Stability Analysis of MNCM Class of Algorithms and two more problems ! EE384Y Project Presentation June 4, 2003 Nima Asgharbeygi

Outline MNCM Class of Algorithms Fluid Analysis of LPF iSLIP Random

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.

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

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).

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.

Outline MNCM Class of Algorithms Fluid Analysis of LPF iSLIP Random

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

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:

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

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

Outline MNCM Class of Algorithms Fluid Analysis of LPF iSLIP Random

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

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).

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].

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

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.

Thank You!