Dynamic Method of Network Coding Based Retransmission for Wireless Multicast

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1 Dynamc Method of Networ Codng Based Retransmsson for Wreless Multcast Houy L State Key Lab of ISN Xdan Unversty X an Chna Emal: lhouy2013@hotmal.com Yng L State Key Lab of ISN Xdan Unversty X an Chna yl@mal.xdan.edu.com Abstract In wreless sngle-hop multcast wth networ codng based Automatc Repeat request (NC-ARQ) scheme, the sender needs to combne one or several lost pacets as one retransmsson pacet to reduce the number of multcast transmssons. Ths paper proposes a new scheme, called Dynamc Networ Codng based ARQ (DNC-ARQ), to mprove the effcency of the multcast transmssons. The man dea s to allow the BS to select the combnaton for the lost pacets n dynamc range: the BS would not retransmt for a lost pacet untl an optmum combnaton s found or the lost pacet acheve the mum delay. We also derve the upper and low bound of DNC-ARQ scheme and gve a practcal algorthm. Both theoretcal analyss and smulaton results confrm the proposed scheme hold more effcent and less decodng delay than the tradtonal NC-ARQ. I. INTRODUCTION In a multcast transmsson system, a sender needs to dssemnate dentcal nformaton to a group of recevers. To ensure relablty, two man approaches are adopted: Forward Error Correcton (FEC) and Automatc Repeat request (ARQ). In multmeda broadcastng and multcast servce (MBMS) system [1], two nds of FEC, Turbo code and Raptor code, wor the physcal layer and applcaton layer respectvely. Both of them need to broadcast the orgnal nformaton and the correspondng redundant nformaton to the recevers. If the amount of the lost data s suffcently small, the recever can recover the lost data relably va usng proper decodng algorthm. When ARQ scheme s employed, the sender wll rebroadcast the lost pacets. However, the transmsson effcency of such system would be very low when dfferent users lost dfferent pacets. In such case, the sender has to broadcast the lost pacet one by one. To ncrease the transmsson effcency, Salm Y. et al. [2] proposed a method to construct the retransmsson pacet by usng the lnear combnaton of several lost pacets. They also proved that the problem of fndng the optmal combnaton of the lost pacets to mnmze the number of retransmsson pacets s a NP-complete problem. S. Katt et al. [3] showed that the lnear combnaton approach for sngle-hop relable multcast can be consdered as a nd of networ codng, named as Opportunstc Networ Codng (ONC). Recently, the combnaton of networ codng wth ARQ scheme has nspred consderable nterests snce ths scheme can utlze the transmsson effcently. In ths paper, we call ths scheme as networ codng based ARQ (NC-ARQ). Nguyen et al. [4] proposed a class of NC-ARQ schemes. In these schemes, the sender wats untl a buffer of pacets have been transmtted before any retransmsson taes place. Durng the retransmsson phase, the sender forms a new pacet by combnng some of the lost pacets. The transmsson effcency of the proposed scheme s also analyzed n [4]. Snce the number of the pacets durng the transmsson phase s fxed, we call ths scheme as fxed-sze networ codng based ARQ (FNC-ARQ). The algorthm on the combnaton of the lost pacets are dscussed n [5] and [6] respectvely. Ch et al. also gave two practcable FNC-ARQ schemes [7]. Both transmsson effcency and transmsson