Distributed Queue-Length based Algorithms for Optimal End-to-End Throughput Allocation and Stability in Multi-hop Random Access Networks

Transcription

1 Distribute Queue-Length base Agorithms for Optima En-to-En Throughput Aocation an Stabiity in Muti-hop Ranom Access Networks Jiaping Liu Department of Eectrica Engineering Princeton University Princeton, NJ 08544, USA Aexaner L. Stoyar Be Labs, Acate-Lucent 600 Mountain Ave, 2C-322 Murray Hi, NJ 07974, USA Abstract We consier a moe of wireess network with ranom (sotte-aoha-type) access an with muti-hop fow routes. The goa is to evise istribute strategies for optima utiity-base en-to-en throughput aocation an queueing stabiity. We consier a cass of queue back-pressure ranomaccess agorithms (QBRA), where actua queue engths of the fows (in each noe s cose neighborhoo) are use to etermine noes channe access probabiities. This is in contrast to previousy propose agorithms, which are purey optimizationbase an obivious of actua queues. QBRA is aso substantiay ifferent from much stuie MaxWeight type scheuing agorithms, aso using back-pressure. For the moe with infinite backog at each fow source, we show that QBRA, combine with simpe congestion contro oca to each source, eas to optima en-to-en throughput aocation, within the network saturation throughput region achievabe by a ranom access. (No en-to-en message passing is require.) This scheme generaizes for the case of aitiona, minimum fow rate constraints. For the moe with stochastic exogenous arrivas, we show that QBRA ensures stabiity of the queues as ong as nomina oas of the noes are within the saturation throughput region. Simuation comparison of QBRA an (queue obivious) optimization-base ranom access agorithms, shows that QBRA performs better in terms of ento-en eays. I. INTRODUCTION In wireess a hoc networks, contention resoution an interference avoiance among inks are among the most important issues, which motivates the extensive stuy of wireess meium access contro (MAC) protocos. The stanar MAC protoco currenty use in IEEE [3] is the Distribute Coorination Function (DCF) with Binary Exponentia Backoff (BEB) mechanism. However, it has been concue by many researchers that DCF an BEB mechanism for contention contro can be inefficient (eg. [6]). Thus, there are significant chaenges in esigning MAC protocos that are both efficient (in terms of throughput, atency, energy consumption, etc.) an aow istribute impementation minimizing signaing (message passing) overhea. It has been shown that the maximum throughput region can be achieve by much stuie MaxWeight -type scheuing agorithms (originay propose in [8]). However, in the context of wireess networks, MaxWeight agorithms This research was in part supporte by the DARPA CBMANET proect typicay nee to be centraize. Ranom access ( Sotte- Aoha-type ) agorithms typicay provie smaer throughput regions, but are simper an more amenabe to istribute impementations. In this paper we consier a moe of ranom access with muti-hop transmissions. Ranom access moes have been wiey aopte in contemporary works [1], [2], [4], [5], [7], [9]. Informay, we can cassify them into two categories: pure optimization-base agorithms (eg. [4], [5], [9]) an ynamic, queue-ength base strategies (eg. [2], [7]). Agorithms of the former type (eg. [4], [5], [9]) sove an optimization probem that aocates network resources (e.g. effective ink throughputs) to satisfy an/or optimize traffic emans of ifferent fows; they require optimization parameters to be a priori specifie an are typicay obivious of the ynamics of actua queues in the network. The atter type agorithms ( [2], [7]), even in cases when they have same optimization obective as the former agorithms, o the optimization by aaptivey responing to actua queueing ynamics; for exampe, they o not nee a priori knowege of traffic fow input rates to achieve queueing stabiity (if such is feasibe). In this paper we propose an stuy a cass of queue backpressure ranom access (QBRA) agorithms for a mutihop network. The agorithms use fow queue ifferentias on the inks to etermine ink access probabiities. (The MaxWeight agorithms use queue ifferentias as we, but in a competey ifferent way.) Our main contributions are as foows. (i) For the probem of utiity-base en-to-en throughput aocation (in Section V), when traffic sources have infinite backog of ata to sen, we prove that QBRA combine with extremey simpe congestion controer at each fow source, soves the probem of weighte proportiona fair (sum-og utiity) en-to-en throughput aocation among the fows. We aso prove an extension of this resut for the case of aitiona minimum fow-rate constraints. (This generaizes an much strengthens the corresponing resut of [2], which appies to singe-hop fows. A further generaization - to more genera utiity functions - is aso possibe; this is subect of future work.) (ii) For the probem of queueing stabiity in a system with stochastic exogenous arrivas (in Section VI), we prove

