Modelling a reliability system governed by discrete phase-type distributions

We present an n-system with one online unit and the others in cold standby. There is a repairman. When the online fails it goes to repair, and instantaneously a standby unit becomes the online one. The operational and repair times follow discrete phase-type distributions. Given that any discrete distribution defined on the positive integers is a discrete phase-type distribution, the system can be considered a general one. A model with unlimited number of units is considered for approximating a system with a great number of units. We show that the process that governs the system is a quasi-birth-and-death process. For this system, performance reliability measures; the up and down periods, and the involved costs are calculated in a matrix and algorithmic form. We show that the discrete case is not a trivial case of the continuous one. The results given in this paper have been implemented computationally with Matlab.

[1]  Nikolaos Limnios,et al.  ASYMPTOTIC PROPERTIES FOR MAXIMUM LIKELIHOOD ESTIMATORS FOR RELIABILITY AND FAILURE RATES OF MARKOV CHAINS , 2002 .

[2]  A. W. Marshall,et al.  Properties of Probability Distributions with Monotone Hazard Rate , 1963 .

[3]  Juan Eloy Ruiz-Castro,et al.  Transient Analysis of a Multi-Component System Modeled by a General Markov Process , 2006, Asia Pac. J. Oper. Res..

[4]  R. P. Gupta,et al.  Classifications of discrete lives , 1992 .

[5]  Valeri Naoumov.  Matrix-Multiplicative Approach to Quasi-Birth-and-Death Processes Analysis , 1996 .

[6]  M. Marseguerra,et al.  Simulation modelling of repairable multi-component deteriorating systems for 'on condition' maintenance optimisation , 2002, Reliab. Eng. Syst. Saf..

[7]  Attahiru Sule Alfa,et al.  Discrete time analysis of a repairable machine , 2002, Journal of Applied Probability.

[8]  Juan Eloy Ruiz-Castro,et al.  Two models for a repairable two-system with phase-type sojourn time distributions , 2004, Reliab. Eng. Syst. Saf..

[9]  Rafael Pérez-Ocón,et al.  A multiple system governed by a quasi-birth-and-death process , 2004, Reliab. Eng. Syst. Saf..

[10]  Srinivas R. Chakravarthy,et al.  Matrix-Analytic Methods in Stochastic Models , 1996 .

[11]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[12]  Leandro Pardo,et al.  Statistical inference for finite Markov chains based on divergences , 1999 .

[13]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .