Intra- and Inter-Session Network Coding in Wireless Networks

Transcription

1 Intra- and Inter-Session Network Coding in Wireless Networks Hulya Seferoglu, Member, IEEE, Atina Markopoulou, Member, IEEE, K K Ramakrisnan, Fellow, IEEE arxiv:857v [csni] 3 Feb Abstract In tis paper, we are interested in improving te performance of constructive network coding scemes in lossy wireless environments We propose I NC - a cross-layer approac tat combines inter-session and intra-session network coding and as two strengts First, te error-correcting capabilities of intra-session network coding make our sceme resilient to loss Second, redundancy allows intermediate nodes to operate witout knowledge of te decoding buffers of teir neigbors Based only on te knowledge of te loss rates on te direct and overearing links, intermediate nodes can make decisions for bot intra-session (ie, ow muc redundancy to add in eac flow) and inter-session (ie, wat percentage of flows to code togeter) coding Our approac is grounded on a network utility maximization (NUM) formulation of te problem We propose two practical scemes, I NC-state and I NC-stateless, wic mimic te structure of te NUM optimal solution We also address te interaction of our approac wit te transport layer We demonstrate te benefits of our scemes troug simulations Index Terms Network coding, wireless networks, error correction, cross-layer optimization I INTRODUCTION Wireless environments lend temselves naturally to network coding (NC), tanks to teir inerent broadcast and overearing capabilities In tis paper, we are interested in wireless mes networks used for carrying traffic from unicast sessions, wic is te dominant traffic today Network coding as been used as a way to improve trougput over suc wireless environments Given tat optimal inter-session NC for unicast is still an open problem, constructive approaces are used in practice [], [], [3], [4], [5] One of te first practical wireless NC systems is COPE [] - a coding sim between te IP and MAC layers tat performs one-op, opportunistic NC COPE codes packets from different unicast sessions, and relies on receivers being able to decode tese using overeard packets Tis way, COPE combines multiple packets by using information on overeard packets wic are excanged troug transmission reports and effectively forwards multiple packets in a single transmission to improve trougput In order for COPE to work in a multiop network, nodes must cooperate to (i) excange information about wat packets tey ave H Seferoglu is wit te Laboratory for Information and Decision Systems (LIDS), Massacusetts Institute of Tecnology seferog@mitedu Mail: 77 Massacusetts Avenue, Room 3-D67, Cambridge, MA 39 A Markopoulou is wit te Electrical Engineering and Computer Science Department, University of California, Irvine atina@uciedu Mail: CalIT Bldg, Suite 4, Irvine, CA 9697 K K Ramakrisnan is wit AT&T Labs Researc kkrama@researcattcom Mail: 8 Park Avenue, Building 3 Floram Park, NJ 793 overeard and also (ii) code so tat all one-op downstream nodes can decode Tis must be done at every op across te pat of a flow and cross-layer optimization approaces can be used [6] to furter boost te performance One important problem tat remains open, and is te focus of tis paper, is COPE s performance in te presence of nonnegligible loss rates Te reason is tat intermediate nodes in COPE require te knowledge of wat teir neigbors ave overeard, in order to perform one-op inter-session NC However, in te presence of medium-ig loss rate, altoug eac node fully cooperates to report wat it as overeard, tis information is limited, possibly corrupted, and/or delayed over lossy wireless cannels COPE turns off NC if loss rate exceeds a tresold wit default value % [] However, tis does not take full advantage of all te available NC opportunities To better illustrate tis key point, let us discuss te following example Example : Let us consider Fig, and focus on te neigborood of node I, ie, only te packets transmitted via I, from A to A and from B to B Tis forms an X topology wic is a well-known, canonical example of one-op opportunistic NC [], [] In te absence of loss, trougput is improved by 333%, because I delivers two packets in tree transmissions (wit NC), instead of four (witout NC) Let us re-visit tis example wen tere is packet loss Assume tat tere is loss only on te overearing link A B, wit probability ρ {A,B } = 3, and all oter links ave no loss In tis case,7% of te packets can still be coded togeter, and trougput can be improved by 6%, wic is still a significant improvement Even at iger loss rate, eg, ρ {A,B } = 5, inter-session coding still improves trougput up to % Tis is under te assumption tat I knows te exact state of A,B, ie, wat packets were overeard, and tus I is able to decide wat packets to code togeter so as to guarantee decodability at te receivers However, at ig loss rates, cooperation among nodes becomes difficult Tis is wy COPE turns off te coding functionality wen loss rate is iger tan a tresold wit default value %, tus not taking full advantage of all coding opportunities We propose a solution to tis problem wit a design wic combines intra- and inter-session NC over wireless mes networks We use intra-session NC to combine packets witin te same flow and introduce parity packets to protect against loss Ten, we use inter-session NC to combine packets from different (already intra-session coded) flows, and tus increase trougput Our approac for combining intra-session wit inter-session NC, wic we refer to as I NC, as two key benefits First, it can correct packet loss and still perform inter-

