The Role of Feedback in the choice between Routing and Coding for Wireless Unicast

Transcription

1 The Role of Feedbac in the choice between Routing and Coding for Wireless Unicast Raarishna Guadi CSL and ECE, University of Illinois, Urbana, USA Laurent Massoulie Thoson Corporate Research, Paris, France Raavarapu Sreenivas CSL and IESE, University of Illinois, Urbana, USA Abstract We consider the benefits of coding in wireless networs, specifically its role in exploiting the local broadcast property of the wireless ediu. We first provide a natural arguent that for unicast, the throughput achieved with networ coding is the sae as that achieved without any coding. This arguent highlights the role of a general ax-flow in-cut duality in such a result, and is ore explicit than previous proofs. This axiu throughput can be achieved either by a flow scheduler with nowledge of the networ topology, or by a blind bac-pressure algorith, however all such policies require dynaic routing decisions which depend critically on rich feedbac signalling inforation. We then investigate how feedbac at a single node affects its throughput to a destination, with fixed rates of its one hop neighbors to the destination. We first analyze static routing policies which are in a sense, feedbac independent. Under such a constraint, we obtain an explicit characterization of the optial policy, which turns out to be a natural generalization of both flooding and traditional routing. For independent losses at the receivers (still possibly allowing for dependencies for transitted by two different nodes to odel interference), the reduction is liited to a constant fraction of the capacity, and could even be arbitrarily close to optial depending on the networ. However, if there are dependencies in the losses seen by receivers fro a single broadcast, the reduction could be arbitrarily bad, even on a 2-hop networ. Our analysis provides several new insights on the scenarios in which coding in the networ can (and can not) be avoided and how it relates to the feedbac. I. INTRODUCTION Networ coding was originally introduced in [1] as a solution to achieve the optial ulticast rate fro a data source to a set of receivers in wired networs. In contrast to store-and-forward, also called routing operation, networ coding perfors recobinations of data pacets at networ nodes, while the forer operation never alters original pacets. Since then, any other applications of networ coding have been identified. In particular it has been considered in the context of wireless networs [3] for unicast counications, where such a wireless setting was identified as an especially good candidate for networ coding because of the local broadcast. The wireless scenario of [3] features lossy transissions (a drawbac of wireless) as well as local broadcast (an advantage of wireless). The experiental evaluations of [3] show benefits of networ coding over routing policies in ters of the transission rates achieved. This leaves soething to be explained because of its contrast with recent wor, [6] and [7], which iply that routing policies can achieve the capacity for unicast counications, in a large class of wireless networ odels. This begs the following question: are there any benefits of networ coding over routing in wireless unicast counications? If so, where do they ste fro, and how large can they be? In a nutshell, our analysis shows that the benefits of coding over routing are dependent on the richness of the feedbac signalling and the networ in question, and quantitatively argues about their critical relation. Our first contribution is a natural arguent that highlights the role of LP duality in understanding why the axiu routing throughput in the case of wireless unicast is the sae as the capacity achieved with coding. As we will see, this iplies that bacpressure routing achieves not only the optial throughput aong routing policies, but also ore generally across policies that involve coding. While being conceptually elegant and distributed, bacpressure policy perfors a highly dynaic routing of each pacet through the networ, and requires exchanging queue lengths fro all neighbors at every step. This otivates an investigation into the fundaental liitations faced by static routing policies which do not need rich feedbac signalling. Our analysis brings forth several new insights on this tradeoff. To begin, let us discuss an extree case (see Fig I): consider distributed agents with one pacet each fro a set of distinct pacets (which all of the have received fro the source) to forward to the destination. With full coordination, of course, they can together cover all the pacets in one transission each. However, if we restrict their choice to be ade in a distributed anner without coordination (note the analogy with the coupon collectors proble), the total nuber of distinct pacets covered by a rando choice at each relay approaches 1 (1 1/) 1 e 1, thus leading to only 63% throughput. Note that 100% throughput could be achieved by either (i) aing coordinated choices aong relay nodes through conferencing aong all the relays, or (ii) by letting each node send an independent rando linear cobination of all pacets. In other words, networ coding sees to be essentially solving a distributed synchronization proble without the need for any feedbac signalling. The above synchronization constraint is odeled as the Feedbac Independent Routing (FIR) restriction (see Defini-

