Analysis and Optimization of Sparse Random Linear Network Coding for Reliable Multicast Services

Transcription

1 1 Analyss and Optmzaton of Sparse Random Lnear Network Codng for Relable Multcast Servces Andrea Tass, Ioanns Chatzgeorgou and Danel E. Lucan arxv: v2 [cs.it] 20 Nov 2015 Abstract Pont-to-multpont communcatons are expected to play a pvotal role n next-generaton networks. Ths paper refers to a cellular system transmttng layered multcast servces to a multcast group of users. Relablty of communcatons s ensured va dfferent Random Lnear Network Codng (RLNC technques. We deal wth a fundamental problem: the computatonal complexty of the RLNC decoder. The hgher the number of decodng operatons s, the more the user s computatonal overhead grows and, consequently, the faster the battery of moble devces drans. By referrng to several sparse RLNC technques, and wthout any assumpton on the mplementaton of the RLNC decoder n use, we provde an effcent way to characterze the performance of users targeted by ultra-relable layered multcast servces. The proposed modelng allows to effcently derve the average number of coded packet transmssons needed to recover one or more servce layers. We desgn a convex resource allocaton framework that allows to mnmze the complexty of the RLNC decoder by jontly optmzng the transmsson parameters and the sparsty of the code. The desgned optmzaton framework also ensures servce guarantees to predetermned fractons of users. The performance of the proposed optmzaton framework s then nvestgated n a LTE-A embms network multcastng H.264/SVC vdeo servces. Index Terms Sparse network codng, multcast communcaton, ultra-relable communcatons, green communcatons, moble communcaton, resource allocaton, LTE-A, embms. I. INTRODUCTION Among the major noveltes lkely to be mplemented n next-generaton networks, there s the possblty of provdng servces characterzed by an avalablty level of almost 100%. In the lterature, that emergng knd of servces s usually referred to as ultra-relable servces [1]. The ultrarelable way of conveyng servces s expected to be greatly useful n a plethora of applcatons, such as relable cloudconnectvty, data harvestng from sensors, professonal communcatons [2]. Among the possbltes, ths paper refers to a system model where a Base Staton (BS transmts, n a multcast fashon, a Pont-to-Multpont (PtM servce to a Multcast Group (MG Ths work s part of the R2D2 project, whch s supported by EPSRC under Grant EP/L006251/1, and by the TuneSCode project (No. DFF granted by the Dansh Councl for Independent Research. A. Tass was wth the School of Computng and Communcatons, Lancaster Unversty, UK. He s now wth the Department of Electrcal and Electronc Engneerng, Unversty of Brstol, UK (e-mal: a.tass@brstol.ac.uk. I. Chatzgeorgou s wth the School of Computng and Communcatons, Lancaster Unversty, Lancaster, UK (e-mal:.chatzgeorgou@lancaster.ac.uk. D. E. Lucan s wth the Department of Electronc Systems, Aalborg Unversty, Aalborg, DK (e-mal: del@es.aau.dk. of users. In partcular, the multcast servce s provded n an ultra-relable way, hence, the servce shall be receved by predetermned fractons of users, and has to meet target temporal constrants. It s worth notng that the possblty of managng ultra-relable multcast applcatons s pvotal, n any Professonal Moble Rado (PMR standard [3]. Even though classc PMR standards, lke Terrestral Trunked Rado (TETRA or Assocaton of Publc-Safety Communcatons Offcals-Project 25 (APCO P25, refer to ad-hoc communcaton protocol stacks, the upcomng evolutons of those standards wll rely on the 3GPP s Long Term Evoluton- Advanced (LTE-A standard and ts extents [4]. As a result, next-generaton PMR standards are expected to enable the deployment of PMR systems over pre-exstng LTE-A networks. In ths paper, we consder a system model where the base staton multcasts a scalable servce composed by one base layer and multple enhancement layers. The base layer provdes a basc reconstructon qualty that s gradually mproved as one or more enhancement layers are progressvely receved. Because of the layered nature of the consdered multcast servce, t s natural to refer to servce relablty constrants, whch mpose that at least a mnmum number of users s able to recover predetermned sets of servce layers, by a gven temporal deadlne. The layered servce approach has been orgnally adopted n vdeo communcatons [5]. However, as dscussed n [1] and [6], the same prncple s lkely to go beyond the tradtonal boundares of multmeda communcatons and be appled n other felds n order to acheve an analog-lke servce degradaton. Because of the ultra-relable nature of the consdered multcast servce, users are requred to acknowledge to the base staton when they successfully recovered one or more servce layers. Even though there exsts Automatc Repeat-reQuest (ARQ [7] and Hybrd ARQ error control protocols [8] sutable for PtM communcatons, the protocol complexty and the requred amount of feedback quckly become ntractable as the number of users ncreases. For these reasons, the relablty of PtM communcatons s ensured va Applcaton Level- Forward Error Correcton (AL-FEC technques based on Luby Transform (LT or low-densty party-check codes. However, as noted n [9], these knd of codes requre large block lengths to operate close to ther capacty, and that could potentally be an ssue, n the case of multmeda communcatons. In addton, the most recent evolutons of LT codes [10] usually rely on fxed degree dstrbuton functons and, hence, the code sparsty cannot be optmzed on-demand. To ths end,

2 2 n order to mtgate those ssues, our system model ensures relablty of multcast communcatons, va Random Lnear Network Codng (RLNC technques [11], [12]. Gven a source message of k source packets to be multcast, the RLNC prncple generates and multcasts a stream of coded packets, where each of them s obtaned as a lnear combnaton of multple source packets. A user recovers the source message as soon as t collects a number of lnearly ndependent coded packets that s equal to k. RLNC schemes have been used n several wreless settngs as a versatle soluton for relable servce delvery [13], [14]. Among the lterature contrbutons, M. Xao et al. [15] refer to a system model where nodes are connected by a network that can be represented by a Drect Acyclc Graph (DAG; that network conssts of one source node and several snks. In [15], the RLNC prncple takes place at the network layer and allows ntermedate nodes to combne several ncomng data flows; relablty of coded packet transmssons s ensured va a channel code operatng at the physcal layer. The sze of coded