Backpressure-based Packet-by-Packet Adaptive Routing in Communication Networks

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1 1 Backpressure-base Packet-by-Packet Aaptive Routing in Communication Networks Eleftheria Athanasopoulou, Loc Bui, Tianxiong Ji, R. Srikant, an Alexaner Stolyar Abstract Backpressure-base aaptive routing algorithms where each packet is route along a possibly ifferent path have been extensively stuie in the literature. However, such algorithms typically result in poor elay performance an involve high implementation complexity. In this paper, we evelop a new aaptive routing algorithm built upon the wiely-stuie back-pressure algorithm. We ecouple the routing an scheuling components of the algorithm by esigning a probabilistic routing table which is use to route packets to per-estination queues. The scheuling ecisions in the case of wireless networks are mae using counters calle shaow queues. The results are also extene to the case of networks which employ simple forms of network coing. In that case, our algorithm provies a low-complexity solution to optimally exploit the routing-coing traeoff. I. INTRODUCTION The back-pressure algorithm introuce in [1] has been wiely stuie in the literature. While the ieas behin scheuling using the weights suggeste in that paper have been successful in practice in base stations an routers, the aaptive routing algorithm is rarely use. The main reason for this is that the routing algorithm can lea to poor elay performance ue to routing loops. Aitionally, the implementation of the back-pressure algorithm requires each noe to maintain perestination queues which can be burensome for a wireline or wireless router. Motivate by these consierations, we reexamine the back-pressure routing algorithm in the paper an esign a new algorithm which has much superior performance an low implementation complexity. Prior work in this area [2] has recognize the importance of oing shortest-path routing to improve elay performance an moifie the back-pressure algorithm to bias it towars taking shortest-hop routes. A part of our algorithm has similar motivating ieas. In aition to provably throughput-optimal routing which minimizes the number of hops taken by packets in the network, we ecouple (to a certain egree) routing an scheuling in the network through the use of probabilistic routing tables an the so-calle shaow queues. The minhop routing iea was stuie first in a conference paper [3] 1 This work was supporte by MURI BAA , ARO MURI, DTRA Grant HDTRA , an NSF grants , an E. Athanasopoulou, T. Ji an R. Srikant are with Coorinate Science Laboratory an Department of Electrical an Computer Engineering, University of Illinois at Urbana-Champaign. ( {athanaso, tji2, rsrikant}@illinois.eu). L. Bui is with School of Engineering, Tan Tao University. ( locbui@ieee.org). A. Stolyar is with Bell Labs, Alcatel-Lucent, NJ, USA. ( stolyar@research.bell-labs.com). an shaow queues were introuce in [4] an [5], but the key step of partial ecoupling the routing an scheuling which leas to both significant elay reuction an the use of per-next-hop queueing is original here. In [4], the authors introuce the shaow queue to solve a fixe routing problem. The min-hop routing iea is also stuie in [6] but their solution requires even more queues than the original backpressure algorithm. Compare to [4], the main purpose of this paper is to stuy if the shaow queue approach extens to the case of scheuling an routing. The first contribution is to come up with a formulation where the number of hops is minimize. It is interesting to contrast this contribution with [6]. The formulation in [6] has the same objective as ours but their solution involves per-hop queues, which ramatically increases the number of queues, even compare to the backpressure algorithm. Our solution is significantly ifferent: we use the same number of shaow queues as the backpressure algorithm, but the number of real queues is very small (per-neighbor). The new iea here is to perform routing via probabilistic splitting, which allows the ramatic reuction in the number of real queues. Finally, an important observation in this paper, not foun in [4], is that the partial ecoupling of shaow back-pressure an real packet transmission allows us to activate more links than a regular back-pressure algorithm woul. This iea appears to be essential to reuce elays in the routing case, as shown in the simulations. We also consier networks where simple forms of network coing is allowe [7]. In such networks, a relay between two other noes XORs packets an broacast them to ecrease the number of transmissions. There is a traeoff between choosing long routes to possibly increase network coing opportunities (see the notion of reverse carpooling in [8]) an choosing short routes to reuce resource usage. Our aaptive routing algorithm can be moifie to automatically realize this traeoff with goo elay performance. In aition, network coing requires each noe to maintain more queues [9] an our routing solution at least reuces the number of queues to be maintaine for routing purposes, thus partially mitigating the problem. An offline algorithm for optimally computing the routing-coing traeoff was propose in [10]. Our optimization formulation bears similarities to this work but our main focus is on esigning low-elay on-line algorithms. Backpressure solutions to network coing problems have also been stuie in [11], [12], [13], but the aaptive routing-coing traeoff solution that we propose here has not been stuie previously. We summarize our main results below. Using the concept of shaow queues, we partially e-

2 2 couple routing an scheuling. A shaow network is use to upate a probabilistic routing table which packets use upon arrival at a noe. The same shaow network, with back-pressure algorithm, is use to activate transmissions between noes; however, first, actual transmissions sen packets from FIFO per-link queues an, secon, potentially more links are activate, in aition to those activate by the shaow algorithm. The routing algorithm is esigne to minimize the average number of hops use by packets in the network. This iea, along with the scheuling/routing ecoupling, leas to elay reuction compare with the traitional back-pressure algorithm. Each noe has to maintain counters, calle shaow queues, per estination. This is very similar to the iea of maintaining a routing table per estination. But the real queues at each noe are per-next-hop queues in the case of networks which o not employ network coing. When network coing is employe, per-previous-hop queues may also be necessary but this is a requirement impose by network coing, not by our algorithm. The algorithm can be applie to wireline an wireless networks. Extensive simulations show ramatic improvement in elay performance compare to the back-pressure algorithm. The rest of the paper is organize as follows. We present the network moel in Section II. In Section III an IV, the traitional back-pressure algorithm an its moifie version are introuce. We evelop our aaptive routing an scheuling algorithm for wireline an wireless networks with an without network coing in Section V, VI an VII. In Section VIII, the simulation results are presente. We conclue our paper in Section IX. II. THE NETWORK MODEL We consier a multi-hop wireline or wireless network represente by a irecte graph G = (N, L), where N is the set of noes an L is the set of irecte links. A irecte link that can transmit packets from noe n to noe j is enote by (nj) L. We assume that time is slotte an efine the link capacity c nj to be the maximum number of packets that link (nj) can transmit in one time slot. Let F be the set of flows that share the network. Each flow is associate with a source noe an a estination noe, but no route is specifie between these noes. This means that the route can be quite ifferent for packets of the same flow. Let b(f) an e(f) be source an estination noes, respectively, of flow f. Let x f be the rate (packets/slot) at which packets are generate by flow f. If the eman on the network, i.e., the set of flow rates, can be satisfie by the available capacity, there must exist a routing algorithm an a scheuling algorithm such that the link rates lie in the capacity region. To precisely state this conition, we efine µ nj to be the rate allocate on link (nj) to packets estine for noe. Thus, the total rate allocate to all flows at link (nj) is given by µ nj := µ nj. Clearly, for the N network to be able to meet the traffic eman, we shoul have: {µ nj } Λ, where Λ is the capacity region of the network for 1-hop traffic. The capacity region of the network for 1-hop traffic contains all sets of rates that are stabilizable by some kin of scheuling policy assuming all traffics are 1-hop traffic. As a special case, in the wireline network, the constraints are: µ nj c nj, (nj). As oppose to Λ, let Υ enote the capacity region of the multihop network, i.e., for any set of flows {x f } Υ, there exists some routing an scheuling algorithms that stabilize the network. In aition, a flow conservation constraint must be satisfie at each noe, i.e., the total rate at which traffic can possibly arrive at each noe estine to must be less than or equal to the total rate at which traffic can epart from the noe estine to : x f I {b(f)=n,e(f)=} + j: µ nj, l:(ln) L µ ln where I enotes the inicator function. Given a set of arrival rates x = {x f } that can be accommoate by the network, one version of the multi-commoity flow problem is to fin the traffic splits µ nj such that (1) is satisfie. However, fining the appropriate traffic split is computationally prohibitive an requires knowlege of the arrival rates. The back-pressure algorithm to be escribe next is an aaptive solution to the multi-commoity flow problem. III. THROUGHPUT-OPTIMAL BACK-PRESSURE ALGORITHM AND ITS LIMITATIONS The back-pressure algorithm was first escribe in [1] in the context of wireless networks an inepenently iscovere later in [14] as a low-complexity solution to certain multi-commoity flow problems. This algorithm combines the scheuling an routing functions together. While many variations of this basic algorithm have been stuie, they primarily focus on maximizing throughput an o not consier QoS performance. Our algorithm uses some of these ieas as builing blocks an therefore, we first escribe the basic algorithm, its rawbacks an some prior solutions. The algorithm maintains a queue for each estination at each noe. Since the number of estinations can be as large as the number of noes, this per-estination queueing requirement can be quite large for practical implementation in a network. At each link, the algorithm assigns a weight to each possible estination which is calle back-pressure. Define the backpressure at link (nj) for estination at slot t to be w nj[t] = Q n [t] Q j [t], where Q n [t] enotes the number of packets at noe n estine for noe at the beginning of time slot t. Uner this notation, Q nn [t] = 0, t. Assign a weight w nj to each link (nj), where (1)

