On priority queues with impatient customers

In this paper, we study three different problems where one class of customers is given priority over the other class. In the first problem, a single server receives two classes of customers with general service time requirements and follows a preemptive-resume policy between them. Both classes are impatient and abandon the system if their wait time is longer than their exponentially distributed patience limits. In the second model, the low-priority class is assumed to be patient and the single server chooses the next customer to serve according to a non-preemptive priority policy in favor of the impatient customers. The third problem involves a multi-server system that can be used to analyze a call center offering a call-back option to its impatient customers. Here, customers requesting to be called back are considered to be the low-priority class. We obtain the steady-state performance measures of each class in the first two problems and those of the high-priority class in the third problem by exploiting the level crossing method. We furthermore adapt an algorithm from the literature to obtain the factorial moments of the low-priority queue length of the multi-server system exactly.      

A preemptive discrete-time priority buffer system with partial buffer sharing

This paper considers a Geo/Geo/1 discrete-time queue with preemptive priority. Both the arrival and service processes are Bernoulli processes. There are two kinds of customers: low-priority and high-priority customers. The high-priority customers have a preemptive priority over low-priority customers. If the total number of customers is equal or more than the threshold (k), the arrival of low-priority customers will be ignored. Hence the system buffer size is finite only for the low-priority customers. A recursive numerical procedure is developed to find the steady-state probabilities. With the aid of recursive equations, we transform the infinite steady-state departure-epoch equations set to a set of (k + 1) × (k + 2)/2 linear equations set based on the embedded Markov Chain technique. Then, this reduced linear equations set is used to compute the steady-state departure-epoch probabilities. The important performance measures of the system are calculated. Finally, the applicability of the solution procedure is shown by a numerical example and the sensitivity of the performance measures to the changes in system parameters is analyzed.     

How pre-emptive priority affects completion rate in an M/M/1 queue with Poisson reneging

Queuing problems in which customers leave the queue without obtaining service (i.e. renege) have many applications. This paper looks at a queue where arrival, service and reneging events are all generated by independent Poisson processes, and customers are selected for service according to priority. A closed-form expression is derived for the probability of completing service for each priority class if high-priority customers always pre-empt low-priority ones. This result has applications in modeling the value of communications between members of an interacting population, such as a formal organization or an online community.     

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.     

Performance comparison of several priority schemes with priority jumps

In this paper, we consider several discrete-time priority queues with priority jumps. In a priority scheduling scheme with priority jumps, real-time and non-real-time packets arrive in separate queues, i.e., the high- and low-priority queue respectively. In order to deal with possibly excessive delays however, non-real-time packets in the low-priority queue can in the course of time jump to the high-priority queue. These packets are then treated in the high-priority queue as if they were real-time packets. Many criteria can be used to decide when packets of the low-priority queue jump to the high-priority queue. Some criteria have already been introduced in the literature, and we first overview this literature. Secondly, we propose and analyse a new priority scheme with priority jumps. Finally, we extensively compare all cited schemes. The schemes all differ in their jumping mechanism, based on a certain jumping criterion, and thus all have a different performance. We show the pros and cons of each jumping scheme.     

Analysis of a discrete-time preemptive resume priority buffer

In this paper, we analyze a discrete-time preemptive resume priority queue. We consider two classes of customers which have to be served, where customers of one class have preemptive resume priority over customers of the other. Both classes contain customers with generally distributed service times. We show that the use of probability generating functions is beneficial for analyzing the system contents and customer delays of both classes. It is shown (theoretically as well as by some practical procedures) how moments and approximate tail probabilities of system contents and customer delays are calculated. The influence of the priority scheduling discipline and the service time distributions on the performance measures is shown by some numerical examples.     

On multiple priority multi-server queues with impatience

We consider Markovian multi-server queues with two types of impatient customers: high- and low-priority ones. The first type of customer has a non-preemptive priority over the other type. After entering the queue, a customer will wait a random length of time for service to begin. If service has not begun by this time he or she will abandon and be lost. We consider two cases where the discipline of service within each customer type is first-come first-served (FCFS) or last-come first-served (LCFS). For each type of customer, we focus on various performance measures related to queueing delays: unconditional waiting times, and conditional waiting times given service and given abandonment. The analysis we develop holds also for a priority queue with mixed policies, that is, FCFS for the first type and LCFS for the second one, and vice versa. We explicitly derive the Laplace–Stieltjes transforms of the defined random variables. In addition we show how to extend the analysis to more than two customer types. Finally we compare FCFS and LCFS and gain insights through numerical experiments.     