2 Fig Example of a unicast flow (from S to R ) traversing multiple wireless ops Eac node performs (intra- and inter-session) NC Te neigborood of I is sown ere in detail (Two unicast flows, S R and S R, meeting at intermediate node I I receives packets a,b from nodes A,B, respectively It can coose to broadcast a, b or a + b in a single transmission to bot receivers Te next opsa,b can decode a+b because tey overear packets b and a transmitted from B,A, respectively) session NC, even in te presence of medium-ig loss rates, tus improving trougput Second, te use of intra-session NC makes all packets in te session equally beneficial Tus, I NC eliminates te need to know te exact packets tat ave been overeard by te neigbors of intermediate node I It is sufficient to know te loss probabilities of overeard and transmitted packets In our sceme, tis information is reported by eac node to te nodes in its neigborood wic makes NC possible even at iger loss rates Adding redundancy in tis setting is non-trivial, since a flow is affected not only by loss on its direct links, but also by loss on overearing links Tis affects te decodability of coded packets Terefore, te amount of redundancy needed to be determined carefully Example - continued: Consider again te neigborood of I in Fig Flow (originated from S ) is affected not only by loss on its own pat B IB, but also by loss on te overearing link A B, wic affects te decodability of coded packet a+b at B In order to protect flow from ig loss rate on te overearing link A B, I may decide eiter to add redundancy on flow, or to not perform coding, or a combination of te two On te oter and, I may also decide to add redundancy on flow (originated from S ), to correct loss on te overearing link A B, tus elping B to receive a and decode a+b Terefore, a number of questions need to be addressed in te design of a system tat combines bot intra- and intersession NC In particular: Q: How to gracefully combine intra- and inter-session NC? We propose a generation-based design, and specify te order we perform te two types of coding Q: How muc redundancy to add in eac flow? We sow ow to adjust te amount of redundancy after taking into account te loss on te direct and overearing links We implement te intra-session NC functionality as a tin layer between IP and transport layer Q3: Wat percentage of flows sould be coded togeter and wat parts sould remain uncoded? We design algoritms tat make tis decision taking into account te loss caracteristics on te direct and overearing links We implement tis and oter functionality (eg, queue management) performed wit or after inter-session NC as a layer between MAC and IP Q4: Wat information sould be reported to make tese decisions? We propose two scemes: I NC-state, wic needs to know te state (ie, overeard packets) of te neigbors; and I NC-stateless, wic only needs to know te loss rate of links in te neigborood Our approac is grounded on a network utility maximization (NUM) framework [7] We formulate two variants of te problem, depending on available information (as in Q4 above) Te solution of eac problem decomposes into several parts wit an intuitive interpretation, suc as rate control, NC rate, redundancy rate, queue management, and sceduling Te structure of te optimal solution provides insigt into te design of our two scemes, I NC-state and I NC-stateless We evaluate our scemes in a multi-op setting, and we consider teir interaction wit te transport layer, including TCP and UDP We propose a tin adaptation layer at te interface between TCP and te underlying coding, to best matc te interaction of te two We perform simulations in GloMoSim [8], and we sow tat our scemes significantly improve trougput compared to COPE Te structure of te rest of te paper is as follows Section II presents related work Section III gives an overview of te system model Section IV presents te NUM formulation and solution Section V presents te design of te I NC scemes in detail Section VI presents simulation results Section VII concludes te paper II RELATED WORK COPE and follow-up work Tis paper builds on COPE, a practical sceme for one-op NC across unicast sessions in wireless mes networks [], wic as generated a lot of researc interest Some researcers tried to model and analyze COPE [9], [], [] Some oters proposed new coded wireless systems, based on te idea of COPE [], [5] In [3], te performance of COPE is improved by looking at its interaction wit MAC fairness Our recent work in [6] improves TCP s performance over COPE wit a NC-aware queue management sceme Tis paper also improves COPE by adding intra-session redundancy wit a cross-layer design and reducing te amount of information tat is needed to be excanged among nodes cooperatively, ie, nodes no longer need to know te exact packets overeard by teir neigbors and can operate only wit knowledge of te link loss rates NUM in coded systems Te NUM framework can be applied in networks, to understand ow different layers and/or modules (suc as flow control, congestion control, routing, etc) sould be restructured wen NC is used Altoug te approac is general, te parts and interpretation of te distributed solution is igly problem-specific For NUM to be successful, te optimization model must be formulated so as to capture and exploit te NC properties Tis is igly non-trivial and problem-specific A body of work as looked at te joint optimization of NC of unicast flows, formulated in a NUM framework

3 Optimal sceduling and routing for COPE are considered in [9] and [], respectively A linear optimization framework for packing butterflies is proposed in [4] A re-transmission sceme for one-op NC is proposed in [4] Forward error correction over wireless for pairwise NC is proposed in [5], [6], wic are also te most closely related formulations to ours Our main differences are tat we consider: (i) multiple flows coded togeter instead of pairwise, (ii) local instead of end-to-end redundancy, and (iii) te effect of losses over direct and overearing links, to generate te rigt amount of redundancy Dealing wit wireless loss Recent studies of IEEE 8b based wireless mes networks [7], [8], ave reported packet loss rates as ig as 5% Dealing suc level of loss in wireless networks is a ard enoug problem on its own, wic is furter amplified by NC Tere is a wide spectrum of well-studied options for dealing wit loss, eg, using redundancy and/or re-transmissions, locally (MAC) and/or end-to-end (transport layer) Local re-transmissions increase end-to-end delay and jitter, wic, if excessive, may cause TCP timeouts or urt real-time multimedia Furtermore, te best re-transmission sceme for network coded packets varies wit te loss probability and it is ard to switc among re-transmission policies wen te loss rate varies over time Re-transmission also requires state syncronization to perform inter-session NC, wic is not