Network Coding in Wireless Networks with Random Access

Transcription

1 ISIT2007, Nie, Frane, June 24 - June 29, 200O Network Coding in Wireless Networks with Random Aess Danail Traskovt, Desmond S. Lun, Ralf Koettert, and Muriel Medardt Broad Institute MIT and Harvard Cambridge, MA dlun@broad.mit.edu tcoordinated Siene Laboratory University of Illinois at Urbana-Champaign Urbana, IL {traskov2, koetter}@uiu.edu +LIDS MIT Cambridge, MA medard@mit.edu Abstrat- We onsider the problem of applying network oding in wireless networks with random medium aess. To optimize the network oding subgraph and the transmission attempt probabilities jointly is a tratable problem only for rather small networks. Therefore, we suggest a suboptimal, yet pratial and deentralized algorithm to ombine network oding with random aess. We illustrate the performane gains of our approah with simulations. I. INTRODUCTION The problem we onsider is to establish a set of multiast or uniast onnetions in a wireless network using network oding. The utility of network oding in suh a setup has been demonstrated e.g. in [1], [2]. There, the authors develop a mathematial framework for omputing a network oding subgraph that an support the desired onnetions and provide simulation results showing signifiant advantages of a network oded approah. As they point out, to inlude the interferene onstraints of the wireless network in the optimization model may pose quite a hallenge. In fat, it is the struture of the medium aess onstraints of the network that essentially determines the diffiulty of the resulting optimization problem. Coneptually, one an view the overall problem, of establishing a set of onnetions in a wireless network with interferene, as two oupled subproblems. Resoure provision (or sheduling) deals with the supply of network resoures suh as interferene-free sets of links, whereas the onern of resoure alloation is how to assign the available resoures to different users. For resoure provision one an onsider a sheduled approah whih effetively eliminates interferene or, on the other hand, an ALOHA-like random medium aess mehanism. For the resoure alloation problem possible hoies are routing as opposed to network oding. The authors in [1], [2] fous on the resoure alloation aspet, assuming that nodes in the network do not experiene interferene. The wireless network is regarded as essentially interferene free, due to an underlying shedule. However, to bring network oding one step loser to pratial implementation, it is neessary to investigate the interplay between these by no means independent problems. The fous of our work is to ombine the network oding subgraph optimization with the resoure provision aspet of the overall problem. This work was supported by the Control-Based Mobile Ad-Ho Networking (CBMANET) Program under DARPA subontrat no , by the National Siene Foundation under grant no. CCR , and by AFOSR. The authors would like to thank R. Srikant his valuable input /07/$25.00 (.)2007 IEEE 2726 The joint resoure provision and alloation strategy we will investigate is random, linear, intra-session network oding ombined with a slotted-aloha random medium aess sheme. To be more preise, network oding takes plae only within the set of pakets of a partiular user (i.e. intra-session) and is applied by forming linear ombinations of pakets with random oeffiients at intermediate nodes of the subgraph. This sheme is easy to implement, it ahieves the apaity for a single onnetion [3], and has a simple flow optimization formulation [4]. Given multiple onnetions, it is reasonable to ask why we restrit ourselves from using network oding ombining pakets of different users (i.e. inter-session). There exists a variety of tehniques for inter-session network oding, suh as [5]-[8], but these are quite omplex and therefore signifiantly further down the road of pratial implementation. In ontrast COPE [9] is a pratial and suessfully implemented network oding protool. However, our fous is on developing mathematial models in terms of optimization of flows and the opportunisti and heuristi nature of COPE is diffiult to apture in suh a framework. Although we use solely intrasession network oding, it is important to keep in mind that, in general, inter-session network oding is neessary to approah the apaity of the network and it is reasonable to expet further gains by applying inter-session oding. At the medium aess layer of the network, we use a random slotted-aloha-like mehanism. The network is