the$r2d2$project:$ network$coding$for$ - lancaster university

The R2D2 project: Network coding for Rapid and Reliable Data Delivery

Dr Ioannis Chatzigeorgiou -‐ [email protected]

School of Computing and Communications

A project funded by EPSRC under the First Grant scheme Web: hKp://www.lancs.ac.uk/~chatzige/R2D2/index.html

University of Liverpool, 11th March 2015

Lancaster University

Aims and Aspira.ons

•  18-‐month EPSRC research project on network error control mechanisms (February 2014 – July 2015).

•  Design novel mathema]cal frameworks -‐ To iden]fy key rela]onships between system and channel parameters, understand network dynamics and op]mise network-‐coded architectures.

•  Inves]gate prac]cal aspects -‐ Ultra-‐reliable communica]ons, delay-‐constrained applica]ons, energy-‐efficient architectures

The R2D2 Team

Ioannis Chatzigeorgiou Principal Inves.gator

Example image

Andrea Tassi Postdoctoral Research Associate

Amjad Saeed Khan PhD Candidate (Oct. 2014 -‐ present)

Andrew Jones MSc by Research

(Sep. 2013 – Aug. 2014)

Research Ac.vi.es

•  Performance modelling of network-‐coded schemes

•  System op]misa]on for layered mul]cast services

•  Design of on-‐the-‐fly rateless decoders

•  Sparse network coding schemes that trade complexity for delay / energy

•  Extension to relay-‐aided transmission

1. Network coding in a nutshell

Random Linear Network Coding


[110100]!


[110100]![000001]!


[110100]![000001]![011000]!


[110100]![000001]![011000]![111111]!


[110100]![000001]![011000]![111111]![010101]!


… [110100]![000001]![011000]![111111]![010101]!


…

Network coding process

Non-‐overlapping window NC


Window 1 Window 2 Window 3


…


Window 1 Window 2 Window 3

Non-‐overlapping window NC •  x = {x1, x2, …, xK} is a layered source message of K source packets,

divided into L service layers (here L=3).

•  Encoding is performed over each service layer independently of the others

•  The source linearly combines the kl data packets composing the l-‐th layer and generates a stream of nl ≥ kl coded packets, where

•  Coefficients gj,i are selected uniformly at random from a finite field of size q, e.g. GF(q).

yj = gj ,i xii=1

kl

!

Non-‐overlapping window NC •  User u can recover the l-‐th layer if kl linearly independent coded packets

from that layer are collected. The probability of this event is

where p is the packet erasure probability, nl,u is the number of transmiKed coded packets of layer l, and r is the number of received coded packets.

•  The probability that user u will recover the first l service layers is

Pl (nl ,u ) = nl ,ur

!"#

$%&r=kl

nl ,u

' pnl ,u(r (1( p)r 1( 1qr(i

!"#

$%&i=0

kl(1

)

DNOW(n1,u ,...,nl ,u ) = Pj (nj ,u )j=1

l

!

Expanding window NC

Window 1

Window 2 Window 3

Expanding window NC

…


Window 1

Window 2 Window 3

Associated with Window 1 Associated with Window 2 Associated with Window 3

Expanding window NC •  Let Kl = k1 + k2 + … + kl denote the size of the l-‐th expanding window,

while Nl,u denotes the number of coded packets transmiKed to user u and associated with expanding window l.

•  The probability of user u recovering the first l service layers (or, equivalently, the l-‐th expanding window), can be expressed as

where rmin, l = Kl – Kl-1 + max(rmin, l-1 – rl-1, 0) and rmin,1 = K1. The probability that Kl out of the r1 + r2 + … + rl received coded packets are linearly independent has been approximated by gl(r1,…,rl).

