Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-15T01:29:02.682Z Has data issue: false hasContentIssue false
  • English
  • Français

New better than used in expectation processes

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

C. Y. Teresa Lam
C. Y. Teresa Lam*
Affiliation:
University of Michigan, Ann Arbor
*
∗ Postal address: Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI 48109, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) properties of Markov renewal processes. We show that a Markov renewal process belongs to a more general class of stochastic processes encountered in reliability or maintenance applications. We present sufficient conditions such that the first-passage times of these processes are new better than used in expectation. The results are applied to the study of shock and repair models, random repair time processes, inventory, and queueing models.

Keywords

NBUE AND NWUE PROCESSESINCREASING MARKOV RENEWAL PROCESSESFIRST-PASSAGE TIMESQUEUEINGRANDOM REPAIR TIME PROCESSESSHOCK AND REPAIR MODELSINVENTORY

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 116 - 128
Copyright
Copyright © Applied Probability Trust 1992 

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

A-Hameed, M. S. and Proschan, F. (1975) Shock models with underlying birth process. J. Appl. Prob. 12, 18–28.Google Scholar
Abdel-Hameed, M. (1984a) Life distribution properties of devices subject to a Levy wear process. Math. Operat. Res. 9, 606–614.Google Scholar
Abdel-Hameed, M. (1984b) Life distribution properties of devices subject to a pure jump damage process. J. Appl. Prob. 21, 816–825.Google Scholar
Block, H. W. and Savits, T. H. (1978) Shock models with NBUE survival. J. Appl. Prob. 15, 621–628.Google Scholar
Block, H. W. and Savits, T. H. (1981) Multidimensional IFRA processes. Ann. Prob. 9, 162–166.Google Scholar
Brown, M. and Chaganty, N. R. (1983) On the first passage time distribution for a class of Markov chains. Ann. Prob. 11, 1000–1008.Google Scholar
ÇlInlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
ÇlInlar, E. (1975a) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
ÇlInlar, E. (1975b) Markov renewal theory: A survey. Management Sci. 21, 727–752.Google Scholar
Derman, C., Ross, S. M. and Scheckner, Z. (1983) A note on first passage times in birth and death and non-negative diffusion processes. Naval Res. Logist. Quart. 30, 283–285.Google Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627–650.Google Scholar
Karasu, I. and Özekici, S. (1989) NBUE and NWUE properties of increasing Markov processes. J. Appl. Prob. 26, 827–834.Google Scholar
Keilson, J. (1979) Markov Chain Models – Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Marshall, A. M. and Shaked, M. (1979) Multivariate shock models for distributions with increasing hazard rate average. Ann. Prob. 7, 343–358.Google Scholar
Marshall, A. M. and Shaked, M. (1983) New better than used processes. Adv. Appl. Prob. 15, 601–615.Google Scholar
Marshall, A. M. and Shaked, M. (1986) NBU processes with general state space. Math. Operat. Res. 11, 95–109.Google Scholar
Ohi, F. and Nishida, T. (1983) Another proof of IFRA property of S. M. Ross' generalized Poisson shock models. Math. Japon. 28, 117–123.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1987) IFRA properties of some Markov jump processes with general state space. Math. Operat. Res. 12, 562–568.Google Scholar
Shanthikumar, J. G. (1984) Processes with new better than used first passage times. Adv. Appl. Prob. 16, 667–686.Google Scholar