1. 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 on Performance Analysis of Optical Packet Switched Networks

2. Surveyed Papers 1. 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.

3. Papers 1 and 2, examples of: “Engineering oriented” analysis of a single switch

4. 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

5. 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 Main performance measure of interest: Two Interacting Processes: Tagged fiber process Wavelength conversion process

6. Some Results

7. Some More Results

8. Approximation Assumptions

9. An Algorithmic Approximate Solution

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

11. 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:

12. A Flavor of the Results

13. Approximation for Symmetric Set

14. Paper 3, an example of: An applied probability paper motivated by optical networks

15. 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. Grochla Outline: 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

16. 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

17. Assume source destination pairs selected 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

18. Simple resulting Markov Chain State space: (distance from destination) Initial distribution Absorbing Transition Matrix: Hitting time of state 0 is the sojourn time (of interest)

19. 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: But is not constant and q depends on the graph

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

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

22. 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

23. Multicommodity Network Flow Problem Approach

24. Full Wavelength Conversion

25. No Wavelength Conversion

26. Sparse Wavelength Conversion

27. Main Idea Choose: Relax: Now we have an LP Main, argument: Solutions are often integer

28. Summary and future directions Papers 1,2: Analysis (exact/approximate) of a single node Paper 3: An example of a nice theoretical paper motivated by this application area Paper 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 control D. Paper 3 shows an example of an application that “housed” a nice theoretical (stochastic order) result

29. THANK YOU