packets and the channel code rate are jontly optmzed to mnmze the end-to-end delay at the network layer. In addton, multple resource allocaton approaches have been proposed to mprove the relablty of layered servces va dfferent RLNC mplementatons [16] [18]. In partcular, [16] consders a mult-hop drected acyclc graph network topology where a scalable servce s multcast to multple recevers. That paper proposes to optmze the communcaton rate on each lnk, n order to mprove relablty. Channel erasures are further mtgated va classc FEC technques. Smlarly to [16], [17] deals wth mult-hop network topologes and layered servces. However, n that case, relablty of end-toend communcatons s mproved va a specfc mplementaton of RLNC, whch acheves a ladder-shaped global codng matrx. Dfferently than [16] and [17], [18] apples RLNC to populate a dstrbuted cachng system, kept by ntermedate network nodes. The communcaton-ends can take advantage of that whle they retreve the desred scalable servce, va a reduced number of Pont-to-Pont sessons. In contrast to [15] [18], ths paper refers to a typcal cellular network topology, where the source node transmts streams of coded packets to a set of users n a multcast fashon. In other words, ths paper adopts RLNC to mprove relablty over a one-hop broadcast network and not as a way to mprove the end-toend communcaton throughout across a mult-hop network topology [12, Ref. [14]-[16]] and [19, Ref. [26]]. We observe that the applcaton of RLNC to one-hop broadcast networks has been also dscussed n [20] and [21]. In both cases, the broadcastng of a set of source packets s splt nto multple stages. Durng the frst stage all the source packets are broadcast by the source node, then, n the followng stages, the source node and/or an ntermedate relay node broadcast streams of coded packets. Both [20] and [21] focus on dfferent forms of Instantly Decodable Network Codes, whch generate coded packets n a determnstc fashon, based on multple user feedback. As a consequence, we observe that the user uplnk traffc can quckly become non-neglgble as the number of users ncreases. Gven that we wll refer to a system model composed by a source node multcastng servces to a multcast group composed by a potentally great number of users, t s not approprate to refer to the strateges as n [20], [21]. On the other hand, we wll refer to classc decodable RLNC strateges (as n [19] that are characterzed by a sgnfcantly smaller user feedback footprnt. Unfortunately, as noted n [22], [23], the flp sde of the consdered RLNC technques s represented by the complexty of the decodng operatons that depends, amongst other code parameters, by the length k of the source message. As noted n [24], [25], the decodng complexty problem can be partally mtgated by the systematc mplementaton of RLNC (SRLNC. However, n case of poor propagaton condtons, the performance of SRLNC concdes wth that of RLNC [26]. Obvously, the more the decodng complexty grows, the more the processng footprnt ncreases and, hence, the battery of moble devces dscharges. For these reasons, ths paper addresses the followng fundamental queston: Is there a way to mnmze the RLNC decodng complexty of ultra-relable layered multcast communcatons wthout alterng the decoder currently onboard moble devces? We wll answer the aforementoned research queston by referrng to multple sparse RLNC technques. As wll be clear n the followng secton, let us ntutvely defne the sparsty of the code as the number of source packets that on average are nvolved n the generaton process of each coded packet [22]. To the best of our knowledge, the general expresson of the decodng complexty as a functon of the source message length and the sparsty s unknown. However, the decodng complexty decreases as the source message gets shorter [23] and/or the sparsty ncreases [22]. Intutvely, as the sparsty ncreases, the nformaton content of each coded packet decreases. Hence, the average number of coded packets needed to recover a source message ncreases as the sparsty grows [22]. That leads us to further refne our research queston as follows: Are there any optmzed sparse RLNC strateges ensurng: ( a reduced decodng complexty, and ( a recovery of the source message wth an average number of coded packet transmssons, whch s close or equal to that provded by non-sparse RLNC technques? The frst contrbuton of the paper s that of provdng an effcent performance modelng of sparse non-systematc and systematc RLNC technques va a unfed theoretcal framework. In partcular, n Secton II, we characterze the user performance n terms of the average number of coded packet transmssons needed to recover a gven servce layer. It s well known n the lterature that an exact expresson for the aforementoned performance ndex s unknown [27] [30]. That s caused by the lack of an analytcal formulaton of the probablty of generatng a full-rank sparse random matrx over a fnte feld [30]. In order to mtgate the aforementoned ssue, X. L et al. [27], [28], proposed a poneerng approach for upper-boundng and lower-boundng the probablty of generatng at random a sparse non-sngular random matrx, based on the zero pattern of the random matrx. Unfortunately, the valdty of the resultng bounds has been proven only for large fnte felds. Apart from that, those bounds cannot be effcently ncorporated nto an optmzaton model meant to be solved on-demand, before startng the transmsson of a

3 3 servce. In fact, the bound expressons nvolve nested sums where each term s a product of several bnomal coeffcents, whch could not be practcally dervable, n the case of large source message lengths (Secton II. Furthermore, t s also not straghtforward to formally prove the convexty of the bounds as n [27], [28], because ther defntons nvolve several nondfferentable ponts. For these reasons, we rely on the results presented n [29] and extended n [30]. However, n [29], [30], authors only provde a lower-bound of the probablty that a sparse (t+1 k matrx s full-rank, gven that the frst t rows are lnearly ndependent, for 0 t (k 1. It s worth mentonng, that the aforementoned result was provded wthout referrng to any communcaton system or codng strategy. By buldng upon that result, we provde an upper-bound for the average number of coded packet transmssons needed to recover a servce layer, va an Absorbng Markov Chan (AMC wth reduced complexty. In partcular, Secton II-B wll show how our performance modelng does not nvolve any explct matrx nverson, whch s a common and computatonally costly step n AMC-based analyss. As wll be clear n the followng sectons, that desrable feature s acheved