Properties of an Aloha-like stability region

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.Comment: 28 pages, 9 figures. Submitted August 15, 2014, revised September 21, 2015 and August 31, 2016, and accepted November 06, 2016 for publication in IEEE Transactions on Information Theory. Preliminary results presented at ISIT 2010, ITA 2010, and ITA 2011. DOI: 10.1109/TIT.2016.2640302. Copyright transferred to IEEE. This is last version uploaded by the authors prior to IEEE proofing proces