Random coding for wireless multicast Brooke Shrader and Anthony Ephremides University of Maryland Joint work with Randy Cogill, University of Virginia May 9, 2008 Performance for fading channels and inter-session coding

Introduction and Motivation Alternatives to overcome channel MAC-layer errors in multicast transmission: repeatedly send packets (ARQ) network coding Previous work: network coding outperforms ARQ for time-invariant channels coding used within (but not between) multicast sessions

For packets form where a i are generated randomly and uniformly from u-ary alphabet and Σ is sum in finite field. If a i = 0 for all i, generate new coefficients. Transmit coefficients a i in packet header Decode: solve a system of linear equations in s i Form random linear combinations of K packets. Random linear coding for multicast

Multicast throughput Let T m denote the number of slots needed for destination m to collect K linearly independent random linear combinations. The multicast throughput is: Difficulty: T m are correlated due to correlation in the random linear combinations sent to different destination nodes. This is true even if the channels to the destination nodes are independent.

Lower bound on multicast throughput Assume: the channels to the M destinations are identically distributed (but not independent). Then the multicast throughput is lower bounded, for any t > 0, as For random variables X 1,X 2,…,X M identically distributed and correlated and for any t > 0,

Our contributions Random linear network coding naturally adapts coding rate to variations in the channel. Coding across sessions means that receivers decode additional packets that aren’t intended for them. We use this bound to quantify the multicast throughput for random linear coding I. over a fading wireless channel where reception probability depends on packet length, overhead, SNR II. across multiple multicast sessions

I: Packet length and overhead Network coding: can approach min-cut capacity in the limit as alphabet size approaches infinity. Random network coding: overhead needed to transmit coefficients of random code. Packet length (symbols per packet & alphabet size) must be sufficiently large in order to: –approach min-cut capacity –ensure small (fractional) overhead Our approach: model the packet erasure probability as a function of packet length (symbols per packet and alphabet size).

I: Packet erasure probability q: Probability that a transmitted packet is successfully received at a destination node. P u : u-ary symbol error probability for modulation scheme, depends on SNR, channel model (e.g., AWGN) Assume that there is no channel coding within packets. For packet to be received, every symbol must be received.

I: Accounting for overhead The multicast throughput is lower bounded, for any t > 0, as Each packet consists of n u-ary symbols. Coding is performed on groups of K packets. ratio of information to information+overhead

I: Fading channel model q G : Probability that a transmitted packet is received in “Good” state q B : Probability that a transmitted packet is received in “Bad” state A packet-erasure version of the Gilbert channel model. The channel to each destination node evolves as a Markov chain with “Good” and “Bad” states.

I: Augmented Markov chain for reception at each destination State (S,j) where S is “Good” or “Bad” state and j=0,1,…K is the number of linearly independent random linear combinations that have been received. P: transition probability matrix Assume q B =0, so initial state is always S=G. Transmission time T 1 : time to reach state (S,K) from (G,0).

I: Multicast throughput versus K M=10, n=250, u=8 QAM modulation over AWGN channel with SNR/bit 3.5 dB in “Good” state and -∞ dB in “Bad” state. Compare to time-invariant channel with probability of reception

II: Coding across multicast sessions One source node K multicast sessions, each with an independent arrival process of equal rate Each session serves M destination nodes Channels to all MK destination nodes are identically distributed with reception probability q Random linear coding: create random linear combinations from the K head-of-line packets, one from each session.

II: Coding across multicast sessions For successful decoding, each destination must decode the packets from all K multicast sessions. Using bound on E[max(X 1,X 2,…X MK )], we bound the throughput as

II: Multicast throughput for coding across sessions K=50, q=0.8 For large number of sessions and receivers per session, coding outperforms retransmissions

Conclusions We provided a lower bound on multicast throughput for random linear coding while accounting for packet length, overhead, SNR, and fading. We demonstrated that random linear coding across multiple multicast sessions often outperforms ARQ. Future work: –incorporate channel coding within packet, study how to allocate coding within and among packets –code over multiple packets from multiple flows