Stability Analysis of MNCM Class of Algorithms and two more problems !

1. Stability Analysis of MNCM Class of Algorithmsand two more problems ! EE384Y Project Presentation June 4, 2003 Nima Asgharbeygi

2. Outline • MNCM Class of Algorithms • Fluid Analysis of LPF • iSLIP Random

3. Introduction • Definition of MNCM : (Tabatabaee et. al. Infocom 2003) A maximal size matching algorithm m belongs to MNCM class iff m contains all nodes with maximum weight. • Node weights: • MNCM includes LPF, MNM and MFM algorithms. • A port-based fluid model proof was represented.

4. Counter Examples • Deterministic arrivals, • Example due Da Chuang • IID Bernoulli arrivals, • Simulation shows instability for uniform traffic. • Counter example: Algorithm:Serveonly if ; otherwise serve some other non-empty VOQ’s to maximize weight of the matching.

5. What’s wrong with the proof? • Lyapunov function: • The issue: • “Due to continuity properties of B(t), for every there exists some such that for all there is always one common index that .” • This is wrong! • An interval of length in continuous time, corresponds to an interval of arbitrarily large length ( ) in discrete time domain. • This is not guaranteed by MNCM (easy to see by a periodic pattern counter example).

6. Important to Remember • To have a valid stability proof, we must ensure that both fluid model policy and the discrete policy always make the same decision; i.e. equivalency of departure processes.

7. Outline • MNCM Class of Algorithms • Fluid Analysis of LPF • iSLIP Random

8. Problem Statement • algorithm definition: • Apply MWM algorithm on these edge weights: Where • This is our famous LPF if . • Not straight forward to use fluid model on original LPF, because of discontinuity of

9. Stability of Fluid Policy • Fluid model weights: • Theorem: This fluid model is weakly stable under MWM policy if for some constants • Proof: Use and show that:

10. Equivalency of Fluid and Discrete Models • How should relate to ensure equivalency? • Recall that • Enough to have • Reasonable to choose

11. Example • Let • Then • Fluid model is based on • Easy to see • So is efficient under general traffic. • LPF is the limiting case of as Uniformity of convergence proves efficiency of LPF under general traffic. 1 z 1 z

12. Outline • MNCM Class of Algorithms • Fluid Analysis of LPF • iSLIP Random

13. Problem Statement • iSLIP Random scheduling algorithm • Wish to find results on stability and convergence of iSLIP-R. Input degree Probability of being empty 1 iteration

14. Approach • The problem is to find • Let • Assume that size of maximal match=N, and initially input i connected to output i (for all i).

15. Approach (continued) • Greedy algorithm: • Pick an available input i with smallest and connect it to a possible output with smallest , (add to ). Repeat until no available input remains. • Theorem: Given and initially input i connected to output i (for all i), the greedy algorithm maximizes E[# of empty output bins].

16. Outline of Proof • The proof is based on the following lemma. • Lemma: If for given the sets maximize , then for any j and k:

17. Results • Need to search for best to maximize E[# of empty output bins]. • I guess it is but yet no proof! • This gives • Therefore, iSLIP-R with only one iteration would be stable by speedup 4 for large N.

18. Thank You!