Waiting and sojourn times in a multi-server queue with mixed priorities

Abstract

We consider a multi-server queue with K priority classes. In this system, customers of the P highest priorities (P < K) can preempt customers with lower priorities, ejecting them from service and sending them back into the queue. Service times are assumed exponential with the same mean for all classes.The Laplace-Stieltjes transforms of waiting times are calculated explicitly and the Laplace-Stieltjes transforms of sojourn times are provided in an implicit form via a system of functional equations. In both cases, moments of any order can be easily calculated. Specifically, we provide formulae for the steady state means and the second moments of waiting times for all priority classes. We also study some approximations of sojourn-time distributions via their moments. In a practical part of our paper, we discuss the use of mixed priorities for different types of Service Level Agreements, including an example based on a real scheduling problem of IT support teams.

Unified and refined analysis of the response time and waiting time in the M/M/<i>m</i> FCFS preemptive-resume priority queue

“…Thus we may call our system an "M/M/m preemptive-resume priority queue with FCFS and LCFD within the same class." The study of response time and waiting time of customers in this model dates back to old days including Brosh [1], Segal [7], Buzen and Bondi [2], Tatashev [10], and Zeltyn et al [13]. The purpose of this paper is to review and refine these works from a unified viewpoint of the first passage time in a birth-and-death process with absorbing states [8].…”

Approximations for the waiting-time distribution in an $$M/PH/c$$ M / P H / c priority queue

“…The authors give an exact method to find per class the steady-state probabilities of the queue length and the number of customers in service. Zeltyn et al (2009) consider a setting with K classes, where classes 1 up to P (P < K ) have preemptive priority over the other (lower priority) classes. The authors give explicit expressions for the LSTs of the waiting times per class.…”

Modeling IT support organizations using multiple-priority queues

“…See for example the works by Wasserkrug et al [12], focusing on shift scheduling for IT support personnel and reporting on the inadequacy of M/M/N queues to model support groups, and by Zeltyn et al [13], developing analytic models for multi-server multi-priority queues with applications in IT support organizations. Our work adopts a different approach, as we are interested in the analysis of transactional logs to development a simulative model of real life IT support groups.…”