Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-17T01:41:40.454Z Has data issue: false hasContentIssue false
  • English
  • Français

General shock models associated with correlated renewal sequences

Published online by Cambridge University Press:  14 July 2016

J. G. Shanthikumar  and
U. Sumita
J. G. Shanthikumar*
Affiliation:
University of Arizona
U. Sumita*
Affiliation:
University of Rochester
*
∗ Postal address: Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, U.S.A.
∗∗ Postal address: The Graduate School of Management, The University of Rochester, Rochester, NY 14627, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we define and analyze a general shock model associated with a correlated pair (Xn, Yn) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level. Two models, depending on whether the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length Yn of the subsequent interval until the next shock, are considered. The transform results, an exponential limit theorem, and properties of the associated renewal process of the failure times are obtained. An application in a stochastic clearing system with numerical results is also given.

Keywords

SHOCK MODELSCORRELATED RENEWAL PROCESSEXPONENTIAL LIMIT THEOREMSTOCHASTIC CLEARING SYSTEM
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983 , pp. 600 - 614
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.)

Footnotes

This research was done while the authors were at Syracuse University and was published as Working Paper No. 82–006.

References

[1] Cohen, J. W. (1977) On up- and downcrossings. J. Appl. Prob. 14, 405–410.CrossRefGoogle Scholar
[2] Epstein, B. (1958) The exponential distribution and its role in life testing. Industrial Quality Control 15, 2–7.Google Scholar
[3] Esary, J. D. (1957) A Stochastic Theory of Accident Survival and Fatality. Ph.D. Dissertation, University of California, Berkeley.Google Scholar
[4] Esary, J. D., Marshall, A. W. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627–650.CrossRefGoogle Scholar
[5] Gaver, D. P. (1963) Random hazard in reliability problems. Technometrics 5, 211–226.CrossRefGoogle Scholar
[6] Harris, R. (1968) Reliability applications of a bivariate exponential distribution. Operat. Res. 16, 18–27.Google Scholar
[7] Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[8] Keilson, J. (1966) A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Statist. 37, 866–870.Google Scholar
[9] Keilson, J. (1979) Markov Chain Models — Rarity and Exponentiality. Springer-Verlag, New York.CrossRefGoogle Scholar
[10] Ross, S. M. (1981) Generalized Poisson shock models. Ann. Prob. 9, 896–898.CrossRefGoogle Scholar
[11] Serfozo, R. and Stidham, S. (1978) Semi-stationary clearing processes. Stoch. Proc. Appl. 6, 165–178.CrossRefGoogle Scholar
[12] Shanthikumar, J. G. (1980) Some analyses on the control of queues using level crossings of regenerative processes. J. Appl. Prob. 17, 814–821.Google Scholar
[13] Shanthikumar, J. G. (1981) On level crossing analysis of queues. Austral. J. Statist. 23, 337–342.Google Scholar
[14] Shanthikumar, J. G. and Sumita, U. (1982) General shock models associated with correlated renewal sequences. Working Paper Series 82–006, Department of Industrial Engineering and Operations Research, Syracuse University.Google Scholar
[15] Stidham, S. (1974) Stochastic clearing systems. Stoch. Proc. Appl. 2, 85–113.CrossRefGoogle Scholar
[16] Stidham, S. (1977) Cost models for stochastic clearing systems. Operat. Res. 25, 100–127.CrossRefGoogle Scholar
[17] Widder, D. V. (1946) The Laplace Transform. Princeton University Press, Princeton, NJ.Google Scholar