A vertex-allocation theorem for resources in queuing networks

A product-form queuing network with multiple open and closed chains is considered. Some of the closed chains, which have a single customer each, require allocation of resources in the network so as to maximize a weighted throughput performance criterion. Chains with more than one customer can be decomposed into many chains of one customer each. It is proved that an optimal allocation of resources lies on a vertex (extreme points) of the set of feasible allocations. This considerably reduces the search space for an optimal allocation. Applications of this result in distributed computing are discussed.