On the exact distribution of a delayed renewal process with exponential sum interarrival times

The paper develops the exact probability functions of a delayed renewal process whose interarrival times are sums of two independent exponential random variables with likely unequal parameters. The initial arrival time is distributed as one of these exponential variables. The impetus for this study comes from the electricity distribution industry whose circuit breaker breakdowns go unnoticed and come to light only at subsequent blackouts. By assuming that a circuit breaker’s life time is exponential and interarrival times of blackouts are also exponential with a likely different parameter, that all variables are independent of each other, and that the circuit breaker is operative at the beginning of the observation period, the present model arises for counting the number of said breakdowns during a given time period. The result is obtained and proved through a recursive application of Laplace transform. Moments of the distribution, and its numerical delineation are given, and an extension to a regular renewal process which counts breakdowns’ detections is also discussed.