Randomized Algorithms for Throughput-Optimality and Fairness in Wireless Networks

Transcription

1 Proceedings o the 45th IEEE Conerence on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006 Randomized Algorithms or hroughput-optimality and Fairness in Wireless Networks Atilla Eryilmaz, Eytan Modiano and Asuman Ozdaglar Laboratory or Inormation and Decision Systems Massachusetts Institute o echnology Cambridge, MA, s: {eryilmaz, modiano, asuman}@mit.edu Abstract In this paper, we study multi-hop wireless networks with general intererence models, and describe a practical randomized routing-scheduling-congestion-control mechanism that is guaranteed to ully utilize the network capacity, and achieve air allocation o the resources. Using the ramework described in this paper, low complexity distributed algorithms can be developed or a large class o intererence models and airness criteria (such as proportional and max-min airness). Earlier distributed algorithms proposed in this context have been highly intererence model dependent, and can guarantee at most 50% utilization o the achievable throughput. his is the irst work that assures 100% utilization and also provides air allocation in multi-hop wireless networks. I. INRODUCION he control o data networks to achieve high throughput and air allocation o the resources among competing users (or lows) is clearly one o the most important problems in data communications. here has been considerable interest and progress in the development o algorithms that address the issues o eiciency and airness or both wireline and wireless networks. In the case o ixed arrival rates (i.e. inelastic traic), assiulas and Ephremides provided in their seminal work [25] a joint routing-scheduling algorithm that can achieve the highest throughput without violating the stability o the network. Such throughput-optimal policies make dynamic routing and scheduling decisions to avoid highly congested nodes. For wireline networks, these policies can be implemented in a distributed ashion by using buer occupancy inormation o only the neighboring nodes. In contrast, in wireless multi-hop networks, there is no known scheme or low complexity implementation o throughput-optimal policies. Many other throughput-optimal algorithms developed later share the same weakness [1, 21, 19, 9]. It is well-recognized that reducing the complexity o the centralized computation is an essential requirement or the development and implementation o throughput-optimal policies. For the case o switches, assiulas has shown how the use o randomized algorithms can reduce the complexity o the centralized computation [24]. One o the main contributions o our work is to extend the use o randomized algorithms or potential distributed implementation o routing and scheduling in the context o multi-hop wireless networks. he issue o air service o elastic traic, where the rate o an elastic low is assumed to be controllable, is irst considered by Kelly et al. [11, 12] in the context o wireline networks. he authors developed de-centralized algorithms that have strong ties to notions in market economics. he main idea behind these algorithms was or each source to measure the congestion level it experiences and to adjust its low rate accordingly with its utility unction in order to ully utilize the resources. hese ideas have been extended to dierent scenarios and algorithms (e.g. [15], see [22] or a review). However, all o these works were developed or wireline networks, assumed ixed routes, and ignored the stochastic nature o the traic. More recently, it has been realized that ideas o low control can be successully utilized together with the dynamic routing and scheduling algorithm described above [13, 23, 7, 18, 8]. It has been shown that mean rates arbitrarily close to the air allocation can be achieved without violating stability. However, the routing and scheduling part o the algorithm inherited the centralized optimization problem o [25], which is impractical to solve or multi-hop wireless networks. he design and analysis o