Joint Congestion Control and Media Access Control Design for Ad Hoc Wireless Networks

Transcription

1 Joint Congetion Control and Media Acce Control Deign for Ad Hoc Wirele Network Lijun Chen, Steven H. Low and John C. Doyle Engineering & Applied Science Diviion, California Intitute of Technology Paadena, CA Abtract We preent a model for the joint deign of congetion control and media acce control (MAC) for ad hoc wirele network. Uing contention graph and contention matrix, we formulate reource allocation in the network a a utility maximization problem with contraint that arie from contention for channel acce. We preent two algorithm that are not only ditributed patially, but more interetingly, they decompoe vertically into two protocol layer where TCP and MAC jointly olve the ytem problem. The firt i a primal algorithm where the MAC layer at the link generate congetion (contention) price baed on local aggregate ource rate, and TCP ource adjut their rate baed on the aggregate price in their path. The econd i a dual ubgradient algorithm where the MAC ub-algorithm i implemented through cheduling linklayer flow according to the congetion price of the link. Global convergence propertie of thee algorithm are proved. Thi i a preliminary tep toward a ytematic approach to jointly deign TCP congetion control algorithm and MAC algorithm, not only to improve performance, but more importantly, to make their interaction more tranparent. Index Term Congetion control, Media acce control, Convex optimization, Cro-layer deign, Dual decompoition, Subgradient method, Ad hoc wirele network. I. INTRODUCTION We conider the problem of congetion control over a multihop wirele ad hoc network. Thi ha been an active reearch area over the pat few year (ee, e.g., [15], [5], [9], [30], [12], [37], [38], [6]) with many facinating and complex iue, involving, e.g., mobility, channel etimation, power control, MAC, routing, etc. Unlike mot of previou work however we focu on the interaction of congetion control at the tranport layer and channel contention at the MAC layer, and ignore all other iue. Our goal i to preent a ytematic approach to jointly deign TCP congetion control algorithm and MAC algorithm, not only to improve performance, but more importantly, to make their interaction more tranparent. Thi i motivated by two obervation. Firt, wirele channel i a hared medium and interference-limited. Link i only a logical concept and link are correlated due to the interference with each other. Under the MAC trategie uch a time-diviion multiple acce and random acce, thee link contend for excluive acce to the phyical channel. Unlike in the wireline network where flow compete for tranmiion reource only when they hare the ame link, here, network layer flow that do not even hare a wirele link in their path can compete. Thu, in ad hoc wirele network the contention relation between link-layer flow provide fundamental contraint for reource allocation. Second, TCP congetion control algorithm can be interpreted a ditributed primaldual algorithm over the Internet to maximize aggregate utility, and a uer utility function i (often implicitly) defined by it TCP algorithm, ee e.g. [18], [22], [21]. Thi erie of work implicitly aume a wireline network where link capacitie are fixed and hared by flow that travere common link. A natural formulation for the joint deign of congetion and media acce control i then the utility maximization framework with new contraint that arie from channel contention. After a brief decription of the interaction between TCP congetion control and MAC in Section II and a brief review of related work in Section III, we explain in Section IV contention graph and introduce contention matrix to model reource contraint in wirele network, and tate our utility maximization problem with MAC contraint. In Section V, we follow [18] and derive a primal algorithm to olve a relaxation of the problem, and prove it global convergence. The algorithm i not only ditributed patially, more interetingly, it decompoe vertically into two protocol layer where the MAC layer at the link generate congetion (contention) price baed on local aggregate ource rate, and TCP ource adjut their rate baed on the aggregate price in their path. Wherea congetion price are generated by AQM (active queue management) algorithm in router in wireline network (e.g. [23]), here they are generated by the MAC layer. We dicu how to deign contention reolution protocol to generate the neceary price. In Section VI, we apply duality theory to derive another decompoition of the ytem problem into congetion control ubproblem and MAC ubproblem. The key idea i to introduce the effective capacity of a link, which i the maximum average data rate a link can achieve without violating chedulability contraint. The Lagrangian of the reulting problem eparate into two maximization ubproblem, one over ource rate, to be olved by TCP, and the other over effective capacity, to be olved by MAC. The introduction of the effective capacity make the primal problem not trictly concave, and hence the dual function non-differentiable. A ubgradient algorithm that generalize the algorithm of [22] i derived to olve the dual problem, and proved to approach arbitrarily cloe to an optimal point tarting from any initial condition. Thi algorithm motivate a joint deign cheme where link-layer flow are cheduled according to congetion price of the link. We illutrate with numerical example of

2 uch a deign. Finally, we conclude in Section VII with limitation of thi paper and poible extenion. II. MOTIVATION TCP wa originally deigned for wireline network, where the link are aumed to be reliable and with fixed capacitie. Thi may not be true for wirele network, where the link are elatic due to the fact that the wirele channel i unreliable (e.g., fading and node mobility) and interference-limited. We need to exploit the interaction between tranport and MAC/phyical layer, in order to improve the performance. Thi paper doe not conider the node mobility or channel fading, but focue on the broadcat and interference-limited nature of wirele channel. In thi context, a fundamental problem i to provide an efficient bandwidth haring mechanim among the competing link-layer flow. Many exiting wirele MAC protocol, uch a ditributed coordination function (DCF) pecified in IEEE tandard[17], are trafficindependent and do not conider the actual requirement of the flow competing for the channel. Thee MAC protocol uffer from the unfairne problem, caued by the location dependency of the contention, and exacerbated by the contention reolution mechanim uch a the binary exponential backoff algorithm adopted in DCF. When they interact with TCP, TCP will further penalize thee flow with more contention. Thi A 1 B 2 C 3 D 4 E Fig. 1. Example of ad hoc wirele network G 6 will reult in ignificant TCP unfairne in ad hoc wirele network [13], [28], [35], [36], [37]. To illutrate thi, conider the example in Fig.1, and aume there are four network-layer flow A B, C D, E F and G H. The flow C D experience more contention and will build up queue fater than the other three flow. TCP will further penalize it by reducing the congetion window more aggreively, and the reulting throughput of flow C D will be much le than that of other flow. In addition to the location dependency of contention, correlation among link i alo the key to undertand the interaction between tranport and MAC layer. In wireline network, link bandwidth i well-defined and link are dijoint reource. But in wirele network, a we mentioned above, link are correlated due to the interference with each other, and network-layer flow, which do not tranvere a common link, may till compete with each other. Thu, congetion i located at ome patial contention region [37]. Conider again the example in Fig.1, and aume there are two network-layer flow A F and G H. Link-layer flow 2, 3, 4 and 6 contend with each other, and congetion i located in the patial contention region denoted by the rectangle. So, unlike wireline network where link capacitie provide contraint for reource allocation, in ad hoc wirele network the contention relation H 5 F between link-layer flow provide fundamental contraint for reource allocation. In thi paper we will model the contention relation between link-layer flow a a flow contention graph (ee, e.g., [25], [11]). Thi contruction capture the location-dependent contention among link-layer flow. Baed on the contention graph, we will ue a contention matrix to mathematically formulate the contention contraint impoed by the MAC layer. We then model the reource allocation for ad hoc wirele network a a concave utility maximization problem with MAC layer contraint, with which we can explicitly exploit the interaction between tranport and MAC layer, and ytematically carry out joint deign of congetion and media acce control. III. RELATED WORK The work in [18], [22], [21], [23] provide a utility-baed optimization framework for internet congetion control. The ame framework ha been applied to tudy the congetion control over ad hoc wirele network (ee, e.g., [6], [38]). In [38], the author tudy congetion control in ad hoc wirele network with primary interference, and formulate rate allocation a a utility maximization problem with time contraint. It aume that the MAC protocol i given, and doe not conider the problem of how the link-layer flow hare the congetion price generated by the contraint. In our work, we will conider the network with both primary and econdary interference, and jointly deign congetion control and MAC. Many cheme have been propoed for fair bandwidth haring at link layer (ee, e.g., [25], [33], [24], [16], [29], [11]). Some of thee cheme try to achieve weighted fairne, but they uually aume the weight are given and do not addre the iue of how to chooe thoe weight. In our work, thee weight or their equivalent are related to the actual flow requirement or the congetion price of the link, which guarantee ome kind of network layer fairne. In [29], the author propoe a maximin fair cheduling which aign congetion-dependent weight to the flow with primary interference and chedule the flow via maximum weighted matching. In [25], [11], the author ue the flow contention graph to characterize the contention among link-layer flow, and propoe utility-baed optimization to achieve MAC layer fairne. We will modify a multiple acce cheme propoed in [25] to implement AQM for congetion control. Alo, ome of our dicuion on the flow feaibility i recaptured from [11] for completene. In [37], the author propoe a neighborhood RED cheme to improve TCP fairne in ad hoc wirele network. Baically, thi cheme aign more hare of congetion price to the flow with le contention to alleviate TCP unfairne. We try to addre the unfairne problem that arie in the MAC layer by uing traffic-dependent MAC cheme. Cro-layer deign in communication network, epecially in wirele network, have attracted great attention recently (ee, e.g., [26] for an overview). Our work belong to the category of cro-layer deign via dual decompoition in optimization framework. Other work that can be put into thi

