Two-Phase Cooperatve Broadcastng Based on Batched Network Code Xaol Xu, Praveen Kumar M. Gandh, Yong Lang Guan, and Peter Han Joo Chong 1 arxv:1504.04464v1 [cs.it] 17 Apr 2015 Abstract In ths paper, we consder the wreless broadcastng scenaro wth a source node sendng some common nformaton to a group of closely located users, where each lnk s subject to certan packet erasures. To ensure relable nformaton recepton by all users, the conventonal approach generally requres repeated transmsson by the source untl all the users are able to decode the nformaton, whch s neffcent n many practcal scenaros. In ths paper, by explotng the close proxmty among the users, we propose a novel two-phase wreless broadcastng protocol wth user cooperatons based on an effcent batched network code, known as batched sparse (BATS) code. In the frst phase, the nformaton packets are encoded nto batches wth BATS encoder and sequentally broadcasted by the source node untl certan termnatng crteron s met. In the second phase, the users cooperate wth each other by exchangng the network-coded nformaton va peer-to-peer (P2P) communcatons based on ther respectve receved packets. A fully dstrbuted and lght-weght schedulng algorthm s proposed to mprove the effcency of the P2P communcaton n the second phase. The performance of the proposed two-phase protocol s analyzed and the channel rank dstrbuton at the nstance of decodng s derved, based on whch the optmal BATS code s desgned. Smulaton results demonstrate that the proposed protocol sgnfcantly outperforms the exstng schemes. Lastly, the performance of the proposed scheme s further verfed va testbed experments. Index Terms BATS code, P2P communcatons, cooperatve broadcastng, schedulng, channel rank dstrbuton. Part of ths work wll be presented n IEEE Internatonal Conference on Communcatons (ICC), London, UK, June 2015. The authors are wth the School of Electrcal and Electronc Engneerng, Nanyang Technologcal Unversty, Sngapore 639798 (emal: {xuxaol, meenaksh, eylguan, ehjchong}@ntu.edu.sg).
2 I. INTRODUCTION Wreless broadcastng, by whch some common nformaton s transmtted from a source node to a set of recever nodes through wreless channels, has a wde range of applcatons [1], such as satellte communcaton, vdeo streamng, and fle dstrbuton. The man desgn crteron for wreless broadcastng systems s to ensure relable nformaton recepton by all users, who may experence qute dfferent channel condtons due to channel fadng, nterference and/or congestons. The two most common approaches for ensurng relable broadcastng are retransmsson and codng [2]. Wth the smple repeat requestretransmsson scheme, the source retransmts the lost packets upon recevng a negatve acknowledgement (NAK) from any of the recevers. Although smple for mplementaton, ths scheme typcally results n poor bandwdth effcency. On the other hand, the codng based approach, such as forward erasure correcton codng, though more effcent, usually ncurs hgh encodng/decodng complexty and severe delays. More recently, network codng based schemes have been proposed to mprove the effcency of retransmsson schemes [3,4]. However, such schemes rely heavly on the prompt and accurate feedbacks from all the recevers, whch are dffcult to be acheved n practcal communcaton systems. All the aforementoned broadcastng schemes assume that there s no cooperaton among the recevers, and hence relable broadcastng can only be acheved va source retransmssons, ether wth or wthout codng. In ths paper, we consder the scenaro where the source node s ntended to broadcast some common nformaton to a group of users that are closely located wthn a small regon. Such a setup models varous practcal communcaton scenaros, e.g., vdeo streamng from a base staton to a group of nearby moble users n cellular networks, communcaton from a source user to a squadron of destnaton users n ad-hoc networks, etc. For such scenaros, the secondary channels between the destnaton users are usually more relable than the prmary channels from the source node, due to the shorter dstance. By explotng ths fact, we propose a two-phase cooperatve broadcastng scheme based on batched sparse (BATS) code, wth lmted source broadcastng n the frst phase and network coded peer-to-peer (P2P) packet exchange n the second phase to acheve relable decodng by all users. Batch sparse (BATS) code s a jont fountan code and network code, frst proposed n [5] for achevng the optmal throughput of wreless erasure networks wth fnte codng length [6]. Compared wth the pure fountan code, BATS code acheves hgher throughput by allowng the ntermedate nodes to reencode the packets. Compared wth random lnear network codng (RLNC) [7], BATS code requres smaller buffer sze and has effcent decodng method based on belef propagaton [8]. Furthermore, snce network codng s only performed wthn the batch, the overhead used to trace network codng coeffcents
3 s much smaller compared wth RLNC. In the frst phase of the proposed protocol, the nformaton packets are encoded nto batches wth BATS encoder. These encoded batches are sequentally broadcasted by the source node untl certan termnatng crteron, whch may be desgned to mnmze the number of source transmssons or the total number of transmssons, s met. For the frst desgn objectve, the source can stop transmsson mmedately when the user group, f allowed to decode cooperatvely, s able to recover the fle, although each ndvdual user stll cannot decode the nformaton based on ther own receved packets. For the second desgn objectve, the number of batches sent by the source s optmzed so that the total number of transmssons n both phases s mnmzed. In the second phase, the users help each other by broadcastng to ther peers va P2P communcatons based on ther respectve receved packets from phase 1. P2P erasure repar has been prevously studed for wreless vdeo broadcastng n Multmeda Broadcast/Multcast Servce (MBMS) applcatons [9], where varous schedulng schemes have been proposed. Later, an adaptve schedulng scheme was proposed n [10] to mprove the effcency of the P2P repar. Gven the global state nformaton of all users, a cooperatve P2P repar (CPR) problem was formulated n [11], whch has been proved to be NP-hard. A suboptmal dstrbuted CPR algorthm was proposed n [12]. However, the proposed scheme stll heavly reles on the exchange of perfect control nformaton, whch s dffcult to be acheved when the nter-user lnks are lossy. Besdes, the effcency of the retransmsson n ths scheme s relatvely low. The CPR algorthm n [12] has also been extended to network coded CPR (NC-CPR) n [13] by applyng random lnear network codng [7] n P2P communcatons. A dstrbuted schedulng scheme was proposed n [13] based on the ntuton that the user wth more nnovatve packets should transmt earler. However, when the number of users and the number of packets ncrease, the transmsson overhead, whch ncludes transmsson of the network codng coeffcents and control nformaton, may overwhelm the gan of the proposed scheme. By ntegratng the dea of NC-CPR [13] and rarest frst schedulng [14], a lghtweght peer schedulng algorthm, termed cooperatve Peer-to-peer Informaton Exchange (PIE), was proposed n [15]. Furthermore, when the users are not fully connected, a cluster based repar, where the users are grouped nto clusters wth one user assgned as the cluster head (CH), have been shown to be more effcent than tradtonal P2P repar [16]. The CH collects the nformaton packets from ts cluster members (CM), whch are then exchanged wth other CHs va P2P communcatons. The tradeoff between the ntra-cluster and nter-cluster repars was studed n [17]. To reduce the network codng overhead, an XOR network codng scheme was proposed to replace the RLNC for P2P repar n [18]. All the exstng schemes [9] [18] assume that the nter-user channels used n the second phase
4 are lossless, so that some state nformaton can be relably exchanged before the startng of the P2P communcatons; otherwse, ther performance may degrade severely f the state/control nformaton s lost. In contrast, the proposed scheme n ths paper s fully dstrbuted, wthout requrng any state nformaton exchange, and hence can be appled to networks wth lossy lnks. Specfcally, n our proposed scheme, each user estmates the usefulness of sendng a coded packet from a partcular batch based on the number of packets receved durng the frst phase and ts own transmsson hstory durng the second phase. In general, the more packets receved by a certan batch n phase 1, the more lkely that a network coded packet generated from ths batch s useful for ts peers. Furthermore, for a gven batch, the usefulness of ts packets decreases wth ts transmssons. The usefulness matrces are generated dstrbutvely by each user at the end of phase 1, based on whch a queue of batch ID wth descendng usefulness s created. When the user has a chance to transmt, a network coded packet generated from the front-most batch n the queue s broadcasted frst. Phase 2 s complete when all the users are able to decode the fle. Wth a good BATS code, the user should be able to decode the fle wth a small overhead, e.g., a fle contanng K packets should be decoded from (1 + η)k receved packets wth η 1. However, the performance of BATS code s largely dependent on a pre-defned degree dstrbuton. The optmal degree dstrbuton can be obtaned as a functon of the channel rank dstrbuton [19]. In wreless erasure networks wth fxed topology, such as those consdered n [5] and [20], the channel rank dstrbuton can be obtaned based on the erasure probablty of each lnk. However, n P2P networks, the channel rank dstrbuton s also affected by the communcaton protocol and the stoppng tme. In ths paper, we analyze the transmt effcency of the proposed cooperatve broadcast protocol and derve the resultng channel rank dstrbuton at the nstance of decodng, based on whch a good BATS code s desgned. Smulaton results show that the proposed two-phase protocol acheves hghly relable broadcastng wth less number of transmssons, compared wth the tradtonal sngle-phase transmssons and the exstng cooperatve broadcast schemes [15,21]. Moreover, snce a large number of transmssons are shfted from the source to the users, where less power s requred per transmsson, the proposed protocol s more power effcent. The performance of the proposed scheme s further valdated expermentally wth a 4-node testbed based on the 802.11g W-F network, where the source and the recevers are connected n ad-hoc mode. It s found that the expermental results match very well wth the analytcal and smulaton results. Furthermore, t s found that the BATS code overhead, desgned based on the estmated channel rank dstrbuton, s less than 1%, whch s qute close to the optmal case wth fxed channel rank dstrbuton [19].
5 The rest of ths paper s organzed as follows. Secton II ntroduces the system model. The proposed two-phase protocol s llustrated and analyzed n Secton III. The effectveness and effcency of the proposed protocol s evaluated n Secton IV. In Secton V, the testbed setup and the expermental results are presented. Fnally, we conclude ths paper n Secton VI. Notatons: Throughout ths paper, random varables are represented by boldface upper-case letters and the probablty of an event s denoted as Pr( ). For a random varable X, we use E[X] to denote ts expectaton. Furthermore, scalars, vectors and matrces are represented by talc, boldface lower- and boldface upper-case letters, respectvely. For a set A, we use A to denote ts cardnalty and use A \ B to denote set subtracton. II. SYSTEM MODEL As shown n Fg. 1, we consder a broadcastng scenaro where a source node s ntends to send some common nformaton to a group of k users, whch are closely located wthn a small regon far away from the source node. We assume that the obstructon and nterference near the source may cause a common packet loss probablty p 0 for all users. In addton, we further assume that the wreless lnk between the source to each user suffers from ndependent 1 and memoryless packet loss wth probablty p 1, and the lnks between the users have erasure probablty p 2, where 0 < p 0, p 1, p 2 < 1. Snce the dstance from the source to the user group s much larger than that between the users, we assume that p 2 p 1. A packet sent by the source node can be successfully receved by each recever wth probablty (1 p 0 )(1 p 1 ), and that sent by one of the users can be receved by ts peers wth probablty 1 p 2. Fg. 1: Wreless broadcastng to a group of closely located users. Intutvely, as the channel between the source and the user group s less relable and much more power s requred to compensate for the path loss over the long transmsson dstance from the source to the 1 The channels are assumed to be ndependent n rch scatterng envronment when the dstance between any par of users s larger than half of the wavelength.
6 users, t s desrable to mnmze the number of transmssons by the source node by explotng the more relable P2P communcaton lnks between the users va user cooperaton. III. PROPOSED TWO-PHASE PROTOCOL WITH BATS CODE In ths secton, we propose a two-phase transmsson protocol based on BATS code and user cooperaton to acheve relable communcaton for the scenaros shown n Fg. 1. A BATS code conssts of an outer code and an nner code, as shown n Fg. 2. The outer code s an extenson of the tradtonal fountan code to matrx form. Specfcally, to apply BATS code, the source node frst obtans a degree d for th batch by samplng a pre-desgned degree dstrbuton Ψ, and then randomly pcks d dstnct nput packets to generate a batch of M fountan-coded packets. The batches are then transmtted sequentally by the source. The nner code of BATS employs RLNC at the ntermedate nodes, whch corresponds to the users n Fg. 1, and only packets wthn the same batch wll be coded together. Hence, the network codng overhead s determned by the batch sze M, whch s usually neglgble compared wth the packet length. Fnally, the nner and outer codes are jontly decoded at the recever usng belef-propagaton (BP) and nactvaton decodng. Fg. 2: Structure of BATS code. Based on BATS code, we propose the followng two-phase transmsson protocol: Phase 1: The fle s dvded nto F packets, whch are encoded wth BATS code of batch sze M at the source node. The batches are broadcasted sequentally to the users untl a stoppng crteron s satsfed. Phase 2: The users help each other by exchangng ther respectve receved packets wth..d. erasures va temporally network-coded peer-to-peer (P2P) transmssons, untl all the users can recover the fle.
7 The effcency of the BATS code s largely dependent on the degree dstrbuton Ψ. In [19], the optmal degree dstrbuton s obtaned by solvng a lnear optmzaton problem based on the fle sze and the channel rank dstrbuton, whch s assumed to be known before transmsson. Snce cooperatve P2P repar wll be used n the network of Fg. 1, the channel rank dstrbuton s affected not only by erasure probablty, but also by schedulng algorthm. To desgn a BATS code for such network, we need to analyze the channel rank dstrbuton observed by each user at the nstance of decodng. In the followng, the proposed protocol s dscussed and analyzed based on two desgn objectves: ) mnmzng the source transmssons; ) mnmzng the total number of retransmssons n both phases. A. Mnmzng the source transmssons Wth optmal BATS code [19], the fle can be recovered f (1+η)F packets are receved, where η 1 for moderate to large F. Denote by X the number of packets receved by the user group. To ensure that the user group can eventually recover the fle va P2P transmssons wth probablty 2 no smaller than (1 ε), we must have Pr(X (1 + η)f ) 1 ε. (1) The probablty that a packet s successfully receved by at least one of the user equals to (1 p 0 )(1 p k 1 ), or equvalently, the effectve erasure probablty s p 1 (1 p 0 )(1 p k 1 ), where k s the number of users n the group. After n batches (or equvalently nm packets) have been sent by the source node, the number of receved packets X s a random varable followng bnomal dstrbuton B(nM, 1 p). As nm s usually large, ths bnomal dstrbuton can be approxmated by the normal dstrbuton N (µ, σ 2 ) wth µ = nm(1 p) and σ 2 = nm(1 p) p. Therefore, (1) can be approxmated as ( ) (1 + η)f µ Q 1 ε, (2) σ where Q(x) = x can be obtaned as 1 2π e τ2 2 dτ denotes the Gaussan Q functon. From (2), the mnmum value for n n F M(1 p) α [ 4 pf 2M(1 p) + α 2 p 2 α p] F M(1 p) α 4 pf 2M(1 p), (3) 2 ε s a small postve number, whch s set to 10 6 for the smulatons n ths paper.
8 where F = (1 + η)f and α = Q 1 (1 ε). The approxmaton n (3) s vald snce α p F. As a result, the mnmum number of batches sent by the source s 2F α 4 pf n l =. (4) 2M(1 p) To mnmze the source transmsson, the source node stops transmsson when the number of batches n reaches the threshold n l gven n (4). Snce the expected number of packets receved by each user s only n l M(1 p 0 )(1 p 1 ), whch s smaller than (1 + η)f, the fle s not yet recoverable by each ndvdual user. In the second phase, the users help each other by broadcastng temporally network-coded packets generated from ther respectve receved packets to ensure that all users can successfully recover the fle eventually. Snce all the users are geographcally separated, they have no knowledge on what packets have been receved by others durng phase 1. Therefore, n phase 2, t s crtcal for each user to ndependently determne whch packets t should send based on ts own receved packets. Denote by N j the set of packets receved by user t j wth batch ndex, where {1,..., n}, j {1,..., k}. Consder two typcal users t j and t j. Wth random lnear network codng, a coded packet generated from t j for batch s useful for user t j f N j \ N j. Furthermore, f