A New Backpressure Algorithm for Joint Rate Control and Routing with Vanishing Utility Optimality Gaps and Finite Queue Lengths

Transcription

1 A New Backpressure Agorithm or Joint Rate Contro and Routing with Vanishing Utiity Optimaity Gaps and Finite Queue Lengths Hao Yu and Michae J. Neey Abstract The backpressure agorithm has been widey used as a distributed soution to the probem o joint rate contro and routing in muti-hop data networks. By controing an agorithm parameter, the backpressure agorithm can achieve an arbitrariy sma utiity optimaity gap. However, this in turn brings in a arge queue ength at each node and hence causes arge network deay. This phenomenon is known as the undamenta utiitydeay tradeo. The best known utiity-deay tradeo or genera networks is [Oɛ), O/ɛ)] and is attained by a backpressure agorithm based on a drit-pus-penaty technique. This may suggest that to achieve an arbitrariy sma utiity optimaity gap, backpressure-based agorithms must incur arbitrariy arge queue engths. However, this paper proposes a new backpressure agorithm that has a vanishing utiity optimaity gap, so utiity converges to exact optimaity as the agorithm keeps running, whie queue engths are bounded throughout by a inite constant. The technique uses backpressure and drit concepts with a new method or convex programming. Index Terms backpressure agorithm; rate contro; routing; utiity-deay tradeo I. INTRODUCTION In muti-hop data networks, the probem o joint rate contro and routing is to accept data into the network to maximize certain utiities and to make routing decisions at each node such that a accepted data are deivered to intended destinations without overowing any queue in intermediate nodes. The origina backpressure agorithm proposed in the semina work [] by Tassiuas and Ephremides addresses this probem by assuming that incoming data are given and are inside the network stabiity region and deveops a routing strategy to deiver a incoming data without overowing any queue. In the context o [], there is essentiay no utiity maximization consideration in the network. The backpressure agorithm is urther extended by a drit-pus-penaty technique to dea with both utiity maximization and queue stabiity [3], [4], [5]. Aternative extensions or both utiity maximization and queue stabiization are deveoped in [6], [7], [8], [9]. The above extended backpressure agorithms have dierent dynamics and/or may yied dierent utiity-deay tradeo resuts. However, a o them rey on backpressure quantities, which are the dierentia backogs between neighboring nodes. It has been observed in [0], [6], [8], [] that the dritpus-penaty and other aternative agorithms can be interpreted The authors are with the Eectrica Engineering department at the University o Southern Caiornia, Los Angees, CA, This work was presented in part at IEEE Internationa Conerence on Computer Communications INFOCOM), Atanta, GA, USA, May, 07 []. This work is supported in part by NSF grant CCF as irst order Lagrangian dua type methods or constrained optimization. In addition, these backpressure agorithms oow certain undamenta utiity-deay tradeos. For instance, the prima-dua type backpressure agorithm in [6] achieves an Oɛ) utiity optimaity gap with an O/ɛ ) queue ength. That is, a sma utiity optimaity gap corresponding to a sma ɛ) is avaiabe ony at the cost o a arge queue ength. The drit-pus-penaty backpressure agorithm [5], which has the best utiity-deay tradeo among a existing irst order Lagrangian dua type methods or genera networks, can ony achieve an Oɛ) utiity optimaity gap with an O/ɛ) queue ength. Under certain restrictive assumptions over the network, a better [Oɛ), Oog/ɛ))] tradeo is achieved via an exponentia Lyapunov unction in [], and an [Oɛ), Oog /ɛ))] tradeo is achieved via a LIFO-backpressure agorithm in [3]. Fundamenta ower bounds on utiity-deay tradeos in [4], [5], [6], [7], [] show that, or various stochastic network settings, a arge queue deay is unavoidabe i a sma utiity optimaity gap is demanded. These works consider certain hard probems with stochastic behavior. It eaves open the question o whether or not perormance can be improved or networks that a outside these hard cases. The current paper investigates network ow probems that can be written as deterministic) convex programs, which are not restricted to the prior ower bounds. We pursue the question o whether or not improved tradeos are possibe. Can optima utiity be approached with constant queue sizes? Recenty, there have been many attempts in obtaining new variations o backpressure agorithms or deterministic network ow probems by appying Newton s method to the Lagrangian dua unction. In the recent work [], the authors deveop a Newton s method or joint rate contro and routing. However, the utiity-deay tradeo in [] is sti [Oɛ), O/ɛ )]; and the agorithm requires a centraized projection step athough Newton directions can be approximated in a distributed manner. Work [8] considers a network ow contro probem where the path o each ow is given and hence there is no routing part in the probem), and proposes a decentraized Newton based agorithm or rate contro. Work [9] considers network routing without an end-to-end utiity and ony shows the stabiity o the proposed Newton based backpressure agorithm. A o the above Netwon s method based agorithms rey on distributed approximations or the inverse o Hessians, whose computations sti require certain coordinations or the oca inormation updates and propagations and do not scae we with the network size. In

