Low Priority Customers Research Articles

Prioritized Retrial Queueing Systems with Randomized Push-Out Mechanism

The article is focused on a single-channel preemptive queuing system. Two stationary Poisson flows of customers are incoming to the system. The first flow has an absolute priority over the second one: a new high-priority customer from the first flow displaces a low-priority one from the service channel and takes its place. The capacity of the system is limited to k customers. There is a probabilistic push-out mechanism in the system: if a new high-priority customer finds that all the places in the queue are occupied, then it has the right to displace one low-priority customer from the queue with probability a. Both types of customers have the same exponentially distributed service times. Customers who failed to enter the system due to the limited size of the queue, as well as those expelled from the queue or service channel when the push-out mechanism is triggered, are not lost immediately, but they are sent to a special part of the system called the orbit and designed to store repeated customers. In orbit, there are two separate unlimited queues, consisting of low-priority and high-priority repeated customers, respectively. If there are no free places in the system, new customers with a probability q are added to the corresponding orbital queue. The waiting time of repeated customers in orbit is distributed according to an exponential law. The parameter of this law may differ for different types of customers. After waiting in orbit, secondary customers try to re-enter the system. The probabilistic characteristics of the described queuing system are calculated by the method of generating functions, previously proposed by the authors for calculating a similar system without repeated customers. This method allows finding the main probabilistic characteristics of distributions for both types of customers. Particular attention is paid to the study of the dependence of the loss probabilities for both types of customers on the parameters of the system, primarily on the push-out probability a, the capacity of the system k, and the probability of repeated circulation (probability of persistence) q. It is shown that the effect of blocking the system and the effect of the linear law of customers’ losses, previously identified in similar problems without repeated customers, remain valid even in the presence of secondary repeated customers. The theoretical results are proved by numerical calculations. The blocking area for the second type of customers was calculated along with the area of linear loss law for both types of customers. We studied the influence of the probability of repeated circulation q on the shape of these areas and on the dependence of the loss probabilities for both types of customers on the push-out probability a.

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Consider a single-server retrial queueing system with non-preemptive priority service, where customers arrive in a Poisson process with a rate of $\lambda_1$ for high-priority customers (class 1) and $\lambda_2$ for low-priority customers (class 2). If a high-priority customer is blocked, they are queued, while a low-priority customer must leave the service area and return after some random period of time to try again. In contrast with existing literature, we assume different service time distributions for the two customer classes. This investigation proposes a stochastic comparison method based on the general theory of stochastic orders to obtain lower and upper bounds for the joint stationary distribution of the number of customers at departure epochs in the considered model. Specifically, we discuss the stochastic monotonicity of the embedded Markov queue-length process in terms of both the usual stochastic and convex orders. We also perform a numerical sensitivity analysis to study the effect of the arrival rate of high-priority customers on system performance measures.

Spreadsheet Simulation of Priority Queues

This paper develops a spreadsheet simulation methodology for teaching simulation and performance analysis of priority queues with multiple servers, without resorting to macros, add-ins, or array formula. The approach is made possible by a “single overtaking” simplifying assumption under which any lower-priority customer may be passed in line by at most one higher-priority customer. By increasing the number of overtaking customers, one at a time, the simulation model is extended to the “multiovertaking” case. These simplifying assumptions make such spreadsheet simulations (of more complex queuing networks) accessible to students, and so the paper includes teaching and learning strategies for the classroom. Performance analysis of single-overtaking versus multiovertaking polices is included. Supplemental Material: The online data is available at https://doi.org/10.1287/ited.2022.0037 .

Analytical Study of Two Serial Channels with Priority and Reneging

Most of the studies of queuing theory, which are useful in our daily life has been investigated by many researchers. The present research is the study of pre-emptive priority queuing system consisting two serial channels in stochastic environment. The impatient behavior of customer’s will be discussed with exponential service distribution and Poisson arrivals. Higher priority customers have pre-emptive priority over the low priority customers. The G.F. technique is used to derive the performance measures of high & low priority queues and assuming FCFS discipline in busy schedule of higher priority class. Also evaluate queue behavior graphically and discussed a special case at the end which shows utilization of channels.

Recursive calculation of moments in priority queues

We reconsider the discrete-time priority queue with two classes. The server serves high-priority customers as long as there are such customers, and only turns to the low-priority customers when there are no high-priority customers. Relying on a multivariate recursive extension of Faà di Bruno's formula, we find recursive equations to calculate the moments of the queue lengths. This allows for calculation of many more moments in much shorter time than conventionally possible.

