Summary: MATHEMATICS OF OPERATIONS RESEARCH
Vol. 25, No. 4, November 2000, pp. 708725
Printed in U.S.A.
PERIODIC ORBITS IN A CLASS OF RE-ENTRANT
IVONNE DIAZ-RIVERA, DIETER ARMBRUSTER, AND TOM TAYLOR
Queue changes associated with each step of a manufacturing system are modeled by constant
vector ˙elds ( uid model of a queueing network). Observing these vector ˙elds at ˙xed events
reduces them to a set of piecewise linear maps. It is proved that these maps show only periodic
or eventually periodic orbits. An algorithm to determine the period of the orbits is presented. The
dependence of the period on the processing rates is shown for a 3(4)-step, 2-machine problem.
1.1. Re-entrant manufacturing systems. A re-entrant manufacturing system (Ku-
mar 1993) is a queueing network modeling a manufacturing process in which parts at
di erent stages of completion await processing by the same machine. Di erent choices of
scheduling policies at such a machine lead to di erent production e ciencies.
In manufacturing processes, such as the production of semiconductor products, there is
also a requirement for release policies. These policies determine how much raw material
needs to be added into the process and when it should be done. An example of a release
policy is one in which a new job is added into the system at the time a job is completely