because of: ( the nature of the aforementoned probablty lower-bound and, ( the way we defned the states of the proposed AMC model. The second contrbuton of the paper s made n Secton III, where we answer to our research queston by buldng upon an effcent user performance characterzaton and proposng a resource allocaton framework for ultra-relable layered multcast servces. The proposed framework ams to maxmze the code sparsty assocated to each servce layer, and hence, the overall decodng complexty s mnmzed. The optmzaton goal s fulflled by a jont optmzaton of both the code sparsty and the Modulaton and Codng Schemes (MCSs used for multcastng each servce layer. In addton, gven the layered nature of the transmtted servces, the optmzaton constrants ensure that the desred number of servce layers are recovered by predetermned fractons of users, wth an average number of coded packet transmssons that s smaller than or equal to a target value. We prove that the proposed resource allocaton framework s convex and can be easly solved. Fnally, we remark that the proposed resource allocaton framework apples for several sparse RLNC technques, n a complete RLNC decoder-agnostc fashon. Even though our analyss deals wth a generc cellular system model, Secton IV nspects the effectveness of the proposed optmzed sparse RLNC technques by referrng to a LTE-A communcaton network. We chose that partcular communcaton standard for two man reasons: ( LTE-A s lkely to play a leadng role n the early-stage deployment of next-generaton networks [31], and ( LTE-A provdes the support to handle PtM communcatons at the rado access and core network level, by means of the evolved Multmeda Broadcast Multcast Servce (embms framework [32]. In the proposed performance nvestgaton, we refer to a MG targeted by non-real tme multmeda multcast servces compressed accordng to the wdely used H.264 vdeo encodng standard. In partcular, we referred to the scalable extenson of H.264, called Scalable Vdeo Codng (H.264/SVC [33]. In lne wth the consdered system model, an H.264/SVC vdeo stream conssts of several layers such that the enhancement layers mprove the reconstructon qualty provded by the base vdeo layer. Fnally, Secton V summarzes the man fndngs of the paper. II. SYSTEM MODEL AND PERFORMANCE CHARACTERISATION We consder a one-hop broadcast communcaton system, whch s composed by one source node and a MG of U users (hereafter called multcast users. In order to mprove the relablty of PtM communcatons, the source transmts data streams encoded accordng to the RLNC prncple. As a consequence, the source node transmts streams of networkcoded packets (henceforth referred to as coded packets to the MG. For the sake of generalty, we assume that the transmsson of a PtM communcaton occurs over a set of orthogonal broadcast erasure subchannels. Each subchannel conssts of basc resource allocaton unts called resource blocks. As mentoned n the prevous secton, our man goal s to desgn a general optmzed servce-provsonng paradgm for ultra-relable multcast servces, wth a reduced decodng computatonal complexty. The followng secton wll also clarfy that the proposed theoretcal modelng (Secton II-B and the resource allocaton procedure (Secton III are easly applcable to any cellular system capable of multcastng multple data streams at the same tme. However, n order to effectvely map user Qualty of Servce (QoS constrants onto typcal system performance metrcs (e.g., delay, packet error rate, etc., we wll refer to an OFDM-based multcarrer communcaton system. In the consdered physcal layer, the downlnk phase s organzed n rado frames. Resource blocks formng each subchannel are transmtted n one or more rado frames. Each frame can be modeled as a frequency tme structure where the frequency and tme domans are dscretzed nto OFDM subcarrers and OFDM symbols, respectvely. Each resource block occupes a fxed tme nterval (ˆτ RB and frequency band,.e., each resource block spans a fxed number of OFDM symbols and OFDM subcarrers. Snce multcast users may experence heterogeneous propagaton condtons, user dversty s exploted by assumng that the subcarrers used n a resource block are selected at random among all the avalable ones [34]. We also assume that users are statc or characterzed by low moblty, hence, the user channel condtons are consdered constant wthn a resource block. Each coded packet s always mapped onto one resource block and transmtted by means of a specfc MCS that s dentfed by an ndex, whch can take M possble values. We denote by p u (m the Packet Error Rate (PER experenced by a multcast useru, and byr(m the number of nformaton bts carred by one resource block, when the MCS wth ndex m s n use. Let us consder two MCSs wth ndexesa and b, where a < b. In our system model we assume that the MCS wth ndex a s characterzed by a smaller modulaton order and/or a lower channel code rate than b. For the same user propagaton

4 4 K 1 K 2 K 3 = K x 1 x 2... x K k 1 k 2 k 3 Fg. 1. Layered source message, n the case of L = 3. condtons, we have p u (a p u (b and r(a < r(b. We also refer to a system where all the resource blocks belongng to the same subchannel shall adopt the same MCS. Coded packets assocated wth a PtM data servce are transmtted va one or more broadcast erasure subchannels. The source node transmts to the MG a layered scalable servce consstng of one basc layer and L 1 enhancement layers. Each layer s characterzed by dfferent prorty levels. The basc layer (also referred to as layer 1 owns the hghest prorty, whch decreases n the case of the enhancement layers (layers 2,...,L. In partcular, layer L s characterzed by the lowest prorty. Because of that, t s natural to defne the level of QoS acheved by a multcast user as the number of consecutve message layers, startng from the base layer, that can be recovered. Hence, a user shall acheve the QoS level l, f all the layers 1,...,l are successfully recovered. For nstance, f a user successfully recovers message layers {1,2,...,l,l+2,l+3,...,L} then layers 2 to l mprove the nformaton provded by layer 1. In that case, the QoS level acheved s equal to l, and layers l+2,...,l do not provde any QoS mprovements, as layer l+1 has not been receved. The consdered mult-layer prncple has been orgnally desgned for vdeo compresson standards. In the case of H.264/SVC [33], t s possble to acheve dfferent knds of vdeo scalablty [5]. Wth the spatal scalablty, the vdeo frame resoluton s gradually ncreased by each layer wth the purpose to ft screens wth dfferent capabltes. In that case, the content provded by layer 1 allows a user, for nstance, to recover a px