2 2 S P(S,i) Fig. 1. relays P(i,D) The relay networ exaple tion 4.1): inforally, this says that the decision of whether a node chooses to forward a pacet it received does not depend on loss events to other nodes. In all exaples of policies that achieve the capacity without any coding, this inforation is iplicitly exploited via extensive feedbac signalling aong nodes. Under the FIR restriction, we show that the optial policies are characterized as tagging policies where each pacet being broadcast is assigned ultiple next hops. Tagging policies can be considered a generalization of both flooding (where every broadcast pacet is routed by everyone that receives it), and traditional routing (where each pacet is routed only by a specific receiver chosen before the broadcast). When transissions are subject to utually independent losses at the receivers (but still possibly allowing for dependencies of losses for pacets transitted by two different nodes), even when restricted to feedbac independent routing, it is possible to achieve at least 63% of the capacity. In fact, as the capacity, C 0, the throughput achieved becoes 100% of C in the liit. For a general feed-forward networ with h + 1 hops in which all nodes are restricted to operate under a siilar routing constraint, we show under a siilar independence condition for lin losses (but allowing dependence across different broadcasts) that the reduction in throughput is lower bounded as f h (C ), where f(x) 1 e x. Thus, for a liited nuber of hops, and when the capacity is low to begin with, one can achieve close to optial throughput without actually aing dynaic routing choices. Lest we get too optiistic about the previous achievability arguent (and iagine that dependencies in lin losses supply iplicit inforation about other lin losses, and thus should allow to iprove over than independent losses), we show a rather surprising counter exaple with dependent lin losses for which static feedbac independent routing capacity is arbitrarily bounded away fro the networ coding capacity even for a 2-hop networ. One of the iplications in this conclusion is that, when feedbac is constrained, coding in the networ could be unavoidable to achieve non vanishing throughput. Related Wor: The erits of routing versus coding have been extensively studied in the context of wireless, both theoretically and by experients. The following two lines of wor deal with the ways in which local broadcast can be exploited: (i) By using Coding: This has been investigated in [3] [5]. The advantage of using networ coding to exploit local broadcast is the lac of a need for sophisticated coordination and/or routing choices aong the nodes. The price to pay for this is the decoding coplexity. (ii) By aing D dynaic routing/flooding choices along with rich feedbac signalling: This was the path taen in [2], [6], [7]. The axiu throughput in an inforation theoretic sense for the Wireless Erasure Networ (WEN) odel was studied in [4] and was shown to be equal to a generalized in cut (GMC). In [7], for the unicast wireless setting, a flooding based policy with networ wide instantaneous broadcast of the identity of each pacet received at the destination was shown to achieve GM C. Using this, [7] aes an iportant observation that coding is in fact not necessary for achieving a throughput equal to the capacity in wireless unicast settings. In [6], bacpressure routing (requiring rich feedbac signalling to perfor routing) was shown to be an optial routing cu scheduling policy in a uch ore general context taing into account interference effects. In general (i.e. for ulticast), [8] proposes the use of feedbac together with networ coding for online decoding. II. THE WIRELESS ERASURE NETWORK (WEN) MODEL Let G (V, E) be a directed graph. Let N (i) {j V : (i, j) E}. We consider an inforation networ that operates on this graph over tie t {0, 1, 2,...}. For each t, a node can broadcast to its neighbors. The networ is subject to probabilistic constraints on the successes of these broadcasts. Specifically, at any given tie, t, for each i V, Z N(i), let χ(i, Z, t) denote a {0, 1} rando variable that represents the following: 1, if broadcast fro node i at tie t χ(i, Z, t) is successful to Z and fails to N (i)/z 0, otherwise The rando variables χ can be arbitrarily correlated across the arguent i, which allows for odeling arbitrary interference constraints, but we will assue that they are stationary across t (not even necessarily independent). Although, in real deployents of wireless networs, the characteristics of the channels ight not quite satisfy the stationarity assuption across t, the odel we adopt provides an excellent coproise between its tractability towards obtaining any non trivial insights, and its faithfulness to reality. Nevertheless, there exists at least one real situation for which our odel is exact: the rando access scheduling odel, where at each tie slot, there is a probability that a given node actually transits, with the transission being successful if and only if no other node in its interference radius is siultaneously active. Note that, for Z, we have: Z N (i) χ(i, Z, t) 1 i, t. We define: c(i, Z) E[χ(i, Z, t)] t. Thus, we are given a networ topology along with c(i, Z) as the capacities. Our analysis will be ipervious to the correlations across i, so we will not explicitly specify the. An interesting special case of the above odel is to have lin (i, j) successful with probability p(i, j), with losses fro i to each of the neighbors in N (i) being independent. Then, we would have: c(i, Z) p(i, j) (1 p(i, j)) j Z j N (i)/z