3 3 w nj is efine to be the maximum back-pressure over all possible estinations, i.e., w nj [t] = max wnj[t]. Let nj be the estination which has the maximum weight on link (nj), nj[t] = arg max {w nj[t]}. (2) If there are ties in the weights, they can be broken arbitrarily. Packets belonging to estination nj [t] are scheule for transmission over the activate link (nj). A scheule is a set of links that can be activate simultaneously without interfering with each other. Let Γ enote the set of all scheules. The back-pressure algorithm fins an optimal scheule π [t] which is erive from the optimization problem: π [t] = arg max c nj w nj [t]. (3) π Γ (nj) π Specially, if the capacity of every link has the same value, the chosen scheule maximizes the sum of weights in any scheule. At time t, for each activate link (nj) π [t] we remove c nj packets from Q n nj [t] if possible, an transmit those packets to Q j nj [t]. We assume that the epartures occur first in a time slot, an external arrivals an packets transmitte over a link (nj) in a particular time slot are available to noe j at the next time slot. Thus the evolution of the queue Q n [t] is as follows: Q n [t + 1] = Q n [t] I { nj [t]=} ˆµ nj [t] + l:(ln) L + j: I { ln [t]=} ˆµ ln [t] I {b(f)=n,e(f)=} a f [t], where ˆµ nj [t] is the number of packets transmitte over link (nj) in time slot t an a f [t] is the number of packets generate by flow f at time t. It has been shown in [1] that the backpressure algorithm maximizes the throughput of the network. A key feature of the back-pressure algorithm is that packets may not be transferre over a link unless the back-pressure over a link is non-negative an the link is inclue in the picke scheule. This feature prevents further congesting noes that are alreay congeste, thus proviing the aaptivity of the algorithm. Notice that because all links can be activate without interfering with each other in the wireline network, Γ is the set of all links. Thus the back-pressure algorithm can be localize at each noe an operate in a istribute manner in the wireline network. The back-pressure algorithm has several isavantages that prohibit practical implementation: The back-pressure algorithm requires maintaining queues for each potential estination at each noe. This queue management requirement coul be a prohibitive overhea for a large network. The back-pressure algorithm is an aaptive routing algorithm which explores the network resources an aapts (4) to ifferent levels of traffic intensity. However it might also lea to high elays because it may choose long paths unnecessarily. High elays are also a result of maintaining a large number of queues at each noe, an each of those queues being large. The queues can be large because, uner back-pressure algorithm, average size of a perestination queue at a noe can grow with the istance from the noe to the estination. Furthermore, large number of queues takes away statistical multiplexing avantage: since only one queue can be scheule at a time, some of the allocate transmission capacity can be left unuse if the scheule queue is too short this can contribute to high latency as well. In this paper, we aress the high elay an queueing complexity issues. The computational complexity issue for wireless networks is not aresse here. We simply use the recently stuie greey maximal scheuling (GMS) algorithm. Here we call it the largest-weight-first algorithm, in short, LWF algorithm. LWF algorithm requires the same queue structure that the back-pressure algorithm uses. It also calculates the back-pressure at each link using the same way. The ifference between these two algorithms only lies in the methos to pick a scheule. Let S enote the set of all links initially. Let N b (l) be the set of links within the interference range of link l incluing l itself. At each time slot, the LWF algorithm picks a link l with the maximum weight first, an removes links within the interference range of link l from S, i.e., S = S\N b (l); then it picks the link with the maximum weight in the upate set S, an so forth. It shoul be notice that LWF algorithm reuces the computational complexity with a price of the reuction of the network capacity region. The LWF algorithm where the weights are queue lengths (not backpressures) has been extensively stuie in [15], [16], [17], [18], [19]. While these stuies inicate that there may be reuction in throughput ue to LWF in certain special network topologies, it seems to perform well in practice an so we aopt it here for simulations. In the rest of the paper, we present our main results which eliminate many of the problems associate with the backpressure algorithm. IV. MIN-RESOURCE ROUTING USING BACK-PRESSURE ALGORITHM As mentione in Section III, the back-pressure algorithm explores all paths in the network an as a result may choose paths which are unnecessarily long which may even contain loops, thus leaing to poor performance. We aress this problem by introucing a cost function which measures the total amount of resources use by all flows in the network. Specially, we a up traffic loas on all links in the network an use this as our cost function. The goal then is to minimize this cost subject to network capacity constraints. Given a set of packet arrival rates that lie within the capacity region, our goal is to fin the routes for flows so that we use as few resources as possible in the network. Thus, we formulate