vdeo stream. By followng the same tran of toughs, the spatal resoluton can be boosted to px and px, by means of layers 1 and 2, and layers 1 to 3, respectvely. It s worth mentonng that our analyss s generc enough to be appled to any layered scalable servce that follows the prevously mentoned herarchcal structure. It s beyond the scope of the paper to provde analytcal and optmzaton frameworks dealng wth the compresson strategy used to generate a scalable servce. For these reasons, the proposed analyss has been made ndependent of the way servce layers are generated and the nature of the adopted servce scalablty. As suggested n [12], [19], we model the transmtted servce as a stream of nformaton messages of the same sze. The scalable nature of the servce s reflected on each message. In partcular, each message conssts of L layers, where layer l s a sequence of b l bts. We remark that coded packets assocated wth dfferent message layers are transmtted by dfferent subchannels. Therefore, the total number of occuped subchannels s L. In the rest of the paper, we wll provde an analytcal framework sutable for optmzng the transmsson of each message and, hence, of the whole layered servce. Each layered message x = {x 1,...,x K } conssts of K source packets, as shown n Fg. 1 for a 3-layer message. In partcular, layer l of x s defned by a fxed number k l of source packets, mplyng that K = L l=1 k l. If the MCS adopted by the subchannel delverng coded packets of servce layerlsm l, the number of bts carred by each resource block wll be equal to r(m l. Hence, we defne k l = b l /r(m l. Wthout loss of generalty we assume that the frst source packets of x belong to the base layer (l = 1, and are progressvely followed by packets defnng the enhancement layers (l = 2,...,L. In the remanng part of the paper, we wll characterze the performance of dfferent network codng strateges. It wll also become clear how the selecton of MCS scheme and sparsty assocated wth each message layer can be jontly optmzed. A. Random Lnear Network Codng Background Let K l = l k t be the number of source packets formng the frst l layers of a source message. In the classc mplementaton of RLNC, the source node lnearly combnes source packets {x } K l =K l 1 +1 formng message layer l, n order to generate a stream {y j } n l j=1 of n l coded packets, where y j = K l =K l 1 +1 c j, x. Each codng coeffcent c j, s unformly selected at random over a fnte feld GF(q of sze q. The codng coeffcents assocated wth y j defne the codng vector c j = (c j,kl 1 +1,...,c j,kl. Snce each codng coeffcent s obtaned by the same Pseudo-Random Number Generator (PRNG, modern NC mplementatons are keen on representng c j by the PRNG seed used to compute the frst codng vector component c j,kl The seed s transmtted along wth the correspondent coded packet. Snce each user s equpped by the same PRNG, t can ncrementally recompute all the codng vector components, startng from the frst one [11], [19]. The RLNC encodng process s then repeated for each message layer l = 1,...,L. A multcast user can recover the source message layer l, f t successfully recevesk l lnearly ndependent coded packets assocated wth that message layer. Unlke classc RLNC, a coded packet stream obtaned by SRLNC assocated wth layerlgeneratesk l systematc packets and one or more coded packets. The systematc packets are dentcal to the source packets {x } K l =K l 1 +1, whle the coded packets are obtaned as n the classc RLNC case. For the sake of the analyss, we defne the codng vector assocated wth systematc packet as a vector where: ( the -th component s equal to 1, and ( all the remanng components are equal to 0. For clarty, we wll refer to a codng vector related to a systematc packet as degenerate codng vector n the rest of the paper. In our system model, we assume that users acknowledge to the source node, over a fully relable channel, the successful recovery of a layer. Furthermore, the source node transmts a message layer untl a predetermned fracton of multcast users has recovered t. Obvously, as wll become clear n Secton III, the transmsson of each layer shall meet a temporal constrant. The sparse versons of both the classc ( and systematc mplementaton of RLNC ( are obtaned as follows. Each component c j, of a non-degenerate codng

5 5 vector assocated wth source message layer l s ndependently and dentcally dstrbuted as follows [28]: p l f v = 0 Pr(c j, = v = 1 p l (1 f v GF(q\{0} q 1 where p l, for 0 < p l < 1, s the probablty of havng c j, = 0. The event c j, 0 occurs wth probablty 1 p l. We remark that the average number of source packets nvolved n the generaton of a non-degenerate coded packet,.e., the sparsty of the code, can be controlled by tunng the value of p l, for any l = 1,...,L. Snce codng vectors are generated at random, there s the possblty of generatng codng vectors where each codng coeffcent s equal to 0. From a system mplementaton perspectve, all-zero coded packets should be dscarded and not transmtted. On the other hand, n the lterature dealng wth the performance characterzaton of RLNC, t s common to nclude the transmsson of all-zero coded packets [35], [36]. In that way, the performance modelng s tractable and keeps a hgher degree of generalty. The same prncple s adopted n ths and the followng sectons. However, Secton IV-A wll show how the proposed analytcal modelng can be appled to a practcal communcaton system where all-zero coded packets are not transmtted. In order to establsh a lnk between the codng schemes presented n [12] and those dscussed n ths paper, the followng sectons wll deal wth the Non-Overlappng Wndow (NOW-RLNC and the systematc NOW-RLNC strateges. We observe that the exact performance model of the Expandng Wndow RLNC (EW-RLNC strategy s unknown, even for the non-sparse case. In fact, [12] proposes an upper-bound to the probablty of recoverng a source message, when the EW- RLNC s used. Snce the reasonng behnd that bound reles on a well-known result of classc non-sparse RLNC [37], ts extenson to the sparse case s not trval. For these reasons, the sparse mplementaton of EW-RLNC s stll an open research ssue. B. Markovan Modellng for Delay Performance In ths paper, user performance wll be expressed n terms of the average number of coded packet transmssons after whch a user u acheves a predetermned QoS level. For ths reason, n the remander of the secton, we focus on user u and model the recovery of message layer l as a Markovan process. In partcular, the user decodng process s modeled va an AMC. Let C u be a matrx assocated wth the user u consstng of k l columns and varable number