A multiple system governed by a quasi-birth-and-death process

The system we consider comprises n units, of which one has to operate for the system to work. The other units are in repair, in cold standby, or waiting for repair. Only the working unit can fail. The operational and repair times follow phase-type distributions. Upon failure, it is replaced by a standby unit and goes to the repair facility. There is only one repairman. When one unit operates the system is up and when all the units are in repair or waiting for repair, the system is down. This system is governed by a finite quasi-birth-and-death process. The stationary probability vector and useful performance measures in reliability, such as the availability and the rate of occurrence of failures are explicitly calculated. This model extends other previously considered in the literature. The case with an infinite number of units in cold standby is also studied. Computational implementation of the results is performed via a numerical example, and the different systems considered are compared from the reliability measures determined.

[1]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[2]  Rommert Dekker,et al.  Preventive maintenance in a 1 out of n system: The uptime, downtime and costs , 1997 .

[3]  Zoltan Papp,et al.  Probabilistic reliability engineering , 1995 .

[4]  R. Natarajan,et al.  A study of an n-unit system operating in a random environment , 1999 .

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

[6]  Vaidyanathan Ramaswami,et al.  A logarithmic reduction algorithm for quasi-birth-death processes , 1993, Journal of Applied Probability.

[7]  Necip Doganaksoy,et al.  Handbook of Reliability Engineering , 1994 .

[8]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

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

[10]  Enrico Zio,et al.  Modeling Components' Degradation Processes by Monte Carlo Simulation , 2001 .

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

[12]  Tae-Jin Lim,et al.  Analysis of system reliability with dependent repair modes , 2000, IEEE Trans. Reliab..

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

[14]  Marvin Rausand,et al.  System Reliability Theory , 2020, Wiley Series in Probability and Statistics.

[15]  R. Yeh.  Optimal inspection and replacement policies for multi-state deteriorating systems , 1997 .

[16]  M. Neuts,et al.  On the use of phase type distributions in reliability modelling of systems with two components , 1981 .

[17]  Bryan L. Deuermeyer,et al.  Utilization of the method of linear matrix equations to solve a quasi-birth-death problem , 1993, Journal of Applied Probability.

[18]  M. Neuts,et al.  Repairable models with operating and repair times governed by phase type distributions , 2000, Advances in Applied Probability.

[19]  A. Birolini.  Quality and reliability of technical systems , 1994 .

[20]  Ruey Huei Yeh.  State-age-dependent maintenance policies for deteriorating systems with Erlang sojourn time distributions , 1997 .

[21]  S. Chakravarthy.  Reliability analysis of a parallel system with exponential life times and phase type repairs , 1983 .

[22]  N. Ravichandran.  Stochastic Methods in Reliability Theory , 1991 .

[23]  Ilya B. Gertsbakh,et al.  Models of Preventive Maintenance , 1977 .

[24]  Vidyadhar G. Kulkarni.  Introduction to matrix analytic methods in stochastic modeling, by G. Latouche and V. Ramaswamy , 1999 .

[25]  I. Gertsbakh.  Reliability Theory with Applications to Preventive Maintenance , 2000 .

[26]  Rommert Dekker,et al.  Applications of maintenance optimization models : a review and analysis , 1996 .