A compound measure of dependability for systems modeled by continuous‐time absorbing Markov processes

The Markov analysis of reliability models frequently involves a partitioning of the state space into two or more subsets, each corresponding to a given degree of functionality of the system. A common partitioning is G ∪ B ∪ {o}, where G (good) and B (bad) stand, respectively, for fully and partially functional sets of system states; o denotes system failure. Visits to B may correspond to, for instance, reparable system downtimes, whereas o will stand for irrecoverable system failure. Let T G and N B stand, respectively, for the total time spent in G , and the number of visits to B , until system failure. Both T G and N B are familiar system performance measures with well‐known cumulative distribution functions. In this article a closed‐form expression is established for the probability Pr[ T G <> t , N B ≤ n ], a dependability measure with much intuitive appeal but which hitherto seems not to have been considered in the literature. It is based on a recent result on the joint distribution of sojourn times in subsets of the state space by a Markov process. The formula is explored numerically by the example of a power transmission reliability model. © 1996 John Wiley & Sons, Inc.