2 contrast, the irst order Lagrangian dua type methods do not need goba network topoogy inormation. Rather, each node ony needs the queue ength inormation o its neighbors. This paper proposes a new irst order Lagrangian dua type backpressure agorithm that is as simpe as the existing agorithms in [5], [6], [8] but has a better utiity-deay tradeo. The new backpressue agorithm achieves a vanishing utiity optimaity gap that decays ike O/t), where t is the number o iterations. It aso guarantees that the queue ength at each node is aways bounded by a ixed constant o the same order as the optima Lagrange mutipier o the network optimization probem. This improves on the utiity-deay tradeos o prior work. In particuar, it improves the steady-state [Oɛ), O/ɛ )] utiity-deay tradeo in [6] and the [Oɛ), O/ɛ)] utiity-deay tradeo o the drit-pus-penaty agorithm in [5], both o which yied an unbounded queue ength to have a vanishing utiity optimaity gap. Indeed, the steady-state utiity-deay tradeo o our agorithm is [0, O)]. The convergence time to reach this imiting perormance is aso aster than prior work. Our agorithm achieves a zero utiity gap and O) queue engths with a ast O/t) convergence rate. The new backpressure agorithm diers rom existing irst order backpressure agorithms in the oowing aspects: ) The backpressure quantities in this paper are with respect to newy introduced weights. These are dierent rom queues used in other backpressure agorithms, but can sti be ocay tracked and updated. ) The rate contro and routing decision rue invoves a quadratic term that is simiar to a term used in proxima agorithms [0]. Note that the beneit o introducing a quadratic term in network optimization has been observed in []. Work [] deveoped a distributive rate contro agorithm or network utiity maximization NUM) probems with given routing paths that can be reormuated as a specia case o the probem treated in this paper. The agorithm o [] considers a ixed set o predetermined paths or each session and does not scae we when treating a typicay exponentiay many) possibe paths o a genera network. The agorithm proposed in [] is not a backpressure type and and uses virtua queues rather than actua queues. It is interesting to note that the virtua queues in [] remain O). However, O) virtua queue size does not impy the achieved utiity has a ast convergence the convergence rate o [] remains an open question). In contrast, the agorithm in the current paper uses both a quadratic term and a modiied weight to get O) actua queue size with a ast O/t) convergence rate. In our conerence version [], the proposed backpressure agorithm has a goba agorithm parameter α, which is required to be chosen based on the tota number o inks and sessions in the network. In this paper, the proposed backpressure A technique in the proo o Prop 6.3. in [] suggests that in the case o ixed path routing o [] and under severa other assumptions that incude the impementation o a cosest-to-source priority rue, there exists a constant that bounds the deviation between virtua and actua queues. The technique in [] reies on a ixed-path assumption and has bounds that can grow at each stage k o the path. agorithm aows each node n to ocay determine its agorithm parameter α n based on the number o oca ink connections. The network agorithm deveoped in this paper is based on our recent work [3] that deveops a new convex programming method with an O/t) error decay or genera constrained convex programs incuding probems that are nonsmooth, not strongy convex and have noninear constraints). Ater our conerence version [] and the arxiv preprint [4] o this paper, Wang and Shro in [5] studied the same joint rate contro and routing probem considered in this paper and propose another backpressure agorithm by using the generaized aternating direction method o mutipiers ADMM) rom [6], [7], [8]. Whie that ADMM approach is very dierent rom ours, the obtained backpressure agorithm in [5] is remarkaby simiar to our agorithm in this paper: It uses identica virtua queues up to a constant scaing) and soves source rate and ink rate subprobems with identica structures. One dierence is the two agorithms use dierent weights or these subprobems. The other main dierence is that at each iteration the agorithm in [5] requires to update ink rates irst and then update source rates based on the new ink rate vaues, whie our agorithm can update ink rates and source rates in parae since these subprobems are uy decouped. This is because the agorithm in [5] is restricted to the sequentia update procedure o ADMM, whie our agorithm uses the parae methods deveoped by us in [3] together with the natura separabiity o the inear node ow baance constraints. The agorithm in [5] achieves source rates that satisy x[t] x, athough a convergence rate or genera utiity unctions is not given. In contrast, our anaysis proves an expicit O/t) optimaity deviation or the time average network utiity. This time average starts at time 0 and provides a perormance guarantee or onine impementation during the entire network operation. However, a remarkabe property o the agorithm in [5] is that when the utiity is smooth i.e., dierentiabe with Lipschitz continuous gradients) and strongy convex, then the agorithm in [5] can ensure the ina vaue o the iterate x[t] converges exponentiay ast to an optima vaue x. It remains a promising research direction to investigate whether the ast ina iteration convergence aso hods or our agorithm, and to investigate the eect o the parae source and ink rate update property and dierent update rues or weight parameters in the subprobems. II. SYSTEM MODEL AND PROBLEM FORMULATION Consider a sotted data network with normaized time sots t {0,,,...}. This network is represented by a graph G = N, L), where N is the set o nodes and L N N is the set o directed inks. Let N = N and L = L. This network is shared by F end-to-end sessions denoted by a set F. For each end-to-end session F, the source node Src ) According to [6], the authors in [5] concude their agorithm has O/t) non-ergodic convergence deined as x[t] x[t ] O/t), or equivaenty, x[t] x[t ] O/ t), in [6]. However, this unusua deinition o non-ergodic convergence rate does not impy any convergence rate o x[t]x or utiity optimaity gap Ux[t])Ux ) and hence does not provide insight into convergence behavior that network optimization concerns. Thus, it has nothing to do with the O/t) convergence to optima utiity we show in the current paper.

