Network Coding Channel Virtualization Schemes for Satellite Multicast Communications
Samah A. M. Ghanem, Ala Eddine Gharsellaoui, Daniele Tarchi, Alessandro Vanelli-Coralli
aa r X i v : . [ c s . I T ] A p r Network Coding Channel Virtualization Schemesfor Satellite Multicast Communications
Samah A. M. Ghanem † , Ala Eddine Gharsellaoui ∗ , Daniele Tarchi ∗ and Alessandro Vanelli-Coralli ∗∗ Department of Electrical and Electronic Engineering, University of Bologna, Italy † Independent Senior Researcher
Abstract —In this paper, we propose two novel schemes to solvethe problem of finding a quasi-optimal number of coded packetsto multicast to a set of independent wireless receivers sufferingdifferent channel conditions. In particular, we propose two net-work channel virtualization schemes that allow for representingthe set of intended receivers in a multicast group to be virtualizedas one receiver. Such approach allows for a transmission schemenot only adapted to per-receiver channel variation over time, butto the network-virtualized channel representing all receivers inthe multicast group. The first scheme capitalizes on a maximumerasure criterion introduced via the creation of a virtual worstper receiver per slot reference channel of the network. The secondscheme capitalizes on a maximum completion time criterion bythe use of the worst performing receiver channel as a virtualreference to the network. We apply such schemes to a GEOsatellite scenario. We demonstrate the benefits of the proposedschemes comparing them to a per-receiver point-to-point adaptivestrategy.
Index Terms —Multicast Communications; Channel Virtualiza-tion; Network Coding; Satellite Communications.
I. I
NTRODUCTION
Multicast communications are fundamental to many prac-tical applications, including Satellite TV broadcast, contentdelivery and interactive communications, multimedia confer-encing, across wired or wireless medium. Network codingis a key enabling technology that offers a unique techniqueto multicast communications. In particular, by mixing theinformation content shared among receivers, higher reliabilityand less delay can be encountered.In wired networks, network coding was shown to achievethe multicast capacity [1]. In [2], it is shown that an explicitconstruction of a code that achieves multicast network capacityis a linear network code. In [3], Random Linear NetworkCoding (RLNC) was proposed for multicast, a distributednetwork coding approach with nodes independently and ran-domly selecting linear coding coefficients from inputs ontooutput links over a Finite Field of known size, which achievescapacity with very high probability.In wireless networks, owing to their broadcast nature, multi-casting on wireless links, suffering different channel behavior,noise, and interference levels, becomes a challenge. Sincethere are no explicit models that express a wireless networkcapacity, this is considered as an open problem; thus, thecharacterization of optimal approaches, that jointly minimizethe system completion time to several entities, is yet an openproblem. This paper goes in this direction by finding the optimal number of coded packets to transmit to all receiversof a multicast wireless network group.Since then, network coding has shown many benefitsin wireless mesh networks. Some practical network codingschemes include COPE, a XOR-form [4], or MORE, anRLNC-form [5], of random mixing of packets, or a combina-tion of both forms [6]. Additionally, in P2P networks, networkcoding shows significant benefits in content distribution [7]and streaming [8]. In [9] multicast network coding capacitywas shown to be inversely proportional to the connectionprobability among the receiving nodes in the multicast. In [10],the authors characterize the expected number of transmis-sions per packet and quantify its gain with network codinganalytically. They have also conjecture that network codingachieves a logarithmic gain in the expected number of trans-missions/retransmissions to multicast compared to an ARQscheme.For line networks [11], the author shows gains of RLNC andadaptive RLNC schemes compared to ARQ for time variantchannels. Then the authors in [12], [13] show that adaptiveRLNC in line networks can achieve different energy and rategains adapting to the channel variation. However, those worksdid not consider how to jointly optimize the transmission toa set of multicast group with different channel variations perreceiver.To the best of our knowledge, this work is the first topropose schemes that can jointly design the number of codedpackets to multicast to a set of multicast group receiversencountering different time variant channels.In this paper, we try to provide in an innovative waysolutions to the question how can we optimize jointly the codedtransmission to a set of receivers in a multicast group?