reliable at all loss rates We follow an alternative approac of local redundancy because we are interested in keeping delay low and we want to eliminate te need for knowing te state of neigbors Tere is extended work on TCP over wireless One key problem is te need to distinguis between wireless and congestion loss and ave TCP react only to congestion; tis is possible eg, troug Explicit Congestion Notification (ECN) Wen re-transmissions exceed te delay budget, end-to-end redundancy may also be used to combat loss on te pat [9] Te error-correcting capabilities of intra-session NC ave recently been used in conjunction wit te TCP sliding window in [] In contrast, we focus on one-op inter-session coding rater tan end-to-end intra-session coding III SYSTEM OVERVIEW We consider multi-op wireless networks, were intermediate nodes perform intra- and inter-session NC (I NC) Next, we provide an overview of te system and igligt some of its key caracteristics A Notation and Setup ) Sources and Flows: Let S be te set of unicast flows between source-destination pairs in te network Eac flow s S is associated wit a ratex s and a utility functionu s (x s ), wic we assume to be a strictly concave function of x s We ave observed troug simulations tat if a network coded packet is lost for one receiver but received correctly for oter receiver(s), it is better to re-transmit te same network coded packet for low loss rates However, it is better to combine te packet wic is lost in te previous transmission wit new packets for ig loss rates ) Wireless Transmission: Packets from a source (eg, S in Fig ) traverse potentially multiple wireless ops before being received by te receiver (eg, R ) We consider a model for interference described in []: eac node can eiter transmit or receive at te same time, and all transmissions in te range of te receiver are considered as interfering We use te following terminology for wireless A yperarc (i,j) is a collection of links from nodei N to a non-empty set of next-op nodes J N A ypergrap H = (N,A) represents a wireless mes network, were N is te set of nodes and A is te set of yperarcs For simplicity, = (i,j) denotes a yperarc, (i) denotes node i and (J) denotes te set of nodes in J, ie, (i) = i and (J) = J We use tese notations intercangeably in te rest of te paper Eac yperarc is associated wit a cannel capacity R Since is a set of links, R is te minimum capacity of all te links in te yperarc, ie, R = min j (J) {R i,j } st i N In te example of node I in Fig, = (I,{B,A }) is one of te yperarcs, and its capacity is min{r {I,B},R {I,A}} Note tat wit bot intra- and inter-session NC, it is possible to construct more tan one code over a yperarc Let K be te set of inter-session network codes over a yperarc S k S be te set of flows coded togeter using code k K and broadcast over Given H, we can construct te conflict grap C = (A,I), wose vertices are te yperarcs of H and edges indicate interference between yperarcs A clique C q A consists of several yperarcs, at most one of wic can transmit witout interference, ie, a transmission over a yperarc interferes wit transmissions over oter yperarcs in te same clique 3) Loss Model: A flow s may experience loss in two forms: loss ρ s over te direct transmission links; or loss ρs,s of antidotes 3 on overearing links Tese two types of loss ave different impact on network coded flows First, let us discuss loss on te direct links A flow s transmitted over yperarc experiences loss wit probability ρ s Tis probability is different per flow s, even if several flows are coded and transmitted over te same yperarc, because different flows are transmitted to different next ops, tus see different cannels For example, in Fig,ρ S (I,{B,A }) is equal to te loss probability over link I A and ρ S (I,{B is,a }) equal to te loss probability over link I B Second, let us discuss te effect of lost antidotes on te overearing link Consider tat flow s is combined wit flow s st s s, and tat some packets of flow s are lost on te overearing link to te next op of s Ten, coded packets cannot be decoded at te next op and flow s loses packets, wit probability ρ s,s For example, in Fig, packets from flow S cannot be decoded (ence are lost) at node A due to loss of antidotes from flows on te overearing linkb A In our formulation and analysis, we assume tatρ s andρs,s Note tat we consider constructive inter-session NC, ie, network codes k K i,j as well as = (i,j) is determined at eac node wit periodic control packet excanges or estimated troug routing table 3 Following te poison-antidote terminology of [4], we call antidotes te packets of flows s tat are coded togeter wit s, and tus are needed for te next op of s to be able to decode Eg, in Fig, a is te antidote tat B needs to overear over link A B, to decode a+b and obtain b

4 are iid according to a uniform distribution However, in our simulations, we consider a Rayleig fading cannel model Te loss probabilities are calculated at eac intermediate node as explained later in tis section 4) Routing: Eac flow s S follows a single pat P s N from te source to te destination, wic is pre-determined by a routing protocol, eg, OLSR or AODV, and given as input to our problem Note tat te nature of wireless networks is time varying, ie, nodes join and leave te system dynamically In suc cases, te routing protocol actively determines new pats wic are used as input to our problem It is not critical tat te pats remain fixed, neiter from a teoretical nor from a practical point of view, as explained in te following sections Also, note tat several different yperarcs may connect two consecutive nodes along te pat We define H s = if s is transmitted troug yperarc using network code k K ; and H s =, oterwise B Intra- and Inter-session Network Coding Next, we give an overview of ow an intermediate node performs intra- and inter-session NC Te implementation details are provided in Section V ) Intra-session Network Coding (for Error Correction): Consider te commonly used generation-based NC [3]: packets from flow s S are divided into generations (note tat we use generation and block terms intercangeably), wit size G s At te source s, packets witin te same generation are linearly combined (assuming large enoug field size) to