DEW(N1,u ,...,Nl ,u ) =

= !N1,ur1

!"#

$%&rl=rmin,l

Nl ,u

'r2=0

N2,u

'r1=0

N1,u

' !Nl ,u

rl!"#

$%&p (Ni ,u(ri )i=1

l' (1( p) rii=1

l' gl (r1,…,rl )

2. Provider-‐centric op.misa.on

Op.misa.on models

•  We consider NOW layered NC and introduce the following indica]on variable

!u ,l = I DNOW(n1,u ,...,nl ,u ) " P̂( )•  Similarly, we use the following indica]on variable for EW layered NC

µu ,l = I ORj=lL

DEW(N1,u ,...,N j ,u ) ! P̂{ }"#

$%

•  Structure of transmission medium:

Frequency bands (sub-‐channels)

Time

Op.misa.on models

Proposed mixed alloca]on (MA) strategies:

minm1,…,mL

n(1,c ) ,...,n(L ,c )

n(l ,c)c=1

C

!l=1

L

!

!u ,lu=1

U

" #U t̂lfor l = 1,...,L

mc!1 " mc

subject to:

(NOW-‐MA)

0 ! n(l ,c)l=1

L

" ! B̂c

minm1,…,mL

N (1,c ) ,...,N (L ,c )

N (l ,c)

c=1

C

!l=1

L

!

µu ,lu=1

U

! "U t̂l

mc!1 " mc

subject to:

(EW-‐MA)

0 ! N (l ,c)

l=1

L

" ! B̂c

for c = 2,...,L

for c = 1,...,L

for l = 1,...,L

for c = 2,...,L

for c = 1,...,L

Network configura.on •  We have considered 80 users equally spaced and placed along the radial

line represen]ng the symmetry axis of one sector of the target cell.

•  Node eNB (eNode-‐B) represents the base sta]on. Users form a Mul]cast Group (MG).

Analy.cal results

The performance of the proposed EW-‐MA is compared to NOW-‐MA and the state-‐of-‐the-‐art Mul]-‐rate Transmission (MrT). The focus is on GF(2).

PSNR layers 1+2+3 PSNR

layers 1+2 PSNR

layer 1

3. User-‐centric op.misa.on

Op.misa.on model •  The objec]ve of the previous alloca]on method was to minimize the

number of transmiKed packets, while mee]ng a minimum set of service level agreements . This is from the point of view of the service provider…

•  Best prac]ce for burglars: To steal objects with the maximum value and the minimum weight. Maximising the profit-‐cost ra]o is the objec]ve of the Unequal Error Protec]on Resource Alloca]on Model (UEP-‐RAM) .

maxm1,…,mLN1,...,NL

!u ,ll=1

L

"u=1

U

" Nll=1

L

"

!u ,lu=1

U

" #U t̂l

l = 1,...,L

l = 1,...,L

0 ! Nl ! N̂l

subject to:

(UEP-‐RAM)

Network configura.on •  LTE-‐Advanced allows mul]ple

con]guous base sta]ons to deliver (in a synchronous fashion) the same services.

•  Top image: we considered a Single-‐Frequency Network (SFN) comprising 4 base sta]ons and 1700 users.

•  BoKom image: The distribu]on of the Signal-‐to-‐Interference-‐plus-‐Noise Ra]o (SINR) in space.

Analy.cal results

The proposed op]misa]on framework based on network coding clearly shows an increase in the coverage of service provider (right-‐hand side of each figure).

Three-‐layered service

UEP-RAM

Four-‐layered service

UEP-RAM

4. Other key findings

Systema.c vs. straighKorward NC •  The transmiKer broadcasts N packets:

–  They could all be linear combina]ons of the K source packets (straight-‐forward NC)

–  The first K transmiKed packets could be un-‐coded and the remaining N-‐K packets could be linear combina]ons (systema]c NC).

•  We proved that:

–  The decoding probability of systema]c RLNC is lower than the decoding probability of non-‐systema]c RLNC.

–  Systema]c RNLC offers the advantage of reduced decoding complexity and progressive packet recovery at the receiver.

