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Some exact results for dams with Markovian inputs

Published online by Cambridge University Press:  14 July 2016

Pyke Tin
Affiliation:
Monash University, Clayton, Victoria
R. M. Phatarfod
Affiliation:
Monash University, Clayton, Victoria

Abstract

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj) where pij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1976 

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References

Khan, M. S. Ali and Gani, J. (1968) Infinite dams with inputs forming a Markov chain. J. Appl. Prob. 5, 7283.Google Scholar
Brockwell, P. J. and Gani, J. (1970) A population process with Markovian progenies. J. Math. Anal. Appl. 32, 264273.Google Scholar
Kuczma, M. (1968) Functional Equations in a Single Variable. Polska Akademia Nauk Monografie Matematyczne, Warszawa.Google Scholar
Lloyd, E. H. and Odoom, S. (1965) A note on the equilibrium distribution of levels in a semi-infinite reservoir subject to Markovian inputs and unit withdrawals. J. Appl. Prob. 2, 215222.Google Scholar
Pakes, A. G. (1973) On dams with Markovian inputs. J. Appl. Prob. 10, 317329.Google Scholar
Phatarfod, R. M., Speed, T. P. and Walker, A. M. (1971) A note on random walks. J. Appl. Prob. 8, 198201.Google Scholar
Phatarfod, R. M. and Mardia, K. V. (1973) Some results for dams with Markovian inputs. J. Appl. Prob. 10, 166180.CrossRefGoogle Scholar
Read, A. H. (1952) The solution of a functional equation. Proc. R. Soc. Edinburgh A 63, 336345.Google Scholar