Blocking probabilities for multiple class batched Poisson arrivals to a shared resource

Abstract

A variant of the Erlang blocking model, in which multiple classes of customers arrive as batched Poisson processes, with arbitrary batch size distributions, is investigated. This model was recently considered by Kaufman and Rege, and they showed that for the partial batch blocking discipline the state distribution is of product form. In this paper the special case of complete sharing of the resource is considered. Asymptotic approximations to the blocking probability for each class of customer are derived, when the capacity of the resource and the traffic intensities are commensurately large. The results are obtained with the help of some contour integral representations, and the use of saddle-point techniques. The main result is a uniform asymptotic approximation to the blocking probability for each class of customer. This approximation is specialized in the overloaded, critically loaded and underloaded regimes. However, the uniform approximation holds across these regimes, where the blocking probabilities may vary by several orders of magnitude. The asymptotic approximations are compared numerically with some exact results, which have appeared in the literature, for the case of single (non-batched) Poisson arrivals. It is found that the uniform approximations are very accurate, and substantially more so than the approximations for the overloaded, critically loaded and underloaded regimes. Moreover, the accuracy of the uniform asymptotic approximations to the blocking probabilities is demonstrated numerically for three examples with geometric batch size distributions for each class.