assumed to operate in time slots and, for every slot, a node is assigned some transmission attempt probability. We assume that in every time slot nodes deide to transmit with these fixed probabilities but otherwise independently of eah other. Random aess is an important medium aess tehnique used in various appliations (see e.g. [10]). One main advantage lies in the simpliity, sine just deiding, whether a vetor of link flows an be supported by a ollision-free transmission shedule is an NP-omplete problem [11]. Furthermore, if nodes are assumed to be mobile, frequent hanges in the network topology have to be taken into aount and a low-omplexity random aess approah is expeted to ope better with the updates due to mobility. Furthermore, our subgraph seletion proedure an be ombined with a sheduled approah in a fairly straightforward fashion. The problem of ross-layer network oding and random aess is also treated in [12], [13]. As it turns out, the problem of finding an optimal network oding subgraph subjet to the random aess interferene onstraints is too diffiult to be tratable in pratial networks

2 'v ISIT2007, Nie, Frane, June 24 - June 29, 2007 Fig. 1. An example of a hypergraph. Here, the node set is JA and the hyperar set is A = {(1, {2, 3}), (2, 3)}. of moderate size. This is largely due to the non-onvexity of the medium aess rate region. Furthermore, in [14], [15] the authors have looked at the problem of ommuniating in multiple aess networks from an information theoreti perspetive and reah similar onlusions regarding the diffiulty of the general problem. We therefore suggest a relaxation of the original problem, whih is suboptimal, yet pratially implementable. Our algorithm an be arried out in a distributed fashion, a highly desirable property in pratial networks, and therefore an serve as the baseline of a protool. The remainder of this paper is organized as follows: In Setion II, we introdue the wireless network model and the network oding sheme. In Setion III, we formulate the joint network oding and random aess problem and disuss its omplexity. Setion IV suggests a relaxation of this problem ombined with an iterative proedure to verify feasibility and assign transmission attempt probabilities. In Setion V, we evaluate the performane gains of our algorithm by means of simulations. Setion VI onludes the paper. II. NETWORK MODEL The wireless network (following the model from [4]) is represented by a hypergraph X = (JV, A), where AV stands for the set of nodes and A is the set of hyperars. A hyperar, our approah to model a broadast hannel, is a pair (i, J), where i is a node and J is a nonempty subset of the node set AV. An example of a hypergraph is depited in Fig. 1. If node i injets a paket on hyperar J it is reeived by some subset K C J, possibly K being the empty set 0. Let Ai (T) be the ounting proess desribing paket injetions on hyperar J and AiJK (T) be the ounting proesses aounting for the pakets reeived preisely by the subsets K. Obviously, we have EKCJ AiJK(T) = Aij(T). We assume that for the injetion proesses time averages exist, i.e. limt"q Ai(T) exists with probability 1, is finite, and equals Zij. Similarly, we define limto AiJK(T) = ZiJK- With these assumptions T Zij = EKCJ ZiJK is the average paket injetion rate on hyperar J. In the remainder of the paper we will assume that ZiJK is proportional to zij and define ZiJK PiJK ZiJ to be the probability that a paket injeted on J is reeived preisely by the subset K. We all the rate vetor (ZiJ)(i,J)C, the network oding subgraph. One the subgraph is omputed our network oding tehnique of hoie will be the well known random network oding [3], [4], some more pratial aspets of whih an be found in [16]. Roughly, that means that a node stores reeived paket 2727 in its memory and upon an opportunity transmits a linear ombination of the stored pakets with oeffiients drawn uniformly from a finite field. Coding is restrited to pakets belonging to the same user (intra-session). {1, 2, 3} III. THE JOINT NETWORK CODING AND RANDOM ACCESS OPTIMIZATION PROBLEM We onsider C multiast sessions, where a multiast session (s, T, R), C 1,... C is haraterized by its soure s, its destination set TC C AV and rate R. Every node t e TC reeives information at the ommon rate R. We seek a network oding subgraph that is able to support these requirements and that is subjet to the multiple aess onstraints of the network. This an be formulated as an optimization problem and translates into the following onstraint