3 3 We will consider a single unicast flow. However, the insights obtained are also general enough for ultiple copeting flows, with each flow assigned a fixed fraction of the lin capacities and a coparison with intra-session networ coding in such a scenario. Thus, without loss of generality we assue a source, S, and destination, D. The source has an infinite set of pacets indexed over the integers, intended for replication at D. For any v V, let α v (t) denote the nuber of distinct pacets that were replicated at node v till tie slot t. Definition 2.1 (A General routing policy, P): A general routing Policy P decides for each node i V, and tie t {0, 1,...}, a pacet to be transitted fro aong the α i (t 1) choices in its possession. Definition 2.2 (Capacity C(P)): The throughput of a policy, P is defined as: C(P) li inf t 0 E[α D (t)] t The capacity is then, C sup P C(P) III. A MAX FLOW CHARACTERIZATION A ey observation is that any policy which uniquely routes each pacet without eeping ultiple copies can be represented by a valid flow on the throughput constrained graph. In the Linear Progra defined below, each feasible solution represents a policy that routes a fraction proportional to r(i, j, Z) of successful broadcasts fro i to Z uniquely to j Z. The ter {Z N (i):j Z} r(i, j, Z) represents the net flow fro i to j. The value of the LP then represents the throughput that is achieved. Definition 3.1 (MFC, the Max Flow Capacity ): Let P denote the set of all S D paths in G. We define the MF C for a given broadcast capacitated graph as the optiu value of the following LP: MF C ax p P x p (1) Subject to: x p 0 p P r(i, j, Z) 0 {(i, j, Z) : (i, j) E, j Z N (i)} x p r(i, j, Z) 0 (i, j) E {p P :(i,j) p} {Z N (i):j Z} r(i, j, Z) c(i, Z) {(i, Z) : Z N (i)} j Z Let x p, r (i, j, Z) denote the optiu solution to the above LP. It is straightforward to show that this capacity can be achieved by the following policy: Definition 3.2 (P fs, the flow splitting policy): Any pacet transitted by node i, and received by the set Z N (i) of its neighbors is routed uniquely to j N (i) with probability r (i,j,z) Z r (i,,z) (thus ensuring that at ost one copy of each distinct pacet is being transitted at any point). We now define a notion of in cut that is generalized to the wireless setting by considering the probability that at least one of the nodes across the cut receives a transission. Definition 3.3 (General Min Cut, GMC): A Cut is a disjoint partition of V into A and Ā with S A and D Ā. The capacity of the cut is then: C(A) c(i, Z) i A,Z N (i),z Ā φ The General Min Cut, is then: GMC in A C(A). It is easy to argue that GMC is an upper bound on the throughput for any schee even with coding, and was in fact shown to be equal to the inforation theoretic capacity of the WEN in [4]. Thus, C GMC. Based on a duality arguent analogous to the classical ax flow in cut theore, we ae the following clai, the proof of which is provided in the appendix: Theore 3.4: GMC MF C C Thus, coding in the networ is not necessary to achieve the optial throughput in unicast, assuing unconstrained feedbac signalling. This fact was also argued by [7] in a closely related continuous tie odel (and conjectured and verified by siulation for the discrete tie odel that we consider). This was accoplished by considering a policy which involves flooding the networ with each pacet until at least one copy reaches the destination and subsequently using a networ wide feedbac signal to delete these copies every tie the destination receives a new pacet. It was shown that such a policy stabilizes the networ for all rates below the GM C using Lyapunov stability arguent. The below corollary shows that a distributed bacpressure schee which routes each pacet to the least loaded neighbor (thus, avoiding ultiple copies), also achieves the inforation theoretic in cut capacity, GMC. This is, in a sense, iplied by theore 3.4 in conjunction with Neely s result of optiality of bacpressure routing schees in a context that also involves power control, with a different interference odel ( [6]). We give an outline of our proof in the appendix. Corollary 3.5: Consider a Marov chain defined on the networ as follows: Each node has a queue of pacets. New pacets arrive to the queue at the source according to a Bernoulli process of rate λ. For any t, i, Z such that χ(i, Z, t) 1, the bacpressure policy routes a pacet fro node i to a node j such that queue size difference is axiized (subject to being positive). If λ < GMC, the Marov Chain thus obtained is stable (positive recurrent). IV. ROUTING AND FEEDBACK While we have so far discussed schees that can achieve the unicast capacity in a wireless networ without the need for coding, eploying coding in the networ can achieve the capacity without using feedbac. This calls for an understanding of the inherent liits to the achievable throughput when we restrict the exploitation of feedbac in the choice of routing policies. The decision of whether to forward a pacet further at a given node should ideally be ade without considering the erasure events on other lins (this is not the case with any of the routing schees we have discussed so far which achieve the axiu throughput). In this context, we first fix