2 that QBRA automaticay (without knowing input rates) ensures stabiity, as ong as nomina ink oas are within the network saturation throughput region. (This generaizes some of the stabiity resuts of [7], which appy to singe-hop fows.) (iii) Finay, we present simuation resuts (in Section VII) showing goo performance of QBRA in terms of en-to-en eays. II. BASIC NOTATION AND DEFINITIONS Typicay, we use bo etters x,y,... to enote vectors, as oppose to scaars x,y. We use the notations R,R + an R ++ for the set of rea, rea non-negative an rea positive numbers, respectivey. Corresponingy, -times prouct spaces are enote as R, R + an R ++. We write x y to enote scaar prouct, an x = x x for the Euciean norm, inucing stanar metric. Carinaity (i.e. the number of eements) of a finite set A is enote by A. We enote [z] + = max{z,0}. We use,,, for componentwise vector inequaities, e.g. x y means x i > y i, i. For any scaar function T : R R, T(x) = (T(x 1 ),,T(x )) an for any subset C R, T(C) = {T(v) : v C}. A. Wireess Network Moe III. SYSTEM MODEL We consier a wireess muti-hop network escribe as a irecte graph G = (N, L), where N is the set of noes an L is the set of the ogica (irecte) communication inks between pairs of noes; t an r are the transmitter an receiver noes of ink, respectivey. There is a finite number of traffic fows, inexe by r R; each fow has fixe source an estination noes, an a fixe route. (We wi use term fow an route interchangeaby, an use inex r either one.) Let L r L enote the set of inks on route r, an inex inks L r from source to estination in an ascening orer as (r,), = 1,2,3,. We aso assume each noe keeps separate queues of ata packets of ifferent fows. Let enote the queue ength of fow r packets ocate in the transmitter noe t of ink L r. To simpify notation, we often write to mean (r,), i.e. for the queue ength of fow r at -th noe in its route. The system operates in iscrete (or, sotte) time t = 0, 1, 2,... In any time sot, each noe may attempt to transmit one packet on (at most) one of its outgoing inks. A packet transmission attempt on a ink is successfu, if it is not interfere by another simutaneous (same sot) transmission; otherwise the transmission fais. The interference moe is same as in [2], [7] (an is somewhat more genera than in [9]). First, any transmission attempt to a noe wi fai if this noe is transmitting. Secon, if there are two or more simutaneous transmissions to the same noe, they a fai. Thir, for each noe n there is the set of noes N n N it interferes with, namey, a transmission to any noe in N n wi fai if noe n transmits. (Note that accoring to our interference moe, n N n an D n N n, where D n N \n is the set of noes m such that noe n has ata to sen. ) In summary, a transmission attempt on ink L is successfu if an ony if no noe in the set {n : n t,r N n } transmits. For each n et us efine S n = { L : r N n }. (This set incues inks originating at n an inks interfere by transmissions from n.) We consier the ink epenence graph as efine in [2], i.e. the irecte graph with vertices being inks L, an the ege from to another vertex L exists if an ony if S t. Throughout the paper we assume the strong connectivity of the ink epenence graph, which assumes that there exists a irecte path between any two vertices. B. Saturation Throughput Region an its Properties Suppose the network empoys a sotte-aoha-type ranom access. Reca that each noe keeps separate queues for the packets of ifferent fows. In each time sot t, noe n attempts a transmission with probabiity P n, an chooses to transmit ata from queue on ink with conitiona probabiity p (r) /P n, where p (r) 0 is efine for each pair (r,) such that n = t an L r. Thus, p (r) is the resuting probabiity of transmission of cass r packets on ink, an P n = p (r) 1, n N. (1) r: L r :n=t We efine P as the set of a feasibe vectors of ink access probabiities p = {p (r), L r, r R}. Obviousy, P = {p [0,1] : P n 1, n N }, (2) where =. Given p P, the transmission attempts by a noes are inepenent, an then the resuting average successfu transmission rate (or, average throughput) aocate to fow r on the ink L r is µ (r) (p) = p (r) n t,r N n (1 P n ). (3) We wi use notation µ(p) = {µ (r) (p), L r, r R}. Definition 1: We efine the system saturation throughput region M as the set of a possibe µ(p), aong with the vectors ominate by them, namey, M = {v [0,1] : p P, s.t. v µ(p)}. (4) We aso efine og-throughput region og M as og M = {u = og v : v M,v R ++} an its Pareto ( north-east ) bounary as [og M] ={u og M: if u u og M, then u = u } Proposition 1 ( [2]): Log-throughput region og M is stricty convex an the bounary [og M] is a smooth ( 1)-imensiona surface in R, which can be parameterize by the vectors of positive ink weights w = {w (r), L r, r R} R ++, as foows. Vector u [og M] if an ony if there exists unique (up to scaing by a positive constant) ink weights vector w R ++ such that u is the unique soution of the probem max w u s.t. u og M,