N j \ N j = m, m useful packets for batch can be generated from t j for t j. Wthout knowng N j, user t j can estmate the value of N j \ N j based on ts own receved packets N j f m out of N j as follows. Specfcally, N j \ N j = m receved packets at user t j are erased at user t j,.e., ( ) Pr N j \ N j = m N j ( N j ) m p m 1 (1 p 1 ) ( N j m), m N j = (5) 0, N j < m M For notatonal convenence, we wll represent above condtonal probablty as Pr(m N j ). Assume that user t j has already sent out u packets generated from batch. Then, the (u + 1)th packet generated from the same batch s stll useful for user t j f ether m u + 1 or at most (m 1) out of u packets are receved by t j. Denote ths event by E u (j), ts probablty of occurrence be estmated as Pr(E u (j) N j ) = M m=u+1 Pr(m N j ) + u m=1 m 1 Pr(m N j ) ( ) u (1 p 2 ) l p (u l) 2. (6) l l=0
9 By symmetry, (6) apples for all the peers of user t j. Hence, Pr(E u(j) N j ) can be used as a metrc to measure the usefulness for user t j to broadcast the (u + 1)th packet generated from batch. In general, (6) s vald for any u 0. However, a user usually wll not send more than M packets from the same batch before decodng. Thus we can calculate the estmaton only up to u = M. Let S j R M n be such usefulness matrx for user t j, wth the (u, )th element equal to Pr(E u(j) N j ). As more phase 2 packets are sent out from a batch, new transmsson s less lkely to be useful. Hence, each column of S j s a monotoncally decreasng vector,.e., S j (u, ) > S j (u + 1, ). Furthermore, f more packets are receved for batch than batch at the end of phase 1, the packet generated from batch s more lkely to be useful than that from batch, expressed mathematcally, f we have N j > N j, then S j (u, ) > S j (u, ), u 0. To maxmze the spectral effcency, a packet that s expected to be more useful should be transmtted wth hgher prorty. To obtan the optmal transmsson order, user t j sorts all the elements n S j n descendng order. Denote the ordered elements by a vector s j R 1 Mn and the column ndex of the ordered elements by a vector v j Z 1 Mn. Then each element of v j represents a batch ID wthn {1,..., n}. User t j sequentally transmts the temporally network-coded packets wth batch ID obtaned from v j. Example 1. For llustraton purpose, we consder a smple setup wth p 0 = 0, p 1 = 0.5, p 2 = 0.1, M = 4 and n = 5. If user t j receves {2, 1, 3, 4, 2} packets at the end of phase 1 for batch 1 to 5, respectvely, the usefulness matrx calculated based on (6) s 0.7500 0.5000 0.8750 0.9375 0.7500 0.3000 0.0500 0.5375 0.7125 0.3000 S j =. (7) 0.0525 0.0050 0.2000 0.3862 0.0525 0.0075 0.0005 0.0448 0.1410 0.0075 [ ] By sortng all the elements n S j, user t j obtans s j = 0.9375 0.8750 0.7500 0.7500 0.7125. [ ] The column ndces of the elements n s j gve the transmsson order as v j = 4 3 1 5 4 3. At the frst tme when user t j can access the channel, t wll send a coded packet generated from all the avalable packets n ts buffer for batch 4, receved n both phases. For the second transmsson, a coded packet from batch 3 wll be sent out, and the process contnues. 1) Estmatng the Total Number of Transmssons n Phase 2: We assume that all the users have equal probablty for transmsson under a multple-access scheme, such as TDMA or CSMA/CA. The
10 transmsson s complete when all the users are able to recover the fle, whch s assumed to be true after T transmssons n total. On average, user t j wll send out T/k coded packets wth batch IDs gven by the frst T/k elements of v j. Among all the T/k packets sent out by t j, only (1 p 2 )T/k wll reach user t j due to packet erasures. Hence, the number of packets receved by user t j from ts (k 1) peers s a functon of T, whch s P (T ) = (1 p 2)(k 1)T k. (8) By symmetry, we may assume that the batch IDs of the receved packets follow a unform dstrbuton. In other words, f we denote by Y 2 the total number of packets receved for a typcal batch n phase 2, then Y 2 s a random number followng bnomal dstrbuton B(P (T ), 1/n),.e., ( ) ( ) P (T ) 1 ( Pr(Y 2 = ) = 1 1 P (T ), = 0,..., P (T ). (9) n n) Denote by Y 1 and Z the number of packets for a typcal batch receved by a sngle user and the user group n phase 1, respectvely. Clearly, we have Y 1 B(M, (1 p 0 )(1 p 1 )),.e., ( ) M Pr(Y 1 = ) = [(1 p 0 )(1 p 1 )] (p 0 + p 1 p 0 p 1 ) (M ), = 0,..., M. (10) Furthermore, Z s a random varable whch satsfes Z Y 1, snce the unon of all sets s always larger than any sngle set. The dstrbuton of Z gven Y 1 can be obtaned as ( ) M Pr(Z = j Y 1 = ) = (1 p k 1 1 ) j p (k 1)(M j) 1, j, = 0,..., M. (11) j Snce only Z packets are avalable for the whole user group, the number of useful packets avalable at any user cannot be larger than Z. Any more packets receved wll be a lnear combnaton of the exstng Z packets. Out of these Z packets, the user already has Y 1 packets receved durng phase 1. Hence, anythng more than (Z Y 1 ) packets receved durng phase 2 wll be redundant. Furthermore, snce random lnear network codng over a suffcently large feld sze s appled, we assume that any packet receved before reachng ts lmt Z s nnovatve. Lemma 1. Let = Z Y 1, we have B(M, p), where p = (1 p k 1 1 )(p 0 + p 1 p 0 p 1 ). Proof: Please refer to Appendx A Based on Lemma 1, the expected number of redundant packets for all batches receved durng phase 2, denoted by R(T ), can be estmated as: P (T ) R(T ) = n l=δ M (l δ) Pr(Y 2 = l, = δ). (12) δ=0
