Encoded packet-assisted Rescue Approach to Reliable Unicast in Wireless Networks

Transcription

1 Encoded pacet-asssted Rescue Approach to Relable Uncast n Wreless Networs Zhheng Zhou, Lang Zhou, Xng Wang, Yuanquan Tan Natonal Key Laboratory of Communcaton Unversty of Electronc Scence and Tech of Chna Chengdu, Chna E-mal: {zhzhou, lzhou, wangxng, tanyuanquan}@uestc.edu.cn Abstract Recently, there has been a growng nterest of usng networ codng to support relable uncast over an error-prone channel. However, prevous networ codng schemes focused only on natve pacets and gnored the role of encoded pacets. In ths paper, we develop an effcent retransmsson approach wth networ codng method, namely Encoded pacet-asssted Rescue (EAR), whch s able to overcome ths lmtaton. Usng the proposed networ codng approach, clents store the overheard encoded pacets and report to the sender. And then, when the sender maes codng decsons, t not only consders natve pacets, but also taes account of encoded pacets. In ths way, more codng opportuntes are emerged,.e. more pacets can be mxed together, resultng n mprovng the retransmsson effcency. Moreover, theoretcal analyss and smulaton results show that comparng wth the exstng schemes, our schemes can greatly reduce the total number of retransmssons. Keywords--Error-prone; Networ Codng; Uncast; Rescue; I. INTRODUCTION It s well-nown that wreless networs are error-prone because of fadng and nterference. Many researches have revealed that IEEE 8.-based wreless mesh networs suffer from severe pacet loss rado n wde condtons []. Hence, automatc repeat request (ARQ) [] technque s used to mae a wreless ln relable and further mprove end-to-end throughput. However, n tradtonal ARQ protocols, whenever a pacet gets erased, the sender smply retransmts t, whch usually consumes a consderable channel capacty. Then, Hybrd ARQ (HARQ) [] protocols that mplement Forward Error Codng (FEC) are ntroduced and have been proven to be more effcent than orgnal ARQ protocols or pure FEC schemes under wde condtons [4]. Recently networ codng (NC) [5] has emerged as a promsng technque to ncrease the networ bandwdth effcency and relablty, snce t enables to mx multple ncomng pacets n a sngle transmsson rather than ust relayng these ncomng pacets to output lns one by one, whch breas wth the conventonal store-and-forward way. Partcularly, authors of COPE [6] presented a practcal XORbased networ codng wth the concept of opportunestc lstenng and opportunstc codng for uncast n wreless mesh networs. It employed cumulatve ACK n ts COPE layer to provde relable transmsson. However, ts performance s only sub-optmal n lossy wreless networs. The reason s that COPE does not consder the ssue of how to effectvely retransmt lost pacets, whch could cause ts performance degradaton under the hgh loss rato. As an elegant relable retransmsson paradgm n one-to-many sngle-hop wreless networs, Nguyen et al. [8] combned networ codng technque wth the tradtonal ARQ technque, NC-ARQ, whch can sgnfcantly enhance the bandwdth effcency. Rozner et al. [9] modfed the retransmsson mechansm n COPE and proposed an effcent retransmsson approach (ER). Wth ER, pacets that need to be retransmtted are encoded together, such that multple pacet-loss for dfferent destnatons can be recovered by one retransmsson. Snce then, a lot of networ codng based retransmsson protocols [-4] have been presented to provde effcent and relable communcatons n lossy wreless networs. Nonetheless, all of these protocols only focal on natve pacets that need to be retransmtted, and none consder the role of encoded pacets, whch contaned these loss natve pacets and have been overheard by clents who aren t ther destnatons. Usng the tradtonal networ codng schemes, when clents overheard encoded pacets, they ust dscard them or smply store them but don t report these nformaton to the sender. Therefore, when the sender maes codng decsons, t can t utlze encoded pacets to explot codng opportuntes, whch s neffcent and expensve. Therefore, we study an effcent retransmsson approach wth networ codng technque, named Encoded pacet- Asssted Rescue (EAR) n ths paper, whch s able to explot the codng opportuntes of both natve and encoded pacets. Our contrbuton s summarzed as follows: We ntroduce a system model of the wreless one-tomany sngle-hop uncast system and then study the theoretcal analyss n terms of the number of retransmssons of the proposed scheme. We defne the weght of pacet-loss pattern, whch helps us to evaluate the EAR s performance more precsely and desgn a optmum codng algorthm to maxmze the beneft of coded pacets, namely to mnmze the total number of retransmssons The remander of ths paper s organzed as follows. In next Secton, we descrbe a system model n the context of wreless uncast scenaro. In Secton III, we present some theoretcal