3 and destination node Dst ) are given but the routes are not speciied. Each session has a continuous and concave utiity unction U x ) that represents the satisaction received by accepting x amount o data or session into the network at each sot. Unike [6], [] where U ) is assumed to be dierentiabe and strongy concave, this paper considers genera concave utiity unctions U ), incuding those that are neither dierentiabe nor strongy concave. Formay, each utiity unction U is deined over an interva domu ), caed the domain o the unction. Denote the capacity o ink as C and assume it is a ixed and positive constant. 3 Deine as the amount o session s data routed at ink that is to be determined by our agorithm. Note that in genera, the network may be conigured such that some session is orbidden to use ink. For each ink, deine S F as the set o sessions that are aowed to use ink. The case o unrestricted routing is treated by deining S = F or a inks. Note that i = n, m) with n, m N, then and C can aso be respectivey written as n,m) and C n,m). For each node n N, denote the sets o its incoming inks and outgoing inks as In) and On), respectivey. Note that x, F and, L, F are the decision variabes o a joint rate contro and routing agorithm. I the goba network topoogy inormation is avaiabe, the optima joint rate contro and routing can be ormuated as the oowing muti-commodity network ow probem: max x,µ ) U x ) ) s.t. x {n=src )} In) On) F, n N \ {Dst )} ) C, L, 3) 0, L, S, 4) = 0, L, F \ S, 5) x domu ), F 6) where { } is an indicator unction; ) represents the node ow conservation constraints reaxed by repacing the equaity with an inequaity, meaning that the tota rate o ow into node n is ess than or equa to the tota rate o ow out o the node since, in principe, we can aways send ake data or departure inks when the inequaity is oose); and 3) represents ink capacity constraints. Note that or each ow, there is no constraint ) at its destination node Dst ) since a incoming data are consumed by this node. The above ormuation incudes network utiity maximization with ixed paths as specia cases. In the case when each session ony has one singe given path, e.g., the network utiity maximization probem considered in [30], we coud modiy the sets S used in constraints 4) and 5) to reect this act. For exampe, i ink is ony used or sessions and, then 3 As stated in [], this is a suitabe mode or wireine networks and wireess networks with ixed transmission power and orthogona channes. S = {, }. Simiary, the case [] where each session is restricted to using inks rom a set o predeined paths can be treated by modiying the sets S accordingy. See Appendix A or more discussions. The soution to probem )-6) corresponds to the optima joint rate contro and routing. However, to sove this convex program at a singe computer, we need to know the goba network topoogy and the soution is a centraized one, which is not practica or arge data networks. As observed in [0], [6], [8], [], various versions o backpressure agorithms can be interpreted as distributed soutions to probem )-6) rom irst order Lagrangian dua type methods. Assumption : Feasibiity) Probem )-6) has at east one optima soution vector [x ), ; µ ], L. Assumption : Existence o Lagrange mutipiers) Assume the convex program )-6) has Lagrange mutipiers attaining the strong duaity. Speciicay, deine convex set C = {[x ; ], L : 3)-6) hod}. Assume there exists a Lagrange mutipier vector λ ), = [λ n ],n N\{Dst )} 0 such that qλ ) = max{) : )-6)} { where qλ) = max [x ;µ ) ] C U x ) n N\{Dst )} λ n ) [ x {n=src )} In) On) ]} is the Lagrangian dua unction o probem )-6) by treating 3)-6) as a convex set constraint. Assumptions and hod in most cases o interest. For exampe, Sater s condition guarantees Assumption. Since the constraints )-6) are inear, Proposition 6.4. in [3] ensures that Lagrange mutipiers exist whenever constraints )-6) are easibe and when the utiity unctions U are either deined over open sets such as U x) = ogx) with domu ) = 0, )) or can be concavey extended to open sets, meaning that there is an ɛ > 0 and a concave unction Ũ : ɛ, ) R such that Ũ x) = U x) whenever x 0. 4 Fact : Repacing inequaity with equaity) I Assumption hods, probem )-6) has an optima soution vector ], L such that a constraints ) take equaities. [x ; µ ), Proo: Note that each can appear on the et side in at most one constraint ) and appear on the right side in at most one constraint ). Let [x ),, µ ], L be an optima soution vector such that at east one inequaity constraint ) ), is oose. Note that we can reduce the vaue o µ on the right side o a oose ) unti either that constraint hods with ), equaity, or unti µ reduces to 0. The objective unction vaue does not change, and no constraints are vioated. We can repeat the process unti a inequaity constraints ) are tight. 4 I domu ) = [0, ), such concave extension is possibe i the rightderivative o U at x = 0 is inite such as or U x) = og x) or U x) = min[x, 3]). Such an extension is impossibe or the exampe U x) = x because the sope is ininite at x = 0. Nevertheess, Lagrange mutipiers oten exist even or these utiity unctions, such as when Sater s condition hods [3].

4 III. THE NEW BACKPRESURE ALGORITHM A. Discussion o Various Queueing Modes At each node, an independent queue backog is maintained or each session. At each sot t, et x [t] be the source session rates; and et [t] be the ink session rates. Some prior work enorces the constraint ) via virtua queues Y n ) [t] o the oowing orm: { Y n ) [t ] = max Y n ) [t] x [t] {n=src )} } [t] [t], 0. 7) In) On) Whie this virtua equation is a meaningu approximation, it diers rom reaity in that new injected data are aowed to be transmitted immediatey, or equivaenty, a singe packet is aowed to enter and eave many nodes within the same sot. Further, there is no cear connection between the virtua queues Y n ) [t] in 7) and the actua queues in the network. Indeed, it is easy to construct exampes that show there can be an arbitrariy arge dierence between the Y n ) [t] vaue in 7) and the physica queue size in actua networks see Appendix B). An actua queueing network has queues Z n ) oowing dynamics: Z n ) [t ] max { Z ) n [t] On) x [t] {n=src )} } [t], 0 In) [t] with the [t]. 8) This is aithu to actua queue dynamics and does not aow data to be retransmitted over mutipe hops in one sot. Note that 8) is an inequaity because the new arrivas rom other nodes may be stricty ess than In) [t] because those other nodes may not have enough backog to send. The mode 8) aows or any decisions to be made to i the transmission vaues [t] in the case that Z ) n [t] On) [t], provided that 8) hods. This paper deveops an agorithm that converges to the optima utiity deined by probem )-6), and that produces worst-case bounded queues on the actua queueing network, that is, with actua queues that evove as given in 8). To begin, it is convenient to introduce the oowing virtua queue equation Q ) n [t ] =Q ) n [t] On) [t] x [t] {n=src )} In) [t] where Q n ) [t] represents a virtua queue vaue associated with session at node n. At irst gance, this mode 9) appears to be ony an approximation, perhaps even a worse approximation than 7), because it aows the Q n ) [t] vaues to be negative. Indeed, we use Q n ) [t] ony as virtua queues to inorm the agorithm and do not treat them as actua queues. However, this paper shows that using these virtua queues to choose the µ[t] decisions ensures not ony that the desired constraints ) are satisied, but that the resuting µ[t] decisions create bounded queues Z n ) [t] in the actua network, where the actua queues evove according to 8). 