set [4] vjkijca J I(i,J)EE,4 E ii < zij, V (i, J) C A, (2) S t'j) < Yij ijk, jek V (i,j) A, K J, tct, C= 1,...,C, v(t,) jeej where we define iaj\f\t}, t Z j (t,) (RC {jt,ia E 1- i (i,i)ea,igi11 0 C T,v = I,...,C, SCj >, 0, V (i, J) C A, j C J, (zij) C Z, V (i, J) C A, (3) t = SC, i = t, else, (4) (5) (6) bijk E PiJL- {LCJILnK 0} Here, bijk stands for the probability that a paket injeted on hyperar (i, J) is reeived by at least one node in K. As disussed Zij is the paket injetion rate on hyperar is the rate alloated for multiast session on (i, J), and y() ths ypra.th vribl (t,) this hyperar. The variable xij node i, the soure of a hyperar, to a node j e J in the destination set of the hyperar. We will therefore, slightly abusing terminology, refer to a variable of this type as a link flow. Constraint (3) defines the feasible rate region of link flow alloations for a hyperar (i, J). Constraints (2)-(5) represents the flow between relate the network oding subgraph (zij) and the link flows (t,) xij. The network oding subgraph is restrited within some onstraint set Z whih is ditated by the medium aess layer of the network. It is the struture of this set that essentially determines the diffiulty of the optimization problem.1 We onsider a random aess mehanism, where in every time slot a node makes a transmission attempt determined by a Bernoulli random variable. Nodes make the transmission 'As we will show in the next setion the onstraints (2)-(5) alone if ombined with a onvex ost funtion define a fairly tratable onvex optimization problem.

3 ISIT2007, Nie, Frane, June 24 - June 29, 200O deisions independent of eah other with the same probability over different slots. We assume that the time granularity of the system is suffiiently high and do not use a bakoff mehanism, suh as binary exponential bakoff. We make no assumptions on the statistial properties of the traffi in the network exept that average rates exist. This is without loss of generality, sine in [17] it is shown that bursty traffi does not derease the apaity region of the multiple aess hannel. To formulate the onstraint set under a random medium aess poliy, let qijj be the probability that node i plaes a paket on hyperar J whih is intended to be reeived by node j e J. In the following we will make the assumption that every node has preisely one outgoing hyperar, whih is not a major restrition and serves mainly to streamline notation. We will therefore write qij suppressing the hyperar index. Let Qi = Eqij (7) jcj be the probability that node i transmits in a slot. If node i transmits a paket on its outgoing hyperar J then it will be reeived by node j C J if and only if 1) it is not lost due to an erasure, 2) node j is not transmitting itself, and 3) none of the nodes with outgoing hyperars ontaining j transmit. As a onsequene of this interferene model for every hyperar (i, J) the link throughputs fij, for all (i,j), j, must be smaller than the apaity where the maximum is taken over the link flows required to serve the different sinks of one multiast session and the summation implies that the traffi demands due to different multiast sessions ontribute additively to the overall network flow2. This is due to the fat that we ode only within one session and different multiast onnetions share the network without network oding ooperation aross them. The medium aess onstraint (6) thus beomes C S maxxt') < q. (I QH)II (1 kgkj\i -Qk) This is an intriguing losed form expression for the problem of jointly seleting a network oding subgraph and assigning optimal transmission attempt probabilities. Unfortunately, it is very diffiult to solve for general networks exept for very small ones. This is due to the non-onvexity of the 2Stritly speaking, (9) as it stands holds in the ase where the network does not experiene erasures. However, to take these into aount is straightforward as we merely have to sale the link flows with a proper set of oeffiients depending on the loss probabilities. the fij (.) 