Waiting time distributions in the accumulating priority queue

Abstract

We are interested in queues in which customers of different classes arrive to a service facility, and where performance targets are specified for each class. The manager of such a queue has the task of implementing a queueing discipline that results in the performance targets for all classes being met simultaneously. For the case where the performance targets are specified in terms of ratios of mean waiting times, as long ago as the 1960s, Kleinrock suggested a queueing discipline to ensure that the targets are achieved. He proposed that customers accumulate priority as a linear function of their time in the queue: the higher the urgency of the customer's class, the greater the rate at which that customer accumulates priority. When the server becomes free, the customer (if any) with the highest accumulated priority at that time point is the one that is selected for service. Kleinrock called such a queue a time-dependent priority queue, but we shall refer to it as the accumulating priority queue. Recognising that the performance of many queues, particularly in the healthcare and human services sectors, is specified in terms of tails of waiting time distributions for customers of different classes, we revisit the accumulating priority queue to derive its waiting time distributions, rather than just the mean waiting times. We believe that some elements of our analysis, particularly the process that we call the maximum priority process, are of mathematical interest in their own right.

Controlling the workload of M/G/1 queues via the q-policy

“…This system is a variant of the M/G/1 queue with accumulating priority, which was recently studied by Stanford, Taylor, and Ziedins [12].…”

“…The above basic model was introduced by Stanford et al [12] in their analysis of a particular multi-class non-preemptive priority system. In order to analyze the M/G/1 queue with accumulating priority, these authors defined a new stochastic process which they called the maximal priority process.…”

“…Furthermore, the accreditation threshold process increases linearly at rate ξ 2 during busy periods. Stanford et al [12] referred to those customers who arrive during busy periods and whose priority overtakes the accreditation threshold as accredited customers.…”

“…In fact, it can be shown that the LST of the busy period is the solution to Eq. (4) with q = ξ 2 /ξ 1 (see Stanford et al [12] and their discussion on accredited busy periods). In addition, using the same argument as in Brill [1], we can show that the steady-state distribution of {M (t), t ≥ 0} when ξ 1 = 1 is equivalent to the steady-state distribution of the workload process {U ξ2 (t), t ≥ 0} of Section 4.…”

Performance analysis of a discrete-time two-class global-FCFS queue with two servers and geometric service times

“…Usually, the resources to be shared are the facilities that are able to deliver the requested services to the customers, or, in queueing language, the "servers" of the queueing system. Very often, the different customer classes are characterized by either their distinct arrival characteristics, their different service requirements, their loss priorities ( [28]), their service priorities ( [21,23,26]), etc. Arriving customers are either accommodated in separate class-specific queues or in one global shared queue, but in general they require some kind of processing from the same servers.…”