Systema.c vs. straighKorward NC

OU: Ordered Un-‐coded transmission, SF NC: Straighlorward Network Coding, Sys. NC: Systema]c Network Coding. The erasure probability is p = 0.1.

10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dec

odin

g P

robab

ilit

y

N

OU Trans., M = 10

SF NC, M = 10

Sys. NC, M = 10

OU Trans., M = 20

SF NC, M = 20

Sys. NC, M = 20

Sparse Random Network Coding •  So far, the entries of the generator matrix were selected uniformly at

random from the elements of a finite field GF(q).

•  What if the probability of selec]ng a non-‐zero element is uniform but the probability of selec]ng the zero element ( ) is higher than 0.5?

•  The service provider can adjust the sparsity of its generator matrix and trade transmit power for reduced decoding complexity.

•  Recent results indicate that the performance gains offered by larger finite fields, e.g. GF(28) as opposed to GF(2), do not jus]fy the increase in decoding complexity.

p!

Sparse Random Network Coding •  ?

p!

τ!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

20

30

40

50

60

70

80

90

100

k! = 10

k! = 30

k! = 50

k! = 70

SimulationsUpper-Bound

•  Results for different numbers of source packets at the transmiKer. Upper bounds (solid lines) are compared to simula]ons (dashed lines).

•  Focus on GF(2). An increase in (probability of selec]ng the zero element) causes an increase in sparsity and a decrease in decoding complexity.

•  For 50 source packets and , ]me to decode=1030 μsec; for , ]me to decode=352 μsec.

p!

p! = 0.5 p! = 0.9

5. Concluding remarks

Random matrices over GF(q)

where � is an element of GF(q): • • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •

!

"

#########

$

%

&&&&&&&&&

x1 x2 x3 x4 x5 … xKxK!1

y1y2y3y4y5 !

yNyN!1

• =

0, p = 1 q1, p = 1 q! !q !1, p = 1 q

"

#$$

%$$

The decoding probability Pd is the probability that the N×K matrix (N ≥ K) is full-‐rank (Trullols-‐Cruces et al., 2011):

Pd = 1! 1qN! j

"#$

%&'j=0

K!1

(

O. Trullols-Cruces, J.M. Barcelo-Ordinas,, M. Fiore, “Exact decoding probability under random linear network coding”, IEEE Communications Letters, vol.15, no.1, pp .67-69, 2011

The expanding window principle

Applica.on: ü  Transmission of

layered informa]on, where ‘nested’ windows carry informa]on of higher/fundamental importance.

• • • • 0 0 0 0 0 0 0 0• • • • 0 0 0 0 0 0 0 0• • • • 0 0 0 0 0 0 0 0• • • • 0 0 0 0 0 0 0 0• • • • • • • • 0 0 0 0• • • • • • • • 0 0 0 0• • • • • • • • 0 0 0 0• • • • • • • • 0 0 0 0• • • • • • • • • • • •• • • • • • • • • • • •• • • • • • • • • • • •• • • • • • • • • • • •

!

"

###############

$

%

&&&&&&&&&&&&&&&

x1 x2 x3 x4 x5 … xKxK!1 …

y1y2y3y4y5 !

!

yNyN!1

The decoding probability Pd has been approximated.

The sliding window principle

Applica.ons: ü  TransmiKers

with a limited buffer size streaming mul]media content.

• • 0 0 0 0 0 0 0 0 0 0• • 0 0 0 0 0 0 0 0 0 0• • • • 0 0 0 0 0 0 0 0• • • • 0 0 0 0 0 0 0 00 0 • • • • 0 0 0 0 0 00 0 • • • • 0 0 0 0 0 00 0 0 0 • • • • 0 0 0 00 0 0 0 • • • • 0 0 0 00 0 0 0 0 0 • • • • 0 00 0 0 0 0 0 • • • • 0 00 0 0 0 0 0 0 0 • • • •0 0 0 0 0 0 0 0 • • • •0 0 0 0 0 0 0 0 0 0 • •0 0 0 0 0 0 0 0 0 0 • •

!