3 category include TCP/IP interaction in [31], joint routing and reource allocation in [34] and joint TCP and power control in [6]. The work on joint congetion control and MAC deign i the firt tep in our attempt to provide a unified framework for ytematically carrying out cro-layer deign through dual decompoition. We will extend the framework to include other layer in the future. IV. SYSTEM MODEL Conider an ad hoc wirele network with a et V of vertice (node) and a et L of logical link. We aume a tatic topology and each link l ha a fixed finite capacity c 0 l packet per econd when active, i.e., we implicitly aume a power control algorithm that maintain a contant data rate in the face of fading and other channel imperfection. Wirele channel i a hared medium and interference-limited. In thi paper, we aume logical link contend for channel acce and the ucceful link tranmit at rate c 0 l for the duration it hold the channel. We will focu on the interaction of MAC and TCP, and characterize the contention relation uing contention graph and contention matrix. The joint MAC and TCP deign i then formulated a a utility maximization problem with the contraint that arie from MAC layer contention. A. Flow Contention Graph and Contention Matrix Wirele node are aumed to be able to communicate with at mot one other node at any given time. Thi follow from the fact that a node cannot tranmit or receive imultaneouly. Link mutually interfere with each other whenever either the ender or the receiver of one i within the interference range of the ender or receiver of the other. Under thee aumption, we can contruct a flow contention graph that capture the contention relation between the link of the network (ee, e.g., [25], [11]). In the contention graph, each vertex repreent an active link, and an edge between two vertice denote the contention between the correponding link: two link interfere with each other and cannot be active at the ame time. An accurate flow contention graph could be contructed baed on the protocol model or phyical SIR model, and alo depend on the the baic multiple acce trategy ued. In practice, when we contruct the flow contention graph, we can aume two link contend with each other if they are within each other carrier ening range. Given a contention graph, we can identify all it maximal clique 1. Maximal clique are local contruction and capture the local contention relation of the flow. Flow within the ame maximal clique cannot tranmit imultaneouly, but flow in different clique may tranmit imultaneouly. For example, Fig. 2 how the flow contention graph that correpond to the ad hoc wirele network of Fig. 1 with 6 active linklayer flow. Flow 1, 2 and 3, which are in the ame clique, cannot tranmit imultaneouly, neither can flow 2, 3, 4 and 6. But flow 1 and 6 can be activated imultaneouly, ince they belong to different clique. Thu, each maximal clique in the contention graph repreent a channel reource with flow 1 A maximal clique of a graph i a maximal complete ubgraph of the graph Fig. 2. Flow contention graph and maximal clique: flow (1, 2, 3) and flow (3, 4, 5) are two maximal clique of ize 3, flow (2, 3, 4, 6) i a maximal clique of ize 4. in the clique contending for excluive acce to the reource [25]. The flow within the ame clique hare the capacity of the clique. A flow may belong to everal clique, and can uccefully tranmit if and only if it i the only active flow in all clique to which it belong. We now conider the problem of determining if a et of link flow are feaible, i.e., whether a chedule can be found to achieve thi et of flow (ee, e.g., [14], [20]). Thi will be the contraint impoed by the MAC layer. Aume that we are given a L-dimenional vector y where y l i the deired flow on link l, in packet per econd. We refer to y a the link-layer flow vector. On average, given link flow y l, the fraction of time required to end thi amount of flow i y l /c 0 l. We refer to y l /c 0 l a the normalized flow rate of link l. Since flow within the ame clique cannot tranmit imultaneouly, we obtain a neceary cheduling contraint: y l 1 l c 0 l where the ummation i over thoe link that belong to the ame clique. We can repreent the cheduling contraint in a compact form by introducing contention matrix. Suppoe the flow contention graph can be decompoed into a et N of maximal clique indexed by n. Each clique n contain a et L n L of link. The et L n define a N L contention matrix F { 1/c 0 F nl = l if l L n 0 otherwie Thu, the above cheduling contraint can be written a Fy 1 (1) where 1 denote a N-dimenional vector with each component being 1. Fig. 3. Ring graph of ize 5: by equation (1) the maximal normalized um rate i 5, but the actual maximal um rate i 2. 2 Since the above decription i a fluid-level decription, i.e., we average the cheduling variable over time, contraint (1)

4 i only a neceary condition for the feaibility of the flow vector y. To illutrate thi, conider the example in Fig.3, where the contention graph i a ring of ize 5. According to the contraint (1), each flow hould attain a normalized rate of 1/2 if the max-min fairne allocation criterion i ued. However, cheduling the link according to the max-min fairne criterion allocate only a rate of 2/5 to each link, ince at anytime at mot two link can tranmit imultaneouly. Given a flow vector y, it i not an eay job to verify it feaibility, ince thi i equivalent to finding a chedule that achieve y. It can be hown that a feaible flow vector mut be a convex combination of the characteritic vector of all independent et of the flow contention graph 2, and that the et of achievable flow vector i a cloed, convex and compact et (ee [1], alo cited in [11]). In addition, contraint (1) i alo a ufficient condition for the feaibility of the flow vector if and only if the contention graph i a perfect graph 3 (ee [1], alo cited in [11]). According to the trong perfect graph theorem [8], [7], a graph i perfect if and only if it ha no induced ubgraph that i iomorphic to an odd hole 4, or it complement. Therefore if there exit odd hole in a contention graph, the um of the normalized flow rate of any clique that include edge of an odd hole hould be reduced. In general, it i hard to tell whether a graph i perfect or not. Such claification may require the global topology information of the graph (e.g., an odd hole can pan the whole graph). Since the algorithm for ad hoc network are deired to be ditributed and depend at mot on local meage paing, we need to trade off the accuracy (and even ome performance optimality) for the implicity of the deign. Hence, we