3 or an equivaent probem max w og v s.t. v M. Moreover, the unique set of access probabiities p such that u = og µ(p) is given by p (r) = w (r) i S n k:i L k w (k) i, (5) where n = t is the transmitter noe of ink. We wi enote by p(w) the function given by (5), an for future reference aopt the convention that p (r) = 0 when w (r) = 0. (This makes p(w) we efine for a w R +, an not ust w R ++, because w (r) > 0 guarantees that the enominator in (5) is positive as we.) The important feature of expression (5) is that the enominator is essentiay the sum of the weights of a inks the transmitting noe n interferes with (pus the ink originating at n itsef), an so noes can compute their access probabiities very efficienty, using imite information exchange within their oca neighborhoo. (See [2], [7] for more etais.) C. Queueing Dynamics The generic queuing ynamics in the ranom access network escribe above is as foows. (We o not iscuss here how new packets arrive in the networks an how access probabiities are set. This wi be specifie ater.) Let A (r) (t) enote the number of (exogenous) ata packet arrivas at the source noe (r,1) of fow r in time sot t, an (t), = 1,..., L r, is the queue ength of type r packets at the (transmitter noe of) ink (r, ) at time t. (Reca convention = (r,).) Then, { (t) + A (r) (t) h (r) (t), = 1 (t + 1) = (t) + h (r) (t) h(r) (t), 1 < < L r where h (r) = 1 if there is a successfu transmission of a fow r packet on ink (r,) in sot t, an h (r) = 0 otherwise. IV. DYNAMIC QUEUE-BACKPRESSURE RANDOM ACCESS In this section we introuce a ynamic istribute agorithm, cae Queue-Backpressure Ranom Access (QBRA), which is the main subect of this work. The agorithm generaizes Queue Length Base Ranom Access (QRA) scheme, introuce in [2], [7] for the specia case of our moe, where a routes have ength one. Uner QRA, noes choose their access probabiities p ynamicay, accoring at time t being a (t). In the (t). (See [2], [7] for more genera weight functions.) Uner QBRA agorithm, noes aso ynamicay choose to formua (5), with ink weights w (r) fixe function of the its current queue ength simpest form, w (r) = access probabiities p accoring to (5), with the weight w (r) of fow r on ink (r,) at time t being set to the current queue ifferentia, efine as foows: (t) = [ ] + (t) +1 (t),1 < Lr, (t), =. (6) As usua, we ientify an (r,), an enote by Q the vector of a in the network. Obviousy, uner QBRA a transmission of a fow r packet at time t on ink (r,) wi not be attempte uness (t) +1 (t) > 0. This easiy impies that if inequaity (t) +1 (t) 1, (7) hos for fow r on ink (r,) at time t = 0, it then hos for a t. In a cases consiere throughout this paper, (7) in fact hos for a fows an inks at time 0 an then for a t. V. UTILITY BASED END-TO-END THROUGHPUT ALLOCATION In this section we stuy the scenario where the sources of a ata fows are saturate, i.e. have infinite amount of ata to sen. Informay, the probem is to aocate throughputs x (r) to fows r aong their respective routes in the network (by setting access probabiities of a noes) in a way that maximizes weighte proportiona fairness obective r θ(r) og x (r), where θ (r) > 0 are fixe weights. This probem was consiere in [9], where two istribute iterative agorithms for setting access probabiities were propose an prove optima; these approaches an resuts were generaize in [4]. However, the soution approaches in [4], [9], base on the the ua an the prima agorithms in convex optimization, both nee en-to-en feeback information to upate variabes maintaine by the noes. This may inuce increase eays ue to the en-to-en signaing aong the route, especiay in arge-scae networks. Moreover, the optimization-base agorithms of [4], [9] are obivious of the actua queueing ynamics in the network, which aso may egrae performance metrics, incuing eays. The purpose of this section is to prove that the above probem can be sove by QBRA agorithm as we. The soution is very simpe. Each fow r source maintains a constant queue ength 1, proportiona to θ(r), at the fow source noe. Then, as we show, the ynamics of the network queues uner QBRA is such that the queue ength converge to the vaues that inuce access probabiities resuting in the optima en-to-en throughput aocation. Since QBRA ony uses oca message passing between neighboring noes, one can say that QBRA provies a more istribute soution to the probem, than those in [9]. The soution provie by QBRA is asymptoticay optima in the foowing sense. Queues at the source noes are maintaine equa to θ (r) /η, where η > 0 is a (sma) scaing parameter. This means that, roughy speaking, parameter η scaes up a queues in the network by arge factor 1/η. The optimaity is achieve when η becomes infinitey sma. Consequenty, our resuts concern fui imits of the queue ength process, which are (roughy) the imits of the process uner ηq(t/η) space an time scaing, as η 0. Finay, in this section we show that QBRA aso soves a more genera probem, with aitiona, minimum en-to-en throughput requirements, x (r) λ (r).

