The stability condition of BMAP/M/$\infty$ queues

TL;DR

This paper establishes a precise stability condition for BMAP/M/∞ queues, showing stability depends on the finiteness of the expected logarithm of batch sizes, and extends this to multiclass queues.

Contribution

It provides a necessary and sufficient stability criterion for BMAP/M/∞ queues based on batch size distribution, a novel theoretical result.

Findings

Stability iff expectation of log batch size is finite

Derived stability condition for multiclass BMAP/M/∞ queues

Theoretical proof of stability criterion

Abstract

This paper considers a BMAP/M/$\infty$ queue with a batch Markovian arrival process (BMAP) and an exponential service time distribution. We first prove that the BMAP/M/$\infty$ queue is stable if and only if the expectation of the logarithm of the batch-size distribution is finite. Using this result, we also present the stability condition for an infinite-server queue with a multiclass batch Markovian arrival process and class-dependent exponential service times.