2 dervatons on the total number of retransmssons wth the proposed EAR schemes. In Secton IV, theory and smulaton results are llustrated. Fnally, concluson s gven n Secton V. II. SYSTEM MODEL AND ASSUMPTIONS If multple flows traverse the same node, we called such node central node (unless otherwse stated we smply call t node C) and there s the opportunty to apply networ codng technques to mprove the overall retransmsson effcency for each flow crossed t. Consequently, we only consder the retransmssons between node C and ts neghbors even for the wheel scenaro n ths paper. And we also consder that the throughput rate of each flow over the networs has already stablzed. Furthermore, we assume node C employs a suffcent large retransmsson buffer to avod too early rescue process. In partcular, each recever would request a dstnct set of loss pacets, whch from node C s pont of vew, corresponds to supportng dfferent uncast sessons. Smlar wth [], we mae the followng assumptons about the wreless uncast retransmsson models. There are N (N >) recevers R ( 6 6 N), and the central node retransmts lost pacets after a fxed tme slot T. We suppose route of each flow would pass through node C to maxmze codng opportuntes [7]. Node C can always now the current pacet-loss states of both natve and encoded pacets at all recevers. Ths can be carred through by usng postve and negatve acnowledgements (ACK/NAKs). To smplfy the analyss, we assume all the feedbac are nstantaneous and relable. Pacet lost rates between node C and each recever are mutually ndependent and follow the Bernoull dstrbuton, where each pacet s lost wth a fxed probablty! ( s the recever s ID, 6 6 N ) at each recever. Let us consder the uncast topology, whch conssts of N recevers. Thereby, f each clent requests K dstnct pacets, then node C needs to successfully delver a total of K N pacets to all of them. To planly represent the pacet-loss state, we defne pacet-loss pattern as follow: Defnton. Pacet-loss pattern ½ P s a row vector that represents the current pacet-loss state of pacet P at all the recevers, thus ts dmenson s equal to the number of the recevers. When pacet P s successfully obtaned by recever R, the th entry n ½ P wll be mared, or else. In ths paper, the pacet-loss pattern relatng to the natve pacets s smply called natve pattern, and coded pattern denotes the one relatng to the encodng pacets. The followng defntons play crucal roles n ths paper. Defnton. The weght of a pacet-loss pattern W (½) s the number of non-zero elements n ½. In partcular, W (½) <N for N-recever scenaros. Remar: Note that, f all the ntended recevers of pacet P correctly receve t, there s no longer a pacet-loss pattern relevant to pacet P, vce versa. Furthermore, node C would retransmt P several tmes to delver t successfully to ts nexthop, for the reason that the wreless channel s error-prone. And the pacet-error pattern ½ P would be altered after each retransmsson of P. The weght of the newer loss pattern (f t exsts), however, s always no lghter than the prevous one, n respect that the recevers who have overheard P would reserve t untl the end of ths retransmsson process. At last, there are no less than one lost pacet relatng to a loss pattern. Thereby, unless otherwse stated, we relate a loss pattern to a set of pacets whch have the same loss state. Then, we defne P ½ as the set of lost pacets whch have the same loss pattern ½ durng the rescue process. Thus, n retransmsson process, f a non-empty set P ½ has been already transferred to the empty set,.e. no loss pacets relevant to ½, we state that the central node has rescued the loss pattern ½ or the set P ½. As we mentoned above, a pacet belongng to P ½ s elmnated from P ½ due to the followng two reasons: ) The ntended recever of ths pacet has receved t. ) The ntended recever s falure to receve t agan, yet the correspondng loss pattern s changed. To sum up, we present the followng theorem. Theorem. The expect number of retransmssons Ð ½ requested by the central node to rescue loss-pattern ½, whose the th r ( r N;r =;:::;N W (½)) entres are equal to, for N-recever scenaro s Or Ð ½ = Q ()! r where Or s the set consstng of the lost pacets whch have the same loss pattern ½ after the orgnal transmsson. And Ð ½!½ pacets n P ½ would be transferred to P ½ after the central node rescued P ½. Ð ½!½ =Ð ½ Prf½ ½g () where Prf½ ½g denotes the probablty that a pacet s transferred from P ½ to P ½. Proof: To smplfy the analyss, we suppose that node C would retransmts P ½ round by round, whch means node C frst transmts the entre pacets belongng to P ½ one by one, and collects ther receve-state to modfy P ½, and then t repeats the these steps for the resdual pacets n P ½ agan and agan untl these s no longer a pacet relatng to ½. We defne P ½ () ( >) as the set relevant to ½ after the th round, and P () ½ = Or. Let random varable Y ( >) represent the cardnalty of P ½ (), and we have Y = Or. For the reason that the delveres are..d. and follow the Bernoull dstrbuton, the random varables Y ( =; ;:::) are..d. and follow the bnomal dstrbuton. Further-more, a pacet s held n P ½ () after the th round, f and only f the recevers R r who lost t before are stll falure to receve t. Hence, we have E[Y ]=E[E[Y Y ]] = Y ( Y! r ) () due to Q! 6, the seres E[Y ] ( >) s convergent, the

