Online network coding for optimal throughput and delay -- the three-receiver case

Online net w ork co ding for optimal throughput and dela y – the three-receiv er case Ja y Kumar Sundarara jan, Dev a vrat Shah, Muriel M ´ edard Abstract F or a pack et erasure broadcast channel with th ree receiv ers, we prop ose a new cod ing algorithm t h at makes use of feedbac k to dynamically adapt the code. O u r a lgorithm is throughput optimal, and we co njecture that it also ac hieves an asymptotically optimal ave rage deco ding delay at the receivers. W e consider heavy t raffic asymptotics, where the load factor ρ approaches 1 from b elo w with either the arriv al rate ( λ ) or the channel parameter ( µ ) b eing fix ed at a num b er less than 1. W e verif y through sim u lations that our algorithm ac h ieves an asymptotically opt imal decod in g dela y of O  1 1 − ρ  . 1. In tro duction Reliable comm unication over pack et er asure c hannels is a well studied problem. Sev eral solutions hav e been pro posed, each with its own requirements, merits and issues. In this w ork, we c onsider a three-r eceiv er pa c ket erasur e broadcast channel with feedback a nd address questio ns of throug hput and deco ding delay a t the rec eiv ers. T o communicate over a pack et er asure br oadcast c hannel, one can use the random linear netw or k co ding solution of [1], where t he sender tr ansmits a random linea r combination of all pack ets that have a rrived so fa r. Digital fountain co des ([2, 3]) form a nother approach to this problem. These solutions use coding to ensure that with high probability , the tr a nsmitted pack et will hav e what we call the i nnov ation guaran tee prop ert y , i.e. , it will b e innovative 1 to every receiver that receives it, except if the receiver already knows as muc h as the sender. Th us, every successful reception brings a unit of new information. Such schemes a c hieve 100% throughput. How ever, b oth foun tain co des and ra ndom linear netw o rk co ding p erform blo ck-based enco ding. In g eneral, the receiver may not be able to extract the original pac kets from the co ded pac kets till the entire blo ck has been received. This lea ds to a deco ding delay , which is a problem for real-time pa c ket streaming applica tions such as video. Ideally we want a c o de that would allow pla yback even befo re the full blo ck is received. In other words, we are in terested in minimizing the av erage p e r-pack e t delay . Rela ted questions have b een studied b y [4], [5], [6 ] and [7 ]. With full feedback, the optimal scheme ov er a p oint-to-point pack e t erasure channel is Automatic Repe a t reQuest (AR Q) – the s ender simply retransmits a pack et upon er asure. This scheme also has the adv a n tage of being a sliding window approach as oppo sed to a block-based approa c h. Although it achiev es 100 % t hroug hput and optimal pack et delay , AR Q do es not extend to bro adcast-mo de links. On the other ha nd, netw o rk co ding readily extends to broadcast-mo de links. Reference [8 ] prop oses a scheme that uses feedba c k to a c knowledge degrees o f freedo m instead o f original pack ets, th us co m bining the b enefits of netw ork co ding and ARQ. This new framework a llo ws the se nder to dyna mically adapt its co de to incorp orate receivers’ s tates of knowledge. This fact was used to design a queue management algorithm called the drop-when-s e e n algorithm that minimizes the sender’s queue size, along with a coding mo dule that provides 100 % throughput. Another related re ference is [9], where the authors combine an acknowledgmen t s c heme with net work co ding. Here, the main fo cus is to maximize the thr oughput. In contrast to these works, our current work fo cuses on achieiving the b est p ossible deco ding delay for all r e ceiv ers, while maintaining optimal thro ughput. By deco ding delay of a r e c eiv er, we mean the time ela ps ed betw een the a rriv al of a pack et in to the sender’s queue and its ge tting deco ded by the receiver under consideration, av er aged over the pack ets in the lo ng run in a pack e t str eaming scenario. This is different fro m but related to the notion of delay used in [7]. F or the spec ia l cas e of a pack et eras ure broadcast channel with only t wo receivers, reference [10 ] prop oses a feedback- based co ding algo rithm that no t only a c hieves 100% throughput, but also guar a n tees that every successful innov ative The authors are i n the Department of Electrical Engineering and Computer Science, at the M assac husetts Institute of T ec hnology , Cam- bridge, MA 02139. Email: { jaykumar, dev avrat, medard } @mit.edu. This m ate rial i s based upon work under sub con tract # 18870740-37362-C issued b y Stanford Universit y and supp orted b y DARP A, and up on w ork under a sub con tract # 060786 issued by BAE Systems National Securit y Solutions, Inc. and supp orted b y DARP A and the Space and Nav al W arfar e System Center (SP A W ARSYSCEN), San Diego under Con tract No. N66001-06-C-2020. 