local meage paing to obtain the expreion in (19). Then, within the time lot, baed on yðpðtþþ, each ource and link ue tandard dualdecompoition-baed algorithm to olve (80) and determine the data rate of each ource in the time lot. Unlike the penalty-function-baed algorithm, thi dual-decompoition-baed algorithm clearly decompoe TCP and MAC layer through the vertical decompoition (80) and (81). However, it need an embedded loop of iteration [i.e., the convergence of a ditributed ubgradient algorithm to olve (80) in each time lot]. Hence, it may require longer convergence time than the penaltyfunction-baed algorithm. Thi comparion between two alternative decompoition, together with yet another decompoition in [169] for the ame NUM formulation, highlight the engineering implication of different decompoition to the time cale of the interaction between functional module. Method 15: Providing Different Timecale of Protocol Stack Interaction Through Different Decompoition Method. D. Cae Study 4: Jointly Optimal Congetion Control, Routing, and Scheduling A generalized NUM i formulated in [17], [35], [36], [82], [83], [99], and [129] where the key additional feature i the optimization over not jut ource rate but alo cheduling of medium acce and the incorporation of cheduling contraint. The tandard dual decompoition decompoe it vertically into ubproblem that can be olved through congetion control, routing, and cheduling. Conider an ad hoc wirele network with a et N of node and a et L of logical link. We aume ome form of power control o that each logical link l ha a fixed capacity c l when it i active. The feaible rate region at the link layer i the convex hull of the correponding rate vector of independent et of the conflict graph. Let / denote the feaible rate region. Let x k i be the flow rate generated at node i for detination k. Weaumethereiaqueuefor each detination k at each link ði; jþ.letfij k be the amount of capacity of link ði; jþ allocated to the flow on that link for final detination k. Conider the following generalized NUM in variable x 0, fij k 0: where n ¼ 0; if P m2l out ðnþ p m 1 1; otherwie U ðx Þ ubject to x k i j:ði;jþ2l f k ij j:ðj;iþ2l f k ji ; 8i; k f 2 / (85) 292 Proceeding of the IEEE Vol.95,No.1,January2007

39 where x i a horthand for x k i. The firt contraint i a flow balance equation: the flow originated from node i for final detination k plu total capacity allocated for tranit flow through node i for final detination k hould be no more than the total capacity going out of node i for final detination k. The econd contraint i on chedulability. The dual problem of (85) decompoe into minimizing the um of the reulting value of the following two ubproblem: D 1 ðlþ :¼ max x 0 D 2 ðlþ :¼ max f k ðu ðx Þ x Þ (86) k ij 0 i i;k j fij k fji k ubject to f 2 /: (87) The firt ubproblem i congetion control where i the congetion price locally at ource ¼ði; kþ. Theecond ubproblem correpond to a joint problem of multipath routing and allocation of link capacitie. Thu, by dual decompoition, the flow optimization problem decompoe into eparate local optimization problem that interact through congetion price. The congetion control problem (86) admit a unique r x ðlþ ¼U 0 1 ð Þ.Thejointroutingandcheduling problem (87) i equivalent to max f k fij k0 ij k i k j i;j k ubject to f 2 /: Hence, an optimal chedule i to have f k ij ¼ c ij, if k ð k i k j Þ and0,otherwie.thimotivatethe following joint congetion control, cheduling, and routing algorithm: Algorithm 5: Joint Congetion Control, Routing, and Scheduling Algorithm 1) Congetion control: the ource of flow et it rate a x ðþ ¼U 0 1 ð Þ. 2) Scheduling: For each link ði; jþ, finddetinationk uch that k 2 arg max k ð k i k j Þ and define w ij :¼ k i k j. Chooe an ~ f 2 arg max f2/ Pði;jÞ2L w ij f ij uch that ~ f i an extreme point. Thoe link ði; jþ with f ~ ij 9 0 will tranmit and other link ði; jþ (with f ~ ij ¼ 0) will not. 3) Routing: over link ði; jþ 2L with f ~ ij 9 0, end data for detination k at full link capacity c ij. 4) Price update: each node i update the price on the queue for detination k according to 2 0 k i ðt þ 1Þ¼ 4 k i xk i ðlðtþþ fji k j:ðj;iþ2l þ fij k j:ði;jþ2l ðlðtþþ 13þ ðlðtþþa5 : (88) The w ij value repreent the maximum differential congetion price of detination k between node i and j, and wa introduced in [138]. The above algorithm ue back preure to perform optimal cheduling and hop-byhop routing. Thi i an illutrating cae tudy on the potential interaction between back-preure-baed cheduling and dual decompoition for protocol tack deign, where the Bpreure[ are the congetion price. Method 16: Maximum Differential Congetion Pricing for Node-Baed Back-Preure Scheduling. There are everal variation that have lead to an array of alternative decompoition, a will be dicued in Section IV-C. For example, intead of uing a dual-driven algorithm, a in [17] and [99], where the congetion control part (Step 1 above) i tatic, the algorithm in [35] and [129] are primal-dual-driven, where the ource congetion control algorithm can be interpreted a an acent algorithm for the primal problem. Method 17: Architectural Implication Due to Dual Decompoition: Aborb Routing Functionality Into Congetion Control and Scheduling. Starting with a different et of optimization variable and a given NUM formulation, another family of variation ue a link-centric formulation, rather than the nodecentric one tudied in thi ection o far. Compared to the dual-decompoition of the link-centric formulation, Algorithm 5 ha the following two major advantage. Firt, it can accommodate hop-by-hop routing without further modeling of multipath routing, and the routing i aborbed into end-to-end congetion control and per-contentionregion cheduling. Second, it only require the congetion price at the ource, rather than the um of congetion price along the path, to accomplih congetion control. However, it alo uffer from a more complicated et of dual variable: each node ha to maintain a per-detination queue. More dicuion on the tradeoff between nodecentric and link-centric formulation can be found in [85]. Now back to generalized NUM (85). It i known that the algorithm converge tatitically to a neighborhood of Vol. 95, No. 1, January 2007 Proceeding of the IEEE 293

40 the optimal point uing contant tepize, in the ene that the time average tend to the optimal value arbitrarily cloely. Specifically, let the primal function (the total achieved network utility) be PðxÞ and let x be the optimum. Let xðtþ :¼ ð1=tþ P t ¼0 xðþ be the running average rate. Similarly, let DðLÞ be the dual objective function, L be an optimal value of the dual variable, and LðtÞ :¼ ð1=tþ P t ¼1 LðÞ be the running average price. Theorem 12: Conider the dual of (85) and uppoe the ubgradient of the dual objective function i uniformly bounded. Then, for any 9 0, there exit a ufficiently mall tepize in (88) uch that lim inf PðxðtÞÞ t!1 Pðx Þ lim up D LðtÞ DðL Þþ: t!1 The mot difficult tep in Algorithm 5 i cheduling. Solving it exactly require a centralized computation which i clearly impractical in large-cale network. Variou cheduling algorithm and ditributed heuritic have been propoed in the context of joint rate allocation, routing, and cheduling. The effect of imperfect cheduling on cro-layer deign have recently been characterized in [83], for both the cae when the number of uer in the ytem i fixed and the cae with dynamic arrival and departure of the uer. IV. DECOMPOSITION METHODS Variou decompoition method have been ued in the lat two ection on horizontal and vertical decompoition. In thi ection, we provide a more comprehenive dicuion on the theory of optimization decompoition, firt on primal and dual decompoition for decoupling contraint, then on conitency pricing for decoupling objective function, and finally on alternative decompoition. A. Decoupling Coupled Contraint The baic idea of a decompoition i to decompoe the original large problem into ubproblem which are then coordinated by a mater problem by mean of ome kind of Table 7 Summary of Main Notation for Section IV ignalling, often without the need to olve the mater problem centrally either [5]. Many of the exiting decompoition technique can be claified into primal decompoition and dual decompoition method. The former (alo called partitioning of variable, decompoition by right-hand ide allocation, or decompoition with repect to variable) i baed on decompoing the original primal problem, wherea the latter (alo termed Lagrangian relaxation of the coupling contraint or decompoition with repect to contraint) i baed on decompoing the dual of the problem. A illutrated in Fig. 15, primal decompoition method have the interpretation that the mater problem directly give each ubproblem an amount of reource that it can ue; the role of the mater problem i then to properly allocate the exiting reource. In computer engineering terminology, the mater problem adapt the licing of reource among competing demand. In dual decompoition method, the mater problem et the price for the reource to each ubproblem which ha to decide the amount of reource to be ued depending on the price; the role of the mater problem i then to obtain the bet pricing trategy. In many cae, it i deirable and poible to olve the mater problem ditributively through meage paing, which can be local or global, implicit or explicit. In ummary, the engineering mechanim realizing dual decompoition i pricing feedback while that realizing primal decompoition i adaptive licing. Note that the terminology of Bprimal-dual[ ha a number of different meaning. For example, Bprimal-dual interior-point method[ i a cla of algorithm for centralized computation of an optimum for convex optimization, and Bprimal-dual ditributed algorithm[ i ometime ued to decribe any algorithm that olve the primal and dual problem imultaneouly. Two other et of related Fig. 15. Schematic illutrating optimization problem decompoition. 294 Proceeding of the IEEE Vol.95,No.1,January2007