of rows. As user u successfully receves a coded packet assocated wth layer l, the correspondng codng vector s extracted and added, as a new row, nto matrx C u. Assume u already receved n l k l coded packets,.e., C u s a n l k l matrx. User u recovers layer l when the rank of C u, denoted by rank(c u, s equal to k l or equvalently when the defect of the matrx, defned as def(c u = k l rank(c u, s zero. For these reasons, we defne a state of the user AMC as follows. Defnton 2.1: The AMC assocated wth user u and message layerls n state s (u,l, f def(c u =, for = 0,...,k l. At frst, when user u has not receved any coded packet or coded packets assocated wth zero-codng vectors, the defect of C u s k l, and hence, the AMC s n state s (u,l k l. The defect progressvely decreases,.e., the ndex of the AMC state decreases, as new lnearly ndependent coded packets are receved. As a consequence, n the case of layer l, we have that the AMC conssts of k l +1 states. Furthermore, n order to defne the probablty transton matrx of the user AMC, we summarze here the proof of the followng lemma, presented n [29, Theorem 6.3]. Lemma 2.1 ([29, Theorem 6.3]: Assume that matrx C u conssts of (t + 1 k l elements, for 0 < t (k l 1, and assume that t out of t+1 rows are lnearly ndependent. The probablty P l,t that matrx C u s not full-rank admts the followng upper-bound: P l,t [ ( max p l, 1 p ] kl t l. (2 q 1 Proof: Wthout loss of generalty, assume that the frst t rows of C u, denoted by C u,1,...,c u,t, are lnearly ndependent. By resortng to basc row-wse operatons, t s possble to transform C u such that the frst t rows and columns of C u defne the t t dentty matrx. Consequently, the frst t rows of the transformed C u generate the same vector space defned by C u,1,...,c u,t. The probablty that C u s not full-rank entrely depends on the last k l t components of the last row C u,t+1 of C u. Hence, the probablty that C u,t+1 does not belong to the vector space defned by C u,1,...,c u,t s at ( least 1 max p l, 1 p l q 1 kl t. That completes the proof. Because of (1, the exact QoS characterzaton s a challengng task [28]. In partcular, to the best of our knowledge, the exact expresson of P l,t s not known. In the rest of the paper, owng to the lack of the exact expresson of P l,t, we use (2 to approxmate P l,t, that s ( P l,t = [max p l, 1 p ] kl t l. (3 q 1 The followng remark s mmedate from (2 and (3. Remark 2.1: Ifp l = q 1, each non-degenerate codng vector s equprobable, for a gven value of k l. Hence, a codng vector belongs to the vector space generated by t lnearly ndependent codng vectors wth probablty P l,t = q t /ql k. Ths result has been dscussed n the lterature [37] but s clearly not applcable to the sparse case, n contrast to (3. It s worth mentonng that the consdered approxmaton (3 collapses to the exact expresson of P l,t and, hence, the relaton P l,t = [max(p l,(1 p l /(q 1] k l t = q t /q k l holds, for p l = q 1. From (3, the transton probablty matrx descrbng the AMC assocated wth user u and message layer l can be derved by the followng lemma. Lemma 2.2: Assume layer l s transmtted over a subchannel whch adopts the MCS wth ndex m. The probablty P (u,l,j

6 6 s (u,l kl P (u,l k l,k l P (u,l k l,k l 1 s (u,l kl 1 P (u,l k l 1,k l 1 P (u,l k l 1,k l 2 P (u,l 2,1 s (u,l 1 s (u,l 0 P (u,l 1,1 P (u,l 1,0 P (u,l 0,0 Fg. 2. State transton dagram for the AMC assocated wth user u and message layer l. of movng from state s (u,l to state s (u,l j s (1 P P (u,l l,kl [1 p u (m] f j = 1,j = P l,kl [1 p u (m]+p u (m f = j (4 0 otherwse. Proof: Snce the user AMC s n state s (u,l, user u has collected k l lnearly ndependent coded packets,.e., rank(c u = k l. As a new coded packet assocated wth layer l s transmtted, we have just two possbltes: The rank of C u s ncreased to k l The coded packet s successfully receved wth probablty 1 p u (m, and t s lnearly ndependent of the prevously receved coded packets wth probablty (1 P l,kl. Ths event occurs wth a probablty equal to (1 P l,kl [1 p u (m]. The rank of C u does not change - That may occur because the coded packet s not successfully receved or because t s lnearly dependent of the prevously receved coded packets. Ths event occurs wth a probablty equal to P l,kl [1 p u (m]+p u (m. From (29, we also understand that the probablty of movng from state s (u,l 0 to another state s zero. Hence, s (u,l 0 represents the so-called absorbng state of the AMC. All the remanng states s (u,l 1,...,s (u,l k l are commonly referred to as transent states [38]. The state transton dagram of the resultng AMC can be represented as reported n Fg. 2. From Lemma 2.2, t drectly follows that the (k l +1 (k l +1 transton matrx T (u,l descrbng the AMC of user u and assocated wth layer l has the followng structure n ts canoncal form [38]: [ ] T (u,l. 1 0 = R (u,l Q (u,l, (5 where Q (u,l s the k l k l transton matrx modelng the AMC process as long as t nvolves only transent states. The term R (u,l s a column vector of k l elements whch lsts all the probabltes of movng from a transent to the absorbng state. From [38, Theorem 3.2.4], let defne matrx N (u,l as N (u,l = (Q (u,l t [ = I Q (u,l] 1. (6 t=0 ElementN (u,l,j at the locaton(,j of matrxn (u,l defnes the average number of coded packet transmssons requred for the process transton from state s (u,l to state s (u,l j, where both s (u,l and s (u,l j are transent states. In partcular, from Lemma 2.2, the followng theorem holds Theorem 2.1 ([38, Theorem 3.3.5]: If the AMC s n the transent state s (u,l, the average number of coded packet transmssons needed to get to state s (u,l 0 s 0 f = 0 = N (u,l (7,j f = 1,...,k l. j=1 From (7 and Theorem 2.1, we prove the followng corollares. Corollary 2.1: In the case of, the average number of coded packets transmssons needed by user u to recover the source message layer l s = τ(u,l k l. Proof: When the source node transmts the very frst coded packet, user u s n state s (u,l k l. That follows from the fact that the source node has not prevously transmtted any coded packets, and, hence, rank(c u s always equal to 0. We remark that, n the case of transmsson, at the end of the systematc phase, user u may have collected one or more source packets, mplyng that def(c u may be smaller than k l. In partcular, f def(c u < k l, the AMC wll start from any of the states s (u,l 0,...,s (u,l k l 1. Corollary 2.2: Consder. If systematc and nonsystematc coded packets assocated wth source message l are transmtted by means of the MCS wth ndex m, the consdered average number of systematc and coded packet transmssons needed to recover layer l s kl = =0 