1 An inno v ative pac ke t is a coded pack et whose coefficient v ector is outside the span of previously received pack ets’ coefficient vec tors. Slot number t Point of arrival Point of departure for physic al queue Point of transmission Time Point wh ere stat e variables are measur ed Point of feedb ack Figure 1: Relative timing of arriv al, service and departure p oints within a slot reception will cause the receiver to deco de a new pack et. W e call this pro perty instantane ous de c o dability . Instantaneous deco dabilit y and 100 % throughput are b oth desirable goals. Howev er , this approa c h do es not extend to the cas e of more than t wo re ceiv ers. With prior knowledge o f the er asure patter n, [7] gives an offline algor ithm that achiev es optimal delay and throughput for the case of three receivers. Howev e r , in the online case, even with only three receivers, [10] shows through an exa mple that it is not p ossible to simultaneously gua rant ee ins tan taneous deco dability as well a s throug hput optimality . In the light of this example, our cur ren t work aims for a relaxed version o f instantaneous deco dability while still retaining the requir emen t of optimal throughput. Our relaxatio n of the c ondition is as follows. Let λ and µ b e the ar riv al rate and the channel quality parameter resp ectiv ely . Let ρ , λ/µ be the lo ad factor. W e consider a symptotics when the load factor on the system tends to 1 ( i.e. , 100%), while k eeping either λ o r µ fixed at a num ber le ss than 1 . The optimal throughput requirement means that the queue of undelivered pac kets is s table for all v a lues of ρ less than 1. Our new requirement on deco ding delay is that the growth o f the av e r age deco ding delay as ρ → 1 should be the sa me a s for the single rec e iver c a se. The exp ected p er-pack et delay of a r eceiv er in a system with more than one receiver is clearly lower bo unded by the corresp onding quantit y for a single-rece iv er system. Thu s, instead of optimal deco ding delay , we aim to guarantee asymptotically optimal deco ding delay . The motiv ation is that in mo st pra ctical systems, delay b ecomes a critical is sue only when the sy stem starts a pproaching its full capa cit y . When the load on the s ystem is well within its capacity , the delay is usually small and hence not an iss ue. F or the case of t wo rece ivers, it can be shown that this rela xed requirement is satisfied b y the sc heme in [1 0] due to the instantaneous deco dability pro perty , i.e. , the sc heme achiev es the asy mptotically optimal av erag e deco ding delay per pack et for the t wo-receiver case. In our current work, we provide a new co ding module fo r the case of three receivers that ac hieves optimal throug hput. W e conjecture tha t at the same time it also achiev es an asymptotically optimal deco ding delay in the following sens e. With a single r eceiv er, the o ptimal sc heme is ARQ with no co ding and we show that this achiev e s an ex p ected p er-packet delay at the sender of Θ  1 1 − ρ  . F or the three-receiver sys tem, w e conjecture that our sc heme also achiev e s a delay of O  1 1 − ρ  , and th us meets the lo wer b ound in an asymptotic s ense. W e have verified this b ehavior through sim ulations for v alues of ρ that ar e v ery close to 1. Our scheme th us achiev es feedback-based control of the deco ding delay , alo ng the lines s ug gested in [11]. W e belie ve that our appro ac h can b e extended to an arbitra r y num b er of receivers as well. 2. Preliminaries 2.1. The setup The setup is the same as in [8]. Time is slotted. P ack ets a rrive in to the sender’s queue a c cording to a Bernoulli pro cess of r a te λ . The sender wan ts to br oadcast this str eam to