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On termination time processes

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Jewgeni H. Dshalalow
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Abstract

In this paper we study the behavior of a delayed compound renewal process, S, about some fixed level, L. Normally, a jump process S increases at random times τ1, τ2, …, in random increments until it crosses L. S would then be terminated in a random number v of phases at time τv. In many applications, a more general termination scenario assumes that S may evolve either through v or σ random phases, whichever of the two is smaller (denoted by T). The number T of actual phases is called the termination index, and we evaluate a joint functional of T, the termination time τ T and the termination level ST. We also seek information about the process S a step before its termination, and derive a joint functional for all relevant processes. Examples of these processes and their applications to various stochastic models are discussed.

Keywords

FIRST-PASSAGE PROBLEMFLUCTUATION THEORYDELAYED RENEWAL PROCESSTERMINATION PROCESSTERMINATION INDEXTERMINATION TIMEFIRST EXCESS LEVELPRETERMINATIONQUEUEINGINVENTORY CONTROLSTOCK MARKETPOWER STATION

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60K15: Markov renewal processes, semi-Markov processes 60K25: Queueing theory
Type
Part 6 Stochastic Processes
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 325 - 336
Copyright
Copyright © Applied Probability Trust 1994 

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References

[1] Abolnikov, L. and Dshalalow, J. (1992) A first passage problem and its applications to the analysis of a class of stochastic models. J. Appl. Math. Stoch Anal. 5, 83–98.Google Scholar
[2] Cohen, J. (1977) On up- and downcrossings. J. Appl. Prob. 14, 405–410.Google Scholar
[3] Dshalalow, J. (1991) On modulated random measures. J. Appl. Math. Stoch. Anal. 4, 305–312.Google Scholar
[4] Dshalalow, J. and Tadj, L. (1993) On applications of first excess level random processes to queueing systems with random server capacity and capacity dependent service time. Stoch. Stoch. Reports. To appear.Google Scholar
[5] Dynkin, E. (1961) Some limit theorems for sums of independent random variables with infinite mathematical expectations. Sel. Trans. Math. Statist. Prob. 1, 171–189.Google Scholar
[6] Shanthikumar, J. G. (1989) Level crossing analysis of priority queues and a conservation identity for vacation models. Nav. Res. Logist. 36, 797–806.Google Scholar
[7] Takács, L. (1978) On fluctuations of sums of random variables. In Studies in Probability and Ergodic Theory. Advances in Mathematics Supplementary Studies Volume 2, ed. Rota, G.-C., pp. 45–93. Academic Press, New York.Google Scholar
[8] Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar