1

Introduction to Network CodingNetwork Coding in Wireless Networks

- The Random Linear Network CodingRandom Linear Network Coding for Time Division DuplexingTime Division Duplexing

Presented by Yung-Cheng Tu

2010/12/24

2

Reference papers

• Daniel E. Lucani, Muriel Médard and Milica Stojanovic1. “Random Linear Network Coding for Time Division Duplexing: When to Stop Talk

ing and Start Listening”, in INFOCOM 2009, April 2009.

2. “Random Linear Network Coding For Time Division Duplexing: Energy Analysis”, in ICC Communication Theory Workshop, June 2009.

3. “Random Linear Network Coding for Time-Division Duplexing: Queueing Analysis”, in IEEE International Symposium on Information Theory (ISIT), July 2009.

4. “Broadcasting in Time-Division Duplexing: A Random Linear Network Coding Approach,” in Workshop on Network Coding (Netcod), June 2009.

5. “Random Linear Network Coding for Time-Division Duplexing: Field Size Considerations”, in IEEE Globecom 2009 Communication Theory Symposium, November-December 2009.

3

About Muriel Médard• A professor in the Department of Electrical Engineering and and Res

earch Laboratory of Electronics , MIT.Homepage: http://www.mit.edu/~medard/

• She has about 20 network coding papers published per year from 2002 to 2010.

• 2009 Communication Society and Information Theory Society Joint Paper Award

– Tracey Ho , Muriel Medard, Rolf Kotter, David Karger, Michelle Effros Jun Shi, Ben Leong,  "A Random Linear Network Coding Approach to Multicast", in IEEE Transactions on Information Theory, vol. 52, no. 10, pp. 4413-4430, October 2006.

• 2009 William R. Bennett Prize in the Field of Communications Networking

– Sachin Katti , Hariharan Rahul, Wenjun Hu, Dina Katabi, Muriel Medard, Jon Crowcroft, "XORs in the Air: Practical Wireless Network Coding", in IEEE/ACM Transactions on Networking, vol. 16, Issue 3, pp. 497-510,June 2008,.

4

Outline

• Brief of Network Coding– XORs (COPE)– Random Linear Network Coding

• Random Linear Network Coding for Time Division Duplexing– Analysis on coding block size– Analysis on energy consumption– Analysis on queueing– Broadcasting– Analysis on field size

• Comments

5

Brief of Network Coding (1)

• The concept of network coding was proposed by Rudolf Ahlswede et al. in 2000.– A router doesn’t just hand off the packets it receives; it mathema

tically combines them into new, hybrid packets.

[1] Rudolf Ahlswede, Ning Cai, Shuo-Yen Robert Li and Raymond W. Yeung, “Network Information Flow,” in IEEE Transactions on Information Theory, Vol. 46, No. 4, July 2000.

S

R1

R2

R3

x1x2x3

x1

x2

x3

S

R1

R2

R3

x1x2x3

y1y2y3

Y=F(X)

Y

x1

x2

x3

x1=D1(Y)

x2=D2(Y)

x3=D3(Y)

Traditional Network Network Coding

6

Brief of Network Coding (2)

– If the combination is done cleverly enough, this makes the whole network more efficient.

– But how that combination should be done, and what kinds of

efficiency gains were possible, were unclear.Without Network Coding With Network Coding

S2

Y

S1

X

a

a b

b

Bottleneck link (X,Y) requires two time units to transmit packets a and b.

(X,Y) transmits ab in a time unit.R1 decodes b from aand ab.R2 decodes afrom band ab.

R1 R2

Send packet a to R1,R2

Send packet b to R1,R2

S2

Y

S1

X

a

a b

b

R1 R2

Send packet a to R1,R2

Send packet b to R1,R2

a

a

b

b

ab

ab ab

7

Brief of COPE (1)• Main idea

– Use the XOR to forward multiple packets in a single transmission in wireless mesh network.

a

SA B

Packet a

Packet b

SA Ba a

b bSA B

a b

ab

(1) A sends a to S(2) S sends a to B(3) B sends b to S(4) S sends b to A

(1) A sends a to S and keeps a(2) B sends b to S and keeps b(3) S sends ab to A and B.

Without Network Coding With Network Coding

8

Brief of COPE (2)• COPE provides a general scheme for inter-session wirel

ess network coding.

• They analyzed the throughput gains of XORs in single-hop wireless network.

• The first deployment of network coding in a wireless network.– COPE can provide several-fold (3-4x) increase in the throughput

of wireless ad hoc networks.

9

Random Linear Network Coding (1)

• Linear Network Coding– Coded packets are the linear combinations of original packs

– Linear Network Coding can achieve the upper bound of networkthroughput obtained by Max-flow Min-cut theorem.