41 terminology have been ued in previou ection. Firt, Bprimal-driven,[ Bdual-driven,[ and Bprimal-dual-driven[ algorithm are ued to differentiate where i the update dynamic carried out: over the primal variable, or over the dual variable, or over both. Second, Bpenaltyfunction-baed algorithm[ refer to thoe ditributed algorithm obtained by moving the coupled contraint to the augmented objective function in the primal problem through a penalty term. Thi i in contrat to Bdualdecompoition-baed algorithm[ that are obtained through dual decompoition. In thi ection, primal and dual decompoition have yet a different et of meaning: decompoing coupling contraint through direct reource allocation and indirect pricing control, repectively. It i alo important to note that a given decompoition method mayinturnleadtomorethanoneditributed algorithm. Primal and dual decompoition leverage decompoability tructure in a given optimization problem to turn it into ubproblem coordinated by a mater problem. Different ditributed algorithm may then be developed baed on the ame decompoition, e.g., depending on the choice of update method (e.g., gradient or cutting plane or ellipoid method), the ordering of variable update (e.g., Jacobi or Gau-Siedel), and the time cale of neted loop. 1) Dual Decompoition of the Baic NUM: We firt illutrate how the dual decompoition approach can be applied to the baic NUM problem to produce the tandard dualdecompoition-baed ditributed algorithm. Aume that the utility function are concave, and poibly linear function. The Lagrange dual problem of (4) i readily derived. We firt form the Lagrangian Lðx; LÞ ¼ U ðx Þþ l l c l 2SðlÞ x 1 A where l 0 i the Lagrange multiplier (i.e., link price) aociated with the linear flow contraint on link l. Additivity of total utility and linearity of flow contraint lead to a Lagrangian dual decompoition into individual ource term Lðx; LÞ ¼ ¼ 2 0 4U ðx L ðx ; q Þþ l l2lðþ l c l l 1 3 Ax 5 þ l c l l where q ¼ P l2lðþ l. For each ource, L ðx ; q Þ¼ U ðx Þ q x only depend on local rate x and the path price q (i.e., um of l on link ued by ource ). The Lagrange dual function gðlþ i defined a the d Lðx; LÞ over x for a given L. ThiBnet utility[ maximization obviouly can be conducted ditributively by each ource x ðq Þ¼argmax½U ðx Þ q x Š; 8: (89) Such Lagrangian r x ðlþ will be referred to a price-baed rate allocation (for a given price L). The Lagrange dual problem of (4) i minimize gðlþ ¼Lðx ðlþ; LÞ ubject to L 0 (90) where the optimization variable i L. Since gðlþ i the pointwie upremum of a family of affine function in L, it i convex and (90) i a convex minimization problem. Since gðlþ may be nondifferentiable, an iterative ubgradient method can be ued to update the dual variable L to olve the dual problem (90) 2 0 l ðt þ 1Þ¼4 l ðtþ ðtþ@ c l 2SðlÞ x ðq ðtþ 13 þ ÞA5 ; 8l (91) where c l P 2SðlÞ x ðq ðtþþ i the lth component of a ubgradient vector of gðlþ, t i the iteration number, and ðtþ 9 0 are tep ize. Certain choice of tep ize, uch a ðtþ ¼ 0 =t, 9 0, guarantee that the equence of dual variable LðtÞ converge to the dual optimal L a t!1. It can be hown that the primal variable x ðlðtþþ alo converge to the primal optimal variable x.foraprimal problem that i a convex optimization, the convergence i toward a global optimum. In ummary, the equence of ource and link algorithm (89), (91) form a tandard dual-decompoitionbaed ditributed algorithm that globally olve NUM (4) and the dual problem (90), i.e., compute an optimal rate vector x and optimal link price vector L.Notethatno explicit ignaling i needed. Thi i becaue the ubgradient i preciely the difference between the fixed link capacity and the varying traffic load on each link, and the ubgradient update equation ha the interpretation of weighted queuing delay update. The general methodology of primal and dual decompoition i now preented. A more comprehenive tutorial can be found in [104]. It turn out that primal and dual decompoition are alo interchangeable through alternative repreentation of the optimization problem. 2) Primal Decompoition: A primal decompoition i appropriate when the problem ha a coupling variable uch that, when fixed to ome value, the ret of the optimization Vol. 95, No. 1, January 2007 Proceeding of the IEEE 295

42 problem decouple into everal ubproblem. Conider, for example, the following problem over y, fx i g: f i ðx i Þ i ubject to x i 2 i ; 8i A i x i y; 8i y 2Y: (92) If variable y were fixed, then the problem would decouple. Thi ugget eparating the optimization in (92) into two level of optimization. At the lower level, we have the ubproblem, one for each i over x i, in which (92) decouple when y i fixed ubject to f i ðx i Þ x i 2 i A i x i y: (93) At the higher level, we have the mater problem in charge of updating the coupling variable y by olving i f i ðyþ ubject to y 2Y (94) where fi ðyþ i the optimal objective value of problem (93) for a given y. A ubgradient for each fi ðyþ i given by If the contraint P i h iðx i Þc were abent, then the problem would decouple. Thi ugget relaxing the coupling contraint in (96) a f i ðx i Þ L T i! h i ðx i Þ c ubject to x i 2 i 8i (97) uch that the optimization eparate into two level of optimization. At the lower level, we have the ubproblem, one for each i over x i, in which (97) decouple i f i ðx i Þ L T h i ðx i Þ ubject to x i 2 i : (98) At the higher level, we have the mater dual problem in charge of updating the dual variable L by olving the dual problem minimize gðlþ ¼ g i ðlþþl T c i ubject to L 0 (99) where g i ðlþ i the dual function obtained a the maximum value of the Lagrangian olved in (98) for a given L. Thi approach i in fact olving the dual problem intead of the original primal one. Hence, it will only give appropriate reult if trong duality hold. A ubgradient for each g i ðlþ i given by i ðyþ ¼L i ðyþ (95) i ðlþ ¼ h i x i ðlþ (100) where L i ðyþ i the optimal Lagrange multiplier correponding to the contraint A i x i y in problem (93). The global ubgradient i then ðyþ ¼ P i iðyþ ¼ P i L i ðyþ. The ubproblem in (93) can be locally and independently olved with the knowledge of y. 3) Dual Decompoition: A dual decompoition i appropriate when the problem ha a coupling contraint uch that, when relaxed, the optimization problem decouple into everal ubproblem. Conider, for example, the following problem: f i ðx i Þ i ubject to x i 2 i 8i h i ðx i Þc: (96) i where x i ðlþ i the optimal olution of problem (98) for a given L. The global ubgradient i then ðlþ ¼ P i iðlþþc ¼ c P i h iðx i ðlþþ. The ubproblem in (98) can be locally and independently olved with knowledge of L. Method 18: Primal and Dual Decompoition for Coupling Contraint. Not all coupling contraint can be readily decompoed through primal or dual decompoition. For example, the feaibility et of SIR in wirele cellular network power control problem i coupled in a way with no obviou decompoability tructure. A reparametrization of the contraint et i required before dual decompoition can be applied [47]. Sometime, the coupling i time-invariant a in ome broadband acce network, and very efficient Btatic pricing[ can be ued to decouple uch Btatic coupling[ [52]. 296 Proceeding of the IEEE Vol.95,No.1,January2007