π (u,l where π (u,l user u starts from state s (u,l π (u,l = ( kl ( k l + (8 s the probablty that the process assocated wth, gven by p u (m [1 p u (m] kl, = 0,...,k l. (9 Proof: Assume that u collects k l out of k l systematc packets. Hence, matrx C u conssts of k l lnearly ndependent rows and, hence, the user AMC s n state s (u,l. In that case, from (7, we have that layer l s recovered, on average, after k l + packet transmssons, namely, k l systematc packets plus coded packets. At the end of the systematc packet transmsson phase, the AMC s n state s (u,l wth probablty ( k l pu (m [1 p u (m] kl, for = 0,...,k l. Hence, the value of s obtaned by smply averagngk l + wth the approprate probablty value of π (u,l, for = 0,...,k l, as provded n (8. III. SPARSE RLNC OPTIMIZATION: MOTIVATIONS AND RESOURCE ALLOCATION MODELS Among the most effectve ways of decreasng the computatonal complexty of the RLNC decodng operatons, we consder the reducton of the number of source packets, and the ncrease of the sparsty of the non-degenerate codng vectors per source message layer. As dscussed n Secton II, we remark that as the MCS ndex m l used to transmt layer l ncreases, the number r(m l of useful bts carred by one

7 7 Avg. Num. of Cod. Packets , kl = 10, kl = 10, kl = 70, kl = 70 pu = 0 pu = (a No. of coded packet transmssons pl Avg. Num. of Decodng Op µs µs µs µs µs kl = 10 kl = µs kl = µs p l µs µs (b No. of decodng operatons Fg. 3. Average number of coded packet transmssons and decodng operatons, for q = 2. Wth regards the scheme, the average number of decodng operatons have been obtaned by consderng p u = 0.1. resource block or, equvalently, formng a coded packet, s lkely to ncrease. Gven that coded and source packets have the same bt sze, the value ofk l s lkely to decrease when m l ncreases. However, as m l ncreases, user PER related to the recepton of subchannell s lkely to ncrease,.e., the fracton of multcast users regardng the recepton of subchannel l as acceptable s lkely to decrease. It s worth notng that both the value of k l and the probablty p l of selectng a codng coeffcent equal to zero determne the average number of coded packet transmssons and the average number of decodng operatons needed to recover layer l. Wth regards to the frst aspect, Fg. 3a shows the value of and τ(u,l S SRLNC as a functon of p l, for q = 2, k l = {10,70} and a packet error probablty p u = {0,0.1}, when or s used. Curves have been obtaned by computer smulatons. More detals about the smulaton envronment wll be gven n Secton IV. In the case of, as dscussed n Secton II-A, coded packets are transmtted after the systematc packets. Obvously, f p u = 0, there s no need of transmttng coded packets as all the systematc packets are successfully receved. That explans the reason way S SRLNC s always equal to k l, for p u = 0. On the other hand, as the value of p u ncreases, the number of coded packets to be transmtted ncreases, as well. We also observe that, for the same value of p u, S SRLNC s smaller than or equal to. That s gven by the fact that, n the case of, there s aways the possblty for a user to collect some systematc packets, whch are obvously lnearly ndependent. Both wth and (for p u > 0, we observe that f p l approaches 1, then the average number of packet transmssons needed to recover layer l ncreases. That s gven by the fact that, codng vectors tend to be composed by allzero. In addton, for a gven value of p l, as k l and/or p u decrease, the value of decreases. Fg. 3b shows the measured average number of decodng operatons ǫ (l and ǫ(l S SRLNC needed to recover layer l, n the and case, respectvely. Results are provded as a functon of p l, for k l = {10,30,70}. Obvously, ǫ (l does not depend on the user PER but just on k l and p l. In ths paper, we wll only refer to the subch. 1 subch. 2 subch. 3 coded packets from layer 1 ˆτRB coded packets from layer 2 resource block Logc Resource Mappng coded packets from layer 3 tme frequency coded packets transmtted across subchannel 1 tme Example of Cyclc Resource Mappng Fg. 4. Logc rado resource mappng (left-hand sde and an example of cyclc resource mappng (rght-hand sde, for L = 3. fundamental fnte feld operatons 1 performed by a network codng decoder based on the Gaussan Elmnaton prncple, whch represent the most computatonally ntensve part of the decodng process [14]. In partcular, the more p l ncreases, the more the codng matrx C u becomes sparser, and, consequently, the Gaussan Elmnaton requres a smaller number of teratons [22]. That behavor s confrmed by Fg. 3b, ǫ (l decreases not only when k l decreases but also when p l ncreases. In the case of, the value of ǫ (l S SRLNC s ndeed affected by the user PER. The more p u ncreases, the more the number of successfully receved systematc packets decreases and, the more the number of coded packets requred to recover the layer ncreases. Hence, that corresponds to an ncrement n the value of ǫ (l S SRLNC. In partcular, Fg. 3b shows the value of ǫ (l S SRLNC, for p u = 0.1. In the case of, n order to establsh a lnk between the average number of decodng operatons and the tme needed to perform that number of decodng operatons on a low-end devce, Fg. 3b also reports the average processng tme, for some (p l,k l pars. We have referred to a Gaussan Elmnaton-based decoder run on a Raspberry P Model B [39]. We note that there exsts a lnear relaton between a reducton n the value of ǫ (l and n the average processng tme. In the rest of the secton, we wll defne a novel optmzaton model amng to jontly optmze the sparsty of the code and the MCS ndex used to multcast each layer of the source message. The proposed model provdes resource allocaton solutons, whch ensure that predetermned fractons of users recover sets of progressve layers, on average, wthn a gven number of packet transmssons. In addton, the proposed model, at the same tme, maxmzes the sparsty and mnmzes the total source message length. A. Proposed Resource Allocaton Models From the logc perspectve, we refer to the rado resource mappng presented n Fg. 4 (left-hand sde. As the resource block s our fundamental resource allocaton unt, the tme duraton of each rado frame shall be an nteger multple of the resource block tme duraton ˆτ RB. Every ˆτ RB seconds, the source mode transmts at most one coded packet per-layer. We remark that the transmsson of a message layer contnues untl the desred fracton of multcast users has recovered t (Secton II-A. As a result, the average number of packet 1 Let a,b,c be three elements n GF(q, we wll consder the followng operatons: a b, a+b, a b, a+(b c and a (b c.

