Consider a Poisson point process with an intensity parameter forming a Markov chain with continuous time and finite state space. A system of ordinary differential equations is derived for the conditional distribution of the Markov chain given observations of the point process. An estimate of the current intensity, optimal in the least-squares sense, is computed from this distribution. Applications to reliability and replacement theory are given. A special case with two states, corresponding to a process in control and out of control, is discussed at length. Adjustment rules, based on the conditional probability of the out of control state, are studied. Regarded as a function of time, this probability forms a Markov process with the unit interval as state space. For the distribution of this process, integro-differential equations are derived. They are used to compute the average long run cost of adjustment rules.

For the single server queueing system, whose distributions of service and inter-arrival times have rational Laplace-Stieltjes transforms, limit theorems are derived for the supremum of the virtual waiting time during k successive busy cycles for k→∞. Similarly, for the supremum of the actual waiting times of all customers arriving in k successive busy cycles. Only the cases with the load of the system less than one and equal to one are considered. The limit distributions are extreme value distributions. The results are obtained by first deriving a number of asymptotic expressions for the quantities which govern the analytic description of the system Km/Kn/1. Using these asymptotic relations limit theorems for entrance times can also be obtained, a few examples are given.

Estimates are given for the rates of convergence in functional central limit theorems for quantities of interest in the GI/G/1 queue and a general multiple channel system. The traffic intensity is fixed ≧ 1. The method employed involves expressing the underlying stochastic processes in terms of Brownian motion using the Skorokhod representation theorem.

Estimates are given for the rates of convergence in functional central limit theorems for the queue length process in a sequence of general multiple channel queues. The situation is considered where the traffic intensity in the nth. queue, ρn, tends to ρ ≧ 1 as n approaches infinity. This extends previous work by the author, [6], in which the traffic intensity was fixed ≧ 1.