4 4 the following optimization problem: min µ nj (5) s.t. x f I {b(f)=n,e(f)=} + {µ nj } Λ. (ln) L µ ln N, n N, µ nj, We now show how a moification of the back-pressure algorithm can be use to solve this min-resource routing problem. (Note that similar approaches have been use in [20], [21], [22], [23], [24] to solve relate resource allocation problems.) Let {q n } be the Lagrange multipliers corresponing to the flow conservation constraints in problem (5). Appening these constraints to the objective, we get ( min q n x f I {b(f)=n,e(f)=} µ Λ µ nj + n, + µ ln (ln) L = min ( µ Λ n, µ nj µ nj ) ( qn q j 1 ) q n x f I {b(f)=n,e(f)=} ). If the Lagrange multipliers are known, then the optimal µ can be foun by solving max µ nj w nj µ Λ where w nj = max (q n q j 1). The form of the constraints in (5) suggests the following upate algorithm to compute q n : [ q n [t + 1] = q n [t] + 1 ( x f I {b(f)=n,e(f)=} M + µ ln µ ) ] + nj (7) (ln) L where 1 M is a step-size parameter. See [25] for etails. Notice that Mq n [t] looks very much like a queue upate equation, except for the fact that arrivals into Q n from other links may be smaller than µ ln when Q l oes not have enough packets. This suggests the following algorithm. Min-resource routing by back-pressure: At time slot t, Each noe n maintains a separate queue of packets for each estination ; its length is enote Q n [t]. Each link is assigne a weight ( 1 w nj [t] = max M Q n[t] 1 ) M Q j[t] 1, (8) where M > 0 is a parameter. Scheuling/routing rule: π [t] arg max π Γ (nj) π (6) c nj w nj [t]. (9) For each activate link (nj) π [t] we remove c nj packets from Q n nj [t] if possible, an transmit those packets to Q j nj [t], where nj [t] achieves the maximum in (8). Note that the above algorithm oes not change if we replace the weights in (8) by the following, re-scale ones: w nj [t] = max (Q n [t] Q j [t] M), (10) an therefore, compare with the traitional back-pressure scheuling/routing, the only ifference is that each link weight is equal to the maximum ifferential backlog minus parameter M. (M = 0 reverts the algorithm to the traitional one.) For simplicity, we call this algorithm M-back-pressure algorithm. The performance of the stationary process which is prouce by the algorithm with fixe parameter M is within o(1) of the optimal as M goes to (analogous to the proofs in [21], [22]; see also the relate proof in [23], [24]): E µ nj [ ] µ nj = o(1), where µ is an optimal solution to (5). Figure 1 illustrates how the M-back-pressure algorithm works in a simple wireline network. All links can be activate simultaneously without interfering with each other. Notice that the backlog ifference of route 1 is 6 an the backlog ifference of route 2 is 4. Because the backlog ifference of route 2 is smaller than M, route 2 is blocke at current traffic loa. The M-back-pressure algorithm will automatically choose route 1 which is shorter. Therefore, a proper M can avoi long routes in when the traffic is not close to capacity. Fig. 1. Picture illustrating link weights uner the M-backpressure algorithm. Although M-back-pressure algorithm coul reuce the elay by forcing flows to go through shorter routes, simulations inicate a significant problem with the basic algorithm presente above. A link can be scheule only if the backpressure of at least one estination is greater than or equal to M. Thus, at light to moerate traffic loas, the elays coul be high since the back-pressure may not buil up sufficiently fast. In orer to overcome all these averse issues, we evelop a new routing algorithm in the following section. The solution also simplifies the queueing ata structure to be maintaine at each noe.

5 5 V. PARN: PACKET-BY-PACKET ADAPTIVE ROUTING AND SCHEDULING ALGORITHM FOR NETWORKS In this section, we present our aaptive routing an scheuling algorithm. We will call it PARN (Packet-by-Packet Aaptive Routing for Networks) for ease for repeate reference later. First, we introuce the queue structure that is use in PARN. In the traitional back-pressure algorithm, each noe n has to maintain a queue q n for each estination. Let N an D enote the number of noes an the number of estinations in the network, respectively. Each noe maintains D queues. Generally, each pair of noes can communicate along a path connecting them. Thus, the number of queues maintaine at each noe can be as high as one less than the number of noes in the network, i.e., D = N 1. Instea of keeping a queue for every estination, each noe n maintains a queue q nj for every neighbor j, which is calle a real queue. Notice that real queues are per-neighbor queues. Let J n enote the number of neighbors of noe n, an let J max = max n J n. The number of queues at each noe is no greater than J max. Generally, J max is much smaller than N. Thus, the number of queues at each noe is much smaller compare with the case using the traitional back-pressure algorithm. In aitional to real queues, each noe n also maintains a counter, which is calle shaow queue, p n for each estination. Unlike the real queues, counters are much easier to maintain even if the number of counters at each noe grows linearly with the size of the network. A back-pressure algorithm run on the shaow queues is use to ecie which links to activate. The statistics of the link activation are further use to route packets to the per-next-hop neighbor queues mentione earlier. The etails are explaine next. A. Shaow Queue Algorithm M-back-pressure Algorithm The shaow queues are upate base on the movement of fictitious entities calle shaow packets in the network. The movement of the fictitious packets can be thought of as an exchange of control messages for the purposes of routing an scheule. Just like real packets, shaow packets arrive from outsie the network an eventually exit the network. The external shaow packet arrivals are general as follows: when an exogenous packet arrives at noe n to the estination, the shaow queue p n is incremente by 1, an is further incremente by 1 with probability ε in aition. Thus, if the arrival rate of a flow f is x f, then the flow generates shaow traffic at a rate x f (1 + ε). In wors, the incoming shaow traffic in the network is (1 + ε) times of the incoming real traffic. The back-pressure for estination on link (nj) is taken to be w nj[t] = p n [t] p j [t] M, where M is a properly chosen parameter. The choice of M will be iscusse in the simulations section. The evolution of the shaow queue p n [t] is p n [t + 1] = p n [t] I { nj [t]=} ˆµ nj [t] + l:(ln) L + j: I { ln [t]=} ˆµ ln [t] I {b(f)=n,e(f)=} â f [t], (11) where ˆµ nj [t] is the number of shaow packets transmitte over link (nj) in time slot t, nj [t] is the estination that has the maximum weight on link (nj), an â f [t] is the number of shaow packets generate by flow f at time t. The number of shaow packets scheule over the links at each time instant is etermine by the back-pressure algorithm in equation (9). From the above escription, it shoul be clear that the shaow algorithm is the same as the traitional back-pressure algorithm, except that it operates on the shaow queueing system with an arrival rate slightly larger than the real external arrival rate of packets. Note the shaow queues o not involve any queueing ata structure at each noe; there are no packets to maintain in a FIFO orer in each queue. The shaow queue is simply a counter which is incremente by 1 upon a shaow packet arrival an ecremente by 1 upon a eparture. The back-pressure algorithm run on the shaow queues is use to activate the links. In other wors, if πnj = 1 in (9), then link (nj) is activate an packets are serve from the real queue at the link in a first-in, first-out fashion. This is, of course, very ifferent from the traitional back-pressure algorithm where a link is activate to serve packets to a particular estination. Thus, we have to evelop a routing scheme that assigns packets arriving to a noe to a particular next-hop neighbor so that the system remains stable. We esign such an algorithm next. B. Aaptive Routing Algorithms Now we iscuss how a packet is route once it arrives at a noe. Let us efine a variable σnj [t] to be the number of shaow packets transferre from noe n to noe j for estination uring time slot t by the shaow queue algorithm. Let us enote by σ nj the expecte value of σ nj [t], when the shaow queueing process is in a stationary regime; let ˆσ nj [t] enote an estimate of σ nj, calculate at time t. (In the simulations we use the exponential averaging, as specifie in the next section.) At each time slot, the following sequence of operations occurs at each noe n. A packet arriving at noe n for estination is inserte in the real queue q nj for next-hop neighbor j with probability Pnj[t] ˆσ nj = [t] (12) k:(nk) L ˆσ nk [t]. Thus, the estimates ˆσ nj [t] are use to perform routing operations: in toay s routers, base on the estination of a packet, a packet is route to its next hop base on routing table entries. Instea, here, the σ s are use to probabilistically choose the next hop for a packet. Packets waiting at link (nj)

