Abstract
The method of Laplace transforms is used to find the distribution function, mean, and variance of the number of renewals of a renewal process whose inter-arrival time distribution has a rational Laplace transform. Where the Laplace transform is not rational, we use the Pade approximation method. We apply our method to certain examples and the results are compared to those reported by other researchers.
Highlights
Renewal and reliability theories are powerful modeling tools in many applications, considering, for example, a series of renewals during a time interval 0,t with the inter-renewal times having certain distributions
The theoretical aspects of renewal theory have been discussed in various books such as Cox [1] and Feller [2,3], not much seem to have been done in applying the theoretical results to practice for lack of availability of computable results
Using Laplace transforms, it is shown that computing the distribution of the number of renewals is straightforward when the LT of the inter-renewal time distribution is rational
Summary
Renewal and reliability theories are powerful modeling tools in many applications, considering, for example, a series of renewals during a time interval 0,t with the inter-renewal times having certain distributions. Using the cubic splining algorithm to compute the recursively-defined convolution integrals that appear in renewal theory, Baxter et al [4] is able to construct some tables for the mean and variance of the number of renewals for different inter-renewal time distributions with varying parameters. Using the root-finding method, Chaudhry [5] develops a unified method to compute the mean and variance of the number of renewals. Though the means and variances are useful for many applications, this paper goes a step further and deals with computing the distribution of the number of renewals from which one can get more information than from the mean and variance. Using the roots method, the results can be first given in an analytically explicit form and used to find the final results
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