three receivers ov er a packet erasure broadcas t channel. The sender has one q ueue with no preset size co nstrain ts. The c hannel a c cepts one packet p er s lot. Each receiver either receives this pack et with no erro rs (with probability µ ) or an erasure o ccurs (with probabilit y (1 − µ )). Era sures o ccur independently acro ss receivers and a cross slo ts and ca n b e detected by rec e iv ers. There is p erfect feedback in each slot. Figure 1 shows the timing of event s in a slo t. In particular, we ass ume that the sender finds out whether the receiv ers received the previous s lot’s transmission b efore selecting the current s lo t’s transmission. 2.2. The lo w er bo und λµ λ + µ λ 0 1 2 3 λµ µ λ + µ λ λµ µ λ + λµ µ λ + µ λ µ λ µ λ µ λ Figure 2: Markov chain for the sender’s queue s ize – sing le receiver case. Here ¯ λ := (1 − λ ) a nd ¯ µ := (1 − µ ). The exp ected p er-packet delay for the sing le receiver case is clea rly a lower b ound for the corr esponding quantit y at one of the r e c eiv ers in a three-receiver system. W e will compute this low er bo und in this section. Figure 2 shows the Marko v chain for the queue s iz e. If ρ := λ µ < 1, then the chain is p ositive recur ren t and its steady state distribution ca n be fo und ([12]). Based on this, the steady state exp ected queue size can b e computed to be ρ (1 − µ ) (1 − ρ ) = Θ  1 1 − ρ  . Now, if ρ < 1 , then the sy stem is stable and Little’s law c an be applied to show that the exp ected p er-pack et delay in the single receiver s y stem is also Θ  1 1 − ρ  . 2.3. Representing kno wledge W e trea t pac kets as vectors ov er some finite field. Throughout this pap er, w e consider a single sour ce that generates a stream of pack ets. The k th pack et that the so ur ce genera tes is said to hav e an index k and is deno ted a s p k . W e assume that the sender uses o nly linear co des, i.e. , the tr ansmission is some linea r combination of pack ets. The linea r combination is uniquely s pecified using the vector of co efficient s used to for m it. With linea r co des, the s tate of knowledge o f a node after receiving some set of linea r c o m binations has a vector spac e s tructure. This is b ecause the node can compute an y linear combination whose c o efficient v ector is within the linear span o f the co efficient vectors of pr e viously received linear combinations. This leads to the following definition of kn ow le dge sp ac e which we res tate from [8]. Definition 1 (Knowledge of a no de) The knowledge of a no de is the set of al l line ar c ombinations of original p ackets that it c an c ompute, b ase d on the informatio n it has r e c eive d so far. The c o efficient ve ctors of these line ar c ombinations form a ve ctor sp ac e c al le d the knowledge spa c e of the no de. The dimension of this ve ctor sp ac e is c al le d the rank of the no de. W e next restate the definition of a no de “seeing” a messag e pack et from [8 ]. A node is said to ha ve se en a message pack et p if it has received enough infor ma tion to compute a linear combination o f the form ( p + q ), where q is itself a linear c om bination inv o lving only pack ets with an index greater than tha t o f p . (Deco ding implies seeing, as we can pick q = 0 .) The num b er of pack ets s een b y a no de is pr ecisely the r ank o f the no de (see [8] for more details ). W e intro duce a new notion of pack ets tha t a no de has “he a rd o f ”. A no de is said to have hear d of a pack et if it knows some linear combination in volving that pack et. 3. The new co ding mo dule Our co ding mo dule works in the Ga lo is field of s ize 3. At the b eginning of every s lot, the mo dule has to decide what linear combination to tra nsmit. Since there is full feedbac k, the mo dule is fully aw a re of the curre nt state of kno wledge of e a c h of the three receiv ers. Thus, it can compute the ra nk of eac h re c eiv er. W e denote the highest rank among the three r eceiv ers as m . Our co ding mo dule main tains a n inv a r ian t that the transmission will never inv olve a packet whose index is greater than ( m + 1). W e denote the rece iver(s) whose r a nk is m as the leader (s). W e consider three cases: 3.1. All three recei v ers are leaders In this case, a ll thr ee receivers hav e a ra nk of m which means each has seen m pack ets. If p m + 1 has not arrived yet, the mo dule do es no thing. Otherw