Inparticular, we try to solve such an open problem due to lack ofavailable models that express a correlated structure, by lookingat the multicast approach through a virtual network thatexpresses an equivalent network of a min-cut-like time variantcapacity. This network virtual link represents the multicastgroup time variant channel and can be exploited to designoptimal or near optimal number of coded packets to transmitto all receivers in a multicast group.We capitalize on the model in [11] for coded packettransmissions over time variant channels in a line network toprovide an approach for the network coding multicast problemthat can express multiple receivers in the multicast group sepa-rately as point-to-point time varying channels. In particular, weropose two schemes for network coding wireless multicast,one that creates a virtual worst channel that intersects withworst case receiver channels, and another that assigns the vir-tual worst channel of the receiver having maximum completiontime. We focus on the GEO satellite application, and considerto multicast a shared content, generated via RLNC, to a set ofreceivers. II. S
YSTEM M ODEL
Consider a downlink multicast over a wireless channel.The GEO satellite, as a source, performs RLNC [3]. Thecoefficients used to generate coded packets are chosen atrandom from a Galois finite field of very large size. Withthis, the probability of generating linearly dependent packetsdecreases with the field size increase. Therefore, we assume avery large field size that allows with almost probability 1 thedecorrelation in the generation. However, it is worth to notethat there yet exists a dependency in the probability of packeterasure due to channel variation over time which inherentlyexists in the Land Mobile Satellite (LMS) channel modelherein considered [14]. Additionally, the erasure probabilityof acknowledgement packets is considered to be zero forsimplicity. After RLNC, the GEO satellite multicasts to agroup of K receivers that should receive common content.Therefore, the per-receiver k will receive a signal modeled as: y k ( t ) = h k ( t ) · A · x ( t ) + n k ( t ) , (1)where x ( t ) and y k ( t ) correspond to the transmit and re-ceive symbols, respectively. By assuming a generalized fadingmodel, where fading is time varying and follows the LMSmodel in a low height building environment [14], A is thetransmitted signal amplitude from the GEO satellite, n ( t ) isthe zero mean complex white Gaussian noise. While t isconsidered to be within interval T = [0 , τ ] . Moreover, a set of K receivers K = { , . . . , K } , in the group of multicast, areassociated to a vector of channel gains of length τ . We assumethat there is no channel coding within the received packets.Thus, for each packet, every symbol needs to be received.Therefore, the corresponding packet erasure probability ofchannel gain h k ( t ) is expressed at time instant t ∈ T as: P e ( h k ( t )) = 1 − (1 − P b ( h k ( t )) B , where P b ( h k ( t )) is the bit error probability for a givenmodulation scheme, considering channel gain h k ( t ) , and B is the number of bits per coded packet.III. C HANNEL V IRTUALIZATION S CHEMES
We propose a novel channel virtualization approach toaddress the multicast modeling problem with time variantchannels. Such virtualization inherently exploits the necessityto represent the wireless network in an equivalent form.Such an equivalent form allows to characterize the capacityof each wireless network by its min-cut [15]. A min-cutmimics the maximum delay or maximum erasures that limitthe information flow in a wireless network. Therefore, thesolution that can be proposed for a virtualized network should , h , h , h , h ... , h , h , h ... , h , h ... AbsorptionState Pe ( h Pe ( h Pe ( h Pe ( h Pe ( h Pe ( h − Pe ( h
0) 1 − Pe ( h
1) 1 − Pe ( h − Pe ( h
1) 1 − Pe ( h
2) 1 − Pe ( h Pe ( h − Pe ( h − Pe ( h − Pe ( hj ) Pe ( h Pe ( h − Pe ( h Fig. 1. Time Varying Channel Model of 3 Packets Transmission in [11] allow receivers with worst channel conditions to yet be ableto decode received information in a reliable way.Due to the broadcast nature of wireless communications,and the impairments associated to challenging wireless envi-ronments, like the satellite communications addressed here, theinterpretation of such approach for time variant channels is torepresent all the wireless links as point-to-point links with areference virtual channel that provides most possible losses interms of packet erasures or in terms of completion time, whereboth are associated to resource wastage in retransmissions orin waiting times until the content is reliably delivered.We capitalize on the coded packet transmission model fortime variant channel proposed in [11] and depicted in Fig. 1to characterize per-receiver completion time. This model isused to characterize per-receiver packet transmission adaptedto its channel variation, and to characterize the networkvirtualized channel and the optimization of coded packets tobe transmitted over it.Therefore, according to [11], the end-to-end completiontime from the GEO satellite source to a single receiver k overa time variant channel is given as; T ( i, h j ) = T d ( N i , h j ) + i X l =1 P N i ( i,h j ) → ( l,h j + Ni ) T ( l, h j + N i +1 ) , (2)with T d ( N i , h j ) = N i T p + T w , where T w is the waiting timefor acknowledgment, T p is the packet time, and N i is thenumber of coded packets to be sent in batches for combining i degrees of freedom or packets. The matrix P is the one steptransition matrix of the proposed model, where: N i Y i =1 P ! ( i,h j ) → ( l,h j + Ni ) = P N i ( i,h j ) → ( l,h j + Ni ) , corresponds to all the transition probabilities over the timeslots from the initial state at h j until the state at h j + N i .he index j + N i + 1 appears in the delay term due to theacknowledgment.In turn, a set of K completion times will be associatedto the receivers, suggesting a unifying approach that allowfor a joint optimization of all the completion times of all thereceivers. Given that such a problem is unsolvable, we proposetwo novel heuristic schemes that allow for near optimal jointoptimization of the number of coded packets to multicast toall receivers. A. Maximum Erasure Scheme (MaxPe)
In this scheme, we propose to represent the receivers’channels in the multicast group by their joint global
MaxPe encountered by each receiver over each time slot in theobservation time window of channel variation. In turn, theoptimization problem of such scheme can be written as: max P e ( h k ( t )) ,...,P e ( h K ( t T )) min N ,...,N i T ( i, h j ) =max P e ( h k ( t )) ,...,P e ( h K ( t T )) min N ,...,N i T d ( N i , h j )+max P e ( h k ( t )) ,...,P e ( h K ( t T )) min N ,...,N i i X l =1 P N i ( i,h j ) → ( l,h j + Ni ) T ( l, h j + N i +1 ) , ∀ k = { , .., K } ∈ τ = { , ..., T } (3) As it is known that the optimization problem is combina-torial and hard, we join the proposed
MaxPe scheme to theheuristic Adaptive Network Coding (ANC) scheme in [11] tosolve the problem of finding the number of coded packets totransmit to all receivers in the multicast group. Therefore, theset of coded packets that will be multicasted from the GEOsatellite to all receivers is guaranteed to be decoded using suchscheme with a design for worst receiver’s channel condition. N ∗ , ..., N ∗ i is found by iterating over the vector of degrees offreedom as: N ∗ i X s = j (1 − P e ( h s )) = i, ∀ P e ( s ) , s.t. P e ( s ) : max { P e ( h k ( t )) , ..., P e ( h K ( t T )) } , ∀ k = { , .., K } ∈ τ = { , ..., T } (4) B. Maximum Completion Time Scheme (MaxCT)
In this scheme, we propose to represent the receiverschannels in the multicast group by the worst receiver’s channelas a reference channel, which encounters
MaxCT or maximumcompletion time to receive and decode reliably all codedpackets from the GEO satellite within the observation timewindow of channel variation. In turn, the optimization problemof such scheme can be written as follows, max k min N ,...,N i T ( i, h j ) = max k min N ,...,N i T d ( N i , h j )+ max k min N ,...,N i i X l =1 P N i ( i,h k ) → ( l,h j + Ni ) T ( l, h k + N i +1 ) , ∀ k = { , .., K } (5) As it is known that the optimization problem is combinato-rial and hard, we resort to the heuristic ANC scheme in [11]to solve the problem of finding the number of coded packetsto transmit to all receivers in the multicast group, therefore,the set of coded packets the GEO satellite will multicast witha guarantee that all receivers will be able to decode underthe design for worst case condition N ∗ , ..., N ∗ i is found byiterating over the vector of degrees of freedom as follows, N ∗ i X s = j (1 − P e ( h s )) = i, ∀ P e ( s ) , s.t. P e ( s ) : { P e ( h k ( t )) , . . . , P e ( h k ( t T )) } ,k : max CT k , ∀ k = { , ..., K } (6)A similar approach of maxCT was proposed to address theXOR network coding multicast scenario for receivers encoun-tering similar erasures in [16], however, the way the comple-tion time is modeled and measured lacks the considerationof packet erasures dependency which is something presentin the model in [11] with time variation. Therefore, besidesthe difference in the coding framework they used, such anassumption is absent in their work.IV. N UMERICAL R ESULTS
In this section, we show numerical results obtained throughcomputer simulations for evaluating the performance of theproposed schemes for coded multicast. The application sce-nario considers one GEO satellite and ten mobile receiversmoving on the Earth ground. Thus, each link is characterizedby a Round Trip Time (RTT) equals . .The receivers are randomly positioned so that each oneexperiences a different channel quality. To this aim we supposethat the receivers are moving in a random direction with aconstant speed equals / s .In addition, the receivers are supposed to be located within aLow Height Building scenario [14], that considers the presenceof three propagation states: line of sight, moderate and deepfading. To this aim, we also suppose that the receivers areequally distributed within the three states.The performance is evaluated in terms of delay, throughputand average number of packets, by considering the on-boardsatellite transmitter multicasts a maximum batch of i equals10 data packets or degrees of freedom (dof), each of size B equals bit . We consider data packets with a duration equalto .
