CWI PNA2, Reading Seminar,Presented by Yoni Nazarathy

EURANDOM and the Dept. of Mechanical Engineering, TU/e Eindhoven

September 17, 2009

An Assortment of Papers onPerformance Analysis of Optical Packet

Switched Networks

Surveyed Papers1. Fixed point analysis of limited range share per

node wavelength conversion in asynchronous optical packet switching systems.N. Akar, E. Karasan, C. Raffaelli,Photon Netw Commun, 2009.

2. Wavelength allocation in an optical switch with a fiber delay line buffer and limited-range wavelength conversion. J. Perez, B. Van Houdt, Telecommun Syst, 2009.

3. Level Crossing Ordering of Markov Chains: Computing End to End Delays in an All Optical Network. A. Busic, T. Czachorski, J.M. Fourneau, K. Grochla, Proceedings of Valuetools 2007.

4. Routing and Wavelength Assignment in Optical Networks.A.Ozdaglar, D. Bertsekas, IEEE/ACM Transactions on Networking 2003.

Papers 1 and 2, examples of:“Engineering oriented”

analysis of a single switch

Paper 1:Fixed point analysis of limited range share per

node wavelength conversion in asynchronous optical packet switching systems.

N. Akar, E. Karasan, C. Raffaelli

Model

•N inputs/outputs•Destinations are uniform 1/N•M wavelengths (K= M N input channels)•R convertors•d Conversion distance•“Far” policy or “Random” policy•Engset Traffic Model:

•ON •OFF

exp( )exp( )

Main performance measure of interest:

blocked ( N, M, , , , )R

P r d policyNM

Two Interacting Processes:• Tagged fiber process• Wavelength conversion process

Some Results

Some More Results

Approximation Assumptions

( ) wavelength channels in use

j(t) ON input wavelength channels

i t

Tagged Fiber

Markov Chain

( ) { ( ), ( )}

0 ,

0 ( , )

X t i t j t

j K

i Min M j

Wavelength Convertor

Markov Chain

( ) { ( ), ( )}

0 ( , ),

0

Y t i t j t

i Min R j

j K

( ) convertors in use

j(t) ON input wavelength channels

i t

An Algorithmic Approximate Solution

(1 )(5)

1

convdirected blocked

convertedblocked

P PP

P

Tagged Fiber

Markov Chain

( ) { ( ), ( )}

0 ,

0 ( , )

X t i t j t

j K

i Min M j

Wavelength Convertor

Markov Chain

( ) { ( ), ( )}

0 ,

0 ( , )

Y t i t j t

j K

i Min R j

directedP convblockedP

Paper 2:Wavelength allocation in an optical

switch with a fiber delay line buffer and limited-range wavelength conversion.

J. Perez, B. Van Houdt

Model•K inputs/outputs•W wavelengths (limited range convertors per link)•Synchronous Operation•FDLs of Duration D, 2D, …, N D per link•Limited Wavelength Conversion•Options for reachable wavelengths:

•Symmetric Set (d)•Fixed Set

•Options for destination wavelength policy•Random•Minimum Horizon (MinH)•Minimum Gap (MinGap)

•Packets arrival process: Discrete Phase Type Renewal•Packet sizes: I.I.D. from general (discrete) distribution

Main performance measure of interest:

blocked (Arrival process, Packet size distribution,

Wavelength pools, Selection policy, N, D)

P

A Flavor of the Results

Approximation for Symmetric Set

Paper 3, an example of: An applied probability paper

motivated by optical networks

Paper 3:Level Crossing Ordering of Markov Chains:

Computing End to End Delays in an All Optical Network.

