Systematic Network Codingwith the aid of a Full-Duplex Relay

June 12, ICCGiuliano Giacaglia, Xiaomeng Shi, MinJi Kim,

Daniel Lucani, Muriel Médard

Introduction• Packet-erasure relay channel• Different frequency assigned for transmission

from sender and relay• Relay can transmit and receive simultaneously

Applications:• Cellular network using relays• Cooperative schemes using multiple interfaces

Conventional network coding solution (throughput optimal)• Systematic network coding @sender• Random linear network coding (RLNC) @relay• Issue: large decoding complexity @ receiver with high packet loss rate P1.

What is the complexity-performance trade off of a systematic relay?• Suboptimal solution from delay perspective• Increases expected number of uncoded packets received

2

Sender Receiver

Relay

P2 P3

P1

Assumptions:• Sender broadcasts M packets• Slotted transmissions• Duplex relay

Solution: Two stage transmission• Systematic stage:

Systematic relay: forwards uncoded pkts, no recodingRLNC relay: performs RLNC using all uncoded pkts in memory

• RLNC Stage: – Sender: broadcasts RLNC pkts– Relay: performs RLNC on all pkts (coded or uncoded) in memory

Transmission terminates when receiver ACKs M dofs received

System Assumptions

3

4

p1

p2

p1

p1

p2

p2

p1 + p2p1 + p2

p1 + p2

p1 + p2

00011

222

System ModelMarkov chain model• State of the network: (i, j, k)

– i = # dof @ receiver– j = # dof @ relay– k = # dofs shared by receiver and relay– (i,j,k) valid if i+j-k ≤ M, i ≤ M, j ≤ M, 0 ≤ k ≤ min(i, j)

• Transmission initiates at (0,0,0) and terminates at (M, j, k)

ik

j

M

State Transition Probabilities

5

Case 1P1(1-P2)(1-P3)

Case 2(1-P1)(1-P2)P3

Case 3(1-P1)(1-P2)(1-P3)

Case 4P1(1-P2)P3

Case 5(1-P1)P2P3

Case 6P1P2 (1-P3)

Case 7(1-P1)P2(1-P3)

Case 8P1P2P3

Transition probabilities for non-systematic case• Combinatorial approach• Two major scenarios

– i+j-k = M Relay and receiver have jointly all dof– i+j-k < M Relay and receiver do not have jointly all dof

• Useful also for second stage of systematic relay

Performance Analysis and Metrics

6

Transition probabilities for systematic relay• Transition probabilities and evolution is different in the systematic stage• In RLNC stage, same as the non-systematic relay case

Metrics• Mean completion time• Expected # uncoded packets received via the systematic relay• Expected # of additional coded packets via the systematic relay

• Decoding complexity @ Receiver on the order of

€

E[Usys −Unon−sys] = MP1 1− P2( ) 1− P3( )

€

O M −U( )3( )

Numerical Results

7

P2P3

P1

• Delay gap < 1dB

• Additional number of uncoded packets grows with P1

Numerical Results

8

P2P3

P1

• Delay gap < 1dB

• The additional number of uncoded packets decreases with P3

Conclusion

9

• Studied the trade-off between performance and complexity– Simple solution @ relay– Used number of uncoded packets as proxy to decoding complexity– Price of complexity is reduced with small cost in delay (<1dB)

• Provided analysis to characterize the system– Markov model– Transition prob required combinatorial approach– Captures dependence between relay and receiver knowledge (critical)

• Future work: multi-hop– Analysis may be hard– Local optimization heuristics– Half-duplex constraints – Judicious feedback use

10

System Model: Systematic Relay

11

• Stage One:– Q(0,0,0)→ (i,j,k) = Prob of being in state (i,j,k) at the end of Stage One.– Result of three independent Bernoulli Trials

• Stage Two: Same system evolution as the non-systematic case

• Expected # uncoded packets received via the systematic relay

• Decoding complexity @ Receiver on the order of O((M-U)3) + O(U(M-U))

Decoding Complexity and Delay

12

Normalized # uncoded packets received via the systematic relay

Normalized Mean Completion Time T/M

P2 P3

P1

Non-System vs. Systematic Relay

Non-systematic: always transmit linear combination of ALL received pkts.

loss

1 0 0 0

0 0 1 0

0 0 0 1

d4,1 d4,2 d4,3 d4,4d5,1 d5,2 d5,3 d5,4

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

c5,1 c5,2 c5,3 c5,4c6,1 c6,2 c6,3 c6,4

Systematic: forwards any received uncoded packets

13

loss

Sender Receiver

Relay

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

c5,1 c5,2 c5,3 c5,4c6,1 c6,2 c6,3 c6,4

d1,1 0 0 0

d2,1 0 d2,3 0

d3,1 0 d3,3 d3,4d4,1 d4,2 d4,3 d4,4d5,1 d5,2 d5,3 d5,4

Sender Receiver

Relay

System Model

14

• State of the network: (i, j, k)– i = # dof @ Receiver– j = # dof @ Relay– k = # dofs shared by Receiver and Relay– (i,j,k) valid if i+j-k ≤ M, i ≤ M, j ≤ M, 0 ≤ k ≤ min(i, j)

• Transmission initiates at (0,0,0)• Transmission terminates at (M, j, k)

• P(i,j,k)→ (i’,j’,k’) = Prob of transiting from state (i,j,k) to state (i’,j’,k’)• Expected time to reach a terminating state from (i,j,k):

• Expected completion time:

ik

j

M