Priority queues

Extensive research has been carried out in the subject of Priority Queues over the past ten years, culminating in the book by Jaiswal [8], in this thesis, certain isolated problems which appear to have been omitted from the consideration of other authors are discussed. The first two chapters are concerned with the question of how priorities should be allocated to customers (or 'units') arriving at a queue so as to minimize the overall meaning waiting time [it is perhaps worth mentioning at the outset that following current usage, the terms 'queueing time' and ‘waiting time' will be used synonymously throughout; both refer to the time a unit waits before commencing service]. In previous treatments of this 'allocation of priorities problem it has always been assumed that on arrival, the service time requirement of a unit could be predicted exactly; the effect of having only imperfect information in the form of an estimated service time is considered here. Chapter l deals with the non-pre-emptive discipline; Chapter 2 with discretionary disciplines. Priority queues in which the arrival epochs of different types of units form independent renewal processes have only been solved under the assumption of random arrivals. However, if the following modified arrival scheme is considered. arrival epochs form an ordinary renewal process, and at any arrival epoch, independently of what happened at all previous epochs, with probability q1 the arrival is a priority unit and with probability q2 a non=priority unit (where ql+q2 =l) then the priority analogues of the ordinary single-server queues E(_b)/G/l and GI/M/1 can be solved (Chapters 3 and 4 respectively)" In conclusion, Chapter 5 is concerned with approximate methods: (v) section 1 is a review of previous work on deriving bounds for the mean waiting time in a GI/G/1 queue, section 2 extends this work to the GI/G/1 priority queue

A Queueing Model for the Ambulance Ramping Problem with an Offload Zone

This work develops a methodology for studying the effect of an offload zone on the ambulance ramping problem using a multi-server, multi-class non-preemptive priority queueing model that can be treated analytically. A prototype model for the ambulance/emergency-department interface is constructed, which is then implemented as a formal discrete event simulation, and is run as a regenerative steady-state simulation for empirical estimation of the ambulance queue-length and waiting-time distributions. The model is also solved by analytical means for explicit and exact representations of these distributions, which are subsequently tested against simulation results. A number of measures of performance is extracted, including the mean and 90th percentiles of the ambulance queue length and waiting time, as well as the average number of ambulance days lost per month due to offload delay (offload delay rate). Various easily computable approximations are proposed and tested. In particular, a closed-form, purely algebraic expression that approximates the dependence of the offload delay rate on the capacity of the offload zone is proposed. It can be evaluated directly from model input parameters and is found to be, for all practical purposes, indistinguishable from the exact result.Comment: 26 pages, 8 figure

Explicit Results for the Distributions of Queue Lengths for a Non-Preemptive Two-Level Priority Queue

Explicit results are derived using simple and exact methods for the joint and marginal queue-length distributions for the M/M/c queue with two non-preemptive priority levels. Equal service rates are assumed. Two approaches are considered. One is based on numerically robust quadratic recurrence relations. The other is based on a complex contour-integral representation that yields exact closed-form analytical expressions, not hitherto available in the literature, that can also be evaluated numerically with very high accuracy

Stochastic queueing-theory approach to human dynamics

Recently, numerous studies have shown that human dynamics cannot be described accurately by exponential laws. For instance, Barabasi [Nature (London) 435, 207 (2005)] demonstrates that waiting times of tasks to be performed by a human are more suitably modeled by power laws. He presumes that these power laws are caused by a priority selection mechanism among the tasks. Priority models are well-developed in queueing theory (e.g., for telecommunication applications), and this paper demonstrates the (quasi-) immediate applicability of such a stochastic priority model to human dynamics. By calculating generating functions and by studying them in their dominant singularity, we prove that nonexponential tails result naturally. Contrary to popular belief, however, these are not necessarily triggered by the priority selection mechanism

An N Server Cutoff Priority Queue Where Customers Request a Random Number of Servers

Consider a multi-priority, nonpreemptive, N-server Poisson arrival queueing system. The number of servers requested by an arrival has a known probability distribution. Service times are negative exponential. In order to save available servers for higher priority customers, arriving customers of each lower priority are deliberately queued whenever the number of servers busy equals or exceeds a given priority-dependent cutoff number. A queued priority i customer enters service the instant the number of servers busy is at most the respective cutoff number of servers minus the number of servers requested (by the customer) and all higher priority queues are empty. In other words the queueing discipline is in a sense HOL by priorities, FCFS within a priority. All servers requested by a customer start service simultaneously; service completion instants are independent. We derive the priority i waiting time distribution (in transform domain) and other system statistics

Analytic Approach to the Non-Preemptive Markovian Priority Queue

Explicit and exact results are obtained for the joint queue-length distribution for the two-level non-preemptive Markovian priority queue. Marginal distributions are derived for the general multi-level problem. The results are based on a representation of the joint queue-length probability mass function as a single-variable complex contour integral, that reduces to a real integral on a finite interval arising from a cut on the real axis. Both numerical quadrature rules and exact finite sums, involving Legendre polynomials and their generalization, are presented for the joint and marginal distributions. A high level of accuracy is demonstrated across the entire ergodic region. Relationships are established with the waiting-time distributions. Asymptotic behaviour in the large queue-length regime is extracted.Comment: 29 pages, 7 figure