3 expect number of retransmssons Ð ½, that node needs C to rescues ½, s Ð ½ = Y + X E[Y ] = = Y (Prf½g) + X Y ( Y! r ) = = Y X (Y! r ) Or = = Q (4)! r Now, let random varable Y ( >) represent the number of pacets that are transferred from P ½ to P ½, ½6=½, after the th round, partcularly, Y =. Random varables Y ( >) are..d as well and follow the bnomal dstrbuton. Then, we have E[Y] =E[E[YY ]] = E[Y ] Prf½ ½g = Y ( Y! r ) Prf½ ½g (5) due to Q! r 6, the seres E[Y ] ( >) s convergent, the number of pacets that are transferred from P ½ to P ½ after node C has rescued ½ s Ð ½!½ = X E[Y ] = = X Y ( Y! r ) ( ) Prf½ ½g = =Ð ½ Prf½ ½g that pacets n P ½ can be relocated to P ½. Obvously, ths theorem s sutable for both natve and coded patterns. Fnally, we ntroduce the followng concept. Defnton. If n pacets respectvely belongng to the sets P ½ ;P ½ ;:::;P ½n ( 6 n 6 N) can be coded together, we state the sets P ½ ;P ½ ;:::;P ½n can be encoded together, and we also say that the correspondng loss patterns,, n can be encoded together. III. PERFORMANCE ANALYSIS In ths secton, we frst revew the avalable wor on applyng networ codng to relable mult-uncast. Then, we study some theoretcal analyss n terms of total number of retransmssons of the tradtonal and the proposed EAR technques for both wheel and sngle-hop scenaros. Before delvng nto detals, we refer the reader to the followng symbols, whch be used n the rest of the paper. P : The set of natve pacets node R requests. ½ : The set of all possble loss patterns relatng to the natve pacets node R requests. ½ ;:::; n : The set of all possble coded patterns whch are n relaton to coded lost pacets, each of whch contans n pacets nodes R ;:::;R n requested separately, where N, 6 6 n, 6 n 6 N. P½ : the set of natve lost pacets whch are reque-sted by node R and have the loss state ½ = ½. P½ ;:::;n : the set of natve lost pacets have the loss state ½ = ½ ;:::; n. (6)!! P P P P P P P 4 P 4 Fgure. Example of NC-ARQ. (a) Two recevers uncast scenaro. (b) Possble retransmsson process: P P, P P, P 4 and P P. We also use notaton ½ P to denote the set of loss patterns relatng to pacets n P. However, when P s suffcently large, we have ½ P = ½ because of the law of large numbers. Thus, unless otherwse stated, we also utlze ½ to denote ½ P. And t s the same to ½ P ;:::;n. A. Overvew The ey dea of Nguyen et al. [8] and Rozner et al. [9] proposed networ codng schemes s to frst buffer the lost pacets n the retransmsson buffer durng some tme, then, nstead of transmttng them one by one, the source combnes the maxmum number of lost pacets wth dstnct ntended recevers nto one pacet and delvers ths coded pacet n one retransmsson. To llustrate how ths codng-base relable retransmsson scheme wors, we tae an example n Fg.. In the example, node C has transmtted P ( 6 6 4; =; ) to recevers R separately. As depcted n Fg (b), pacets P, P, P and have been falure to be receved by ther respectve P ntended recever but been correctly overheard by the other one. Tradtonally, each one of P, P, P and P s retransmtted one by one and each one of them has only one ntended recever. Wth the NC-ARQ scheme, node C can combne P wth P ; to Pen = P P, whch s avalable for both R and R. Smlarly, we can do the same operaton for P and P. Even so, P 4 stll has to be sent alone, snce recever R doesn t obtan t,.e. R s unable to remove P4 from the coded pacet contanng t to extract the ntended pacet. Furthermore, f P and P are lost agan, node C wll XOR them together for next retransmsson. From ths example we can explctly see that, by explorng the broadcast nature of wreless and mxng lost pacets tog-ether, the total number of retransmssons can be effectvely reduced. Authors of [] evaluated the performance of ths method for sngle-hop scenaros, and got the followng theorem. Theorem. Usng the NC-ARQ technque, when the number of pacets to be sent s suffcently large, the average number of retransmssons NUNC for the N-recever uncast sngle-hop scenaro s NUNC = Q NX N =! (7) N! = where! s the pacet loss rato between central node and recever R, and! 6! f N. In partcular, for the -recever scenaro,

4 UNC = μ!! +! (8)!! B. EAR performs on the N-recever uncast scenaro In ths subsecton, we evaluate the performance of the proposed EAR approach for the uncast topology. We frst present how to use pacet-loss pattern to dentfy the codng opportuntes over retransmsson process. Theorem. Let R ½ denote the set consstng of node(s) whch s/are the ntended recever(s) for pacets n P ½. In rescue process, the loss pacets P, P,,P n ( 6 n 6 N), each of whch can be natve or coded pacet, wth respect to the patterns ½ ;:::;½ n can be coded together f and only f R ½ \ R ½ = Á; 6= and only the th entry of the th column of matrx F =[½ T ::: ½ T n ] T s equal to, where R R ½, [;n] and [; S n = ]. R½ Proof: Frst of all, let us assume that there are two loss patterns ½ and ½ ( 6= ; ; n) whch satsfy R ½ \ R ½ =fr cap g. Suppose P contans the natve pacet P cap and the correspondng pacet n P s P cap. Under these assumptons, there would be two possbltes:. P cap 6=P cap, obvously, R cap cannot decode such coded pacet;. P cap = P cap, then R cap cannot obtan ths pacet, for the reason P cap P cap =. Nether of them s expected by us. Hence, there must be R ½ \ R ½ = Á; 6=. Wthout loss of generalty, let us consder node R. To nsure that the node R get ts correspondng pacet P, t must have correctly overheard the pacets P ;=;:::;n. So based on Defnton, the