Loss-Cognizant Pricing in Feed-Forward Networks with Greedy Users

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1 Loss-Cognizant Picing in Feed-Fowad Netwoks with Geedy Uses Ashaf Al Daoud and Muat Alanyali Depatment of Electical and Compute Engineeing Boston Univesity, Boston, MA {ashaf, Abstact We conside feed-fowad packet switched netwoks with fixed outing and andom congestion losses. Uses of the netwok ae assumed to be geedy in the sense that each use inceases its taffic ate in popotion to pofit pe packet that it geneates. Using a deteministic fluid model, we popose pepacket picing policies to contol packet geneation ates of uses. The esulting ates ae shown to stabilize at local maxima of two suitable choices of social welfae in the netwok. We give decentalized algoithms fo netwok links to compute thei pices in a stateless fashion based on piggybacking contol messages on acknowledgements. The obtained analytical conclusions ae veified via packet-level simulations. I. INTRODUCTION Steep incease in volume and divesity of netwok applications within the last decade has led to netwok achitectues that ae based on Active Queue Management (AQM). AQM boadly efes to involvement of outes in congestion contol, eithe by discading packets at the onset of congestion o by explicitly signalling congestion to taffic souces via packet making. Thee has been a suge of impotant ecent wok on netwok achitectues based on AQM, establishing the pomising potential of such achitectues when coupled with socially esponsible end-use behavio such as that of TCP. While an ovewhelming faction of the cuent Intenet taffic is egulated via TCP, tends in high-bandwidth applications that use non-ate-adaptive tanspot potocols aise new issues concening efficiency of netwok opeation [1]. An appealing famewok to exploe possible opeating egimes of netwoks that cate to non-tcp uses is povided by game theoy wheein such uses ae typically modeled as non-coopeative paticipants shaing a common esouce. In this pape we take a simila appoach in that we conside non-coopeative uses. Howeve athe than esoting to nomal-fom games and static notions of equilibium such as Nash equilibia, we assume a paticula dynamical behavio patten fo uses that amounts to geediness, and seek a systematic appoach to manipulate that behavio to dive the netwok to efficient opeating egimes. We conside a collection of communication links shaed by autonomous packet souces that adjust thei ates of packet geneation dynamically. Packets of each souce follow a fixed oute identified by an odeed sequence of links. Each link in the netwok dops packets pobabilistically depending on its cuent load, and chages each admitted packet a cetain amount that is detemined dynamically. The peceived utility of a souce is assumed to be elated to the diffeence between its goodput, i.e. the ate of packets that each thei destination without being dopped, and the total pice it pays to the netwok pe unit time due to admitted packets. This assumption is motivated by intepeting each packet to geneate evenue fo its souce if it is deliveed to its destination, in which case the afoementioned diffeence is popotional to the expected pofit pe packet eleased into the netwok. The main hypothesis of the pape is a specification of individual souce behavio that amounts to inceasing the packet geneation ate when the instantaneous pofit pe packet is positive and deceasing it othewise. Such a souce might, fo example, epesent a sevice povide that adopts a geedy stategy to maximize its cumulative pofit. The objective of this pape is to intoduce guiding pinciples fo picing netwok links to maximize appopiate measues of social welfae fo uses, when abitay ate-based packet dopping functions ae employed at links. We conside two social welfae functions, namely the sum of goodputs and the sum of log-goodputs on account of fainess, and identify dynamic picing schemes that ende local maxima of these functions asymptotically stable. We also povide a pactically appealing appoach fo detemining optimal pices that involves only local messaging ove outes and that admits convenient implementation if acknowledgements tavel back on the same path as thei associated packets. We finally employ packet-level simulations to veify the picing guidelines that ae obtained though the afoementioned fluid-level analysis. Main contibutions of this pape ae: a) Joint consideation of packet dopping and picing: The two AQM functionalities that ae typically consideed in isolation ae consideed hee in one pespective, leading to nontivial foms of picing fo geneal packet dopping functions at links. b) Dynamic famewok: We conside an analytic famewok based on an aguable model of use dynamics that leads to pedictions on stability of possible equilibia. c) Distibuted and stateless implementation: The picing ules developed hee ae stateless in the sense that no pe-flow infomation is needed in thei computation o implementation. Moe specifically, each link chages the same amount to all admitted packets egadless of the owning flow, and computation of link pices admit distibuted implementation that does not equie exchanging explicit pe-flow infomation within the netwok.