4 A. Probem Formuation The probem is to operate our ranom access network in a way such that the average en-to-en fow throughputs x (r) maximize r θ(r) og x (r), where θ (r) > 0 are fixe parameters, whie keeping a the queues in the network stabe. This in particuar means that we want x (r) to be those given (as x (r) = 1 ) by a soution of the foowing optimization probem for the average ink-fow throughputs v: max v M subect to θ (r) og 1, v(r), = 2,..., L r, r R, (8) (Here we use notationa convention = (r,), an we use simiar ones ater in the paper.) Since any optima soution to (8) must be such that v 0, probem (8) can be equivaenty written in terms of og-throughputs u = og v: max u og M subect to θ (r) u (r) 1, u (r) u(r), = 2,..., L r, r R. (9) Since region og M is stricty convex by Proposition 1, the optima soution u to (9) is unique. (An then v such that u = og v is the unique soution of (8). Moreover, again using Proposition 1 an consiering the Lagrangian for the probem (9) (given in (15) beow), it is easy to estabish the foowing facts. The optima ink throughputs aocate to each fow aong its route are a equa: u (r) 1 =... = u (r), r R; (10) the optima ua variabes, = 2,..., L r, r R, corresponing to the inequaity constraints in (9), are unique an are such that: = θ (r) > 2 >... > > 0, r R. (11) We wi put by convention 1 = θ (r), an use vector notation q = {, L r, r R}. We aso enote by q 0 the vector with components { (r) q +1, 1 <,, =. (12) Again consiering the Lagrangian for probem (9), it is easy to see that u = arg max u og M q u, which means that q is nothing ese but the (unique) vector of weights, which resuts in the optima rates v, i.e. v = µ(p( q)). B. Appication of QBRA agorithm. Fui Limit The appication of QBRA to sove probem (8) is as foows. A (sma) parameter η > 0 is fixe. Each fow r source maintains a constant queue ength 1 = θ (r) /η, at the fow source noe. (Here is the integer part of of a number.) It can aways o that, because the source has infinite amount of ata, an it can simpy a a new packet in the queue after each successfu transmission from it. Otherwise, the QBRA in the network works exacty as efine earier. Without oss of generaity, we can assume that at time t = 0, the reations (7) ho (for exampe, a queues on each route are 0, except the first queue), an so (7) hos for a t. We consier the fui imit asymptotic regime. Namey, we ook at a sequence of system, with parameter η 0. For each system we consier the space-time rescae queueing process ηq(t/η) in continuous time t 0, an then consier the process-eve imit of those, as η 0. The foowing fact, prove essentiay same way as the anaogous resut in [7], roughy speaking says that any imiting process is concentrate on the famiy of (continuous) traectories q(t), t 0, cae fui sampe paths, an escribes their basic properties. (We omit the proof here - it foows essentiay same argument as that use for anaogous resut in [7].) Proposition 2 (Fui Limit): The sequence of rescae processes ηq(t/η), t 0, can be constructe on a common probabiity space in a way such that, with probabiity 1, the sequence of reaizations has a subsequence converging (uniformy on compact sets) to a Lipschitz continuous traectory q(t), t 0, cae fui sampe path (FSP). The famiy of FSPs satisfies, in particuar, the foowing properties. For each r, θ (r) (t) 1 For each r an 1 < L r, { t q(r) (t) = (t) q(r) (t)... q(r) (t) 0. (13) [ where v(t) is such that 2 (t) v(r) (t), > 0, (t) v(r) (t)] +, = 0, v(t) arg max z M q(t) og z, (14) with the vector of queue ifferentias q(t) 0 being efine anaogousy to (12). Note that orering property (13) is the imit version of (7), an that the key property (14) foows from the fact that QBRA uses queue ifferentias as ink weights to set access probabiities via (5). We enote by D the set of a possibe FSP states q(t), i.e. those satisfying inequaities (13), an by D the subset of those q D with at east one zero component = 0. C. Asymptotic Optimaity Given the properties of optima prima an ua soutions to probem (9), u an q, respectivey, it foows immeiatey that the stationary traectory q(t) q satisfies a the FSP properties escribe in Proposition 2. Moreover, it is easy to see (anaogousy to the way it is one in [2] for a simper moe) that any stationary traectory q(t) q D, satisfying FSP properties in Proposition 2, must be such that q = q, because then q satisfies KKT conitions for probem (9). This to some egree motivates the foowing main resut of this section.