is e, since all transmissions hav e in volv ed only the set of packets up to p m + 1 , there is exactly one unseen pack et for each receiver within this set. This could be a different pack et for each of the three r eceiv ers. The co ding mo dule selects a linear com bination that if received successfully by any receiver, will rev eal to that receiver its unseen pack et, thereby guaranteeing innov a tion. This is done by simply forming a linear com bination in volving only the unseen pack ets of the three rece iv ers. It can b e verified that with a field of size 3, it is always p ossible to choose co efficien ts such that innov ation is guara n teed for all three receivers. S 1 S 2 S 3 S 4 S 4 S 5 S 6 S 7 S 8 S 9 H 1 H 2 D 1 D 2 U Figure 3: Sets used by the co ding mo dule 3.2. There are tw o le aders It ca n b e shown by induction that at all times, at lea st one leader would have decoded all pack ets from 1 to m . Now, when ther e are tw o leader s, if ex a ctly one lea der has dec o ded all pack ets 1 to m , then the co ding mo dule perfor ms the op erations o f ca se 3, trea ting this lea de r a s the unique leader. If b oth leaders have decode d pack e ts 1 to m , then mo dule do es the following. If p m + 1 has no t arrived y et, the mo dule tra nsmits the o ldest undeco ded pack et of the non-leader (if there are packets that the non-lea der has he a rd of but not yet deco ded, then they are pr eferred). Suppose p m + 1 has arrived. Now, if it has alrea dy b een deco ded b y the no n-leader, then the mo dule s ends the sum of p m + 1 and the oldest undeco ded pack et of the non-lea de r (ag ain, if there are pack ets that the no n-leader has heard of but not yet deco ded, then they are pre ferred). Otherwise, the mo dule sends p m + 1 by itself. 3.3. Unique leader In this ca se, the mo dule co mputes the following sets for the tw o non-lea de r s: H 1 := Set of packets heard o f by first non-leader H 2 := Set of packets heard o f by seco nd no n-leader D 1 := Set of pack ets deco ded by first non-lea der D 2 := Set of pack ets deco ded by second non- le a der Note that D 1 ⊆ H 1 and D 2 ⊆ H 2 . W e also define a universe se t U c o nsisting of pack ets p 1 to p m , and also p m + 1 if it has arrived. In this setting, the following sets partition the universe (r efer to Figur e 3): • S 1 = D 1 ∩ D 2 • S 2 = D 1 ∩ ( H 2 \ D 2 ) • S 3 = D 2 ∩ ( H 1 \ D 1 ) • S 4 = ( H 1 \ D 1 ) ∩ ( H 2 \ D 2 ) • S 5 = D 1 \ H 2 • S 6 = D 2 \ H 1 • S 7 = ( H 1 \ D 1 ) \ H 2 • S 8 = ( H 2 \ D 2 ) \ H 1 • S 9 = U \ ( H 1 ∪ H 2 ) The co ding mo dule picks a linear combination de p ending on which of these sets p m + 1 falls in, as follows: Case 1 – p m + 1 has not arrive d: Chec k if S 4 is non-empty . If it is, then send the oldest packet in S 4 . Otherwise, chec k if b oth S 2 and S 3 are non-empty . If they a re, pick the oldest pa cket from each, and s e nd their sum. If no t, tr y the following pairs of sets: S 3 and S 5 , e ls e S 2 and S 6 , e ls e S 5 and S 6 . If none o f these pa irs of sets work, then send the olde s t pack et in S 7 if it is non-empty . If not, tr y S 8 , S 9 , S 2 , S 3 , S 5 and S 6 in that order. If all of these are empty , then s end nothing. Case 2 – p m + 1 ∈ S 1 : This is identical to case 1, except that p m + 1 m ust also b e added to the linea r combination that case 1 suggests. Case 3 – p m + 1 ∈ S 2 : Send p m + 1 added to ano ther packet. The other pack et is chosen to b e the oldest pa c ket in the first non-empty set in the following sets, tested in this par ticular o rder: S 3 , S 4 , S 6 , S 8 , S 7 and then S 9 . Case 4 – p m + 1 ∈ S 3 : This is similar to the S 2 case (using symmetry) – test S 2 , S 4 , S 5 , S 7 , S 8 and then S 9 . Case 5 – p m + 1 ∈ S 4 : Send p m + 1 as it is. Case 6 – p m + 1 ∈ S 5 : Send p m + 1 added to ano ther packet. The other pack et is chosen to b e the oldest pa c ket in the first non-empty set in the following sets, tested in the following order: S 3 , S 6 , S 4 , S 8 , S 7 and then S 9 . Case 7 – p m + 1 ∈ S 6 : This is similar to the S 5 case (using symmetry) – test S 2 , S 5 , S 4 , S 7 , S 8 and then S 9 . Case 8 – p m + 1 ∈ S 7 : Send p m + 1 as it is. Case 9 – p m + 1 ∈ S 8 : Send p m + 1 as it is. Case 10 – p m + 1 ∈ S 9 : Send p m + 1 as it is. In all these cases, the co efficien t for the chosen packets must b e selected to be either 1 or 2, in such a w ay that the resulting linear combination is innov ative to any receiver that receives it, except if the receiver already knows all that the sender knows. It can b e shown that s uc h a choice is always p ossible with a field o f size 3. Remark 1 W e conjecture based on the simulations tha t the a lgorithm maintains the follo wing in v ar ian t – a t most one of H 1 \ D 1 and H 2 \ D 2 is no n- empt y at any given time. If proved to b e true, this observ a tion can be used to simplify the algorithm. 