Properties of an Aloha-like stability region

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.