On the first time a separately maintained parallel system has been down for a fixed time

Abstract

AbstractConsider a system consisting of n separately maintained independent components where the components alternate between intervals in which they are “up” and in which they are “down”. When the ith component goes up [down] then, independent of the past, it remains up [down] for a random length of time, having distribution Fi[Gi], and then goes down [up]. We say that component i is failed at time t if it has been “down” at all time points s ϵ[t‐A.t]: otherwise it is said to be working. Thus, a component is failed if it is down and has been down for the previous A time units. Assuming that all components initially start “up,” let T denote the first time they are all failed, at which point we say the system is failed. We obtain the moment‐generating function of T when n = l, for general F and G, thus generalizing previous results which assumed that at least one of these distributions be exponential. In addition, we present a condition under which T is an NBU (new better than used) random variable. Finally we assume that all the up and down distributions Fi and Gi i = l,….n, are exponential, and we obtain an exact expression for E(T) for general n; in addition we obtain bounds for all higher moments of T by showing that T is NBU.