Preemptive Priority Discipline Research Articles

Equilibrium analysis of a partially observable priority queue

This study considers a single-server Markovian queue with preemptive priority discipline. In front of a server, two queues are formed: a normal queue for ordinary customers and a priority queue for priority customers. When a customer arrives, they can observe the length of the normal queue, but not the priority queue. By considering the length of the normal queue and the expected waiting time in the priority queue, an arriving customer will choose either the priority queue by paying a priority fee or the normal queue for free. We investigate the customers’ equilibrium strategies. Specifically, we find the necessary and sufficient conditions for a threshold-type strategy to be an equilibrium. We also investigate the stability of equilibria. In addition, we examine the effect of information disclosure regarding the length of the priority queue on equilibrium behavior.

N , n -Preemptive-Priority M/G/1 Queues with Finite and Infinite Buffers

In this paper, we analyze an M/G/1 priority queueing model with finite and infinite buffers under the N , n -preemptive priority discipline, under which preemption decisions are made based on the number of high-priority customers. This priority queueing model can be used for the performance analysis of communication systems accommodating delay- and loss-sensitive packets simultaneously. To analyze the proposed model, we extend the method of delay cycle analysis and develop a queue length version of it for finite-buffer queues. Throughout our analysis, we demonstrate that by the proposed method the analysis of the complex priority queueing model can be reduced to that of simple delay cycles, so two different preemption modes of the queueing model can be dealt with in a unified way. The numerical study reveals that adjusting the decision variables N and n allows us to fine-tune system performance for different classes of customers, and N operates as a primary control variable, regardless of the preemption mode and service-time distributions.

M/G/1 Preemptive Priority Queues With Finite and Infinite Buffers

Recently, M/G/1 priority queues with a finite buffer for high-priority customers and an infinite buffer for low-priority customers have applied to the analysis of communication systems with two heterogeneous traffics : delay-sensitive traffic and loss-sensitive traffic. However, these studies are limited to M/G/1 priority queues with finite and infinite buffers under a work-conserving priority discipline such as the nonpreemptive or preemptive resume priority discipline. In many situations, if a service is preempted, then the preempted service should be completely repeated when the server is available for it. This study extends the previous studies to M/G/1 priority queues with finite and infinite buffers under the preemptive repeat-different and preemptive repeat-identical priority disciplines. We derive the loss probability of high-priority customers and the waiting time distributions of high- and low-priority customers. In order to do this, we utilize the delay cycle analysis of finite-buffer M/G/1/K queues, which has been recently developed for the analysis of M/G/1 priority queues with finite and infinite buffers, and combine it with the analysis of the service time structure of a low-priority customer for the preemptive-repeat and preemptive-identical priority disciplines. We also present numerical examples to explore the impact of the size of the finite buffer and the arrival rates and service distributions of both classes on the system performance for various preemptive priority disciplines.

A Geo/Geo/1 Inventory Priority Queue with Self Induced Interruption

This paper considers discrete time (s, S) inventory Geo/Geo/1 priority queue formed by the customers’ induced interruption during service. An arriving customer enters into a high priority queue of infinite capacity. During his/her service, the service may be interrupted due to own reasons with some probability. The interrupted customer is moved to a lower priority queue of infinite capacity. An item in the inventory is supplied whenever a customer leaves from high priority queue after interrupting/completing the service. The customer in the lower priority queue is served according to preemptive priority discipline. Inter-arrival time, service time and lead time are considered to be geometrically distributed. We use the matrix analytic method to analyse the model. The necessary and sufficient condition for the stability of the system is obtained. The marginal distributions of both higher and lower priority queue lengths are studied. Numerical experiments are incorporated to draw special attention to the importance of the model.

Analysis of a MAP/PH/1 Queue with Discretionary Priority Based on Service Stages

In this paper, we study a MAP/PH/1 queue with two classes of customers and discretionary priority. There are two stages of service for the low-priority customer. The server adopts the preemptive priority discipline at the first stage and adopts the nonpreemptive priority discipline at the second stage. Such a queuing system can be modeled into a quasi-birth-and-death (QBD) process. But there is no general solution for this QBD process since the generator matrix has a block structure with an infinite number of blocks and each block has infinite dimensions. We present an approach to derive the bound for the high-priority queue length. It guarantees that the probabilities of ignored states are within a given error bound, so that the system can be modeled into a QBD process where the block elements of the generator matrix have finite dimensions. The sojourn time distributions of both high and low priority customers are obtained. Some managerial insights are given after comparing the discretionary priority rule with the preemptive and nonpreemptive disciplines numerically.

Analysis of the Geom/Geom/1 Queue under $(N, n)$-preemptive Priority Discipline

In this paper, we consider a Geom/Geom/1 discrete time queue with (N, n)-preemptive priority discipline. Under this discipline, each time the number of high-priority customers in the system reaches the upper threshold N(N ‚ 1), the service of the low-priority customer is preempted. This preempted service is restarted as soon as the number of high-priority customers in the system decreases to a certain lower

Formula omitted]-preemptive priority queues

In this paper, we propose a new priority discipline, called the ( N , n ) -preemptive priority discipline. Under this discipline, the preemption of the service of a low-class customer is determined by two thresholds N and n of the queue length of high-class customers. We consider M/G/1 priority queueing models with two customer classes under two different ( N , n ) -preemptive priority disciplines and derive the Probability Generating Functions (PGFs) of queue lengths of each class. We also show that, under the ( N , n ) -preemptive priority discipline, the QoS for high-class customers can be controlled within a certain bound, regardless of the restarting mode of a preempted service.

