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 GℝBℝ{ω}, where G (good) and B (bad) stand, respectively, for fully and partially functional sets of system states; ω denotes system failure. Visits to B may correspond to, for instance, reparable system downtimes, whereas ω will stand for irrecoverable system failure. Let TG and NB stand, respectively, for the total time spent in G, and the number of visits to B, until system failure. Both TG and NB are familiar system performance measures with well-known cumulative distribution functions. In this article a closed-form expression is established for the probability Pr[TG>t, NB•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.
