Two service units with interference in the access to servers

Abstract

We examine the service mechanism of two queueing models with two units in tandem. In the first model, customers who complete service in Unit 1 must wait in an intermediate buffer until the ongoing service in Unit II ends. In the second model, jobs can be pre-positioned in an intermediate buffer to await service in Unit II. Under the assumption of phase-type service times, the steady-state regime of the service system is studied in detail.The models are inspired by the gas pump model of A.B. Clarke and by phenomena observed in cafeteria lines and certain manufacturing systems. However, their primary interest may lie in the methodology of their exceptionally tractable analysis. We derive formulas for the throughput and other quantities by using the familiar PH-formalism. These formulas turn out to be unusually transparent and have probabilistic interpretations that do not depend on the PH assumptions. These interpretations therefore also hold for general service time distributions. The methodology is general and can be applied to other systems with interactions between servers. The models also present interesting algorithmic problems of didactic interest.