th entry n ½ ;=;:::;n s equal to,.e. the th column of matrx F s only the frst entry that s equal to. It s the same to other recevers. On the other hand, f the th column of matrx F s only the th entry that s equal to, where R R ½. It means that R has already obtaned the pacets P t ;t=;:::;n;t6=,.e. t enable to elmnatep t from the coded pacet. Under the assumpton, all pacets except P can be removed by R. Clearly, there are the same results for all other recevers. Therefore, based on the code rule of COPE [], all the pacets can be mxed together to one pacet. Defnton 4. The set CG = f½ ;½ ;:::;½ g ; 6 6 N (9) where error patterns ½ ;½ ;:::;½ can be coded together, s called the Code Group. Partcularly, though there are no less than one code group nterrelatng wth a error pattern, f all the th ( = S n = R ½ ) columns of matrx F must be zero n Theorem 5, and then each combnaton of loss patterns fulfllng the theorem s unque,.e f ½ CG and ½ CG, then CG = CG, and acheve ther maxmum codng opportuntes. In ths way, all the code groups we mentoned are lmted to ths restrcton n the rest of ths paper. In last sub-phase, we ntroduce the NC-ARQ approach. However, t only consdered the natve pacets and gnored the encoded pacets, whch contan codng opportuntes and have P P P P P pacets to be sent s suffcently large, the average number of P Fgure. Three recevers uncast scenaro. Let s tae an example n Fg. to llustrate how to utlze them to mprove the retransmsson effcency. In ths example, node C has transmtted P ( =; ; ) to recevers R separately. The receve-state of pacets P, P and P s depcted n Fg 6(a). Due to the assumpton, only P and P can be XORed together. Therefore, node C mxes them together and forwards encoded pacet P P to the downstream nodes. However, P P would be lost agan over the nosy channel, f ts receve-state s the showed n Fg. 6(b), then the best codng decson would be to combne P P and P together, and not to delver them respectvely. From ths example we can explctly see that, by consderng the encoded pacets generated n rescue process, the number of retransmssons would be effectvely reduced. Lemma. Usng the NC-ARQ technque, for N- recever scenaro, f ½ r and ½r can be coded together and P½ r 6 P½ r,! r 6! r, then the entre pacets n P½ r can be XORed wth the ones belongng to P r ½ n recovery process, and we say that ½ r s domnated by ½r. Ths relaton s denoted by ½ r  ½ r. In partcular, node C need to retransmt Ð r ½ tmes to rescue ½ r. And Ð Natve natve pacets would be sent alone. Ð r ½ = P r ½ ()! r r Ð Natve = P r P ½ ½ (! r ) ()! r potental to enhance the retransmsson effcency as well. We use nducton method to prove the lemma. Interested readers can fnd detals of the proof n the Appendx. Theorem 4. Usng the EAR method, when the number of retransmssons EUNC for the N-recever uncast sngle-hop scenaro s EUNC = Q NX N =! N = Q N =! () where! s the pacet loss rato between central node and recever R, and! 6! f N. Proof: Frst, we consder the -recever scenaro. Let ½ =[ ], ½ =[ ], ½ =[ ] and ½ =[ ]. Let random varable X, X, X and X, separately, denote the number of lost pacets relevant to loss pattern ½, ½, ½ and ½ after K transmssons. X ( =; ; ; ) follow the bnomal dstrbuton. As we mentoned above, node C would rescue ½ and ½ frst. Then based on Theorem, we have

5 Ð ½ =Ð ½ = K!!!! () Ð ½!½ =Ð ½! (! ) (4) Ð ½!½ =Ð ½ (! )! (5) Snce Ð ½!½ 6Ð ½!½ and E[X ]6E[X ], then E[X ] +Ð ½!½ =m 6m =E[X ]+Ð ½!½. Based on Lemma, we drectly get ½ Â ½, and Ð ½ = m! (6) Combnng () and (6), the expected number of retransmssons to successfully delver K encoded pacets to R and R s gven by Ð all =Ð ½ +Ð ½ +Ð ½ = K! + K!! (7)!!! The proof for N(>) recevers scenaro wll be found n the Appendx. Remar: Though no encoded pacets would be utlzed n -recever uncast scenaro, the EAR approach also outperforms the NC-ARQ technque as shown n Fg., where! =!. The reason s that the EAR approach not only studes the potental of encoded pacets, but also consders that loss pacets should synchronously update ther loss-state over rescue process, whch would result n more codng opportuntes. The average number of retransmsson EAR NC-ARQ Pacet loss rate Fgure. The average number of retransmssons on -recever scenaro. Note that, n proof we suppose each coded pacet can be dvded nto several natve pacets, all of whch are the same receve-state and would be recovered separately, even the recevers cannot decode t. However, ths hypothess does clearly not hold n actual applcatons. As a result, some encoded pacets are unable to be mxed wth any other pacet. We call them the unwanted pacets. We stll tae the - recever uncast scenaro to llustrate how t happens. The zero weght patterns and CG = f½ ;½ ;½ g don t produce undecodabe pacets n each node. Consequently, there are only three codng combnatons that would create unwanted pacets, meanwhle, though each coded pacet has sx possble loss-state, only three of them as shown n Fg. 4(a) would generate undecodable pacets n unntended node. Based on Theorem, the potental codng schemes for these patterns are depcted n Fg. 4(b). ½ ½ ½ + ½ + ½ + ½ + ½ + ½ + ½ ½ ½ + ½ + ½ + ½ + ½ + ½ ½ (a) ½ ½ ½ + ½ + ½ + ½ + (b) ½ + ½ + ½ + ½ ½ ½ ½ Fgure 4. Example of the nfluence of unwanted pacets. (a) The effectve coded patterns. (b) The codng schemes for coded patterns. To smplfy analyss, we assume! =! =! =!. In the lght of Theorem, we have N = P ½ = P ½ = K!