43 B. Decoupling Coupled Objective Examining the dual decompoition of the baic NUM reveal the following reaon why ditributed and end-toend algorithm can olve the baic NUM (4): 1) Separability in objective function: The network utility i a um of individual ource utilitie. 2) Additivity in contraint function: The linear flow contraint are umming over the individual flow. 3) P Interchangeability of ummation index: l P l 2SðlÞ x ¼ P x P l2lðþ l. 4) Zero duality gap. Property 3 i trivial. When Property 2 i violated, decompoition i much harder and uually involve ome reparametrization of the contraint et. When Property 4 doe not hold, recent work have provided three alternative olution, a will be outlined in Section V-E. For cae where Property 1 fail, recent progre on coupled utility formulation ha been made [23], [130]. In many communication ytem, utilitie are indeed coupled. An example of the cooperation model can be found in network where ome node form a cluter and the utility obtained by each of them depend on the rate allocated to other in the ame cluter (thi can be interpreted a a hybrid model of elfih and nonelfih utilitie). An example of the competition model i in wirele power control and DSL pectrum management, where the utilitiearefunctionofsirthataredependentonthe tranmit power of other uer. The generalized NUM problem conidered in thi ubection i k U k ubject to x k 2 k 8k K k¼1 x k ; fx l g l2lðkþ h k ðx k Þc (101) where the (trictly concave) utilitie U k depend on a vector local variable x k and on variable of other utilitie x l for l 2LðkÞ (i.e., coupled utilitie), LðkÞ i the et of node coupled with the kth utility, the et k are arbitrary convex P et, and the coupling contraining function k h kðx k Þ i not necearily linear, but till convex. Note that thi model ha two type of coupling: coupled contraint and coupled utilitie. One way to tackle the coupling problem in the utilitie i to introduce auxiliary variable and additional equality contraint, thu tranferring the coupling in the objective function to coupling in the contraint, which can be decoupled by dual decompoition and olved by introducing additional conitency pricing. It i reaonable to aume that if two node have their individual utilitie dependent on each other local variable, then there mut be ome communication channel in which they can locally exchange pricing meage. It turn out that the global link congetion price update of the tandard dualdecompoition-baed ditributed algorithm i not affected by the local conitency price update, which can be conducted via thee local communication channel among the node. The firt tep i to introduce in problem (101) the auxiliary variable x kl for the coupled argument in the utility function and additional equality contraint to enforce conitency k U k x k ; fx kl g l2lðkþ ubject to x k 2 k 8k h k ðx k Þc k x kl ¼ x l ; 8k; l 2LðkÞ: (102) Next, to obtain a ditributed algorithm, we take a dual decompoition approach by relaxing all the coupling contraint in problem (102) U k x k ; fx kl g l2lðkþ þl T c k k þ G T kl ðx l x kl Þ k;l2lðkþ h k ðx k Þ ubject to x k 2 k 8k x kl 2 l 8k; l 2LðkÞ (103) where L are the congetion price and the G kl are the conitency price. By exploiting the additivity tructure of the Lagrangian, the Lagrangian i eparated into many ubproblem where maximization i done uing local variable (the kth ubproblem ue only variable with the firt ubcript index k).theoptimalvalueof(103)fora given et of G kl and L define the dual function gðg kl g; LÞ. The dual problem i then minimize fg kl g;l gðfg kl g; LÞ ubject to L 0: (104) It i worthwhile noting that (104) i equivalent to minimize L minimize fg kl g gðfg kl g; LÞ ubject to L 0: (105) Problem (104) i eaily olved by imultaneouly updating the price (both the congetion price and the! Vol. 95, No. 1, January 2007 Proceeding of the IEEE 297

44 conitency price) uing a ubgradient algorithm. In problem (105), however, the inner minimization i fully performed (by repeatedly updating the fg kl g) for each update of L. Thi latter approach implie two time cale: a fat time cale in which each cluter update the correponding conitency price and a low time cale in which the network update the link price; wherea the former approach ha jut one time cale. Therefore, in problem (101), where the utilitie U k are trictly concave, the et k are arbitrary convex et, and the contraining function h k are convex, can be optimally olved by the following ditributed algorithm: link update the congetion price (the following vector equation can be carried out by each link autonomouly a before): " Lðt þ 1Þ ¼ LðtÞ 1 c!# þ h k ðx k Þ (106) k where 1 i the tepize; the kth node, for all k, update the conitency price (at a fater or ame time cale a the update of LðtÞ) a G kl ðt þ 1Þ ¼G kl ðtþ 2 ðx l ðtþ x kl ðtþþ; l 2LðkÞ (107) where 2 i the tepize, and then broadcat them to the coupled node within the cluter; and the kth node, for all k, locally olve the problem x k ;fx kl gr ubject to U k x k ; fx kl g l2lðkþ L T h k ðx k Þ k 0 1 T A x k G T kl x kl l:k2lðlþ G lk l2lðkþ x k 2 k x kl 2 l 8l 2LðkÞ (108) where fx kl g l2lðkþ are auxiliary local variable for the kth node. Summarizing, all the link mut advertie their local variable x k (not the auxiliary one x kl ); congetion price L are updated a before, each link can update the correponding G kl (with knowledge of the variable x k of the coupled link) and ignal it to the coupled link; each link can update the local variable x k a well a the auxiliary one x kl. The only additional price due to the coupled utilitie i limited ignaling between the coupled link within each cluter. Method 19: Uing Conitency Pricing to Decouple Coupled Utility Objective Function. C. Alternative Decompoition Decompoition of a generalized NUM ha ignificant implication to network protocol deign along two direction: vertical (functional) decompoition into layer and horizontal (geographical) decompoition into ditributed computation by network element. There are many way to decompoe a given NUM formulation along both direction, providing a choice of different ditributed algorithm and layering cheme. A ytematic exploration of alternative decompoition i more than jut an intellectual curioity, it alo derive different network architecture with a wide range of poibilitie of communication overhead, computation load, and convergence behavior. Thi ha been illutrated through ome cae tudie in Section III-C and III-D. Alternative horizontal decompoition (i.e., ditributed control acro geographically diparate network element) ha been tudied in [104], with application to reourcecontrained and direct-control rate allocation, and rate allocation among QoS clae with multipath routing. Recent reult on alternative vertical decompoition for a given NUM model (i.e., modularized control over multiple functional module or layer) catter in an increaingly large reearch literature. For example, on the topic of joint congetion control and multipath traffic engineering, different decompoition have been obtained in[34],[46],[49],[59],[69],[84],and[101].onthetopic of joint congetion control, routing, and cheduling, different decompoition have been obtained in [1], [17], [36], [83], and [129]. On the topic of joint congetion control and random acce, different decompoition have been obtained in [74], [143], and [169]. On the topic of rate control for network coding-baed multicat, different decompoition have been obtained in [6], [15], [86], [151], [152], and [157]. Some of thee have been briefly dicued in Section III. A ytematic treatie on thi variety of vertical decompoition i an intereting reearch direction that will contribute to a rigorou undertanding of the architectural choice of allocating functionalitie to control module. One of the technique that lead to alternative of ditributed architecture i to apply primal and dual decompoition recurively, a illutrated in Fig. 16. The baic decompoition are repeatedly applied to a problem to obtain maller and maller ubproblem. For example, conider the following problem over y, fx i g which include both a coupling variable and a coupling contraint f i ðx i ; yþ ubject to x i 2 i ; 8i h i ðx i Þc i i A i x i y; 8i y 2Y: (109) 298 Proceeding of the IEEE Vol.95,No.1,January2007

45 One way to decouple thi problem i by firt taking a primal decompoition with repect to the coupling variable y and then a dual decompoition with repect to the coupling contraint P i h iðx i Þc. Thi would produce a two-level optimization decompoition: a mater primal problem, a econdary mater dual problem, and the ubproblem. An alternative approach would be to firt take a dual decompoition and then a primal one. Another example that how flexibility in term of different decompoition i the following problem with two et of contraint: where gðl; MÞ i the dual function defined a ( gðl; MÞ ¼ up f 0 ðxþ i f i ðxþ ) i h i ðxþ : x2 i i (112) Then, gðlþ i convex and a ubgradient, denoted by L ðlþ, igivenby i ðlþ ¼ f i ðx ðl; M ðlþþþ (113) f 0 ðxþ ubject to f i ðxþ 0; 8i h i ðxþ 0; 8i: (110) One way to deal with thi problem i via the dual problem with a full relaxation of both et of contraint to obtain the dual function gðl; MÞ. At thi point, intead of minimizing g directly with repect to L and M, it can be minimized over only one et of Lagrange multiplier firt andthenovertheremainingone:min L min M gðl; MÞ. Thi approach correpond to firt applying a full dual decompoition and then a primal one on the dual problem. The following lemma [105] characterize the ubgradient of the mater problem at the top level. Lemma 3: Conider the following partial minimization of the dual function: gðlþ ¼inf gðl; MÞ (111) M where x ðl; MÞ i the value of x that achieve the upremum in (112) for a given L and M, andm ðlþ i the value of M that achieve the infimum in (111). Alternatively, problem (110) can be approached via the dual but with a partial relaxation of only one et of contraint, ay f i ðxþ 0, 8i, obtaining the dual function gðlþ to be minimized by the mater problem. Oberve now that in order to compute gðlþ for a given L, the partial Lagrangian ha to be d ubject to the remaining contraint g i ðxþ 0, 8i, for which yet another relaxation can be ued. Thi approach correpond to firt applying a partial dual decompoition, and then, for the ubproblem, another dual decompoition. On top of combination of primal and dual decompoition, there can alo be different ordering of update, including the choice of parallel (Jacobi) or equential (Gau-Siedel) update [5]. When there are more than one levelofdecompoition,andalllevelconductometype of iterative algorithm, uch a the ubgradient method, convergence and tability are guaranteed if the lower level mater problem i olved on a fater time cale than the higher level mater problem, o that at each iteration of a Fig. 16. Schematic illutrating multilevel decompoition. Vol. 95, No. 1, January 2007 Proceeding of the IEEE 299