9) In short, our agorithm can be aithuy impemented with respect to actua queueing networks, and converges to exact optimaity on those networks. The next emma shows that i an agorithm can guarantee virtua queues Q n ) [t] deined in 9) are bounded, then actua physica queues satisying 8) are aso bounded. Lemma : Consider a network ow probem described by probem )-6). For a L and F, et [t], x [t] be decisions yieded by a dynamic agorithm. Suppose Y n ) [t], Z n ) [t], Q n ) [t] evove by 7)-9) with initia conditions Y n ) [0] = Z n ) [0] = Q n ) [0] = 0. I there exists a constant B > 0 such that Q n ) [t] B, t, then we aso have ) Z n ) [t] B On) C or a t {0,,,...}. ) Y n ) [t] B On) C or a t {0,,,...}. Proo: ) Fix F, n N \ {Dst )}. Deine an auxiiary virtua queue Q n ) [t] that is initiaized by Q n ) [0] = B On) C and evoves according to 9). It oows that Q n ) [t] = Q n ) [t]b On) C, t. Since Q n ) [t] B, t by assumption, we have Q n ) [t] On) C On) [t], t. This impies that Q ) n [t] aso satisies: { } Q n ) [t ] = max Q n ) [t] [t], 0 On) x [t] {n=src )} In) [t], t 0) which is identica to 8) except the inequaity is repaced by an equaity. Since Z n ) [0] = 0 < Q n ) [0]; and Q n ) [t] satisies 0), by inductions, Z n ) [t] Q n ) [t], t. Since Q n ) [t] = Q n ) [t] B On) C, t and Q n ) [t] B, t, we have Q n ) [t] B On) C, t. It oows that Z n ) [t] B On) C, t. ) The proo o part ) is simiar and is in Appendix C. B. The New Backpressure Agorithm In this subsection, we propose a new backpressure agorithm that yieds source session rates x [t] and ink session rates [t] at each sot such that the physica queues or each session at each node are bounded by a constant and the time average utiity satisies t t U x [t]) U x ) O/t), t where x are rom the optima soution to )-6). Note that Jensen s inequaity urther impies that U t t x [τ] ) U x ) O/t), t The new backpressure agorithm is described in Agorithm. Simiar to existing backpressure agorithms, the updates in Agorithm at each node n are uy distributed and ony depend on weights at itse and its neighbor nodes. Unike existing backpressure agorithms, the weights used to update

5 decision variabes x [t] and [t] are not the virtua queues Q n ) [t] themseves, rather, they are augmented vaues W n ) [t] equa to the sum o the virtua queues and the amount o net injected data in the previous sot t. In addition, the updates invove an additiona quadratic term, which is simiar to a term used in proxima agorithms [0]. Note that the source rate update in the second buet) and the ink rate update in the third buet) are uy decouped and hence can be perormed in parae or in a swapped order. Agorithm The New Backpressure Agorithm Let α n > 0, n N be constant parameters. Initiaize x [] = 0, [] = 0, F, L and Q ) n [0] = 0, n N, F. At each time t {0,,,...}, each node n does the oowing: For each F, i node n is not the destination node o session, i.e., n Dst ), then deine weight W n ) [t]: W ) n [t] =Q n ) [t] x [t ] {n=src )} [t ] In) On) [t ]. ) I node n is the destination node, i.e., n = Dst ), then deine W n ) [t] = 0. Notiy neighbor nodes nodes k that can send session data to node n, i.e., k such that S k,n) ) about this new W n ) [t] vaue. For each F, i node n is the source node o session, i.e., n = Src ), choose x [t] as the soution to max x U x ) W ) n [t]x α n x x [t ] ) ) s.t. x domu ) 3) For a n, m) On), choose { [t], F } as the n,m) soution to the oowing convex program: ) max W n [t] W m ) [t] ) µ ) n,m) n, m) ) ) α n α m µ n,m) µ ) n,m) [t ]) 4) s.t. n,m) C n,m) 5) n,m) 0, S n,m) 6) n,m) = 0, S n,m) 7) For each F, i node n is not the destination o, i.e., n Dst ), update virtua queue Q ) n [t ] by 9). C. Amost Cosed-Form Updates in Agorithm This subsection shows the decisions x [t] and [t] in Agorithm have either cosed-orm soutions or amost cosed-orm soutions at each iteration t. Lemma : Let ˆx x [t] denote the soution to )-3). ) Suppose domu ) = [0, ) and U x ) is dierentiabe. Let ψx ) = U x ) α n x α n x [t ] W n ) [t]. I ψ0) 0, then ˆx = 0; otherwise ˆx is the root to the equation ψx ) = 0 and can be ound by a bisection search. ) Suppose domu ) = 0, ) and U x ) = w ogx ) or some weight w > 0. Then: ˆx = α nx [t ] W n ) [t] 4α n [t] α n x [t ]) 8α n w 4α n Proo: Omitted or brevity. The probem 4)-7) can be represented as oows by eiminating n,m), S n,m), competing the square and repacing maximization with minimization. Note that K = S n,m) F.) min s.t. W ) n K z k a k ) 8) k= K z k b 9) k= z k 0, k {,,..., K} 0) Lemma 3: The soution to probem 8)-0) is given by z k = max{0, a k θ }, k {,,..., K} where θ 0 can be ound either by a bisection search See Appendix D) or by Agorithm with compexity OK og K). Proo: A simiar probem where 9) is repaced with an equaity constraint is considered in [3]. The optima soution to this quadratic program is characterized by its KKT condition and a corresponding agorithm can be deveoped to obtain its KKT point. A compete proo is presented in Appendix D. Agorithm Agorithm to sove probem 8)-0) ) Check i K k= max{0, a k } b hods. I yes, et θ = 0 and z k = max{0, a k}, k {,,..., K} and terminate the agorithm; ese, continue to the next step. ) Sort a a k, {,,..., K} in a decreasing order π such that a π) a π) a πk). Deine S 0 = 0. 3) For k = to K Let S k = S k a k. Let θ = S k b k. I θ 0, a πk) θ > 0 and a πk) θ 0, then terminate the oop; ese, continue to the next iteration in the oop. 4) Let z k = max{0, a k θ }, k {,,..., K} and terminate the agorithm. Note that step 3) in Agorithm has compexity OK) and hence the overa compexity o Agorithm is dominated by the sorting step ) with compexity OK ogk)). IV. PERFORMANCE ANALYSIS OF ALGORITHM A. Preiminaries This subsection introduces acts rom convex anaysis and some useu additiona notation. Deinition Strongy Concave Functions): Let Z R n be a convex set. Function is said to be strongy concave on

