Re-direction in queueing networks with two customer types: The inter-departure analysis

Abstract

Re-direction occurs when a customer arriving at a station in a queuing network has to be re-directed to a downstream station to complete service. Re-direction is extremely common in practice and occurs for a variety of reasons, ranging from incorrect initial station assignment to cases where the initial station only provides part of the service. Gatekeeper stations (e.g., information desks) is a special case of re-direction. We consider re-direction in a queueing network consisting of single-server stations serving two customer types with different service time requirements. The behavior of such queueing networks is quite complex: even when all external arrivals and all services are Markovian, the customers’ inter-departure distribution, and hence their arrival process to downstream stations, is non-Markovian. Thus, product-form representation does not hold for such networks. Our analysis focuses on the key building block: the inter-departure process from a station serving two distinct customer types and routing them to two different downstream service paths. Using a novel approach, we obtain a very accurate phase-type representation of the inter-departure process under equilibrium. We show that the resulting methodology has significant advantages over both simulation modeling (our method is much faster) and the available approximation techniques (our method is more accurate). Finally, we demonstrate an interesting phenomenon: even when the station merely re-directs one of the customer types (providing no service and seemingly useless waits), it can serve as a “regulator”, reducing the variability of the downstream arrival process. We show that, under some conditions, this can improve the overall system performance.