delay are analyzed va computer smulaton.the comparson between the FEC scheme wth fountan code and the FNC-ARQ scheme s conducted by Nguyen et al. [8]. It s demonstrated that FNC- ARQ performs better than FEC scheme wth fountan code n the case of small scale of users and vce versa n the case of large scale. Meanwhle, some Networ Codng based HARQ (NC-HARQ) schemes have also been explored [9][10]. In the FNC-ARQ scheme, all orgnal pacets are transmtted by buffer and buffer. So the sender can t select combnaton for the lost pacet between dfferent buffers. That mght loss some transmsson effcency. We manage to desgn a new scheme to mze the effcency of every retransmsson pacet and eep the decodng delay at the same level wth FNC-ARQ scheme. So we buld the math model of NC-ARQ wth the scenaro of Wreless Sngle-Hop Relable Multcast (WSHRM). Then we summarze some characterstcs of NC- ARQ and propose a new class of NC-ARQ: Dynamc Networ Codng based ARQ (DNC-ARQ) whch replaces the fxed-sze buffer wth a dynamc lst. We also desgn a codng selecton algorthm for DNC-ARQ and show that DNC-ARQ scheme has better transmsson effcency than FNC-ARQ. II. MATH MODEL A wreless sngle-hop multcast transmsson system contans two parts: one s the Base Staton (BS), the sender; and the other s a group of User Equpment (UE), the recevers. Let M be the number of UEs n the group. The UE group can be wrtten as U = {U 1,..., U M }.Let H denote the number of retransmsson pacets. And the L orgnal pacets can be wrtten as P = {P1 0,..., PL 0 }. In dfferent NC-ARQ scheme, these L + H pacets are transmtted n dfferent order. We

2 wrte j th pacet broadcasted by the BS as P j = c 1,j P 0 1 c 2,j P c L,j P 0 L where 1 j L + H. It can be easly seen that c,j can be ether 0 or 1. The pacet P j can be represented by φ j = [c 1,j, c 2,j,..., c L,j ] whch s called as the networ codng vector (NCV). If P j s an orgn pacet, only one element n the NCV s 1 and all other element are 0. Thus we have: c 0 j = L g=1 c g,j = 1.If P j s a retransmsson pacet, we have:c 0 j = L g=1 c g,j 1. The pacet P j receved by U can be also consdered as a vector φ j,. If U receve the pacet P j successfully, φ j, = φ j. Otherwse, φ j, = 0. We call φ j, as j th decodng vector of U. After the P j has been transmtted, the BS holds j NCVs. We can wrte them as a matrx: Φ j = [φ 1,..., φ j ] T We call ths matrx as networ codng matrx for j th slot. Lewse, the U holds j decodng vectors at j th slot. We can wrte them as a decodng matrx of U for current slot: Φ j, = [φ 1,,..., φ j, ] T When P j+1 has been transmtted, the networ codng matrx of BS s updated as: Φ j+1 = [Φ j, φ j+1 ] T, and decodng matrx for U s updated as: Φ j+1, = [Φ j,, φ j+1, ] T. III. DNC-ARQ SCHEME In ths chapter, we frstly analyze the characterstcs of the optmal retransmsson pacets. And then based on these characterstcs, we propose the DNC-ARQ scheme. An example s also gven to show the dfference between DNC-ARQ and FNC-ARQ. A. Characters Based on the WSHRM model, we frst defne a parameter to descrbe the effcency of one retransmsson pacet. Defnton 1. Sngle Pacet Effcency (SPE) Supposng the NCV of P j s φ j, the appendx set of φ j s: θ j = { Φ j 1, s lnearly ndependence of φ j } And the SPE of the pacet can be expressed as: µ j = θ j /M Obvously, we have θ j M and µ j 1, whch means the most effcency pacet s lnearly ndependent of these decodng