Sojourn times in Markov processes for power transmission dependability assessment with MatLab

Abstract

In Markov reliability analysis, systems which ultimately suffer irrecoverable failure are modelled by absorbing chains whereas repairable systems are modelled by irreducible chains. In the former case, the state space is partitioned as P = U ∪ D ∪ { ω} where U and D denote the sets of system ‘up’ and ‘down’ states respectively and ω stands for ‘system failure’. For repairable systems, we have P = U ∪ D and the system alternates between the states in U and D idefinitely. It is customary to describe the transient behaviour of such systems by quantities such as reliability and point and interval availability. Two recent papers by Rubino and Sericola [J. Appl. Prob., 26, 1989, pp 744–756, and Dependable Computing and Fault-Tolerant Systems, 4, 1991, pp 239–254] have provided an alternative framework to characterise the behaviour of such systems via the sequence of sojourn times T U,1, T U,2, … in the set of operational states U . In a recent paper [Stoch. Proc. Appl., 39, 1991, pp 287–299], the author established a closed-form expression for the Laplace transform of the joint distribution of any finite collection of sojourn times in U and D under the semi-Markov assumption. This representation is used here to establish three results for continuous-parameter Markov chains. For absorbing chains, a closed-form expression is obtained for the cdf of the maximum duration of all working periods until final breakdown, i.e., max{T U,1, T U,2, … }. In addition to this new result, we also provide a new unified derivation of two known results in reliability analysis: the cdf of the l -th cumulative sojourn time in U , i.e., T U,1 + … + T U, l and for the absorbing model, the cdf of the total up time until system failure, i.e., T U,1 + T U,2 + … . The Macintosh version of the MatLab matrix computation package is a suitable tool for the concise implementation of the formulae. We demonstrate its relevant features by studying the transient behaviour of the Markov model of a three-unit parallel power transmission system. Future research directions are also indicated.