Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T20:01:43.406Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beneš, V. E. (1957) On queues with Poisson arrivals. Ann. Math. Statist. 28, 670–677.CrossRefGoogle Scholar
Cox, D. R. and Isham, V. (1978) Series expansions for the properties of a birth-process of controlled variability. J. Appl. Prob. 15, 610–616.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Davis, M. H. A. (1984) Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J.R. Statist. Soc. B 46, 353–388.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Gnedenko, B. V. and Kovalenko, I. N. (1968) Introduction to Queueing Theory. Israel Program for Scientific Translations, Jerusalem.Google Scholar
Hasofer, A. M. (1964) On the single-server queue with non-homogenous Poisson input and general service time. J. Appl. Prob. 1, 369–384.Google Scholar
Le Minh, D. (1978) The discrete-time single-server queue with time-inhomogenous compound Poisson input and general service-time distribution. J. Appl. Prob. 15, 590–601.CrossRefGoogle Scholar
Neuts, M. F. (1984) Matrix-analytic methods in queueing theory. European J. Operat. Res. 15, 2–12.Google Scholar
Ott, T. J. (1977a) The covariance function of the virtual waiting-time process in an M/G/1 queue. Adv. Appl. Prob. 9, 158–168.CrossRefGoogle Scholar
Ott, T. J. (1977b) The stable ?/G/1 queue in heavy traffic and its covariance function. Adv. Appl. Prob. 9, 169–186.CrossRefGoogle Scholar
Reynolds, J. F. (1975) The covariance structure of queues and related processes –a survey of recent work. Adv. Appl. Prob. 7, 383–415.Google Scholar
Takács, L. (1955) Investigation of waiting-time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hungar. 6, 101–129.CrossRefGoogle Scholar
Yevdokimova, G. S. (1974) The distribution of waiting time in the case of a periodic input flow. Engineering Cybernet. 12(3), 81–85.Google Scholar