2 d) Geneality of topology: The topologies consideed hee ae abitay subject to a feed-fowad popety, which is descibed fomally late in the text. The main Theoems , howeve, apply to geneal topologies if a cetain system of equations can be shown to admit a unique and smooth solution. The est of the pape is oganized as follows: Section II gives a sketch of pio wok elated to the viewpoint taken hee. The consideed netwok model is intoduced in Section III. Section IV identifies picing schemes that stabilize the netwok aound local maxima of cetain social welfae definitions unde a fluid model fo the evolution of geedy use behavio. This section also intoduces a local message passing algoithm fo distibuted computation of optimal pices in feed-fowad netwoks. We veify the esults and the fluid model via packet-level simulations in Section V, and conclude with final emaks in Section VI. II. RELATED WORK Sevice disciplines based on Weighted Fai Queueing impose fainess iespective of flow chaacteistics, albeit at a high opeational complexity due to thei stateful natue. Basic stateless AQM schemes such as Dop Tail and Random Ealy Detection (RED) have tendency to potect less aggessive flows, and othes have been poposed and analyzed to futhe accentuate that featue, e.g. [2] [7]. Although they offe a clea account of fainess fom the pespective of socially esponsible flows, these esults povide limited insight on how capacity is allocated among aggessive flows. Social welfae optimization has been extensively consideed unde TCP-like end-use behavio. In [8] Kelly decomposed maximization of social welfae into two poblems to be solved by the netwok and the use, whose solutions led to a picing intepetation of distibuted flow contol. A dynamic solution scheme that lends itself to implementation via explicit congestion notification (ECN) was given in [9] and it was shown to convege to optimal solutions when uses employ a cetain behavio akin to the additive incease-multiplicative incease (AIMD) featue of TCP. Othe schemes that ae based on negotiations between the uses and the netwok wee consideed by [10], [11]. Game theoetic pespectives to the picing intepetation wee ecently given by [12], [13], which established bounds on inefficiency of Nash equilibia when uses anticipate consequences of thei own bids. In [14] a dynamic viewpoint was adopted in the social welfae poblem of [8], and seveal end-use behavios wee investigated in model that was based on ECN and that did not diffeentiate between dopped and maked packets. An optimization setting with both congestion elated and andom packet losses was consideed by [15] fo geneal topologies and ound tip delays. In [15] the authos showed that unde had capacity constaints, that is when no packet is lost at a link unless the aival ate exceeds capacity, congestion-elated packet losses can be nealy eliminated povided that uses employ TCP-type flow contol algoithms. The model theein concens a diffeent use behavio than consideed hee and it applies with a special packet dopping Fig. 1. A feed-fowad netwok with fou links and fou outes. function that admits accuate appoximations when thinning of taffic in upsteam links is ignoed. In contast, opeating egimes of a given netwok entail non-negligible packet losses unde the individual use stategies consideed in the pesent pape. Thee is a substantial body of game-theoetic appoaches that boadly models esouce shaing in a netwok as a nomalfom game, and focus on Nash equilibia (see, fo examples, [16] [20] and [21] fo a geneal suvey). An adequate suvey of the impotant wok theein is beyond the scope of this pape. Pehaps among those papes that ae moe elevant to the paticula viewpoint of the pesent pape is [19], which consideed a queue employing AQM with selfish TCP-type uses that choose thei AIMD paametes to achieve high goodputs. Recently [20] consideed a simila poblem in which the assumptions on the use algoithms in [19] ae elaxed, and investigated packet dopping functions that admit Nash equilibia. In contast to these papes, we conside hee a dynamic fomulation that enables addessing stability of equilibia. We also conside moe geneal topologies than a single link. Ou appoach is closely elated to the techniques of [22] that chaacteizes shadow pices fo a social welfae function in the context of finite capacity cicuit-switched netwoks. [22] consideed an appoximate fomulation (namely, the Elang fixed point) of call blocking in such a netwok, and detemined the sensitivity of social welfae to call aival ates. The sensitivity coefficient, efeed to as the shadow pices, ae shown to have an appealing additive fom. A distibuted method fo thei computation is discussed in detail. We conside hee packet-switched netwoks and geedy use behavio theein, and povide a computation scheme that entails little communication ovehead fo feed-fowad netwoks. III. PROBLEM SETUP We conside a packet-switched netwok with finite set K of links. A oute in this netwok is an odeed subset of links. Let R be a collection of outes. We intepet each oute R as a netwok use which geneates packets that tavese links in the ode they appea in the odeed tuple. Letaouted netwok (K, R) efe to a collection of links and outes. Definition 3.1: (Feed-Fowad Netwok) A outed netwok is feed-fowad if its links can be indexed such that each oute goes though links that ae monotonically ascending in indices.