5 Theorem 1: Every FSP is such that q(t) q as t an, consequenty, v(t) v. The convergence is uniform on a FSP. Theorem 1 basicay says that, when parameter η > 0 is sma, then regaress of the initia state of the queues, the queues converge to an stay cose to the vaues which resut (via QBRA rue for access probabiity assignment) in the optima en-to-en throughput aocation. The key iea of the proof of Theorem 1 is containe in the foowing Lemma 1, which states that essentiay (up to some technica work we wi o beow) the Lagrangian of the convex optimization probem (9), L(q,u) = θ (r) u (r) L r 1 =2 ( ) u (r) u(r), (15) = q u, (16) (where, by convention, q is such that 1 = θ (r) for a r) can serve as a Lyapunov function to prove the convergence. Lemma 1: For any FSP at any time t such that q(t) D \ D, the foowing hos. The vaue of v(t), an then u(t) = og v(t), is efine by (14) uniquey, an moreover u(t) = arg max u og M q(t) u an u(t) [og M]. Consequenty, (q(t),u(t)) is a smooth function of time in a neighborhoo of t; by (16) L(q(t),u(t)) is the vaue of the convex probem ua to (9) at point q(t), an θ (r) u (r) 1 L(q(t),u(t)) 0; (17) function L(q, u) is smooth in a neighborhoo of (q(t), u(t)), an has zero partia graient on prima variabes u at (q(t),u(t)): Finay, u L(q(t),u(t)) = 0. (18) t L(q(t),u(t)) = =2 ) ( (t) ( (t) u (r) ) (t) u(r) (t) (19) 0, (20) an the inequaity in (20) is strict uness q(t) = q. Proof: If q(t) D \ D, then, by Proposition 1, in the neighborhoo of this point the epenence of v(t) on q(t) is given by the (expicit) smooth function v(p(q)). Obviousy, the epenence u = og v is smooth as we. Then, a the properties escribe in the emma easiy foow, using in particuar the smoothness of the bounary [og M]. Inequaity (20) hos because each ifference (t) (t) obviousy has same sign as u (r) (t) u(r) (t); a such ifferences cannot be simutaneousy equa to 0 uness q(t) = q, because otherwise a stationary traectory sitting at a point ifferent from q wou exist. In aition to the key Lemma 1, we nee some auxiiary resuts to prove Theorem 1. Lemma 2: For any FSP an any time t 0, there exists an arbitrariy cose to t time s > t, such that q(s) 0, i.e. q(s) D \ D. Proof: Let us ca a ink-route pair (,r) such that R a virtua ink. For a given FSP, et us ca (,r) a zero (resp. non-zero ) virtua ink at time t if = 0 (resp. > 0). Suppose there are some zero virtua inks at time t. (If not, the emma statement is trivia.) Since traectory q( ) is continuous, to prove statement of the emma it wi suffice to show that there exists time s > t, arbitrariy cose to t, such that at east one virtua ink which was zero at t becomes non-zero at s. Consier two cases. Case (a): Suppose on one of the routes r, there is a non-zero virtua ink foowe by a zero one, that is (t) > 0 an (t) = 0. (This is the situation where a transmission on the -th ink kis a simutaneous transmission on the 1- th ink.) Then, it is easiy seen from (14) that (t) = 0, (t) > 0, an both these functions are continuous in time at t. Then, by (14), has positive, boune away from 0, erivative in the interva (t,t + ǫ), with sma ǫ > 0. In the same time interva, aso by (14), the erivative of +1 is upper boune by an arbitrariy sma δ > 0, if we choose sma enough ǫ > 0. (If = L r, then +1 (t) 0 by convention.) These facts mean that (s) > 0 for a s (t,t + ǫ). We are one with case (a). Case (b) = [NOT Case (a)]: At time t, aong each route r, there is a (possiby empty) sequence of zero virtua inks at the beginning, foowe by the (efinitey non-empty) sequence of non-zero virtua inks unti the en of the route. In this case, there is at east one zero virtua ink, et it be the -th ink on route r, such that it either shares a ink with a non-zero virtua ink, or it interferes with transmissions on a non-zero virtua ink. (The atter observation uses strong connectivity of the ink epenence graph.) Either way, (t) = 0 an it is continuous at time t. For the first non-zero virtua ink on this route, say m-th with m >, v m (r) (t) > 0 an is continuous at t. Then, using (14) we easiy see that, in a sma interva (t,t + ǫ), s [q(r) +1 (s) q(r) m (s)] < C, for some C > 0 inepenent of ǫ; an in the same interva s q(r) (s) > δ, where δ > 0 can be mae arbitrariy sma by choosing sma ǫ. We concue that for any s (t,t+ǫ), we must have (s) > (s) for at east one, + 1 m, an therefore one of the virtua inks from -th to m 1-th must be non-zero at time s. Lemma 3: For any FSP, q(t) 0 for a t > 0. Proof: In view of Lemma 2, it suffices to show that if q(t) 0 for t = s > 0, then this hos for a t s as we. Suppose not, an τ, s < τ <, is the first time after s when q(t) hits set D. This means that there exists

