A dependability measure for Markov models of repairable systems: Solution by randomization and computational experience

Abstract Irreducible, continuous-time Markov models for reliability analysis are considered whose finite state space is partitioned as G ∪ B , where G and B stand for the set of system up (‘good’) and down (‘bad’) states, respectively. For a fixed length of time t 0 > 0, let T G ( t 0 ) and N B ( t 0 ) stand, respectively, for the total time spent in G and the number of visits to B during [0, t 0 ]. The dependability measure considered here is P ( T G ( t 0 ) > t , N B ( t 0 ) ≤ n ), i.e., the probability that during [0, t 0 ] the cumulative system up-time exceeds t ( t 0 ) and the system does not suffer more than n failures. Using the randomization technique and some recent tools from the theory of sojourn times in finite Markov chains, a closed form expression is obtained for this dependability measure. The scope of the practical computational utility of this analytical result is explored via its MatLab implementation for the Markov model of a system comprising two parallel units and a single repairman.

[1]  Masaaki Kijima Some Results for Uniformizable Semi-Markov Processes , 1987 .

[2]  G. Zyskind Introduction to Matrices with Applications in Statistics , 1970 .

[3]  Donald Gross,et al.  The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes , 1984, Oper. Res..

[4]  A. Csenki The joint distribution of sojourn times in finite semi-Markov processes , 1991 .

[5]  Attila Csenki The number of working periods of a repairable Markov system during a finite time interval , 1994 .

[6]  Edmundo de Souza e Silva,et al.  Calculating Cumulative Operational Time Distributions of Repairable Computer Systems , 1986, IEEE Transactions on Computers.

[7]  Attila Csenki The joint distribution of sojourn times in finite Markov processes , 1992, Advances in Applied Probability.

[8]  Raymond A. Marie,et al.  The uniformized power method for transient solutions of Markov processes , 1993, Comput. Oper. Res..

[9]  Gerardo Rubino,et al.  Interval Availability Analysis Using Operational Periods , 1992, Perform. Evaluation.

[10]  Edmundo de Souza e Silva,et al.  Calculating availability and performability measures of repairable computer systems using randomization , 1989, JACM.

[11]  Micha Yadin,et al.  Randomization Procedures in the Computation of Cumulative-Time Distributions over Discrete State Markov Processes , 1984, Oper. Res..

[12]  B. Sericola Closed form solution for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period , 1990 .

[13]  Derek Ray,et al.  A Primer of Reliability Theory , 1990 .

[14]  Attila Csenki Some new aspects of the transient analysis of discrete-parameter Markov models with an application to the evaluation of repair events in power transmission , 1991 .

[15]  Donald Gross,et al.  Multiechelon repairable‐item provisioning in a time‐varying environment using the randomization technique , 1984 .