4 4 the vector of rates that the one hop neighbors can support to the destination siultaneously and study how feedbac constrained routing affects the overall throughput fro the source. The ost obvious visualization of this is a networ in which the source and destination are assisted by relays which do not counicate within theselves (Figure I). Let c(z) E[χ(S, Z, t)] for Z []. Let p(s, i) Z []:i Z c(z) denote the probability that the a pacet transitted fro S is received at i successfully. We shall let p(i, D) denote the long ter throughput that relay i can support to the sin D. Given a generalized routing/flooding policy, we shall use the {0, 1} rando variables, r i (p), ri (p) which denote the following events: 1) {r i (p) 1} Pacet transitted in tie slot p fro the source (which shall henceforth be referred to as pacet p) was received successfully by relay i. We ay assue that the source attepts distinct broadcast of distinct pacets at each instance without any loss of generality (by eploying appropriate source coding, or because it has an unliited strea of useful pacets). 2) {ri (p) 1} Pacet p is routed to D via relay i. Note that: (1) {ri (p) 1} {r i(p) 1} (2) It is possible that ri (p) 1 for ultiple i We will now explicitly describe what we ean by feedbac independent routing. Definition 4.1 (Feedbac Independent Routing (FIR)): p {0, 1,...}, A, B [] such that A B φ, the given routing policy satisfies the F IR restriction if, conditioned on {r i (p) 1 i A}, the following two collections of rando variables: {ri (p)} i A and {r i (p)} i B, are utually independent. This condition is essentially equivalent to assuing a lac of feedbac to the broadcasting node. Technically, lac of feedbac is sufficient but not entirely necessary to satisfy this. While this distinction is subtle, it ight be noted that one does not violate FIR by using rudientary feedbac for purposes other than routing. Nevertheless, source coding is a convenient way to copletely eliinate feedbac. As for the first hop nodes, we erely consider the as blac-boxes that support soe arbitrary vector of siultaneous rates to the destination. For a 2-hop relay networ, these siultaneous rates could be achieved by eploying any capacity achieveing schee for a single erasure channel, i.e. either by (1) feedbac to the relays, or (2) Forward Erasure Coding (FEC) at each relay. 1 We have thus far discussed 3 policies which achieve capacity without any coding: (1) the flow splitting policy, P fs defined in 3.2, (2) the bacpressure policy in lea 3.5 and (3) the policy of [7] but none of the satisfies the above constraint and involve a heavy interaction between feedbac and routing, unlie networ coding. 1 The ey aspect that distinguishes such FEC fro networ coding is that in the case of FEC, the destination has to be able to decode the data being encoded by each relay independently fro the transissions received fro that specific relay alone, whereas with networ codes, the relay only needs to collect the pacets fro all relays and jointly decode the. A. Characterization of the Optial Policies Definition 4.2: The Capacity with Feedbac Independent Routing, C F IR is defined as the axiu throughput as in def 2.1, restricted to policies P satisfying def 4.1. For FIR, one extree is to tag each pacet with a single relay (this is how routing is done in practice in the protocol). This could be suboptial because it does not exploit the local broadcast advantage. The other extree is to flood every pacet to all relays. This could be suboptial when the relays do not have enough capacity to forward all pacets they received to the destination. They will then have to ae distributed decisions on which pacets to forward fro aong the received pacets, leading to redundant transissions and a hence a decreased throughput. As we will show, when we restrict to FIR, the axiu throughput is achieved within a subclass of policies that we shall call the tagging policies. A tagging policy assigns to each pacet, a subset of relays, Z [], which is independent of any feedbac. We represent the fraction of pacets that are tagged with Z as t(z). A relay i that receives a pacet successfully routes the pacet without dropping it if and only if i Z. These pacets are retransitted until the destination receives the. The capacity of such policies can be expressed via the below LP, where each feasible solution corresponds to a specific tagging policy. The last constraint in the LP states that the arrival rate of pacets to any relay s queue has to be less than its forwarding capacity to the destination. Definition 4.3 (Tag Capacity, TC): TC is defined as the optiu value of the following LP with variables t(z) 0 where Z []. T C ax t(z)c(z ) (2) Subject to: p(s, i) Z,Z []:Z Z φ Z [] Z []:i Z t(z) 1; (3) t(z) < p(i, D) i [] (4) All pacets that reach a relay successfully and have the relay in the tag will eventually