matrxes of all UEs. The defnton also suggests that a pacet s conceved to be more effcent than the other f ts nformaton s meanngful to more UEs. The concept that a pacet s conceved to be meanngful to a UE means that ths pacet can t be expressed by a combnaton of those pacets receved successfully by ths UE. We gve the followng defnton to descrbe the most effcent pacet. Defnton 2. Perfect Match Pacet (PMP) We call P j as perfect match pacet, f t satsfes: θ j = M Fg. 1. An example of dynamc lst Defnton 3. Maxmum Match Pacet (MMP) Supposng that the BS has transmtted j 1 pacets ncludng t j orgnal pacets, we can wrte another auxlary set: ϑ j = { Ran(Φ j 1, ) < t j }.We call P j as mum match pacet, f t satsfes: θ j = ϑ j Apparently, the SPE of a PMP s 1. And that of a MMP s µ = ϑ j /M 1. The orgn pacet transmtted n frst tme can be consdered as a PMP. But retransmsson pacet may nether a PMP nor a MMP. B. Process of DNC-ARQ To mze the SPE of the retransmsson pacet, we propose a new class NC-ARQ, Dynamc Networ Codng based ARQ (DNC-ARQ) scheme. Instead of employng the fxed sze buffer, the DNC-ARQ adopt a dynamc lst whch s created and updated by the BS. The BS retransmts a pacet only f the pacet s a PMP or a MMP. We frstly ntroduce the concepton of dynamc lst and then show the process of DNC-ARQ scheme. The dynamc lst consst of several dynamc chans, each of whch contans the basc nformaton of certan orgn pacet. The dynamc chan for Pj 0 can be wrtten as followng: CH j = {NP j, ST j, E j }, {TN j, } where NP j s the ID of orgnal pacet Pj 0. In ths paper, we let NP j = j. ST j s the survve tme of Pj 0. And {TN j,} s the ID set of those UEs whch can t recover Pj 0. Supposng j orgnal pacets and h retrans-pacets have been transmtted, the set {TN j, } can be wrtten as {TN j, } = { Φ h+j, (:, j) = 0, 1 M} Let E j denote the element number of {TN j, }. We have E j = {TN j, }. When all UEs can recover the orgnal pacet Pj 0 ( E j = 0), the BS wll delete the CH j from the dynamc lst. Fg.2. llustrates an example for the dynamc lst. The decodng delay s also consdered n dynamc lst. We denote the mum affordable decodng delay of the system as NH unt tme. ST j s ntalzed as 1, whch means the

3 orgnal pacet Pj 0 has been just transmtted by BS. ST j wll be ncreased by 1, once he BS multcast orgn or retransmsson pacet. When ST j s approachng to NH, the BS wll multcast retransmsson pacets to mae all UEs recover Pj 0 as soon as possble. In ths way, we control the decodng delay of DNC-ARQ scheme n an acceptable level. Now, we descrbe the DNC-ARQ scheme step by step: Step1: Intalzng dynamc lst. The BS multcast ρ orgnal pacets one by one, and then ntal the dynamc lst accordng to the feedbac message from M UEs. The dynamc lst created by the BS can be wrtten as: DL = [CH 1 ;...; CH ρ ] = [{NP 1, ST 1, E 1 }{TN 1, };...; {NP ρ, ST ρ, E ρ }{TN ρ, }] where ρ = NH/2.And then BS delete the CH λ whose E λ = 0 (1 λ ρ). Step2: Multcastng pacet. There are two cases n ths step: multcastng an orgnal pacet or a retransmsson pacet. We suppose that DL n current slot nclude y dynamc chans, whch can be wrtten as [CH x ;...; CH x+y ]. If ST x < NH-1 and BS can t fnd a PMP for {TN j, }, the BS multcast the next orgnal pacet. Otherwse, the BS multcast a retransmsson pacet. If a PMP s found for {TN j, } whle ST x < NH 1, the BS wll multcast the PMP to UEs. If ST x NH 1, the