Conditional PASTA

Let Y be a stochastic process representing the state of a system and N a doubly stochastic Poisson process whose intensity varies with the state of a random environment represented by a stochastic process X. In this context a generalization of “PASTA” (Poisson Arrivals See Time Averages) is shown to be valid. Various applications of the result are given

The queue M|G|1 with Markov modulated arrivals and services

We study an M|G|1 queue in which both the arrival rate and the service time distribution depend on the state of an underlying finite-state Markov chain. The solution is obtained by a matrix factorization method. This leads to results for waiting times and queue lengths both at arrival epochs and in continuous time. A numerical algorithm for the calculation of several quantities of interest is described and some numerical examples are given

Stochastic analysis of a dependent parallel system

This article discusses the stochastic behaviour of a two-unit parallel redundant repairable system with statistically dependent units. Important performance measures for the system, namely reliability, mean time to system failure, availability, stationary availability, joint availability and interval reliability are obtained in an explicit form. The transient behaviour of the system is characterised for a wide class of repair time distributions. The lifetimes of the units are modelled as bivariate exponential to capture the statistical dependence of the units. The article concludes with a detailed investigation of the stochastic point process induced by entries to various states, which correspond to the number of failed components in the system