67 ms .The proposed coded multicast schemes are evaluated, andthe results are compared with two benchmark schemes: theper-receiver ANC scheme, and per-receiver non-adaptive NCscheme, for time variant channels, proposed in [11]. a) Maximum Erasure Scheme:
This scheme considersthat we adapt to the worst channel conditions among allthe receivers at any time instant. This corresponds to avirtual channel composed by the channel behavior havinginstantaneously the maximum erasure, and, hence, the higherattenuation. Such assumption can be seen as a virtual channel D e l a y ( s e c ) • Rec. 1 • Rec. 2 • Rec. 3 • Rec. 4 • Rec. 5 • Rec. 6 • Rec. 7 • Rec. 8 • Rec. 9 • Rec. 10NC Virtual ReceiverNC ReceiversANC Virtual ReceiverANC ReceiversANC Receivers(MaxPe)
Fig. 2. Performance in terms of delay for the Maximum Erasure scheme byconsidering 10 receivers at different E b /N values. associated to the worst conditions among all in the multicastgroup.Notice that in the figures, a certain color is used to expressa certain receiver, and the same line type is used to expressthe scheme: dotted with/without circle represents the non-adaptive NC, dashed with/without rectangle represents theANC, and the dotted-dashed represents the joint ANC with MaxPe scheme for coded multicast.In Fig. 2 the delay performance is shown comparing the
MaxPe policy adopting ANC, with the two benchmark ANCand NC schemes. Moreover, as a reference, the performancefor the virtual receivers, employing NC, and ANC are alsoshown as a reference. It is possible to notice that the re-ceivers are grouped into three main groups, reflecting thethree different propagation conditions: line-of-sight, moderateshadowing, and deep fading. As expected, it is possible to seethat NC encounters higher delays w.r.t. ANC for all receivers.Additionally, the receivers adopting the NC schemes selectedby the Virtual receivers show a delay gain with respect to theperformance obtained by using a pre-receiver policy. This canbe observed by looking to the values in Tab. I, where, amongother performance indicators, the average delay results for allthe considered
Eb/N values are reported for the 10 receivers.Moreover, it is possible to notice that, due to their worstcase condition design criterion, the NC Virtual Receiver and
ANC Virtual Receiver experience the highest delay among allNC and ANC receivers.Moreover, the following main observations are drawn: • Receivers under line of sight and moderate fading, aregaining the most in terms of delay. For instance, Tab. Iemphasizes the gain of receiver 5 equals .