distributed implementations o the above cross-layer approach attracted a lot o attention rom the community. In particular, [14, 27] provided algorithms that guarantee 50% utilization o the stability region or node-exclusive-spectrum-sharing (NESS) intererence model, where each easible allocation orms a matching 1 o the graph. It has been shown in [3] that 33% can be guaranteed even when the algorithm operates in a totally asynchronous manner (see [2] or a deinition). Distributed implementations exist or this particular intererence model i a signiicant portion o the capacity is sacriiced. For other general intererence models, it has been shown that the amount o sacriice is even greater. For example, [28] and [4] consider the two-hop intererence model 2 and show that the guaranteed raction o the capacity region drops with the increasing number o neighbors. In particular, or a grid topology, only 12.5% o the capacity region is achieved, which is very discouraging. 1 In a matching, no two links that are incident on the same node can be active simultaneously. 2 In the two-hop intererence model, a transmission over a link (n, m) is successul i and only i all the neighbors o n and m are silent at the time /06/$ IEEE. 1936

2 Our main contributions in this work are listed next. By extending the approach in [24] to the multi-hop wireless network scenario, we prove that the capacity region o the network can be ully utilized with a practical schedulingrouting algorithm. hus, no raction o the throughput need to be sacriiced or a practical implementation. In the case o elastic traic, we propose a cross-layer mechanism that is composed o a decentralized congestion controller operating in parallel with the practical schedulingrouting algorithm discussed in the previous item. We prove that this mechanism asymptotically achieves air division o the resources across the lows. All o our analysis considers a generic intererence model that can be applied to a large set o graph theoretic collision models considered in the literature. It is important to note that the proposed mechanism can be implemented with low complexity distributed algorithms or speciic intererence models o practical interest (see [6]). he rest o the paper is organized as ollows. In Section II, we describe the system model. hen, in Section III, we describe the practical algorithm and show its throughputoptimality. Section IV introduces the congestion control mechanism or elastic traic and establishes its air characteristics. We provide our concluding remarks in Section V, and some o the proos in the Appendix. II. SYSEM MODEL Consider a ixed wireless network that is represented by a graph G =(N, L), where N denotes the set o ixed nodes and L denotes the set o undirected links. We assume a time slotted system with synchronized nodes, where each slot is just long enough to accommodate a single packet transmission. Suppose there is a set F o end-to-end lows traversing the network, where each low F is given by the node pair (s(),d()). Here, s() denotes the source node o low and d() denotes the destination node. We allow or multiple routes between each source-destination pair. We consider both the inelastic and elastic traic cases. Associated with each destination a buer is maintained at each node. We let q [t] denote the length o the queue at node n keeping packets destined or node d at the beginning o slot t. Deinition 1 (Stability). A given queue is called stable i E[q [ ]] <, where q [ ] denotes the random variable with distribution given by the steady-state distribution o {q [t]}. he network is stable i all queues are stable; and unstable otherwise. We consider a general intererence model ormulation that contains all o the graph theoretic collision models considered in the context o scheduling (e.g. NESS [20, 14, 27, 3], or two-hop intererence models [28, 4]). We say that two links interere i their concurrent transmissions collide, and assume that i two interering links are activated in a slot, both o the transmissions ail. We use I(l) Lto denote the set o links