Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-30T21:11:47.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Sojourn times, exit times and jitter in multivariate Markov processes

Published online by Cambridge University Press:  01 July 2016

J. Keilson
J. Keilson*
Affiliation:
University of Rochester
Get access Rights & Permissions [Opens in a new window]

Abstract

To treat the transient behavior of a system modeled by a stationary Markov process in continuous time, the state space is partitioned into good and bad states. The distribution of sojourn times on the good set and that of exit times from this set have a simple renewal theoretic relationship. The latter permits useful bounds on the exit time survival function obtainable from the ergodic distribution of the process. Applications to reliability theory and communication nets are given.

Keywords

MARKOV PROCESSESCHAINSRENEWAL THEORYRELIABILITY THEORYSOJOURN TIMESEXIT TIMESRARE EVENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 4 , December 1974 , pp. 747 - 756
Copyright
Copyright © Applied Probability Trust 1974 

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] Barlow, R. and Proschan, F. (1965) Mathematical Theory of Reliability. John Wiley, New York.Google Scholar
[2] Birnbaum, Z. W., Esary, J. D. and Saunders, S. C. (1961) Multicomponent systems and structures and their reliability. Technometrics 3, 55–77.CrossRefGoogle Scholar
[3] Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.CrossRefGoogle Scholar
[4] Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
[5] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. John Wiley, New York.Google Scholar
[6] Keilson, J. (1965) A review of transient behavior in regular diffusion and birth-death processes. Part II. J. Appl. Prob. 2, 405–428.CrossRefGoogle Scholar
[7] Keilson, J. (1966) A technique for discussing the passage time distribution for stable systems. J. R. Statist. Soc. B 28, 477–486.Google Scholar
[8] Keilson, J. (1966) A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Statist. 37, 866–870.CrossRefGoogle Scholar
[9] Keilson, J. (1972) A threshold for log-concavity for probability generating functions and associated moment inequalities. Ann. Math. Statist. 43, 1702–1708.CrossRefGoogle Scholar
[10] Keilson, J. (1973) Convexity and complete monotonicity in queueing distributions and associated limit behavior. Proceedings of Conference on Mathematical Methods in Queueing Theory, University of Western Michigan, Kalamazoo, May 1973. To appear.Google Scholar
[11] Keilson, J. (1973) Monotonicity and convexity in system survival functions and metabolic disappearance curves. Center for System Science Report 73–10, University of Rochester. Proceedings of Conference of Reliability and Biometry, Florida State University, July 1973. To appear.Google Scholar
[12] Keilson, J. and Steutel, F. (1974) Mixtures of distributions, moment inequalities, and measures of exponentiality and normality. Ann. Probab. 2, 112–130.CrossRefGoogle Scholar
[13] Kingman, J. F. C. (1968) Markov population processes. J. Appl. Prob. 6, 1–18.CrossRefGoogle Scholar
[14] Prabhu, N. U. (1965) Queues and Inventories. John Wiley, New York.Google Scholar