01 ms . • The receivers under deep fading, i.e., receivers 7 to10, still enjoy gains compared to the case without thevirtualization scheme albeit in a limited way.In Fig. 3, the throughput performance is shown, in terms ofdelivered packets per second. Similar to the delay performance T h r oughpu t ( pa ck e t s / s e c ) • Rec. 1 • Rec. 2 • Rec. 3 • Rec. 4 • Rec. 5 • Rec. 6 • Rec. 7 • Rec. 8 • Rec. 9 • Rec. 10NC Virtual ReceiverNC ReceiversANC Virtual ReceiverANC ReceiversANC Receivers(MaxPe)
Fig. 3. Performance in terms of throughput for the Maximum Erasure schemeby considering 10 receivers at different E b /N values. it is possible to notice that the receivers employing the MaxPe policy are gaining compared to the receivers employing a per-receiver ANC scheme. This has been highlighted in Tab. Iwhere the average throughput for the 10 receivers is reported.In general, the throughput performance of
MaxPe receiverscan show noticeable gains with respect to the per-receiverANC scheme at the moderate SNR values.Finally, the performance in terms of average number ofpackets is shown in Fig. 4. Those ave. no. of packets aredrawn from evaluating the delay encountered at zero waitingtime [12], when the system is adapted to the virtualized chan-nel. Thus, the variance between the virtualization schemes isnegligible, or small compared to no-virtualization. The gain inthe delay and throughput performance at moderate SNR in castto an increase in the average number of transmitted packets.Moreover, it is worth to observe that the cost associated tolarger batches of coded packets, appears as a reward in thedelay due to less retransmissions and less RTTs. b) Maximum Completion Time Scheme:
The perfor-mance evaluation of this virtualization scheme is appliedconsidering the channel time-variant vector of the receiversuffering the maximum completion time among all receiversin the multicast group as the channel of a virtual receiver.Such virtual receiver becomes the reference receiver to allother receivers in the multicast group to which the codedtransmission will be designed or adapted.In Tab. I, it is possible to notice that, in the considered sce-nario, receiver 9 suffers the highest completion time. Hence,other receivers are supposed to utilize the ANC transmis-sion strategy of receiver 9 and optimize their transmissionof coded packets according to it. The performance of the maxCT scheme is compared with per-receiver NC and ANCbenchmark schemes.Fig. 5, illustrates the delay performance of the proposed
MaxCT scheme and the benchmark schemes. Receiver is the one used as the reference receiver for the multicast ABLE IS
UMMARY TABLE OF THE PERFORMANCE FOR
GEO
SATELLITE RESULTS .Rec. Channel Delay [ ms ] Throughput [packet/ s ] Ave. No. of packets Delay gain Throughput gainNo. gain[ dB ] Withoutvirtual. MaxPe MaxCT Withoutvirtual. MaxPe MaxCT Withoutvirtual. MaxPe MaxCT MaxPe MaxCT MaxPe MaxCT1 0,16 301,01 290,35 290,30 35,39 36,66 36,67 16 19 19 10,66 10,71 1,27 1,272 0,09 304,01 293,77 293,72 35,12 36,38 36,39 16 19 19 10,24 10,29 1,26 1,273 0,22 302,18 288,00 287,95 35,34 36,84 36,85 15 19 19 14,18 14,23 1,50 1,514 -0,95 330,62 319,02 318,97 32,99 34,40 34,41 19 21 21 11,59 11,64 1,41 1,425 -1,00 339,19 323,18 323,14 32,24 34,06 34,07 20 21 21 16,01 16,06 1,82 1,836 -0,83 331,96 321,21 321,15 32,89 34,23 34,24 19 21 21 10,75 10,80 1,34 1,357 -2,19 371,92 367,28 369,45 29,73 30,18 29,98 24 24 24 4,64 2,47 0,46 0,258 -2,26 375,03 372,46 375,03 29,46 29,68 29,46 24 24 24 2,57 0,00 0,22 0,00 -2,29 375,17 372,41 375,17 29,47 29,69 29,47 24 24 24 2,76 0,00 0,23 0,0010 -2,27 375,05 372,02 374,75 29,46 29,71 29,48 24 24 24 3,03 0,30 0,25 0,02 A v e r age nu m be r o f pa ck e t s • Rec. 1 • Rec. 2 • Rec. 3 • Rec. 4 • Rec. 5 • Rec. 6 • Rec. 7 • Rec. 8 • Rec. 9 • Rec. 10NC Virtual ReceiverNC ReceiversANC Virtual ReceiverANC ReceiversANC Receivers(MaxPe)
Fig. 4. Performance in terms of average number of packets for the MaximumErasure scheme by considering 10 receivers at different E b /N values. group. Similar to the previous set of results, the receivers aregrouped into three main groups representing the three channelscenarios: line of sight, moderate fading, and deep fading.Additionally, it is possible to see that using a receiver asa reference for setting up a multicast strategy allows to gainwith respect to the per-receiver strategy. This has been clearlydemonstrated as delay gains in Tab. I.However, we observe for the receivers suffering deep fad-ing (receivers 7-10), that MaxPe starts outperforming in de-lay/throughput gains the