6 a subset of virtua inks which simutaneousy become zero at time τ. However, consiering the vaues of (t) for t cose to τ, an essentiay repeating the argument in the proof of Lemma 2, we can show that for at east one of those inks, (t) must in fact be increasing for such t - a contraiction. Proof of Theorem 1. Accoring to Lemma 2, for any FSP at any time t > 0, we are in the conitions of Lemma 1. In particuar, this means that the uniform boun (17) ho. Thus, to prove the uniform convergence q(t) q it remains to show that the negative erivative tl(q(t),u(t)), given by (19), is boune away from zero as ong as q(t) is outsie of an ǫ-neighborhoo of q. This is obvious if vaues of q(t) are confine to a compact set, not intersecting with D. To show that it is sti the case within the entire set D\ D, it remains to observe the foowing. If point q approaches an arbitrary point a D, the erivative tl at q approaches, because for at east one virtua ink, the corresponing (see (19)) approaches 0 whie is not, or vice versa. (Here, again, we essentiay repeat the argument of Lemma 2.) D. Generaization: Systems with Minimum Fow Rate Requirements In practica systems, a minimum rate ower boun is often require on the en-to-en throughput to guarantee Quaity of Service of the ata transfers. Accoringy, Theorem 1 can be generaize to incue such aitiona constraints. More precisey, suppose that, aitionay, the en-to-en throughput aocate to each fow r nees to be at east λ (r) 0. Formay, the more genera optimization probem (which we wi write irecty in terms of og-throughputs u = og v, as in (9)) is max u og M subect to θ (r) u (r) 1, u (r) u(r), = 2,..., L r, r R, og λ (r) u (r) 1, r R. (21) We wi assume that probem (21) is feasibe, an moreover a inequaity constraints can be satisfie as strict inequaities. (This wi be referre to as the feasibiity conition.) Then, there exists a unique optima soution u such that (10) hos, an the optima ua soution y (r),, = 2,..., L r, r R, where y (r) are the uas corresponing to the minimum throughput constraints. The generaize version of (11) is: 1 θ (r) + y (r) > 2 >... > > 0, r R, (22) q 0 is efine as in (12), an, again, u = arg max u og M q u. Appication of QBRA in this case uses a virtua queue Y (r), maintaine by each fow r source noe. Tokens are ae to Y (r) at the average rate λ (r) (tokens/sot); one token is remove from it (if there is any) in every sot when a packet of fow r is successfuy transmitte (from source noe). As oppose to the previous situation, source noe uses not the constant vaue θ (r) /η as the queue ength 1, but rather the variabe 1 (t) = θ(r) /η +Y (r) (t). Otherwise, the QBRA in the network works exacty same way as before. An FSP now contains aitiona component y (r) (t) for each r, which is a imit of ηy (t/η), an it satisfies conition { λ (r) t y(r) 1 (t), q(r) > 0, (t) = [λ (r) 1 (t)]+, = 0, in aition to (14). If we enote, by convention, 1 (t) θ (r) + y (r) (t), then the key conition (14) etermining v(t) sti hos. The generaization of Theorem 1 is the foowing. Theorem 2: Assume the feasibiity conition. Then, uniformy on a FSP with initia states q(0) within arbitrary fixe compact set, q(t) q as t an, consequenty, v(t) v. Theorem 2 both generaizes an much strengthens a resut of [2], which appies to QBRA in a system with one-hop routes an states ony that if convergence q(t) q hos then q = q. Proof of Theorem 2 is carrie out anaogousy to that of Theorem 1. We o not provie etais here, ust the foowing key points. The Lagrangian in this case, which serves as a Lyapunov function in the proof, is: L(q,u) = θ (r) u (r) L r 1 =2 y (r) ( og λ (r) u (r) 1 ( ) u (r) u(r) = q u y (r) og λ (r), (23) where, by convention, for those fows r with λ (r) = 0, we have y (r) 0 an y (r) og λ (r) = 0. For each FSP, the bouns (17) generaize as θ (r) u (r) 1 L(q(t),u(t)) L(q(0),u(0)) y (r) (0)og λ (r). (24) As in the proof of Theorem 1, an important intermeiate step is showing that q(t) 0 for a t > 0 - this is one anaogousy to the arguments in Lemmas 2 an 3. VI. STOCHASTIC STABILITY OF A NETWORK WITH EXOGENOUS ARRIVALS We now turn to a version of our moe, where fow sources o not have an infinite suppy of ata to sen, but rather there is a ranom process of exogenous arrivas to a the first queue 1 at the fow source noe. For simpicity et us assume that each such arriva process A (r) (t), t = 1,2,... is i.i.. with the average rate λ (r) = E[A (r) (t)] > 0, an a arriva processes are inepenent. (The i.i.. an )