11 For smplcty, we assume that Y 2 and are ndependent bnomal random varables. Hence, (Y 2 ) can be approxmated as a Gaussan random varable dstrbuted accordng to N (µ R, σr 2 ), where µ R = P (T ) n M p and σ2 R = P (T ) ( n 1 1 n) + M p(1 p). Hence, (12) can be vewed as the postve expectaton of a Gaussan varable, whch can be computed as x R(T ) = n exp ( (x µ R) 2 0 2πσR 2 2σ R ) dx σ 2 ( ) ( = n R 2π exp µ2 R 2σR 2 + µ R nq µ ) R. (13) σ R Denote by D j the number of nnovatve packets receved by user t j at the end of phase 2, j = 1,..., k. For smplcty, we assume D j N (µ D, σd 2 ). Followng smlar analyss gven above, we have µ D = (1 p 0 )(1 p 1 )nm + P (T ) R(T ), (14) where P (T ) s gven n (8) and R(T ) s gven n (13). Snce R(T ) s usually much smaller than P (T ) and the number of transmssons durng phase 1, t can be gnored when computng σd 2. Hence, we have σ 2 D nm(1 p 0 )(1 p 1 )(p 0 + p 1 p 0 p 1 ) + Phase 2 s complete when the last user s able to decode the fle,.e., T (k 1) (1 p 2 )p 2. (15) k mn{d 1,..., D k } (1 + η)f. (16) Based on the approxmaton n [22], (16) can be explctly expressed as ( ) 0.625 µ D + σ D Φ 1 (1 + η)f, (17) k + 0.25 where Φ( ) s the cumulatve dstrbuton functon (cdf) of the standard normal dstrbuton N (0, 1); µ D and σ D are functons of T, as gven n (14) and (15), respectvely. Hence, we can obtan the stoppng tme T from (17). Intutvely, the more packets transmtted, the more nnovatve packets wll be receved. Hence, the left-hand-sde of (17) s monotoncally ncreasng wth T ; thus (17) has a unque soluton. The numercal soluton of (17) can be effcently obtaned by bsecton method, as shown n Algorthm 1. 2) Estmatng the Rank Dstrbuton: In conventonal drected acyclc networks consdered n [5,19], the rank dstrbuton of the batches s determned by the network topology and the erasure probablty of each lnk. However, for the network shown n Fg. 1 wth cycles, the rank dstrbuton s affected by the schedulng scheme n phase 2 transmsson. Snce rank dstrbuton s an mportant parameter for
12 Algorthm 1 T (n, M, k, F, p 0, p 1, p 2 ) Intalze: ( ) p = (1 p k 1 1 )(p 0 + p 1 p 0 p 1 ); β = Φ 1 0.625 k+0.25 T l = 0 T u = nm f l = µ D (T l ) + σ D (T l )β F f u = µ D (T u ) + σ D (T u )β F whle f u f l > 1 do T = Tl+Tu 2 f = µ D (T ) + σ D (T )β F f f > 0 then else T u = T ;f u = f T l = T ;f l = f end f end whle T = T u desgnng BATS code, t s crucal to get a good estmaton for t before transmsson, whch s pursued n ths subsecton. Followng smlar assumptons as n (12), the rank of a typcal batch for a user s r f ether of the followng two events occur: ) the user group has more than r packets for ths batch, but the user only receves r,.e., Z > r and Y 1 + Y 2 = r; ) the user receves more than r packets, but only r out of them are nnovatve,.e., Y 1 +Y 2 r and Z = r. Hence, at the end of the transmssons, the probablty that a batch has rank r, r {0, 1,..., M}, for a user s gven by Pr(r) = Pr(Z > r, Y 1 + Y 2 = r) + Pr(Z = r, Y 1 + Y 2 r) r M = Pr(Z = j Y 1 = ) Pr(Y 1 = ) Pr(Y 2 = r ) =0 j=r+1 r P + Pr (Z = r Y 1 = ) Pr(Y 1 = ) Pr(Y 2 = j), (18) =0 j=r where each ndvdual probablty Pr(Y 2 ), Pr(Y 1 ) and Pr(Z Y 1 ) are gven n (9),(10) and (11), respec-
13 tvely, wth T obtaned from Algorthm 1. Snce the number of source transmssons s set to the mnmum, the decodng usually occurs only when T s suffcently large. In ths case, we may assume that Y 1 + Y 2 > Z and Pr(r) Pr(Z = r),.e., where p = p 0 + p k 1 p 0p k 1. Pr(r) = ( ) M (1 p) r p M r, (19) r Example 2. Assume that a fle contanng 1600 packets s to be transmtted from the source to three users through the network shown n Fg. 1 wth p 0 = 0.05, p 1 = 0.5 and p 2 = 0.1. A batch code wth batch sze M = 16 s used to correct the erasures. The batch overhead s assumed to be 1% and hence the mnmum number of batches sent by the source s computed from (4) to be 129. The analytcal rank dstrbuton s plotted together wth the smulated rank dstrbuton for the three users n Fg. 3. It s observed that the analytcal rank dstrbuton gven n (18) matches qute well wth the smulaton results. Furthermore, the approxmated rank dstrbuton gven n (19) s also of suffcent accuracy. Hence, we can desgn good BATS code based on the rank dstrbuton gven n (18) or (19). Fg. 3: Comparson between analytcal and smulated rank dstrbutons.
