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The virtual waiting-time and related processes

Published online by Cambridge University Press:  01 July 2016

D. R. Cox  and
Valerie Isham
D. R. Cox*
Affiliation:
Imperial College, London
Valerie Isham*
Affiliation:
University College London
*
∗Postal address: Department of Mathematics, Imperial College, Huxley Building, Queen&s Gate, London SW7 2BZ, UK.
∗∗Postal address: Department of Statistical Science, University College London, Gower St, London WC1E 6BT, UK.
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Abstract

The virtual waiting-time process of Takács is one of the simplest examples of a stochastic process with a continuous state space in continuous time in which jump transitions interrupt periods of deterministic decay. Properties of the process are reviewed, and the transient behaviour examined in detail. Several generalizations of the process are studied. These include two-sided jumps, periodically varying ‘arrival’ rate and the presence of a state-dependent decay rate; the last case is motivated by the properties of soil moisture in hydrology. Throughout, the emphasis is on the derivation of simple interpretable results.

Keywords

TAKÁCS PROCESSSHOT-NOISE PROCESSSOIL MOISTURE MODEL
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 558 - 573
Copyright
Copyright © Applied Probability Trust 1986 

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