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A storage model in which the net growth-rate is a Markov chain

Published online by Cambridge University Press:  14 July 2016

P. J. Brockwell
P. J. Brockwell*
Affiliation:
Michigan State University
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Abstract

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).

Keywords

STORAGE PROCESSMARKOVIAN GROWTH-RATESTOCHASTIC INTEGRALDISCONTINUOUS MARKOV PROCESSTIME TO EMPTINESSLIMITING DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 1 , March 1972 , pp. 129 - 139
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Ali Khan, M. S. and Gani, J. (1968) Infinite dams with inputs forming a Markov chain. J. Appl. Prob. 5, 72–83.CrossRefGoogle Scholar
[2] Ali Khan, M. S. (1970) Finite dams with inputs forming a Markov chain. J. Appl. Prob. 7, 291–303.CrossRefGoogle Scholar
[3] Brockwell, P. J. and Gani, J. (1970) A population process with Markovian progenies. J. Math. Anal. Appl. 32, 264–273.CrossRefGoogle Scholar
[4] Doob, J. L. (1953) Stochastic Processes. John Wiley and Sons, Inc., New York.Google Scholar
[5] Finch, P. D. (1963) A limit theorem for Markov chains with continuous state space. J. Aust. Math. Soc. 3, 351–358.CrossRefGoogle Scholar
[6] Gaver, D. P. and Miller, R. G. Jr. (1962) Limiting distributions for some storage problems. Studies in Applied Probability and Management Science. Ed. Arrow, , Karlin, and Scarf, , 110–126.Google Scholar
[7] Keilson, J. and Subba Rao, S. (1970) A process with chain dependent growth rate. J. Appl. Prob. 7, 699–711.CrossRefGoogle Scholar
[8] Keilson, J. and Wishart, D. M. G. (1964) A central limit theorem for processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 61, 173–190.CrossRefGoogle Scholar
[9] Keilson, J. and Wishart, D. M. G. (1965) Boundary problems for additive processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 63, 187–193.CrossRefGoogle Scholar
[10] Lloyd, E. H. (1963) Reservoirs with serially correlated inflows. Technometrics 5, 85–93.Google Scholar
[11] Miller, R. G. Jr. (1963) Continuous time stochastic storage processes with random linear inputs and outputs. J. Math. Mech. 12, 275–291.Google Scholar
[12] Moyal, J. E. (1957) Discontinuous Markov processes. Acta Math. 98, 221–264.CrossRefGoogle Scholar
[13] Odoom, S. and Lloyd, E. H. (1965) A note on the equilibrium distribution of levels in a semi-infinite reservoir subject to Markovian inputs and unit withdrawals. J. Appl. Prob. 1, 215–222.CrossRefGoogle Scholar
[14] Weldon, K. L. M. (1969) Stochastic storage processes with multiple slope linear inputs and outputs. Technical Report No. 9, Statistics Department, Stanford University, California.Google Scholar