A Method for Predicting System Downtime

A system whose components, upon failure, are repaired or replaced is considered. Only two system states, the ``operating'' state and the ``failed'' state, are distinguished. The system is by defined a reliability network and by the failure rate and repair rate of each component. The time to failure and the time to repair of the components are assumed to be exponentially distributed. A criterion of system worth is the random variable ``downtime,'' denoted by D(t), which is defined as the time the system is down during the time interval (0, t). The following questions are answered: 1) What is the distribution function of D(t)? 2) What are the mean and the variance of D(t)? 3) What is the asymptotic behavior of D(t) for large values of t? 4) How can one make approximate probability statements about D(t)? It is shown that the beta distribution is a suitable approximation for the conditional distribution of D(t)/t, given that at least one failure has occurred, and that for t greater than 20 mean failure times the distribution of D(t) is practically normal.