2728 onstraint set whih, to the best of our knowledge, annot be overome with onvexifying transformations. In an apparently similar setting, the authors in [18] have suessfully applied a logarithmi transformation to exploit the fat that the RHS of (10) is log-onave (i.e. its logarithm is onave). However, they onsider joint medium aess and end-to-end rate ontrol only. We on the other hand are also interested in finding a subgraph to ode over. In this ase, this and related well known transformations fail to yield a onvex problem formulation. IV. A RELAXATION OF THE JOINT OPTIMIZATION PROBLEM The joint optimization problem is very diffiult and intratable for all but rather small network instanes. We therefore suggest a simple relaxation that is easily implementable not only with a manageable amount of omplexity but also in a distributed fashion. To that end, we drop the random aess onstraints and solve the remaining optimization problem whih is the seletion of the network oding subgraph. One we have omputed the required link throughputs fij to support the given demands, we an use the iterative proedure from [10] to either find transmission attempt probabilities qij supporting the desired link throughputs or to rule out their feasibility. In ase we enounter an infeasible set of demands, we redue the rates (aording to some heuristi rule), reompute the required link throughputs and hek again for feasibility. Consider to that end the following linear program, where for brevity we have omitted the qualifying onditions: ij = qij(i - Qj) 1 (1 - Qk), (8) minimize A ZiJ kkjk\i (i, J)GA where Ki C AV is the set of all nodes whose hyperars ontain subjet to node j. The link throughput fij between node i and j is x i< < zij, C=l fi; = E: max x(tjj): (9) E ti') ()iyi(j) < bijk, jek S S (t,) {JI(i,J)EA4 jej (RC Z( t,) j E zjtsic t- {ii (j,i)ea,igl 1 0 (t, - R (1 1) (12) t = s, i = t, else, (13) (14) The ost funtion we have hosen to minimize is the total number of transmissions neessary to support a given rate, however any onvex separable funtion of the network oding subgraph3 an easily be aommodated in our framework. This optimization problem has a deentralized solution. As shown in [19], for a single session, the multiast subgraph seletion an be arried out in a distributed fashion. Under our assumptions on the ost funtion and the relaxed onstraint set it is readily seen that minimizing Z(iJ), Zij is equivalent to minimizing jc7 Z.j) Yi( ( Furthermore, the different 3That means a funtion of the form f(z) =(i,j)v4 fij(zij). are inreasing and onvex funtions. where

4 Lt ISIT2007, Nie, Frane, June 24 - June 29, 2007 multiast sessions are not oupled through any onstraints, whih implies that a solution to the subgraph optimization problem an be obtained by solving the different sessions oneby-one and then adding the resulting link flows. Clearly, in this ase the distributed algorithm from [19] applies. After having omputed the network oding subgraph, we an determine the required link throughputs fij using (9). To find transmission attempt probabilities qij yielding the desired link throughputs we an use the iterative proedure from [10], whih onsists of the following steps: 1) Initialize: qij [O] = 0. 2) Perform the iteration: Qi [n] = 1: qij [n], Pij [n] = (I1- Qj [n] ) (]-Qk [n]), kg Kj \i Pij Inl] Path oding/sheduling ^ Our algorithm 0.2 Link-by-link retransm./random aess 0.18 o o: (15) Fig. 2. Maximal rate of a uniast onnetion as a funtion of the number of nodes. The number of neighbors is upper bounded by 5. (16) (17) If the iteration onverges, then it is guaranteed that the transmission attempt probabilities qij support the required link throughputs. If, on the other hand, at some point any of the Qi[n] exeeds 1 the link throughputs are infeasible. Then we have to redue the end-to-end rates, ompute a new subgraph and repeat the feasibility iteration loop. Note that at every step of this iteration all required variable updates an be performed with information gathered within a one hop neighborhood, implying that also this omponent of our algorithm is deentralized and, ombined with the above subgraph optimization, yields a fully distributed approah. Our algorithm is suboptimal sine we separated the original problem into a subgraph seletion problem and a multiple aess problem and solved them more or less independently. However, simulation results illustrate that our approah shows signifiant gains. This is the fous of the next setion. V. SIMULATION RESULTS To get an idea of the performane gains of our approah we ondut simulations over random network topologies. We assume that a number of nodes are uniformly sattered over a square region with unit node density. Two nodes are in radio range if their distane is below some threshold, the radius of onnetivity, whih we take to be 1.8. Transmissions are subjet to erasures whih may be due to distane attenuation and Rayleigh fading. When a node transmits another node in radio range at distane d will reeive the