(!) P ½ =! P ½ = P ½ = P ½ = P ½ = P ½ = P ½ and we carry out P ½ + =P ½ + =P ½ + =P ½ + =P ½ + =P ½ + N =P + ½ =P + ½ =P + ½ = K!4 (!) (! )(! ) Clearly, CG =f½ + ;½ + ;½ + g and CG =f½ + ;½ + ; ½ + g would not create unwanted pacets, for the reason that the expect number of retransmssons requested to rescue each error pattern s the same. Moreover, f N >N, then ½ + Â ½, ½+ Â ½ and ½+ Â ½, and the remanng pacets n P½, P½ and P½, respectvely, are the same sze based on Lemma. Snce ½, ½ and ½ can be XORed together, CG = f½ + ; ½ g, CG 4 = f½ + ;½ g and CG 5 = f½ + ;½ g would not produce unwanted pacets ether. On the contrary, f N <N, then the resdual pacets n P½ +, P½ + and P½ + have to be recovered separately. However, the pacets n P + ½ and P + ½ are not unwanted pacets, n respect that they stll combne wth the pacets requested by node R. Therefore, only the one n P + ½ s called unwanted pacet. The expect number of delveres to recover them s Ð= K!(!)(! +! ) (! )(! (8) ) where! +! >. Though Theorem 4 only provdes the upper bound of the performance of the EAR approach on N-recever sngle-hop uncast scenaro, the smulaton results ndcate that the nfluence of the unwanted pacets s small and the performance degradaton t caused s consdered neglgble.

6 Updatng algorthm: : ½ P = : Trumph = : Transform = false 4: for recevers = to N do 5: f R has obtaned P correctly then 6: f R s one of the destnaton of P then 7: Trumph = Trumph + 8: f Trumph s equal to NC(P ) then 9: delete P from the retransmsson queue P : f P s a coded pacet then : delete the correspondng natve pacets contaned n P : end f : return 4: end f 5: set the th entry of ½ P equal to 6: else 7: f R has gan P for the frst tme then 8: set the th entry of ½ P equal to 9: end f : Transform = true : end f : end for : f Transform s true and P s a coded pacet then 4: delete the correspondng natve pacets contaned n P 5: end f Fgure 5. Updatng algorthm for each loss pacet Eventually, node C must update the receve-state of loss pacets synchronously to effectvely mplement EAR. Thus, after C has receved N feedbac from R, =; :::;N for a pacet P, t wll execute the followng updatng algorthm, where NC(P ) denotes the number of the natve pacets contaned n P. Notce that when a coded pacet s lost by ts ntended recevers and ts loss-state s changed, then node C would only reserve these pacets and dscard the correspondng natve pacets comprsng t to save memory. IV. EXPERIMENTAL RESULTS AND DISCUSSIONS In ths secton, retransmsson gan s used to evaluate the retransmsson effcency of dfferent approaches by varyng the number of recevers and bt error rate (BER) under both uncast and wheel topology. The BER at each recever are mutually ndependent and follow the Bernoull dstrbuton. We defne the retransmsson gan as the total number of retransmssons usng a typcal retransmsson algorthm, whch s the HARQ or NC-HARQ technques, dvded by the total number of retransmssons usng our algorthm. A hgher retransmsson gan s preferred snce t ndcates fewer retransmssons. In the followng smulate, the pacet sze s to be 5 bytes and data s encoded wth RS (, 8, 4). We use CRC-6 for error detecton n all the smulatons. We record the total number of retransmssons over a mass of experments. In the nterest of space and clarty, we only present the average retransmsson gans n the followng smulaton results. Frst, we compare the performance of proposed EAR approach wth the NC-HARQ technque and the HARQ technque. Fg. 6 shows the effect of dfferent BERs on the retransmsson gan for the -recever uncast scenaros, respectvely. BERs between the sender and all ts recevers are the same, and vared from 4 to :5 n 5 4 ncrements. As expected, the smulaton results support our theoretcal dervatons. Though n large BERs regon the unwanted pacets would brng down the performance of the EAR approach as we dscussed before, the smulaton results for uncast scenaro show such declne s so small as to be neglectable. Furthermore, an nterestng phenomenon would be observed s that the retransmsson gans for uncast scenaro ncrease wth BER, and are very close to under the lght BERs. The reason s that the more awful nose the channels encounter, the more pacets are lost, that s to say the more encoded pacets would be generated from whch the EAR protocol enables to explore the codng opportuntes, yet NC-ARQ cannot. Then, we compare the performance of proposed EAR approach wth the NC-HARQ technque versus the number of clents for uncast and wheel topology. In the experment, BERs between the sender and all ts recevers are set equal to and. Fg. 7 shows the mpact of dfferent number of recevers, whch s vared from to 5, on the retransmsson gan. As seen, the retransmsson gans hardly. -recever Retransmsson gan recever Sm. 5-recever 5-recever Sm Bt error rate - Fgure 6. retransmsson gan versus BER for theory and smulaton. Retransmsson gan Theorem(.) Smulaton(.) Theorem(.) Smulaton(.) The number of recevers Fgure 7. Retransmsson gan versus the number of recevers for theory and smulaton.