reach D because of the constraint in Eq (4) (which iplies that the queue of pacets at each relay is stable). Since the tags are chosen independent of the losses, the probability that a pacet is successfully transitted to soe relay which is also included in its tag is Z Z φ t(z)c(z ), which can be readily shown to be equal to the throughput of the policy as per definition 2.1. Lea 4.4: C F IR T C Proof: Oitted. The arguent involves observing that any feasible solution to the LP gives us a tagging schee whose throughput is equal to the value of the LP. Rearably, these policies will now also be shown to be optial in general under the FIR constraint, thus giving us an explicit characterization of C F IR in the for of the below theore, whose proof is provided after a brief discussion. Theore 4.5: C F IR T C, iplying T C C F IR

5 5 S S D Cut Value 1 Cut Value ¾ Cut Value 1 (a) Exaple 1 3/8 D 3/8 Cut Value 7/8 Cut Value ¾ Cut Value ¾ (b) Exaple 2 Fig. 2. Exaples P ( A B ) for an event B with positive probability. It is easily verified that t(z) 0 and Z [] t(z) 1. We will now verify that the third constraint (Eq (4)) holds for any i []. p(i, D) E[ 1{ri (p) 1}] P (r i (p) 1)P p(s, i) p(s, i) p(s, i) P r i (p) 1 r i (p) 1 r i (p) 1 r i (p) 1 ( ri P (p) 1 ) (using def.4.1) r j (p) 1 j [] ( r ) j (p) 1 j Z and 0 j []/Z P r j (p) 1 j [] Z i p(s, i) Z i t(z) (by definition of t(z) in Eq (5)) In general, it is not obvious if C F IR is strictly less than the general in cut capacity, GMC. Indeed, in any cases, these two quantities atch, iplying that in such cases, not only do we not need any coding, but the capaciy can be achieved by optiized tagging schees that satisfy FIR. For exaple, consider a networ with two relays with success probabilities p(s, 1), p(s, 2), p(1, D), p(2, D) where lin losses for the sae transission are independent. Exaple 1: If p(s, 1), p(s, 2), p(1, D), p(2, D) are all 1/2, we have: C({1}) C({2}) C({1, 2}) 1/4. Fro the Fig. 2(a), GMC 3/4 and fro the LP (Eq (2)), we can calculte that C F IR 3/4. In fact, a flooding policy achieves this rate of 3/4. So there is no reduction in the capacity under FIR. Exaple 2: Consider an exaple where the cuts are ore finely balanced: p(s, 1), p(s, 2) 1/2; and p(1, D), p(2, D) 3/8. Again, GMC 3/4, as explained in Fig 2(b). Set t({1}) x, t({2}) y, t({1, 2}) z. Then: C F IR Max ( 1 2 x y + 3 4z) Subject to: x, y, z 0, x + y + z 1, x + z 3/4, y + z 3/4. The optial value can be verified to be 5/8 < 3/4 GMC. This optiu throughput of 5/8 under FIR is achieved by routing 25% of the pacets exclusively to relay 1, 25% exclusively to relay 2 and the reaining 50% of the pacets to both the relays. Proof of Theore 4.5: Consider any arbitrary policy. We will show that the expected nuber of distinct pacets that reach the destination in tie slots can not be ore than T C. as long as FIR in definition 4.1 is satisfied, thus iplying that the C F IR is at ost T C. Define: t(z) 1 r P i (p) 1 i Z and 0 i []/Z r i (p) 1 p [] (5) where we use the standard convention of writing P (A B) P (B) as We now calculate the nuber of distinct pacets replicated at D in tie slots. E[α D ()] P ({ {ri (p) 1}}) Z [] Z [] i [] ( ) P r i (p) 1 i Z and 0 i []/Z ( ri (p) 1 for soe i [] ) P r i (p) 1 i Z and 0 i []/Z ( ri (p) 1 for soe i Z c(z)p r i (p) 1 i Z and 0 i []/Z ( i []/Z, {r i (p) 0} {ri (p) 0}) r c(z)p i (p) 1 for soe i Z r i (p) 1 i [] Z [] (using FIR definition 4.1) ( r c(z) P i (p) 1 i Z ) and 0 else r i (p) 1 i [] Z Z Z φ c(z) t(z) Z [] Z,Z []:Z Z φ Z []:Z Z φ t(z)c(z ) B. Analysis of the Loss of throughput In this section, we will use the theory fro previous sections to obtain results on the throughput attainable under FIR. We will first consider the case where the lin losses fro a given transitter seen at its various receivers are independent. Note )

6 6 that this involves no assuptions on interference between two different transissions. As a critical tool, we will use a flooding policy, which is defined below. Definition 4.6 (Flooding Policy, P F ): Each relay blindly chooses each received pacet fro the source to be forwarded to the destination with probability in(1, p(i, D)/p(S, i)). In other words, the relay effectively aes a uniforly rando selection of a p(i, D)/p(S, i) fraction of its received pacets to be forwarded to the destination whenever p(i, D) p(s, i). We naed the above policy as flooding in a general sense, because every pacet that is successfully received is considered for being forwarded at any relay. It is possible that the total nuber of received pacets at each relay could be ore than the rate it can support to the destination, in which case, P F essentially aes a rando selection of pacets corresponding the axiu throughput it can support. The throughput of P F can be calculated by evaluating the probability that a pacet transitted by the source in any given tie slot reaches the Destination along at least one of the relays. Since these events are independent under our current hypothesis, we clai: Lea 4.7: C(P F ) 1 (1 in(p(s, i), p(i, D))) i [] Proof: Oitted. The arguent relies on calculating the probability that any given pacet reaches the destination via at least one of the relays. Using this flooding policy, we now prove a lower bound on C F IR for a networ with arbitrary fixed rates