8 8 transmssons can be easly related to the average tme needed to recover a layer. Even though all the resource blocks formng the same subchannel are mapped onto tme contguous OFDM symbols, they could span a dfferent set of OFDM subcarrers every ˆτ RB seconds. For nstance, subchannels could cyclcally span dfferent frequency sub-bands, as shown n Fg. 4 (rght-hand sde. In that way, the transmsson of the same subchannel across the same set of OFDM subcarrers s avoded. Hence, users experencng poor channel condtons across specfc OFDM subcarrers wll not always be prevented from recevng the same message layer. In order to optmze m l and, ndrectly, k l, the knowledge of the user propagaton condtons s requred. Obvously, the exact propagaton condtons are unknown to the source node. However, modern communcatons standards allow users to perodcally provde feedback about ther average channel condtons across the whole transmsson band 2. Generally, the PER experenced by u s consdered acceptable f t s smaller than or equal to a threshold ˆp. In the rest of the paper, we wll refer to the prncple adopted by the LTE-A standard, where any user u provdes as propagaton condton feedback the greatest MCS ndex M u such that p u (M u ˆp, defned as [32]: M u ={m m [1,M] p u (m ˆp p u (m+1 > ˆp}. (10 For these reasons, f layer l s transmtted wth MCS ndex m l M u, p u (m l wll be equal to or smaller than ˆp. Gven the aggregate nature of the user channel feedback, relaton p u (M u ˆp s to be consdered vald across the whole system band. Hence, the noton of M u s ndependent to the way subchannels are actually transmtted across each frame. Owng to the lack of knowledge of the user PER, durng the resource allocaton phase, the source node approxmates the user PER as { p u (m l ˆp f ml M u = (11 1 otherwse. In the case of, the proposed Sparsty-Tunng (ST resource allocaton model s defned as follows: ST s.t. max p 1,...,p L p 1 (12 m 1,...,m L ( U l l l δ τ (u,t ˆτ t Û t, l = 1,...,L u=1 (13 q 1 p l < 1 l = 1,...,L (14 m l {1,...,M} l = 1,...,L (15 where objectve functon (12 maxmzes the 1-norm of vector p = {p 1,...,p L }, whch can be equvalently expressed as L l=1 p l. Term δ(t s an ndcaton functon that s equal to 1 f statement t s true, otherwse t s equal to 0. Parameters ˆτ l and Ûl represent the maxmum number of coded packet transmssons needed to recover (on average message 2 3GPP LTE and LTE-A standards refer to ths knd of user channel feedback as wdeband Channel Qualty Indcators [32]. layer l and the mnmum number of users that shall recover layer l, respectvely. For these reasons, the left-hand sde of constrant (13 represents the number of multcast users that can recover layers 1,...,l, on average, n at most l ˆτ t coded packet transmssons. As a result, constrant (13 ensures that the number of multcast users acheve QoS level l s at least equal to l t=0ût. Snce user u can only acheve QoS level l f all the layers 1,...,l have been recovered, t would be pontless to recover layer l before layer l 1. For the same reasons, there s no pont n havng stuatons where the fracton of users recoverng layer l s greater than the fracton of users recoverng l 1. Hence, t s reasonable to assume that the relatonsûl 1 Ûl and m l 1 m l hold, for l = 2,...,L. Furthermore, constrant (14 avods both dense codng vectors (.e., p l < q 1 and all-zero codng vectors (.e., p l = 1. Then constrant (15 remarks that varable m l can only take values n range 1,..., M. The ST problem can also be defned for the case of by smply replacng n constrant (13 the term τ (u,t wth τ(u,t. We observe that the selecton of parameters ˆτ l and Ûl, for l = 1,...,L, allow the ultra-relable servce to be delvered, by meetng the Servce Level Agreements (SLAs between the servce provder and the users. In our case, SLAs mposes the mnmum fracton of users that shall acheve target QoS levels and the maxmum tme needed (on average to do so. Because of constrant (13, the ST problem presents vast couplng constrants among the whole set of optmzaton varables. In spte of the apparent optmzaton complexty, we wll show that the ST problem can be effcently solved, both n the case of and, by decomposng t nto subproblems of a reduced complexty. In order to do so, t s worth solvng the Layer Sparsty Maxmzaton (LSM problem assocated wth user u, MCS ndex m and layer l. We wll eventually refer to the LSM problem to solve the SM problem. In partcular, the LSM problem s defned as follows: LSM-(l, u, m max p l p l (16 s.t. ˆτ l (17 q 1 p l 1 (18 From Corollary 2.1, we have that s defned as a sum of terms from matrx N (u,l. In the followng, we equvalently rewrte constrant (17 n order to avod the explct nverson of I Q (u,l n (6, and we prove the convexty of LSM- (l,u,m. We defne thek l k l matrxw (u,l asw (u,l = I Q (u,l. From (6, we have that N (u,l = (W (u,l 1. Let Q (u,l,j be the (,j-th element of matrx Q (u,l. From (29 and (5, we have that Q (u,l s a non-negatve lower-trangular matrx wth the followng structure: Q (u,l = Q (u,l 1, Q (u,l 2,1 Q (u,l 2, Q (u,l k l,k l 1 Q(u,l k l,k l. (19

9 9 Hence, for and j = 1,...,k l, element (,j of W (u,l s (1 p u (m(1 P l,kl f j = 1 W (u,l,j = (1 p u (m(1 P l,kl f = j (20 0 otherwse. From (6, the followng relaton holds: W (u,l N (u,l = I. (21 Relaton (21 defnes a set of k l dsjont parametrc systems of equatons, where p l s the system parameter and the elements of N (u,l are the system unknowns. System s, for s = 1,...,k l, conssts of k l s+1 equatons. In partcular, the -th equaton of system s, for = s,...,k l, s defned as: j=s W (u,l,j N (u,l j,s = δ( = s. (22 From (19, (20, the soluton of system s can be expressed as { N (u,l [(1 pu (m(1 P,s = l,kl s] 1 f = 1,...,k l (23 0 otherwse. As a result we can prove the followng lemma. Lemma 3.1: The LSM-(l, u, m problem s convex. In addton, the optmum soluton of the problem s the real root of k l [(1 p u (m(1 P l,kl ] 1 ˆτ l = 0, (24 =0 whch s greater than or equal to q 1 and smaller than 1. Proof: From Corollary 2.1 and (23, we have that can be equvalently rewrtten as kl = [(1 p u (m(1 P l,kl ] 1. (25 =0 Snce we refer to the approxmaton as n (3, P l,kl s the non-negatve power of a pontwse maxmzaton of two convex functons. Hence, P l,kl s convex wth respect to p l. Consder functon (1 p u (m(1 P l,kl of (25. Snce P l,kl s convex, functon (1 p u (m(1 P l,kl s concave and, hence, [(1 p u (m(1 P l,kl ] 1 s convex. As a result,, expressed as n (25, s a non-negatve weghted sum of convex functons, whch s a convex functon. For these reasons, t follows that the LSM-(l,u,m problem s convex [40]. From (25, we rewrte constrant (17 as kl =1 [(1 p u(m(1 P l,kl ] 1 ˆτ l. Because of the convexty of LSM-(l,u,m, we have that the optmum soluton of the problem s gven by the real root of (24, whch belongs to [q 1,1. The LSM-(l, u, m problem can be adapted to the S- SRLNC case by smply replacng constrant (17 wth ˆτ l. The resultng optmzaton problem can be solved as follows. Lemma 3.2: In the case, the resultng LSM- (l,u,m problem s convex, and ts optmal soluton s the real root, greater than or equal to q 1 and smaller than 1, of the followng equaton: k l =0 k l + =1 π (u,l (k l + π (u,l [(1 p u (m(1 P l,kl j] 1 ˆτ l = 0.