14 B. Mnmzng the total number of transmssons It s observed that f the source sends a few more batches than the threshold gven n (4), the total number of transmssons n both phases may be sgnfcantly reduced. The optmal number of batches to be transmtted by the source, denoted as n, can be found by solvng the followng optmzaton problem n = arg mn(t + nm) n subject to: (13)-(15),(17) Snce the constrants n (20) s non-convex, a closed-form soluton does not exst n general. We propose to solve for n by exhaustvely searchng all possble values of n snce the search space s not large, as explaned next. Frst, the mnmum value of n, denoted by n l, s gven by (4). Furthermore, the maxmum value of n, denoted by n u, s the number of batches sent out by the source when all the users are able to decode the fle wthout requrng phase 2 transmssons. To fnd n u, we further denote by X j the number of packets receved by user t j, for j = 1,..., k, after the source sent out n u batches,.e., Mn u packets. X j are ndependent and dentcally dstrbuted random varables accordng to B(Mn u, (1 p 0 )(1 p 1 )). The source transmsson stops when all the users are able to recover the fle,.e., when (20) mn {X j} (1 + η)f. (21) j {1,...,k} Snce Mn u 1, the bnomal dstrbuton can be approxmated by the normal dstrbuton N ( µ, σ 2 ), where µ = Mn u (1 p 0 )(1 p 1 ) and σ 2 = Mn u (p 0 + p 1 p 0 p 1 )(1 p 0 )(1 p 1 ). The statstcal mean of mn{x j } can be approxmated as [22] [ ] E mn {X j} j {1,...,k} ( ) 0.625 µ + σφ 1, (22) k + 0.25 By substtutng (22) nto (21), we can solve for n u as 2F + ˆpβ 2 + 4ˆpβ n u = 2 F + ˆpβ 4, (23) 2M(1 ˆp) ( ) where F = (1 + η)f, ˆp = p 0 + p 1 p 0 p 1 and β = Φ 1 0.625 k+0.25. For all the nteger values wthn [n l, n u ], the correspondng phase 2 transmssons T can be found from Algorthm 1, and hence the total number of transmssons nm + T can be computed. The value of n that leads to the mnmum number of transmssons s returned as n. Example 3. If a fle contanng 5000 packets s to be sent to a group of k = 5 recevers through the network shown n Fg. 1, where p 0 = 0.05, p 1 = 0.5, p 2 = 0.1. Assume that a BATS code wth batch sze M = 16 s used and the degree dstrbuton s well desgned such that the codng overhead s
15 mantaned wthn 1%. From (4) and (23), the mnmum and maxmum value of n can be computed to be 351 and 673, respectvely. For all the ntegers n between, we can fnd the correspondng total number of transmssons requred, whch s plotted n Fg. 4, and the the optmal number of batches s n = 402. Fg. 4: Total number of transmssons (phase 1 and 2) versus number of batches. Compared wth the sngle phase transmssons where the number of batches sent by the source s set to the maxmum value n u, the proposed cooperatve broadcast scheme wth n batches saves 1284 transmssons. Furthermore, compared wth the one targetng at mnmum source transmssons,.e., wth n = 351, 512 transmssons are saved by makng the source transmt 51 more batches wth n = 402. When the number of batches sent s larger than the mnmum value, the users do not have to receve all the nnovatve packets from ts peers before decodng the fle. Hence, the resultng channel rank dstrbuton wll not follow that gven n (19). However, the estmated dstrbuton gven n (18) stll holds. Hence, we can fnd the optmal degree dstrbuton for the BATS code accordngly. Example 4. Consder the same network settngs as that n Example 3. The analytc channel rank dstrbuton gven n (18) s compared wth smulated dstrbuton n Fg 5. The smulated rank dstrbuton s averaged over the 5 users at the moment when all the users can recover the fle. Eventually, the estmated dstrbuton matches well wth the smulated one. The slght dscrepancy for hgh-rank batches s manly
16 due to the assumpton that all batches are sent wth equal probablty n phase 2. The expermental results n Secton V show that the performance degradaton of the BATS code due to ths small error of channel estmaton s neglgble. Fg. 5: Comparson between the analytcal and smulated channel rank dstrbutons. IV. PERFORMANCE EVALUATION Ths secton s devoted to evaluatng the effectveness, effcency, and robustness of the proposed 2-phase cooperatve broadcastng schemes. Frst, the transmsson effcency of the proposed 2-phase scheme wth optmal number of batches s compared wth the coded sngle-phase transmssons and the cooperatve P2P nformaton exchange (PIE) ntroduced n [15,21]. Then, the computatonal overhead of the proposed scheme s analyzed. Fnally, we also nvestgate the robustness of the proposed scheme aganst unknown number of users. A. Transmsson Effcency 1) Comparson wth Sngle-Phase Transmsson: Frst, we compare the proposed two-phase broadcastng scheme wth the tradtonal sngle-phase transmsson, where the source keeps transmttng the packets untl all the users can recover the fle. Assume that optmal erasure code, such as Raptor code [23], has been appled so that a user can recover the fle after recevng (1 + η)f packets.
17 Note that the proposed two-phase scheme n Secton III-B reduces to the coded sngle-phase transmsson when n obtaned from (20) s equal to the upper bound, n u, gven n (23). Moreover, the proposed scheme should outperform the sngle-phase broadcast f the nter-user lnks are better than the lnks between the source and the users, and/or the number of users s suffcently large. Example 5. Assume a fle contanng F = 2000 packets s to be dstrbuted by the source node to a group of k users. The codng overhead η s set to be 1% n for both BATS code and the erasure code. The total number of transmssons requred for all users to recover the fle s plotted aganst the number of users for both the sngle-phase transmsson and the proposed two-phase broadcast scheme wth optmal number of batches. Fg. 6: Comparson of the proposed two-phase cooperatve broadcast wth sngle-phase transmsson. It s observed from Fg. 6 that the total number of transmssons ncreases wth the number of users n the sngle phase scheme. In contrast, wth the proposed two-phase scheme, t decreases wth the number of users due to spatal dversty gan. Furthermore, when the nter-user lnks have the same packet loss rate as the lnks between the source and the users,.e., p 1 = p 2, the proposed two-phase cooperatve broadcast reduces to the sngle-phase transmsson as the optmal number of batches n obtaned from (20) equals to n u n (23). When p 2 < p 1, the proposed two-phase scheme outperforms the sngle-phase
