A semi-product-form for a pair of queues with finite batches: equilibrium state probabilities and response time densities

A Markovian network of two queues, with finite size batch Poisson arrivals and departures, is solved approximately, but to arbitrary accuracy, for its equilibrium state probabilities. Below a pair of thresholds on the queue lengths, a modification of the Spectral Expansion Method is used to construct a semi-product-form at all lengths of one queue in a finite lattice strip defined by the threshold of the other queue. No additional special arrival streams are required, for example at empty queues, from which it is already known that a product-form can be constructed. Hence the first exact closed form solution for the equilibrium probabilities in an unmodified Markovian queueing network with batches is obtained, the only constraint being finiteness of the batches. The method is illustrated numerically, first in a tandem network and then in a two-node network with feedback. Simulation results confirm convincingly the precision of the method, and partial batch forwarding and discarding are thereby compared quantitatively. Response time distributions are derived using the generating function method, which complements well the semi-product form of the equilibrium state probabilities. Again, agreement with simulation is excellent. The regenerative simulation method was used, so that no warm-up period was needed, and the statistical estimates for the 95% confidence bands are explained for the different cases of state occupancy and response time prediction at equilibrium