6 Z with moduus α i there exists a constant α > 0 such that z) α z is concave on Z. By the deinition o strongy concave unctions, it is easy to show that i z) is concave and α > 0, then z) α z z 0 is strongy concave with moduus α or any constant z 0. The next emma shows that a strongy concave unction deviates at east quadraticay rom its maximum vaue. Lemma 4 Coroary in [3]): Let Z R n be a convex set. Let unction be strongy concave on Z with moduus α and z opt be a goba maximum o on Z. Then, z opt ) z) α zopt z or a z Z. Deine coumn vector y = [x ; ], L, which aggregates a optimization variabes in probem )-6). For each F, n N \ {Dst )}, deine coumn vector { y n ) [x = ; ] In) On) i n = Src ), [ ) ] In) On) ese, which is composed o the contro actions appearing in each constraint ); and introduce a unction with respect to y n ) as g n ) y n ) ) = x {n=src )} ) In) Thus, constraint ) can be rewritten as On) g ) n y ) n ) 0, F, n N \ {Dst )}. Note that each vector y n ) is a subvector o y and has ength d n where d n is the degree o node n the tota number o outgoing inks and incoming inks) i node n is the source o session ; and has ength d n i node n is not the source o session. Note that components in dierent vector variabes y n ) can overap. The vector variabes y and y n ) are introduced ony to simpiy notation. The virtua queue update equation 9) can be rewritten as: Q ) n [t ] = Q ) n [t] g ) n y ) n [t]). 3) The weight update equation ) can be rewritten as: Deine W ) n [t] = Q ) n [t] g ) n y ) n [t ]). 4) Lt) = n N\Dst ) Q ) n [t] ) 5) and ca it a Lyapunov unction. In the remainder o this paper, doube summations are oten written compacty as a singe summation, e.g., ) = ). n N\Dst ) Deine the Lyapunov drit as, n N\Dst ) [t] = Lt ) Lt). The oowing emma oows directy rom equation 3). Lemma 5: At each iteration t {0,,...} in Agorithm, the Lyapunov drit is given by [t] =, n N\Dst ) Q n ) [t]g n ) yn[t]) ) g n y n[t]) ) ). 6) Proo: Fix F and n N \ Dst ), we have ) Q n [t ] ) ) Q n [t] g n ) y ) Q ) n [t] ) Q ) n [t] ) a) = n [t]) ) =Q n ) [t]g n ) yn[t]) ) g n y n[t]) ) where a) oows rom 3). By the deinition o [t], we have [t] = Q ) n [t ] ) ) Q n [t] ) ) a) =, n N\Dst ), n N\Dst ) where a) oows rom 7). Q ) n [t]g ) n y n[t]) g ) n y n[t]) ) ) 7) B. Intuitions o Agorithm Reca y = [x ; ], L. Deine hy) = U x ). 8) The intuition behind our approach comes rom examining the drit-pus-penaty expression rom prior Lyapunov based backpressure type agorithms as in [5]. With [t] given in 6) being a change o the Lyapunov unction, prior backpressure agorithms woud yied V hy[t]) [t] =V hy[t]) Q[t] T gy[t]) gy[t]) } {{ } where V > 0 is an agorithm parameter and Q[t] and gy[t]) are vectors that concatenate the individua Q n ) [t] and g n ) ) components. Prior agorithms seek to choose y[t] to maximize the irst two terms on the right-hand-side o this expression over 3)-6); the quadratic term marked by an underbrace is bounded by a constant B and resuts in a B/V error which can be made sma by increasing the V parameter). One coud eiminate the B term by changing the agorithm to optimize a three terms incuding the gy[t]) term) but this woud be a nonseparabe optimization that is as diicut as the origina probem )-6). Our insight is to approximate this quadratic term as [gy[t ])] T [gy[t])], incude this easy-to-optimize term in the maximization step, and then compensate or the approximation error by introducing proxterms y n ) [t] y n ) [t ] together with a strong concavity argument which we can rigorousy estabish even though our utiity unction is not necessariy strongy concave). Since this approach eiminates the B constant, we no onger need the V parameter to be arge, and so we use V =. At each time t, consider choosing a decision vector y[t] that incudes eements in each subvector y n ) [t] to sove probem 9)-30). The expression 9) is a modiied drit-pus-penaty expression. Unike the standard drit-pus-penaty expressions B

7 rom [5], the above expression uses weights W n ) [t], which augments each Q n ) [t] by g n ) y n ) [t ]), rather than virtua queues Q n ) [t]. It aso incudes a prox -ike term that penaizes deviation rom the previous y[t ] vector. This resuts in the nove backpressure-type agorithm o Agorithm. Indeed, the decisions in Agorithm were derived as the soution to probem 9)-30). This is ormaized in the next emma. Lemma 6: At each iteration t {0,,...}, the action y[t] jointy chosen in Agorithm is the soution to probem 9)- 30). Proo: The proo invoves coecting terms associated with the x [t] and [t] decisions. See Appendix E or detais. Furthermore, the next emma reates hy ) and hy[t]) yieded by action y[t] that aggregates a contro actions jointy chosen in Agorithm at each iteration t {0,,...}. Lemma 7: Let y = [x ), ; µ ], L be an optima soution to probem )-6) given in Fact, i.e., g n ) ), y n ) = 0, F, n N \ Dst ). I α n d n ), n N, where d n is the degree o node n, then the action y[t] = [x [t]; [t]], L jointy chosen in Agorithm at each iteration t {0,,...} satisies hy[t]) hy ) Φ[t] Φ[t ] [t] where Φ[t] = ),,n N αn {n Dst )} y n y n ) [t] ), α n {n=dst )} In)µ [t]) ) and hy) is deined in 8). Proo: See Appendix F. It remains to show that this modiied backpressure agorithm eads to undamentay improved perormance. C. Utiity Optimaity Gap Anaysis Deine coumn vector Q[t] = [ Q n ) [t] ],n N\{Dst )} as the stacked vector o a virtua queues Q n ) [t] deined in 9). Note that 5) can be rewritten as Lt) = Q[t]. Deine vectorized constraints ) as gy) = [g n ) y n ) )],n N\Dst ). Lemma 8: Let y = [x ), ; µ ], L be an optima soution to probem )-6) given in Fact, i.e., g n ) ), y n ) = 0, F, n N \ Dst ). I α n d n ), n N in Agorithm, where d n is the degree o node n, then or a t, t hy[τ]) thy ) ζ Q[t]. where ζ = Φ[] = ),,n N αn {n Dst )} y n ), α n {n=dst )} In)µ ) ) is a constant. Proo: By Lemma 7, we have hy[τ]) hy ) Φ[τ] Φ[τ ] [τ], τ {0,,..., t }. Reca [τ] = L[τ ] L[τ]. Summing over τ {0,,..., t } yieds t hy[τ]) t thy ) t ) Φ[τ] Φ[τ ] [τ] =thy ) Φ[t ] Φ[] L[t] L[0] a) thy ) Φ[] Q[t] where a) oows rom the act that Φ[t] 0, t, L[t] = Q[t] and L[0] = 0. The next theorem summarizes that Agorithm yieds a vanishing utiity optimaity gap that approaches zero ike O/t). Theorem : Let y = [x ), ; µ ], L be an optima soution to probem )-6) given in Fact, i.e., g n ) ), y n ) = 0, F, n N \ Dst ). I α n d n ), n N in Agorithm, where d n is the degree o node n, then or a t, we have t t U x [τ]) U x ) t ζ, where ζ is a constant deined in Lemma 8. Moreover, i we deine x [t] = t t x [τ], F, then U x [t]) U x ) t ζ. Proo: Reca by 8) that hy) = U x ). By Lemma 8, we have t U x [τ]) t U x ) ζ Q[t] a) t U x ) ζ. where a) oows rom the trivia act that Q[t] 0. Dividing both sides by a actor t yieds the irst inequaity in this theorem. The second inequaity oows rom the concavity o U ) and Jensen s inequaity. D. Queue Stabiity Anaysis Lemma 9: Let Q[t], t {0,,...