4. The in tuition b ehi n d the algorithm The main idea behind the algorithm is to first o f all guarantee inno v ation. It can be shown that the linear combination computed by this co ding mo dule is indeed innov a tiv e to any receiver that receives it. In addition to this r equiremen t how ever, the module also tries to cause each receiver that has a successful reception to deco de as many pack e ts as possible. An interesting prop erty of this algo rithm is that the transmitted linear com bination always ha s at most t w o undeco ded pack ets inv o lved in it from any receiv er’s p oint of view. In other words, every transmission is essentially either a n unco ded pack et or the sum of t wo pack ets. This prop erty leads to a nice structure in the knowledge space of the r eceivers, using which, we present a strategy to control the exten t to which pack ets get mixed with each other, thereby controlling the deco ding delay . In order to explain this s tructure, we define the following r elation. The gro und set G of the relation co n tains a ll pack ets that hav e arrived at the sender so far, alo ng with a fictitious all-zero pack et tha t is known to all receivers even befo re tr ansmission b egins. The r elation is defined with res pect to a sp ecific r eceiv er. T wo pa c kets p x ∈ G and p y ∈ G are defined to b e rela ted to each o ther if the receiver knows at lea st one o f p x + p y and p x + 2 p y . Now, a packet added with tw o times the sa me pack et gives 0 which is tr ivially known to the receiver. Hence, the relation is reflexive. It is a lso symmetric since addition is a commutativ e op eration. Now, for an y p x , p y , p z in G , if a receiver knows p x + α p y and p y + β p z , then it can compute either p x + p z or p x + 2 p z by canceling out the p y , for α = 1 or 2 and β = 1 or 2. Therefore the relation is also transitive and is thus an equiv alence relatio n. It defines a par titio n on the ground set, namely the equiv alence cla sses, which provide a structured wa y to r epresent the knowledge of the node. It ca n b e seen that the class con taining the all-zero pa c ket is precisely the set of deco ded pac kets ( D 1 or D 2 ). Pack ets that have not b een inv olved in any of the successfully rece iv ed linear combinations so far will form singleton equiv a lence classes. These co r resp ond to the pack ets that the rece iver ha s not hear d of ( U \ H 1 or U \ H 2 ). W e s a y a clas s is nontrivial if it is neither a s ingleton class nor the class of deco ded pack ets. Thus, nontrivial class e s contain the pack ets that hav e b een heard of but not deco ded. Revealing any pack et in a class will reveal the entire class to the no de. T he num b er of nontrivial classes is th us the n umber of pack ets that the no de needs to know in order to deco de all pack e ts it has heard of. This n umber is thus a measure of how far aw ay a no de is fro m de c o ding a ll packets it has hear d o f. W e call this num b er the deficit of the n o de . F or instance, revealing a pa c ket from H 1 \ D 1 will allow the en tire class containing that pack et to b e deco ded by receiver 1. The a lgorithm ensures that a pack et from H 1 \ D 1 or H 2 \ D 2 is revealed whenever p ossible, a s oppo s ed to a pack et that the receiver has not heard of. This ensures that the deficit is r educed whenever po ssible. As a result, the deficit drops to zero fre q uen tly , thereby causing the no de to deco de pa c kets. 5. P erformance of the algorithm 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 1/(1− ρ ) Decoding delay per packet (in time slots) Figure 4: Deco ding delay as loa d approaches capacity 5.1. Throughput optimalit y The algorithm has bee n designed in such a wa y that innov ation is guara n teed to a ll the r e ceiv ers whenever po s sible. Pac ket p m + 1 is alwa ys included in the linear combination if it has arrived, in