3 In paticula the outed netwok of Figue 1 is feed-fowad. The following definition, which involves no ambiguity in a feed-fowad outed netwok, will be useful in the exposition: Definition 3.2: Given links j, k K and a oute R, the symbol j k indicates that oute taveses link j befoe it taveses link k. We also identify the links on a oute (but not thei ode) by the incidence matix A =[A j ] K R such that { 1 if link j is on oute A j = 0 othewise. Fo each R let v denote the ate at which use geneates packets. In late sections we will focus on time vaying packet geneation ates, howeve fo notational convenience suppose fo the moment that each v is constant. Let also the collection of all offeed ates be denoted by v = {v : R}. Suppose that each packet that aives at link j is dopped andomly, whee the dopping pobability is detemined by the total aival ate of packets at the link. Namely, F (y) is the dopping pobability at a link given that the aival ate to the link is y, whee the dopping function F : R + [0, 1] is non-deceasing and continuously diffeentiable. The packet dopping ates at links ae detemined by all aival ates v collectively. Since taffic aiving to a given link is thinned due to dopped packets at othe links, this elation takes on a paticulaly appealing fom povided that a link independence assumption holds. Namely, let ρ k (v) and B k (v) denote the total packet aival ate and the dopping pobability at link k espectively, so that in paticula B k (v) =F (ρ k (v)). We will assume that blocking occus independently at netwok links; and thus ρ k (v) = A k v (1 B j (v)), k K. (1) j:j k Though its justification fo geneal topologies might be involved, we will see late via packet-level simulations that this assumption leads to an accuate model of packet dopping fo feed-fowad netwoks. Unde equality (1) the packet dopping pobabilities {B 1 (v),b 2 (v),,b K (v)} satisfy: B k (v) =F A k v (1 B j (v)), k K. (2) j:j k The functional fom of these equalities emind of the Elang fixed point in the context of cicuit-switched netwoks [23], though thee ae diffeences due to the paticula mechanics of the two settings. Existence of a solution to (2) is guaanteed via Bouwe s fixed point theoem. Whethe these equalities identify the dopping pobabilities uniquely in geneal topologies is not clea, since the uniqueness conclusion of the cicuitswitched analogue [23] does not cay ove to the pesent case immediately. Fo feed-fowad netwoks, howeve, an affimative answe can be obtained with little effot. Namely, by Definition 3.1 all aivals to the link with the smallest index ae exogenous; in tun packet dopping pobability, and so the ate of eleased packets fom the link, can be detemined in tems of the offeed ates v = {v 1,v 2,v R }. Induction on the link indices now yields that the dopping pobability and the packet output ate of each link can be detemined in tems of the ate of exogenous packet aivals to the link, as well as the packet output ates of all othe links with smalle indices. In summay the system of equalities (2) has a unique solution {B 1 (v),b 2 (v),,b K (v)} fo feed-fowad netwoks. Since F is diffeentiable by hypothesis, this solution vaies smoothly in the offeed packet ates v. When thee is a unique vecto {B 1 (v),b 2 (v),,b K (v)} that solves equalities (2) we epesent the goodput of each use R by λ (v). Namely, in that case, λ (v) =v (1 B j (v)). (3) j In the netwok achitectue consideed hee, uses obtain pofit fo successfully deliveed packets while they ae chaged fo using netwok esouces. Namely, each use obtains a evenue of w units fo each packet that eaches the destination without being dopped at any of the links on the associated oute. Hence fo a fixed set of offeed ates v the use geneates evenue at ate w λ (v) pe unit time. On the othe hand, the use is chaged at a cetain ate ĉ (v) units pe geneated packet, which geneically depends on all packet geneation ates. We conside uses who aim to maximize thei long-tem pofit in a geedy manne. Moe specifically, use inceases its packet geneation ate v in popotion to the aveage pofit it gains pe packet that it geneates. The evolution of use ates ae modeled via the following system of diffeential equations: v = h w (1 B j (v)) ĉ (v), R, (4) j whee h is a positive constant fo each use. Note that the fist tem in paenthesis on the ight hand side is the aveage evenue pe packet geneated by use ; hence the tem in the paenthesis is the expected pofit pe geneated packet. Fo notational convenience we have suppessed the dependence on time of the ates v. In the exteme case when no picing is enfoced (that is, ĉ (v) =0fo all v) use makes positive expected pofit pe geneated packet as long as its packets have a positive pobability of eaching thei destination; theefoe the use will have incentive to incease its packet ate until the netwok finally entes a state of congestion collapse and blocks all packets. An obsevation with simila flavo was made by [20] fo a single link. An appopiate picing policy should stabilize the dynamical system (4) aound opeating points that ae favoable with espect to a suitable measue of social welfae. In this pape we sepaately conside two definitions of social