} be the virtua queues in Agorithm. For any t, t Q[t] = gy[τ]) Proo: This emma oows directy rom the act that Q[0] = 0 and queue update equation 9) can be written as Q[t ] = Q[t] gy[t]). The next theorem shows the boundedness o a virtua queues Q n ) [t] in Agorithm. Theorem : Let y = [x ), ; µ ], L be an optima soution to probem )-6) given in Fact, i.e., g n ) ), y n ) = 0, F, n N \ Dst ), and λ be a Lagrange mutipier

8 max y hy), n N\Dst ) W ) n [t]g ) n y ) n ) α n y ) n y ) n [t ] ), n=dst ) α n In) [t ]) 9) s.t. 3)-6) 30) vector given in Assumption. I α n d n ), n N in Agorithm, where d n is the degree o node n, then F, n N \ {Dst )}, we have Q ) n [t] Q[t] λ ζ, t where ζ is a constant deined in Lemma 8. Proo: Let qλ) = sup y C { hy) λ T gy) } be the Lagrangian dua unction deined in Assumption. For a τ {0,,..., }, by Assumption, we have hy ) = qλ ) a) hy[τ]) λ,t gy[τ]) where a) oows rom the deinition o qλ ). Rearranging terms yieds hy[τ]) hy ) λ,t gy[τ]), τ {0,,...}. Fix t > 0. Summing over τ {0,,..., t } yieds t t hy[τ]) thy ) λ,t gy[τ]) t =thy ) λ,t gy[τ]) a) = thy ) λ,t Q[t] b) thy ) λ Q[t] where a) oows orm Lemma 9 and b) oows rom Cauchy-Schwarz inequaity. On the other hand, by Lemma 8, we have t hy[τ]) thy ) ζ Q[t]. Combining the ast two inequaities and canceing the common terms yieds Q[t] ζ λ Q[t] Q[t] λ ) λ ζ Q[t] λ λ ζ a) Q[t] λ ζ where a) oows rom the basic inequaity a b a b or any a, b 0. Thus, or any F and n N \ {Dst )}, we have Q ) n [t] Q[t] λ ζ. This theorem shows that the absoute vaues o a virtua queues Q ) n [t] are bounded by a constant B = λ ζ rom above. By Lemma and discussions in Section III-A, the actua physica queues Z n ) [t] evoving via 8) satisy Z n ) [t] B On) C, t. This is summarized in the next coroary. Coroary : Let y = [x ), ; µ ], L be an optima soution to probem )-6) given in Fact, i.e., g n ) ), y n ) = 0, F, n N \ Dst ), and λ be a Lagrange mutipier vector given in Assumption. I α n d n ), n N in Agorithm, where d n is the degree o node n, then a actua physica queues Z n ) [t], F, n N \ {Dst )} in the network evoving via 8) satisy Z n ) [t] 4 λ ζ C, t. On) where ζ is a constant deined in Lemma 8. Deine vector x = [x ] and x[t] = [x [t]] where x and x [t] are deined in Theorem. Note that i each U x ) is strongy concave with respect to x, then x is unique by strong ), concavity. However, [µ ] L is not necessariy unique.) In this case, Coroary shows x[t] yieded by Agorithm converges to the unique maximizer x. Coroary : I the conditions in Theorem hod and each U x ) is strongy concave with respect to x, then Agorithm guarantees x[t] x as t. Proo: Assume each U x ) is strongy concave with respect to x with moduus c. Let c = min {c }. By Assumption, we have y = argmax y C {hy) λ,t gy)}. Reca that hy) = U x ) and gy) are separabe since they can be written as the sum o scaar unctions in terms o x and. Thus, x ), and [µ ] appear separaby and maximize in the et-side o 3) where each x satisying 6) maximizes ), a strongy concave part and each vector [µ ] satisying 3)-5) maximizes a concave part. Deine y[t] = t t y[τ]. Note that y[t] satisies 3)-6) since each y[τ] is generated by Agorithm. By Lemma 4, or a t, U x ) λ,t gy ) U x [t]) λ,t gy[t]) a) b) = c) = c x x [t]) U x [t]) λ,t gy[t]) c x[t] x U x [t]) λ,t t gy[τ]) c t x[t] x t=0 U x [t]) t λ,t Qt) c x[t] x 3)

9 where a) oows rom c = min {c }; b) oows rom the inearity o g ) and the deinition o y[t]; and c) oows rom Lemma 9. Reca that λ,t gy ) = 0 by strong duaity o convex programs Assumption ). Thus, 3) impies c x[t] x U x ) U x [t]) t λ,t Qt) a) U x ) U x [t]) t λ Qt) b) t ζ t λ λ ζ) where a) oows rom the Cauchy-Schwarz inequaity; and b) oows rom Theorem, which impies U x [t]) U x ) t ζ, t, and Theorem, which impies Qt) λ ζ, t. Taking imits t on both sides yieds that x[t] x as t. E. Perormance o Agorithm Theorems and together impy that Agorithm with α n d n ), n N can achieve a vanishing utiity optimaity gap that decays ike O/t), where t is number o iterations, and guarantees the physica queues at each node or each session are aways bounded by a constant that is independent o the utiity optimaity gap. This is superior to existing backpressure agorithms rom [6], [5], [] that can achieve Oɛ) utiity gaps ony at the cost o O/ɛ ) or O/ɛ) queue engths. To obtain a vanishing utiity gap, existing backpressure agorithms in [6], [5], [] necessariy yied unbounded queues. Note that to achieve Oɛ) utiity gaps, existing backpressure agorithms in [6], [5], [] need to choose an agorithm parameter V = O/ɛ), which in turn eads to OV ) or OV) queue engths. In act, the O/ɛ ) queue bound in the prima-dua type backpressure agorithm [6] is given by V λ B where λ is the Lagrangian mutipier vector attaining strong duaity and B is a constant determined by the probem parameters. A recent work [33] aso shows that the O/ɛ) queue bound in the backpressure agorithm rom drit-pus-penaty is o the order V λ B where B is aso a constant determined by the probem parameters. Since λ is a constant vector independent o V and V = O/ɛ), both agorithms are caimed to have O/ɛ ) or O/ɛ) queue bounds. By Coroary, Agorithm guarantees physica queues at each node are bounded by 4 λ B 3, where B 3 is constant given a probem. Thus, the constant queue bound guaranteed by Agorithm is typicay smaer than the OV ) or OV) queue bounds rom [6] and [33] even or a sma V. A sma V corresponds to a poor utiity perormance or the backpressure agorithms in [6], [5].) V. NUMERICAL EXPERIMENT In this section, we consider a simpe network with 6 nodes and 8 inks as described in Figure. This network has two sessions: session rom node to node 6 has utiity unction ogx ) and session rom node 3 to node 4 has utiity unction.5 