generate G s network coded packets Eac intermediate node along te pat of flow s adds P s parity packets, depending on te loss rates of te links involved in tis op At te next op, it is sufficient to receive G s out of G s + P s packets Te same process is repeated at every intermediate node until te receiver receives G s error-free packets, wic can ten be decoded and be passed on to te application Tere are many ways to generate parities (P s ) in practice We use generation based intra-session NC [3] for tis purpose Altoug one could use various coding tecniques, suc as Reed-Solomon or Fountain codes, using intra-session NC as several advantages First, it as lower computational complexity Second, in systems like COPE tat already implement inter-session NC, it is natural to incrementally add intrasession NC functionality Moreover, in tis setting, op-by-op intra-session coding (in wic redundant packets are generated at eac op) is clearly a better coice tan end-to-end coding for dealing wit loss In terms of performance, op-by-op coding acieves iger end-to-end trougput (tanks to introducing less redundancy tan end-to-end coding), witout adding ig complexity (and tus delay) to te intermediate nodes Furtermore, in terms of system implementation, our op-by-op sceme requires minimal modifications on top of te inter-session NC, wic is already implemented ) Inter-session Network Coding (for Trougput): After an intermediate node as added redundancy (P s ) to flow s, it treats all (G s +P s ) packets as indistinguisable parts of te same flow Inter-session NC is applied on top of te already intra-coded flows, as a tin layer between MAC and IP (similar Fig Operations taking place at end-points and intermediate nodes to COPE), sown in Fig We design two scemes, I NCstate and I NC-stateless, depending on te type of information tat is needed to make network coding decisions We define as state of a node te information about wic exact packets ave been overeard at tat node I NC-state: First, we assume tat intermediate nodes use COPE [] for inter-session coding Eac node i listens all transmissions in its neigborood, stores te overeard packets in its decoding buffer, and periodically advertises te content of tis buffer to its neigbors Wen a nodeiwants to transmit a packet, it cecks or estimates te contents of te decoding buffer of its neigbors If tere is a coding opportunity, te node combines te relevant packets using simple coding operations (XOR) and broadcasts te combination to J Te content of te decoding buffers needs to be excanged, in order to make NC decisions, ie, state syncronization is required I NC-stateless: Second, we design an improved version of COPE, wic no longer requires state syncronization Te key idea is to exploit te fact tat te redundancy already introduced by intra-session coding makes all G s +P s packets in a generation equally important 4 In tis improved sceme, eac nodei still listens to all transmissions in its neigborood and stores te overeard packets 5 Te node periodically advertises te loss rate for eac received and overeard flow, wic is ten provided as input to te intra-session NC module to determine te amount of redundancy needed In particular, te loss rates are calculated at eac intermediate node as one minus te ratio of correctly received packets over all te packets in a generation Also, te loss rate over overearing links is calculated as effective loss rate Eg, in Fig, te loss rate at node A is calculated as follows If G S +P S packets are sent by B and at least G S packets are received at A, ten te loss rate is set to If G S u packets are received by A suc tat u G S, ten te loss rate is set to u/g S Te loss rates calculated for eac generation are advertised to oter nodes in te neigborood Ten, eac node calculates 4 It no longer matters wic exact packets a node as As long as a node as any G s out of G s +P s, it can decode wit ig probability As long as it knows te percentage of received packets it can make coding decisions 5 Note tat wen inter-session network coded packets are overeard, tey are not stored in te decoding buffer, but discarded

5 its loss probabilities (ρ s and ρs,s ) as weigted average of te loss rates it as received In summary, tere is a synergy between intra- and intersession NC Intra-session makes te process sequence agnostic, wic allows inter-session coding to operate using only information about te loss rates, not about te identity of te packets Te loss rates can be used as input for tuning te amount of redundancy in intra-session NC In terms of implementation, te two modules are separable: an intermediate node first performs intra-session, ten inter-session NC IV NETWORK UTILITY MAXIMIZATION FORMULATION A I NC-state Sceme ) Formulation: Our objective is to maximize te total utility function by optimally coosing te flow rates x s at sources s S, as well as te following variables at te intermediate nodes: te fraction α s (or traffic splitting parameters, following te terminology of [4]) of flows intersession coded using code k K over yperarc ; and te percentage of time τ eac yperarc is used max x,α,τ U s (x s ) s S st Hs αs x s ρ s + A,k K,s S k (J) A H s αs x s ρs,s R τ, α s =, s S,i P s k K s S k τ γ, C q A () k K C q Te first constraint is te capacity constraint for eac flow s S k It is well-known, [5], tat NC allows flows tat are coded togeter in code k K, to coexist, ie, eac ave rate up to te rate allocated to tat code k Te rigt and side, R τ, is te capacity of yperarc ; τ is te percentage of time yperarc can be used for transmitting te k-t network code τ is determined by sceduling in te tird constraint, taking into account interference: all yperarcs in a clique interfere and sould time-sare te medium Terefore, te sum of te time allocated to all yperarcs in a clique sould be less tan an over-provisioning factor, γ Te second constraint is te flow conservation: at every node i on te pat P s of source s, te sum of α s over all network codes and yperarcs sould be equal to Indeed, wen a flow enters a particular node i, it can be transmitted to its next op j as part of different network coded and uncoded flows Te first constraint is key to our work because it determines ow to deal wit loss on te direct (ρ s ) and