Exact computation of time and call blocking probabilities in large, multi-traffic, multi-resource loss systems

We consider a model in which discrete resources of finite capacity are completely shared amongst customers pertaining to K different classes. For each class, customers arrive according to a Bernoulli-Poisson-Pascal (BPP) process. The amount of resource-units each customer requests and the average holding time may be different for each customer class. A customer whose request cannot be satisfied is blocked, and its request is lost. This model has a particular relevance for teletraffic theory of multi-rate, multi-cast, circuit-switching networks. The joint probability distribution function (PDF) of the number of occupied resource-units, is found using a well-known recursion formula. However, the formula suffers from numerical instability in certain situations. The first contribution of this paper is an algorithm to work around the stability problems to make it applicable in practice. Previous articles on the subject concentrated on using the occupancy PDF for finding time blocking probabilities. Those that deal with call blocking probabilities suggest a method that is either approximative or adds considerable computation time. The second contribution is a recursion that calculates these probabilities with a negligible computational overhead and no additional storage requirements. Finally, the call blocking recursion is compared to other, approximative call blocking algorithms with regard to accuracy, computation time and storage requirements.