will not verify whether a graph i perfect or not, but reduce the um of the normalized rate of a clique to enure flow feaibility. Determining exactly by how much we hould reduce the um rate i difficult and alo depend on the baic fairne criterion we chooe. In thi paper, we will not further dicu thi iue, but aume a maximal clique um rate vector ε. Thevalueof ɛ will depend on local topology of the contention graph. Thu, the contraint impoed by the MAC layer can be written a Fy ε (2) We will ee later that we do not need to know the value of ε, ince in the joint deign in ection V we will relax the contraint (2), and in the joint deign in Section VI thi contraint can be replaced with the contraint (1) with ome additional contraint on the value that y can take. Note that the contention graph and contention matrix i a rather general contruction. It include wireline network a a pecial cae where the contention matrix F i a L L identity matrix, ince there i no interference among the link. It can be ued to characterize the interference relation among wirele and wired link in hybrid wireline-wirele network. It can 2 An independent et of a graph i a ubet of the vertice uch that no two vertice in the ubet are adjacent. 3 A graph i perfect if for every induced ubgraph it chromatic number i equal to the clique number of the graph [8]. 4 A hole i a graph induced by a chordle cycle of length at leat 4. A hole i odd if it contain an odd number of vertice [7]. alo be modified to characterize the contention relation in the frequency-diviion or other trategie for channel acce. B. Problem Formulation Aume the network i hared by a et S of ource indexed by. Each ource ue a et L L of link. The et L define an L S routing matrix { 1 if l L R l = 0 otherwie We will fix the routing matrix R and focu on congetion control. Each ource attain a utility U (x ) when it tranmit at rate x packet per econd. We aume U i continuouly differentiable, increaing, trictly concave, and unbounded a x 0. Our objective i to chooe ource rate x o a to [18], [22], [21]: U (x ) (3) max x 0 ubject to FRx ε (4) The contraint (4) follow from (2) with y = Rx. A unique maximizer exit, ince the objective function i trictly concave and feaible et i convex and compact. We can ee the ytem problem (3)-(4) from two complement perpective. On one hand, it i a utility-baed congetion control problem with the MAC layer contraint. A uch, the congetion price are not decided by the link capacity, but determined by the contention region. In other word, the MAC layer impoe the ultimate contraint to the achievable rate. On the other hand, it i a media acce control problem, which i to allocate phyical bandwidth to each link, with the objective of maximizing aggregate end uer utilitie. A uch, the reulting MAC protocol i traffic-dependent and will allocate more bandwidth to the link with more contention to alleviate flow congetion. Solving the ytem problem (3)-(4) directly require coordination among poibly all ource and i impractical in real network. According to the theory of convex optimization, ditributed algorithm can be derived by conidering it relaxation and dual problem. In the next two ection, we will olve thee two problem and give them different interpretation in the context of joint deign of congetion control and media acce control. V. JOINT DESIGN I: GENERATING CONGESTION PRICE DIRECTLY FROM THE MAC LAYER In thi ection, a primal algorithm i derived by olving the relaxation of the ytem problem (3)-(4), firt propoed in [18]. Baed on the algorithm, we propoe a traffic-dependent cheme for media acce control and generate congetion price directly from the MAC layer. A. Primal Algorithm and It Convergence Intead of olving the ytem problem (3)-(4), let u conider it relaxation: max x 0 V (x) (5)

5 with zn(x) V (x) = U (x ) λ n (v)dv (6) n 0 where z n (x) = l F nlr l x i normalized um rate of clique n for given ource rate x, and λ n ( ) i the penalty function, which can be interpreted a the price for ending traffic at normalized rate z n on clique n. We further aume λ n ( ) i a non-negative, non-decreaing, continuou function, and not identically zero. Term c 0 l c l y l x z n λ n p l R F γ t,γ Π TABLE I SUMMARY OF MAIN NOTATION Definition capacity of link l when active effective capacity of link l aggregate flow on link l ource rate of ource normalized um rate of clique n price of clique n congetion price of link l routing matrix contention matrix tepize feaible rate region Lemma 1: Under the above aumption, the function V (x) defined in (6) i trictly concave. Thu, the problem (5) admit a unique olution in the interior of the feaible et. Proof: Let zn(x) f(x) = λ n (v)dv n 0 Since λ n ( ) i non-decreaing, for any x, x 0 n zn(x) f(x) f( x) = λ n (v)dv n z n( x) λ n (z n ( x)) l F nl R l (x x ) = (x x ) f ( x) x Thu, according to the firt-order condition of convexity for differentiable function [4], f(x) i a convex function and f(x) i a concave function. Since U ( ) i trictly concave, V (x) i the um of a trictly concave function and a concave function. Thu, V (x) i trictly concave. Note that V (x) a x 0 or a x for any S. So, the problem (5) admit a unique olution that i in the interior of the convex et x 0. The optimal ource rate atify V =0, S x which give U (x ) nl λ n (z n (x))f nl R l =0, S Define q = nl λ n(z n )F nl R l. Applying the gradient method to (5) (6), we obtain the following congetion control algorithm ẋ = κ (U (x (t)) q (t)), S (7) where κ i a poitive. Note that the primal algorithm (7) i completely ditributed. Here, the aggregate normalized price q (t) i a feedback ignal ource oberve. A dicued in [18], λ n (z n ) can be interpreted a a congetion (contention) price that meaure the degree of contention in clique n when the total normalized flow through the clique i z n. Hence, q (t) meaure the degree of contention in all the clique that contain any link in ource path (a larger