46 mater problem all the problem at a lower level have already converged. If the update of the different ubproblem operate on imilar time cale, convergence of the overall ytem may till be poible but require more proof technique [5], [118]. Method 20: Partial and Hierarchical Decompoition for Architectural Alternative of the Protocol Stack. A a more concrete example, conider the following pecial cae of NUM in variable ðx; yþ: U i ðx i Þ i ubject to f i ðx i ; y i Þ0; 8i y i 2Y i ; 8i h i ðx i ; y i Þ0 (114) i where x model the performance metric that uer utilitie depend on and y model ome reource that are globally coupled (the third contraint above) and have impact on performance (the firt contraint above). Thi problem ha application in ditributed waterfilling algorithm in DSL pectrum management and ditributed power control algorithm in wirele cellular network, and can be decompoed in at leat even different way following three general approache below. Each decompoition reult in a new poibility in triking the mot appropriate tradeoff between computation and communication. 1) A primal decompoition approach. Problem (114) decouple if the y i arefixed.wecandecompoe the original problem into the mater problem over y ~U i ðy i Þ i ubject to y i 2Y i 8i h i ðy i Þ0 (115) where each ~U i ðy i Þ i the optimal objective value of the ubproblem over x i i the local function U i, f i and the local et i )and the correponding y i (given by the mater problem). Once each ubproblem i olved, the optimal value U i ðy i Þ and poibly a ubgradient can be communicated to the mater problem. In thi cae, the mater problem need to communicate to each of the ubproblem the available amount of reource y i allocated. 2) A full dual decompoition approach with repect to all coupling contraint f i ðx i ; y i Þ0 and P i h iðy i Þ0. The mater dual problem i to minimize gðl;þ (117) over L, 0, where gðl;þ i given by the um of the optimal objective value of the following ubproblem over ðx i ; y i Þ for each i U i ðx i Þ i f i ðx i ; y i Þ h i ðy i Þ ubject to x i 2 i : (118) Each of the ubproblem can be olved in parallel and only need to know it local information and the Lagrange multiplier i and (given by the mater problem). Once each ubproblem i olved, the optimal value and poibly a ubgradient (given by f i ðx i ; y i Þ and h i ðy i Þ) can be communicated to the mater problem. In thi cae, the mater dual problem need to communicate to each of the ubproblem the private price i and the common price. 3) A partial dual decompoition approach only with repect P to the global coupling contraint i h iðy i Þ0. The mater dual problem over 0 minimize gðþ (119) where gðþ i given by the um of the optimal objective value of the following ubproblem for all i: ubject to U i ðx i Þ x i 2 i f i ðx i ; y i Þ0: (116) ubject to U i ðx i Þ h i ðy i Þ x i 2 i f i ðx i ; y i Þ0: (120) Each of the ubproblem can be olved in parallel and only need to know it local information (i.e., Each of the ubproblem can be olved in parallel and only need to know it local information and 300 Proceeding of the IEEE Vol.95,No.1,January2007

47 Table 8 Summary of Signalling Between the Mater Problem and the Subproblem in the Three Decompoition Conidered the Lagrange multiplier (given by the mater problem). Once each ubproblem i olved, the optimal value and (poibly) a ubgradient, given by h i ðy i Þ, can be communicated to the mater problem. In thi cae, the mater dual problem need to communicate to each of the ubproblem imply the common price. We conider two of the metric that can be ued to compare alternative decompoition: the tradeoff between local computation and global communication through meage paing, and the convergence peed. Theamountofignallingofthethreedecompoition method i ummarized in Table 8. In thi particular problem, Approach 1 require the larget amount of ignalling in both direction. Approach 3 only require a ingle common price from the mater problem to the ubproblem and a ingle number from each ubproblem to the mater problem. Approach 2 i intermediate, requiring the ame amount a Approach 3 plu an additional individual price from the mater problem to each ubproblem. For a particular intance of problem (114), Fig. 17 how the convergence behavior in a numerical example for variou ditributed algorithm obtained by adding the choice of Jacobi or Gau-Siedel update order on top of the three Table 9 Summary of Main Notation for Section V approache in Table 8. The top two curve correpond to variation under Approache 3 and 2, repectively. Implicit in all decompoition i a choice of particular repreentation of the contraint. There are everal way to obtain different repreentation of the ame NUM problem. One way i through ubtitution or change of variable [47]. The econd way i to ue a different et of variable, e.g., node-centric veru link-centric formulation in III-D. A third way i to group the variable in different order. For example, in [16], it i hown that the joint TCP and MAC deign problem may be formulated a maximizing network utility ubject to the contraint that FRx c, wheref i a contention matrix and R i the routing matrix. Then depending whether we firt group Rx or FR in the contraint, the Lagrange dual variable we introduce are different, correponding to either drawing a diviion between TCP and MAC or not. In general, ince every individual contraint (e.g., capacity of a link) give rie to a correponding Lagrange multiplier, we have the omewhat urpriing conequence that the pecific dual problem will be different depending on how the primal contraint are written, even redundant contraint that may change the dual problem propertie. A ummarized in Fig. 18, each repreentation of a particular NUM may lead to different decompoability tructure, different decompoition approache, and the aociated different ditributed algorithm. Each of thee algorithm may repreent a different way to modularize and ditribute the control of network, and differ from the other in term of engineering implication (more dicuion in Section V-C). Fig. 17. Different peed of convergence for even different alternative of horizontal decompoition in a numerical example of (114). Each curve correpond to a different decompoition tructure. V. FUTURE RESEARCH DIRECTIONS Depite the wide range of progre made by many reearcher over the lat everal year, there are till a variety of open iue in the area of BLayering a Optimization Decompoition.[ Some of thee have formed a et of Vol. 95, No. 1, January 2007 Proceeding of the IEEE 301

48 more in print, baed on different phyical model, e.g., node-excluive, SIR-baed, and capture model. Mot of the deign have focued on the optimal meage paing acro layer and theoretically motivated choice of parameter uch a tepize. A ytematic tudy on uboptimal meage paing heuritic and practical guideline in chooing algorithmic parameter would help characterize the tradeoff between complexity and uboptimality gap. Fig. 18. Each alternative problem repreentation may lead to a choice of ditributed algorithm with different engineering implication. widely recognized and well-defined open problem, which i in fact a ign indicating the maturity of a reearch area. Thi ection highlight ome of the main iue for future reearch and their recent development. Aide from technical challenge in proving deirable propertie, uch a global aymptotic convergence for different ditributed algorithm under arbitrary delay [106], [107], we will claify the array of open iue in ix group. A. Modeling and Complexity Challenge Firt of all, there are emantic functionalitie, uch a eion initiation and packet reordering, that we do not explicitly model. BGP in IP protocol and a variety of wirele ad hoc network routing protocol are yet to be fully incorporated in modeling language of NUM. Much further work alo remain to be done to model utility function in pecific application, epecially inelatic, realtime application uch a VoIP [79] and treaming media where the notion of fairne may alo need to be reconidered [136]. In a more refined phyical/link layer model, the option of forwarding rather than re-encoding at intermediate node mut be conidered, a well a per-hop retranmiion cheme through ARQ. More NUM model are alo needed to explicitly incorporate robutne with repect to algorithmic error, network failure [160], multiple et of contant parameter, and uncontrolled tochatic perturbance. Several important module commonly encountered in many cae of BLayering a Optimization Decompoition[ till do not have imple, ditributed olution. An important topic i on ditributed and uboptimal cheduling algorithm that have low patial complexity (the amount and reach of explicit meage paing do not grow rapidly with network ize) and temporal complexity (the amount of backoff needed doe not grow rapidly with network ize) and can till guarantee certain throughput-delay performance. There have been many recent reult on thi topic, e.g., [12], [14], [80], [121], [149], [150], [165], and many B. Reearch Iue Involving BTime[ Utility function are often modeled a function of equilibrium rate. For application involving real-time control or multimedia communication, utility hould intead be a function of latency or even the entire vector of rate allocation through the tranient. How to uch utility function remain an under-explored topic. Different function in each layer operate on time cale that may differ by everal order-of-magnitude different. For example, the application layer time cale i determined by the uer behavior, the tranport layer time cale by the round-trip-time in travering the network, and the phyical layer time cale by the phyic of the tranmiion medium. Iterative algorithm themelve alo have a time cale of operation determined by their rate of convergence, which i often difficult to bound tightly. Furthermore, characterizing tranient behavior of iterative algorithm remain a challenging and underexploredtopicinthiarea.forcertainapplication,ifthe reource allocation (e.g., window ize, SIR) for a uer drop below a threhold during the tranient, the uer may be diconnected. In uch cae, the whole idea of equilibrium become meaningle. Invariance during tranient [41], intead of convergence in the aymptote, become a more ueful concept: how fat can the algorithm get cloe enough to the optimum and tay in that region? Uually the overall ytem performance derived out of a modularized deign determine Bhow cloe i cloe enough[ for each module tranient. C. Alternative Decompoition Even a different repreentation of the ame NUM problem may change the duality and decompoability tructure even though it doe not change the optimal olution. It remain an open iue how to ytematically explore the pace of alternative vertical and horizontal decompoition, and thu the pace of alternative network architecture, for a given et of requirement on, e.g., rate of convergence, ymmetry of computational load ditribution, and amount of explicit meage paing. An intellectually bold direction for future reearch i to explore if both the enumeration and comparion of alternative decompoition, horizontally and vertically, can be carried out ytematically or even be automated. To enumerate the et of poible decompoition and the aociated et of ditributed algorithm, we have to 302 Proceeding of the IEEE Vol.95,No.1,January2007