(26 j=1 Proof: From Corollary 2.2 and (23, expressed as { = k l π(u,l 0 k l + π (u,l (k l + + π (u,l =1 can be [(1 p u (m(1 P l,kl j] }. 1 (27 j=1 Lkewse Lemma 3.1, s convex because t s the non-negatve weghted sum of convex functons. Then the proof follows exactly the same reasonng as n the proof of Lemma 3.1. Once more, consder the ST problem and the followng remark. Lemma 3.3: Constrant (13 of the ST problem can be equvalently rewrtten as U u=1 ( δ ˆτ l Ûl, l = 1,...,L. (28 or restated for the case, n a smlar way. Proof: From Secton III-A, relaton τ (u,t ˆτ t shall hold, for at leastût users. Hence, the complete statement of the argument of functonδ( n (13 s equvalent to the followng system of nequaltes l l τ (u,t ˆτ t (29 τ (u,t ˆτ t, for t = 1,...,l. We observe that the frst nequalty s made redundant by the remanng ones. Hence, (13 can be rewrtten as ( U l l δ τ (u,t ˆτ t Û t, for l = 1,...,L, u=1 (30 where the leftmost term stll counts exactly the same number of users achevng QoS level l as n (13. Consder layer t, t shall be receved by at least Ût users, for t = 1,...,L. Hence, the complete statement of (30, for a gven l, s ( U l l δ τ (u,t ˆτ t Û t, u=1 U ( δ τ (u,t ˆτ t Ût, for t = 1,...,l. u=1 (31 We remark that relatons Ûl 1 Ûl and m l 1 m l hold, for l = 2,..., L. In addton, from the consdered PER model (11, we have that the set of users achevng QoS level

10 10 l entrely contans those achevng QoS levels 1,...,l 1. Hence, the frst nequalty of (31 s made redundant by the followng ones. That completes the proof. Ths proof can be smlarly restated for the case. From Lemma 3.3, ST can be decomposed nto L ndependent optmzaton problems ST-(1,..., ST-(L, where the ST- (l problem: ( refers to the vdeo layer l, ( has the goal of maxmzng p l, and ( refers to just the constrants of ST that are related to layer l. ST-(l problem can be solved as follows. Remark 3.1: From Lemmas 3.1 and 3.2, we have that andτ(u,l are non-decreasng functons wth respect to p l, for q 1 p l < 1. In addton, for a gven value of p l, we remark that as m l ncreases, the value of wll decrease as well (Secton II. Hence, ST-(l s solved by the par (m l,p l characterzed by the greatest values of m l and p l such that relatons ˆτ l or ˆτ l hold, for at least Ûl users. In partcular, ST-(l can be solved by resortng to LSM problems as follows. For any m l = 1,...,M and l = 1,...,L, let U ml sgnfy the set of users such that M u m l. 1. Let us solve LSM-(l,u,m, for a user u U ml and m = m l. Let p l,m l be the optmum soluton of LSM- (l,u,m l. If s n use then the value of p l,m l s derved as provded by Lemma 3.1. On the other hand, f s n use then we wll refer to Lemma 3.2, for the computaton of p l,m l. Snce p u (m s approxmated as n (11, the soluton p l,m l wll always be the same, for every user n U ml. 2. For any m l = 1,...,M such that U ml Ûl and an optmum soluton p l,m l exsts, the par (m l,p l,m l s an optmum soluton of ST-(l. Among the optmum solutons of problem ST-(l, we choose the par(m l,p l,m l assocated wth the greatest MCS ndex,.e., we consder the soluton that ensures the smallest value of k l (see Secton II. The process s repeated to solve any problem ST-(l, for l = 1,...,L and, hence, to solve problem ST. We observe that, for a gven value of m l, the par (m l,p l,m l may not exst. That can happen because: ( the value of ˆτ l s too small and the average number of coded packet transmssons always exceed ˆτ l, for q 1 p l < 1, and/or ( the target user coverage Ûl s too bg (constrant (28 s not met, gven the overall user propagaton condtons and, hence, the MCSs that can be used n a consdered scenaro. For these reasons and, n partcular, from Lemmas 3.1, 3.2 and 3.3, t s mmedate to prove the followng theorem. Theorem 3.1: Both n the and cases, the resource allocaton soluton of ST problem derved by Remark 3.1, for anyl=1,...,l, s optmal and characterzed by the greatest MCS ndexes,.e., the derved optmal soluton ensures the smallest values of k l, for l = 1,...,L. IV. NUMERICAL RESULTS A. Assessment of the Performance Model We recall from Secton II-B that we mtgated the lack of an accurate expresson of the probablty P l,t that a sparse kl = 10 kl = 30 kl = 50 kl = 70 Smulatons Upper-Bound p l kl = 10 kl = 30 kl = 50 kl = 70 Smulatons Upper-Bound (a q = p l (b q = 2 8 Fg. 5. Average number of coded packet transmssons vs. the average number of coded packet transmssons obtaned by referrng to the approxmaton as n (3, for q = 2 and 2 8. random (t+1 k l matrx s not full-rank over GF(q, gven that the frst t rows are lnearly ndependent. In partcular, we upper-bounded the value of P l,t by referrng to the approxmaton n (3. Hence, the average user delay values (Corollary 2.1 and (Corollary 2.2 are expected to be greater than or equal to the correspondent average user delay values obtaned va computer smulatons. In ths paper, all the computer smulatons rely on the encoders and decoders provded by the Kodo C++ network codng lbrary [14]. Fg. 5 refers to a scenaro, where a source message of k l {10,30,50,70} source packets s transmtted to a user by means of, over a fully relable channel (.e., the user PER s equal to 0. In partcular, Fg. 5a compares, for q = 2, the value of as n Corollary 2.1 wth that obtaned by smulatons, as a functon of the probablty p l of selectng a zero codng coeffcent. Fg. 5b reports the same performance comparson, n the case of q = 2 8. Fgs. 5a and 5b show that, for p l = q 1, smulaton and our theoretcal upper-bound of overlap. In fact, from Remark 2.1, n that case, (3 no longer s an approxmaton. However, the gap between the theoretcal upper-bound and smulaton results ncreases, as p l becomes larger than q 1. Let us focus on such that p l q 1, regardless of the value of q, we observe that the performance gap between the theoretcal upper-bound and smulaton results manly depends only on the value of k l and p l. On the other hand, for large values of k l (such as, k l 50 and p l (p l 0.93, the value of the performance gap, normalzed wth respect to k l, s almost constant and equal to In other words, the mpact of q on the performance gap s not pvotal and, at the same tme, t s manly proportonal to k l. Gven that the smulaton results reported n Secton IV-B refer to values of k l and p l n the aforementoned ranges, that gves a clear upper-bound of the mpact of our approxmaton onto the dsplayed performance, on a layer-bass. Snce can