6 6 A. Exponential Averaging To compute ˆσ nj [t] we use the following iterative exponential averaging algorithm: ˆσ nj[t] = (1 β) ˆσ nj[t 1] + β σ nj[t], (13) where 0 < β < 1. Fig. 2. Probabilistic splitting algorithm at Noe n. are transmitte over the link when that link is scheule (See Figure 2). The first question that one must ask about the above algorithm is whether it is stable if the packet arrival rates from flows are within the capacity region of the multi-hop network. This is a ifficult question, in general. Since the shaow queues are positive recurrent, goo estimates ˆσ nj [t] can be maintaine by simple averaging (e.g. as specifie in the next section), an therefore the probabilities in (12) will stay close to their ieal values P nj = σ nj. k:(nk) L σ nk The following theorem asserts that the real queues are stable uner the aitional assumption that the routing probabilities Pnj are fixe at their ieal values P nj (as oppose to being upate via (12), which is what the actual algorithm oes). Theorem 1: Suppose the routing probabilities are fixe at P nj. Assume that there exists a elta such that {x f (1+ϵ+δ)} lies in Γ. Let a f [t] be the number of packets arriving from flow f at time slot t, with E(a f [t]) = x f an E(a f [t]) <. Assume that the arrival process is inepenent across time slots an flows (this assumption can be consierably relaxe). Then, the Markov chain, jointly escribing the evolution of shaow queues an real FIFO queues (whose state inclue the estination of the real packet in each position of each FIFO queue), is positive recurrent. Proof: The key ieas behin the proof are outline. The etails are similar to the proof in [4] an are omitte. The average rate at which packets arrive to link (nj) is strictly smaller than the capacity allocate to the link by the shaow process if ε > 0. (This fact is verifie in Appenix A.) It follows that the flui limit of the real-queue process is same as that of the networks in [26]. Such flui limit is stable [26], which implies the stability of our process as well. VI. IMPLEMENTATION DETAILS The algorithm presente in the previous section ensures that the queue lengths are stable. In this section, we iscuss a number of enhancements to the basic algorithm to improve performance. B. Token Bucket Algorithm Computing the average shaow rate ˆσ nj [t] an generating ranom numbers for routing packets may impose a computational overhea of routers which shoul be avoie if possible. Thus, as an alternative, we suggest the following simple algorithm. At each noe n, for each next-hop neighbor j an each estination, maintain a token bucket rnj. Consier the shaow traffic as a guiance of the real traffic, with tokens remove as shaow packets traverse the link. In etail, the token bucket is ecremente by σnj [t] in each time slot, but cannot go below the lower boun 0: r nj[t] = max{r nj[t 1] σ nj[t], 0}. When rnj [t 1] σ nj [t] < 0, we say that σ nj [t] r nj [t 1] tokens (associate with bucket rnj ) are waste in slot t. Upon a packet arrival at noe n for estination, fin the token bucket rnj which has the smallest number of tokens (the minimization is over next-hop neighbors j), breaking ties arbitrarily, a the packet to the corresponing real queue q nj an a one token to the corresponing bucket: rnj [t] = r nj [t 1] + 1. (14) To explain how this algorithm works, enote by σ nj the average value of σnj [t] (in stationary regime), an by η n the average rate at which real packets for estination arrive at noe n. Due to the fact that real traffic is injecte by each source at the rate strictly less than the shaow traffic, we have η n < j σ nj. (15) For a single-noe network, (15) just means that arrival rate is less than available capacity. More generally, it is an assumption that nees to be prove. However, here our goal is to provie an intuition behin the token bucket algorithm, so we simply assume (15). Conition (15) guarantees that the token processes are stable (that is, roughly, they cannot runaway to infinity) since the total arrival rate to the token buckets at a noe is less than the total service rate an the arrivals employ a join-the-shortest-queue iscipline. Moreover, since rnj [t] are ranom processes, the token buckets will hit 0 in a non-zero fraction of time slots, except in some egenerate cases; this in turn means that the arrival rate of packets at the token bucket must be less than the token generation rate: η nj < σ nj, (16) where ηnj is the actual rate at which packets arriving at n an estine for are route along link (nj). Inequality (16) thus escribes the iea of the algorithm. Ieally, in aition to (16), we woul like to have the ratios ηnj / σ nj to be equal across all j, i.e., the real packet arrival

7 7 rates at the outgoing links of a noe shoul be proportional to the shaow service rates. It is not ifficult to see that if ε is very small, the proportion will be close to ieal. In general, the token-base algorithm oes not guarantee that, that is why it is an approximation. Also, to ensure implementation correctness, instea of (14), we use r nj [t] = min{r nj [t 1] + 1, B}, (17) i.e., the value of r nj [t] is not allowe to go above some relatively large value B, which is a parameter of the orer of O(1/ϵ). Uner normal circumstances, r nj [t] hitting ceiling B is a rare event, occurring ue to the process ranomness. The main purpose of having the upper boun B is to etect serious anomalies when, for whatever reason, the conition (15) breaks for prolonge perios of time such situation is etecte when any r nj [t] hits the upper boun B frequently. C. Extra Link Activation Uner the shaow back-pressure algorithm, only links with back-pressure greater than or equal to M can be activate. The stability theory ensures that this is sufficient to rener the real queues. On the other han, the elay performance can still be unacceptable. Recall that the parameter M was introuce to iscourage the use of unnecessarily long paths. However, uner light an moerate traffic loas, the shaow back-pressure at a link may be frequently less than M, an thus, packets at such links may have to wait a long time before they are processe. One way to remey the situation is to activate aitional links beyon those activate by the shaow back-pressure algorithm. The basic iea is as follows: in each time slot, first run the shaow back-pressure algorithm. Then, a aitional links to make the scheule maximal. If the extra activation proceure epens only on the state of shaow queues (but beyon that, can be ranom an/or arbitrarily complex), then the stability result of Theorem 1 still hols (with essentially same proof). Informally, the stability prevails, because the shaow algorithm alone provies sufficient average throughput on each link, an aing extra capacity oes not hurt ; thus, with such extra activation, a certain egree of ecoupling between routing (totally controlle by shaow queues) an scheuling (also controlle by shaow queues, but not completely) is achieve. For example, in the case of wireline networks, by the above arguments, all links can be activate all the time. The shaow routing algorithm ensures that the arrival rate at each link is less than its capacity. In this case the complete ecoupling of routing an scheuling occurs. In practice, activating extra links which have large queue backlogs leas to better performance than activating an arbitrary set of extra links. However, in this case, the extra activation proceure epens on the state of real queues which makes the issue of valiity of an analog of Theorem 1 much more subtle. We believe that the argument in this subsection provies a goo motivation for our algorithm, which is confirme by simulations. D. The Choice of the Parameter ε From basic queueing theory, we expect the elay at each link to be inversely proportional to the mean capacity minus the arrival rate at the link. In a wireless network, the capacity at a link is etermine by the shaow scheuling algorithm. This capacity is guarantee to be at least equal to the shaow arrival rate. The arrival rate of real packets is of course smaller. Thus, the ifference between the link capacity an arrival rate coul be proportional to epsilon. Thus, epsilon shoul be sufficiently large to ensure small elays while it shoul be sufficiently small to ensure that the capacity region is not iminishe significantly. In our simulations, we foun that choosing ε = 0.1 provies a goo traeoff between elay an network throughput. In the case of wireline networks, recall from the previous subsection that all links are activate. Therefore, the parameter epsilon plays no role here. VII. EXTENSION TO THE NETWORK CODING CASE In this section, we exten our approach to consier networks where network coing is use to improve throughput. We consier a simple form of network coing illustrate in Figure 3. When i an j each have a packet to sen to the other through an intermeiate relay n, traitional transmission requires the following set of transmissions: sen a packet a from i to n, then n to j, followe by j to n an n to i. Instea, using network coing, one can first sen from i to n, then j to n, XOR the two packets an broacast the XORe packet from n to both i an j. This form of network coing reuces the number of transmissions from four to three. However, the network coing can only improve throughput only if such coing opportunities are available in the network. Routing plays an important role in etermining whether such opportunities exist. In this section, we esign an algorithm to automatically fin the right traeoff between using possibly long routes to provie network coing opportunities an the elay incurre by using long routes. Fig. 3. Network coing opportunity. A. System Moel We still consier the wireless network represente by the graph G = (N, L). Let x f be the rate (packets/slot) at which packets are generate by flow f. To facilitate network coing, each noe must not only keep track of the estination of the packet, but also remember the noe from which a packet was receive. Let µ lnj be the rate at which packets receive from either noe l or flow l, estine for noe, are scheule over link (nj). Note that, for compactness of notation, we allow l in the efinition of µ lnj to enote either a flow or a noe. We assume µ lnj is zero when such a transmission is not feasible,