49 take into account that tranformation of the problem (e.g., change of variable) may lead to new decompoability tructure, or turn a eemingly nondecompoable problem into a decompoable one [47]. Thi would open the door to even more choice of modularized and ditributed network architecture with different propertie. To compare a variety of ditributed algorithm, the following metric all need to be conidered: peed of convergence, the amount and ymmetry of meage paing for global communication, the ditribution of local computational load, robutne to error, failure, or network dynamic, the impact to performance metric not directly incorporated into the objective function (e.g., uerperceived delay in throughput-baed utility maximization formulation), the poibility of efficient relaxation and imple heuritic, and the ability to remain evolvable a the application need change over time. Some of the above metric have no quantitative unit of meaurement, uch a evolvability. Some do not have a univerally agreed definition, uch a the meaure of how ditributed an algorithm i. 11 Some are difficult to analyze accurately, uch a the rate of convergence. Application context lead to a prioritization of thee poibly conflicting metric, baed on which, the Bbet[ decompoition can be choen from the range of alternative. Summarizing, there are three tage of conceptual undertanding of a decompoition-theoretic view of network architecture: Firt, modularized and ditributed network architecture can be rigorouly undertood a decompoition of an underlying optimization problem. Second, there are in fact many alternative of decompoition and therefore alternative of network architecture. Furthermore, we can ytematically explore and compare uch alternative. Third, there may be a methodology to exhautively enumerate all alternative, to quantify variou comparion metric, and even to determine a priori which alternative i the bet according to any given combination of comparion metric. Many iue in the third tage of the above lit remain open for future reearch. D. Stochatic NUM Stochatic theory of communication network ha a long and rich hitory. However, many key problem in thi area remain open depite decade of effort. In hi eminal paper [64], Kelly et al. ued a determinitic fluid model to remove packet level detail and microcopic queuing dynamic, followed by an optimization/game/controltheoretic approach. By auming determinitic fluid with infinite backlog, we can avoid the difficulty of coupling acro link in a general queuing network, and are oftenabletoobtaininightonthetructureofnetwork reource allocation and functionality allocation. An analogy can be drawn with Shannon eminal work in 1948 [120]. By turning the focu from deign of finiteblocklength code to the regime of infinite-blocklength code, thu enabling the law of large number to take effect, Shannon provided architectural principle (e.g., ource-channel-eparation) and etablihed fundamental limit (e.g., channel capacity) in hi mathematical theory of communication. Since then, the complicating iue aociated with the deign of practical code have been brought back into the framework. In the framework of BLayering a Optimization Decompoition,[ it i time to incorporate tochatic networkdynamic,ateion,packet,channel,andtopology level, back to the generalized NUM formulation. Thi lead to challenging model of queuing network. For example, ervice rate of queue are determined by ditributed olution to NUM, while parameter of NUM formulation are in turn tochatically varying according to tate of the queue. A combination of tochatic network control and optimization-baed reource allocation raie challenging new quetion, including tochatic tability, average cae performance, outage performance, and, eventually, the ditribution of attained utility (or other QoS parameter uch a delay) a induced by the ditribution of the tochatic model. Stochatic tability i the mot baic and well-explored quetion in thi area: under what condition will a certain ditributed algorithm of an NUM problem remain tochatically table, in the ene that the number of flow and the total queue length in the network remain finite? Sometime, determinitic optimization formulation can be derived from the limiting regime of tochatic network optimization, a in the cae of ocial welfare maximization for lo network in [110] and [111]. Stochatic model arie at four different level due to many reaon: Seion level (alo referred to a flow level, connection level, or end-uer level): flow arrive with finite workload and depart after finihing the workload, rather than holding infinite backlog and taying in the network forever. For certain combination of the model of arrival proce, utility function, contraint et, and time-cale eparation, 12 reearcher have etablihed that the (tochatic) tability region of the baic NUM i the larget poible, which i the capacity region formed by the fixed link capacitie in the determinitic NUM formulation. Thi mean that atifying the contraint in a determinitic formulation i both neceary and ufficient for tochatic tability. 11 A common unit i the amount and reach of explicit meage paing needed, and how they grow a the network grow. 12 Here time-cale eparation mean that the reource allocation algorithm converge before the number of eion change. 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50 Table 10 State-of-the-Art in Stochatic NUM Packet level: packet of each flow arrive in burt, and at a microcopic level go through probabilitic marking, and interact with uncontrolled flow uch a UDP-baed multimedia flow and web mice traffic. There have been at leat three et of reult that appeared over the lat everal year. The firt et how many-flow aymptotical validation of fluid model (jutifying the tranition from microcopic to macrocopic model) [2], [30], [119], [139], and analyze the interaction between congetion-controlled flow and uncontrolled flowundervariouqueueregimeanddifferent time-cale aumption on flow random variation [31], [113], [137], [164], [166]. The econd et tranlate on off HTTP eion utility into tranport layer TCP utility (mapping from microcopic to macrocopic model) [13]. The third et demontrate convergence behavior for tochatic noiy feedback [170] (characterizing the impact of microcopic dynamic to macrocopic propertie). Channel level: network tranmiion condition are time-varying rather than fixed. Channel variation offer both the challenge to prove tability/ optimality for exiting algorithm and the ability to do opportunitic tranmiion and cheduling. We focu on the firt et of iue in thi urvey. 13 For example, in [17], tability and optimality are etablihed for dual-decompoition-baed algorithm under channel-level tochatic for any convex optimization where the contraint et ha the following tructure: a ubet of the variable lie in a polytope and other variable lie in a convex et that varie according to an irreducible, finitetate Markov chain. Topology level: Topology of wirele network can change due to mobility of node, leep mode, and battery power depletion. Solving generalized NUM problem over network with randomly varying topology remain an under-explored area with little known reult on model or methodologie. The problem i particularly challenging when the topology level tochatic dynamic i determined by battery uage, which i in turn determined by the olution of the NUM problem itelf. 13 Reader are referred to [85] for a urvey of the econd et of iue. A hown in Table 10, where we ue a ytem of zero to three tar to roughly repreent the tate of our undertanding of the ubject (from almot no reult to complete characterization), much remain to be explored in thi long over-due union between ditributed optimization of network and tochatic network theory. In the ret of thi ection, we briefly ummarize ome of the recent reult in the firt column of the table. 1) Seion Level Stochatic: Conider eion level dynamic characterized by the random arrival and departure of eion. For each type r, uppoefornow that flow arrive and depart according to a Poion proce with intenity r, and the ize of the flow to be tranmitted i exponentially ditributed with mean 1= r. The traffic load i r ¼ r = r.letn r be the number of ongoing flow, i.e., the number of type r flow in the network. It i a Markov proce with the following tranition rate: N r ðtþ!n r ðtþþ1, with rate r ; N r ðtþ!n r ðtþ 1, with rate r x r ðtþn r ðtþ. The tochatic verion of the baic NUM formulation naturally become the following problem over x: ubject to N r U r ðx r Þ r R lr x r N r c l ; 8l: (121) r Obviouly, the definition of problem (121) depend on fn r g, whoe tochatic variation are in turn determined by the olution to problem (121). For problem (121), it i hown [7], [29], [94], [162] that the tochatic tability region i the interior of feaibility rate region formed by the fixed link capacitie, i.e., the following condition i ufficient to guarantee tochatic tability of the baic NUM for for -fair utility function with 9 0: RR G c: (122) Thee reult aumed time-cale eparation. However, in many practical network, eion level tochatic operate on a fat time cale, with the arrive-and-depart proce of flow varying contantly. Hence, intantaneou convergence of the rate allocation algorithm may 304 Proceeding of the IEEE Vol.95,No.1,January2007

