On a Generalized Foster-Lyapunov Type Criterion for the Stability of Multidimensional Markov chains with Applications to the Slotted-Aloha Protocol with Finite Number of Queues

TL;DR

This paper extends a stability criterion for multidimensional Markov chains and applies it to analyze the stability of the slotted-Aloha protocol with finite queues, providing new sufficient conditions for stability and instability.

Contribution

It generalizes a positive recurrence criterion for Markov chains and applies it to derive linear stability conditions for the slotted-Aloha protocol without needing joint queue statistics.

Findings

Derived a linear stability condition based on arrival rates.

Provided sufficient conditions for protocol instability.

Extended the positive recurrence criterion to multidimensional Markov chains.

Abstract

In this paper, we generalize a positive recurrence criterion for multidimensional discrete-time Markov chains over countable state spaces due to Rosberg (JAP, Vol. 17, No. 3, 1980). We revisit the stability analysis of well known slotted-Aloha protocol with finite number of queues. Under standard modeling assumptions, we derive a sufficient condition for the stability by applying our positive recurrence criterion. Our sufficiency condition for stability is linear in arrival rates and does not require knowledge of the stationary joint statistics of queue lengths. We believe that the technique reported here could be useful in analyzing other stability problems in countable space Markovian settings. Toward the end, we derive some sufficient conditions for instability of the protocol.