On the joint right-tail distribution of the forward and backward recurrence times in a delayed renewal process

In this paper, we study the joint distribution of the forward and backward recurrence times in a delayed renewal process, as well as their marginal distributions. We obtain several exact results and bounds for these quantities. Some of these bounds are “general,” in the sense that the bounds are valid for any arbitrary distributions of the inter-arrival times, and some are based on aging properties of the distributions of the interarrival times of the renewals. Finally, several numerical examples are presented to illustrate the results.