Closed Exponential Networks of Queues with Saturation: The Jackson-Type Stationary Distribution and Its Asymptotic Analysis

Two models of a closed queueing network with saturation are proposed. Given that the “reversibility” condition holds in the first model, the stationary distribution is shown to be of product-form for either of them. Queueing networks with the space of admissible states generated by limited capacities of servers is the most important special variant of the general scheme. The asymptotic analysis of the stationary distribution in the case of a large number of customers is given and shows that a special nonlinear programming problem must be solved to obtain parameters of the limiting distribution. In particular, saturation probabilities are asymptotically expressed through the Lagrangian multipliers dual to corresponding linear restrictions of queue-sizes for servers with limited capacities.