Algorithms for Queueing Systems with Reneging and Priorities Modeled as Quasi-Birth-Death Processes

We consider a Markovian multiserver queueing system with two customer classes, preemptive priorities, and reneging. We formulate this system as an infinite level-dependent quasi-birth-death process (LDQBD). We introduce an algorithm that endogenously truncates the level and calculates lower and upper bounds on stationary probabilities of this LDQBD such that the gap between the bounds can be any desired amount. Our algorithm can be applied to any LDQBD for which the rate matrices become elementwise nonincreasing above some level. This appears to be the first algorithm that provides bounds on stationary probabilities for an infinite-level LDQBD. To obtain these bounds, the algorithm first obtains lower and upper bounds on the rate matrices of the LDQBD using a novel method, which can be applied to any LDQBD. We then extend this algorithm to approximate performance measures of the system of interest and calculate exact lower and upper bounds for those that can be expressed as probabilities, such as the probability that an incoming low-priority customer will wait. We generate a wide range of instances with up to 100 servers and compare the solution times and accuracy of our algorithm with two existing algorithms. These numerical experiments indicate that our algorithm is faster than the other two algorithms for a given accuracy requirement. We investigate the impact of changing service rates on the proportion of low-priority customers served and their wait time, and we demonstrate how ignoring one of these measures can possibly mislead decision makers. Summary of Contribution: We contribute to operations research by modeling a practically important queueing system and developing an algorithm to accurately compute performance measures for that system. We also contribute to computer science by providing error and complexity analysis for the algorithm to solve a broad class of two-dimensional Markov chains with infinite state space.

A Fluid-Diffusion-Hybrid Limiting Approximation for Priority Systems with Fast and Slow Customers

The paper “Fluid-Diffusion-Hybrid (FDH) Approximation” proposes a new heavy-traffic asymptotic regime for a two-class priority system in which the high-priority customers require substantially larger service times than the low-priority customers. In the FDH limit, the high-priority queue is a diffusion, whereas the low-priority queue operates as a (random) fluid limit, whose dynamics are driven by the former diffusion. A characterizing property of our limit process is that, unlike other asymptotic regimes, a non-negligible proportion of the customers from both classes must wait for service. This property allows us to study the costs and benefits of de-pooling, and prove that a two-pool system is often the asymptotically optimal design of the system.

MMAP/(PH,PH)/1 Queue with Priority Loss through Feedback

In this paper, we consider two single server queueing systems to which customers of two distinct priorities (P1 and P2) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. The probability of a P1 customer to feedback is θ on completion of his service. The feedback (P1) customers, as well as P2 customers, join the low priority queue. Low priority (P2) customers are taken for service from the head of the line whenever the P1 queue is found to be empty at the service completion epoch. We assume a finite waiting space for P1 customers and infinite waiting space for P2 customers. Two models are discussed in this paper. In model I, we assume that the service of P2 customers is according to a non-preemptive service discipline and in model II, the P2 customers service follow a preemptive policy. No feedback is permitted to customers in the P2 line. In the steady state these two models are compared through numerical experiments which reveal their respective performance characteristics.

Time dependent retrial queueing model with orbital search under non-preemptive priority services

Transient solution of Single server Non-preemptive priority Retrial model with orbital search is studied using eigenvalues and eigenvectors. In this model, customers are arriving in Poisson process. Arrival rates of low priority and high priority customers are respectively λ 1 and λ 2 . The service times for low and high priority customers follow exponential distribution with parameters µ 1 and µ 2 respectively. Customer finding the system busy, on arrival, goes to the orbit and form a virtual queue. In this model of orbital search, when the server becomes free, he has two options. Either the server search for the customer in the orbit with probability p to provide service for a customer in the orbit or remains idle with the probability 1-p. Whenever the server is free, customer from orbit try for service, under classical retrial policy with rate σ which follows Poisson process. In this paper, transient solution of average number of customers in the orbit and high priority queue, the probability of server being idle, the probabilities of server being busy with low and high priority customers for various values of λ 1, λ 2, µ 1 , µ 2 , σ, p and t are estimated.

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.

Queueing System with Two Input Flows, Preemptive Priority, and Stochastic Dropping

We consider a single-line queuing system with an infinite buffer that receives two Poisson flows of customers with different intensities. Customers of the first type have preemptive priority over customers of the second type. In addition, at the time of the end of servicing, a high-priority customer with some probability can drop all low-priority customers in the queue. Serving both types of customers has an exponential distribution with different parameters. We show expressions for calculating stationary probabilities in this system, the probability of servicing a low-priority customer in terms of the generating function, and a formula for the average number of customers of the second type.

Solving a bi-objective mixed-model assembly-line sequencing using metaheuristic algorithms considering ergonomic factors, customer behavior, and periodic maintenance

In today’s competitive world, companies must maintain their customers and attract new ones. Hence, they paid a great attention paid to mixed model assembly lines (MMAL). In this study, a two-step framework was developed to investigate and optimize customer relationships and the sequence of orders in an MMAL. First, based on customers past behavior, they were grouped into three clusters with high, normal, and low priority. Then, an optimal sequence was defined using a mathematical model. The objectives of the sequence were maximizing, first, the satisfaction of customers with high priority and, second, profits. Moreover, orders for low priority customers could be rejected. A multi-objective tabu search algorithm was proposed to solve the sequencing problem and then compared with non-dominated sorting genetic algorithm II and multi objective simulated annealing. The results indicated that this new algorithm is superior to others. We also developed an algorithm for the integration of periodic maintenance with sequencing of orders. The results suggested that the lack of this integration causes non-optimal sequences.

A queueing inventory system with threshold based priority service and a flexible server

In this paper, we analyze two-server queueing - inventory system which consists of two-parallel queues and two category of customers i.e., high and low priority customers. The high priority customer generates demand for an item whereas the low priority customers arrive only for repair (not demanding items). Server 1 offers his service to first queue which involves only high priority customers and server 2 is a flexible server who offers his service to both the queues. When there is no availability of service in both the queues, server 1 will be at idle state whereas server 2 goes on vacation. We find a priority service rule to minimize the loss probability of low priority customers. The low priority customers will be served on threshold-based service such that a threshold L which is placed in the second queue and the service starts only if the number customers in the second queue exceeds the threshold level. The limiting probability distribution of the four random variables is computed in steady state. Also, some significant change in measures of the system are derived. Finally, the suitable cost function is defined and significant numerical examples are projected.

Inventory rationing on a one-for-one inventory model for two priority customer classes with backorders and lost sales

In this study, we are primarily motivated by the research problem of recognizing heterogeneous customer behavior towards waiting for order fulfillment under the threshold rationing policy (also known as the critical level policy), and aim to find its effect on system stock levels and performance measures. We assume a continuous review one-for-one ordering policy with generally distributed lead times. In the first model, we consider the case in which the low-priority customer class exhibits zero patience for waiting if the demand is not satisfied immediately (a lost sale), whereas the demand of the high-priority customer class can be backordered. This is the first study in the literature to consider this model. We provide an exact analysis for the derivation of the steady-state probability distribution and the average infinite horizon cost per unit time. We then develop an efficient optimization procedure to minimize the average expected cost rate. We also determine the forms of the optimal solutions for the two service level optimization models that are common in practice. In the second model, we study the opposite case in which the high-priority customer class exhibits zero patience for waiting. We establish a theoretical basis for the rationale of using the Continuous-Time Markov Chain (CTMC) approach as an approximation. We show that under certain assumptions, the steady-state probabilities of the system with generally distributed lead times are identical to the steady-state probabilities of the CTMC system with the same mean. This result enables us to link the dynamics of the studied model to the CTMC model, which may open new doors for future research.

Strategic Behavior of Customers and Optimal Control for Batch Service Polling Systems with Priorities

During the past few years, batch service systems have attracted considerable attention due to their wide area of applications. In this present paper, we study a special batch service polling system (the so-called Israeli queue) with priorities. Different from the previous papers which focus on the performance analysis, we aim to investigate the strategic behavior of customers and optimal design for the underlying queueing model. By considering two levels of information (observable and unobservable) provided upon customers’ arrival, we, respectively, derive the equilibrium strategies of high-priority and low-priority customers, regarding the joining or balking dilemma. We also present some numerical examples to reveal the impacts of several parameters on the equilibrium strategies, together with some intuitive explanations. Finally, we formulate the revenue function of the service provider and present the Particle Swarm Optimization algorithm to seek the optimal service prices for the high-priority and low-priority customers to maximize the service provider’s revenue under the two levels of information.

Delay cycle analysis of finite-buffer M/G/1 queues and its application to the analysis of M/G/1 priority queues with finite and infinite buffers

In this paper, we develop a new type of delay cycle analysis for single-class finite-buffer M/G/1 queues, and then analyze M/G/1 priority queues with a finite buffer for high-priority customers and an infinite buffer for low-priority customers using the proposed delay cycle analysis. For this type of priority queues, we derive the waiting time distributions of high- and low-priority customers, as well as the loss probability of high-priority customers, under the nonpreemptive and preemptive-resume priority disciplines. Compared with M/G/1 priority queues with infinite buffers for both classes, the priority discipline has more significant impacts on system performance in the priority queues in question, especially on the queue waiting time and stability conditions of low-priority customers. This is explored in theoretical propositions, as well as in numerical examples.

Queuing System with Two Types of Customers and Dynamic Change of a Priority

The use of priorities allows us to improve the quality of service of inhomogeneous customers in telecommunication networks, inventory and health-care systems. An important modern direction of research is to analyze systems in which priority of a customer can be changed during his/her stay in the system. We considered a single-server queuing system with a finite buffer, where two types of customers arrive according to a batch marked Markov arrival process. Type 1 customers have non-preemptive priority over type 2 customers. Low priority customers are able to receive high priority after the random amount of time. For each non-priority customer accepted into the buffer, a timer, which counts a random time having a phase type distribution, is switched-on. When the timer expires, the customer with some probability leaves the system unserved and with the complimentary probability gains the high priority. Such a type of queues is typical in many health-care systems, contact centers, perishable inventory, etc. We describe the behavior of the system by a multi-dimensional continuous-time Markov chain and calculate a number of the stationary performance measures of the system including the various loss probabilities as well as the distribution function of the waiting time of priority customers. The illustrative numerical examples giving insights into the system behavior are presented.

(M, MAP)/(PH, PH)/1 Queue with Non-preemptive Priority and Working Vacation Under N-Policy

In this paper we consider two single server queueing models with non-preemptive priority and working vacation under two distinct N-policies. High priority (type I) customers are served even in vacation mode whereas low priority (type II) customers are served only when the server comes to normal mode of service. Type I customers have only a limited waiting space L whereas type II customers have unlimited capacity. The two distinct N-policies are as described below: In model I, while service of type I customers are in progress in vacation mode (working vacation), if the number of such customers present in the system hits N ($$\le L$$) or the vacation timer (clock) expires, whichever occurs first, the server is switched on to normal mode. In model II, switching the server to normal mode from vacation mode occurs as soon as the accumulated number (those served out plus those present in the system) of type I customers during that working vacation hits N or the vacation timer expires, whichever occurs first. Type I customers arrive according to a Poisson process whereas type II customer’s arrival is governed by Markovian Arrival Process. Service time of type I and type II customers follow distinct phase type distributions. At a service completion epoch, finding the system empty, server takes an exponentially distributed working vacation. During working vacation, type I customers are served at a reduced rate. On vacation expiration, the service of the type I customer already in service, will start from the beginning in the normal mode of service. We analyze these models in steady state to compute the distribution of duration of service time continuously in slow mode, expected number of returns to 0 type I customer state, starting from 0 type I customer state during vacation mode of service before the arrival of a type II customer, the distribution of a p-cycle in normal mode, LSTs of busy cycle, busy period of type I customers generated during the service time of a type II customer and LSTs of waiting time distributions of type I and type II customers. We compare these models in steady state by numerical experiments to identify the superior model.

Transient Analysis of M[X1],M[X2]/G1,G2/1 retrial queueing system with priority services, working breakdown, start up/close down time, Bernoulli vacation, reneging and balking

This paper considers M[X1],M[X2]/G1,G2/1 general retrial queueing system with priority services. Two types of customers from different classes arrive at the system in different independent compound Poisson processes. The server follows the pre-emptive priority rule subject to working breakdown, startup/closedown time and Bernoulli vacation with general (arbitrary) vacation periods. After completing the service, if there are no priority customers present in the system the server may go for a vacation or close down the system. On completion of the close down, the server needs some time to set up the system. The priority customers who find the server busy are queued in the system. A low-priority customer who find the server busy are routed to a retrial (orbit) queue that attempts to get the service. The system may breakdown at any point of time when it is in operation. However, when the system fails, instead of stopping service completely, the service is continued only to the high priority customers at a slower rate. We consider balking to occur to the low priority customer while the server is busy or idle, and reneging to occur at the high priority customers during server’s vacation, start up/close down time. Using the supplementary variable technique, we derive the joint distribution of the server state and the number of customers in the system. Finally, some performance measures and numerical examples are presented.

Delay analysis of a two-class priority queue with external arrivals and correlated arrivals from another node

Calculation of the delay distribution of (low-priority) customers in priority queues is difficult, especially when the arrival process is correlated. In this paper, we find the generating functions of the delays of high- and low-priority customers in a discrete-time queue where the arrivals are generated by a superposition of two processes, an independent one and a correlated one that can bring an extra customer in the buffer during a slot. The latter process can be seen as a simplified model for the output process of another priority queue. As a consequence, our results can be used to estimate the delay in the second stage of a tandem queueing system with an extra exogenous arrival process. A tandem priority queue is not of product-form type and its analysis is known to be a hard problem. This paper can lead to a first approximate analysis that can later be extended to incorporate the correlation in the output process of a priority queue more accurately.