New Approximations for the Event Count Distribution for Superimposed Renewal Processes at the Time Origin with Application to the Reliability of New Series Systems

When n independent identical renewal processes are superimposed, the number of events in (0, s ) is approximately Poisson distributed for large n . For small to moderate n , this approximation is inaccurate. If no component process has more than one event in (0, s ), the probability of exactly r events for the superimposed process is given by a binomial-like expression where the p and q do not sum to one. Several approximations have been derived from this observation and compared to the exact probability for small n , when the underlying distribution of time between events is gamma. One of these binomial approximations gives excellent results for small n . The following application is discussed. When k new series systems, each consisting of m identical components, are tested for time s and failed components are replaced upon failure, a superimposed renewal process results. To design an acceptance sampling scheme for new series systems using exact probability computations is cumbersome. The Poisson approximation is convenient, but may give misleading estimates of the O.C. curve for small n . The new binomial approximation is a compromise: It is easier to use for test design than the exact computation and more accurate than the simpler Poisson approximation.