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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
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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.
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Papers 1 and 2, examples of:“Engineering oriented”
analysis of a single switch
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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
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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
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Some Results
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Some More Results
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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
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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
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Paper 2:Wavelength allocation in an optical
switch with a fiber delay line buffer and limited-range wavelength conversion.
J. Perez, B. Van Houdt
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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
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A Flavor of the Results
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Approximation for Symmetric Set
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Paper 3, an example of: An applied probability paper
motivated by optical networks
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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
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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
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• 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
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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)
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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
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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
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Paper 4, an example of:A paper that deals with network wide (global)
optimization
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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
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Multicommodity Network Flow Problem Approach
Flow on link
Set of linkslf l
L
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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
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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
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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
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Main IdeaChoose:
Relax:
Now we have an LP
Main, argument: Solutions are often integer
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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
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THANK YOU