On computations of the mean and variance of the number of renewals: A unified approach

This paper presents a unified approach to finding both the mean (the so-called renewal function) and variance of the number of renewals in continuous time. Basically, it deals with three mutually exclusive situations depending on whether the inter-renewal time distribution has a closed-form (i) rational Laplace-Stieltjes transform (L-ST); (ii) irrational L-ST or (iii) it cannot be represented by a closed-form L-ST. Explicit closed-form expressions are obtained (in terms of roots) for both the mean and the variance. Asymptotic expressions in terms of roots are also obtained for large t. Numerical results are discussed for a variety of inter-renewal time distributions and their accuracies have been checked against the well-known asymptotic results and also against other numerical results available in the literature. The method discussed here can be employed for a variety of other problems occurring in areas such as queueing, inventory and reliability. The real strength of the method lies in the fact that it gives excellent results particularly in cases (i) and (ii) mentioned above.