Performance of service policies in a specialized service system with parallel servers

This paper considers a scheduling problem occurring in a specialized service system with parallel servers. In the system, customers are divided into the “ordinary” and “special” categories according to their service needs. Ordinary customers can be served by any server, while special customers can be served only by the flexible servers. We assume that the service time for any ordinary customer is the same and all special customers have another common service time. We analyze three classes of service policies used in practice, namely, policies with priority, policies without priority and mixed policies. The worst-case performance ratios are obtained for all of these service policies.     

Mixed gated/exhaustive service in a polling model with priorities

In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers.     

The MAP/(PH/PH)/1 queue with self-generation of priorities and non-preemptive service

Customers arriving according to a Markovian arrival process are served at a single server facility. Waiting customers generate priority at a constant rate γ$\gamma$; such a customer waits in a waiting space of capacity 1 if this waiting space is not already occupied by a priority generated customer; else it leaves the system. A customer in service will be completely served before the priority generated customer is taken for service (non-preemptive service discipline). Only one priority generated customer can wait at a time and a customer generating into priority at that time will have to leave the system in search of emergency service elsewhere. The service times of ordinary and priority generated customers follow PH-distributions. The matrix analytic method is used to compute the steady state distribution. Performance measures such as the probability of n consecutive services of priority generated customers, the probability of the same for ordinary customers, and the mean waiting time of a tagged customer are found by approximating them by their corresponding values in a truncated system. All these results are supported numerically.     

A semi-preemptive priority scheduling discipline: Performance analysis

In this paper, we present an in-depth analytical study of a semi-preemptive priority scheduling discipline. This discipline eliminates the deficits of both the full- and non-preemptive versions. Under the non-preemptive category, in particular, higher-priority customers may have to wait even when the service of a lower-priority customer has just started, while under the full-preemptive discipline, the almost completed service of a lower-priority customer may be interrupted due to the arrival of higher-priority customers, possibly causing a large extra delay. For fixed low-priority service times, the semi-preemptive priority scheduling discipline shows a performance gain of up to 6% compared to the full- and non-preemptive versions.     

Die nicht signalgesteuerte Nebenstraßenzufahrt als verallgemeinertes M/G/1-Warteschlangensystem

This paper deals with the queuing of side-street vehicles at high-priority traffic streams of unsignalized intersections. Among other results there are obtained the solutions for the mean delay of a low-priority vehicle and the probability of the waiting system being empty. In doing so, the system under study is reduced to a M/G/1 queuing system, where the service times of those vehicles that have to queue are different from the service times of those cars that arrive to find the system empty. The exact solutions are compared with commonly used approximate solutions and evaluated numerically. Thereby important information on the quality of those approximate solutions is obtained.     

Time-dependent analysis of an <Emphasis Type="Italic">M</Emphasis> / <Emphasis Type="Italic">M</Emphasis> / <Emphasis Type="Italic">c</Emphasis> preemptive priority system with two priority classes

We analyze the time-dependent behavior of an M / M / c priority queue having two customer classes, class-dependent service rates, and preemptive priority between classes. More particularly, we develop a method that determines the Laplace transforms of the transition functions when the system is initially empty. The Laplace transforms corresponding to states with at least c high-priority customers are expressed explicitly in terms of the Laplace transforms corresponding to states with at most \(c - 1\) high-priority customers. We then show how to compute the remaining Laplace transforms recursively, by making use of a variant of Ramaswami’s formula from the theory of M / G / 1-type Markov processes. While the primary focus of our work is on deriving Laplace transforms of transition functions, analogous results can be derived for the stationary distribution; these results seem to yield the most explicit expressions known to date.     

Delay analysis of a discrete-time multiclass slot-bound priority system

This paper introduces a new priority mechanism in discrete-time queueing systems that compromises between first-come-first-served (FCFS) and head-of-line priority. In this scheduling discipline—which we dubbed slot-bound priority—customers of different priority classes entering the system during the same time-slot are served in order of their respective priority class. Customers entering during different slots are served on a FCFS basis. In this paper we study the delay in an N-class discrete-time queueing system under slot-bound priority. General independent arrivals and class-specific general service time distributions are assumed. Expressions for the probability generating function of the delay of a random type-j customer are derived, from which the respective moments are easily obtained. The tail behaviour of these distributions is analyzed as well, and some numerical examples show the effect slot-bound priority can have on the performance measures.     