overearing (ρs,s ) links and ow large a fraction (α s ) of flow rate (x s) to code in te k-t code over yperarc Let us discuss te left and side in more detail 6 6 Note tat our formulation as two novel aspects, compared to prior work, wic allow us to better andle loss and parities First, we allow for flows coded togeter to ave different rates (in te first constraint in Eq ()) Second, we allow for loss rates of eac link to be specified separately, even for links in te same yperarc Te first term refers to te direct link of flow s H s αs x s is te fraction of flow rate x s allocated to code k and yperarc It is scaled by ρ s to indicate tat we use redundancy to protect against loss tat flow s experiences wit probability ρ s (Hs αs x s)/(ρ s ) is te total rate of flow s, including data and redundancy Te second term refers to loss on te overearing links Hs αs x s ρs,s is te amount of redundancy (via intra-session coding) added by te intermediate node on flow(s) s to protect flow s against loss of antidote packets Tese antidotes come from oter flows (s K ) tat are coded togeter wit flow s, reac te next op for flow s troug te overearing links, and are needed to decode intersession coded packets Example - continued In Fig, let us consider flow from B to B, as te flow of interest Te intermediate nodei adds redundancy to S to protect against loss rate ρ S (I,{B on,a }) te direct link I B It also adds redundancy to flow to protect against loss rate ρ S,S (I,{B,A }) of antidotes coming to B from flow over te overearing link A B ) Optimal Solution: To solve Eq () we follow a similar approac proposed in [36] First, we relax te capacity constraint in Eq (), and we ave te Lagrangian function: L(x,α,τ,q) = U s (x s ) ( H s q s α s x s ρ s s S Ak K s S k + ) Hα s s R τ, () were q s is te Lagrange multiplier, wic can be interpreted as te queue size for k-t network code at yperarc for flow s We define ρ s,s = if s = s, s,s S and we rewrite k K s S k as s S k K s S k Te Lagrange function is L(x,α,τ,q) = s S (U s(x s )x s A k K s S k H s αs ((qs )/( ρ s )+ s S k q k K s s S k q s R τ,s ρs ))+ A It can be decomposed into several intuitive parts (rate control, traffic splitting, sceduling, and queue update), eac of wic solves te optimization problem for one variable Rate Control First, we solve te Lagrangian wrt x s : x s = (U s )( i P s Q s i), (3) were (U s ) is te inverse function of te derivative of U s, and Q s i is te occupancy of flow s at node i and expressed as Q s i = (J) A k K s S k H s αs Qs, (4) were Q s is te queue size of flow s associated wit yperarc and network code pair {, k}: Q s = qs ρ +,s s s S k {s} qs ρs (5) Traffic Splitting Second, we solve te Lagrangian forα s At eac node i along te pat (ie, i P s ), te traffic splitting problem can be expressed as follows: min α st (J) A (J) A k K s S k α s Hs Qs k K s S k α s = (6)

6 Let us assume tat E i [Q(t)] is te maximal Q i (t) at time t suc tat Q i (t) = (t) wit A i (t) ϕ A i (t)hs Qs A i (t) := {ϕ = {(J),k} αs > or H s Qs Q i (t),(j) N st A,k K } At eac node i, te amount of traffic splitting factor α s for flow s over yperarc and code k follows; α s = κ i[e i [Q] H s Qs ]+ α were κ, s i is a positive constant, and [b] + z = b if z and [b] + z = if b and z = s It can be seen tat (J) A k K s S k α = and s (J) A k K s S k α Hs Qs Also, s k K s S k α H s Qs = only if s α = (J) A wic is possible only if H s Qs Q i, and α s (Hs Qs Q i ) = Te structure of te optimal solution of Eq (6) (ie, α s = κ i [E i [Q] H s Qs ]+ α s ) as te following interpretation: te iger te loss rate of antidotes on overearing links ρ s,s, te iger Q s, and te smaller αs Tis means tat flow s sould code fewer packets wit packets from flow(s) s in code k, wen antidotes from s are likely to be lost Example - continued: In Fig, tis means tat I sould combine fewer packets from te two flows if tere is loss on te overearing link A B In te extreme case were loss rate is over te link A B, inter-session coding sould be turned off At te oter extreme, were tere is no loss, te two flows sould always be combined Sceduling Tird, we solve te Lagrangian for τ Tis problem is solved for every yperarc and every clique for te conflict graps in te ypergrap max τ A k K s S k q s R τ st C q k K τ τ, C q A (7) Let us assume tat Q = R s S k q s, and E C q [Q(t)] is te minimal Q Cq (t) at time t suc tat; Q Cq (t) = Q (t) wit A C q := {φ = {} τ > A Cq (t) φ A Cq or Q (t) Q Cq (t), A,k K } At eac clique C q, te fraction of te time τ tat is allocated to yperarc, and code k is as follows; τ = ε Cq [Q E Cq [Q]] + τ, were ε Cq is a positive constant and [b] + z = b if z and [b] + z = if b and z = It can be seen tat C q k K τ = and C q k K τ Q Also, C q k K τ Q = only if τ = wic requires tat Q = Q Cq or τ = and Q < Q Cq Queue Update We find te Lagrange multipliers (queue sizes) q s, using te gradient descent: q s (t+) = {qs (t)+c t{ Hs αs xs + ρ s Hs αs x s ρs,s R τ }} + (8) were t is te iteration number, c t is a small constant, and te + operator makes te Lagrange multipliers positive q s is interpreted as te queue for flow s allocated for te k- t network code over yperarc A Indeed, in Eq (8), q s is updated wit te difference between te incoming (H s αs x s)/( ρ s ) + Hs αs x s ρs,s and outgoing R τ traffic rates at 7 B I NC-stateless Sceme Te second term in Eq () describes te redundancy added by node i to protect flow s from loss of antidotes on te overearing link An implicit assumption was tat node i knows wat antidotes are available at te next op and uses only tose packets for inter-session coding However, tis knowledge can be imperfect, especially in te presence of loss Here, we formulate a variation of te problem, were suc knowledge is not necessary Instead, node i needs to know only te loss rate on all te links to te next op for flow s (eg, in Fig for flow (S ), tese are links I B and A B ) We replace te capacity constraint in Eq () wit: H s αs xs ρ s + H s αs x s ρs,s ρ s R τ (9) and tis is A,k K,s S k Te oter constraints remain te same as in Eq () Te difference from Eq () is in te second term, related to te overeard packets at te next op Any fraction of flow s added as redundancy to flow s, as well as overeard packets from s in te next op, elp to decode