ogx ). The og utiities are widey used as metrics o proportiona airness in data networks [9].) The routing path o each session is arbitrary as ong as data can be deivered rom the source node to the destination node. For simpicity, assume that each ink has capacity. The optima source session rate to probem )-6) is x =. and x =.8 and ink session rates, i.e., static routing or each session, is drawn in Figure. Session : ->6 3 3 Session : 3->4 Fig.. A simpe network with 6 nodes, 8 inks and sessions. Session : -> Session : 3->4 Fig.. The optima routing or the network in Figure. To compare the convergence perormance o Agorithm and the backpressure agorithm in [5] with the best utiitydeay tradeo among a existing backpressure agorithms), we run both Agorithm with α n = dn ), n N and the backpressure agorithm in [5] with V = 500 This backpressure agorithm [5] yieds O/V) utiity gaps and OV) queue engths, where V is the agorithm parameter) to pot Figure 3. Since each session has a og utiity unction, which is strongy convex, by Coroary, the running averages o source rates yieded by Agorithm converge to the optima source rates. In act, our simuation resuts in Figure 3 show that per iteration source rates x[t] without averaging) aso converge to the optima source rates. It can be observed rom Figure 3 that Agorithm converges to the optima source rates aster than the backpressure agorithm in [5]. The backpressure agorithm in [5] with V = 500 takes around 500 iterations to converge to source rates cose to.,.8) whie Agorithm ony takes around 400 iterations to converges to.,.8) as shown in the zoom-in subigure at the top right corner.) In act, the backpressure agorithm in [5] with V = 500 can not converge to the exact optima source session rate.,.8) but can ony converge to its neighborhood with a distance gap determined by the vaue o V. This is an eect rom the undamenta [O/V), OV)] utiity-deay tradeo o the backpressure agorithm in [5]. In contrast, Agorithm can

10 eventuay converge to the exact optima source session rate.,.8). A zoom-in subigure at the bottom right corner in Figure 3 veriies this and shows that the source rate or Session in Agorithm converges to. whie the source rate in the backpressure agorithm in [5] with V = 500 osciates around a point sighty arger than.. Source Rates Backpressure in Neey 00 Session : 3-> New Backpressure: Agorithm Session : -> Iterations Fig. 3. Convergence perormance comparison between Agorithm and the backpressure agorithm in [5] with V = 500. Coroary shows that Agorithm guarantees each actua queue in the network is bounded by constant 4 λ ζ On) C. Reca that the backpressure agorithm in [5] can guarantee the actua queues in the network are bounded by a constant o order V λ. Figure 4 pots the sum o actua queue ength at each node or Agorithm and the backpressure agorithm in [5] with V = 0, 00 and 500. Reca a arger V in the backpressure agorithm in [5] yieds a smaer utiity gap but a arger queue ength.) It can be observed that Agorithm has the smaest actua queue ength see the zoom-in subigure) and the actua queue ength o the backpressure agorithm in [5] scaes ineary with respect to V. Network Sum Physica Queue Length Backpressure in Neey 00 V=500) Backpressure in Neey 00 V=00) Backpressure in Neey 00 V=0) New Backpressure: Agorithm Iterations Fig. 4. Actua queue ength comparison between Agorithm and the backpressure agorithm in [5]. VI. CONCLUSION This paper deveops a new irst-order Lagrangian dua type backpressure agorithm or joint rate contro and routing in muti-hop data networks. The new backpressure agorithm can achieve vanishing utiity optimaity gaps and inite queue engths. This improves the state-o-the-art [Oɛ), O/ɛ )] or [Oɛ), O/ɛ)] utiity-deay tradeo attained by existing backpressure agorithms [6], [0], [8], []. APPENDIX A NETWORK UTILITY MAXIMIZATION WITH PREDETERMINED MULTI-PATH Consider the muti-path network utiity maximization in [] where each session has mutipe given paths. Let x be the tota source rate o each session F. Let P be the set o paths or session. The ink session rate becomes a vector, j) = [µ ] j P. Note that mutipe paths or the same session are aowed to overap.) Deine S ) as the set o paths or session that are aowed to use ink. Note that S ) are determined by the given paths or each session. That is, i path j or session uses ink, then j S ) ; i no given path or session uses ink, then S ) =. The muti-path network utiity maximization probem can be ormuated as oows: max U x ) s.t. x {n=src )} In) µ, j) j P j P µ, j) C, L, µ, j), On) j P F, n N \ {Dst )}, j) µ 0, L, F, j S ),, j) µ = 0, L, F, j P \ S ) x domu ), F The above ormuation is in the orm o probem )-6) except that the variabe dimension is extended. In this case, Agorithm deveoped in Section III-B to sove probem )-6) can be adapted to sove the above muti-path network utiity maximization probem by repacing with, j) j P µ in updates ) and 9); and repacing subprobem 4)-7) with max µ ) n, m) s.t. ) W n [t] W m ) [t] ) µ, j) n,m) j P ) α n α m n,m) C n,m) j P, j) µ n,m), j) µ n,m) j P µ, j) n,m) 0, F, j S ) m,n) = 0, F, j S ) m,n), µ, j) n,m) [t ]) which again has the same structure as subprobem 4)-7) except that the variabe dimension is extended.

11 APPENDIX B AN EXAMPLE ILLUSTRATING THE POSSIBLY LARGE GAP BETWEEN MODEL 7) AND MODEL 8) Consider a network exampe shown in Figure 5. The network has 3k nodes. Ony node 0 is a destination and ony nodes a i, b i, i {,,..., k} can have exogenous arrivas. Assume a ink capacities are equa to and the exogenous arrivas are periodic with period k, as oows: Time sot : One packet arrives at node a. Time sot : One packet arrives at node a. Time sot k: One packet arrives at node a k. Time sot k : One packet arrives at node b. Time sot k : One packet arrives at node b. Time sot k: One packet arrives at node b k. Under dynamics 7), each packet arrives on its own sot and traverses a inks o its path to exit on the same sot it arrived. The queue backog in each node is 0 or a time. Under dynamics 8), the irst packet arrives at time sot to node a. This packet visits node a at time sot, when the second packet aso arrives at a. One o these packets is deivered to node a 3 at time sot 3, and another packet aso arrives to node 3. The nodes {,..., k} do not have any exogenous arrivas and act ony to deay the deivery o a packets rom the a i