Blocking in a shared resource environment with batched Poisson arrival processes

We study a variant of the multi-dimensional Erlang blocking model in which customer arrivals require multiple servers and are modelled as batched Poisson processes with arbitrary batch size distribution. We show that the state distribution for the partial batch blocking discipline is a product form, and for the special case of the complete sharing policy, we obtain a one-dimensional recursion for the aggregate state distribution. When the batch size distribution is geometric, our recursion is identical to a recursion due to Delbrouck. This correspondence provides a new interpretation for Delbrouckʹs recursion, which was introduced to model peaked stream blocking. We show that this correspondence is due to an equivalence between a batched Poisson blocking model with geometrically distributed batches and a blocking model with a linear state dependent arrival process.