7 The average overhead. (bytes / pacet) 5 4 Scheme A(.) Scheme B(.) Scheme A(.) Scheme B(.) Scheme A(.5) Scheme B(.5) overcome the dsregard for the role of encoded pacets, an encoded pacet-asssted rescue approach (EAR) based on XOR networ codng has been proposed. The core features of the EAR approach are ) explore the codng opportuntes of encoded pacet; ) synchronously updatng the receve-state of loss pacets. Under the access pont model, some analytcal results are derved for the retransmsson eff-cency over the uncast scenaros. The theoretcal and smu-laton results show that the proposed EAR approach always outperforms the NC- HARQ technque n terms of the retran-smsson effcency for a typcal range of wreless networ condtons The number of recevers Fgure 8. The average overhead ncrease wth the number of clents on the lght BER, whereas the gans sgnfcantly advanced upon the heavy BER wth the slowng growth rate. The reason s that the probablty that a loss pacet s successfully receved at lease at one clent ncreases wth the number of the clents, and then more loss pacets are relate to one node, namely the growth rate of the total number of retransmssons s gettng slower and slower, when the number of recevers goes up. Fnally, to ensure each ntended recever s able to successfully decode a coded pacet, node C should add the nformaton, whch specfes whether t conssts of encoded pacets and whch encoded pacets are embraced, wthn the NC header. To acheve ths goal, there are two schemes. Scheme A: node C drectly records all the natve pacets that construct the coded pacets, regardless whether any one of them s or was a loss pacet. We utlze a -byte hashvalue to dentfy a pacet s source address and sequence number. Thereupon, n the worst condton, node C has to use K (! + +! N ) bytes to record these pacets. Scheme B: node C employs bt map to regster these natve pacets. In ths scenaro, C has to eep 9-byte for each destnaton, owng to the randomcty of loss-state of each pacet. Hence, n the worst condton, node C needs to append 9 N bytes. Fg. 8 shows the average length of header that records whch pacets n the combned pacets for both schemes. Pacet loss rate between the sender and all ts recevers are the same and equal to.,. and.5, respectvely. Through the fgure, we see that scheme A has much smaller overhead than scheme B. And we further notce that the overhead ncreases wth the number of recevers, but decreases when BERs rses. Even so, comparng to the length of the data pacets, the overhead assocated wth scheme A s no more than 5%, whch can be neglected. V. CONCLUTIONS In ths paper, we addressed the problem of exstng NCbased relable transmsson schemes for wreless mult-uncast system such as WF and WMAX networs. In order to REFERENCES [] D. Aguayo, J. Bcet, S. Bswas, G. Judd, and R. Morrs. Ln-level Measurements from an 8.b Mesh Networ, n Proc. of ACM SIGCOMM 4, August 4, pp. -. [] M. Naoh, S. Sampe, N. Mornaga, and Y. Kamo, ARQ schemes wth adaptve modulaton/tdma/tdd systems for wreless multmeda communcaton servces, The 8th IEEE Internatonal Symp. on PIMRC 97, Sept. 997, pp [] D. J. Costello, J. Hagenauer, H. Ima, and S. B. Wcer, Applcaton of error-control codng, IEEE Transactons on Informaton Theory, vol. 44, Oct. 998, pp [4] Jung-Fu Cheng, Codng performance of hybrd ARQ schemes, IEEE Trans. on Conmmun., vol. 54, no.6, June 6, pp [5] R. Ahlswede, N. Ca, S.-Y. R. L, and R. W. Yeung, Networ nformaton flow, IEEE Transactons on Informaton Theory, vol.46, no., July, pp [6] S. Katt, H. Rahul, W. Hu, D. Katab, M. Medard, and J. Crowcroft, XORs n The Ar: Practcal Wreless Networ Codng, n Proc. of ACM SIGCOMM 6, July 6. [7] S. Sengupta, S. Rayanchu and S. Baneree, An Analyss of Wreless Networ Codng for Uncast Sessons: The Case for Codng-Aware Routng, n Proc. of IEEE INFOCOM 7, May 7, pp [8] D. Nguyen, T. Nguyen and B. Bose, Wreless broadcastng usng networ codng, n Thrd Worshop on Networ Codng, Theory, and Applcatons, Jan. 7. [9] E. Rozner, A. P. Iyer, Y. Mehta, L. Qu, and M. Jafry, ER: effcent retransmsson scheme for wreless lans, n Proc. of the 7 ACM CoNEXT conference, Dec. 7, pp. -. [] T. Tran, T. Nguyen, B. Bose, and V. Gopal A Hybrd Networ Codng Technque for