"

#################

$

%

&&&&&&&&&&&&&&&&&

x1 x2 x3 x4 x5 … xKxK!1 …

y1y2y3y4y5 !

!

yNyN!1

The decoding probability Pd is under inves]ga]on.

The random block-‐angular matrix

Applica.ons: ü  Mul]ple access

relay networks (MARC)

ü  Distributed data storage.

• • • • 0 0 0 0 0 0 0 0• • • • 0 0 0 0 0 0 0 0• • • • 0 0 0 0 0 0 0 00 0 0 0 • • • • 0 0 0 00 0 0 0 • • • • 0 0 0 00 0 0 0 • • • • 0 0 0 00 0 0 0 0 0 0 0 • • • •0 0 0 0 0 0 0 0 • • • •0 0 0 0 0 0 0 0 • • • •• • • • • • • • • • • •• • • • • • • • • • • •• • • • • • • • • • • •

!

"

###############

$

%

&&&&&&&&&&&&&&&

x1 x2 x3 x4 x5 … xKxK!1 …

y1y2y3y4y5 !

!

yNyN!1

The decoding probability Pd has been computed (Ferreira et al., 2013).

P. J. S. G. Ferreira, B. Jesus, J. Vieira, A. J. Pinho, “The rank of random binary matrices and distributed storage applications”, IEEE Communications Letters, vol.17, no.1, pp.151-154, 2013

Sparse random matrices

where � is an element of GF(q): • • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •• • • • • • • •

!

"

#########

$

%

&&&&&&&&&

x1 x2 x3 x4 x5 … xKxK!1

y1y2y3y4y5 !

yNyN!1

• =

0, p!1, p = 1! p!( ) q !1( )" "q !1, p = 1! p!( ) q !1( )

"

#

$$

%

$$

The decoding probability Pd has been bound from above (Blömer et al., 1997). We are not aware of exact expressions.

J. Blömer, R. Karp, E. Welzl, “The rank of sparse random matrices over finite fields”, Random Struct. Algorithms, 10 (1997), pp. 407-419.

Publica.ons – Project outcomes u  A. Tassi, I. Chatzigeorgiou and D. Vukobratović, “Resource alloca]on framework for network-‐

coded layered mul]media mul]cast services”, to appear in IEEE Journal on Selected Areas in Communica.ons (JSAC).

²  A. Khan and I. Chatzigeorgiou, “Performance analysis of random linear network coding in two-‐source single-‐relay networks”, to appear in Proc. IEEE Int. Conf. on Commun. (ICC), Workshop on Coopera]ve and Cogni]ve Mobile Networks, London, UK, June 2015.

²  A. Tassi, I. Chatzigeorgiou, D. Vukobratović and A. L. Jones, “Op]mized network-‐coded scalable video mul]cas]ng over eMBMS networks”, to appear in Proc. IEEE Int. Conf. on Commun. (ICC) , London, UK, June 2015.

²  A. L. Jones, I. Chatzigeorgiou and A. Tassi, “Binary systema]c network coding for progressive packet decoding”, to appear in Proc. IEEE Int. Conf. on Comm. (ICC) , London, UK, June 2015.

²  A. L. Jones, I. Chatzigeorgiou and A. Tassi, “Performance assessment of fountain-‐coded schemes for progressive packet recovery”, in Proc. 9th Int. Symp. on Comm. Systems, Networks & Dig. Sig. Proc. (CSNDSP), Manchester, UK, July 2014.

²  I. Chatzigeorgiou, “Condi]ons for coopera]ve transmission on Rayleigh fading channels”, in Proc. 80th IEEE Vehicular Tech. Conference (VTC), Vancouver, Canada, September 2014.

Project Website hKp://www.lancs.ac.uk/~chatzige/R2D2/index.html

the$r2d2$project:$ network$coding$for$ - lancaster university

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