Discrete-time queues with discretionary priorities

In this paper, we consider a discrete-time two-class discretionary priority queueing model with generally distributed service times and per slot i.i.d. structured inputs in which preemptions are allowed only when the elapsed service time of a lower-class customer being served does not exceed a certain threshold. As the preemption mode of the discretionary priority discipline, we consider the Preemptive Resume, Preemptive Repeat Different, and Preemptive Repeat Identical modes. We derive the Probability Generating Functions (PGFs) and first moments of queue lengths of each class in this model for all the three preemption modes in a unified manner. The obtained results include all the previous works on discrete-time priority queueing models with general service times and structured inputs as their special cases. A numerical example shows that, using the discretionary priority discipline, we can more subtly adjust the system performances than is possible using either the pure non-preemptive or the preemptive priority disciplines.

Strong approximations for multiclass feedforward queueing networks

This paper derives the strong approximation for a multiclass queueing network,where jobs after service completion can only move to a downstream service station. Job classes are partitioned into groups. Within a group, jobs are served in the order of arrival; that is, a first-in first-out (FIFO) discipline is in force, and among groups, jobs are served under a preassigned preemptive priority discipline. We obtain the strong approximation for the network through an inductive application of an input–output analysis for a single-station queue. Specifically, we show that if the input data (i.e., the arrival and the service processes) satisfy an approximation (such as the functional law-of-iterated logarithm approximation or the strong approximation), then the output data (i.e., the departure processes) and the performance measures (such as the queue length, the workload and the sojourn time processes) satisfy a similar approximation. Based on the strong approximation, some procedures are proposed to approximate the stationary distribution of various performance measures of the queueing network. Our work extends and complements the existing work of Peterson and Harrison and Williams on the feedforward queueing network. The numeric examples show that strong approximation provides a better approximation than that suggested by a straightforward interpretation of the heavy traffic limit theorem.

Message delay analysis for a TDMA scheme operating under a preemptive priority discipline

A TDMA (time-division multiple-access) access-control scheme operating under a preemptive message-based priority discipline is considered and analyzed. The system considered is general, so that an arbitrary number of slots can be allocated to different stations, on a uniformly (equally) spaced basis, in each frame. Under the assumptions of a Poisson message arrival stream and random message lengths governed by a general distribution, it is shown how the Laplace-Stieltjes transform of the actual message waiting time and the delay distributions can easily be obtained, for each priority class, at each of the network stations. Using these results, explicit formulas for any moment of the message delay for any priority class can be obtained. In particular, expressions for the first two moments of the message waiting time and for the average delay of class-k messages are derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Closed two node priority queueing networks

Analytical results for priority queueing networks are extremely rare. Morris (1981) gave the exact solution for closed two node preemptive priority networks with negative exponential service time distributions which were independent of state. In this paper these results are extended to the case of state dependent service rates. Morris also gave an approximate solution to the system with non-preemptive priority at one node. The exact solution is presented and a comparison given between the exact and approximate solution. Also obtained are the exact solutions for two node systems with priorities reversed, state dependent service parameters and generalised service time distributions. Finally, a matrix geometric solution is obtained for two node systems with pre-emptive priority discipline at one node and any non-batch servicing queue discipline at the other.

Characterization and Optimization of Achievable Performance in General Queueing Systems

This paper considers general (single facility) queueing systems with exponential service times, dealing with a finite number J of distinct customer classes. Performance of the system, as measured by the vector of steady state expected sojourn times of the customer classes (the performance vector) may be controlled by adopting an appropriate preemptive priority discipline. We show that the performance space, the set of performance vectors which are achievable under some preemptive work conserving rule, is a polyhedron described by 2J − 1 (in)equalities. The special structure of this polyhedron nevertheless allows for efficient procedures to minimize any separable convex function of the performance vector. Linear objectives are shown to be minimized by absolute priority rules, thus generalizing a well known result for M/M/1 systems. We also show that each point in the performance space may be achieved by a specific randomization of at most J + 1 absolute priority rules.

A note on multi-queue scheduling of two tasks

Optimal schedules in a central server model, consisting of a `CPU' server and m `I/O' servers, are obtained for the case of two customers. Each customer has exponential service time distributions with a distinct mean at the CPU server and with the same mean at any I/O servers. The performance measure considered is an arbitrarily weighted sum of the CPU utilization for each customer. It is shown that the optimal CPU and I/O schedules are either one of preemptive priority disciplines depending on certain conditions; the conditions are also obtained. A proof is given to the conjecture made by Spirn on the model and the results of Spirn's are generalized.

Preemptive Priority Assignment in Multichannel Systems

Customers of different priorities arrive at a system in accordance with a Poisson process. The customers are serviced by c service stations using a preemptive priority discipline. When the servicing times are exponentially distributed, this paper derives expressions for the expected waiting time for inception of service, as well as the lower and upper bounds for the expected staying time and expected number of customers of each priority class (1 ≦ p ≦ r) in the system and in the line.