Low Priority Customers Research Articles

Waiting time analysis of a two-stage queueing system with priorities

A two-stage queueing system with two types of customers and non-preemptive priorities is analyzed. There is no waiting space between stages and so the blocking phenomenon is observed. The arrivals follow a Poisson distribution for the high priority customers and a gamma distribution for the low priority customers, while all service times are arbitrarily distributed. We derive expressions for the Laplace transform of the waiting time density of a low priority customer both in the transient and the steady state.

Workload in queues having priorities assigned according to service time

System designers often implement priority queueing disciplines in order to improve overall system performance; however, improvement is often gained at the expense of lower priority cystomers. Shortest Processing Time is an example of a priority discipline wherein lower priority customers may suffer very long waiting times when compared to their waiting times under a democratic service discipline. In what follows, we shall investigate a queueing system where customers are divided into a finitie number of priority classes according to their service times.

On a Preemptive Markovian Queue with Multiple Servers and Two Priority Classes

We consider a queueing system with multiple servers and two classes of customers operating under a preemptive resume priority rule. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. In a convenient state space representation of the system, we obtain the matrix equation for the two-dimensional, vector-valued generating function of the equilibrium probability distribution. We give a rigorous proof that, by successively eliminating variables from the matrix equations, a nonsingular, block tridiagonal system of equations is obtained for the set of (m + 1)(m + 2)/2 constants that describe the probabilities of the states of the system when no customers are awaiting service. The mean waiting time of the low priority customers is shown to be given by a simple formula in terms of the (known) waiting time of the high priority customers and the expected number of low priority customers in the queue when no high priority customers are waiting.

The preempt-resume M/G/1 queue with feedback and clocked schedules

An M/G/1 queueing system with two priority classes of traffic and Bernoulli feedback for the low priority class is analyzed. For the low priority customers, the distributions of the ergodic time to completion, the first pass time, and the number in the system are first found in the transform domain for the M/G/1 system and both the mean and the variance are displayed. The distributed Poisson approximation is then used to calculate the corresponding quantities when the high priority customers arrive at clock ticks.

Analysis of a finite capacity non preemptive priority queue

In this paper a finite capacity priority queue with multiple servers and non preemptive service discipline is analysed. The arrival and service processes were assumed Markovian and only two priority classes were considered. An analytical solution for blocking probabilities for the customers in the two classes is obtained under the assumption that a high priority customer upon arrival can displace a low priority customer from the waiting line if the queue is full.

Evaluation of the total time in system in a preempt/resume priority queue via a modified Lindley process

A Poisson stream of arrival rate λI and service-time distribution AI(x) has preempt/resume priority over a second stream of rate λII and distribution AII(x). Abundant theoretical results exist for this system, but severe numerical difficulties have made many descriptive distributions unavailable. Moreover, the distribution of total time in system of low-priority customers has not been discussed theoretically. It is shown that the waiting-time sequences of such customers before first entry into service is a Lindley process modified by replacement. This leads to the total time distribution needed. A variety of descriptive distributions, transient and stationary, is obtained numerically via the Laguerre transform method.

Evaluation of the total time in system in a preempt/resume priority queue via a modified Lindley process

A Poisson stream of arrival rate λI and service-time distribution A I(x) has preempt/resume priority over a second stream of rate λII and distribution A II(x). Abundant theoretical results exist for this system, but severe numerical difficulties have made many descriptive distributions unavailable. Moreover, the distribution of total time in system of low-priority customers has not been discussed theoretically. It is shown that the waiting-time sequences of such customers before first entry into service is a Lindley process modified by replacement. This leads to the total time distribution needed. A variety of descriptive distributions, transient and stationary, is obtained numerically via the Laguerre transform method.

The existence of waiting time moments for single server queues with mixed renewal and poisson inputs

The following queuing system is considered: Two independent recurrent input streams (streams 1 and 2) arrive at a server. It is assumed that stream 1 is of Poisson type. Three priority disciplines are studied in case that customers of type 1 have priority: head-of-the-line, preemptive-resume, and preemptive-repeat discipline. For all three cases, the limiting distribution function of actual waiting times of low-priority customers is considered, and conditions are given for the existence of moments related to these limiting distributions.