inter-session coded packets ofswit flows To protect transmissions of tese elping fractions (H s αs x s ρs,s ) from being lost on te direct link to te next op of flow s (eg, from I to B ), we add redundancy to matc te loss rate of tat direct link (ρ s in general, ρs {I,B } in te example) Tis is wy te term H s αs x s ρs,s is divided by ρs Te solution of tis optimization problem also decomposes into rate control, traffic splitting, and sceduling problems, wic correspond to Eq (3), (6), and (7), respectively Q s needs to be updated: Q s = qs ρ + q s s s S k {s} ρs,s ρ s Te Lagrange multiplier is updated as follows; () q s (t+) = {qs (t)+c t{ Hs αs xs ρ + s H s αs x s ρs,s ρ R s τ }} + () We provide te convergence analysis of our solution in Appendix A 8 We first give te proof of convergence, ten we verify te convergence troug numerical calculations 7 Note tat te queue update in Eq (8) can be re-written as; q s = γ [ Hs αs xs ρ s + Hs αs x s ρs,s R τ ] + q s, were γ is a positive constant 8 We do not claim tat te solution of our network utility maximization problem is te optimal solution to te general intra- and inter-session NC problem over wireless networks Tis is well-known, open problem [6], [7], [8] Even witout an optimal, closed form solution, tere is still value in using te structure of te solution to design mecanisms tat perform well in practice, as we sow troug te numerical and simulations results in te next sections

7 V SYSTEM IMPLEMENTATION We propose practical implementations of te I NC-state and I NC-stateless scemes (Fig ), following te NUM formulation structure A Operation of End-Nodes At te end nodes, tere is an adaptation layer between transport and intra-session NC wic as two tasks: (i) to interface between application and intra-session NC; and (ii) to optimize te reliability mecanism at te transport layer Task (i): At te source, te adaptation layer sets te generation (block) size G s G s is set according to application; eg, media transmission requirements for UDP, or set equal to TCP congestion window for TCP applications and canges over time Te adaptation layer receives G s original packets p,p,,p G s from te transport layer of flow s and generates G s intra-session coded packets; a = p, a = p + p,, a G s = p ++p G s We call tis coding incremental additive coding We cose te incremental additive coding to avoid introducing coding delays (ie, our algoritm does not need to wait G s packets to encode packets) as proposed in [] Te intra-session eader includes te block id, packet id, block size, and coding coefficients At te receiver side, te reverse operations are performed Task (ii): To furter optimize te interaction between I NC and transport, particularly TCP, we keep track of and acknowledge te number of received packets in a generation, rater tan teir sequence numbers (note tat tis part is not needed for UDP protocol) Tis idea is similar to te use of end-to-end FEC and intra-session NC tat make TCP sequence agnostic [9], [9], [] Eg, if a receiver receives te first packet labeled wit block id g s =, ten it generates an ACK wit block id g s = and packet id η s = Te uncoded packets, p,p,,p G s, are stored in a buffer at te source for TCP ACK adaptation Eg, if an ACK for block id g s = and packet id η s = is received by te source, ten te TCP adapter matces tis ACK to packet p and informs TCP tat packet p is ACK-ed As long as te TCP receiver transmits ACKs, te TCP clock moves, tus improving TCP goodput After te ACK wit te block and packet ids is transmitted by te TCP receiver, te packet is stored at te receiving buffer Wen te last packet from a generation is received, ten packets are decoded and passed to te application B Operation of Intermediate Nodes An intermediate node needs to take a number of actions wen it receives (Alg ) or transmits (Alg ) a packet ) Receiving a packet and intra-session network coding: Buffer packets A node i may receive a packet from iger layers or from previous ops In te latter case, if te received packet is inter-coded, it is decoded and te packet wit destination to tis node is stored (or is passed to transport if it is te last op) If it is not te last op, a packet a l {a,a,,a G s} is stored in te output queue Q i In addition to te pysical output queue Q i, te node i keeps track of several virtual queues; Q s per (flow, Algoritm Node i processes packet a l from flow s : Read te information: packet a l, from flow s (generation size G s ) : Insert a l into te pysical output queue Q i 3: Determine {,k } and label a l wit {,k } pair and s 4: Update q s,k (using Eqs (8) and ()) and qs,k 5: Calculate Q s,k (using Eqs (5) and ()) and Qs i 6: G s,k = Gs,k + 7: if G s packets from flow s are received at node i ten 8: Calculate te number of parities P s,s, Ps,s 9: Create P s,s parities from s and Ps,s parities from s : Label all generated parities wit {} pair and s (using Eq (4)) yperarc, code) Te packet a l is labeled wit (,k,s), wic essentially indicates weter and ow to code tis packet according to te traffic splitting in Eq (6): we pick {,k } = argmin {H s Qs }, randomly breaking ties Note tat tis labeling is local at te node, and does not introduce any transmission overead Note tat H s is te indicator weter flow s is transmitted over yperarc wit code k Tis indicator is determined at eac node using a routing table wic as a data structure to determine te next ops (note tat pats do not need to be known by te sources or any node in te system) Basically, if a packet from flowsis able to reac to te next op determined by te routing table wen it transmitted over yperarc and wit code k, ten te indicator is set to, oterwise We also note tat in tis system, as long as pats remain fixed for longer (at least longer tan a time required to transmit a packet) time periods, we can see more benefit from NC, because eac node will learn wic flows can be network coded and estimate te loss rates better as time gets longer However, even in te extreme case in wic pats cange very fast (say for example at every packet transmission), our system works well, but it does not fully exploit NC opportunities, since it cannot estimate weter NC is possible or not However, it works not worse tan a system witout NC Terefore, I NC is designed to adapt to