BS wll see a MMP for {TN j, } and multcast t to UEs. Both the PMP and the MMP can be wrtten as: P r = φ r (1)P 0 1 φ r (2)P φ r (L)P 0 L where φ r s the NCV of the retransmsson pacet. Step3: Recevng feedbac. The BS receves ACK/NACK message from UEs, and create a success set γ s to mar the ID of UEs from whch BS receves ACK message. Supposng the BS receves n ACKs from UEs, the set can be wrtten as: γ s = [TN 1 ac,..., TN n ac]. Step4: Updatng dynamc lst. Correspondng to the step 2, there are two cases n ths step: updatng dynamc lst for orgnal pacet or for retrans-pacet. In both two case, the BS wll let all ST from CH x to CH x+y ncreased by 1 at frst. For orgnal pacet, the BS wll create a new dynamc chan CH cs = {NP cs, 1, E cs }{TN cs, } where cs [x+y+1,..., x+y-1+st x+y ], NP cs = cs, {TN cs, } = { / γ s, 1 M}. For retransmsson pacet, the BS deletes some elements n {TN x, },..., {TN x+y, }, accordng to the success set γ s and networ codng vector φ r. More specfcally, the TN j, s deleted both from dynamc lst and γ s, f φ r (j) = 1 and TN j, γ s. After updatng the dynamc lst, the BS removes those dynamc chan whose E = 0 and goes bac to step 2 untl L orgnal have been recovered by M UEs. C. Example Here we tae an example to llustrate the dfference of the two schemes. Fg.2. s called as the feedbac matrx of NC- ARQ scheme. Any lne of ths matrx note the ACK/NACK Fg. 2. Feedbac Matrx of NC-ARQ schemes message of one UE for all orgnal pacets, where 1 means ACK and 0 means NCAK. We have 10 orgnal pacets and 4 UEs n ths example. In FNC-ARQ scheme wth buffer sze 5, the BS frstly multcast the pacets from P 1 to P 5 one by one, and then multcasts the followng retransmsson pacets to ensure all UEs recover ther lost pacets: R 1 = P 1 P 2 P 3, R 2 = P 4, R 3 = P 5 Then the BS multcast the pacets from P 6 to P 10 one by one, and then multcast the retransmsson pacets: R 4 = P 6, R 5 = P 7, R 6 = P 10.Obvously, the number of the retransmsson pacets n a buffer s equal to the largest number of lost pacets for one UE n the buffer. And the average SPE of those 6 retransmsson pacets s In DNC-ARQ scheme where the survve tme s equal to 5, we could fnd two prefect match retransmsson pacets (the yellow and orange area) and two mum match retransmsson pacets (the blue and gray area).they are: R 1 = P 1 P 2 P 3, R 2 = P 4 P 6, R 3 = P 5 P 7, R 4 = P 10. And the average SPE of those 4 retransmsson pacets s In order to explan the dfference between the FNC-ARQ and DNC-ARQ, we dvde the matrx above nto two buffers, each of whch has sze 5. In buffer 1, UE 1, UE 2, UE 3 and UE 4 loss 1,1,1,3 pacets respectvely. Thus three retransmsson pacets are need to recover all lost pacets n buffer 1. They are the pacets: R 1, R 2 and R 3. In buffer 2, UE 1, UE 2, UE 3 and UE 4 loss 1,3,3,1 pacets respectvely. But R 2 recover P6 0 for UE 1, UE 2, UE 3. R 3 recover P7 0 for UE 2, UE 3. So the largest number of lost pacets s 1 and just one retransmsson pacets s needed to buffer 2. It s pacet R 4. As we have seen n ths example, the DNC-ARQ scheme also follow the law that the number of retransmsson pacets n a buffer s equal to the largest number of lost pacets for one UE n the buffer. But n DNC-ARQ scheme, the retransmsson pacet could be a combnaton of lost pacets from dfferent buffers. That means the retransmsson pacets of last buffer mght recover some lost pacets n ths buffer. So n DNC- ARQ scheme, we count the largest number of lost pacets after subtractng those recovered pacets. D. Maxmum match algorthm Prefect match for {TN j, } means we could fnd a networ codng vector whch satsfes: { 1 f s = x φ r (s) = (3.1) 0 f s < x or s > x + y