4 welfae, namely W (v) and W (v) with W (v) = w λ (v) (5) W (v) = w log λ (v), (6) and seek to identify picing schemes {ĉ (v) : R} that dive the ates (4) towads local maxima of these functions. IV. MAIN RESULTS In this section we obtain cetain popeties of the two social welfae functions (5) (6) that inspie explicit foms fo the oute costs ĉ (v). We also intepet these popeties in ode to decompose oute costs into pe-packet pices imposed by individual links, and to develop distibuted netwok algoithms that allow computation of the ight pe-packet pices adopted by the links. A. Picing Policy fo Social Welfae Definition W (v) We stat by adapting techniques of [22] to the packetswitched setting of the pesent pape. Towads this end, we augment the collection of outes by adding a single-link oute {j} fo each link j, such that v j = w j =0. It is clea that this modification does not affect neithe the goodputs on othe outes no the social welfae in the netwok, howeve it facilitates esults that lend themselves to intuitive intepetations. The following theoem is an adaptation of [22, Theoem 2.7]. The poof of this and seveal othe theoems ae omitted in the inteest of space as well as the focus of the pesent pape. These poofs can be found in [24]. Theoem 4.1: Fo each use W(v) = w (1 B j (v)) v j (1 B j (v)) (1 B l (v))c j (v), (7) j whee fo each link j l:l j 1 W(v) c j (v) = (1 B j (v)). (8) v j Suppose that fo each instantaneous value v of the packet geneation ates, the value c j (v) defined by equation (8) is known at link j. Suppose futhe that the link chages c j (v) fo each packet that it admits, iespective of the souce of the packet. Fo a given use whose oute taveses link j, link j is offeed v l:l j (1 B l(v)) packets pe unit time fom that use, afte thinning of taffic at upsteam links of oute. The link admits a faction (1 B j (v)) of such packets; theefoe it chages use at a ate of v (1 B j (v)) l:l j (1 B l(v))c j (v) units pe unit time. The total ate of payment that use makes to netwok links is then v j (1 B j(v)) l:l j (1 B l(v))c j (v); intun the expected cost pe geneated packet by that use is ĉ (v) = (1 B j (v)) (1 B l (v))c j (v). (9) j l:l j In view of the following theoem, the picing scheme descibed above stabilizes packet geneation ates of uses at local maxima of the social welfae function W (v): Theoem 4.2: Unde the link picing scheme (9), the function W : R R R defined by (5) is a Lyapunov function fo the system of diffeential equations (4). Connected set of local maxima of W that ae contained in compact sets ae asymptotically stable. Poof: Note that equations (4), (7), and (9) lead to the equality W(v) v = h ; v in tun Ẇ (v) = W(v) v = ( ) 2 W(v) h 0. (10) v v The last inequality above is due to the positivity of h, R, and it is satisfied with equality if and only if W(v) v =0fo all. Hence est points of (4) ae local extema of W, and local minima of W ae not stable because W (v) is non-deceasing in time due to (10). The desied conclusion is obtained by application of LaSalle s invaiance pinciple [25]. Remak 4.1: A close inspection of equation (8) yields that cost c j imposed by link j is non-negative. To see this, notice that the tem (1 B j (v)) 1 is non-negative and that the tem W(v) v k is always non-positive because any incease in the ate v j of the fictitious oute j does not incease the social welfae since w j =0by definition. Remak 4.2: Given the pe-packet pice, the picing policy descibed above is stateless in the sense the same instantaneous link pice is applied to all admitted packets egadless of the owne flows. Remak 4.3: Unde the poposed picing policy, a use may be chaged fo a packet at seveal links but may not obtain a evenue because the packet may be dopped befoe eaching its destination. In tun, making the link pice on an admitted packet does not guaantee that the souce of the packet eventually leans the costs associated with that packet. Howeve uses need to know only the aveage cost pe geneated packet in ode to implement the ate update (4). This quantity can be estimated via the infomation obtained fom the packets that each thei destination, povided that the ates v do not change significantly until a easonably lage numbe of packets ae geneated. Remak 4.4: Fom a packet-wise implementation viewpoint it is clea that the packet geneation ates need to change slowly with espect to the time scales of seveal implicit aveaging schemes that ae involved in pactical ealization. Fo example, each link j needs to estimate its offeed load ρ j (v) to apply the ight packet dopping pobability fo a given value of v. It is aguable that such equiements ae not binding estictions fo high speed netwoks. B. Decentalized Computation of Pices We next focus on dynamic computation of the pe-packet pices c j (v) at each link j. Thoughout this section packet