1 1 1 2 2

3 5 1 6 3 1 5 2 6 4 1( ) ( )

Y X X

Y X X

2 3 1 4 2

3 5 1 6 3 1 5 2 6 4 1( ) ( )

Y X X

Y X X

Coded packet Y1 Coded packet Y2

Coded packet Y3

10

Random Linear Network Coding (2)

• Linear Network Coding (cont’d)– Recv 1 and Recv 2 can decode X1 and X2 only if

and is nonsingular.

– The structure of coded packets

– A receiver must get M packets with independent coefficient vectors to decode packets.

1 2

5 1 6 3 5 2 6 4

3 4

5 1 6 3 5 2 6 4

Coefficient vector

Coefficient vector

11

Random Linear Network Coding (3)

• How to select the coefficients ξ ? Randomly Select

Coefficients are chosen uniformly at random from a finite field Fq

(Fq is the set of integers from 0 to q-1, where q=2g )

If q is large, then the probability of that two coefficient vectors are dependent is small.

– In the simple case of only one source and destination.• M original packets are coded to N packets. (N ≥ M)

• The expected number of successfully received packets before having M linearly independent combinations, is

S Rx1x2xM

y1y2yNEncode

12

Random Linear Network Coding for Time Division Duplexing (1)

• The original network coding research considers a channel with no erasures and, therefore, no need for feedback.

• For networks with packet erasures, two approaches have been used.– Transmitting coded data packets until the receiver sends an ACK.

(In TDD mode, senders must stop and wait for ACK per-packet.) Codes data packets in a block-by-block fashion using ACKs to indi

cate successful transmission of each block.

13

Random Linear Network Coding for Time Division Duplexing (2)

• Why the network coding is better?– In unicast, the sender can reduce the retransmissions by sendin

g more coded packets.

S Rx1x2x4 x3

y1y2y4encode y3y5

y1y2y4 y3y5 x1x2x4 x3

decode

14

Random Linear Network Coding for Time Division Duplexing (3)

• Why the network coding is better? (cont’d)– In multicast, the sender only needs to know the number of

packets that its receivers requires to decode packets, rather than which packets are required by the receivers.

S

R1

R2

x1x2x3x4

x1x2x3x4

x1x2x3x4

R1 and R2 mayloss different packets

x1x2x3x4

Retransmit 4 packets

S

R1

R2

y1y2y3y4

y1y2y3y4

y1y2y3y4

Continue to transmit the other 2 coded packets

y5y6

Traditional Network Network Coding

15

Random Linear Network Coding for Time Division Duplexing (4)

• The work focus on the problem of transmitting M data packets through a link using random linear network coding.

• The system transmits Ni coded packets (CP), and waits to receive an ACK packet that updates the value of i to j, at which point it will transmit Nj coded packets.

• The system will keep transmitting and stopping to update i, until i = 0.

i: the number of coded packets that the receiver requires to decoded.Ni: the number of packets that the sender will send

16

Random Linear Network Coding for Time Division Duplexing (5)

• Every time the system stops to wait for an ACK, it incurs in an additional delay.

• We want to minimize the number of stops required to complete transmission of the M packets.

• There is a natural trade off in the choice of the Ni’s.

– if the Ni’s are too small given the channel conditions, the system will have to transmit more ACK packets.

– if the Ni’s are too large, the receiver will have decoded the M packets before the transmitter stops.

The communication process

17

When To Stop Talking And Start Listening?[Infocom 2009]

• How to estimated the Ni?

• This paper uses a Markov chain to estimate the expected the completing time given the Ni.

18

When To Stop Talking And Start Listening?(cont’d)

• The expected completing time:

, where

• The objective is to minimize the value of TM.

Find the N1,N2,…NM

to solve the minimizationproblem.

19

When To Stop Talking And Start Listening?(cont’d)

• Instead of using the previous approach, we perform a search for the optimal values Ni using integer values.

• In particular, the search method for the optimal value can be made much simpler by exploiting the recursive characteristic of the problem.

20

When To Stop Talking And Start Listening?(cont’d)

• Compare the utilization with other schemes– Optimal TDD (n=the bits of data in a coded packets)

– Go-back-N (GBN) [15]

– Selective Repeat (SR) [15]

[15] Ozugur, T., Naghshineh, M., Kermani, P., Copeland, J. A.,“On the performance of ARQ protocols in infrared networks”, Int. Jour. Commun. Syst., vol. 13, pp. 617-638, 2000

21

• Parameters: g=20 bits, nack=100 bits, n=10000 bits, h=80 bits, Trt=0.25 ms

The utilization of network coding is much better for high packet error probability.

*This is numerical results, not simulations or experiments.

22

• The expected completing time of transmitting 10 packets with different Ni.

The TDD optimal scheme and the network coding full duplex optimal scheme have similar performance over a widerange of block error probabilities.

*This is numerical results, not simulations or experiments.

Full Duplex Optimal:• The sender transmits coded packets back-to-back until an ACK packet• Once the M packets have been decoded. The receiver transmits ACK packets each of duration Tack.

23

Energy Analysis [ICC 2009]

• Let Ep be the transmission energy of a coded packet, and Eack be the transmission energy of an ACK packet.

