Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T14:42:47.825Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-stationary distribution of a two-unit warm-standby redundant system

Published online by Cambridge University Press:  14 July 2016

S. Kalpakam  and
M. A. Shahul Hameed
S. Kalpakam*
Affiliation:
Indian Institute of Technology, Madras
M. A. Shahul Hameed*
Affiliation:
Indian Institute of Technology, Madras
*
∗ Postal address: Department of Mathematics, Indian Institute of Technology, Madras 600 036, India.
∗ Postal address: Department of Mathematics, Indian Institute of Technology, Madras 600 036, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper establishes the existence of the limit as t → ∞ of the distribution of the residual lifetime at time t of a two-unit warm-standby reliability system supported by a single repair facility, conditioned on the event that the system has not been down at any time in (0, t). The conditioned limit distribution is proved to be exponential irrespective of the distributions of the lifetimes and repair times of the individual units, provided that their Laplace transforms are rational functions of their arguments.

Keywords

QUASI-STATIONARY DISTRIBUTIONWARM STANDBYREDUNDANT SYSTEMs-IDENTICAL UNITSONLINE FAILURERESIDUAL LIFETIMERELIABILITY FUNCTION
Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 2 , June 1983 , pp. 429 - 435
Copyright
Copyright © Applied Probability Trust 1983 

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

Cavender, J. A. (1978) Quasi stationary distribution of birth and death processes. Adv. Appl. Prob. 10, 570–586.Google Scholar
Daley, D. J. (1969) Quasi-stationary behaviour of left-continuous random walk. Ann. Math. Statist. 40, 532–539.Google Scholar
Darroch, J. N. and Seneta, E. (1965) On quasi-stationary distributions in absorbing discrete time finite Markov chains. J. Appl. Prob. 2, 88–100.CrossRefGoogle Scholar
Darroch, J. N. and Seneta, E. (1967) On quasi-stationary distributions in absorbing continuous time finite Markov chains. J. Appl. Prob. 4, 192–196.CrossRefGoogle Scholar
Kyprianou, E. K. (1971) On the quasi-stationary distribution of virtual waiting time in queues with Poisson arrivals. J. Appl. Prob. 8, 494–507.Google Scholar
Kyprianou, E. K. (1972a) On the quasi stationary distributions of the GI/M/1 queue. J. Appl. Prob. 9, 117–128.Google Scholar
Kyprianou, E. K. (1972b) On the quasi-stationary distributions of queues in heavy traffic. J. Appl. Prob. 9, 821–831.CrossRefGoogle Scholar
Mandl, P. (1960) On the asymptotic behaviour of the probabilities within groups of states of a homogeneous Markov process (in Russian). Casopis Pest. Mat. 85, 448–456.CrossRefGoogle Scholar
Seneta, E. and Vere-Jones, D. (1967) On quasi-stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar
Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.CrossRefGoogle Scholar
Widder, D. V. (1971) An Introduction to Transform Theory. Academic Press, New York.Google Scholar