8 8 i.e., when n is not the source noe or is not the estination noe of flow l, or if (ln) or (nj) is not in L. At noe n, the network coing scheme may generate a coe packet by XORing two packets receive from previous-hop noes l an j estine for the estination noes an respectively, an broacast the coe packet to noes j an l. Let µ, n jl enote the rate at which coe packets can be transferre from noe n to noes j an l estine for noes an, respectively. Notice that, ue to symmetry, the following equality hols µ, n jl =, µ n lj. Assume µ, n jl to be zero if at least one of (nl), (ln), (nj) an (jn) oesn t belong to L. Note that µ lnj = 0 when = l or = n, an µ, n jl = 0 when = n or = n. There are two kins of transmissions in our network moel: point-to-point transmissions an broacast transmissions. The total point-to-point rate at which packets receive externally or from a previous-hop noe are scheule on link (nj) an estine to is enote by µ nj,pp = µ lnj + µ lnj, l:l F l:l N an the total broacast rate at which packets scheule on link (nj) estine to is enote by µ nj,broa = µ, n jl. l:l j The total point-to-point rate on link (nj) is enote by µ nj,pp = µ nj,pp an the total broacast rate at which packets are broacast from noe n to noes j an l is enote by µ n jl = µ, n jl. Let µ be the set of rates incluing all point-to-point transmissions an broacast transmissions, i.e., µ = {{µ nj,pp } (nj), {µ n jl } (n jl) }. The multi-hop traffic shoul also satisfy the flow conservation constraints. Flow conservation constraints: For each noe n, each neighbor j, an each estination, we have µ nj,pp + µ nj,broa µ njk + µ, j kn, (18) k k:k n where the left-han sie enotes the total incoming traffic rate at link nj estine to, an the right-han sie enotes the total outgoing traffic rate from link nj estine to. For each noe n an each estination, we have µ fnj, (19) x f I {b(f)=n,e(f)=} where I enotes the inicator function. j N B. Links an Scheules We allow broacast transmission in our network moel. In orer to efine a scheule, we first efine two kins of links: the point-to-point link an the broacast link. A point-to-point link (nj) is a link that supports point-to-point transmission, where (nj) L; A broacast link (n lj) is a link which contains links (nl) an (nj) an supports broacast transmission. Let B enote the set of all broacast links, thus (n lj) B. Let L be the union of the set of the point-to-point links L an the set of the broacast links B, i.e., L = L B. We let Γ enote the set of links that can be activate simultaneously. By abusing notation, Γ can be thought of as a set of vectors where each vector is a list of 1 s or 0 s where a 1 correspons to an active link an a 0 correspons to an inactive link. Then, the capacity region of the network for 1- hop traffic is the convex hull of all scheules, i.e., Λ = co(γ ). Thus, µ Λ. C. Queue Structure an Shaow Queue Algorithm Each noe n maintains a set of counters, which are calle shaow queues, p ln for each previous hop l an each estination, an p 0n for external flows estine for at noe n. Each noe n also maintains a real queue, enote by q lnj, for each previous hop l an each next-hop neighbor j, an q 0nj for external flows with their next hop j. By solving the optimization problem with flow conservation constraints, we can work out the back-pressure algorithm for network coing case (see the brief escription in Appenix B). More specifically, for each link (nj) L in the network an for each estination, efine the back-pressure at every slot to be wnj [t] = max l:(ln) L or l=0 w lnj[t] where wlnj [t] = p ln[t] p nj [t] M, an lnj [t] = arg max l:(ln) L or l=0 w lnj[t]. (20) For each broacast at noe n to noes j an l estine for an, respectively, efine the back-pressure at every slot to be w, n jl [t] = w lnj[t] + w jnl[t]. (21) The weights associate with each point-to-point link (nj) L an each broacast link (n jl) are efine as follows w nj [t] = max {w nj[t]}, w n jl [t] = max n jl [t]},, {w, with nj [t] = arg max {w nj[t]}, {, } n jl [t] = arg max n jl [t]}., {w, The rate vector µ [t] at each time slot is chosen to satisfy { µ [t] arg max µ nj,pp w nj [t] µ Γ + } µ n jl w n jl [t]. (n jl) B (22)

