On the Distribution and Prediction of Excess Time
This paper considers a system whose components, upon failure, are repaired or replaced. Only two system states, the “operating” state and the “failed” state are distinguished. The time to failure and the time to repair of the system are assumed to be random variables with general distribution functions. A criterion of system worth is the random variable “excess time,” denoted by B(t), and defined as the total time the system is down for t units of time spent in the operating state. The following questions are answered in this paper: (a) What is the distribution function of B(t)? (b) What are the moments of B(t)? (c) What is the asymptotic behavior of B(t) for large t? (d) How can one make approximate probability statements about B(t)? It is shown that the gamma distribution is a suitable approximation for the conditional distribution of B(t), given that at least one failure has occurred, and that for t greater than 20 mean failure times the distribution of B(t) is practically normal.