Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T14:18:39.798Z Has data issue: false hasContentIssue false
  • English
  • Français

On termination time processes

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Jewgeni H. Dshalalow
Get access Rights & Permissions [Opens in a new window]

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 

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] 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