5 geneation ates v ae assumed to change on a slowe timescale compaed to speed at which pices ae computed. Hence in the exposition of the intoduced algoithms these ates ae taken as constants, though the algoithms adopt to dynamic ates povided that these ates change slowly. The chaacteization (8) of the link pices gives little computational insight; so we stat with a moe convenient epesentation of link pices. Theoem 4.3: Let F denote the deivative of the packet dopping function F, and let γ k (v) =F (ρ k (v)) fo each link k. Then c k (v) identified by (8) has the following altenate epesentation: c k (v) =γ k (v)(1 B k (v)) 2 ( v j:k j c j (v)(1 B j (v)) A k λ (v)w m:m j (1 B m (v)).(11) The epesentation (11) identifies the infomation that is needed fo computation of c k (v) but not locally available at link k. In ode to develop a distibuted computation scheme fo c k (v), we eaange (11) as follows. Let, fo each link j and oute, D j (v) be the ate of taffic on oute that is admitted by link j. In paticula D j (v) =v (1 B j (v)) so that equality (11) takes the fom c k (v) =γ k (v)(1 B k (v)) 2 A k λ (v)w m:m j j:k j (1 B m ); c j (v)d j (v). (12) In view of equality (12), the non-local infomation equied to compute the pe-packet cost c k (v) involves the following: 1) The goodput of each oute passing though link k, (i.e. λ fo each with A k =1), 2) Pe-packet pices c j (v) at each downsteam link j, (i.e. j such that k j), 3) Packet admission ate fo each oute passing though link k on each downsteam link j (i.e. D j such that k j). The est of this section focuses on two distibuted algoithms that exploit the epesentation (11) in computing c k (v). In explaining the mechanics of these algoithms we note that each link needs infomation available at downsteam links in ode to compute its pe-packet pice; hence that infomation needs to flow in the opposite diection to packet flow. Hee we give low-ovehead implementations of the equied infomation exchange among links, povided that acknowledgement (ACK) packets take the same outes as thei associated data packets in evese diection. The poposed messaging algoithms ae based on piggybacking cetain infomation bytes in ACK packets. Two sepaate data fields need to be eseved on 2 R = {1,2} 1 = (1,3,5) 2 = (2,3,4) 1 Z C 3 D 3,2 + C 4 D 4,2 Z 3 X w 2 λ 2 ACK 2 X C w λ 3 D 3,1 + C 5 D 5,1 1 1 ACK 1 Z C 5 D 5,1 Z 5 C 4 D 4,2 4 X w 1 λ 1 ACK 1 X w 2 λ 2 ACK 2 Fig. 2. Illustation of the picing algoithm A in a outed netwok with 5 links and 2 outes. Links 4 and 5 compute thei own pe-packet pices locally and convey elevant infomation to link 3 via acknowledgements. When link 3 eceives one acknowledgment fom each of these links, it extacts the infomation on the acknowledgment packets, computes its own cost, and updates both acknowledgment packets to allow pice computation at upsteam links. each ACK packet fo this pupose. We shall call these fields X and Z, and teat them as vaiables that ae updated at evey link visited by the ACK packet. Befoe giving a complete specification of the fist algoithm we intoduce its undelying mechanism in infomal tems. Note that due to the feed-fowad stuctue of the netwoks we conside, thee exists at least one link that does not have any downsteam links. Clealy the link that has the highest index accoding to Definition 3.1 satisfies this popety. Such a link, say link K, can compute its pe-packet cost c K (v) based on only local infomation, namely γ K (v), B K (v) and D K (v) fo outes with A K = 1 (note that D K (v) = λ (v) fo such outes). To illustate computation of pices at othe links, let us specialize to the topology of Figue 2. Hee two links 4 and 5 do not have any downsteam links; and theefoe they can detemine thei own costs as outlined above. Once links 4 and 5 convey espectively the local quantities w 2 λ 2 (v), c 4 (v)d 4,2 (v) and w 1 λ 1 (v), c 5 (v)d 5,1 (v) to link 3, this link can compute its pice based on additional local infomation. As fo link 1, the non-local infomation equied to compute c 1 (v) is compised of w 1 λ 1 (v) and c 3 (v)d 3,1 (v)+c 5 (v)d 5,1 (v), which can both be constucted with the infomation available at link 3. Link3mayconve- niently convey these values to link 1 via ACK packets. The following algoithm, which uns autonomously on each link, fomalizes this pocedue fo a geneal feed-fowad topology. Algoithm A. (Instantiation at link k) Vaiables: ϕ and θ,ξ fo each oute that taveses link k. Thevalueofϕ is an estimate of optimal pice. Initialize: Compute local quantities γ k (v),b k (v), and D k (v) fo each oute such that A k =1. Fo each eceived ACK(X, Z): Let be the oute that owns ACK(X, Z).