q (t) indicate a greater degree of contention). The congetion control mechanim for each ource i to adjut it rate x (t) according to the network contention it perceive. In the next ubection, we will deign a MAC protocol to generate thee contention price in a ditributed manner. The following theorem, following [18], how that the primal algorithm (7) i globally table, i.e., the unique olution to problem (5) i a table point, to which all trajectorie converge. Theorem 2: Starting from any initial rate x(0) 0, the congetion control algorithm (7) will converge to the unique olution of the problem (5). Proof: From lemma 1, V (x) i a trictly concave function, and problem (5) admit a unique olution x. Further V = V ẋ = κ (U x (x ) q ) 2 0 Note that V > 0 for x x and i equal zero for x = x. Thu, V (x(t)) i trictly increaing with t, unle x(t) =x. More preciely, chooe V (x ) V (x) a a Lyapunov function for ytem (7). By Lyapunov theorem [19], the trajectorie of (7) converge to x, tarting from any initial condition x(0). Note that algorithm (7) olve the ytem problem (3)-(4) only approximately. By chooing appropriate price function λ n ( ), the optimal olution can be guaranteed to atify the contraint (4), and even olve the ytem problem (3)-(4) exactly [32]. In practice, the price function λ n ( ) determine the efficiency of the congetion control cheme, a we will further dicu in the next ubection. B. Generating Congetion Price from the MAC Layer Unlike the price function in wireline network which i a function of aggregate flow rate of the link [18], [22], [21], the price function λ n ( ) i required to be a function of the normalized um rate z n of clique n. Thi i conitent with the fact that, in wirele network, link i only a logical concept and the contention region i the reource that flow hare and contend for acce. However, the clique i only a virtual entity and no centralized controller exit to monitor it congetion tatu, how can we implement the congetion price? We need to deign an active queue management cheme where each logical link generate or hare a portion of

6 the congetion price uch that their ummation i equal to λ n (z n ) for clique n. Oberve that a imilar problem appear in cheduling flow over ad hoc wirele network, and that each logical link will get the right portion of the congetion price automatically if the link are granted channel acce according to the flow requirement. We propoe a multiple acce cheme and generate congetion price directly from it. In multiple acce protocol, contention reolution i uually achieved through two mechanim: peritence and backoff [25]. In the peritence mechanim, each contending node or link-layer flow maintain a peritence probability and contend for the channel with thi probability. In the backoff mechanim, each contending node or link-layer flow maintain a backoff window and wait for a random amount of time bounded by the backoff window before a tranmiion. When multiple imultaneou tranmiion caue colliion, the peritence probability or backoff window i adjuted appropriately o that colliion are reduced. Thu, the peritence probability and backoff window are function of the etimated contention, and different contention reolution algorithm differ in term of how they adjut thee parameter in repone to colliion and ucceful tranmiion. In our problem, the normalized um rate z n = l F nlr l x i the natural meaure of the contention in clique n. Thu, the deign of multiple acce i to adjut peritence probability or/and backoff window according to z n. The intuition behind thi i the ame with that behind congetion control algorithm (7), which ugget that we can jointly deign congetion control and media acce control, and generate congetion price directly from the MAC layer. Note that the normalized flow rate F nlr l x i the fraction of time that i required to tranmit the amount of flow y l = R lx, and the normalized um rate of a clique mut not exceed 1 (ee contraint (1)). It ha a natural interpretation a a probability. Thu, in our propoed cheme, we approximate the normalized flow rate y l /c 0 l a a peritence probability with which the flow l contend for the channel. Furthermore, ince each flow l contend for the channel with the probability y l /c 0 l, the flow hould contend for the channel in the ame way after they decide to contend, conitent with the fact that the congetion price i a function of the normalized um rate. Thi implie that all flow hould have the ame backoff window. To be more pecific, define p l = min{ y l, 1}, and let w c 0 l denote the backoff window. The joint deign of congetion control and media acce control work a follow: each linklayer flow y l will contend for the channel with probability p l when it ene the channel i idle. If it decide to contend for the channel, it randomly chooe a waiting time B l from the interval [0,w] uniformly. After the waiting time, the flow ene the channel and acquire the channel if it i idle. If either the channel i buy or there i colliion, the flow will drop or mark the packet a the congetion ignal. Upon receiving the congetion ignal, the ource will adjut it rate according to algorithm (7). We can ee that the bandwidth i allocated in proportional to the normalized flow rate of each link. Thu, we obtain a traffic-dependent multiple acce cheme. Note that link needn t know explicitly flow contention graph and the clique they belong to. But, in order to be conitent with the derivation and convergence analyi of the primal algorithm, the congetion price λ n of clique n mut be a function of the normalized um rate z n. Unfortunately, the propoed MAC cheme i very difficult to analyze. For the imple cae with no backoff, i.e., w =0, under the aumption of Poion arrival proce, the above cheme doe generate approximately the right price function λ n =1 e zn z n e zn Thi price i jut the probability when there are two or more packet, and can be readily derived following