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Repairable models with operating and repair times governed by phase type distributions

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Marcel F. Neuts ,
Rafael Pérez-Ocón  and
Inmaculada Torres-Castro
Marcel F. Neuts*
Affiliation:
The University of Arizona
Rafael Pérez-Ocón*
Affiliation:
Universidad de Granada
Inmaculada Torres-Castro*
Affiliation:
Universidad de Extremadura
*
∗ Postal address: Department of Systems and Industrial Engineering, The University of Arizona, Tucson, Arizona 85721, USA. Email address: marcel@sie.arizona.edu
∗∗ Postal address: Departamento de Estadística e Investigaciónes Operationales, Universidad de Granada, Campus de Fuente Nueva s/n, E-18071 Granada, Spain. Email address: rperezo@goliat.ugr.es
∗∗∗ Postal address: Escuela Politécnica, Departmento de Matemáticas, Avenida de la Universidad, s/n, Universidad de Extremadura, 10071 Cáceres, Spain. Email address: inmatorres@unex.es
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Abstract

We consider a device that is subject to three types of failures: repairable, non-repairable and failures due to wear-out. This last type is also non-repairable. The times when the system is operative or being repaired follow phase type distributions. When a repairable failure occurs, the operating time of the device decreases, in that the lifetimes between failures are stochastically decreasing according to a geometric process. Following a non-repairable failure or after a previously fixed number of repairs occurs, the device is replaced by a new one. Under these conditions, the functioning of the device can be modelled by a Markov process. We consider two different models depending on whether or not the phase of the operational system at the instants of failure is remembered or not. For both models we derive the stationary distribution of the Markov process, the availability of the device, the rate of occurrence of the different types of failures, and certain quantities of interest.

Keywords

Phase type distributionrate of occurrence of failuresMarkov processesreplacement

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 468 - 479
Copyright
Copyright © Applied Probability Trust 2000 

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