A Two Priority Queue with Crossover Feedback

SSRN Electronic Journal, 2000

In this article we give a new derivation for the waiting time distributions in an M/M/c queue with multiple priorities and a common service rate by using elementary lattice paths counting. An advantage of the approach is that it does not require inversion of the Laplace-Stieltjes transform.

European Journal of Operational Research, 2006

The dual queue consists of two queues, called the primary queue and the secondary queue. There is a single server in the primary queue but the secondary queue has no service facility and only serves as a holding queue for the overloaded primary queue. The dual queue has the additional feature of a priority scheme to help reduce congestion. Two classes of customers, class 1 and 2, arrive to the dual queue as two independent Poisson processes and the single server in the primary queue dispenses an exponentially distributed service time at the rate which is dependent on the customerÕs class. The service discipline is preemptive priority with priority given to class 1 over class 2 customers. In this paper, we use matrix-analytic method to construct the infinitesimal generator of the system and also to provide a detailed analysis of the expected waiting time of each class of customers in both queues.

Probability in the Engineering and Informational Sciences, 2013

In this paper, we study three delay systems where different classes of impatient customers arrive according to independent Poisson processes. In the first system, a single server receives two classes of customers with general service time requirements, and follows a non-preemptive priority policy in serving them. Both classes of customers abandon the system when their exponentially distributed patience limits expire. The second system comprises parallel and identical servers providing the same type of service for both classes of impatient customers under the non-preemptive priority policy. We assume exponential service times and consider two cases depending on the time-to-abandon distribution being exponentially distributed or deterministic. In either case, we permit different reneging rates or patience limits for each class. Finally, we consider the first-come-first-served policy in single and multi-server settings. In all models, we obtain the Laplace transform of the virtual waiting time for each class by exploiting the level-crossing method. This enables us to compute the steady-state system performance measures.

Cybernetics and Systems Analysis, 2013

An algorithmic approach to study queuing models with common finite and infinite buffer and jump priorities is developed. It is assumed that upon arrival of a low-priority call, one call of such kind can be transferred with some probability to the end of the queue of high-priority calls. The transition probability depends on the state of the queue of heterogeneous calls. The algorithms are proposed to calculate the quality of service metrics of such queuing models.

Applied Mathematics & Information Sciences, 2018

A single server queueing system is considered in which two types of customers arrive according to independent Poisson processes. Customers of type 1 are of priority nature and the other customers of type 2 are of non-priority. Type 1 customers have nonpreemptive priority over type 2 customers. Assuming that service times for both types of customers have exponential distribution with mean 1/µ, we obtain explicit expressions for the transient solution for the state probability distribution. We deduce the steady-state joint distribution of the number of customers of type 1 and customers of type 2 and also obtain performance measures of the system.

Queueing Systems, 1991

In this paper, a K class M/G~1 queueing system with feedback is examined. Each arrival requires at least one, and possibly up to K service phases. A customer is said to be in class k if it is waiting for or receiving its k th phase of service. When a customer finishes its phase k ~ K service, it either leaves the system with probability Pk, or it instantaneously reenters the system as a class k +1 customer with probability (1--pk). It is assumed that PK =1. Service is non-preemptive and FCFS within a specified priority ordering of the customer classes. Level crossing analysis of queues and delay cycle results are used to derive the Laplace-Stieltjes Transform (LST) for the PDF of the sojourn time in classes 1,..., k; k ~ K.

In this paper we analyze an M/M/1 queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments, we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an M/M/1 queueing system with non-preemptive priorities between customer classes.

This paper studies the M/M/1/K queue under nonpreemptive service priority discipline. The paper shows the state transition diagram of the Markov chain and presents the state balance equations, from which the stationary queue length distribution and other measures of interest can be obtained.

HAL (Le Centre pour la Communication Scientifique Directe), 2016

Kleinrock (1964) proposed a queueing discipline for a single-server queue in which customers from different classes accumulate priority as linear functions of their waiting time. When the server becomes free, it selects the waiting customer with the highest amount of accumulated priority at that instant, provided that the queue is nonempty. For such a queue, Kleinrock developed a recursion for calculating the expected waiting time of customers from each class. More recently, Stanford, Taylor and Ziedins (2014) took another look at this queue, which they termed the Accumulating Priority Queue (APQ), and derived the waiting time distributions for each class. Kleinrock and Finkelstein (1967) also studied an accumulating priority system in which customers' priorities increase as a power-law function of their time in the queue. They established that it is possible to associate a particular linear accumulating priority queue with such a power-law accumulating priority queue, in such a way that the expected waiting times of customers from the different classes are preserved. In this paper, we extend their analysis to characterise the class of nonlinear accumulating priority queues for which an equivalent linear APQ can be found, in the sense that the waiting time distributions for each of the classes are identical in both the linear and nonlinear systems.

Queueing Systems - Theory and Applications - QUESTA, 2002

In this paper, we present a performance analysis of a 2-dimensional preemptive priority queueing system with state-dependent arrivals. Using a Markovian formulation we first compute the steady state distribution for the queue length of both classes. Then, waiting times and busy periods are characterized through (i) first and second moments and (ii) the approximation of their cumulative distribution functions (cdf) and Laplace–Stieltjes transforms (LST). We derive these approximations connecting bounds in the Laplace domain with bounds on the original time domain. We also, study the behavior of the inter-departure time for each class. Finally, we conclude the paper with a set of computational experiments testing our results.