On equilibrium probabilities in a class of two station closed queueing networks

Due to the finite state space of closed exponential queueing networks, one can obtain the equilibrium probability distribution by directly solving the global balance equations. However, as the number of trapped customers increases, the state space grows and it becomes practically impossible to obtain a solution. Alternatives include performance bounds, approximations and simulations, however these do not completely characterize the steady state behavior. By focusing our attention on a class of two station closed reentrant queueing networks, we obtain closed form expressions for the equilibrium probability distribution that are computationally independent of the number of trapped customers. The computational complexity depends only on the network structure. By considering the last buffer first served (LBFS) policy, we can reduce the state space to a three dimension rectangle whose height increases with the number of trapped customers. By recognizing a sense of causality in the balance equations, we are able to employ z-transform techniques to obtain an explicit solution for the equilibrium probabilities. Several examples, including the closed Lu-Kumar network under LBFS are studied to demonstrate the approach. The networks identified represent the only class of non-product form queueing networks which, to our knowledge, possess an explicit equilibrium probability distribution.