A conservation law for single-server queues and its applications

Abstract

We establish a conservation law forG/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in aG/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in aGI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

Extremal properties of the FIFO discipline in queueing networks

“…Considerable attention has been paid to finding optimal 968 RHONDA RIGHTER AND J. GEORGE SHANTHIKUMAR service disciplines for such models, although most work has been limited to singlestage systems. See for example Chang and Yao (1990), Halfin and Whitt (1989), Hirayama and Kijima (1989), Righter and Shanthikumar (1989), Righter et al (1990), Shanthikumar and Sumita (1987), Yamazaki and Sakasegawa (1987), and Yamazaki et al (1991). Also see Towsley and Baccelli (1991) for a tandem queue example.…”