fro the first hop forward. This arguent explicitly bounds the loss of throughput because of the redundant transissions arising out of aing distributed decisions at each of the relay nodes with arbitrary in cuts when we have independence aong lin successes. The bound we provide also iplies that when the capacity is low, the flooding is alost optial for independent losses to the first hops. We will provide the proof on a 2-hop networ for easier visualization, but the sae clai holds for any fixed rates fro the first hop to the destination. Theore 4.8: Consider the relay networ of figure I with independent losses fro the source to the relays. C F IR 1 e C Thus, C F IR /C 1 e 1. It follows fro this bound that, as C 0, C F IR /C 1 Proof: We will specifically show that C(P F ) 1 e GMC, which in turn iplies the theore since P F clearly satisfies FIR (definition 4.1), and GMC C. Recall that: C(P F ) 1 i [] (1 in(p(s, i), p(i, D))) Applying the definition of GMC to the networ under consideration, we see that GMC in 1 p(s, i)) + A [] i []/A(1 p(i, D) i A Let A be the arg in over A [] for the above equation, so that: S P(S,i) A* P(i,D) Fig. 3. An Illustration of a general in cut and the set defining A in this in cut. GMC 1 (1 p(s, i)) + p(i, D) (6) i []/A i A Consider any i A. By definition of A, we have: 1 (1 p(s, j)) + p(j, D) j []/A j A 1 (1 p(s, i)) (1 p(s, j))+ p(j, D) p(i, D) j []/A j A which iplies: Thus: p(i, D) p(s, i) j []/A (1 p(s, j)) p(s, i) (7) C(P F ) 1 (1 in(p(s, i), p(i, D))) 1 i [] (1 in(p(s, i), p(i, D)) i A 1 (1 p(i, D)) i A ( p(i, D) p(s, i) i A fro Equation 7) 1 ( 1 x e x x 0) i A e p(i,d) 1 e i A p(i,d) 1 e GMC ( i A p(i, D) GMC fro Equation 6) The above analysis characterizes the degradation through the effects of feedbac at a single node, given the rates achievable fro the subsequent hops in any general networ. We next build upon this arguent to obtain a bound for arbitrary feedforward networs where every node is restricted to static routing and without any assuption of achievable rates fro the one hop neighbors. D

7 7 Theore 4.9: For a general feedforward networ with h+1 hops, it is possible to achieve a throughput of at least f h (C ) where f(x) 1 e x and C is the in cut capacity. Proof Outline: First, we note that feedforward networs can be reduced to general layered networs where only nodes at subsequent levels are connected, but introducing duy nodes with unit capacity lins. For such a layered networ, we will adopt the convention that the sin is at level 0, and the source is at level h+1. We will define (i, j) to be the node indexed j at level i. In this notation, the source is assued to be (h + 1, 0), and the sin is (0, 0). Given any policy, P, we define P(i, j) as the rate at which distinct pacets are streaing to the sin through (i, j). For exaple, P(h+1, 0) is the throughput of the policy. We define, ψ, a generalized flooding policy satisfying FIR and copare it with the splitting policy, P fs in Definition 3.2: A node b queues the pacets that it receives fro node a in proportion to the rate r (a, b). Z N (a) r (a, b, Z) with r as in definition 3.2. Note that this is a static calculation and involves no feedbac unlie P fs or the bacpressure policy, which requires queue length inforation to decide every routing step. For such a policy, we ae the following clai, which can be shown using induction on i. Clai 1: ψ(i, j) f i 1 (P fs (i, j)) The clai follows easily for i 1. Assue that it holds for soe i. We will loo at a node (i + 1, j) which is connected to (i, j 1 ),..., (i, j ). Consider two different 2-hop networs both with (i+1,j) as the source and (i, j 1 ),..., (i, j ) as relays with the sae lin characteristics as the feedforward networ for the first hop. To describe the second hop capacities, we will use the assuption that both P fs and ψ choose the sae fraction of the incoing capacities fro various nodes for the forward throughput. Let α t be the fraction of incoing rate that is chosen by node (i, j t ) for the throughput fro (i + 1, j). For the second hop, the first and second networs have capacities (α 1 ψ(i, j 1 ),..., α ψ(i, j )) and (α 1 P fs (i, j 1 ),..., α P fs (i, j )) respectively. Let C1, C2 be the respective GM C s. Now, we can use theore 4.8 to argue that ψ(i + 1, j) f(c1 ). Using the induction hypothesis, we now that α t ψ(i, j t ) α t f i 1 (P fs (i, j )). It can be shown under this condition (using lea 6.2 of the appendix) that C1 f i 1 (C2 ), which iplies that ψ(i + 1, j) f(f i 1 (C2 )) f i (C2 ) f i (P fs (i, j)), thus copleting the induction hypothesis. Fro the above result, one wonders if we are always assured of a constant factor reduction capacity in general, without either feedbac or networ coding on at least bounded diaeter networs. Naively speaing, this sees plausible because correlations in the losses can only supply the relays with iplicit inforation that ight help the synchronize their routing ore efficiently while satisfying FIR. This intuition is flawed, as the below counter exaple shows that C F IR 0 even though the capacity C 1. This counter exaple could also have practical relevance since it can be otivated as an extree case of a situation where ost of the tie (with probability p to be explicated later), the broadcast channel is bad enough to be actually useful