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Interval availability, defined as the fraction of time that a system is operational during a period of time $[ {0,T} ]$ , is an important indicator of system performance, especially for industries with a Service Level Agreement, e.g., telecommunication industry, computer industry, etc. Most existing methods to compute interval availability are based on numerical simulations. In this paper, we present a new analytical solution for interval availability of Markov repairable systems. Three interval availability indexes, i.e., interval availability, interval availability in a general interval, and interval availability in multiple intervals, are considered. The interval availability indexes are derived based on aggregated stochastic processes and the results are obtained in closed form using Laplace transforms. A numerical example is presented and the results are compared with those of Monte Carlo simulation. The developed methods are applied to calculate the interval availability of a fault-tolerant database system from the literature.

Here we present the structure functions for the linear and circular $m$ -consecutive- $k$ , $l$ -out-of- $n$ :F (G) systems, which consists of $n$ linearly or circularly ordered components such that the system fails (works) iff there are at least $m$ times $l$ -overlapping runs of consecutive $k$ failed (working) components. The duality of these F and G systems is obtained, and the proofs are given. The generalized reliability formulae for the systems are obtained by using the finite Markov chain imbedding approach for statistically independent and Markov-dependent cases. Some miscellaneous results on the reliability of these systems and numerical examples are provided to illustrate the results obtained in the paper.

Since Birnbaum importance was introduced in 1969, there have been more than twenty kinds of importance measures so far. Among the various measures, Birnbaum importance plays an extremely important role because many importance measures have been defined under its illumination and have relationships with it. A lot of work has been done for Birnbaum importance in consecutive- $k$ systems since the systems were introduced. Because the problems in practice are increasingly complicated, in 2007, Zhao proposed consecutive- $k$ systems with sparse $d$ , which is an extension of the current consecutive- $k$ systems. In this paper, we study Birnbaum importance for linear consecutive- $k$ -out-of- $n$ systems with sparse $d$ . Some equations on Birnbaum importance are proposed. With these equations, the ranking of components in the system on the basis of Birnbaum importance is given; and then some patterns of ranking are presented. Finally, two numerical examples are given to illustrate the results obtained in this paper.

Availability is an important measure of performance for repairable systems because it can describe the probability of a repairable system being in working states. Single point and interval availability measurements (availabilities) have been widely used, but they cannot meet the requirements of all applications. In this paper, multi-point, multi-interval, and mixed multi-point-interval availabilities are introduced, which are extensions of the single point and interval availabilities. The multi-interval availability formula was given by Csenki in 2007 by using the induction method in which the multi-interval availability was called the joint interval reliability, but a new simple technique is provided to present the formula of multi-interval availability in this paper. We also consider the steady-state situations for all new availabilities. For Markov repairable systems, the calculation formulae are presented for the new availabilities, and some properties of new availabilities are discussed too. Finally, some special cases and numerical examples are given to illustrate the results obtained in the paper.