that interere with link l. ypically, this set will contain links rom the local neighborhood o l. As an example, the matching constraint o the NESS model implies that or link l =(n, m), I(l) :={l L: l i(n) or l i(m)}, where i(n) denotes the set o links that are incident to node n. We will prove our results or the general model. We use π = { π (n,m) to denote a link activation }(n,m) L (or allocation) vector, and Π denote the easible set o allocations that complies with the intererence constraints. An allocation is easible i and only i no two links in the set interere with each other. As an example, or the NESS intererence model, Π corresponds to the set o matchings o G. We introduce the notation π(n,m) d to distinguish packets destined or dierent nodes: at any given slot, π(n,m) d [t] {0, 1} is 1 i link (n, m) serves a packet destined or node d in that slot, and 0 otherwise. Notice that π (n,m) [t] = d N πd (n,m) [t], or all t. Also note that it is suicient to restrict our attention to policies that sets π(n,m) d [t] to zero whenever q [t] = 0, or all (n, m) L, because any other policy that sets π(n,m) d [t] =1when q [t] =0can be replaced by a policy with π(n,m) d [t] =0without aecting the evolution o the queue-lengths. hen, we can write the evolution o a particular queue, say q, as q [t +1]= (q [t] πout(n) d [t]+ x [t]+πinto(n) d [t]), (1) S where x [t] is the number o exogenous low- arrivals to the network in slot t, and S { F: s() =n, d() =d} denotes the set o lows that start at node n and are destined to node d. Also, πinto(n) d [t] k:(k,n) L πd (k,n)[t] is a shorthand or the number o packets entering node n that are destined or node d. Similarly, πout(n) d [t] is the number o packets leaving node n and are destined or node d. We set q d,d [t] =0 t. Deinition 2 (Capacity (Stability) Region). Let G = (N, L) be a given network and Π be the set o easible allocations. he capacity (or stability) region Λ o the network is given by the set o vectors r =(r ) F or which there exists π d() (n,m) 0, or all (n, m) Land F, such that both the low conservation constraints at the nodes and the easibility constraints are satisied, as given below: (C1) For all n Nand F, we have 3 r 1 s()=n + π d() (k,n) = π d() (n,m), k:(k,n) L (C2) [ π (n,m) ](n,m) L Conv(Π). 4 m:(n,m) L It is shown in [25, 19] that Λ is the set o mean arrival rates or which there exists a policy that stabilizes the network. III. SABILIY FOR INELASIC RAFFIC he ocus o this section is throughput-optimality under stability or the inelastic traic scenario. We provide simple 3 We use 1 A as the indicator unction o event A. 4 Conv(A) denotes the convex hull o set A, which is the smallest convex set that includes A. he convex hull is included due to the possibility o timesharing between easible allocations. 1937

3 routing-scheduling mechanisms that can support any mean arrival rate in the capacity region without violating stability (such mechanisms are said to be throughput-optimal [25, 21, 19, 7]). In earlier work [24], assiulas used randomized schemes to provide a low complexity stabilizing algorithm or switches using a centralized controller. In this section, we will extend the use o randomized throughput-optimal schemes or multi-hop networks with general intererence models. In particular, we show that practical algorithms satisying several simple conditions can be designed to achieve ull utilization o the capacity o multi-hop wireless networks with general intererence models. o this end, we introduce the ollowing notation or link weights: w (n,m) [t] =w (m,n) [t] max d q [t] q m,d [t]. his is also reerred to as the dierential backlog 5 over link (n, m) and can be interpreted as a measure o the importance o the link. Consider the ollowing allocation vector π w [t] arg max π Π l L w l [t]π l arg max(w[t] π). (2) π Π his allocation rule is called the back-pressure policy. Once the π w is determined according to (2), only the commodity that maximizes the dierential backlog over link (n, m) is served over that link at the chosen rate. I the backpressure policy