to a very sall set of relays but occasionally experiences high strength, in which case ost of the relays receive the broadcast. We ae use of the characterization of C F IR fro theore 4.5 in showing this fact. Definition 4.10 ( Confusion Networ of size ): When the source broadcasts, with probability p, exactly one of the relays receive the pacet, and with probability 1 p, all relays receive the pacet, where Ω( 1 ) 1 p o(1). More explicitly, we can choose p 1 1 and then have: p if Z 1 c(z) 1 p if Z [] 0 otherwise and for the relay to destination, we have: p(i, D) 1/ i [] Theore 4.11: For the Confusion networ, C 1, but li C F IR 0 Proof: It is easy to verify that GMC 1. We shall now derive an upper bound on the optiu value of the LP in definition 4.3 which tends to 0 as. This iplies the clai because of theore 4.5. Since c(z) depends only on Z, and the LP 2 is syetric over i, one can set without loss of generality, t(z) φ( Z ), in which case: Z,Z []:Z Z φ t(z)c(z ) φ()( Z :Z [] φ c(z )) φ() (1 p + p ) The constraint in Eq (3) becoes: φ() 1 (8) As for the constraint in Eq (4), we have: p(s, i) c(z) p + 1 p and Z []:i Z t(z) Z []:i Z 1 φ() 1 1 Thus the second constraint becoes: φ() p + (1 p) φ() (9)

8 8 Thus, T C (1 p) φ() (1 p + p ) ( ) φ() + p ( p 1 p + ( fro Eqs (8), (9)) p + (1 p) O(1/ ) ( since p 1 1 ) V. CONCLUSION ) φ() While networ coding is necessary to achieve the axiu throughput for ulticast connections, this is not the case with wireless unicast. Rather, networ coding is a convenient way to solve the distributed routing proble without having to depend on feedbac signalling to ae coplicated routing choices for achieving the axiu throughput. In this context, we analyzed a relay networ and quantitatively characterized the liitations of static routing policies that operate in a feedbac independent anner. Our characterization allows for explicitly identifying situations when there is no loss of throughput by restricting to such siple routing policies. Further, we show that the reduction in the throughput is controlled when the lin losses for a given transission are independent, and could even be inial when the capacity is low, on a general feedforward networ. At worst, this is 63% and gets progressively close to 100%, as the capacity itself goes to 0. Thus, in such a situation, networ coding delivers no benefit over sipler blind routing policies in the liit of unreliable counication. Nevertheless, highly correlated losses could lead to an unbounded loss even on a 2-hop networ. In such a situation, networ coding ight be unavoidable if we need to be conservative with the feedbac. REFERENCES [1] R. Ahlswede, N. Cai, S. Li, and R. Yeung. Networ inforation flow. IEEE Trans. on Inforation Theory, 46: 4: , July [2] S. Biswas and R. Morris. Opportunistic routing in ulti-hop wireless networs. ACM SIGCOMM, [3] S. Chachulsi, M. Jennings, S. Katti, and D. Katabi. Trading structure for randoness in wireless opportunistic routing. ACM SIGCOMM, [4] A. Dana, R. Gowaiar, R. Palani, and M. Effros. Capacity of wireless erasure networs. IEEE Trans. on Inforation Theory, 52: 3: , March [5] D. Lun, M. Medard, R. Koetter, and M. Effros. On coding for reliable counication over pacet networs. Physical Counication, 1:3 20, March [6] M. Neely and R. Urgaonar. Optial bacpressure routing for wireless networs with ulti-receiver diversity. AD HOC NETWORKS (ELSE- VIER), 7: 5: , July [7] B. Sith and B. Hassibi. Wireless erasure networs with feedbac. On Arxiv: [8] J. Sundararajan, D. Shah, and M. Medard. Feedbac-based online networ coding. On Arxiv: April [9] L. Tassiulas and A. Ephreides. Stability properties of constrained queueing systes and scheduling policies for axiu throughput in ultihop radio networs. IEEE Trans. on Autoatic Control, 37: , Deceber VI. APPENDIX Proof of Theore 3.4: Consider the dual progra for the LP at Eq (1) with dual variables b(i, Z) for each i V, Z N (i) and y(i, j) for each (i, j) E. We have: DUAL in c(i, Z)b(i, Z) (10) Such that: y(i, j) 1 p P (11) {(i,j) p} y(i, j) + b(i, Z) 0 {(i, j, Z) : (i, j) E, j Z N (i)} y(i, j) 0 (i, j) E b(i, Z) 0 (i, Z) : i V, Z N (i) (12) Consider the incut as described in (3.3), and let A, Ā denote this cut. Let y (i, j) 1 if i A, j Ā and 0 otherwise. Siilarly, let b (i, Z) 1 if i A, Z Ā φ and 0 otherwise. Then, it can be verified that this defines a feasible solution to the dual LP above, and that c(i, Z)b (i, Z) GMC. Thus, GMC DUAL (13) Now consider an integral constrained version of the above dual: DUAL in c(i, Z)b(i, Z) (14) Such that: y(i, j) 1 p P (15) {(i,j) p} y(i, j) + b(i, Z) 0 {(i, j, Z) : (i, j) E, j Z N (i)} (16) y(i, j) {0, 1} (i, j) E b(i, Z) {0, 1} (i, Z) : i V, Z N (i) (17) Let y, b define the optial solution to the above integral constrained LP. Then, define A {i V : a path, p, fro S to i s.t. {(i,j) p} y (i, j) 0}. Then, clearly, D / A because of constraint (15), and thus A defines an (S, D) cut. Furtherore, by interpreting (16) and (14), it can be verified that C(A ) DUAL. This iplies that GMC DUAL (18) To suarize, we can so far clai: MF C DUAL C GMC DUAL (19) If we are able to argue that the constrained LP indeed achieves the optiu (the details of this are given in Lea 6.1), we would then have DUAL DUAL, iplying that all quantities in Eq (19) are the sae. Lea 6.1: DUAL DUAL Proof: We use an analogous arguent eployed in showing the corresponding stateent for the classical ax flow in cut theore. The arguent considers a probability distribution