paket orretly if -yd2 > /3, where 'yis a unit mean exponential variable and Q = 1/4 is our hosen SNR threshold, otherwise the paket is lost ompletely. We also take into aount interferene by assuming nodes to operate half-duplex (i.e. they annot transmit and reeive at the same time) and furthermore if in a given time slot a node is in radio range of more than one transmitting neighbor then it annot reover any of the pakets Path oding/sheduling Our.. algorithm... :Link-by-link retransm./random aess 0.18 o, 16 'r 0.14 U E a Fig. 3. Maximal rate of a uniast onnetion as a funtion of the number of nodes. The number of neighbors is upper bounded by 6. In Fig. 2 and Fig. 3, we are interested in the maximal rate of a single point-to-point onnetion. The soure is taken to be the node with the smallest x-oordinate and the destination the one with the highest x-oordinate. The differene between the two figures is merely the limit on the number of neighbors, with Fig. 3 having slightly riher onnetivity. We simulate the following shemes: 1) Path oding and sheduling: The shortest path4 from soure to sink is hosen and nodes on the path employ a random linear ode (e.g. the oding sheme from [1]) to ompensate for erased pakets. This is a form of network oding. Alternatively, one ould employ a FECode on every link and let intermediate nodes deode and re-enode pakets. However, suh an approah seems prohibitive due to large delays espeially on longer paths. An optimal ollision-free shedule is used whih, for the speial ase of a single onnetion on a path rather than a subgraph, is not a diffiult task. 4See also [20] for an interesting approah using multiple paths.

5 ISIT2007, Nie, Frane, June 24 - June 29, F Ir E 0.09 E i 0.08 E LPath oding/sheduling *Our algorithm Fig. 4. Maximal ommon rate of four simultaneous uniast onnetion as a funtion of the number of nodes. The number of neighbors is upper bounded by 6. 2) Subgraph oding and random aess heuristi: This is our algorithm as desribed in Setions III and IV. Note, that in ontrast to the three other tehniques, we ode over a subgraph rather than over a path. 3) Path oding and random aess: The shortest path between soure and destination is hosen and the same oding sheme from the first approah is used on the path. However, instead of sheduling we use random aess with optimal transmission attempt probabilities. 4) Link-by-link retransmission and random aess: The shortest path between soure and destination is hosen and nodes are required to aknowledge reeived pakets. If a paket or an aknowledgement is lost a retransmission is triggered, however we do not expliitly use negative aknowledgements (NACK's). It is important to keep in mind that approahes (1)-(3) use network oding, while approah (4) does not. Therefore, although approah (3) operates on a path rather than on a subgraph it apitalizes already signifiantly on network oding gains. Approah (2) goes one step further by hoosing a subgraph rather than a path to ode over and therefore utilizes the overhearing of pakets by other neighbors, a phenomenon alled the wireless broadast advantage [19]. We observe, that our suboptimal network oding algorithm is able to lose muh of the gap between random aess and a sheduled approah. The gains are more pronouned in Fig. 3 where the network onnetivity is riher. Another experiment we onduted is to use the same network topology and setup simultaneously four uniast onnetions sharing the network. Fig. 4 shows the maximal ommon rate at whih they an transmit and, again, the urves illustrate the gains due to network oding. Together, these experiments give a ompelling reason to expet onsiderable advantages by using network oding. In fat, together with the simulation results in [2], [21], [22], it is reasonable to investigate further the appliation of network oding in wireless networks VI. CONCLUSION We have suggested a ombined network oding and random aess strategy whih shows signifiant performane gains in simulations. It an be implemented in a distributed fashion whih is a basi requirement if it is to be used as the basis of a protool. What is not yet lear in partiular is the delay harateristis of pakets in atual network operation. This is an important aspet in networking and requires more investigation. REFERENCES [1] D. S. Lun, M. Medard, R. Koetter, and M. Effros, "On oding for reliable ommuniation over paket networks," Tehnial report #2741, MIT LIDS, January [2] D. S. Lun, M. 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