Limit theorems for single server queues with mixed renewal and poisson inputs

The following queuing system is considered:Two independent recurrent input streams (streams 1 and 2) arrive at a server. It is assumed that stream 1 is of Poisson type. Three priority disciplines are studied in case that these customers have priority:head-of-the-line, preemptive-resume, and preemptive-repeat discipline. Formulas derived for the limiting distribution functions of the actual and the virtual waiting time of low priority customers and of the number of these customers in the system, by using of independences of certain random processes when the time tends to infinity.

A M/G/1 Priority Queue

A M/G/1 queue with discretionary priority is considered. There are two priority classes. An arriving high priority customer interrupts a low priority customer in service, if the remaining service time of the latter exceeds a constant d. Otherwise it is not found necessary to make an interruption. The equilibrium joint distribution of the number of type 1 and type 2 customers present is studied, using remaining service time of both the customer in service and the interrupted one (if any) as supplementary variables. The generating functions for the distributions of queue length (both at an arbitrary time and at various imbedded points) are obtained. It is observed that there are simple connections between these generating functions. The Laplace- Stieltjes transforms of the waiting time and some related variables follow easily. The means of all these variables are also calculated.

A limit theorem for priority queues in heavy traffic

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

A limit theorem for priority queues in heavy traffic

A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

A Priority Queue with Low-Priority Arrivals General

This paper considers a single-server queue with two priority classes of customers. High-priority arrivals are assumed to be Poisson, but interarrival times for low-priority customers are allowed to have a general distribution. Under either the head-of-the-line or preemptive-resume discipline, a formal solution is obtained for the Laplace-Stieltjes transform of the steady-state distribution of the low-priority waiting time. A limit theorem for the low-priority waiting time is also obtained under one heavy-traffic condition for which a steady state does not exist.

Technical Note—A Particular Two-Priority Queue

This note describes a two-priority queue where the low-priority customers arrive in what is essentially a steady stream. It is a useful model in certain inventory problems.

Queue Length Dependent Priority Queues

Priority rules that are mixtures of pre-emption and postponable rules are analyzed. Whether a pre-emption occurs is made to depend on some factor in addition to priority class. A lower priority customer is pre-empted if and only if the queue length of higher priority customers is N, a decision parameter. The stochastic model (without priorities) is that of the M/G/1 queue. First moment expressions (e.g., expected number of customers in the system) in the steady state case are obtained for each priority class, using the concept of work conservation. A linear cost model is introduced which is a function of expected waiting time and expected number of pre-emptions. By considering a parametric class of rules determined by the decision parameter, the problem of finding an optimal rule is formulated.

Parametric Priority Rules: An Approach to Optimization in Priority Queues

This paper analyzes priority rules that are mixtures of preemption and postponable rules. Whether a preemption occurs is made to depend on some factor in addition to priority class; dependence on the completed portion of service (age) of the lower-priority customer in service and the number of prior preemptions of the lower-priority customer in service are each considered. The stochastic model (without priorities) is that of the M/G/1 queue. The paper obtains first-moment expressions (e.g., expected number of customers in the system) in the steady-state case for each priority class as a function of the particular factor being considered. It then introduces a linear cost model that is a function of expected waiting time and expected number of preemptions, and formulates the problem of finding an optimal rule from each class by considering a parametric class of rules determined by each factor. In each case, conditions are sought for the global optimality of a rule over a class of rules, but results are obtained only after restrictions (e.g., monotone failure rates) on service distributions are imposed.

On Discretionary Priority Queueing

AbstractA priority regime is envisaged for single queuing systems composed of two customer populations with Poisson arrivals which is intermediate between the two extreme doctrines: a) head of the line priority, b) pre‐emptive priority. The state of intermediacy is represented by discretionary powers vested in the server to interrupt recently initiated ‐ and not to interrupt almost completed ‐ service to a low priority customer upon the arrival of a high priority customer. For the case of constant service times the discretionary rule is defined and the ensuing queueing characteristics analyzed; in particular, the average total queue lengths of both high and low priority customers are derived for two different cases: a) the resume situation where service renewed to a low priority customer starts at the point of interruption; b) the repeat situation where service given to a low priority customer before an interruption, is completely lost. Optimisation procedures are outlined and for the resume situation a simple optimal discretionary rule is obtained.