9 9 By running the shaow queue algorithm in network coing case, we get a set of activate links in L at each slot. Next we escribe the evolution of the shaow queue lengths in the network. Notice that the shaow queues at each noe n are istinguishe by their previous hop l an their estination, so p ln only accepts the packets from previous hop l with estination. The similar rule shoul be followe when packets are raine from the shaow queue p ln. We assume the epartures occur before arrivals at each slot, an the evolution of queues is given by [ p ln [t + 1] = p ln [t] µ nj,pp[t]i {l=l nj,= nj } j N ] + µ n jl [t]i {{, }={, } n jl } N j N + k N + k N ˆµ kln[t]i {k=l ln,= ln } (23) N ˆµ, l nk [t]i {{, }={, } l nk } + â f [t]i {b(f)=n,e(f)=,l=0}, where ˆµ kln [t] is the actual number of shaow packets scheule over link (ln) an estine for from the shaow queue p kl at slot t, ˆµ, l nk [t] is the actual number of coe shaow packets transfere from noe l to noes n an k estine for noes an at slot t, an â f enotes the actual number of shaow packets from external flow f receive at noe n estine for. D. Implementation Details The implementation etails of the joint aaptive routing an coing algorithm are similar to the case with aaptive routing only, but the notation is more cumbersome. We briefly escribe it here. 1) Probabilistic Splitting Algorithm: The probabilistic splitting algorithm chooses the next hop of the packet base on the probabilistic routing table. Let Plnj [t] be the probability of choosing noe j as the next hop once a packet estine for receives at noe n from previous hop l or from external flows, i.e., l = 0 at slot t. Assume that Plnj [t] = 0 if (nj) L. Obviously, j N P lnj [t] = 1. Let σ lnj [t] enote the number of potential shaow packets transferre from noe n to noe j estine for whose previous hop is l uring time slot t. Notice that the packet comes from an external flow if l = 0. Also notice that σlnj [t] is contribute by shaow traffic point-to-point transmission as well as shaow traffic broacast transmission, i.e., σlnj[t] = µ nj,pp [t]i {l=lnj [t],= nj [t]} + µ n jl [t]i {{, }={, } n jl [t]}. N We keep track of the the average value of σlnj [t] across time by using the following upating process: ˆσ lnj[t] = (1 β)ˆσ lnj[t 1] + βσ lnj[t], (24) where 0 β 1. The splitting probability Plnj [t] is expresse as follows: Plnj[t] ˆσ lnj = [t] (25) k N ˆσ lnk [t]. 2) Token Bucket Algorithm: At each noe n, for each previous-hop neighbor l, next-hop neighbor j an each estination, we maintain a token bucket rlnj. At each time slot t, the token bucket is ecremente by σlnj [t], but cannot go below the lower boun 0 : r lnj[t] = max{r lnj[t 1] σ lnj[t], 0}. When rlnj [t 1] σ lnj [t] < 0, we say σ lnj [t] r lnj [t 1] tokens (associate with bucket rlnj ) are waste in slot t. Upon a packet arrival from previous hop l at noe n for estination at slot t, we fin the token bucket rlnj which has the smallest number of tokens (the minimization is over next-hop neighbors j), breaking ties arbitrarily, a the packet to the corresponing real queue q lnj, an a one token from the corresponing bucket: E. Extra link Activation rlnj [t] = r lnj [t] + 1. Like the case without network coing, extra link activation can reuce elays significantly. As in the case without network coing, we a aitional links to the scheule base on the queue lengths at each link. For extra link activation purposes, we only consier point-to-point links an not broacast. Thus, we scheule aitional point-to-point links by giving priority to those links with larger queue backlogs. VIII. SIMULATIONS We consier two types of networks in our simulations: wireline an wireless. Next, we escribe the topologies an simulation parameters use in our simulations, an then present our simulation results. A. Simulation Settings 1) Wireline Setting: The network shown in Figure 4 has 31 noes an represents the GMPLS network topology of North America [27]. Each link is assume to be able to transmit one packet in each slot. We assume that the arrival process is a Poisson process with parameter λ, an we consier the arrivals that come within a slot are consiere for service at the beginning of the next slot. Once a packet arrives from an external flow at a noe n, the estination is ecie by probability mass function ˆP n, = 1, 2,...N, where ˆP n is the probability that a packet is receive externally at noe n estine for. Obviously, : n ˆP n = 1, an ˆP nn = 0. The probability ˆP n is calculate by ˆP n = J + J n (J k + J n ), k:k n where J n enotes the number of neighbors of noe n. Thus, we use ˆP n to split the incoming traffic to each estination base on the egrees of the source an the estination.

10 10 Fig. 4. Sprint GMPLS network topology of North America with 31 noes.[27] algorithm, we choose ε = Figure 6 shows elay as a function of the arrival rate lamba for the three algorithms. As can be seen from the figure, simply using a value of M > 0 oes not help to reuce elays without extra link activation. The reason is that, while M > 0 encourages the use of shortest paths, links with back-pressure less than M will not be scheule an thus can contribute to aitional elays. Because we exaggerate the shaow traffic by a factor of ε, the throughput region of the algorithm without extra link activation is smaller than the throughput region of the traitional back-pressure algorithm. 2) Wireless Setting: We generate a ranom network with 30 noes which resulte in the topology in Figure 5. We use the following proceure to generate the ranom network: 30 noes are place uniformly at ranom in a unit square; then starting with a zero transmission range, the transmission range was increase till the network was connecte. We assume that each link can transmit one packet per time slot. We assume a 2-hop interference moel in our simulations. By a k-hop interference moel, we mean a wireless network where a link activation silences all other links which are k hops from the activate link. The packet arrival processes are generate using the same metho as in the wireline case. We simulate two cases given the network topology: the no coing case an the network coing case. In both wireline an wireless simulations, we chose β in (13) to be 0.02, an we use probabilistic splitting algorithm for simulations except Figure 12. Fig. 6. The impact of the parameter M an extra link activation in Sprint GMPLS network topology. We also compare the elay performance of PARN with that of the shortest path routing in Figure 7. For each pair of source an estination, we fin a shortest path between them by using Dijkstra s algorithm. When the arrival rate λ < 0.38, the ifference between the average packet elays of PARN an the shortest path routing is very small. This implies that PARN can obtain similar elay performance as the shortest path routing at light traffic. However, the shortest path routing can only achieve about 60% of the capacity region of the network. Fig. 5. Wireless network topology with 30 noes. B. Simulation Results 1) Wireline Networks: First, we compare the performance of three algorithms: the traitional back-pressure algorithm, the basic shaow queue routing/scheuling algorithm without the extra link activation enhancement an PARN. Without extra link activation, to ensure that the real arrival rate at each link is less than the link capacity provie by the shaow Fig. 7. The elay performance of PARN an shortest path routing. Next, we stuy the impact of M on the performance on

11 11 PARN. Figure 8 shows the elay performance for various M with extra link activation in the wireline network. The elays for ifferent values of M (except M = 0) are almost the same in the light traffic region. Once M is sufficiently larger than zero, extra link activation seems to play a bigger role, than the choice of the value of M, in reucing the average elays. The wireline simulations show the usefulness of the PARN algorithm for aaptive routing. However, a wireline network oes not capture the scheuling aspects inherent to wireless networks, which is stuie next. elay (ms) Wireless 30 Noes, 2 Hop Interference Moel, No Coing No Coing M=0 No Coing M=2 No Coing M=4 No Coing M= lamba (packets/slot) Fig. 9. Packet elay as a function of λ uner PARN in the wireless network uner 2-hop interference moel without network coing. region of the no coing case), network coing increases elays slightly. We believe that this is ue to fact that packets are store in multiple queues uner network coing at each noe: for each next-hop neighbor, a queue for each previous-hop neighbor must be maintaine. This seems to result in slower convergence of the routing table. Fig. 8. Packet elay as a function of λ uner PARN in Sprint GMPLS network topology. 2) Wireless Networks: In the case of wireless networks, even with extra link activation, to ensure stability even when the arrival rates are within the capacity region, we nee ε > 0. We chose ε = 0.1 in our simulations ue to reasons mentione in Section VI. In Figure 9, we stuy wireless networks without network coing. From the figure, we see that the elay performance is relatively insensitive to the choice of M as long as it is sufficiently greater than zero. However, M oes play an important role because it suppresses the search of long paths when the traffic loa is not high. Extra link activation can be use to ecrease elays significantly for M > 0 especially in light traffic region. In Figures 10 an 11, we show the corresponing results for the case where both aaptive routing an network coing are use. Comparing Figures 9 an 10, we see that, when use in conjunction with aaptive routing, network coing can increase the capacity region. We make the following observation regaring the case M = 0 in Figure 11: in this case, no attempt is mae to optimize routing in the network. As a result, the elay performance is very ba compare to the cases with M > 0 (Figure 10). In other wors, network coing alone oes not increase capacity sufficiently to overcome the effects of back-pressure routing. On the other han, PARN with M > 0 harnesses the power of network coing by selecting routes appropriately. Next, we make the following observation about network coing. Comparing Figures 10 an 11, we notice that at moerate to high loas (but when the loa is within the capacity elay (ms) Wireless 30 Noes, 2 Hop Interference Moel, Network Coing Network Coing M=2 Network Coing M=4 Network Coing M= lamba (packets/slot) Fig. 10. Packet elay as a function of λ uner PARN for M > 0 in the wireless network uner 2-hop interference moel with network coing. Finally, we stuy the performance of the probabilistic splitting algorithm versus the token bucket algorithm. In our simulations, the token bucket algorithm runs significantly faster, by a factor of 2. The reason is that many more calculations are neee for the probabilistic splitting algorithm as compare to the token bucket algorithm. This may have some implications for practice. So, in Figure 12, we compare the elay performance of the two algorithms. As can be seen from the figure, the token bucket an probabilistic splitting algorithms result in similar performance. Therefore, in practice, the token bucket algorithm may be preferable. IX. CONCLUSION The back-pressure algorithm, while being throughputoptimal, is not useful in practice for aaptive routing since the

