1. Analysis of Maximum Size Matching scheduling algorithm (MSM) in input- queued switches under uniform traffic Neda Beheshti and Mohsen Bayati {nbehesht, bayati} @stanford.edu EE384Y Project May 25, 2004

2. Outline Motivation Previous work The Lyapunov function and Foster’s Criteria Final words References EE384Y Project May 25, 2004

3. The Lyapunov function Consider the following function EE384Y Project May 25, 2004

4. It can be written in matrix form: For example if N = 2, assuming: Then we would have: EE384Y Project May 25, 2004

5. Foster’s criteria Consider the following random variables: No let’s denote arrival and departure vectors with A(n) and D(n), hence: Now looking at the last two terms the second one is a bounded function, so in order to check Foster’s criteria we need to show the first one can become very negative as queue sizes get large. Call this term B.

6. Arrival traffic is uniform with rate, hence: We also have: So B is now equal to: Now using we only need to show for all i,j:

7. Using symmetry we only need to show the inequality for one queue say Now let and then the inequality reduces to: Consider the following three types of maximum size matchings: a) The black edge + k green edges for then b) Two red edges + k greed edges: for then c) One red edge + k green edges: for then For a, b we have: For any type c matching, MSM chooses a type c matching with equal probability. (Because we assume MSM chooses one of maximum matchings at random) And for any type a there can be at most N-k+1 different type c hence:

8. Is L(n) non-negative ? Answer: For N = 2,3,4 yes. So it’s a Lyapunov function. When N = 2, positive semidefinite matrix Similarly for N=3, 4 we can write P as sum of a positive semidefinite matrix and a matrix whose entires are nonnegative, and this is enough to show L(n) is not negative.

9. Final word We showed MSM is stable under uniform traffic for and input queued switches. For arrival traffic we only need the following assumptions.