Technical Note—Approximations to the Renewal Function m(t)

Abstract

Models of queuing, inventory, and reliability processes often have a useful renewal process imbedded in the fundamental stochastic process. The number of renewals is sufficient to determine performance measures such as the total cost, shortages, etc. The limit theorems of renewal theory are unsatisfactory in obtaining the expected values of these performance measures over a finite time horizon. This paper compares an accurate numerical technique for calculating the expected number of renewals with an approximation that uses the asymptotic expansion of the dominating residues of the Laplace transform M(θ). Furthermore, when a parameter of the renewal process is uncertain except for its Bayesian prior distribution, an approximation that uses a modified exponential renewal process appears better.