51 not hold. The above reult i extended in [81] and [126] to the cae without time-cale eparation aumption: [81] tudie -fair utilitie uing a dicrete-time model and how that there i an upper bound on the tep ize that would guarantee tochatic tability, and [126] how imilar reult for -fair utilitie uing the fluid limit model. Other recent extenion include the following. Uing the technique in [28] and [163] relaxe the aumption of Poion arrival, by tudying a general tationary and a burty network model. Recently, [68] tudie a model that include flow of two type, file tranfer and treaming traffic, and generalize the congetion control problem with convex contraint. And [8] correlate the utility maximization to claical queueing theory, and tudie everal typical utility function and the tability condition. Then tochatic tability for any trictly concave maximization over general convex contraint without time-cale eparation i recently reported. An important and often invalid aumption in all thee tability reult i that file ize (or workload) follow exponential ditribution. In [66], a proper fluid model i formulated for exponentially ditributed workload to tudy the Binvariant tate[ a an intermediate tep for obtaining diffuion approximation for all 2ð0; 1Þ. In [45], the fluid model i etablihed for -fair rate allocation, 2ð0; 1Þ, under general ditributional condition on arrival proce and ervice ditribution. Uing thi fluid model, they have obtained characterization of Binvariant tate,[ which led to tability of network under -fair allocation, 2ð0; 1Þ, when the network topology i a tree. For general network topologie, three recent preprint have tackled thi difficult problem of tochatic tability under general file-ize ditribution, for pecial cae of utility function: firt, [11] etablihe tability for max-min fair (correponding to ¼1) rate allocation, then [93] etablihe tability for proportional fair (correponding to ¼ 1) rate allocation for Poion arrival and phae-type ervice ditribution. Finally, uing the fluid model in [45] but under a different caling, [22] etablihe the tability of -fair rate allocation for general file-ize ditribution for a continuum of : ufficiently cloe to (but trictly larger than) zero, and a partial tability reult for any 9 0 fair allocation policy. 2) Packet Level Stochatic: Randomne at packet level may be a reult of probabilitic marking of certain AQM cheme. It can alo model Bmice[ traffic which i not captured in the tandard NUM model. In [2], a detailed tochatic model i preented for N TCP Reno ource haring a ingle bottleneck link with capacity Nc implementing RED. They how that a the number of ource and the link capacity both increae linearly, the queue proce converge (in appropriate ene) to a determinitic proce decribed by differential equation a uually aumed in the congetion control literature. Even though thee reult are proved only for a ingle bottleneck node, they provide a jutification for the popular determinitic fluid model by uggeting that the determinitic proce i the limit of a caled tochatic proce a the number of flow and link capacitie cale to infinity [139]. Other convergence reult are hown in [30] and [119]: the determinitic delay differential equation model with noie replaced by it mean value i accurate aymptotically in time and the number of flow. Becaue uch convergence i hown aymptotically in time (except in the pecial cae of log utility [119] where it i hown for each time), the trajectory of the tochatic ytem doe not converge to that of the determinitic ytem in the manyflow regime [30]. However, [30] how that the global tability criterion for a ingle flow i alo that for the tochatic ytem with many flow, thu validating parameter deign in the determinitic model even when network have packet level tochatic dynamic. Stochatic tability of greedy primal-dual-driven algorithm, a combination of utility maximization and maximum weight matching, i hown in [129] for dynamic network where traffic ource and router are randomly time-varying, interdependent, and limited by intantaneouly available tranmiion and ervice rate. Beide packet-level tochatic dynamic, there i alo burtine at the application level. Reference [13] conider it effect on TCP. It how that the utility maximization at the tranport layer induce a utility maximization at the application layer, i.e., an objective at the application layer i implemented in the tranport layer. Specifically, conider a ingle link with capacity Nc (bit) hared by N HTTP-like flow. Each flow alternate between think time and tranfer time. Duringtheperiod of a think time, a flow doe not require any bandwidth from the link. Immediately after a period of think time, the ource tart to tranmit a random amount of data by a TCP connection. The tranfer time depend on the amount of tranfer and the bandwidth allocation to thi flow by TCP. The number of active flow i random, but at any time, the active flow hare the link capacity according to TCP, i.e., their throughput aggregate utility ubject to capacity contraint. Aume there are a fixed number of flow type. Then it i hown in [13] that the average throughput, i.e., the throughput aggregated over active flow of each type normalized by the total number of flow of that type, alo olve a utility maximization problem with different utility function a the TCP utility function. Yet another line of recent work [170] tudie the impact of tochatic noiy feedback on dual-decompoition-baed ditributed algorithm, where the noie can be due to probabilitic marking, dropping, and contention-induced lo of packet. When the gradient etimator i unbiaed, it i etablihed, via a combination of the tochatic Lyapunov Stability Theorem and local analyi, that the iterate generated by ditributed NUM algorithm converge with Vol. 95, No. 1, January 2007 Proceeding of the IEEE 305

52 probability one to the optimal point, under tandard technical condition. In contrat, when the gradient etimator i biaed, the iterate converge to a contraction region around the optimal point, provided that the biaed term are aymptotically bounded by a caled verion of the true gradient. Thee reult confirm thoe derived baed on determinitic model of feedback with error [18], [95]. In the invetigation of the rate of convergence for the unbiaed cae, it i found that, in general, the limit proce of the interpolated proce baed on the normalized iterate equence i a tationary reflected linear diffuion proce, not necearily a Gauian diffuion proce. 3) Channel Level Stochatic: Model in [17], [35], [83], [99], and [128] conider random channel fluctuation. For example, tability of primal-baed algorithm under channel variation i etablihed in [128]. In [17], the channel i aumed to be fixed within a dicrete time lot but change randomly and independently acro lot. Let hðtþ denote the channel tate in time lot t. Correponding to the channel tate h, the capacity of link l i c l ðhþ when active and the feaible rate region at the link layer i /ðhþ. We further aume that the channel tate i a finite tate proce with identical ditribution qðhþ in each time lot. Define the mean feaible rate region a ( / ¼ r : r ¼ ) qðhþrðhþ; rðhþ 2/ðhÞ : (123) h The joint congetion control, routing, and cheduling algorithm dicued in Section III-D can be directly applied with the chedulable region / instep2replaced by the current feaible rate region /ðhðtþþ. Itiprovedin [17] that the price LðtÞ form a table Markov proce, by appealing to the generalized NUM (85) with the rate region / replaced by the mean rate region / U ðx Þ ubject to x k i j:ði;jþ2l f k ij j:ðj;iþ2l f k ji ; 8i; j; k f 2 /: (124) Moreover, the primal and dual value along the trajectory converge arbitrarily cloe to their optimal value, with repect to (124), a the tepize in the algorithm tend to zero. For generalized NUM problem, [17] etablihe the tability and optimality of dual-decompoition-baed algorithm under channel-level tochatic for any convex optimization where the contraint et ha the following tructure: a ubet of the variable lie in a polytope and other variable lie in a convex et that varie according to an irreducible, finite-tate Markov chain. Algorithm developed from the determinitic NUM formulation and requiring only the knowledge of current network tate remain tochatically table and optimal (in the expectation, with repect to an optimization problem whoe contraint i replaced by the average contraint et under the given channel variation). E. Nonconvex NUM It i widely recognized that the waterhed between eay and hard optimization problem i convexity rather than linearity. Nonconvex optimization formulation of generalized NUM often appear, in at leat four different form. Firt i nonconcave utility, uch a the igmoidal utility that are more realitic model in application including voice. Second i nonconvex contraint et, uch a lower bound on SIR a a function of tranmit power vector, in the low-sir regime of interference-limited network. Third i integer contraint, uch a thoe in ingle-path routing protocol. Fourth i convex contraint et requiring a decription length that grow exponentially with the number of variable, uch a certain chedulability contraint in multihop interference model. In general, uch nonconvex optimization i difficult in theory and in practice, even through centralized computation. In particular, nonconvex optimization problem often have nonzero duality gap. A nonzero duality gap mean that the tandard dual-decompoition-baed ditributed ubgradient algorithm may lead to uboptimal and even infeaible primal olution and intability in cro layer interaction. Bounding, etimating, and reducing the reulting duality gap remain a challenging tak. Sometime, thee very difficult problem can be tackled through a combination of well-etablihed and more recent optimization technique, e.g., um-of-quare programming [108], [109] and geometric-ignomial programming [19], [32], although ome apect of thee technique are not well undertood (e.g., convergence to global optimality in ignomial programming). A an example, conider nonconvexity in the objective function. There have been three recent approache to olve nonconcave utility maximization over linear contraint: Reference [77] propoe a ditributed admiion control method (for igmoidal utilitie) called the Belf-regulating[ heuritic, which i hown to avoid link congetion caued by igmoidal utilitie. Reference [48] determine optimality condition for the dual-decompoition-baed ditributed algorithm to converge globally (for all nonlinear utilitie). The engineering implication i that appropriate proviioning of link capacitie will enure global convergence of the dual-decompoitionbaed ditributed algorithm even when uer utility function are nonconcave. 306 Proceeding of the IEEE Vol.95,No.1,January2007