7 inepenence assumptions can be greaty reaxe. Aso, it is not an accient that here we use the same symbo λ (r) for the input rate as we use use for the minimum rate boun in Section V-D; the reason wi become cear shorty.) Consier such a network uner QBRA ranom access scheme. The question is uner which conitions the queueing process Q(t), t = 0,1,2,..., in the network is stabe. If we assume (for further simpicity) that P{A (r) (t) = 0} > 0 for each r, then it is cear that Q(t) is a countabe state space, irreucibe, aperioic Markov chain. By stabiity we unerstan its ergoicity. Note that, without oss of generaity we can assume that the queueing orer reations (7) ho aong each route at a times. The main resut of this section is the foowing Theorem 3: Suppose input rates λ (r) > 0, r R, satisfy the feasibiity conition, as given in Section V-D. Then the network queueing process is stabe. This theorem generaizes to the muti-hop setting one of the stabiity resuts in [7], which appy to the singe-hop system. (It shou be note that our proof, outine beow, is substantiay ifferent from that in [7], even though both use fui imits.) We wi use the fui imit technique to estabish Theorem 3. (See [7] for an appication of the technique to a ranom-access system, an references therein to a genera theory.) With this technique, we ook at the fui imit, efine anaogousy to the way escribe in Section V-B. (It is important to emphasize that the QBRA agorithm in the network oes not use parameter η in any way. This is true for the use of QBRA in Section V as we, but there traffic sources use parameter η to ecie when to sen new packets. In this section, parameter η is ony use to efine the fui imit asymptotic regime.) In our case, the FSPs turn out to satisfy the same properties as those for the FSPs in Section V-D, but speciaize to the case θ (r) = 0 for a r. (This is not an accient - it is easy to observe that if in Section V-D we were to assume that a θ (r) = 0, then the behavior of each virtua queue Y (r) there wou be anaogous to the behavior of actua queue 1 in this section setting.) Then, accoring to fui imit technique, to prove Theorem 3 it suffices to prove the foowing Theorem 4: There exists T > 0 such that, uniformy on a FSPs with q(0) = 1, we have q(t) = 0 for a t T. Proof of Theorem 4 is, again, anaogous to the proof of the convergence resuts in Theorems 1 an 2. We omit fu etais, but the key points are as foows. Since a θ (r) = 0, an consequenty y (r) (t) 1 (t), the Lagrangian in (23) speciaizes to L r ( ) L(q,u) = u (r) u(r) =2 ( ) 1 og λ (r) u (r) 1 Fow 3 1(3,1) 1 2 2(1,4) Fig. 1. 4(3,2) 3(1,3) 3 4 Fow 1 Fow 2 7(2,1) 5(3,3) 6(2,2) A 6-noe a-hoc network = q u 6 8(1,2) 9(1,1) 1 og λ (r). (25) This Lagrangian is use as Lyapunov function, an for each FSP we have the bouns 0 L(q(t),u(t)) 1 (0)og λ(r). (26) In particuar, if q(0) = 1, L(q(t),u(t)) r og λ(r). Using arguments anaogous to those in Lemmas 2 an 3, we can show that for a t > 0 a subset of components of q(t) cannot hit 0, uness a components hit 0 simutaneousy; this impies that q(t) 0 for a 0 < t < t, where t is the first, possiby finite time when q(t) = 0, an then q(t) = 0. For a 0 < t < t, we have, anaogousy to (19), t L(q(t),u(t)) = ( ) ( ) λ (r) 1 (t) og λ (r) u (r) 1 (t) L r ( ) ( ) (t) v(r) (t) u (r) (t) u(r) (t) (27). =2 Finay, we observe that the RHS of (27) not ony is nonpositive, but in fact boune away from 0 by a negative constant ǫ, uniformy on a possibe u [og M]. Thus, L(q(t),u(t)), an then q(t), must hit 0 within a uniformy boune time. The fact that q(t) cannot eave 0 after first hitting it easiy foows. VII. NUMERICAL EXAMPLE In this section we investigate performance of QBRA via a numerica exampe. We consier a simpe 6-noe, 3-route a hoc network as shown in Figure 1, which has the same network topoogy as the secon exampe in [9]. The noes are abee from 1 to 6, an the inks are abee from 1 to 9 an the route-ink pair as in the format (r,), i.e. 5(3,3) enotes the ink 5 in L which is the 3r ink of route 3. The interference moe is that each noe interferes with the reception at its 1-hop neighbors; for exampe, N 2 = {1,2,3}, N 3 = {2,3,4,5,6}. The behavior of QBRA for the optima en-to-en fow throughput aocation, with an without minimum throughput constraints is as preicte by Theorems 1 an 2 in Section V. We o not present those simuations to save space. Instea 5

