Geometric Approximations of Some Aloha-like Stability Regions

TL;DR

This paper introduces geometric ellipsoid bounds for a complex Aloha stability region, simplifying membership checks and analyzing the convexity of control sets for fixed arrival rates.

Contribution

It proposes ellipsoid approximations as inner and outer bounds for a key Aloha stability region, and studies the convexity of control sets stabilizing fixed arrival rates.

Findings

Ellipsoids can serve as easy-to-check bounds on the Aloha stability region.

The set of controls stabilizing a fixed arrival rate is convex.

Conjectured bounds could simplify stability analysis in Aloha networks.

Abstract

Most bounds on the stability region of Aloha give necessary and sufficient conditions for the stability of an arrival rate vector under a specific contention probability (control) vector. But such results do not yield easy-to-check bounds on the overall Aloha stability region because they potentially require checking membership in an uncountably infinite number of sets parameterized by each possible control vector. In this paper we consider an important specific inner bound on Aloha that has this property of difficulty to check membership in the set. We provide ellipsoids (for which membership is easy-to-check) that we conjecture are inner and outer bounds on this set. We also study the set of controls that stabilize a fixed arrival rate vector; this set is shown to be a convex set.