pat canges and to exploit NC benefit if possible Update Virtual Queue Sizes Wen packet a l is selected to be transmitted wit te k -t network code over yperarc, te virtual queues; Q s,k and qs,k sould be updated q s,k is updated according to Eqs (8) and () Qs,k is calculated according to Eq (5) for I NC-state and Eq () for I NC-statelessQ s i is calculated according to Eq (4) Ten, te number of packets G s,k from te same generation tat are allocated to,k pair is incremented: G s,k = Gs,k + G s is set to for eac new generation Generate Parities After G s packets from a generation of flow s are received at node i, P s parity packets are generated via intra-session NC (wic is performed according to random linear NC [3]) and labeled wit information (s,, k) Tere are two types of parities P s,s = Gs ρs /(ρs ) parities are added on flow s s virtual queue to correct for loss during direct transmission to te next op over yperarc = G s,s ρs, s S k parities are added on P s,s te virtual queues of oter flows s tat are inter-session coded togeter wit s Tis is to elp te next op for s to decode despite losses on te overearing link

8 Algoritm Node i transmits a packet : Select {,k } pair tat maximizes Q = R ( s S k q s ) : Initialize: ξ = 3: for a l Q i do 4: if a l is labeled wit {,k } AND flow id label of a l is different from a l ξ ten 5: if I NC-state AND ξ a l is decodable OR I NC-stateless ten 6: Insert packet to ξ 7: Network code (XOR) all packets in ξ 8: Broadcast te network coded packet over yperarc 9: Update q s,k, s S k : Re-calculate Q = R ( s S q s k ) and Qs i (using Eq (4)) Tese parity packets are for I NC-state For I NC-stateless P s,s is te same, but,s Ps = G s,s ρs /( ρs ), ie, additional redundancy is used to protect parity packets from loss on te direct link ) Transmitting a packet and inter-session network coding: We consider te 8 MAC Wen a node i accesses a cannel, {,k } is cosen to maximize Q = R ( s S k q s ) according to Eq (7), randomly breaking ties Altoug te pair {,k } determines te yperarc, code and flows to be coded togeter in te next transmission, te specific packets from tose flows still need to be selected and coded We call tese packets te set ξ, and select tem using te procedure specified in Alg 9 To acieve tis, we first initialize te set of network coded packets ξ = For eac packet a l Q i, ceck weter a l is labeled wit {,k } If it is, ten we ceck weter its flow id label already exists in one of te packets in ξ, ie, anoter packet from te same flow as already been put in ξ If not, tere is one more ceck for I NC-state for decodability at te next ops of all packets in te network code, based on reports or estimates of overeard packets in te next ops, similarly to [] If te packet is decodable wit some probability larger tan a tresold (default value is ) ten, a l is inserted to ξ In I NC-stateless, te packet a l is inserted to ξ witout cecking te decodability, wic is ensured troug te additional redundancy packets Tis is te strengt of I NC-stateless: it eliminates te need to excange detailed state, wic is costly and unreliable at ig loss rates After all packets in Q i are cecked, te labels (,s) of te packets in ξ, inter-session NC eader is added, and coded (XORed) and broadcast over After a coded packet is transmitted, te virtual queues are updated according to Eqs (8), () Te queues Q,k and Q s i are calculated according to Eqs (5), (), (4) We note tat in bot I NC-state and stateless, packets are network coded if some conditions are satisfied However, if tese conditions are not met, a packet witout NC is still transmitted, because at least one packet is inserted in ξ (Alg ) Tus, we do not delay any packets in our scemes Yet, delaying packets may create more NC opportunities and tere is a tradeoff between delay and trougput Tese issues ave 9 Te inter-session NC eader includes te number of coded packets togeter, next op address, and te packet id s Note tat tis eader as well as te IP eader of eac packet are not network coded Note tat I NC may cause re-ordering at te receiver, but since we already implemented intra-session NC, and made TCP receiver sequence agnostic in tis term, out of packet delivery is not a problem for TCP been considered in some previous work [3], [3] However, tis is an aspect ortogonal to te focus of I NC (wic is te synergy between inter- and intra-session NC) and can potentially be combined wit it 3) Keeping Track of and Excanging State Information: For I NC-state, intermediate nodes need also to keep track of and excange information wit eac oter, so as to enable te intra- and inter-session NC modules to make teir redundancy and coding decisions and to provide reliability An approac similar to COPE is used: ACKs are sent after te reception and successful decoding of a packet Information about overeard packets is piggy-backed on te ACKs Wit I NC-stateless, we only need neigbors to excange information about te loss rates at te neigboring nodes Information about te loss rates as well as te number of received packets at a generation is reported troug control packets for every generation In order to provide reliability, we consider re-transmissions In I NC-state, a packet is removed from te output queue only after an ACK related to te packets is received Oterwise, te packet is re-transmitted after a round trip time In I NCstateless, packets are removed from te output queue wen a control packets is received and confirms te successful transmission of all packets of te corresponding generation Oterwise, a number of intra-session coded packets from te generation wic are missing at te receiver are generated from te packets kept in te queue and transmitted 4) Congestion Control and Queue Management: End-toend congestion control (ie, rate control) is given by Eq (3) in wic if U s (x s ) = log(x s ), ten x s = / ( i P s Qi) s Tis means tat flow rate x s is inversely proportional wit increasing queue size over te pat of flow s Tis beavior is similar to TCP s end-to-end congestion control algoritm, were congestion at a node may result in one or more packets may be dropped from te buffer at tis node TCP reacts to packet drops by reducing its rate Tus, TCP reduces its flow rate wen queue size increases Tis gives us intuition tat TCP