On the cycle maximum of birth-death processes and networks of queues

TL;DR

This paper investigates the maximum number of customers in birth-death processes and queue networks, providing stochastic compactness results and asymptotic characterizations for cycle maxima in various queueing models.

Contribution

It introduces new stochastic compactness results for sample maxima in birth-death processes and characterizes cycle maxima in Kelly-Whittle queue networks using asymptotic analysis.

Findings

Sample maxima are stochastically compact in positive recurrent processes.

Conditioned maxima are stochastically compact in transient processes.

Explicit asymptotic expressions enable cycle maximum evaluation in queue networks.

Abstract

This paper considers the cycle maximum in birth-death processes as a stepping stone to characterisation of the cycle maximum in single queues and open Kelly-Whittle networks of queues. For positive recurrent birth-death processes we show that the sequence of sample maxima is stochastically compact. For transient birth-death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact. We show that the Markov chain recording the total number of customers in a Kelly-Whittle network is a birth-death process with birth and death rates determined by the normalising constants in a suitably defined sequence of closed networks. Explicit or asymptotic expressions for these normalising constants allow asymptotic evaluation of the birth and death rates, which, in turn, allows characterisation of the cycle maximum in a single busy cycle, and convergence of the sequence of sample maxima for Kelly-Whittle networks of queues.