Bounds for the distribution and moments of the forward and backward recurrence times in a renewal process

Abstract

We obtain a variety of lower and upper bounds for the bivariate tail of the joint distribution between the (forward and backward) recurrence times in a renewal process. Some of these bounds are valid generally, some others in the case where the interarrival distribution $F$ exhibits monotone aging. We also give various bounds for the moments of the recurrence times. Our results improve upon existing bounds for these quantities in the literature. We also discuss the case where the aging of $F$ is non-monotonic. Finally, we present numerical examples to illustrate the performance of the new bounds.