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A note on a one-compartment model with clustering

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

James G. Booth
James G. Booth*
Affiliation:
The Australian National University
*
∗Postal address: Centre for Mathematics and its Applications, ANU, GPO Box 4, Canberra, ACT 2601, Australia.
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Abstract

The classical one-compartment model with no input or pure death process is shown to be a limiting case of a ‘binomial cascade' model which has the same mean and in which particles exit the compartment in binomial clusters. The transition probabilities of the binomial cascade process are derived in closed form. The model is easily modified to allow Poisson input into the compartment. Distributional results are given for this model also. In particular, it is shown that the M/M/∞ queue is a limiting case.

Keywords

MARKOV PROCESSM/M/∞ QUEUEPOISSON PROCESSSTOCHASTIC COMPARTMENTAL MODELWEAK CONVERGENCE

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 535 - 542
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

On leave from the Department of Statistics, University of Florida, Gainesville, FL 32611, USA.

References

Ball, F. and Donnelly, P. (1988) A unified approach to variability in compartmental models. Biometrics 44, 685–694.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Loève, M. (1963) Probability Theory, 3rd edn. Van Nostrand, Princeton, NJ.Google Scholar
Matis, J. H. and Hartley, H. O. (1971) Stochastic compartment analysis: model and least squares estimation from time series data. Biometrics 27, 77–102.Google Scholar
Purdue, P. (1981) Variability in a single compartment system: a note on Barnard's model. Bull. Math. Biol. 43, 111–116.Google Scholar
Saunders, R. (1975) Conservative processes with stochastic rates. J. Appl. Prob. 12, 447–456.Google Scholar