Resource-Sharing Models with State-Dependent Arrivals of Batches

Abstract

We recently developed a new algorithm for calculating the blocking probability of each class in resource-sharing models with upper limit and guaranteed minimum sharing policies as well as the standard complete-sharing policy. These models may have multiple resources and multiple classes, with each class requiring multiple units from each resource. These models may also have state-dependent arrival and service rates. Our new algorithm is based on calculating normalization constants appearing in the product-form steady-state distributions by numerically inverting their generating functions. In the present paper we provide the basis for extending the algorithm to resource-sharing models with batch arrival processes. The batch sizes are mutually independent random variables with distributions depending on the class. We show that the steady-state distribution of the resource-sharing model has a product form for both complete-batch blocking and partial-batch blocking, and we derive the generating functions of the normalization constants for partial-batch blocking. We primarily focus on the Bernoulli-Poisson-Pascal (BPP) special case in which the batches have linear state-dependent arrival rates, which includes finite-source inputs and Poisson inputs for the batches as special cases. With batches, we require exponential service times, but when there are state-dependent arrivals of single customers (no batches), the service-time distributions can be general. By considering state-dependent arrivals for the batches, multiple resources and noncomplete-sharing policies, our treatment extends recent results for resource-sharing models with batch arrivals by van Doom and Panken, by Kaufman and Rege and by Morrison. Even for the batch models previously considered, our algorithm is faster than recursive algorithms when the model is large. We also provide a new derivation of the product-form steady-state distributions that helps explain why service-time insensitivity does not hold when there are batches.