nodes. It oows that the ink rom node k to node 0 wi send exacty one packet over each sot t {k, k,..., k k}. Simiary, the ink rom b k to 0 sends exacty one packet to node 0 over each o these same sots. Thus, node 0 receives packets on each sot t {k, k,..., k k}, but can ony output packet per sot. The queue backog in this node grows ineary and reaches k at time k k. Thus, the backog in node 0 can be arbitrariy arge when k is arge. This exampe demonstrates that, even when there is ony one destination, the deviation between virtua queues under dynamics 7) and actua queues under dynamics 8) can be arbitrariy arge, even with an outdegree o at most and an in-degree o at most. a a a k- a k b b b k- b k 0 Fig. 5. An exampe iustrating the possiby arge gap between queue mode 7) and queue mode 8) k APPENDIX C PROOF OF PART ) IN LEMMA Fix F, n N \ {Dst )}. By 0), Q ) n [t ] = max { Q ) n [t] = max In) a) max [t] On) { Q ) n [t] x [t] {n=src )} On) [t], 0 } x [t] {n=src )} In) [t], x [t] {n=src )} { Q ) n [t] x [t] {n=src )} On) } [t], 0 In) [t] In) [t] } [t] where a) oows rom the act that [t], x [t],,, t are non-negative. Note that the right side o the above equation is identica to the right side o 7) except that Y n ) t) is rewritten as Q n ) t). Since Y n ) 0) = 0 < Q n ) 0), by induction, we have Y n ) [t] Q n ) [t], t. Since Q n ) [t] = Q n ) [t]b On) C, t and Q n ) [t] B, t, we have Q n ) [t] B On) C, t. It oows that Y n ) [t] B On) C, t. APPENDIX D PROOF OF LEMMA 3 Note that probem 8)-0) satisies Sater s condition. So the optima soution to probem 8)-0) is characterized by KKT conditions [34]. Introducing Lagrange mutipiers θ R or inequaity constraint K k= z k b and ν = [ν,..., ν K ] T R K or inequaity constraints z k 0, k {,,..., K}. Let z = [z,..., z K ]T and θ, ν ) be any prima and dua pair with the zero duaity gap. By KKT conditions, we have zk a k θ νk = 0, k {,,..., K}; K k= zk b; θ 0; θ K k= zk b) = 0; zk 0, k {,,..., K}; ν k 0, k {,,..., K}; νk z k = 0, k {,,..., K}. Eiminating νk, k {,,..., K} in a equations yieds θ a k zk, k {,,..., K}; K k= zk b; θ 0; θ K k= zk b) = 0; zk 0, k {,,..., K}; z k a k θ )zk = 0, k {,,..., K}. For a k {,,..., K}, we consider θ < a k and θ a k separatey: ) I θ < a k, then θ a k zk hods ony when z k > 0, which by zk a k θ )zk = 0 impies that z k = a k θ. ) I θ a k, then zk > 0 is impossibe, because z k > 0 impies that zk a k θ > 0, which together with zk > 0 contradicts the sackness condition zk a k θ )zk = 0. Thus, i θ a k, we must have zk = 0. Summarizing both cases, we have zk = max{0, a k θ }, k {,,..., K}, where θ is chosen such that K k= zk b, θ 0 and θ K k= zk b) = 0. To ind such θ, we irst check i θ = 0. I θ = 0 is true, the sackness condition θ K k= zk b) is guaranteed to hod

12 and we need to urther require K k= zk = K k= max{0, a k } b. Thus θ = 0 i and ony i K k= max{0, a k } b. Thus, Agorithm check i K k= max{0, a k } b hods at the irst step and i this is true, then we concude θ = 0 and we are done! Otherwise, we know θ > 0. By the sackness condition θ K k= zk b) = 0, we must have K k= zk = Kk= max{0, a k θ } = b. To ind θ > 0 such that Kk= max{0, a k θ } = b, we coud appy a bisection search by noting that a zk are decreasing with respect to θ. Another agorithm o inding θ is inspired by the observation that i a j a i, i, j {,,..., K}, then z j z i. Thus, we irst sort a a k in a decreasing order, say π is the permutation such that a π) a π) a πk) ; and then sequentiay check i k {,,..., K} is the index such that a πk) θ 0 and a πk) θ < 0. To check this, we irst assume k is indeed such an index and sove the equation k j= a πj) θ ) = b to obtain θ ; Note that in Agorithm, to avoid recacuating the partia sum k j= a πj) or each k, we introduce the parameter S k = k j= a πj) and update S k incrementay. By doing this, the compexity o each iteration in the oop is ony O).) then veriy the assumption by checking i θ 0, a πk) θ 0 and a πk) θ 0. The agorithm is described in Agorithm and has compexity OK ogk)). The overa compexity is dominated by the step o sorting a a k. APPENDIX E PROOF OF LEMMA 6 The objective unction 9) can be rewritten as a) = hy), n N\Dst ) α n,n=dst ) In) U x ) W ) n [t]g ) n y ) n ) α n y n ) y n ) [t ] ) [t ]) W n ) [t] x {n=src )} In) µ ) ) On) µ ), n N\Dst ), n N\Dst ) α n x x [t ]) {n=src )} α n, In) n N\Dst ) α n, On) n N\Dst ) α n,n=dst ) In) [t ]) [t ]) [t ]) b) = U x ) W ) Src ) [t]x α Src ) x x [t ]) ) ) W n [t] W m ) [t] ) n,m) n,m) L n,m) L α n α m ) n,m) µ ) n,m) [t ]) 3) where a) oows rom the act that y n ) y n ) [t ] = x x [t ]) {n=src )} In) [t ]) On) [t ]) ; and b) oows by coecting each inear term and each quadratic term [t ]). In the ast step b), the quadratic term n,m) Lα n α m ) n,m) µ ) n,m) [t ]) is obtained by simpy coecting each quadratic term terms in step a); and the inear term n,m) L W ) m [t] ) n,m) [t ]) in the ast 3 ) W n [t] is obtained by noting each ink session rate appears twice with opposite signs in the summation term,n N\{Dst )} W n ) [t] x {n=src )} In) On) ) uness ink ows into Dst ) and recaing that W ) Dst ) = 0, F. Note that equation 3) is now separabe or each scaar x and vector [ n,m) ]. Thus, probem 9)-30) can be decomposed into independent smaer optimization probems in the orm o probem )-3) with respect to x, and in the orm o probem 4)-7) with respect to n,m), F. APPENDIX F PROOF OF LEMMA 7 Deinition Lipschitz Continuity): Let Z R n be a convex set. Function : Z R m is said to be Lipschitz continuous on Z with moduus β i there exists β > 0 such that z ) z ) β z z or a z, z Z. The oowing act summarizes the Lipschitz continuity o each unction g ) n ). Fact : Each unction g n ) continuous with respect to vector y ) n ) deined in ) is Lipschitz with moduus β n d n. where d n is the degree o node n. Proo: This act can be easiy shown by noting that each g n ) y n ) ) is a inear unction with respect to vector y n ) and has at most d n non-zero coeicients that are equa to ±. Now, we are ready to present the main proo. Note that W n ) [t] appears as a known constant in ). Since U x ) is concave and W n ) [t]x is inear, it oows that ) is strongy concave with respect to x with moduus α n. Since x [t] is chosen to sove )-3), by Lemma 4, F, we have U x [t]) W ) Src ) [t]x [t] α n x [t] x [t ]) } {{ } 33)-et U x ) W ) Src ) [t]x α nx x [t ]) α n x x [t]). 33) } {{ } 33)-right