18 transmsson, e.g., wth k = 9, the proposed scheme saves 108 transmssons when p 1 = 0.4, p 2 = 0.2, and t saves 563 transmssons when p 1 = 0.5, p 2 = 0.1. 2) Comparson wth cooperatve peer-to-peer nformaton exchange (PIE) [15]: PIE s an effcent peer schedulng algorthm for network codng enabled wreless networks, ntroduced n [15,21]. However, phase 1 transmsson n PIE s uncoded, and hence the resdual loss s nevtable. To make a far comparson, we assume that the fle s frstly encoded wth some good erasure code so that the recever can recover the fle upon recevng (1 + η)f packets. The coded packets s then dvded nto blocks of sze M and sequentally broadcasted by the source node. Furthermore, snce PIE s desgned only for lossless nter-user channels, when there are erasures n phase 2,.e., p 2 > 0, the schedulng algorthm wll termnate before the the data block can be decoded. To acheve relable communcatons n phase 2, addtonal retransmssons are requred, whch results n some performance degradaton. In the followng example, we compare the performance of the proposed cooperatve two-phase scheme wth PIE, n favor of the latter by gnorng transmsson overhead used for exchangng the state nformaton. Example 6. Consder the network shown n Fg. 1, where the source node s ntended to send a fle wth F = 2000 packets to a group of k users. The lnks between the source node wth the users are assumed to have ndependent erasures wth probablty p 1. The total number of transmssons requred for all the users to recover the fle wth PIE and wth the proposed scheme are compared n Fg. 7(a), under the assumpton of lossless phase 2 lnks,.e., p 2 = 0. It s observed that the proposed scheme outperforms PIE when p 1 = 0.2. Ths s consstent wth expectaton because complete repar s not necessary wth proposed scheme n phase 2, due to the applcaton of BATS code, but when a packet s mssed by n PIE, t must cost a retransmsson to repar. When p 1 = 0.5, the receved packets among dfferent users have more dverse erasures, hence PIE, whch has a centralzed schedulng algorthm, slghtly outperforms the proposed scheme, at the cost of addtonal control complexty and state nformaton exchange. When there are erasures n phase 2 channels,.e., p 2 = 0.1, the proposed scheme outperforms PIE for both p 1 = 0.2 and p 1 = 0.5, as shown n Fg. 7(b). B. Computatonal Overhead The desgn of the proposed 2-phase cooperatve broadcastng scheme nvolves determnng the number of batches, the degree dstrbuton of the BATS code, schedulng of each users and BATS decodng. Frst, the mnmum number of batches can be obtaned from equaton (4) drectly and the optmal number of
19 Fg. 7: Comparson of the proposed scheme wth PIE. batches s determned by solvng the optmzaton n (20), whch s of complexty O( K M log 2 K). When solvng for the number of batches, we can also obtan the estmated number of transmssons of phase 2. Then, the correspondng channel rank dstrbuton h can be drectly computed from (18). Based on h, the optmal degree dstrbuton for the BATS code can be obtaned by solvng a lnear optmzaton problem formulated n [19]. All these computatons can be carred out off-lne, whch wll not cause any communcaton delay. On-the-fly computatons ncludes the schedulng and BATS decodng. The proposed schedulng algorthm s completely dstrbuted, whch conssts of computng and sortng the usefulness matrx, wth complexty O(M n log M). BATS decodng s based on belef propagaton and nactvaton decodng, wth complexty O(K(M 2 + ML)) [5], where L s the packet length. C. Robustness In certan scenaros, the number of users k may be unknown. The exstng P2P repar schemes n [9] [18] are not applcable for such scenaros snce they requre the state nformaton of all the users for desgnng the schedulng algorthms. In contrast, the proposed two-phase cooperatve broadcastng scheme s fully dstrbuted, hence applcable for the case of unknown number of users. Due to ncreasng dversty gan wth k, the proposed scheme, desgned for a network wth k users, allow a network wth unknown extra users to recover the fle, at the cost of some performance degradaton. Denote by L(k) the mnmum number of transmssons requred to delver a fle from the source to a group of k users wth the proposed two-phase protocol. Under the same setup, another network wth k
20 users, where k k, should also be able to recover the fle after L(k) transmssons because addtonal users brng more space dversty. In other words, the performance degradaton can be bounded by the dfference between the mnmum number of transmssons,.e., L(k) L(k ). Snce the batches are chosen wth equal probablty n phase 2, wth fxed number of batches, the channel rank dstrbutons at the moment of decodng are almost the same for dfferent number of users. Hence, the redundant transmssons caused by extra users n the second network s manly due to the sub-optmal choce of n, whch s much lower than the bound L(k) L(k ). Example 7. Consder the network shown n Fg. 1, wth p 0 = 0.05, p 1 = 0.5 and p 2 = 0.1. A fle contanng K = 2083 packets s to be transmtted to k users wth batch sze 16. If the proposed scheme s desgned for k = 3, the optmal number of batches s set as n = 200 and the degree dstrbuton of BATS code s desgned based on the estmated channel rank dstrbuton for k = 3. In Fg. 8, the number of requred transmssons for networks wth k 3 (based on the two-phase scheme desgned for k = 3) s compared wth the deal case (where the proposed two-phase cooperatve broadcast scheme s desgned for the exact k users). It s observed that the performance degradaton ncreases when k gets larger, as expected. However, the degradaton s stll margnal n proporton, for example, the degradaton s less than 5% even when the networks has 3 tmes more users than t was desgned for. Fg. 8: Mnmum number of transmssons versus number of users k.