• Using the same Markov chain state diagram in [Infocom 2009]• The expected energy for completing the transmission given that the

Markov chain is in state i:

• Objective: Find the N1,N2…,NM, to minimize EM.

24

Energy Analysis (cont’d)

• Numerical results– Consider the transmission power is normalized, i.e., P=1.

(Eo=P*Tp, Eack=P*nack/R)

– Other parameters used:• M = 10, packet size n = 10; 000 bits, R = 1:5 Mbps, h = 80 bits,

• g = 20 bits, nack = 100 bits.

– Energy consumption and Completion time with different schemes:

25

Queueing Analysis [ISIT 2009]

• We study the scenarios for random linear network coding for TDD channels when the data packets arrive randomly at the source node according to a Poisson process.

• Queue model

– If the buffer has fewer than m data packets, the system will wait until m packets arrive before providing service.

– If the buffer contains more than K packets, the system will service exactly K packets.

– If the buffer has M packets with m ≤ M ≤ K, then the system will service M packets.

26

Queueing Analysis (cont’d)

• We can use the bulk queueing model M/G(m,K)/1 developed in [7] in to study the problem.– In [7], the transition probability of the number of packets in the queue is

presented.

• Assume that the number of coded packets to be sent back-to-back (Ni) are chosen to minimize the mean transmission time.

– Expected service time of each block transmission can obtained.[Infocom 2009]

[7] Bar-Lev, S. K., Parlar, M., Perry, D., Stadje, W., Van der Duyn Schouten, F. A., “Applications of bulk queues to group testing models with incomplete identification”,

European Journal of Operational Research, no. 183, pp. 226-237, 2007.

27

Queueing Analysis (cont’d)

• Numerical results– The mean queue size and mean batch size for different (m,K)

with λ = 30 PACKET/S, B = 30

m=minimum coding group size, K=maximum coding group size.Previous works assumed that the coding group size M is fixed.This work tells us we should code all packets in queue togetheronce the system idles.

Smaller K, longer queue

Smaller m, shorter queue

28

Broadcasting in Time-Division Duplexing [Netcod 2009]

• The time window allocated to the system to transmit N, coded packets in broadcasting scheme

Allocate a time period to each receiver’s ACK.

29

Broadcasting in Time-Division Duplexing (cont’d)

• Transmission begins with M information packets, which are encoded into NM random linear coded packets.

• Each ACK informs the transmitter how many dofs are missing, say i1, i2,…,iN for receivers 1,2, ..., N, respectively.

• The transmitter then sends Ni coded packets, where i=maxj=1,2,…N ij.

• If an ACK is lost, the transmitter assumes the previous state for• the corresponding receiver.• This process is repeated until all M packets have been decoded suc

cessfully by all receivers.

30

Broadcasting in Time-Division Duplexing (cont’d)

• Markov chain for the case of N = 2 receivers and a block size of M = 3.

• If we consider independent packet erasure channels for each of the receivers, the transition probability is

Equal to the transition probability P(s’i|Ni) in [infocom 2009]

31

Broadcasting in Time-Division Duplexing (cont’d)

• Similar to the work in [infocom 2009]– Estimate the expected completing time of each state.

– Find a Ni for each state to minimize expected completing time.

• Comparison schemes– Worst link channel

• In this heuristic we approximate the system as a link to the receiver with the worst channel, i.e. Pe = maxj Pej.

– Broadcast with Round Robin in Full Duplex Channel (RR Full Duplex):

– Broadcast with Round Robin in TDD (RR TDD) • the transmitter broadcasts all M packets back-to-back,

• If there are nodes that have not acknowledged the block of packets, the transmitter repeats the process, i.e. sends all M packets.

Why not compare with Selective Repeat?

32

Broadcasting in Time-Division Duplexing (cont’d)

• Numerical results

33

Field Size Considerations [Globecom 2009]

• This work considers the broadcasting TDD scheme.• The objective is also to minimize the value of the

expected transmission time TM.

• The previous works assumed the field size q is given.

• This work analyze the mean completing time of optimal TDD with different field size q.

q=2g

34

Field Size Considerations (cont’d)

• Numerical results– Mean completion time for the TDD scheme with two receivers.

– parameters R = 1.5 Mbps, h = 80 bits, nack = 100 bits.

35

• Summary– The authors proposed a network coding scheme for TDD to handl

e the retransmission when packets are lost. – They analyzed the one-hop performance of this scheme

in terms of • Block size • Energy• Queue length• Field size

• Other issues of network coding– Extension of XORs (routing, scheduling)– Load balance (Multi-path routing, bandwidth allocation)– Security and reliability

36

• Only numerical results.• The improvement of network coding seems slight, especi

ally when packet error rate is low.– Prof. Muriel Médard tries to perform the network codin

g in underwater network in recent years.

• The analyses are worth learning.• The scheme can be extended to multi-hop network. But t

he analyses will become much hard.• How to extend to TDMA in mesh network is a problem.

Comments