imilar analyi carried out for Aloha [2]. For the general cae with backoff, we have not yet obtained an explicit price function. We can alo implement active queue management through deigning other kind of traffic-dependent multiple acce cheme. In practice, different deign will give different price function, which in turn will determine the performance of the congetion control cheme. VI. JOINT DESIGN II: SCHEDULING LINK-LAYER FLOWS ACCORDING TO CONGESTION PRICE In thi ection, a dual algorithm i derived by olving the dual problem of the ytem problem (3)-(4)[22], [23]. The olution to the dual problem motivate a cheme for media acce control in which link-layer flow are cheduled according to congetion price. A. Dual Algorithm and It Convergence The ytem problem (3)-(4) doe not involve explicitly the variable for link. We now introduce an auxiliary variable c, which i a L-dimenional vector with each component c l interpreted a effective or average capacity of link l. Conider the following problem: max U (x ) (8) x 0,c l 0 ubject to Rx c & Fc ε (9) The firt contraint ay that the aggregate ource rate at any link l doe not exceed the effective link capacity. The econd contraint ay that the effective link capacitie atify the MAC layer contraint. It i eay to how that thi problem i equivalent to the ytem problem (3)-(4). Conider the dual problem min D(p) (10) p 0 with partial dual function D(p) = max U (x ) p T (Rx c) (11) x 0,c l 0 ubject to Fc ε (12) where we relax only the contraint Rx c by introducing Lagrange multiplier p. The maximization problem in (11) can be decompoed into the following two ubproblem D 1 (p) = max x 0 U (x ) p T Rx (13)

7 and D 2 (p) = max c 0 pt c ubject to Fc ε (14) The firt ubproblem i jut TCP [22], [23], and the econd one i the cheduling which i to maximize the weighted um of effective link capacitie with the congetion price a the weight. Thu, by dual decompoition, the flow optimization problem decompoe into eparate local optimization problem of tranport and link layer, repectively, and thee two layer interact through the congetion price. Note that the objective function U (x ) i not trictly concave with repect to variable (x, c), hence the dual function D(p) might not be differentiable. Indeed, the problem (13) admit a unique maximizer ( ) x (p) =U 1 p l R l (15) and D 1 (p) i differentiable, but problem (14) may have multiple maxima and D 2 (p) i a piecewie linear function and not differentiable. Thu, D(p) i not differentiable at every point p [3], and we cannot ue the uual gradient method, which are developed for differentiable problem, to olve the dual problem. Here we will olve the dual problem uing ubgradient method. Suppoe c(p) i a maximizer of the problem (14), i.e., c(p) arg max c 0 pt c ubject to Fc ε (16) then g(p) =c(p) Rx(p) (17) i a ubgradient 5 of dual function D(p) at point p. To ee thi, conider any two point p and p, by definition D( p) = max U (x ) p T (Rx c) x 0,c l 0 ubject to Fc ε hence D( p) U (x (p)) p T (Rx(p) c(p)) = D(p)+( p T p T )(c(p) Rx(p)) Thu, by the ubgradient method [3], we obtain the following algorithm for price adjutment for link l p l (t +1)=[p l (t)+γ t ( R l x (p(t)) c l (p(t)))] + (18) where γ t i a poitive calar tepize, and + denote the projection onto the et R + of non-negative real number. (15), (16) and (18) are the congetion control algorithm. The algorithm ha a nice interpretation in term of law of upply and demand and their regulation through price. Eq.(18) ay that, if the demand R lx (p(t)) for bandwidth at link l exceed the upply c l, the price p l will rie, which will in turn decreae the demand (ee eq. (15)) and increae upply 5 Given a convex function f : R n R, a vector d R n i a ubgradient of f at a point u R n if f(v) f(u)+(v u) T d, v R n. l (ee eq. (16)). Alo, note that equation (15) and (18) are completely ditributed. We will tudy the ditributed olution to problem (14) in the next ubection. Subgradient may not be a direction of decent at point p, but make an angle le than 90 degree with all decent direction at p. The new iteration may not improve the dual cot for all value of the tepize. There exit many reult on the convergence of the ubgradient method [27], [3]. For contant tepize, the algorithm i guaranteed to converge to within a range of the optimal value 6. For diminihing tepize, the algorithm i guaranteed to converge to the optimal value. For our purpoe, we would like an aynchronou implementation of the ubgradient algorithm, and thu a contant tepize i deired. Note that the dual cot will uually not monotonically approach the optimal value, but wander around it under the ubgradient algorithm. The uual criterion for tability and convergence i not applicable. Here we define convergence in a tatitical ene. Definition 3: Let p denote an optimal value of the dual variable. The algorithm (15), (16) and (18) with contant tepize i aid to converge tatitically to p, if for any given δ > 0 there exit a tepize γ uch that 1 t lim up t t D(p(τ)) D(p ) δ. The following theorem guarantee the tatitical convergence of the ubgradient method. Clearly, an optimal value p exit. Theorem 4: Let p be an optimal price. Let γ denote the contant tepize. If the norm of the ubgradient i bounded, i.e., there exit G uch that g(t) 2 G for all t, then the algorithm (15), (16) and (18) converge tatitically to within γg 2 /2 of the optimal value. Proof: By equation (18), we have p(t +1) p 2 2 = [p(t) γg(p(t))] + p 2 2 p(t) γg(p(t)) p 2 2 = p(t) p 2 2 2γg(p(t)) T (p(t) p ) +γ 2 g(p(t)) 2 2 p(t) p 2 2 2γ(D(p(t)) D(p )) +γ 2 g(p(t)) 2 2 where the lat inequality follow from the definition of ubgradient. Applying the inequalitie recurively, we obtain p(t +1) p 2 2 p(1) p 2 2 2γ D(p )) + γ 2 Since p(t +1) p 2 2 0, wehave 2γ t (D(p(τ)) t g(p(τ)) 2 2 t (D(p(τ)) D(p )) 6 The gradient algorithm with contant tepize converge to the optimal value, provided the tepize i mall enough.