MaxCT scheme, and the gain collapseas the receiver has worst channel attenuation. This is due tothe fact that the
MaxPe leads to a design of larger batches ofcoded packets than
MaxCT , however, in the rounded averageto the nearest integer (in Tab. I), the two virtualization schemeshave roughly similar coded packets.Similarly, in Fig. 6, it is possible to see that by usingthe
MaxCT scheme it is possible to gain with respect to theper-receiver ANC in terms of throughput. A similar trend isobserved for receivers with deep fading with respect to thethroughput.Fig. 7 and Tab. I, show clearly an increase in the numberof average coded packets for the
MaxCT scheme. The more D e l a y ( s e c ) • Rec. 1 • Rec. 2 • Rec. 3 • Rec. 4 • Rec. 5 • Rec. 6 • Rec. 7 • Rec. 8 • Rec. 9 • Rec. 10NC Virtual ReceiverNC ReceiversANC Virtual ReceiverANC ReceiversANC Receivers(MaxCT)
Fig. 5. Performance in terms of delay for the Maximum Completion Timescheme by considering 10 receivers at different E b /N values. the receiver suffers deep fading, the more is such increase.However, as we discussed earlier, the MaxCT outperforms the
MaxPe until the receivers encounter very deep fading wherethe later outperforms. In general, worth to observe that the costin terms of resources comparing both virtualization schemes,is almost negligible.Finally, Tab. II provides a summary of comparison of bothvirtualization schemes on their virtual channels. In particu-lar, comparing non-adaptive schemes to adaptive ones using
MaxPe and
MaxCT , we can see a gain of .
85 ms and .
48 ms on the max mean completion time, respectively.This bear witness on the gains introduced from making a care-ful design of the adaptive coded packets, while guaranteeingthrough both virtualization schemes to have the receivers ableto decode reliably.We can see that the cost of adaptation paid is a maximumof 10 packets over non-adaptive ones for both virtualizationschemes, a cost worth to be paid to serve almost half themaximum delay. This is again resorted that a transmissionof longer adaptive batches is associated with less numberof RTTs. 40 means that we need to transmit in average abatch of 40 coded packets to receive reliably 10. However,
ABLE IIP
ERFORMANCE TABLE OF THE PROPOSED VIRTUAL CHANNELS FOR
GEO
SATELLITE . Virtual Virtual Virtual receiver Non-adaptive Network Coding (NC) Adaptive Network Coding (ANC)channel receiver channel gain [ dB ] Delay [ ms ] Thr. [packet/ s ] No. of packets Delay [ ms ] Thr. [packet/ s ] No. of packetsscheme No. Max. Min. Max. Min. Max. Min. Max. Min. Max. Min. Max. Min. Max. Min.MaxPe [1,2,...,10] -2,25 -2,55 1143,25 245,50 40,73 8,75 30 10 504,4 245,5 40,73 19,83 40 10MaxCT 9 -2,15 -2,55 1127,88 245,50 40,73 8,87 30 10 504,4 245,5 40,73 19,83 40 10 T h r oughpu t ( pa ck e t s / s e c ) • Rec. 1 • Rec. 2 • Rec. 3 • Rec. 4 • Rec. 5 • Rec. 6 • Rec. 7 • Rec. 8 • Rec. 9 • Rec. 10NC Virtual ReceiverNC ReceiversANC Virtual ReceiverANC ReceiversANC Receivers(MaxCT)
Fig. 6. Performance in terms of throughput for the Maximum CompletionTime scheme by considering 10 receivers at different E b /N values. A v e r age nu m be r o f pa ck e t s • Rec. 1 • Rec. 2 • Rec. 3 • Rec. 4 • Rec. 5 • Rec. 6 • Rec. 7 • Rec. 8 • Rec. 9 • Rec. 10NC Virtual ReceiverNC ReceiversANC Virtual ReceiverANC ReceiversANC Receivers(MaxCT)
Fig. 7. Performance in terms of average number of packets for the MaximumCompletion Time scheme by considering 10 receivers at different E b /N values.
30 average no. of coded packets for the non- adaptive schemedoes not mean 10 data packets are guaranteed to be received,as there might not be sufficient degrees of freedom to decodeall packets. V. C
ONCLUSIONS
We propose two network channel virtualization schemesto address the network coding for multicast with time vari-ant channels. The proposed schemes rely on representingthe multicast network with a worst performing virtual line network in packet erasure or completion time. The proposedvirtualization schemes prove improvements when compared toper-receiver optimization, with and without adaptation, whileassuring reliable reception by all receivers in the multicastgroup. Future research will consider correlation structure ofthe underlying multicast.R
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