Sngle-Hop Wreless Networs, IEEE Journal on Selected Areas n Com., vol. 7, no. 5, June 9, pp [] Z. Zheng, and P. Snha, XBC: XOR-based buffer codng for relable transmssons over wreless networs, n: Proc. IEEE BROADNETS, Sep. 7, pp [] M. Ghader, D. Towsley, and J. Kurose, Relablty Gan of Networ Codng n Lossy Wreless Networs, n Proc. of IEEE INFOCOM, Aprl 8, pp [] A. A. Yazd, S. Sorour, S. Valaee, and R. Y. Km Optmum Networ Codng for Delay Senstve Applcatons n WMAX Uncast, n Proc. of INFOCOM 9, Aprl 9, pp [4] H. Sofane, M. Patrc, and R. Davd, On the Impact of Random Losses on TCP Performance n Coded Wreless Mesh Networs, n Proc. of IEEE INFOCOM, March. APPENDIX Proof of lemma : We stll suppose that the central node retransmts P½ r round by round. We defne P½ r () and P½ r () ( >) as the loss pacets sets relevant to ½ r and ½r after the th round and we get P½ r () = P½ r, P½ r () = P r ½. Let random varable Y and Z (>) represent the cardnalty of P½ r () and P½ r (), furthermore, we defne Y = P½ r,

8 Z = P½ r. As the delveres are..d. and follow the Bernoull dstrbuton, random varables Y and Z ( = ; ;:::) are..d too and follow the Bnomal dstrbuton. Due to Y 6 Z, the entre pacets n P½ r () s able to be combned wth the ones belongng to P½ r () n next round. After the frst round, Y =! r Y and m =Z Y pacets n P½ r () are not delvered,.e. Z =! r Y +m. Because of! r 6! r, we explctly have Y 6 Z. It means that the entre pacets wthn P½ r () can be mxed wth the ones belongng to P½ r () over the next round n the same way. Now, we suppose that Y n 6 Z n (n > ). Then after the n th round, node R s falure to receve Y n+ =! r Y n pacets, so the central node has to retransmt these pacets n the next round. Moreover, m n = Z n Y n pacets n P½ r (n) are not delvered,.e. Z n+ =! r Y n +m n. Clearly, we have Y n+ 6 Z n+, whch ndcates that we are able to combne the entre pacets n P½ r (n +) wth the ones belongng to P½ r (n +). Then we deduce ½  ½ by nducton, and wor out the expect number of retransmssons requred to rescue ½. Ð Code = X Y = P ½ (9) =! r In partcular, Ð Natve natve pacets relevant to ½ would be delvered alone. Ð Natve = Z = Z + X (! r )Y = = P ½ P ½ (! r ) ()! r The lemma has been proved. Proof of theorem 7: Frst, we separate each coded pattern nto the correspondng natve patterns, n respect that a coded pattern can be consdered as the set consstng of loss pacet(s) requested by dstnct node(s) wth the same errorstate. For example, f a encoded pacet P n P r ;r ;r ½ contans loss pacets P r and P r, then we put P r and P r nto P½ r r and P ½ respectvely, where ½ =½ =½. After that, we cancel P from P r ;r ;r ½ and repeat these steps untl P r ;r ;r ½ = Á. In ths way, we only consder natve pacets n the followng proof. Second, let denote the set that contans the error patterns whch belong to ½ ( <) and the th entry s nonzero, partcularly = Á. Let denote the subset of ½ consstng of all those error patterns whose the th ( >) entres are zero, specally = ½ ª, where ½ s zero vector, and N = ½ N. Then the set U = [ s called the prmary set for node R,. Clearly, U \ U = Á, 6= and S N = U = S N = ½. Based on the above concepts and Theorem 5, for any loss pattern ½ n, there s a unque code group, CG = f½ ª ; ½ ;:::;½ w w μ U, where w = W (½ ), N and h ; ½. Furthermore, only the and entres of vector are equal to, and,! 6!, thereby Pr ½ ª > Pr ½ ª. We defne the loss pacets set P (n) ½ relatng to ½ CG after the n th round, and P () ½ = P ½. Let random varable Y n = P ½ (n) (n>). Snce the delveres are..d. and follow the Bernoull dstrbuton, random varables Y n (n =; ;:::) are..d and follow the Bnomal dstrbuton. Then usng the EAR approach, when the ntended node(s) s/are falure to receve a encoded pacet and some clent overheard t for the frst tme, these contaned natve pacets would be elmnated from ther orgnal sets and transferred to the matched sets respectvely. Therefore, after the n th round, Y n+ = Y n Pr ½½ª pacets have to be retransmtted n the next round. Notce that t dffers from the result n Lemma. Ths s due to the fact that the NC-ARQ technque s only nterest n whether the ntended nodes have already correctly obtans the coded pacet, yet the EAR approach not merely taes care of t, but also consders how to combne more requested pacets together whether they are encoded pacets or not. Then to rescue ½, node C s expected to retransmt Ð ½ tmes, Ð ½ = X Y KP ½ n = = Pr () ½½ª and Ð ½!