Approximations in Renewal Theory

Abstract

A renewal process [N(t), t ≥ 0] with interarrival times Xi, for i ≥ 1 and renewal function m (t) is considered. Let Gn, λ denote the gamma distribution with parameters n and λ–that is, dGn, λ(x) = λε–λx(λx)n-1/(n – 1)In Section 1 we show how m(t) can be approximated by f m(s) dGn, n/t(s). In addition, we show that these approximations constitute an increasing sequence of lower bounds when the interarrival distribution has the decreasing failure rate property. In Section 2 we show how the integrated renewal function can be approximated in a similar fashion by a decreasing sequence of upper bounds. In Section 3 we consider the problem of approximating the residual life (also called excess life) and the renewal age distribution and their means, and in Section 4 we consider the distribution of N(t). Finally, in Section 5 we remark on the relationship between our approximations and the Feller technique for inverting a Laplace transform.