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 <inline-formula> <tex-math notation="LaTeX">$\Lambda $ </tex-math></inline-formula>. Although testing membership in <inline-formula> <tex-math notation="LaTeX">$\Lambda $ </tex-math></inline-formula> 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, 1) an equivalence between membership in <inline-formula> <tex-math notation="LaTeX">$\Lambda $ </tex-math></inline-formula> and the existence of a positive root of a given polynomial, 2) a method to construct a vector of contention probabilities to stabilize any stabilizable arrival rate vector, 3) the volume of <inline-formula> <tex-math notation="LaTeX">$\Lambda $ </tex-math></inline-formula>, 4) explicit polyhedral, spherical, and ellipsoid inner and outer bounds on <inline-formula> <tex-math notation="LaTeX">$\Lambda $ </tex-math></inline-formula>, and 5) characterization of the generalized convexity properties of a natural “excess rate” function associated with <inline-formula> <tex-math notation="LaTeX">$\Lambda $ </tex-math></inline-formula>, including the convexity of the set of contention probabilities that stabilize a given arrival rate vector.

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