Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 4, Pages 357-370
Two service units with interference in the access to servers
1Universidad Carlos III de Madrid, Dpto. de Estadistica y Econometria, C/ Madrid 126, Getafe, Madrid 28903, Spain
2The University of Arizona, Dept. of Systems and Industrial Eng., Tucson 85721, AZ, USA
Received 1 December 1998; Revised 1 July 1999
Copyright © 1999 Rosa E. Lillo and Marcel F. Neuts. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 -formalism. These formulas
turn out to be unusually transparent and have probabilistic interpretations
that do not depend on the 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.