Class clustering destroys delay differentiation in priority queues

This paper considers a discrete-time priority queueing model with one server and two types (classes) of customers. Class-1 customers have absolute (service) priority over class-2 customers. New customer batches enter the system at the rate of one batch per slot, according to a general independent arrival process, i.e., the batch sizes (total numbers of arrivals) during consecutive time slots are i.i.d. random variables with arbitrary distribution. All customers entering the system during the same time slot (i.e., belonging to the same arrival batch) are of the same type, but customer types may change from slot to slot, i.e., from batch to batch. Specifically, the types of consecutive customer batches are correlated in a Markovian way, i.e., the probability that any batch of customers has type 1 or 2, respectively, depends on the type of the previous customer batch that has entered the system. Such an arrival model allows to vary not only the relative loads of both customer types in the arrival stream, but also the amount of correlation between the types of consecutive arrival batches. The results reveal that the amount of delay differentiation between the two customer classes that can be achieved by the priority mechanism strongly depends on the amount of such interclass correlation (or, class clustering) in the arrival stream. We believe that this phenomenon has been largely overlooked in the priority-scheduling literature.     

Analysis of preemptive loss priority queues with preemption distance

In a queueing system with preemptive loss priority discipline, customers disappear from the system immediately when their service is preempted by the arrival of another customer with higher priority. Such a system can model a case in which old requests of low priority are not worthy of deferred service. This paper is concerned with preemptive loss priority queues in which customers of each priority class arrive in a Poisson process and have general service time distribution. The strict preemption in the existing model is extended by allowing the preemption distance parameterd such that arriving customers of only class 1 throughp — d can preempt the service of a customer of classp. We obtain closed-form expressions for the mean waiting time, sojourn time, and queue size from their distributions for each class, together with numerical examples. We also consider similar systems with server vacations.     

A preemptive repeat priority queue with resampling: Performance analysis

In this paper, we analyze a discrete-time preemptive repeat priority queue with resampling. High-priority packets have preemptive repeat priority, and interrupted low-priority packets are subjected to independent retransmission attempts. Both classes contain packets with generally distributed transmission times. We show that the use of generating functions is beneficial for analyzing the system contents and packet delay of both classes. The influence of the priority scheduling on the performance measures is illustrated by some numerical examples. This work has been supported by the Interuniversity Attraction Poles Programme–Belgian Science Policy.     

Optimal server scheduling in nonpreemptive finite-population queueing systems

We consider a finite-population queueing system with heterogeneous classes of customers and a single server. For the case of nonpreemptive service, we fully characterize the structure of the server's optimal service policy that minimizes the total average customer waiting costs. We show that the optimal service policy may never serve some classes of customers. For those classes that are served, we show that the optimal service policy is a simple static priority policy. We also derive sufficient conditions that determine the optimal priority sequence.     

A Priority Tandem Queue with No Intermediate Buffer

Priority queues are important in modelling and analysis of manufacturing systems, and computer and communication networks. In this paper, a priority tandem queueing system with two stations in series is studied. There is no intermediate buffer between the two stations, and the lack of buffers may cause blocking at the first station. K types of customers arrive at the system according to Poisson processes. The expected delay in the system for each type of customer is obtained when all the customers have the same service time distribution at the second station. Two cases are studied in detail when service times are either all exponentially distributed or all deterministic.     

Effect of global FCFS and relative load distribution in two-class queues with dedicated servers

In this paper, we investigate multi-class multi-server queueing systems with global FCFS policy, i.e., where customers requiring different types of service—provided by distinct servers—are accommodated in one common FCFS queue. In such scenarios, customers of one class (i.e., requiring a given type of service) may be hindered by customers of other classes. The purpose of this paper is twofold: to gain (qualitative and quantitative) insight into the impact of (i) the global FCFS policy and (ii) the relative distribution of the load amongst the customer classes, on the system performance. We therefore develop and analyze an appropriate discrete-time queueing model with general independent arrivals, two (independent) customer classes and two class-specific servers. We study the stability of the system and derive the system-content distribution at random slot boundaries; we also obtain mean values of the system content and the customer delay, both globally and for each class individually. We then extensively compare these results with those obtained for an analogous system without global FCFS policy (i.e., with individual queues for the two servers). We demonstrate that global FCFS, as well as the relative distribution of the load over the two customer classes, may have a major impact on the system performance.