is applied at every time slot, it is known to be throughput-optimal. However, it requires a centralized controller that knows w[t] at every time-slot, and also communicates the allocation vector π w [t] instantly to all the nodes o the network. hese requirements make the implementation o this algorithm impractical or the multihop wireless network scenario. he idea is to use a random algorithm, instead o the optimal one described in (2), which yields a easible allocation π[t] Π at every time slot, which is not necessarily equal to π w [t], but has a positive probability δ>0 o being equal to π w [t]. hus, we have P ( π[t] = π w [t]) δ, or all w[t] and t. (3) Once the allocation π[t] is chosen, the actual allocation π[t] is determined according to the ollowing evolution. { π[t] i w[t] π[t] w[t] π[t] π[t +1]= (4) π[t] otherwise he above randomized algorithm was introduced in [24] in the context o switches, where there exists a centralized scheduler. A similar approach has been used in developing a distributed implementation or networks restricted to singlehop communication with matching constraints and Bernoulli arrival processes [17]. As we will prove shortly, it turns out that the two conditions (3) and (4) are suicient to achieve throughput-optimality in a more general setting. he algorithm updates its allocation vector only i the proposed allocation o the randomized algorithm yields a better objective unction. Such an algorithm can be divided into two 5 his deinition o dierential-backlog is slightly dierent rom the ones in the literature, which is due to the assumption o undirected links here. parts: PICK and COMPARE, where PICK chooses a easible schedule π[t] satisying (3), and COMPARE communicates the relevant weight inormation to other nodes in the network to perorm (4). Distributed implementations o both PICK and COMPARE can be developed or a given intererence model (see [6] or an example). heorem 1. Assume that x [t] is independent and identically distributed (i.i.d.) 6 or all t and with E[x 2 [1]] A<. hen, or any mean arrival vector λ := (E[x [1]]) interior(λ), the above randomized policy is stabilizing. Proo: We start by noting that y[t] (q[t],π[t]) orms a Markov Chain. hen, we study the mean drit o the ollowing Lyapunov unction o the state y =(q,π): V (y) = ( ) 2 q 2 + w l (( π w ) l π l ) n N = q q }{{} V 1 (y) d N l L +(w ( π w π)) 2 }{{} V 2 (y) he proo uses the ollowing two key lemmas the proos o which are moved to Appendix A and B. Lemma 1. Let 1 (y[t]) E[V 1 (y[t + 1]) V 1 (y[t]) y[t]], then or some γ>0and c 1 <, we have 1 (y[t]) γ V 1 (y[t])+2 V 2 (y[t]) + c 1. Lemma 2. Let 2 (y[t]) E[V 2 (y[t + 1]) V 2 (y[t]) y[t]], then or some δ>0 and c 2,c 3 <, we have 2 (y[t]) δv 2 (y[t]) + c 2 V2 (y[t]) + c 3. Proo o heorem 1 (Continued): Combining these lemmas, we upper-bound (y[t]) E[V (y[t + 1]) V (y[t]) y[t]] by γ V 1 (y[t]) δv 2 (y[t])+(2+c 2 ) V 2 (y[t]) + c 1 + c 3. hus, i we consider the scenario where V (y[t]) B or a suiciently large B<, then we can guarantee that (5) (y[t]) ɛ V 1 (y[t]) + ˆB, (6) or some ɛ>0 and ˆB <. his is true because in this scenario, since we have V 1 (y[t]) B V 2 (y[t]), we can write E[V (y[t + 1]) y[t]] V (y[t]) γ 2 V1 (y[t]) γ 2 B V2 (y[t]) δv 2 (y[t])(7) +(2 + c 2 ) V 2 (y[t]) + c 1 + c 3.(8) Notice that the sum o the expressions in (7) and (8) can be made negative by choosing B large enough. Hence (6) holds when we let ɛ = γ/2 and choose ˆB large enough. he proo o positive recurrence o the chain ollows rom Foster s Criteria (c. [22] or [16]). Let us use y[ ] to denote the random variable to which {y[t]} t converges. Due to the -ergodic theorem o [16], the drit condition o (6) 6 he assumption o i.i.d. arrivals is not critical to the analysis. he same results continue to hold or processes with mild ergodicity properties ([10]). 1938