53 Fig. 19. Three major type of approache when tackling nonconvex NUM: Go 1) around, 2) through, or 3) above nonconvexity. Reference [40] develop an efficient but centralized method to compute the global optimum (for a wide cla of utilitie that can be tranformed into polynomial utilitie), uing the um-of-quare method. However, no ditributed verion of thi method are available. A illutrated in Fig. 19, there are at leat three very different approache to tackle the difficult iue of nonconvexity in either the objective function or the contraint et: Go around nonconvexity: dicover a change of variable that turn the eemingly nonconvex problem into a convex one, determine condition under which the problem i convex or the KKT point i unique [75], [124], or make approximation to make the problem convex. A popular example of tackling nonconvexity i the application of geometric programming to communication ytem [19]. In ome problem, an appropriate change of variable turn an apparently nonconvex problem into a convex one [131]. Go through nonconvexity: ue ucceive convex relaxation (e.g., um-of-quare, ignomial programming), utilize pecial tructure in the problem (e.g., difference of convex function, generalized quai-concavity), or leverage marter branch and bound method. Go above nonconvexity: oberve that optimization problem formulation are induced by ome underlying aumption on what the architecture and protocol hould look like. By changing thee aumption, a different, much eaier-to-olve or eaier-to-approximate NUM formulation may reult. We refer to thi approach a deign for optimizability [51], which concern with electively perturbing ome underlying aumption to make the reulting NUM problem eaier to olve. Thi approach of changing a hard problem into an eaier one i in contrat to optimization, whichtrieto olve a given, poibly difficult NUM problem. A recent ucceful example of Bdeign for optimizability[ i on intra-domain routing in the Internet [159]. F. Quantifying Network -itie A we draw thi paper toward the end, it i important to ak why hould network operator optimize performance in the firt place? Indeed, optimality i not the key point. Optimization i ued here primarily a a modeling language and a tarting point to develop and compare architectural choice, rather than jut defining a particular point of operation at global optimum. Suboptimal, but imple (low patial-temporal complexity) algorithm can be ued in variou module (e.g., the cheduling module). A long a the uboptimality gap i bounded and the network architecture i Bgood,[ then the Bdamage[ from the uboptimal deign in one layer can be contained at the ytem level. Similarly, tochatic dynamic may alo wah away the wort cae and even be beneficial to the average ytem performance [83], again provided that the network architecture i appropriately deigned. In uch cae, it i alo neceary to tudy the meaning of utilityuboptimality in term of degradation to fairne, ince x% of optimality lo may not imply x% degradation of fairne. In fact, even quantified metric of unfairne are not well-etablihed. Protocol and layered architecture are not jut for maximizing the efficiency of performance metric, uch a throughput, latency, and ditortion, but alo for enuring ecurity and for maximizing the important yet fuzzy metric of robutne in operation, uch a evolvability, calability, availability, diagnoability, and manageability. Interaction among layer introduce the rik of loing robutne againt unforeen demand ariing over time or ignificant growth over pace. Indeed, under the metric of deployment cot and operation cot, the ucce of packet network come down to the calability and evolvability of TCP/IP and the way control i modularized and ditributed. Depite their importance in practical network operation, thee network -itie remain a fuzzy or even illdefined notion, and a quantified foundation for them i long overdue [24]. Intuitively, Bdeign by decompoition[ enhance calability and evolvability, but may preent rik to manageability uch a diagnoability and optimizability. The benefit and rik arie together, in part becaue layering mean that each layer i limited in what it can do Vol. 95, No. 1, January 2007 Proceeding of the IEEE 307

