Computing the Distribution for the Number of Renewals with Bulk Arrivals

The distribution of the number of renewals for bulk arrivals in continuous time is calculated using an algorithm employed through MAPLE software. These numerical results are acquired by considering rational as well as nonrational Laplace transforms and Padé-approximated Laplace transforms for the distributions of interrenewal times. Further, through the use of Laplace transforms an elegant solution to determine the asymptotic results for the first and second moments of the number of bulk renewals is presented. These derivations help validate the numerical results and are an extension of previous work by the authors regarding single-arrival renewal theory in discrete time.