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

This paper considers the cycle maximum in birth–death processes as a stepping stone to characterisation of the asymptotic behaviour of the maximum number of customers 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.