4 In fact, α s a dscrete random varable whose probablty densty functon depends on the pacet loss probablty p and L. We can denote as: α f α (x, p, L) Fg. 3. Maxmum/Perfect Macth Algorthm θ r = M, where θ r s appendx set of φ r (3.2) Maxmum match for {TN j, } means we could fnd a Networ codng vector whch satsfes (3.1) and θ r = ϑ r, (3.3) Supposng DL ncludes y dynamc chans: [CH x ;...; CH x+y ], the procedure of MMA s shown n Fg.3. IV. THEORETICAL ANALYSIS The transmsson effcency, decodng delay and decodng complexty are three most mportant factors n WSHRM system. A good NC-ARQ scheme s the one that can mze the transmsson effcency, uder the lmt of transmsson delay and decodng complexty. We theoretcally analyze both transmsson effcency and decodng delay of DNC-ARQ n ths secton. A. transmsson effcency Defnton 4. Transmsson Bandwdth(TB) s the rato between the number of pacets needed to be transmtted and the number of the orgnal pacets: TB = (L + H)/L (4.1) Obvously, the TB s the bandwdth that should be occuped to multcast L pacets to M UEs. TB s also adopted by many other research about NC-ARQ scheme. Lemma 1. Supposng α 1,..., α M s the number of lost pacets for UE 1,..., UE M respectvely, H (α 1,..., α M ). Proof: For an arbtrary user equpment UE, the decodng matrx for L+H slot s Φ L+H, = [φ 1,,..., φ L+H, ]. UE loss α pacets, whch means α decodng vectors of φ L+H, are equal to 0. Because UE can recover L orgnal pacets, ran(φ L+H, ) L. We have: L + H α L. Thus, H (α 1,..., α M ) (4.2) The equalty s hold, f the NCV n networ codng matrx Φ L+H satsfy: φ j (s) can be 1 or 0, for any j [1,..., L + H] and any s [1,..., L]. Accordng to [4], the probablty densty functon of α s: p x (1 p ) N x CN z Cz 1 x 1, wthx N z=0 f α (x, p, L) = p x (1 p ) N N C y N Cz 1 x 1, wthx > N z=0 (4.3) where x s the value of α, p s the pacet loss probablty of UE. In deal crcumstance, H = M =1 α. So the H can be also consder as a dscrete random varable. And we can ts probablty densty functon H = M α f H (y, p, M, L) =1 M y M y 1 = f α (x, p, L) f α (x, p, L) =1 x=0 =1 x=0 (4.4) where y s the value of H, p = [p 1,..., p M ], and M s the number of UE, L s the number of orgnal pacets. Theorem 1. The upper bound of TB wth DNC-ARQ s: where H f H(y, p, M, NH) TB DN C 1 + E(H ) / NH (4.5) Proof: To analyze convenently, we devce the L orgnal pacets n l buffers,each of whch holds NH orgnal pacets. We frstly consder the crcumstance wthout nteracton between dfferent buffers. For -th buffer, the number of lost pacets of UE s noted as α. Supposng that the BS multcast H retransmsson pacets for th buffer, we haveh (α ). In DNC-ARQ scheme, the BS can search combnaton for lost pacets n last NH-1 and next NH-1 pacets. So all NCVs n ths buffer satsfes equalty condton of (4.2). Thus we get probablty densty functon H H = M α f H (y, p, M, NH) =1 M y M y 1 = f α (x, p, NH) f α (x, p, NH) =1 x=0 =1 x=0 In ths way, we can get the TB n ths crcumstance: TB = 1 + H / L =1 l =1 H = 1 + lm L L = 1 + E(H ) / NH = 1 + yf H (y, p, M, NH) y=0 when L (4.6) (4.7) The (4.7) s also the lmt of TB wth FNC-ARQ scheme.