4 is equivalent to E[ V 1 (y[ ]] <. Invoking the deinition o V 1 ( ) results in the intended stability result. heorem 1 shows that under very mild conditions, i a randomized scheduler can be ound that satisies (3), and the control in (4) can be perormed, then the stability will be achieved or any stabilizable incoming traic. IV. FAIR SERVICE OF ELASIC RAFFIC In the previous section, we proved the throughput optimality o a randomized scheme or inelastic traic, provided that the exogenous arrival rates are stabilizable. hus, with that scheme, even i stability is achieved or a given mean arrival vector, the system may be seriously underutilized. Ideally, it would be desired to know the stability region o the network and choose the arrival rates to ully utilize the network resources. However, as noted in [26], the task o determining whether a given set o arrival rates is in the characterized stability region becomes an intractable problem as the network size grows. In this section, we extend the throughput-optimal algorithm so that the available capacity o the network is ully utilized and airly shared among the lows in a completely decentralized and dynamic ashion without the need to explicitly characterize the stability region. o that end, we change the considered traic model rom inelastic to elastic, where in the latter case the mean arrival rates o the sources can be modiied. o deine airness, as is standard in the recent literature (e.g. [11, 12, 15, 22, 7, 18, 13]), we use a utility unction, U ( ), o mean arrival rates that is a measure o low s preerences. hroughout, we will assume that this unction is concave and non-decreasing. We call the allocation, x, that satisies x arg max U (x ) (9) x Λ F the air allocation. Notice that this is the allocation that maximizes the aggregate utility o the network. It is known that by deining U ( ) appropriately dierent airness criteria o interest, such as proportional or max-min airness, can be achieved (see [22] or an extensive review). Next, we describe the so called Dual Congestion Control mechanism that is implemented at the source o each low in a completely decentralized ashion. Variations o this mechanism are studied recently in the literature [7, 18, 14, 23]. DUAL CONGESION CONROL MECHANISM: Assume that every low has access to the its entry point queue-length inormation, i.e. low knows q b(),e() [t] or all t. hen, at the beginning o each time slot t, low generates x [t] packets satisying { ( ) } x [t] = min U 1 qb(),e() [t],m, (10) K where M and K are positive scalars. he policy is easy to implement at each source because it only requires the queue length o the buer at the source, and the individual utility unction o the low. Note that the only common inormation required at all the sources is K. Once K is determined, the low control mechanism can operate in a completely decentralized ashion in parallel with the randomized routing-scheduling algorithm described in Section III. heorem 2. For some inite constants C 1,C 2 we have q C 1 K (11) n N d N U ( x ) U (x ) C 2 (12) K F F 1 1 where x lim E[x [t]], and similarly or q. t=0 Proo: We start by introducing a relaxed version o the optimization problem (9): or any ε>0, let x (ε) arg max U (x ), x Λ(ε) where Λ(ε) ={x Λ:[x + ε] Λ}. Λ(ε) is a subset o the capacity region Λ with an ε-strip o its positive surace is deleted. It is not diicult to see that x (ε) x as ε 0. We use the same Lyapunov unctions V (y),v 1 (y) and V 2 (y) introduced in Section III (c.. (5)) to prove the theorem. he ollowing theorem studies the single-step mean drit o V 1 (y). he proo o this lemma is provided in Appendix C. Lemma 3. For some constants ɛ>0,c 1 <, 1 (y[t]) ε q [t]+2k E[U (x [t]) y[t]] 2K U (x (ε)) ε V1 N 2 (y[t]) + 2 V 2 (y[t]) + c 1 Proo o heorem 2 (Continued): In what ollows, we will omit [t] or convenience. By combining the result o Lemma 3 with that o Lemma 2, we can write: (y[t]) ε q +2K U (x ) 2KE[ U (x (ε)) y]+c 1 + c 3 V1 (y) ε N 2 δv 2 (y) + (2 + c 2 ) V 2 (y) (13) (a) c 4 ε q +2K{ E[U (x ) y] U (x (ε))} where the inequality (a) ollows rom the act that (13) is upper-bounded as argued in the proo o heorem 1. By taking expectations o both sides o the last inequality and summing over t =0, 1,, 1, we get 1 E[V (y[ ]) V (y[0])] c 4 ε E[q [t]] t=0 1 +2K E[U (x [t])] 2K t=0 U (x (ε)). (14) 1939

