Simple product-form bounds for queueing networks with finite clusters

Queueing networks are studied with finite capacity constraints for clusters of stations. First, by an instructive tandem cluster example it is shown how a product-form modification method for networks with finite stations can be extended to networks with finite clusters. Next, a general result is established by which networks with finite clusters can be studied at cluster level by merely keeping track of the total number of jobs at these clusters, that is, by regarding these clusters as aggregate stations. This result is of practical interest to conclude simple performance bounds at global network level. A number of illustrative examples with numerical support are provided.