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

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

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### Stability Analysis of MNCM Class of Algorithmsand 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,
- 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.

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

- Apply MWM algorithm on these edge weights:

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.

- Pick an available input i with smallest and connect it to a possible output with smallest ,
- 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.

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