Some bounds for the renewal function and the variance of the renewal process

Renewal equations are frequently encountered in several applications when regenerative arguments are used in modelling. Since, these equations usually do not have analytical so-lutions, bounds have a great practical importance. The aim of this paper is to present some new bounds for the renewal function and the variance of the renewal process. A general lower bound for the renewal function is given, which is a refinement of all known lower bounds and attains the well-known asymptotic result of Asmussen (2003). Also, some new tighter upper and lower bounds for the renewal function, which are based on several re-liability classifications of the distribution of the inter-arrival times, are given. Improved upper and lower bounds for the variance of the renewal process are also presented. Fi-nally, several numerical examples are given to illustrate the effectiveness of the proposed new bounds.(c) 2022 Elsevier Inc. All rights reserved.