5 When the nteracton between the buffer and buffer are taen nto consderaton, some lost pacets n ths buffer may be recovered by the retransmsson pacets of the last buffer. After removng the recovery pacets due to ( 1)-th buffer, the number of lost pacets n th buffer can be rewrtten as β. As the example of DNC-ARQ n secton 3, we can wrte the β as: β = α ε (β 1 β 1 ) (4.8) where ε s a random varable from 0 to 1, β 1 (β 1 ). And then we have: TB DN C = 1 + (β ) / L =1 ( (β = 1 + E ) ) when L NH ( (α 1 + E ) ) = 1 + E( ) H NH HN Theorem 2. The low bound of TB wth DNC-ARQ s: where H f H (y, p, M, L) = TB DN C 1 + H / NH (4.9) Proof: Accordng to (4.8), the TB wth DNC-ARQ would reach mnmum f ε = 1, we have: So, β β β 1 β α (β 1 = α 1. (β 2 β 1 β 2 α 2 (β 1 β 1 ) β 1 α 1 ) β 2 ) can be rewrtten as: { τ=1 ατ 1 τ=1 βτ,wth 2 α,wth = 1 Here α τ f α(x, p, NH). We can get β from (4.10): β 1 (α 1 ) β 2 β 3. β Thus, we have: (α 2 + α 1 α) 1 ( α 3 + α 2 + α 1 (α 2 + α 1 ) ) ( τ=1 (β ) = β 1 α τ α τ ) τ=1 (4.10) A A 1 (4.11) where A f α (x, p, NH), A 1 f α (x, p, ( 1)NH), and A 0 = 0. Comparng to the expresson of (4.4), and we can wrte: H A = M =1 A f H (y, p, M, NH) M y M y 1 = f α (x, p, NH) f α (x, p, NH) =1 x=0 Thus, we get the low bound of TB: TB DN C = = 1 + =1 ( (β ) / L =1 A =1 x=0 A 1 ) / L ( H A 1 + H A 2 H A H A l H A l 1 = 1 + H A l /L here, H A l f H (y, p, M, L) When L +, we have: )/ L TB DN C = 1 / (1 p ) when, L + (4.12) B. Decodng Delay Defnton 5. Maxmum Decodng Delay The mum decodng delay of a NC-ARQ scheme can be wrtten as D = L ( ) µ(j) υ(j) j=1 where, Φ[µ(j), j] = Φ[υ(j), j] = 1, and Φ[, j] = 0, for [1,..., υ(j)] or [1,..., µ(j)] In DNC-ARQ scheme, the mum decodng delay occurred n the followng crcumstance: a lost pacet couldnt fnd ts prefect match whle ts survve tme s smaller than NH. Supposng t need h retransmsson pacets to ensure all UE can recover ths lost pacet.so the MTD of DNC-ARQ scheme can be wrtten as D = NH + h We can regard the lost pacet as a buffer wth sze=1, thus E(D) = NH + E(h ), h f H (y, p, M, 1) (4.13) We consder mum decodng delay wth FNC-ARQ scheme n the same way. In FNC-ARQ scheme, networ codng matrx of can be dvded nto submatrxes. Soppusng that the buffer sze s N, the mum decodng delay wth FNC-ARQ scheme can be wrtten as D = N + h But here, h f H(y, p, M, N), we have E(D ) = NH + E(h ) (4.14) Due to E(h ) > E(h ) when N > 1, we can get E(D) < E(D ) when NH = N (4.15)

6 The nequalty above suggest that: the decodng delay of DNC- ARQ scheme s smaller (n the math expectaton mean) than the decodng delay of FNC-ARQ, when the mum survve tme of DNC-ARQ s equal to the buffer sze of FNC-ARQ. V. SIMULATION AND RESULTS In ths secton, we gve numercal results to verfy the theoretcal analyss for DNC-ARQ scheme. The upper and low bound can be calculated by formula (4.5) and (4.9) when the pacet loss probablty pe, the number of UEs M, and the mum survve tme HN are gven. To obtan TB of DNC- ARQ schemes wth the proposed MMA, we assume the BS multcasts 500,000 orgnal pacets. The Fg.4 and Fg.5 show the smulaton results and the theoretcal result of the transmsson bandwdths of DNC-ARQ schemes for the scenaro consstng of 10 UEs. The average