6 1) Update θ,ξ : { w D θ k (v) if k is the last link on X else, { 0 if k is the last link on ξ Z else. 2) Estimate Pice: ϕ γ k (v)(1 B k (v)) 2 3) Update ACK fields: 4) Send out ACK(X, Z). X θ, Z ξ + ϕd k (v). A k (θ ξ ). The algoithm can be shown by induction to lead to identification of optimal pices in a feed-fowad netwok. We give hee a sketch of this agument. We stat by noting that fo the link with the highest index, say K, the initial value of ϕ above is equal to the optimal pe-packet pice c K (v) fo link K. Note also that ϕ fo that link neve changes. As the induction hypothesis suppose that all links whose indices ae stictly lage than j have thei ϕs equal to thei optimal pices. Then the same condition holds tue fo link j afte eceiving at least one ACK packet fo each one of the outes that taveses link j. Algoithm A is decentalized in the sense that each link computes its own pe-packet cost using infomation extacted fom the ACK taffic. Howeve, notice that the algoithm equies each link to keep pe-oute infomation; namely each link k measues D k and maintains θ fo each oute that taveses the link. This stateful natue of the algoithm leads to peviously alluded scalability issues and theeby beats the pupose of stateless AQM. A stateless countepat of the algoithm can be implemented as illustated next. Let M j (v) to be the poduct of admission pobabilities of downsteam links with espect to link j on oute. In paticula M j (v) = (1 B m (v)); m:j m so that equation (11) takes the fom c k (v) =γ k (v)(1 B k (v)) 2 A k λ (v) w j:k j c j (v). (13) M j (v) This epesentation is appealing fom the viewpoint of stateless opeation because the non-local infomation needed fo computation of the link pice is conveniently lumped into a summation. Namely, the last sum in (13) is enough to compute the cost, and knowledge of individual tems theein is not necessay. The algoithm intoduced next exploits this obsevation to achieve stateless opeation. The main featues of the algoithm ae as follows: 1) A special field X is eseved on each ACK packet. If the ACK belongs to a packet of oute then at link j on this oute the value of X is equal to M j.this popety is maintained by enfocing each link to modify each fowaded ACK by multiplying the X value of the ACK with the instantaneous admission pobability of the link. 2) A sepaate filed Z is eseved on each ACK to assume c j(v) M j(v) the value w j:k j at each link j that the oute of the associated packet taveses. This popety is maintained as follows: The last link on oute, say link s, sets this field to w c s (v). Each upsteam link j extacts the quantity M j fom the field X on the same ACK and subtacts the quantity cj(v) M j the field Z. 1 ( fom the value of ) c j(v) M j(v) 3) The tem A kλ (v) w j:k j is a weighted sum of goodputs of the outes tavesing link k. The tems in the paenthesis ae eadily available in the Z fields of ACK packets, and each goodput λ (v) is the ate of ACK packets fom that oute; hence the weighted sum can be measued without computing the individual tems. Multiplying this value with the local quantity γ k (v)(1 B k (v)) 2 yields the cost c k (v) accoding to (13). The following is a fomal desciption of the algoithm: Algoithm B. (Instantiation at link k) Vaiables: µ, τ, ϕ. Paamete: α (0, 1). Initialize: Compute local quantities γ k (v),b k (v). Set µ = τ =1. Fo each eceived ACK(X, Z): 1) Update µ, τ: Let t o be the time elapsed since last ACK aival. 2) Estimate Pice: µ αµ +(1 α)z τ ατ +(1 α)t o. ϕ = γ k (v)(1 B k (v)) 2 µ τ. 3) Update ACK fields: { 1 Bk (v) if k is the last link on X X(1 B k (v)) else. { w ϕ if k is the last link on Z Z ϕ X else, 4) Send out ACK(X, Z). Note that in the above specification µ tacks the expected value of the field Z (pe ACK packet, ove all outes), wheeas τ 1 tacks the cumulative aival ate of ACKs, which is the 1 To be pecise, the costs c j (v) should be intepeted as cost estimates except fo the highest index link. Coectness poof of the algoithm entails an induction agument simila to that outlined fo Algoithm A, which is not epeated hee.

7 total goodput of all outes that tavese link k. Hence if α is easonably close to 1, one should expect µ A k λ (v) A k λ (v)z k τ 1 A k λ (v), c j(v) M j(v) whee Z k = w j:k j is the common value of Z in ACK packets of oute at link k. This justifies the pice estimate ϕ of the algoithm. Finally note that Algoithm B is stateless in the sense that the links do not need to save infomation specific to diffeent outes. In paticula, links pocess each ACK packet in the same manne egadless of its souce. C. Fainess Maximizing the total goodput in the netwok should be expected to favo shote outes. In this section we adopt a fainess viewpoint and conside maximizing the social welfae function W defined by (6). Hee new picing ules ae developed fo this social welfae definition and thei decentalized implementation is sketched. Note that although implicitly elated, maximizes of W ae geneally not popotionally fai [16] due to the paticula elation between the goodputs and the packet geneation ates. We next give a theoem that is analogous to Theoem 4.2, and then intepet it to outline a picing scheme akin to that of the pevious sections. Theoem 4.4: Fo each oute W (v) = w (1 B j (v) v j (1 B j (v)) (1 B l (v)) c j (v) j w (λ (v) 1) λ (v) whee fo each link k l:l j (1 B j (v)) (14) j c k (v) =γ k (v)(1 B k (v)) 2 ( v j:k j c j (v)(1 B j (v)) A k w m:m j (1 B m (v)) (15). Remak 4.5: (Route costs) An adaptation of Theoem 4.2 establishes stability of local maxima of W unde the dynamical system (4), povided that link picing is done so that ĉ (v) = (1 B j (v)) (1 B l (v)) c j (v) j l:l j + w (λ (v) 1) λ (v) (1 B j (v)). (16) j Fig. 3. The 9-link feed-fowad netwok consideed in the numeical study. Remak 4.6: (Pe-packet pices) The ight hand side of equation (14) diffes fom that of equation (7) in its last tem. In tun, the picing policy descibed in Section IV-A (though with the pe-packet pices identified by (15)) can be applied hee with a slight modification that last link on each oute chages an additional amount w(λ(v) 1) λ (v) pe admitted packet. Remak 4.7: (Computation of pices) Equation (15) diffes fom equation (11) only in the fist tem in the ight-most paenthesis. Namely the expession λ (v)a k w in (11) is eplaced by A k w. Accodingly, algoithm A poposed in Section IV-B is applicable to the pesent setting without egisteing λ (v) on acknowledgements. Instead, link k can compute A kw by summing evenue pe packet w of each oute that passes though. V. NUMERICAL RESULTS AND SIMULATIONS In this section we numeically veify the analytical esults obtained in pevious sections, on the nine-link feed-fowad netwok shown in Figue 3. We assume that each link implements the ate-based AQM stategy that dops packets accoding to the function F illustated in Figue 4. 2 The paticula paametes used hee fo F ae: min th = 10 packets/ms, max th =50packets/ms, and p max =.9. Fist, we numeically solve fo the tajectoies of the dynamical system (4) via time-discetization, unde the two picing function defined by equalities (9) and (16). Figue 5(a) illustates the tajectoies fo the picing function (9) that was shown to locally maximize the social welfae W.Hee the tajectoies ae stated with the abitaily chosen initial condition v 1 = 1,v 2 = 5,v 3 = 3,v 4 = 4,v 5 = 1,v 6 = 5 (packets/ms). The convegence of packet geneation ates leads to the convegence of the esulting social welfae of oughly 38.5 as shown in Figue 6. An exhaustive seach on a fine gid coveing the egion [0,max th ] K (with K = 9 in the paticula example) in the domain of W suggests that this value is a global maximum fo pactical puposes. Ou 2 To comply with diffeentiability equiement of F the cones in Figue 4 need to be smooth. Howeve, this modification does not affect the epoted esults because the opeating points in the examples ae sufficiently fa fom the cones.

8 Locally maximizing packet geneation ates Social Welfae v 1 v 2 v 3 v 4 v 5 v 6 λ log(λ) TABLE I LIMITING PACKET GENERATION RATES IN THE NINE-LINK NETWORK SHOWN IN FIGURE 3 FOR THE PACKET DROPPING PROBABILITY FUNCTION OF FIGURE 4. Fig. 4. The ate-based packet dopping pobability function used in the numeical study. Pesented esults wee obtained with p max =.9, min th = 10 packets/ms and max th =50packets/ms. Social Welfae Time (sec) Fig. 6. A ealization of the total numbe of packets that ae deliveed to thei destinations in the netwok of Figue 3. The limit point of the social welfae λ along tajectoies of the fluid model (4) is oughly seach, as well as the appaent symmeties in the netwok, also suggests existence of multiple (local and global) extema fo W. We obtained simila tajectoies and esults unde the picing function (16) fo the social welfae function W.Two stable egimes of packet geneation ates obseved in those simulations ae listed in Table I. Next we tun to the fluid model (4) and examine its suitability via packet level simulations. Fo this pupose we again use the nine-link netwok shown in Figue 3. We conside a discete-time setting in which uses geneate independent Poisson taffic steams, with ates modulated by aveage pofit pe geneated packet. It is assumed that each packet visits the links on its oute within the same time slot; in paticula each packet is eithe dopped o it eaches its destination within the same slot it is geneated. Dopping decisions ae taken on a pe-packet basis. Links estimate thei aival ates via aveaging, choose the appopiate packet dopping pobabilities via the function F depicted in Figue 4, and employ the distibuted algoithm given in Section IV-B in ode to detemine the optimal pices. Namely, the simulation involves the following components: 1) Packet geneation: At the beginning of time slot n, each use {1, 2,...,6} injects a andom numbe of packets chosen accoding to a Poisson distibution whose ate is equal to v (n), i.e. the value of packet geneation ate at the beginning of the slot. 2) Packet dopping/fowading: Each link k computes its load (i.e. the total aival ate of packets pe time slot) via exponential aveaging. Namely, if the link eceives a total of a k (n) packets in slot n, then the estimate ˆρ k (n) of the mean numbe of packet aivals pe slot computed ecusively via ˆρ k (n) =β ˆρ k (n 1) + (1 β)a k (n), whee β is a constant such that 0 <β<1. 