54 (optimization variable in a decompoed ubproblem) and what it can oberve (a ubet of contant parameter and variable in other decompoed ubproblem). Throughout Section II and III, we have illutrated how the framework of BLayering a Optimization Decompoition[ help make the deciion of what each module hould control and oberve. Still, quantifying network -itie, and trading-off network -itie with performance metric, in layered protocol tack deign remain a challenging long-term reearch direction. We may alo carry the intellectual thread from Bforward-engineering[ (olving a given problem) to Brevere-engineering[ (findingtheproblembeingolved by a given protocol) one tep further to Bdeign for optimizability.[ The difficulty of olving a particular et of ubproblem may illutrate that the given decompoition wa conducted poibly in a wrong way and ugget that better alternative exit. In ummary, in order to fulfill the long-term goal of providing a imple, relevant abtraction of what make a network architecture Bgood,[ the framework of BLayering a Optimization Decompoition[ need to move away from the retriction of one decompoition fit all, away from the focu on determinitic fluid model and aymptotic convergence, and even away from the emphai on optimality, and intead toward Boptimizability.[ VI. CONCLUSION We provide a urvey of the recent effort to etablih BLayering a Optimization Decompoition[ a a common Blanguage[ for ytematic network deign. BLayering a Optimization Decompoition[ i a unifying framework for undertanding and deigning ditributed control and crolayer reource allocation in wired and wirele network. It ha been developed by variou reearch group ince the late 1990, and i now emerging to provide a mathematically rigorou and practically relevant approach to quantify the rik and opportunitie in modifying the exiting layered network architecture or in drafting cleanlate architecture. It how that many exiting network protocol can be revere-engineered a implicitly olving ome optimization-theoretic or game-theoretic problem. By ditributively olving generalized NUM formulation through decompoed ubproblem, we can alo ytematically generate layered protocol tack. There are many alternative for both horizontal decompoition into diparate network element and vertical decompoition into functional module (i.e., layer). A variety of technique to tackle coupling and nonconvexity iue have become available, which enable development of ditributed algorithm and proof of global optimality, repectively. Many uch technique are becoming tandard method to be readily invoked by reearcher. BLayering a Optimization Decompoition[ provide a top-down approach to deign layered protocol tack from firt principle. The reulting conceptual implicity tand in contrat to the ever-increaing complexity of communication network. The two cornertone for the rigor and relevance of thi framework are Bnetwork a optimizer[ and Blayering a decompoition.[ Together, they provide a promiing angle to undertand not jut what Bwork[ in the current layered protocol tack, but alo why it work, what may not work, and what alternative network deigner have. Realizing the importance of functionality allocation and motivated by the view of Barchitecture firt,[ we hope that there will be continuou progre toward a mathematical theory of network architecture. h Acknowledgment The author would like to gratefully acknowledge their coauthor in the reearch area urveyed here: L. Andrew, M. Breler, L. Chen, H. Choe, J.-Y. Choi, M. Fazel, D. Gayme, P. Hande, J. He, S. Hegde, J. Huang, C. Jin, J.-W. Lee, J. Lee, L. Li, Y. Li, J. Liu, F. Paganini, D. Palomar, J. Pongajapan, J. Rexford, D. Shah, R. Srikant, C. W. Tan, A. Tang, J. Wang, D. Wei, B. Wydrowki, and J. Zhang. They alo thank many colleague for illuminating dicuion, epecially on network architecture and layering, with D. Acemoglu, V. Chan, D.-M. Chiu, A. Ephremide, N. Gerhenfeld, A. Goldmith, B. Fry, M. Johanon, K. Kar, F. P. Kelly, P. R. Kumar,. Lin, Z. Liu, H. Newman, D. O Neill, A. Proutiere, C. Ramming, S. Shakkottai, N. Shroff, J. Wieeltheir,. Wang, Z.-K. Wang, L. iao, E. Yeh, Y. Yi, AT&T Lab, Bell Lab, and the Cico ARTI group, on optimization theory, with S. Boyd, D. Gao, A. Ozdaglar, and P. Parrilo, and on network -itie, with J. Kuroe, R. Lee, D. Towley, and D. u. REFERENCES [1] M. Andrew, BJoint optimization of cheduling and congetion control in communication network,[ in Proc. CISS, Mar [2] F. Baccelli, D. R. McDonald, and J. Reynier, BA mean-field model for multiple TCP connection through a buffer implementing RED,[ INRIA, Tech. 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Netw., Feb. 2008, to be publihed. [49] J. He, M. Breler, M. Chiang, and J. Rexford, BToward multi-layer traffic engineering: Optimization of congetion control and routing,[ IEEE J. Sel. Area Comm., 2007, to be publihed. [50] J. He, M. Chiang, and J. Rexford, BTCP/IP interaction baed on congetion price: Stability and optimality,[ in Proc. IEEE ICC, Jun [51] J. He, J. Rexford, and M. Chiang, BDeign principle of manageable network,[ Princeton Univerity Computer Science, Technical Report TR , Oct [52] R. Cendrillon, J. Huang, M. Chiang, and M. Moonen, BAutonomou pectrum balancing for digital ubcriber line,[ in IEEE Tran. Sign. Proc., 2007, to be publihed. [53] J. Huang, Z. Li, M. Chiang, and A. K. Kataggelo, BPricing-baed rate control and joint packet cheduling for multi-uer wirele uplink video treaming,[ in Proc. IEEE Packet Video Workhop, Apr [54] Internet 0 Meeting Material. [Online]. Available: cba.mit.edu/event/04.09.i0/. [55] V. Jacobon, BCongetion avoidance and control,[ in Proc. ACM SIGCOMM, Aug [56] C. Jin, D.. Wei, and S. H. Low, BFAST TCP: Motivation, architecture, algorithm, and performance,[ in Proc. IEEE INFOCOM, Mar [57] C. Jin, D.. Wei, S. H. Low, G. Buhrmater, J. Bunn, D. H. Choe, R. L. A. Cottrell, J. C. Doyle, W. Feng, O. Martin, H. Newman, F. Paganini, S. Ravot, and S. Singh, BFAST TCP: From theory to experiment,[ IEEE Netw., vol. 19, no. 1, pp. 4 11, Jan./Feb [58] B. Johanon, P. Soldata, and M. Johanon, BMathematical decompoition technique for ditributed cro-layer optimization of data network,[ IEEE J. Sel. Area Commun., vol. 24, no. 8, pp , Aug [59] S. Kandula and D. Katabi, BWalking the tightrope: Reponive yet table traffic engineering,[ in Proc. ACM SIGCOMM, Aug [60] K. Kar, S. Sarkar, and L. Taiula, BOptimization baed rate control for multipath eion,[ in Proc. Int. Teletraffic Congre, Dec [61] VV, BAchieving proportional fairne uing local information in Aloha network,[ IEEE Tran. Autom. Control, vol. 49, no. 10, pp , Oct [62] D. Katabi, M. 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58 ABOUT THE AUTHORS Mung Chiang (Member, IEEE) received the B.S. (Hon.) degree in electrical engineering and in mathematic, and the M.S. and Ph.D. degree in electrical engineering from Stanford Univerity, Stanford, CA, in 1999, 2000, and 2003, repectively. He i an Aitant Profeor of Electrical Engineering, and an affiliated faculty member of the Program in Applied and Computational Mathematic at Princeton Univerity, Princeton, NJ. He conduct reearch in the area of optimization of communication ytem, theoretical foundation of network architecture, algorithm in broadband acce network, and tochatic model of communication. Dr. Chiang ha been awarded a Hertz Foundation Fellowhip, and received the Stanford Univerity School of Engineering Terman Award for Academic Excellence, the SBC Communication New Technology Introduction contribution award, the National Science Foundation CAREER Award, and the Princeton Univerity Howard B. Wentz Junior Faculty Award. He i the Lead Guet Editor of the IEEE JOURNAL OF SELECTED AREAS IN COMMUNICATIONS, SPECIAL ISSUE ON NONLINEAR OPTIMIZATION OF COMMUNI- CATION SYSTEMS, a Guet Editor of the IEEE TRANSACTIONS ON INFORMATION THEORY and IEEE/ACM TRANSACTIONS ON NETWORKING, JOINT SPECIAL ISSUE ON NETWORKING AND INFORMATION THEORY, aneditorofieeetransactions ON WIRELESS COMMUNICATIONS, the Program Co-Chair of the 38th Conference on Information Science and Sytem, and a co-editor of Springer book erie on Control and Optimization of Communication Sytem. He i a co-author of IEEE GLOBECOM Bet Student Paper Award, and one of hi paper become the Fat Breaking Paper in Computer Science in 2006 by ISI citation data. Steven H. Low (Senior Member, IEEE) received the B.S. degree from Cornell Univerity, Ithaca, NY, and the Ph.D. degree from the Univerity of California, Berkeley, both in electrical engineering. He i a Profeor of Computer Science and Electrical Engineering at California Intitute of Technology (Caltech), Paadena. He wa with AT&T Bell Laboratorie, Murray Hill, NJ, from 1992 to 1996, the Univerity of Melbourne, Autralia, from 1996 to 2000, and wa a Senior Fellow of the Univerity of Melbourne from 2000 to He i a member of the Networking and Information Technology Technical Adviory Group for the U.S. Preident Council of Advior on Science and Technology (PCAST). Hi interet are in the control and optimization of network and protocol. Dr. Low wa a co-recipient of the IEEE William R. Bennett Prize Paper Award in 1997 and the 1996 R&D 100 Award. He wa on the editorial board of IEEE/ACM TRANSACTIONS ON NETWORKING from 1997 to 2006 and on that of Computer Network Journal from 2003 to He i on the editorial board of ACM Computing Survey, NOW Foundation, and Trend in Networking, and i a Senior Editor of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS. A. Robert Calderbank (Fellow, IEEE) i a Profeor of Electrical Engineering and Mathematic at Princeton Univerity, Princeton, NJ, where he direct the Program in Applied and Computational Mathematic. He joined Bell Laboratorie a a member of Technical Staff in 1980, and retired from AT&T in 2003 a Vice Preident of Reearch. He ha reearch interet that range from algebraic coding theory and quantum computing to the deign of wirele and radar ytem. Dr. Calderbank erved a Editor in Chief of the IEEE TRANSACTIONS ON INFORMATION THEORY from 1995 to 1998, and a Aociate Editor for Coding Technique from 1986 to He wa a member of the Board of Governor of the IEEE Information Theory Society from 1991 to He wa honored by the IEEE Information Theory Prize Paper Award in 1995 for hi work on the Z4 linearity of Kerdock and Preparata Code (joint with A. R. Hammon Jr., P. V. Kumar, N. J. A. Sloane, and P. Sole), and again in 1999 for the invention of pace time code (joint with V. Tarokh and N. Sehadri). He i a recipient of the IEEE Millennium Medal, and wa electedtothenationalacademyofengineeringin2005. John C. Doyle received the B.S. and M.S. degree in electrical engineering from Maachuett Intitute of Technology in 1977 and the Ph.D. degree in mathematic from the Univerity of California, Berkeley, in He i the John G. Braun Profeor of Control and Dynamical Sytem, Electrical Engineering, and Bio Engineering at California Intitute of Technology (Caltech), Paadena. Hi early work wa in the mathematic of robut control, LQG robutne, (tructured) ingular value analyi, H-infinity plu recent extenion. He coauthored book and oftware toolboxe currently ued at over 1000 ite worldwide, the main control analyi tool for high performance commercial and military aeropace ytem, a well a many other indutrial ytem. Early example indutrial application include-29,f-16l,f-15smtp,b-1,b-2,757,shuttleorbiter,electric power generation, ditillation, catalytic reactor, backhoe lopefinihing, active upenion, and CD player. Current reearch interet are in theoretical foundation for complex network in engineering, biology, and multicale phyic. Dr. Doyle group led the development of the open ource Sytem Biology Markup Language (SBML) and the Sytem Biology Workbench (SBW), the analyi toolbox SOSTOOLS, and contributed to the theory of the FAST protocol that hattered multiple world land peed record. Prize paper include the IEEE Baker, the IEEE Automatic Control Tranaction Axelby (twice), and the AACC Schuck. Individual award include the AACC Eckman and the IEEE Control Sytem Field and Centennial Outtanding Young Engineer Award. He ha held national and world record and championhip in variou port. 312 Proceeding of the IEEE Vol.95,No.1,January2007