mimics te rate control part of te decomposed solution Tis intuition as been validated in [7], [33], [34], [35] Similarly, we consider tat TCP already mimics te structure of te rate control part in Eq (3) Terefore, upon congestion at nodei, te per-flow queue sizesq s i are compared and te last packet from flow s aving te largest Q s i is dropped from te queue; in case of a tie, an incoming packet is dropped We do not make any additional updates to TCP s end-to-end congestion control algoritm Also, we do not implement any end-to-end congestion control mecanism for UDP Our goal is to keep UDP as it is (witout any end-to-end control) and sow te effectiveness of I NC-state and I NCstateless wen tere is no end-to-end control Example : Let us re-visit te X-topology from Fig, sown again for convenience in Fig 3, and illustrate ow we perform intra- and inter-session NC under sceme I NCstateless Te loss probabilities over te direct (I B ) and In our implementation, te loss probabilities are calculated as weigted average of te loss rates Te weigted average is calculated over a window of samples Te last samples are ordered suc tat te newest sample is te first sample, and te oldest sample is te t sample Eac sample is given a weigt inversely proportional to its sample number

9 (a) Intra-session coding (b) Inter-session coding Fig 3 Example of coding (under sceme I NC-stateless) at node I in te X-topology Tere is loss only on two links: te direct link I B (wit probability 5) and te overearing link A B (wit probability 5) overearing (A B ) links are assumed 5 and 5 In Fig 3(a), we describe intra-session NC Let us assume te generation size of S is G S = 4 and S is G S = Te packets transmitted by A, B are a,a,a 3,a 4 and b, respectively Note tat tere is only one option for intersession NC, ie, to XOR packets from te two flows, tus tere exists only one possible network code k = over yperarc = (I,{B,A }) All packets are labeled wit tis information and teir flow ids Te labeled packets are a S,aS,aS 3,aS 4 and b S Parities are generated as follows Since G S I,{B,A } = 4 and GS I,{B,A } =, te number of parities is P S,S I,{B,A } =, PS,S I,{B,A } =, PS,S I,{B =,A } (tus generating one parity from flow S and labeling it wit S, ie, b S ), and PS,S I,{B,A } = (tus generating two parities from flow S and labeling tem wit S, ie, a S 5,aS 6 ) In Fig 3(b), we describe inter-session NC Node I performs inter-session NC and transmits packets according to Alg : it XORs packets from te two queues, for S,S, and broadcasts over te yperarc (I,{B,A }) In particular, it transmits te, and as following packets: a S bs, as bs, as 3 as 5 a S 6 A receives and decodes all te packets B receives 4 3 packets on te average over overearing link A B and receivespackets over transmission link IB Five received packets allows B to decode all five packets a,a,a 3,a 4,b, so b is successfully decoded A Simulation Setup VI PERFORMANCE EVALUATION We used te GloMoSim simulator [8], wic is well suited for simulating wireless environments We considered various topologies: X topology, sown in part of Fig and repeated in Fig 4(a); te cross-topology wit four end-nodes generating bi-directional traffic, wit one relay sown in Fig 4(b); te weel topology sown in Fig 4(c); and te multi-op topology sown in Fig In X, cross, and weel topologies, te intermediate node I is placed in a center of of circle wit radius 9m over m m terrain and all oter nodes A, B and etc are placed around te circle In te multiop topology of Fig, two X topologies are cascaded and te distance between consecutive nodes is set to 9m Te topology is over a 8m 3m terrain We also considered various traffic scenarios: FTP/TCP and CBR/UDP TCP and CBR flows start at random times witin te first 5sec and are on until te end of te simulation wic is 6sec Te CBR flow generates data packets at every ms IEEE 8b is used in te MAC layer, wit te addition of te pseudo-broadcasting mecanism, as in COPE [] In terms of wireless cannel, we simulated te two-ray pat loss model and a Rayleig fading cannel wit average cannel loss rates,,3,4,5 % We ave repeated eac 6sec simulation for seeds Cannel capacity is Mbps, te buffer size at eac node is set to packets, packet sizes are set to 5B, te generation size is set to 5 packets for UDP flows and to te TCP window size for TCP flows We compare our scemes (I NC-state and I NC-stateless) to no network coding (nonc), and COPE [], in terms of total transport-level trougput (added over all flows) B Simulation Results TCP Traffic In Fig 5, we present simulation results for two TCP flows in X topology sown in Fig 4(a) to illustrate te key intuition of our approac Consider, for te moment, tat loss occurs only on one link, eiter (a) te overearing link A B or (b) te direct link I B Te first case is depicted in Fig 5(a) Loss on te overearing link does not affect te uncoded streams, tus te trougput of TCP+noNC does not cange wit loss rate Wen NC is employed, reports carrying information about overeard packets may be delivered late to intermediate node I Tus, tere are some instances tat intermediate node sould make a decision even if it does not ave te exact knowledge In tis case, I makes a decision probabilistically Specifically, if decoding probability exceeds some tresold (% in our simulations), I codes packets However, some of tese packets may not be decodable at te receiver It is wy te performance of TCP+COPE and TCP+I NC-state reduce wit increasing loss rate and equals to te trougput of TCP+noNC after % loss rate (NC is turned off after % loss rate) However, TCP+I NC-state is still better tan TCP+COPE, because wen it makes probabilistic NC decision (wen loss rate is less tan %), it adds redundancy considering te loss rate over te overearing link Tis improves trougput, because adding redundancy using intra-session NC makes all packets equally beneficial to te receiver and te probability of decoding inter-session network coded packets increases TCP+I NCstateless outperforms oter scemes over te entire loss range For example, if tere is no loss, I NC-stateless still brings te benefit due to eliminating ACK packets and using less overead to communicate information (ie, COPE and I NCstate excanges te information about te overeard packets,