21 On the other hand, f the proposed scheme s desgned for a k larger than the actual value, the users may not be able to recover the fle. Hence, when the exact value of k s unknown, the proposed two-phase scheme should be desgned for the mnmum expected value. V. EXPERIMENTAL RESULTS Ths secton evaluates the performance of the proposed two-phase broadcast protocol over a 4-node testbed based on IEEE802.11g wreless network, n order to valdate the analytcal results n Secton III. The testbed conssts of 3 laptops as recevers and one desktop as the source. We use the HP ProBook 430G1 laptops whch has nbult W-F and HP Z210 desktop computer whch uses PROLNK USB W-F dongle WG2000/R. The operatng systems used n the laptops and the desktop are Ubuntu 13.10 and Ubuntu 12.04, respectvely. The source and recevers are connected n Ad-Hoc mode. A pcture fle of sze 2.1MB s dstrbuted from the source to all the three recevers. The fle s dvded nto 2083 packets, each of 1000 bytes. In phase 1, the source broadcast the coded packets to all ts recevers, whle n phase 2 the recevers exchange network-coded packets usng broadcast transmsson. Snce the transmssons are carred out n broadcast mode, UDP s used as the transport layer protocol. A. Channel Characterzaton The IEEE 802.11 MAC operates n two modes, namely uncast and broadcast. Our testbed operates n the broadcast mode. There are two nherent problems n ths mode: poor relablty and lack of back-off [24]. Snce there are multple recevers n broadcast communcaton, t s unclear who should ACK. In the absence of ACK, there wll be no retransmsson. Furthermore, the broadcast sender cannot sense the medum and t wll keep transmttng durng collson wthout backng off, causng collson. These lead to packet loss at the recevers. There are two knds of packet loss n 802.11 broadcast: correlated loss and uncorrelated loss. The correlated loss refers to the common loss experenced by all users, whch s manly caused by collsons. On the other hand, the uncorrelated loss refers to the ndependent packet erasures at the recevers, whch s manly caused by nterference and nose [25].
22 B. Expermental Results The correlated erasure probablty for phase 1 s measured to be 5%,.e., p 0 = 0.05. The uncorrelated loss 3 s set to 0.5,.e., p 1 = 0.5. Wth network codng, most of the packets sent out durng phase 2 transmssons are nnovatve for the correspondng recevers, unless t reachs the lmt determned by the total number of receved packets n phase 1. Each user s able to recover the fle upon recevng a suffcent number of nnovatve packets. Hence, t s unnecessary to dfferentate the correlated and ndependent packet loss n phase 2. The erasure probablty for phase 2 s measured to be around p 2 = 0.2, whch s manly due to congeston 4. A BATS code over GF (2 8 ) s used to encoded the packets nto batches of sze M = 16. The analytcal channel rank dstrbutons are derved wth BATS code overhead set as η = 1%. Based on the analyss presented n Secton III, the mnmum number of batches sent by the source s 167. To mnmze the total number of transmssons, the optmal number of batches sent by the source s 211. The number of nnovatve packets receved s plotted aganst the number of transmssons n Fg. 9 and Fg. 10 for n = 167 and n = 211, respectvely. Note that a packet s called nnovatve, f t s not a lnear combnaton of the prevously receved packets. On the other hand, f a receved packet s a lnear combnaton of the prevous packets wthn the batch, ths packet s vewed as redundant packet. For the case wth n = 167, user 1, user 2 and user 3 recover the fle after recevng 2432, 2455 and 2395 packets, respectvely. The total number of transmssons made by the network n both phases s 4939. The number of redundant packets receved s measured to be 340 on average, whch s 15% of the total number of transmssons n phase 2. The redundance s close to the estmated value 325 computed from (12). The number of nnovatve packets used for decodng by the three users are 2091, 2088 and 2083, respectvely, whch means that the overhead of the BATS code s mantaned wthn 0.4%. Ths small overhead valdates our channel rank estmaton made n (18), based on whch the BATS code s desgned. For the case wth n = 211, the number of phase 2 transmssons s measured to be 968, whch s close to the estmated value 956. The redundant packets receved n phase 2 s 13 on average, whch s 1.34% of all phase 2 transmssons. Furthermore, the fle s recovered from 2091, 2086 and 2090 packets at user 3 The nstant packet loss rate changes over tme due to nterference and changng envronment. Hence, the average packet loss rate over the transmsson of entre fle s used. 4 802.11 broadcast mode lacks congeston control mechansm. The erasure probablty for phase 2 transmsson can be reduced f some congeston control mechansm, such as the pseudo-broadcast n [24], s appled.
23 Fg. 9: Number of nnovatve packets receved versus number of transmssons for n = 167. Fg. 10: Number of nnovatve packets receved versus number of transmssons for n = 211. 1, user 2 and user 3, respectvely. Hence, the codng overhead for the BATS code s mantaned wthn 0.5%. The small overhead of BATS code desgned based on the analytcal channel rank dstrbuton further valdates our analytcal results presented n Secton III. Compared wth the case wth n = 167, 704 more packets are transmtted by the source n phase 1. However, the total number of transmssons
24 n both phases s reduced from 4939 to 4344. VI. CONCLUSION In ths paper, we have proposed a fully dstrbuted two-phase cooperatve broadcastng scheme based on BATS code to acheve relable communcaton from the source node to a group of users. Wth the proposed scheme, the number of source transmssons s reduced by ntroducng user cooperatons n phase 2. Furthermore, the total number of retransmssons may also be reduced when the nter-user channels exploted n phase 2 have less erasures than the phase 1 channel from the source to the users. The performance of the proposed two-phase scheme has been analyzed and valdated through smulatons and experments. When the power or bandwdth at the source s lmted, we propose to apply the twophase cooperatve broadcast scheme wth mnmum number of batches. Otherwse, the proposed scheme wth the optmal number of batches should be appled to mnmze the total number of transmssons, and hence the communcaton delay. When global state nformaton s avalable, the proposed two-phase protocol can be further mproved by optmzng ts schedulng algorthm. APPENDIX A PROOF OF LEMMA 1 Snce = Z Y 1, ts probablty dstrbuton can be calculated as Pr( = δ) = = M δ =0 M δ =0 Pr(Z = + δ Y 1 = ) ( M δ ) (1 p k 1 1 ) δ p (k 1)(M δ) 1 ( M ) [(1 p 0 )(1 p 1 )] (p 0 + p 1 p 0 p 1 ) M (24)
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