Duality and equivalencies in closed tandem queuing

Equivalence relations between closed tandem queueing networks are established. Four types of models are under consideration : single-server infinite-capacity buffer queues, infinite-server queues with resequencing, single-server unit-capacity buffer queues with blocking before service, and single-server unit-capacity buffer queues with blocking after service. Using a customer/server duality we show that in a network consisting of single-server infinite-capacity-buffer queues, customer-dependent service times and server-dependent service times yield equivalent performance characteristics. We further show that for closed tandem queueing networks, a system consisting of single-server infinite-capacity-buffer queues (res. infinite-server queues with resequencing) and a system consisting of single-server unit-capacity-buffer queues with blocking before service (res. blocking after service) have equivalent performance behaviors. As applications of these equivalence properties, we obtain new results on the analysis of symmetric closed tandem networks, where all the service times are independent and identically distributed random variables. In particular, we obtain a closed-form expression for the throughput of networks with unit-capacity-buffer queues and blocking before service when the service times are exponentially distributed. We also prove the monotonicity of throughput (of queues) with respect to the number of queues and number of customers in these models. This last property in turn implies the existence of nonzero asymptotic throughput when the number of queues and number of customers go to infinity