½ pacets s transformed from P ½ to P ½. Ð ½!½ =Ð Pr ½ ½ ª () Note that ths result s the same as Theorem. On the other hand, some loss pacets relatng to ½ U t (t 6 ) would be relocated to P ½ (½ CG), f Prf½½ g >. And based on the defnton of the sets and, there s always an error pattern ½ x = ½ x Ut (t<) and ½ y enables to be combned wth ½ y. Because the th entry of ½ x s zero and of ½ y s non-zero, then f½ xg\f½ yg = Á and f½ xg[f½ yg=. Due to ½ x = ½ x, then we have Prf½ xg=prf½ x g, Prf½ x½ zg=prf½ x ½ z g ( ½ z =½ z ), x g, namely Ð ½ = x > Ð ½ x!½. In addton, we also have Pr ½ ª y =Pr ½ ª y Pr ½ ª ½ y =Pr ½ ª () ½ y P ½ x = P ½ x and Prf½ ½ xg>prf½ ½ Ð ½ x and Ð ½ x!½ for the reason ½ ½ = ½ y ½ y. Hence, we dvde the loss pacet set nto three parts P½ =Or½ [ S ½ (Ð x ½ x!½ )ª [ S ½ (Ð y ½ y!½ )ª P½ =Or½ [ S (Ð ½ x ½ x!½ ) ª [ S ½ (Ð y ½ y!½ ) ª Due to Or½ = K Prf½ g > Or ½ = K Prf½ g, we have Or½ [ S ½ (Ð ½ )ª x x!½ > Or ½ [ S (Ð ½ x ½ x!½ ) ª. Now, we assume W (½ )=, then S ½ y (Ð ½ y!½ ) ª =

9 S ½ (Ð ½ )ª =Á, thereby P y y!½ ½ =max fp ½g. If W (½ ) ½CG =, then Ð ½ y!½ 6= unless W (½ y)=. Based on (), () and (), we get Ð ½ y!½ > Ð ½ y!½ for each par (½ y ;½ y ), so P ½ =max fp ½g. Suppose P½ ½CG =max fp ½g ½CG s vald f W (½ )=n (n<). Then for W (½ )=n, there are [ (Ð ½ x!½ )ª = [ n (Ð ½ (w))ª w= x!½ [ ½ x ½ y (Ð ½ y!½ ) ª = [ n w= (Ð ½ y!½ (w)) ª where Ð ½!½ (w) s the set consstng of all the Ð ½!½ ( W (½) =w). Because of the assumpton, for each par (½ y; ½ y ), there arew (½ y) <n, and we have P ½ y > P ½ y, then accordng to (), () and (), we stll get Ð ½ y!½ > Ð ½ y!½. It s able to deduce Ð ½ x!½ (w) > Ð ½ y!½ (w), w [;n ]. Therefore, P½ =max fp ½g. ½CG Then by nducton, the equaton P½ =max fp ½g s ½CG avalable at all tme for each ½, where CG contans ½. Based on (), we have Ð ½ > Ð ½, namely after the central node retransmts Ð ½ tmes, CG has been already rescued, n other words CG s domnated by ½. Consequently, U Á and to rescue U s equvalent to recover loss pacets n P ½, ½. Furthermore, we defne that when the loss-state of a pacet n Or ½ (½ ) s altered, but stll belongs to, then ths pacet would be retaned n Or ½. Accordngly, rescue Or ½ means to delver a pacet n Or ½ untl ts loss-state ddn t belong to. In ths way, to recover P s equvalent to rescue Or. Now, let us proof that the expect number of retransmssons requred to rescue Or ½ ( ½ ) under the above defnton s Ð Or ½ = K Prf½g Q N =! (4) Frst, let ½ denote the pattern that ½ and W (½ )=, then Eq. (4) s obvously true for ½. Then Let ½ (t) denote the vector the t th entry s zero and W (½ (t)) =. Based on Theorem, to rescue loss pacets n Or ½ (t) relatng to ½ (t) (for smply, we also say that to rescue ½ (t) n Or ½ (t)), the expect number s Ð ½ (t) = K Prf½ (t)g Q! N t =! and to rescue ½ n Or ½ (t), the number s Ð ½ (t)!½ = Ð ½ (t) (! t) Q N =! Q N =! Snce ½ (t) can only be convert to ½, the number requred to rescue Or ½ (t) s Ð Or ½ (t) =Ð ½ (t) +Ð ½ (t)!½ = K Prf½ (t)g Q N =! Now, suppose, ths thess s true for any ½ n, W (½ n )= n +>. Then let ½ n (T ), T = ft :::;t n g, denote the pattern the T th entres are zero and W (½ n )= n. Usng Theorem, to rescue ½ n (T ) n Or ½n (T ), the number s Ð ½ n(t ) = K Prf½ n (T )g Q n =! t Q N =! and Ð ½n(T )!½ x(t ) =Ð ½ n (T ) Prf½ x(t )½ n (T )g pacets transformed to ½ x (T ), 6 x<n T =ft ;:::; t x g. If we operate these pacets as well as Or ½ x (T ), then n the lght of assumpton, the requred number to rescue them s Therefore, we have Ð Or ½n(T ) = X Ð Or ½x(T ) = Ð ½ n(t )!½ x(t ) Q N =! Ð Or +Ð ½ x(t ) ½x(T ½ ) n(t ) = Ð ½ n(t ) Q N =! X Prf½ x ½ n g +Ð ½ n(t ) ½ x(t ) = Ð ½ n(t ) ( Q n =! t ) Q N =! Q N =! +Ð ½ n(t ) = K Prf½ n(t )g Q N =! By nducton the proposton s proofed. Moreover, Ð =Ð Or = K P ½ Prf½g Q N =! = K Q N =! Q N =! then we fgure out the expected number of retransmssons requred to recover the entre loss pacets, NX NX NX Ð all = Ð U = Ð = = = = Dvdng KN, then gves the theorem. K Q N =! Q N =! (5)