12 12 Fig. 11. Packet elay as a function of λ uner PARN for M = 0 in the wireless network uner 2-hop interference moel with network coing. Fig. 12. Comparison of probabilistic splitting an token bucket algorithms uner PARN in the wireless network uner 2-hop interference moel without network coing. elay performance can be really ba. In this paper, we have presente an algorithm that routes packets on shortest hops when possible, an ecouples routing an scheuling using a probabilistic splitting algorithm built on the concept of shaow queues introuce in [5], [3]. By maintaining a probabilistic routing table that changes slowly over time, real packets o not have to explore long paths to improve throughput, this functionality is performe by the shaow packets. Our algorithm also allows extra link activation to reuce elays. The algorithm has also been shown to reuce the queueing complexity at each noe an can be extene to optimally trae off between routing an network coing. APPENDIX A THE STABILITY OF THE NETWORK UNDER PARN Our stability result uses the result in [26] an relies on the fact that the arrival rate on each link is less than the available capacity of the link. We will now focus on the case of wireless networks without network coing. All variables in this appenix are assume to be average values in the stationary regime of the corresponing variables in the shaow process. Let σ nj enote the mean shaow traffic rate at link (nj) estine to. Let µ nj an αn(1 + ε) enote the mean service rate of link (nj) an the exogenous shaow traffic arrival rate estine to at noe n. Notice that ε comes from our strategy on shaow traffic. The flow conservation equation is as follows: αn(1 + ε) + σ ln = σ nj, n, N. (26) l:(ln) L j: The necessary conition on the stability of shaow queues are as follows: σ nj µ nj. (27) N Since we know that the shaow queues are stable uner the shaow queue algorithm, the expression (27) shoul be satisfie. Now we focus on the real traffic. Suppose the system has an equilibrium istribution an let λ nj be the mean arrival rate of real traffic at link (nj) estine to. The splitting probabilities are expresse as follows: P nj = σ nj, where n. (28) k N σ nk Thus, the mean arrival rates at a link satisfy traffic equation: λ nj = αn P nj + λ P ln nj, (nj) L, N, (29) l:(ln) L where n. The traffic intensity at link (nj) is expresse as: ρ nj = 1 λ µ nj. (30) nj N Now we will show ρ nj < 1 for any link (nj) L. Let λ nj = σ nj /(1 + ε) for every (nj) L, an substitute it into expression (29). It is easy to check that the caniate solution is vali by using expression (26). From (27), the traffic intensity at link (nj) is strictly less than 1 for any link (nj) L : ρ nj = 1 λ 1 nj = σ nj < 1. (31) µ nj (1 + ε) µ nj N N Thus we have shown that the traffic intensity at each link is strictly less than 1. The wireline network is a special case of a wireless network. Substitute the link capacity c nj for µ nj an set ε to be zero, an stability follows irectly. The stability of wireless networks with network coing is similar to the case of wireless network with no coing. APPENDIX B THE BACK-PRESSURE ALGORITHM IN THE NETWORK CODING CASE Given a set of packet arrival rates that lie in the capacity region, our goal is to fin routes for flows that use as

