Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T16:48:53.975Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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 

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

[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