pacet loss probablty s 10% n Fg.4 and 30% n Fg.5 The low bound of DNC-ARQ(the blue lne) s equal to 1/(1 p) = 1/0.9 = approxmatvely n Fg.4 and 1/(1 p) = 1/0.7 = 1.43 n Fg.5. The low bound of DNC-ARQ n both two fgure are almost horzontal. The DNC-ARQ scheme (the blac lne) s always under the low bound of FNC-ARQ (the red lne). The TB of both are decreasng and convergng to the blue lne wth ncreasng of HN. To compare the TB of our proposed technques and the correspondng theoretcal bound. Fg.6 and Fg.7 show the varance of thetb wth pacet loss probablty at M = 5 and M = 50 respectvely. In ths scenaro, all UEs have the same the pacet loss probablty at every slot n multcast, and the mum survve tme s 20. All of the three lne s ncreasng wth pe. And only the blue lne stay unchanged n two fgures. Comparng all result above, we get the followng conclusons: 1) The TB of DNC-ARQ scheme s always under the low bound of FNC-ARQ, and that testfy our theoretcal analyss: the low bound of FNC-ARQ s also the upper bound of DNC-ARQ. 2) Increasng of NH have sgnfcant mprovement for TB of DNC-ARQ, and when NH s gettng larger t s convergng to the low bound. 3) The low bound of DNC-ARQ only depend on pacet loss probablty of UEs, and t approxmates to 1/(1 p) when the number of orgnal pacets s suffcently large. 4) The performance of DNC-ARQ s affected by number of UEs when NH s not suffcently large. It s less effcency when UEs s gettng more. REFERENCES [1] Mchael Luby, Tago Gasba, Thomas Stochammer, and Mar Watson, Relable Multmeda Download Delvery n Cellular Broadcast Networs, IEEE Transacton on Broadcastng, VOL. 53, NO. 1, PP , MAR [2] Shu L.,Phlp S.Y., A hybrd ARQ scheme wth party retransmsson for error control of satellte channels, IEEE transactons on communcatons, Vol.30(7) PP.78 92, [3] Salm Y.El Rouayheb, Mohammad Asad R.Chaudhry, and Alex Sprntson, On the Mnmum Number of Transmssons n Sngle-Hop Wreless Codng Networs, IEEE Informaton Theory Worshop, Lae Tahoe, Calforna, PP , Sep [4] Dong Nguyen,Tuan Tran,Thnh Nguyen,and Bella Bose, Wreless Broadcastng Usng Networ Codng, IEEE Transactons On Vehcular Technology, Vol.58, NO.2, PP , FEB Fg. 4. Fg. 5. Fg. 6. Fg. 7. TB of DNC-ARQ wth M=10,pe=0.1 TB of DNC-ARQ wth M=10,pe=0.3 TB of DNC-ARQ wth M=5,N=20 TB of DNC-ARQ wth M=50,N=20 [5] Qang.Hu, Jun Zheng, An effcent pacet selecton algorthm for networ codng based multcast retransmsson n moble communcaton networs, IEEE ICCT, Jnan,Chna,2011. [6] Qurensh J., Chuan Heng Foh, Janfe Ca, An effcent networ codng based retransmsson algorthm for wreless multcast, IEEE 20th Internatonal Sympo, Toyo,Japan,2009. [7] Kaa Ch, Xaohong Jang, Susumu Horguch, Networ codngbased relable multcast n wreless networ,computer Networ, Vol.54, PP , [8] H.D.T. Nguyen, L. Tran, E. Hong, On transmsson effcency for wreless broadcast usng networ codng and fountan codes, IEEE Communcatons Letters, Vol.15(5), PP , [9] Smeh Sorour, S. Valaee, An Adaptve Networ Coded Retransmsson Scheme for Sngle-Hop Wreless Multcast Broadcast Servces, ACM Trans. On Networng, Vol.19, No.3, Jun [10] K. Ch, X. Jang, S. Horguch, Bloc-level pacet recovery wth networ codng for wreless relable multcast, Computer Networ, Vol.57 PP , 2010.