order to guarantee innov a tion to the lea de r . If bo th the other r eceivers hav e als o not deco ded it, then sending p m + 1 by itself sa tisfies the innov atio n gua rant ee. This happ ens in cases 5, 8, 9 and 10. If howev er , so me receiver has already dec oded it as in the o ther c a ses, then another pack et is included in the linear combination that the receiver has not yet decoded, ther eb y ensuring innov ation. While choosing such a pack e t, preference is given to pack ets that the receiver has hear d of, as revealing such a pack et will ca use several pack ets to b e deco ded a t once. If p m + 1 has not yet arrived, then the leader is alrea dy sa tis fied. F o r the other tw o receivers, the transmission is selected in such a wa y that it sim ultaneously reveals an undeco ded pack et to b oth of them whenever p ossible. W e can show tha t in all these cas es, over a field of size 3, the coefficients ca n also b e c hosen car efully to guarantee innov ation for all those who receive the linear combination successfully . This discussion is summarized in the following theorem. Theorem 1 The c o ding mo dule satisfies t he innovation guar ante e pr op erty. This mea ns that the algor ithm a c hieves o ptimal throughput, i.e. , for a ll ρ < 1 , the deco ding delay and the queue at the sender will remain sta ble. 5.2. Deco ding dela y W e now study the delay exp erienced by an arbitrary arriv al before it g ets decoded by one of the re ceiv ers, say re c eiv er 1. W e consider a system w he r e µ is fix ed at 0.5. The v alue of ρ is v a ried as follows: 0 .95, 0 .97, 0.98 , 0.9 9 and 0.99 5. W e plot the exp ected deco ding delay p er pack et av eraged across the three rec e iv ers, as a function of  1 1 − ρ  in Figur e 4 . W e also plot the log of the same qua n tities in Figure 5. The v a lue of the de lay is av eraged ov er 50 0 000 time slots for the firs t three p oint s and 10 6 time slots for the last tw o p oints. Figure 4 shows that the growth of the exp ected deco ding delay is linear in  1 1 − ρ  as ρ a pproaches 1. Fig ure 5 confirms this b ehavior – we can see that the slo pe on the plot o f the lo garithm of these quantities is indeed close to 1. This observ a tion leads to the following co njecture: Conjecture 1 F or the n ew ly pr op ose d c o ding mo dule, t he exp e cte d de c o ding delay p er p acket fr om one p articular r e c eiver’s p oint of view gr ows as O  1 1 − ρ  , which is asymptotic al ly optimal. 5.3. Queue size The queue up date rule is as follows – the sender drops a packet if all the receivers hav e deco ded it. This means that by Little’s law, the ex pected queue siz e will b e pr o por tional to the time a packet sp ends in the s ystem befor e it is deco ded. 2.5 3 3.5 4 4.5 5 5.5 6 2.5 3 3.5 4 4.5 5 5.5 6 log(1/(1− ρ )) log(Decoding delay) Figure 5: Deco ding delay – lo g-log plot Thu s, if the e xpected deco ding dela y is indeed O  1 1 − ρ  as conjectured ab o ve, then the drop- when-deco de d q ue ue up date rule will ensure that the exp ected q ueue size a t the sender is O  1 1 − ρ  , which is asymptotica lly optimal. 6. Conclusions F or a three receiver pack et era sure broadca st c hannel with feedba c k, we hav e pr opos ed a new co ding scheme that makes use of the feedback to dynamically adapt the co de. As a rgued earlier , Θ  1 1 − ρ  is an a symptotic low er bo und on the deco ding delay . W e hav e observed through sim ulations that this lo wer b ound seems to b e achiev ed by our scheme, which would imply the as ymptotic optimality of our c o ding mo dule in terms of deco ding delay . W e conjecture that this is indeed tr ue. All these delay b enefits ar e obtained without co mpromising on throughput. If the conjecture is true, then the exp ected queue size of undeco ded pack ets is a lso O  1 1 − ρ  , which is asymptotica lly o ptimal. Thus, our s cheme also simplifies the q ueue mana gemen t at the sender. In the future, we wish to extend this approach to an ar bitr ary num b er of r e ceiv ers. Also, w e wish to make the a lgorithm r obust to delays and erasures in the feedback. References [1] D. S. Lun, M. M ´ edard, and M. Effros, “On co ding for relia ble communication o ver pack et netw orks,” in 42nd Annual Al lert on Confer enc e on Commun ic ation, Contr ol, and Computing , Septem b er – O ctober 2004. [2] M. 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