Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-09-09T08:24:19.720Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Abstract

A sequence of random variables η 0, η 1, …, ηn , … is defined by the recurrence formula ηn = max (η n–1 + ξn , 0) where η 0 is a discrete random variable taking on non-negative integers only and ξ 1, ξ 2, … 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

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 4 , December 1975 , pp. 830 - 844
Copyright
Copyright © Applied Probability Trust 1975 

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.)

Article purchase

Temporarily unavailable

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