Properties of an Aloha-like stability region

TL;DR

This paper investigates the stability region of the slotted Aloha protocol, providing new theoretical insights, bounds, and properties that enhance understanding of network stability and control.

Contribution

It introduces novel results including polynomial root conditions, construction methods for contention probabilities, and geometric bounds on the stability region.

Findings

Equivalence between stability membership and polynomial roots

Method to construct contention probabilities for stabilization

Explicit bounds on the stability region volume

Abstract

A well-known inner bound on the stability region of the finite-user slotted Aloha protocol is the set of all arrival rates for which there exists some choice of the contention probabilities such that the associated worst-case service rate for each user exceeds the user's arrival rate, denoted $\Lambda$. Although testing membership in $\Lambda$ of a given arrival rate can be posed as a convex program, it is nonetheless of interest to understand the properties of this set. In this paper we develop new results of this nature, including $i)$ an equivalence between membership in $\Lambda$ and the existence of a positive root of a given polynomial, $ii)$ a method to construct a vector of contention probabilities to stabilize any stabilizable arrival rate vector, $iii)$ the volume of $\Lambda$, $iv)$ explicit polyhedral, spherical, and ellipsoid inner and outer bounds on $\Lambda$, and $v)$ characterization of the generalized convexity properties of a natural ``excess rate'' function associated with $\Lambda$, including the convexity of the set of contention probabilities that stabilize a given arrival rate vector.