13 13 few resources as possible. Thus, we formulate the following optimization problem for the network coing case. min µ nj,pp + µ (n jl) (32) s.t. (nj) L (n jl) L µ nj,pp + µ nj,broa k x f I {b(f)=n,e(f)=} µ njk + j N k:k n µ fnj µ, j kn Let {q nj } an {q 0n } be the Lagrange multipliers corresponing to the flow conservation constraints in problem (32). Appening the constraints to the objective, we get min µ nj,pp + [ q nj µ Λ (nj) L µ n jl + (n jl) L (nj) L µ nj,pp + µ nj,broa µ njk ] µ, j kn (33) k k:k n + [ q 0n x f I {b(f)=n,e(f)=} ] µ fnj n, j N ( = min µ ( µ Λ lnj qln q nj 1 ) (nj) L l:(ln) L ( qln q nj + q jn q nl 2 ) µ, n jl (n jl) L,j<l, (nj) L + n, µ ( fnj q0n q nj 1 ) q 0n x f I {b(f)=n,e(f)=} ). If the Lagrange multipliers are known, then the optimal µ can be foun by solving max µ nj,pp w nj + µ n jl w n jl (34) where µ Λ (nj) L (n jl) L,j<l w nj = max {w nj}, w n jl = max, {w, n jl }, w, n jl = w lnj + w jnl w nj = max l:(ln) L or l=0 w lnj w lnj = q ln q nj 1. Similar to the upate algorithm of q n in (7), we can erive the upate algorithm to compute q nj : [ q nj [t + 1] = q nj [t] + 1 ( µ M nj,pp + µ nj,broa µ njk ) µ, j kn (35) k k:k n + 1 ( x f I {b(f)=n,e(f)=} µ ) ] + fnj M j N By choosing 1 M to be the step-size parameter, Mq nj looks very much like a queue upate equation. Replacing Mq nj by p nj, we get (20)-(23). It can be shown using the results in [21], [22] that the stochastic version of the above equations are stable an that the average rates can approximate the solution to (32) arbitrarily closely. REFERENCES [1] L. Tassiulas an A. Ephremies, Stability properties of constraine queueing systems an scheuling policies for maximum throughput in multihop raio networks, IEEE Transactions on Automatic Control, pp , December [2] M. J. Neely, E. Moiano, an C. E. Rohrs, Dynamic power allocation an routing for time varying wireless networks, IEEE Journal on Selecte Areas in Communications, vol. 23, no. 1, pp , January [3] L. Bui, R. Srikant, an A. L. Stolyar, Novel architectures an algorithms for elay reuction in back-pressure scheuling an routing, in Proceeings of IEEE INFOCOM Mini-Conference, April [4], A novel architecture for elay reuction in the back-pressure scheuling algorithm, IEEE/ACM Trans. Networking, vol. 19, no. 6, pp , December [5], Optimal resource allocation for multicast flows in multihop wireless networks, Philosophical Transactions of the Royal Society, Ser. A, vol. 366, pp , [6] L. Ying, S. Shakkottai, an A. Rey, On combining shortest-path an back-pressure routing over multihop wireless networks, in Proceeings of IEEE INFOCOM 2009, April [7] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Mear, an J. Crowcroft, XORs in the air: Practical wireless network coing, in ACM SIG- COMM Computer Communication Review, vol. 36, 2006, pp [8] M. Effros, T. Ho, an S. Kim, A tiling approach to network coe esign for wireless networks, in Information Theory Workshop, [9] H.Seferoglu, A.Markopoulou, an U.Kozat, Network coing-aware rate control an scheuling in wireless networks, in Special Session on Network Coing for Multimeia Streaming, ICME, Cancun, Mexico, June [10] S. B. S. Sengupta, S. Rayanchu, An analysis of wireless network coing for unicast sessions: The case for coing-aware routing, in Proc. IEEE INFOCOM, Anchorage, Alaska, May [11] T. Ho an H. Viswanathan, Dynamic algorithms for multicast with intra-session network coing, IEEE Transactions on Information Theory, February [12] A. Eryilmaz an D. S. Lun, Control for inter-session network coing, in Proceeings of the Workshop on Network Coing, Theory an Applications (NetCo), January [13] L. Chen, T. Ho, S. H. Low, M. Chiang, an J. C. Doyle, Optimization base rate control for multicast with network coing, in Proc. IEEE INFOCOM, Anchorage, Alaska, May [14] B. Awerbuch an T. Leighton, A simple local-control approximation algorithm for multicommoity flow, in Proc. 34th Annual Symposium on the Founations of Computer Science, [15] A. Dimakis an J. Walran, Sufficient conitions for stability of longest-queue-first scheuling: Secon-orer properties using flui limits, Avances in Applie Probability, June [16] C. Joo, X. Lin, an N. B. Shroff, Unerstaning the capacity region of the greey maximal scheuling algorithm in multi-hop wireless networks, in Proc. IEEE INFOCOM, [17] A. Brzezinski, G. Zussman, an E. Moiano, Enabling istribute throughput maximization in wireless mesh networks - a partitioning approach, in Proc. ACM Mobicom, Sep [18] M. Leconte, J. Ni, an R. Srikant, Improve bouns on the throughput efficient of greey maximal scheuling in wireless networks, in Proc. ACM MobiHoc, [19] B. Li, C. Boyaci, an Y. Xia, A refine performance characterization of longest-queue-first policy in wireless networks, in Proc. ACM MobiHoc, [20] X. Lin an N. Shroff, On the stability region of congestion control, in Proceeings of the Allerton Conference on Communications, Control an Computing, [21] M. J. Neely, E. Moiano, an C. Li, Fairness an optimal stochastic control for heterogeneous networks, in Proceeings of IEEE INFO- COM, [22] A. L. Stolyar, Maximizing queueing network utility subject to stability: Greey primal-ual algorithm, Queueing Systems, vol. 50, no. 4, pp , 2005.

14 14 [23] A. Eryilmaz an R. Srikant, Fair resource allocation in wireless networks using queue-length-base scheuling an congestion control, in Proceeings of IEEE INFOCOM, 2005, revise version to appear in IEEE/ACM Transactions on Networking. [24], Joint congestion control, routing an mac for stability an fairness in wireless networks, in Proc. International Zurich Seminar on Communications, [25] X. Lin, N. Shroff, an R. Srikant, A tutorial on cross-layer optimization in wireless networks, IEEE Journal on Selecte Areas in Communications, [26] M. Bramson, Convergence to equilbria for flui moels of FIFO queueing networks, Queueing Systems: Theory an Applications, vol. 22, pp. 5 45, [27] Sprint IP network performance, available at R. Srikant (S 90-M 91-SM 01-F 06) receive his B.Tech. from the Inian Institute of Technology, Maras in 1985, his M.S. an Ph.D. from the University of Illinois in 1988 an 1991, respectively, all in Electrical Engineering. He was a Member of Technical Staff at AT&T Bell Laboratories from 1991 to He is currently with the University of Illinois at Urbana-Champaign, where he is the Freric G. an Elizabeth H. Nearing Professor in the Department of Electrical an Computer Engineering, an a Research Professor in the Coorinate Science Lab. He was an associate eitor of Automatica, the IEEE Transactions on Automatic Control, an the IEEE/ACM Transactions on Networking. He has also serve on the eitorial boars of special issues of the IEEE Journal on Selecte Areas in Communications an IEEE Transactions on Information Theory. He was the chair of the 2002 IEEE Computer Communications Workshop in Santa Fe, NM an a program co-chair of IEEE INFOCOM, His research interests inclue communication networks, stochastic processes, queueing theory, information theory, an game theory. Eleftheria Athanasopoulou (M 02) receive her Diploma egree in Electrical an Computer Engineering from the University of Patras in 2000 an her M.S. an Ph.D. egrees in Electrical an Computer Engineering from the University of Illinois at Urbana-Champaign in 2002 an 2007, respectively. She was then a post-octoral research associate at the University of Illinois at Urbana-Champaign an the Coorinate Science Lab. Her research interests inclue wireline an wireless communication networks, stochastic moels, iscrete event systems, an failure iagnosis of ynamic systems an networks. Loc X. Bui receive the B.Eng. egree in Electronics an Telecommunications from the Posts an Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam, in 2003, an the M.S. an Ph.D. egrees in Electrical an Computer Engineering from the University of Illinois at Urbana-Champaign in 2006 an 2008, respectively. From October 2008 to March 2010, he was with Airvana Inc., where he was a Senior Software Engineer. an then a Senior Sustaining Engineer. From April 2010 to September 2011, he was a Postoctoral Scholar in the Department of Management Science an Engineering, Stanfor University. From October 2011 to January 2012, he was a Visiting Fellow in the Department of Electrical Engineering, Technion - Israel Institute of Technology. He is currently a Lecturer of Electrical Engineering in the School of Engineering, Tan Tao University. His research interests inclue communication networks, wireless communications, game theory, an machine learning. Tianxiong Ji (M 07) receive his B.Eng. an M.S. egrees in Electrical Engineering from Tsinghua University, Beijing, China in 2005 an 2007, respectively an his Ph.D. in Electrical an Computer Engineering from the University of Illinois at Urbana-Champaign in He has been working at Google Inc. since December His research interests inclue wireless networks, queuing theory, ata center networks, an wireless communication. Alexaner Stolyar is a Distinguishe Member of Technical Staff in the Inustrial Mathematics an Operations Research Department of Bell Labs, Alcatel-Lucent, Murray Hill, New Jersey. He receive Ph.D. in Mathematics from the Institute of Control Sciences, USSR Acaemy of Science, Moscow, in Before joining Bell Labs in 1998, he was with the Institute of Control Sciences (Moscow), Motorola (Arlington Heights, IL) an AT&T Labs-Research (Murray Hill, NJ). His research interests are in stochastic processes, queueing theory, an stochastic moeling of communication an service systems. He is an associate eitor of Operations Research; Queueing Systems - Theory an Applications; an Avances in Applie Probability.

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