3) Pice update: In computing the pe-packet link pices, each link employs Algoithm A specified in Section IV- B. Namely, the link chages ϕ amount pe accepted packet, whee ϕ is a dynamic vaiable whose value changes in each time slot as descibed in Section IV-B. We note that while in the algoithm signalling is caied out though acknowledgements, we did not implement the acknowledgement mechanism in the simulations but athe assumed that links chage pices that ae computed based on the vaiables of next downsteam links at the pevious time slot. 4) Rate update: At the end of each time slot, uses aveage thei pofits ove the numbe of geneated packets and accodingly update thei taffic ates. Namely the packet geneation ate of oute at slot n +1, v (n +1),is detemined by v (n +1)=v (n)+hˆp (n) whee h is a positive scala constant, ˆp (n) is an estimate of the expected pofit pe geneated packet obtained via ˆp (n +1)=β ˆp (n)+(1 β)y (n), and Y (n) is the pofit pe packet that is geneated in slot n. The step-size paamete h and the aveaging paamete β used in estimating the dynamic quantities detemine the time scales of espectively the ate adaptation and pice computation. In the epoted simulations

9 Taffic Rate V(t) V1 V2 V3 V4 V5 V6 Taffic Rate V (packets/ms) V1 V2 V3 V4 V5 V Time (t) Time (sec) (a) (b) Fig. 5. Tajectoies of taffic ates fo the netwok in Figue 3 unde the dopping pobability function of Figue 4. a) Numeical solution of the fluid model (4) with h =1fo all outes. b) Packet-wise simulation as explained in Section V. In picing, the social welfae taken to be the aggegate goodput of all uses. While the time-scale and the vaiation of the sample paths in (b) depend on the step-size paametes h, β used in the simulations, the natue of the limiting ates and the value of the social welfae theein wee invaiant. we have chosen h =0.1 and 1 β =0.2; hence the estimation pocedues in the netwok ae actually only slightly faste than the use adaptation. In the simulation study the social welfae function is taken as W (v) with w =1fo all, that is the aggegate goodput in the netwok. Sample paths of the packet geneation ates that ae obtained via simulations ae shown in Figue 5(b). Evolution of the instantaneous value of the social welfae, namely the numbe of packets deliveed to thei destinations, is plotted in Figue 6. One time slot of the simulation model was taken to be 1ms. Note that in steady state oughly half of the geneated packets each thei destination. VI. CONCLUSION We consideed picing in packet switched netwoks in which links employ AQM mechanisms that entail ate-based packet dopping. At each link on its way a packet is eithe dopped o admitted and is chaged a cetain amount. Packets that each thei destinations geneate evenue fo thei souces. Uses of the netwok employ fixed outing and dynamically detemine thei flow ates in a geedy fashion to maximize thei pofit. We adopt a fluid model as an abstaction of this use behavio and obtain picing guidelines that ae applicable at the packet level. The analysis of this model is intepeted to devise scalable picing algoithms that locally maximize suitable definitions of social welfae in the fluid analogue. The analytical conclusions of the pape ae veified via packet-level simulations. Although substantially moe detailed simulation settings ae possible, the pesent esults suppot effectiveness of the applied technique in developing picing policies. The conclusions of the pape ae limited to feed-fowad netwoks fo which the fluid model is shown to be well-defined. A conclusive statement about possible ambiguity of the model fo geneal topologies is pesently not available, and applicability of the pesented techniques in geneal topologies is an open issue. ACKNOWLEDGMENT This wok was suppoted in pat by NSF CAREER Pogam though gant ANI/CNS REFERENCES [1] S. Floyd and K. Fall, Pomoting the use of end-to-end congestion contol in the Intenet, IEEE/ACM Tansactions on Netwoking, vol.7, pp , [2] S. Floyd and V. Jacobson, Random ealy detection gateways fo congestion avoidance, IEEE/ACM Tansactions on Netwoking, vol. 1, [3] I. Stoica, S. Shenke, and H. Zhang, Coe-stateless fai queueing: Achieving appoximately fai bandwidth allocations in high-speed netwoks, Poceedings of ACM/SIGCOMM, [4] W. Feng, K. Shin, D. Kandlu, and D. Saha, Stochastic fai blue: A queue management algoithm enfocing fainess, Poceedings of IEEE INFOCOM, [5] D. Lin and R. Mois, Dynamics of andom ealy detection, ACMSIG- COMM, pp , [6] R. Pan, P. Pabhaka, and K. Psounis, CHOKe a stateless active queue management scheme fo apoximating fai bandwidth allocation, Poceedings of IEEE INFOCOM, pp , Apil [7] A. Tang, J. Wang, and S. Low, Undestanding CHOKe: Thoughput and spatial chaacteistics, Poceedings of IEEE INFOCOM, vol. 1, pp , Apil [8] F. P. Kelly, Chaging and ate contol fo elastic taffic, Euopean Tansactions on Telecommuncations, vol. 8, pp , [9] F. Kelly, A. Maulloo, and D. Tan, Rate contol fo communication netwoks: Shadow pices, popotional fainess and stability, Jounal of the Opeational Reseach Society, vol. 49, pp , [10] S. H. Low and D. E. Lapsley, Optimization flow contol I: Basic algoithm and convegence, IEEE/ACM Tansactions on Netwoking, vol. 7, no. 6, pp , [11] R. J. La and V. Ananthaam, Utility-based ate contol in the Intenet fo elastic taffic, IEEE/ACM Tansactions on Netwoking, vol. 10, no. 2, pp , Apil 2002.

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