A. Busic, T. Czachorski, J.M. Fourneau, K. GrochlaOutline:•The main (theoretical) result proved is a stochastic order relation between the hitting time of a given state of two Markov chains•Applied to networks with no-buffers and deflection routing:

• Formulating a simple model on a hyper-cube topology• Using the main result to formulate a stochastic order between a hyper-cube model and more general models• Using the main result to prove convergence of a fixed-point algorithm for obtaining the “deflection probability” using mean-value analysis

Deflection routing on a Hyper-cube

• Topology: Hyper-cube of dimension n• Typical node: • Directed edge between x and y if differ by one coordinate• nodes and directed edges•In degree = out degree = n•Diameter = n•On route from x to destination y, all directions with are “good”•At distance k, there are k good directions

1:{ ,..., } x {0,1}n ix x x

2n 2nn

i ix y

• Assume source destination pairsselected uniformly•Assume packets are independent•Select with uniform distribution a direction among the good ones (assume routing is uniform)•Two phases:

1. Route packets which “got their routing choice”2. Send to directions still available after first phase

(THIS IS DEFLECTION)•All packets are equivalent, so consider an arbitrary packet in an arbitrary switch (all switches are equivalent)•Denote the deflection probability at an arbitrary switch: p

Routing Rule and Assumptions

Simple resulting Markov Chain

( 5)

1 0

1 0

3 31 04 42 21 04 4

1 11 04 41 0

n

p p

p pR

p p

p p

0

1 !0, ( ,1),..., ( , ) , with ( , )

2 1 !( )!n

nC n C n n C n k

k n k

{0,..., }S nState space: (distance from destination)

Initial distribution

Absorbing Transition Matrix:

Hitting time of state 0 is the sojourn time (of interest)

Assumptions:

General Graphs (not just Hyper-Cubes)

• Symmetrical (all links are full-duplex)• Observe: Distance to destination after deflection can only change by (-1,0,+1)• Traffic is uniform, choice of links are uniform• Many symmetries so that modeling by states that denote the distance to destination works

Resulting Markov Chain:•State {0,…,m} is distance from destination•At node i, rejection with probability (before it was constant)•If rejection (w.p. ) we have•As a result, again tri-diagonal structure:

, 1

,0

,1

1 : 1 (1 )

:

1 : ( )

i i

i i

i i

i i p q

i i p q

i i p q i m

ip

ip , 1 ,0 ,1( , , )i i iq q q

But is not constant and q depends on the graphip

Stochastic Bounds on Sojourn Times

Application idea: now use Corollary 2 to bound general graphs with the hyper-cube (which can be calculated more easily)

Main Resul

t

Second Application: proving convergence of a fixed-point iteration algorithm for approximating p using mean value approximations

Paper 4, an example of:A paper that deals with network wide (global)

optimization

Paper 4:Routing and Wavelength Assignment

in Optical Networks.A.Ozdaglar, D. Bertsekas.

•Routing and Wavelength Assignment Problem (RWA):•A “circuit switching” oriented paper (not OPS)•Two light paths that share a physical link can not use the same wavelength on that link.•Without converters: have to use same wavelength along whole light path•Typically minimize number (or probability) of blocked calls or (as in this paper) – minimize concave functions of flows•Static vs. Dynamic•Typically hard integer programs (NP – Complete) or intractable dynamic programs

•In this paper:•Formulate LP problems which typically yield integers

Multicommodity Network Flow Problem Approach

Flow on link

Set of linkslf l

L

Full Wavelength Conversion

{ | }

Set of (origin,destination) pairs

Set of paths that the pair may use

C Set of wavelengths available on each link

Decision variables: { | , }, flows of path p

demans

p

w l pp l p

W

P

x W p P

r f x

No Wavelength Conversion

{ | }

Decision variables: { | , , }

indicator of wavelength c is used by path p

cp

cp

cl p

p l p c C

x W p P c C

x

f x

Sparse Wavelength Conversion

,

,

,{ | }

Decision variables: { | , , , }

indicator of wavelength c is used by path p on link

cp l

cp l

cl p l

p l p c C

x W p P l L c C

x l

f x

Main IdeaChoose:

Relax:

Now we have an LP

Main, argument: Solutions are often integer

Summary and future directions

Papers 1,2: Analysis (exact/approximate) of a single nodePaper 3: An example of a nice theoretical paper

motivated by this application areaPaper 4: Network wide optimization (centrally controlled).

Note: there are many papers (and even a book) in this direction

Possible Future Directions:A.In the flavor of papers 1 and 2, many other possible

configurations (~15 papers). Can be collected into a summarizing work

B.How to expand (A) to the network level, similar to the “hard” step from a single server queue to a queuing network

C.Network level stochastic analysis (simulation) and controlD.Paper 3 shows an example of an application that “housed”

a nice theoretical (stochastic order) result

THANK YOU