8 t p(1) p γ 2 g(p(τ)) 2 2 p(1) p tγ 2 G 2 From thi inequality we obtain 1 t D(p(τ)) D(p ) p(1) p tγ 2 G 2 t 2tγ Thu lim up t 1 t t D(p(τ)) D(p ) γg2 2 (19) i.e., the algorithm converge tatitically to within γg 2 /2 of the optimal value. The aumption of bounded norm for ubgradient g(p) i reaonable, ince c i finite and we can alo enforce an upper bound to x. We ee that, by chooing appropriate value of the tepize, the algorithm can approach the optimal value arbitrarily cloe within a finite number of tep. The ytem decribed by equation (15), (16) and (18) i a hybrid ytem. Although Theorem 4 guarantee that it dynamic i bounded in an average ene, it i untable in the trict ene. It may have complex behavior uch a limit cycle, i.e., it may go through an ergodic equence. The reaon for thi intability i that the dual function i nondifferentiable or nonmooth. One way to avoid intability i to add ome regularization term, uch a trictly convex/concave term, to make the dual function differentiable. For example, in our problem we can add a concave utility V l (c l ) to each link l. The reulting ytem i table but may not maximize the end uer utilitie. So, there exit a tradeoff between tability and end uer utility maximization (ee alo [31]). However, in our problem the ocillatory behavior in the teady tate correpond to the cheduling proce. B. Scheduling Link-layer Flow according to Congetion Price Scheduling i to decide which link and when to tranmit, which i equivalent to chooing an independent et of flow contention graph to be active at each time lot. However, olving problem (14) cannot guarantee that we obtain a rate vector correponding to an independent et. Recall that the reaon why contraint (1) may not be a ufficient condition i that it i a fluid level decription. However, when the flow vector y i uch that each component y l take value at 0 or c 0 l while atifying contraint (1), it i alo feaible. Such a flow vector correpond to an independent et of flow contention graph. Thu, we propoe to replace the contraint in the problem (14) with Fc 1, and olve the following cheduling problem with an additional dicrete contraint max c 0 ubject to Fc 1 c l =0or c 0 l, p T c (20) l L Having done that, we need to clarify with repect to which ytem problem the above algorithm converge. To ee thi, we firt repreent an independent et i a a L-dimenional rate vector r i with { rl i c 0 = l if l i 0 otherwie The feaible rate region Π at the link-layer i then defined to be the convex hull of thee rate vector [1] Π:={r : r = a i r i,a i 0, a i =1} i i It i eay to verify that olving problem (20) i equivalent to olving the following problem max c 0 ubject to p T c c Π Thu, the whole joint congetion control and cheduling algorithm i to olve the following ytem problem U (x ) max x 0 ubject to Rx c & c Π Note that the original problem (8)-(9) i a relaxation to the above problem. We now come to olve the problem (20). If the contention graph i perfect, all the extreme point of contraint Fc 1 are independent et. In thi ituation, we can jut olve the problem (20) by neglecting the dicrete contraint, which ha the ame optimal olution a the original dicrete problem. Thi i imilar to what happen in network flow optimization problem [3]. When the contention graph i not perfect, not all the extreme point of Fc 1 are independent et. In thi ituation, we will firt olve the relaxed problem without dicrete contraint, and then round up the olution to the nearet independent et, ince the objective function p T c i continuou with repect to c. Although the computational complexity of linear programming i polynomial, the known algorithm for general linear programming are not uitable for large cale optimization problem uch a thoe in network. Intead, an efficient, ditributed algorithm with only local information i required for thee ytem. In our problem, we aume that each link only know it own weight and the contraint it i involved in. We will again ue dual decompoition and ubgradient method to obtain a ditributed algorithm to olve problem (20). Note that by olving the dual problem we obtain the optimal dual variable, but the optimal primal variable i not immediately available and need to be recovered with care. One imple way to obtain feaible primal olution i to add a mall regularization term to the primal function. Here, we add a mall quadratic term to the objective function, and maximize p T c δc T c where δ i a mall poitive number. A δ approache zero, the olution obtained approache an exact olution to the original problem. Thi approach i cloely related to penalty

9 and augmented Lagrangian method for olving the dual of a convex program [3]. Conider the dual problem with min L(λ) (21) λ 0 L(λ) = max p T c δc T c λ T (Fc 1) c 0 The gradient algorithm to the dual problem (21) i [( c l (t) = p l ) + λ n F nl /(2δ)] (22) n [ ( + λ n (t +1) = λ n (t)+β F nl c l (t) 1)] (23) where β i a poitive tepize. The convergence analyi of uch algorithm i well-known [3]. Let Ō denote the maximal ize of clique, and N the larget number of clique that contain the ame link. The range of the tepize with which the algorithm converge can be defined a in [22]: 0 <β< 4δ Ō N After obtaining a value of c l, link l round it up to c 0 l or 0, whichever i cloer. Thi doe not guarantee that the reulting c i optimal or even an independent et all the time, but we can ue the notion of ɛ-ubgradient 7 to analyze the effect of error [3]. Theorem 5: Suppoe at each iteration t a ɛ t -ubgradient i ued. Aume that ɛ t ɛ for all t or lim t ɛ t ɛ, then under the ame aumption a in Theorem 4 the algorithm (15), (16) and (18) converge tatitically to within γg 2 /2+ɛ of the optimal value. Proof: We kip the detail, ince it i the ame a the proof of Theorem 4 except that we ue ɛ-ubgradient here. To derive a ditributed algorithm for cheduling, we have aumed that each link know it own contraint. In order to achieve thi, each link will collect it local flow information 8, contruct it local contention graph and decompoe it into a et of maximal clique. Since the clique i only a virtual entity, the price adjutment algorithm (23) for a clique will be carried out by the link within the clique. To be able to calculate new price for a clique, each link