9 9 on the set of all possible cuts and argues that the expected value of GMC thus obtained is no ore than DUAL, which in turn iplies that DUAL DUAL copleting the proof. Consider the dual LP at Eq (10), and let y (i, j), b (i, Z) denote the optial solution which achieves DUAL. Consider a graph with edge lengths given by y (i, j) and let d(i) length of the shortest path fro S, with edge lengths given by y (i, j). Let λ (0, 1) be chosen uniforly and define A {i V : d(i) λ} This defines a cut with probability 1, since d(s) 1 fro Eqn 11. Then: E[C(A )] E[ c(i, Z)] E[ i A,Z N (i),z A φ c(i, Z)1{i A, Z N (i), Z Ā φ}] c(i, Z)P ( i A, j Ā for soe j Z ) c(i, Z)P (d(i) λ, d(j) > λ for soe j Z) c(i, Z)(ax j Z d(j) d(i)) + ( λ is unifor) c(i, Z)(ax j Z y (i, j)) ( traingle inequality ) c(i, Z)b (i, Z) ( fro Eq. (12)) DUAL Proof Outline for Corollary 3.5: The proof follows along the lines of [9] using a quadratic Lyapunov function and Foster s condition to show that the Marov Chain is positive recurrent. Pacets are injected at the source node according to a Bernoulli i.i.d. process with ean λ < GMC. We shall use the potential, V (t) n i1 q2 i (t), where i represents an index over the n nodes in the networ and q i (t) represents the nuber of pacets that are held at node i. Let {F t } t 0 be a filtration adapted to the queue length process. By Foster s theore (see [9] and the references therein), a sufficient condition for stability is to show that that the lyapunov drift E[V (t+1) V (t)/f t ] < 0 when V (t) is sufficiently large. Let q(t) denote the n diensional row vector with q i (t) being the i th eleent. Let R be the adjacency atrix of diension n L (where L E, the nuber of lins) for the given graph (i.e., the (i, l) th eleent, r(i, l) is 1( 1) iff the lin l starts(ends) at i, and 0 otherwise). Let E(t) denote the L diensional indicator vector bdenoting the subset of lins on which pacets were routed by the bacpressure policy, and let A(t) denote the arrival process indicator vector,i.e. the eleent of A(t) corresponding to the source is the bernoulli Rando variable with ean λ and all other eleents are 0. Then, the queue lengths evolve according to q(t + 1) q(t) + R E + A(t). Thus (with. denoting the usual dot product of vectors): V (t + 1) V (t) q(t + 1).q(t + 1) q(t).q(t) (R E(t) + A(t)).(2q(t) + RE(t) + A(t)) (n + 1) 2 + 2(q(t).RE(t) + q(t).a(t)) Since the first ter above is constant over tie, showing that the second ter has a large negative drift for large queue lengths is sufficient (since large potential iplies large queue length under connectivity assuptions on the graph). Let W(t) denote the L diensional vector where the l th eleent denotes the queue length difference for the l th lin. Then, q(t).re(t) W(t).E(t). Hence, we only need to argue that E[ W(t).E(t) + q(t).a(t)/f t ] has a negative drift. E[q(t).A(t)/F t ] q(t).a where A is a vector where the eleent corresponding to the source is λ and all other eleents 0. Since λ < GM C, it follows fro the Max flow interpretation of theore 3.4 that there exist variables r (i, j, Z) satisfying the constraints of LP at Eq (1), and an ɛ > 0 such that q(t).a (1 ɛ)w(t).f, where the coponent of f corresponding to lin l (i, j) is fl {Z N (i):j Z} r (i, j, Z). Let e (t) denote the indicator vector of diension L, for the routing selected by the flow splitting policy, P fs given in Definition 3.2. Then, we clai that W(t).f E[W(t).e (t)/f t ]. Thus, E[ W(t).E(t) + q(t).a(t)/f t ] E[ W(t).E(t) + (1 ɛ)w(t).e (t)/f t ] E[ W(t).E(t) + W(t).e (t)/f t +/F t ] ɛe[w(t).e (t)/f t ]. Here, F t + denotes the siga algebra that contains inforation about the successes on lins at tie t in addition to the queue length process until tie t, and thus, F t F t + F t+1. The first ter is non-positive because the bacpressure policy iniizes W(t).E(t) aong all possible options conditioned on the inforation available on the lin successes and the queue lengths (which is what conditioning on F t + denotes). The second ter taes arbitrarily large negative values for large V (t) for any fixed ɛ > 0 and thus, we have the required negative drift. Lea 6.2: Following the notation used in Section IV, consider a 2 hop relay networ with paraeters c(z) for the first hop and r i : i [] as the rates for the second hop. Let the in cut of this networ be C1. Replace the second hop rates by α i g(r i /α i ) where g(x) f n (x) for soe n with f(x) 1 e x. The new in cut C2 g(c1 ) Proof: First we note that for each α > 0, for the given g, we have α i g(r i /α) g(r i ), which can again be shown using induction on the exponent of f corresponding to the given g. Therefore, it is sufficient to show that the networ with second hop capacities, g(r i ) has a incut, C g(c1 ). Let A be the set defining the in cut for the networ defining C1 in the sae sense as it was used in proof of theore 4.8. It can further be shown that g(x x n ) g(x 1 ) +... g(x n ) by using induction on the exponent of f in the representation of g. Using this fact (and denoting Θ 1 i []/A (1 p(s, i)) for convenience below), we then have: C Θ + i A g(r i ) g(θ) + i A g(r i ) g(θ + i A r i ) g(c1 )