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Cooperative Communications in Networks: Random coding for wireless multicast

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Cooperative Communications in Networks:Random coding for wireless multicast

Performance for fading channels and inter-session coding

Brooke Shrader and Anthony Ephremides

University of Maryland

October, 2008

Joint work with Randy Cogill, University of Virginia

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 ai are generated randomly and uniformly from u-ary

alphabet and Σ is sum in finite field.

If ai = 0 for all i, generate new coefficients.

Random linear coding for multicast

Form random linear combinations of K packets.

Transmit coefficients ai in packet header

Decode: solve a system of linear equations in si

Previous workStable throughput in random access

- Setting: 2 sources, 2 destinations
- If user n has a packet to send, it transmits in a slot with probability pn

Random linear coding

Retransmit packets

- Stable throughput: region of (λ1, λ2) for which there exists (p1, p2) such that both queues remain finite

Previous workStable throughput in random access

Increasing K

Random linear coding approaches capacity as K→∞.

Previous workQueueing delay in random linear coding

Policy: fix K, the number of packets in coding

Time spent waiting to form a group of K

Multicast: 3 destinations, q=0.9

Previous workQueueing delay in random linear coding

Policy: let k, number of packets in coding, vary according to traffic load

1 ≤ k ≤ K

Unicast: 1 destination, q=0.5

As K →∞, delay performance approaches retransmission delay

Rateless 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 analyze the multicast throughput for random linear coding
- over a fading wireless channel where reception probability depends on packet length, overhead, SNR
- across multiple multicast sessions

Let Tm denote the number of slots needed for destination

m to collect K linearly independent random linear combinations.

The multicast throughput is:

Difficulty: Tm 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.

Assume: the channels to the M destinations are identically distributed (but not independent).

For random variables X1,X2,…,XM identically distributed

and correlated and for any t > 0,

Then the multicast throughput is lower bounded, for any t > 0, as

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).

Assume that there is no channel coding within packets. For packet to be received, every symbol must be received.

q: Probability that a transmitted packet is successfully received at a destination node.

Pu: u-ary symbol error probability for modulation scheme,

depends on SNR, channel model (e.g., AWGN)

Each packet consists of nu-ary symbols.

Coding is performed on groups of K packets.

The multicast throughput is lower bounded, for any t > 0, as

ratio of information to information+overhead

The channel to each destination node evolves as a Markov chain with “Good” and “Bad” states.

qG: Probability that a transmitted packet is received in “Good” state

qB: Probability that a transmitted packet is received in “Bad” state

A packet-erasure version of the Gilbert channel model.

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 qB=0, so initial state is always S=G.

Transmission time T1: time to reach state (S,K) from (G,0).

Compare to time-invariant channel with probability of reception

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.

- 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.

For successful decoding, each destination must decode the packets from all K multicast sessions.

Using bound on E[max(X1,X2,…XMK)], we bound the throughput as

Coding across sessions (lower bound)

--- Retransmissions (upper bound)

For large number of sessions and receivers per session, coding outperforms retransmissions

K=50, q=0.8

- Combine network coding with back-pressure algorithm for congestion control
- Packet erasure models for networks
- erasure probability → 1 as packet “length” grows

- Cooperative techniques in which relay nodes utilize idle slots