Queues with Lévy input and hysteretic control

We consider a (doubly) reflected Lévy process where the Lévy exponent is controlled by a hysteretic policy consisting of two stages. In each stage there is typically a different service speed, drift parameter, or arrival rate. We determine the steady-state performance, both for systems with finite and infinite capacity. Thereby, we unify and extend many existing results in the literature, focusing on the special cases of M/G/1 queues and Brownian motion. © The Author(s) 2009

Examples of fitting structured phase–type distributions

A sub–class of phase–type distributions is defined in terms of a Markov process with sequential transitions between transient states and transitions from these states to absorption. Such distributions form a very rich class; they can be fitted to data, and any structure revealed by the parameter estimates used to develop more parsimonious re–parametrizations. Several example data sets are used as illustrations. Copyrigh

Markov death process modelling and analysis of binary data

It is shown that any discrete distribution with finite support has a representation in terms of a general Markov death process with transition rates μi (i≥0), the binomial distribution corresponding to a linear sequence of these μi. Accordingly, log-linear forms for μi/i will provide generalisations of the binomial distribution. Such modelling is illustrated with reference to published data-sets on surviving foetuses in animal pregnancies, where models are constructed which fit the data reasonably well and offer useful interpretations in terms of the actual process of foetal death

Diffusion approximations for stochastic compartmental models

A diffusion approximation is developed for one-compartment systems with nonlinear elimination rates. Both impulsive input and continuous infusion are considered. Explicit expressions for the mean and autocovariance functions of the contents of the compartment as time elapses are given

A structured compartmental model for drug kinetics

A compartmental model with a structure that describes drug kinetics by incorporating the ideas of diffusion and gamma distributed clearance times is proposed. The equations describing this model may be solved by elementary numerical techniques, and the model is shown to fit a data set describing renal gentamicin concentrations, better than a simple power-function model

Phase-type distributions for failure times

Phase-type distributions describe the random time taken for a Markov process to reach an absorbing state. In the context of component failure, sequential movement through the transient states (phases) of such a system could describe the ageing process with movement out of these states (absorption) corresponding to failure. Thus, the lifetime of a component is the absorption time and the probability distribution of these times can be written in terms of the solution of a system of differential equations for which there are many convenient computational algorithms. A variety of different distributions is possible by varying the parameters of the process and hazard rates of various shapes can be constructed, allowing different patterns of variation in observed data to be modelled. These distributions are applied to some industrial data-sets and further features of the processes discussed

Analysis of recreational fish catches – dealing with highly skewed distributions with many zeros

Data from surveys of recreational anglers fishing on three estuaries in eastern Australia reveal highly skewed distributions of catches with many zeros. Such data may be analysed using a two component approach involving a binary (zero/non-zero catch) response and the non-zero catches. A truncated regression model was effective in analysing the non-zero catches. Covariates were incorporated in the modelling, and their critical assessment has led to improved measures of fishing effort for this recreational fishery

Compartmental Modelling of Equipment subject to partial repair

A semi-Markov model for equipment that can be repaired and returned to service in a less than new state is developed, where several less than new states are possible before the equipment finally fails. A subclass of phase-type distributions is used to model the times spent in each of these states. Then, exploiting the phase-type structure, the composite distribution of the entire lifetime of the equipment can be constructed. Data from railway wagon wheel-sets are used to illustrate this modelling and data analysis. (C) 2000 Elsevier Science Ltd. All rights reserved