ON MEASURE OF SYSTEM INTERVAL AVAILABILITY

Abstract

Let [script S] be a system that can be in one of two states, up or down. We can interpret the up state as the working state and the down state as the nonworking state for system [script S]. Assume the system [script S] is up at time t = 0. Denote the elapsed time from zero until [script S] enters down state by U₁, and the elapsed time from then until [script S] is up again by D₁. The interval [0,U₁ + D₁) is called cycle 1, and, at that point, cycle 2 starts and the sequence is repeated so on and so forth. In general, the up time and down time in the jth cycle are denoted by U_j and D_j, respectively, for j ≥ 1. We can introduce a binary process {X(t), t ≥ 0} to describe the state of [script S] at time t using X(t) = 1(0) to indicate [script S] is up (down). Many problems require finding the probability that [script S] is up at time t, that is, P(X(t) = 1). However, an even more interesting question to answer is what the probability is that [script S] is in the up state in an interval of length w starting at time t, that is, P(X(s) = 1, t ≤ s ≤ t + w). Due to the complexity of this problem, no explicit expressions are available for most systems even in the case where (U_j, D_j), j ≥ 1 are i.i.d. Fortunately, in practice engineers are more interested in the long-run properties of the above two probabilities.