8 Q (1) Q (2) Q (3) QBRA OPT λ (1) (a). Route 1: Q (1) = Q (1) 1 + Q (1) 2 + Q (1) 3 + Q (1) QBRA OPT λ (2) (b). Route 2: Q (2) = Q (2) 1 + Q (2) 2. QBRA OPT λ (3) (c). Route 3: Q (3) = Q (3) 1 + Q (3) 2 + Q (3) 3. Fig. 2. Comparison of the QBRA scheme an the optimization-base scheme on queueing performance: λ (1) = λ (2) = λ (3). we show simuation resuts for the system with exogenous arrivas, comparing queueing performance of QBRA with performance of optimization base agorithms (et s refer to them as OPT), such as that in [9]. First, we want to emphasize that, even when QBRA an OPT are appie to sove the same probem, such as (9), there are significant ifferences between them: QBRA upates network variabes base on queue-engths, whie OPT are obivious of the rea queues; QBRA can be impemente by noes exchanging queueing information within oca neighborhoos, whie OPT require en-to-en message passing aong each fow route. When we tak about proviing queueing stabiity in a system with exogenous arrivas, the ifference is even more pronounce: OPT wou require estimation of the fow input rates to be use in the appropriate optimization probem to cacuate ink access probabiities resuting in sufficient ink throughputs aong each route; QBRA nee not know or estimate input rates an ensures stabiity automaticay (when feasibe). We simuate a system with exogenous (i.i.. Poisson) arrivas with equa rates for a fows, λ (1) = λ (2) = λ (3) an scae them up to observe the changes of the queue engths. The QBRA works exacty as specifie in Section VI. An OPT agorithm that we simuate works as foows: we a priori pre-cacuate ink access probabiities so that the resuting en-to-en rates x (r) provie to the fows are maxima, subect to x (1) = x (2) = x (3). (In other wors, we preten that an optimization base agorithm is run a priori to cacuate appropriate access probabiities.) OPT is obivious of the queue engths, except if there is no packets at a ink, the ink oes not attempt transmission. We stuy the tota average queue ength of each fow r (which by Litte aw is proportiona to the en-to-en queueing eay): = Q(r). Figure 2 compares Q (1),Q (2),Q (3) uner the QBRA an OPT. It shows that the average queues uner QBRA are significanty ower than the optimization-base agorithm. An intuitive expanation of this is that QBRA better aapts to the current queue ength in the network. REFERENCES [1] N. Abramson. The ALOHA system - another aternative for computer communications. In Proceeings of AFIPS Conference, voume 37, pages , [2] P. Gupta an A. Stoyar. Optima throughput aocation in genera ranom-access networks. In Proceeings of the 38th Conference on Information Sciences an Systems, March [3] IEEE. Wireess an meium access contro (mac) an physica ayer (phy) specificationsieee stanar , June IEEE Stanar [4] J. W. Lee, M. Chiang, an R. A. Caerbank. Jointy optima congestion an contention contro in wireess a hoc networks. IEEE Communications Letters, 10(3): , [5] J. W. Lee, M. Chiang, an R. A. Caerbank. Utiity-optima ranomaccess contro. IEEE Transactions on Wireess Communications, 6(7): , Juy [6] T. Nanagopa, T.-E. Kim, X. Gao, an V. Bharghavan. Achieving mac ayer fairness in wireess packet networks. In Proceeings of ACM MobiCom, pages 87 98, [7] A. Stoyar. Dynamic istribute scheuing in ranom access networks, Submitte. [8] L. Tassiuas an A. Ephremies. Stabiity properties of constraine queueing systems an scheuing poicies for maximum throughput in mutihop raio networks. IEEE Transactions on Automatic Contro, 37: , [9] X. Wang an K. Kar. Cross-ayer rate contro for en-to-en proportiona fairness in wireess networks with ranom access. In Proceeings of MobiHoc, 2005.