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A storage process by semi-Markov input

Published online by Cambridge University Press:  01 July 2016

Lajos Takács
Lajos Takács*
Affiliation:
Case Western Reserve University, Cleveland, Ohio
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Abstract

A sequence of random variables η0, η1, …, ηn, … is defined by the recurrence formula ηn = max (ηn–1 + ξn, 0) where η0 is a discrete random variable taking on non-negative integers only and ξ1, ξ2, … ξn, … is a semi-Markov sequence of discrete random variables taking on integers only. Define Δ as the smallest n = 1, 2, … for which ηn = 0. The random variable ηn can be interpreted as the content of a dam at time t = n(n = 0, 1, 2, …) and Δ as the time of first emptiness. This paper deals with the determination of the distributions of ηn and Δ by using the method of matrix factorisation.

Keywords

SEMI-MARKOV PROCESSESMATRIX FACTORISATIONSTORAGE PROCESSESDISTRIBUTION OF THE CONTENTDISTRIBUTION OF THE TIME OF FIRST EMPTINESS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 4 , December 1975 , pp. 830 - 844
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Birkhoff, G. D. (1913) A theorem on matrices of analytic functions. Mathematische Annalen 74, 122–133.Google Scholar
[2] Gani, J. (1969) Recent advances in storage and flooding theory. Adv. Appl. Prob. 1, 90–110.Google Scholar
[3] Gani, J. and Matthews, J. (1973) Recent results in first emptiness problems of storage. Inventory Control and Water Storage, Colloquia Mathematica Societatis János Bolyai No. 7. North Holland, Amsterdam, pp. 73–82.Google Scholar
[4] Gohberg, I. C. and Krein, M. G. (1958) Systems of integral equations on a half line with kernels depending on the difference of arguments. (Russian) Uspehi Mat. Nauk 13, 2, 3–72. (English translation: American Mathematical Society Translations (2) 14 (1960), 217–287.) Google Scholar
[5] Masani, P. (1956) The Laurent factorization of operator-valued functions. Proc. London Math. Soc. (3) 6, 59–69.Google Scholar
[6] Miller, H. D. (1962) A matrix factorization problem in the theory of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc. 58, 268–285.Google Scholar
[7] Miller, H. D. (1962) Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc. 58, 286–298.Google Scholar
[8] Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. John Wiley and Sons, New York.Google Scholar
[9] Takács, L. (1975) Combinatorial and analytic methods in the theory of queues. Adv. Appl. Prob. 7, 607–635.Google Scholar
[10] Wiener, N. (1955) On the factorization of matrices. Comment. Math. Helv. 29, 97–111.Google Scholar
[11] Wiener, N. and Masani, P. (1957) The prediction theory of multivariate stochastic processes. I–II. Acta Mathematica 98, 111–150 and 99, (1958) 93–137.CrossRefGoogle Scholar