need to exchange new flow rate information, which i calculated by link uing algorithm (22), with all it contending flow within one hop. Thi can be done by periodically broadcating the flow rate information. In order for thi joint deign to work, we require that cheduling be carried out at a much fater time cale than congetion control. Within a time interval γ, the MAC layer hould be able to decide which link to tranmit and then finih the tranmiion. The time cale matching problem i difficult 7 Given a convex function f : R n Rand ɛ 0, a vector d R n i a ɛ- ubgradient of f at a point u R n if f(v) f(u) ɛ+(v u) T d, v R n. 8 Thi can be achieved by paively litening to other link broadcating flow information or actively ending inquiring meage to other link to ak for flow information. l to olve for cro-layer deign in general. The key to olving thi iue i to be able to deign fat, efficient algorithm. For example, in our joint deign we can carry out cheduling by heuritically identifying the et of concurrently active link that can achieve the maximization in (14) approximately (ee, e.g., [10]). C. A Numerical Example To illutrate the characteritic of the joint congetion control and cheduling algorithm (15), (16) and (18), and their implication for the algorithm implementation in ad hoc wirele network, we conider a imple example with the network in Fig. 1. We aume that all the link have the ame capacity when active. We further aume c 0 l =1, l L, and that all network layer flow have the ame utility U (x )=log(x ). Fig. 4. G 1 A B C D E F 2 Ad hoc wirele network with three network layer flow. Suppoe there are three network layer flow G H, A B and D F in the network a hown in Fig. 4, with the rate denoted by x 1, x 2 and x 3. We imulate the algorithm (15), (16) and (18) with different choice of tepize γ. The left panel of Fig. 5 how the evolution of dual function with the tepize γ =0.1. We can ee that the dual function approache the optimal very fat, but not monotonically. It will ocillate around the optimal. A we have dicued before, thi ocillating behavior mathematically reult from the nondifferentiability of the dual function and phyically can be interpreted a correponding to the cheduling proce. The Dual Function D(p) Normalized Time Normalized Source Rate H Flow 1 Flow 2 Flow Normalized Time Fig. 5. The evolution of dual function and ource rate with tepize γ =0.1. The optimal flow rate are (1/3,1/9,1/3). right panel of Fig. 5 how the evolution of ource rate of each

10 flow. Similarly, the flow rate approach the primal optimal very fat, but not monotonically. We alo note that the performance of the algorithm i much better than the bound γ/2 pecified in Theorem 4. Thu, we can ay that, if a protocol i deign baed on thi algorithm, it will likely converge fat. The choice of the tepize γ i important. It characterize the optimality of the algorithm, a hown in Theorem 4. Fig. 6 how the evolution of the dual function and ource rate with the ame initial tate but different tepize γ =0.5. Compared with the cae with tepize γ =0.1, it almot ha the ame convergence peed, but with a bigger ocillation. Note that, near the primal optimal, the flow rate ocillate between the feaible et and non-feaible et of the contraint (4). The bigger ocillation mean that the network will be underloaded and overloaded more often. Thu it will ha poorer performance uch a lower throughput. So, a maller tepize lead to a better performance. Dual Function D(p) Normalized Time Normalized Source Rate Flow 1 Flow 2 Flow Normalized Time Fig. 6. The evolution of dual function and ource rate with tepize γ =0.5. The optimal flow rate are (1/3,1/9,1/3). However, the tepize γ alo pecifie an upper bound for the length of time lot ued in the cheduling. A we mentioned before, within time interval γ the MAC layer hould decide which link to tranmit and then finih the tranmiion. So, the tepize cannot be too mall. Thu, there exit a tradeoff between congetion control, which prefer a maller tepize, and the cheduling, which prefer a larger tepize. In practice, the tepize hould take value of order of from m to ten of m. In all the imulation, we ue ditributed algorithm (22)-(23) to olve the cheduling in (16). To evaluate the performance of our cheduling algorithm, we alo ue a linear programming oftware to olve the cheduling. We do not find any ditinguihable difference between the imulation uing the linear programming oftware and the algorithm (22)-(23). Our imulation are baed on ideal implementation of the algorithm (15), (16) and (18). In it practical implementation in ad hoc wirele network, we need to take into conideration uch iue a the ignaling overhead, the propagation delay, and the time ued to make cheduling deciion, etc. To deign a practical protocol baed on thi algorithm will be one of our future work. VII. CONCLUSION We have preented a model for the joint deign of congetion control and media acce control for ad hoc wirele network, where the reulting algorithm are to olve a utility maximization problem with contraint that arie from contention for the wirele channel. We have derived two algorithm that are not only ditributed patially, but more interetingly, they decompoe vertically into two protocol layer where TCP and MAC jointly olve the ytem problem. The firt i a primal algorithm which motivate a joint deign where the multiple acce cheme i traffic dependent and the congetion price are generated directly from the MAC layer. The econd i a ubgradient algorithm for the dual problem and it motivate a joint deign where link-layer flow are cheduled according to the congetion price of the link. Thi paper i a preliminary tep toward a ytematic approach to jointly deign TCP congetion control algorithm and MAC algorithm, not only to improve performance, but more importantly, to make their interaction more tranparent. Much work remain. Firt it would be intereting to derive a formal MAC protocol in our joint deign I, prove that it generate correct price, and analyze it dynamic propertie. Second, for our joint deign II, we will need a fater and more efficient algorithm to olve the cheduling problem if it i to be applied to broadband wirele environment. 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