5 Since V (y) 0 or all y, we can re-arrange the terms in (14) to obtain the ollowing inequality. 1 1 t=0 E[q [t]] V (y[0]) ε + c 4 +2M F K. ε he proo o (11) is complete when we let, and deine C 1 2M F ε. I, on the other hand, we re-arrange the terms in (14) dierently, we can get 1 1 E[U (x [t])] U (x (ε)) V (y[0]) 2K c 4 2K. t=0 Hence, the proo o (12) is complete when we use Jensen s inequality to write U ( x ) 1 1 t=0 E[U (x [t])], and deine C 2 c4 2. V. CONCLUSIONS In this work, we provided a ramework or the design o practical cross-layer algorithms or multi-hop wireless networks that are throughput-optimal and air. Previously proposed low complexity implementations or multi-hop wireless networks have been greedy in nature. Despite their ease o implementation, such policies have been shown in recent works to have poor throughput perormance. he greatest weakness o greedy policies is their lack o memory. In this work, we considered the generalization o a practical randomized policy irst introduced in [24] or switches to multi-hop networks. We proved that with the use o a small memory unit at the nodes, this policy achieves 100% o the available capacity o the network. We also considered the case o elastic lows and provided a decentralized congestion control algorithm that works in parallel with the randomized algorithm. We showed that the resulting cross-layer algorithm allocates resources airly across users. We note that the extra overhead that will appear in our randomized algorithm can be reduced at the expense o larger delay, but without sacriicing rom throughput. Also, our initial indings suggest that the additional complexity is comparable to the complexity o the greedy policies. In our companion paper [6], we show that the development and analysis o low complexity, distributed PICK and COMPARE algorithms (as described in Section III) or the two-hop intererence model. APPENDIX A. Proo o Lemma 1: Proo: We start by arranging the terms o 1 (y[t]). 1 (y[t]) = E[(q[t +1] q[t]) (q[t +1]+q[t]) y[t]] (a) c 1 +2 q [t](λ + π d into(n) [t] π d out(n) [t]) (15) +2 q [t](πinto(n) d [t] πd out(n) [t] π d into(n) [t]+ π d out(n) [t]), (16) where inequality (a) is due to the act that E[x 2 [t]] A, and that πinto(n) d [t] and πd out(n) [t] are both upper-bounded by the maximum degree o G. In (15) and (16), the optimal allocation π w is added and subtracted. For notational convenience, we use λ S λ. Also, π d into(n) and π d out(n) are deined similarly to πinto(n) d and πd out(n). Since the arrival rate vector λ is known to be within the stability region Λ, we can upper-bound the second expression in (15) by γ V 1 (y[t] (e.g. see [5]). Next, notice that (16) = 2w[t] ( π w [t] π[t]) 2 V 2 (y[t]). Combination o these results yields the proo. B. Proo o Lemma 2: Proo: We start by analyzing the conditional expectation E[V 2 (y[t + 1]) y[t]] (1 δ)e[(w[t +1] ( π w [t +1] π[t + 1])) 2 y[t],π[t +1] π w [t + 1]]. Next, we write the evolution o the weight vector w[t] as w[t +1] =w[t]+r[t] z[t], where r[t] denotes the vector o packets entering the queues and z[t] is the vector o packets leaving the queues. We can describe these vectors more precisely, but this is not necessary or the proo. he only important actor is the boundedness o them. Now, we can write E[V 2 (y[t + 1]) y[t]] (1 δ)e[(w[t] ( π w [t +1] π[t + 1]) (17) +( π w [t +1] π[t + 1]) (r[t] z[t])) 2 y[t],π[t +1] π w [t + 1]](18) It is not diicult to upper-bound the expression in (18) with a inite constant. o bound (17), we make two observations: (i) w[t] π w [t +1] w[t] π w [t] due to the deinition o the optimal allocation vector π w in (2). (ii) w[t] π[t +1] w[t] π[t] due to the update rule (3) o the randomized algorithm. By combining these observations, we can ind constants c 2,c 3 < that satisy E[V 2 (y[t + 1]) y[t]] (1 δ)v 2 (y[t]) + c 2 V2 (y[t]) + c 3, which completes the proo. C. Proo o Lemma 3: Proo: he proo starts with the ollowing upper bound on 1 (y[t]) (c. (15) and (16)): 1940

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