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.     

The discrete-time preemptive repeat identical priority queue

Priority queueing systems come natural when customers with diversified delay requirements have to wait to get service. The customers that cannot tolerate but small delays get service priority over customers which are less delay-sensitive. In this contribution, we analyze a discrete-time two-class preemptive repeat identical priority queue with infinite buffer space and generally distributed service times. Newly arriving high-priority customers interrupt the on-going service of a low-priority customer. After all high-priority customers have left the system, the interrupted service of the low-priority customer has to be repeated completely. By means of a probability generating functions approach, we analyze the system content and the delay of both types of customers. Performance measures (such as means and variances) are calculated and the impact of the priority scheduling is discussed by means of some numerical examples.     

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 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.      

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.     

Analysis of a finite buffer model with two servers and two nonpreemptive priority classes

In this paper, we analyze a finite buffer queueing model with two servers and two nonpreemptive priority service classes. The arrival streams are independent Poisson processes, and the service times of the two classes are exponentially distributed with different means. One of the two servers is reserved exclusively for one class with high priority and the other server serves the two classes according to a nonpreemptive priority service schedule. For the model, we describe its dynamic behavior by a four-dimensional continuous-time Markov process. Applying recursive approaches we present the explicit representation for the steady-state distribution of this Markov process. Then, we calculate the Laplace–Stieltjes Transform and the steady-state distribution of the actual waiting times of two classes of customers. We also give some numerical comparison results with other queueing models.     

Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm-Liouville problems

The differential quadrature method (DQM) and the Boubaker Polynomials Expansion Scheme (BPES) are applied in order to compute the eigenvalues of some regular fourth-order Sturm-Liouville problems. Generally, these problems include fourth-order ordinary differential equations together with four boundary conditions which are specified at two boundary points. These problems concern mainly applied-physics models like the steady-state Euler-Bernoulli beam equation and mechanicals non-linear systems identification. The approach of directly substituting the boundary conditions into the discrete governing equations is used in order to implement these boundary conditions within DQM calculations. It is demonstrated through numerical examples that accurate results for the first kth eigenvalues of the problem, where k = 1, 2, 3, … , can be obtained by using minimally 2(k + 4) mesh points in the computational domain. The results of this work are then compared with some relevant studies.     

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.     

Mixing of non-preemptive and preemptive repeat priority disciplines

A single server dispenses service to m priority classes. The arrival process of the ith class, i = 1, 2,…,m, is homogeneous Poisson. Service times of each class are independent, identical, arbitrarily-distributed random variables with a finite second moment. The smaller the index of a class, the higher its priority degree. For i < j, class i has preemptive priority over j if and only if j ? i > d (where d is a predetermined non-negative integer), and non-preemptive priority otherwise. An interrupted service is resumed when the system contains no costomers with higher priority, preemptive and non-preemptive. Within each priority class the FIFO rule is obeyed.The preemptive regimes analyzed are repeat with and without resampling. For a k-customer, k = 1, 2,…,n, steady-state Laplace-Stieltjes transforms, and expectations of the waiting time and the time in the system are calculated.     

On the steady-state queue size distribution of the discrete-timeGeo/G/1 queue with repeated customers

In this paper, we study the steady-state queue size distribution of the discrete-timeGeo/G/1 retrial queue. We derive analytic formulas for the probability generating function of the number of customers in the system in steady-state. It is shown that the stochastic decomposition law holds for theGeo/G/1 retrial queue. Recursive formulas for the steady-state probabilities are developed. Computations based on these recursive formulas are numerically stable because the recursions involve only nonnegative terms. Since the regularGeo/G/1 queue is a special case of theGeo/G/1 retrial queue, the recursive formulas can also be used to compute the steady-state queue size distribution of the regularGeo/G/1 queue. Furthermore, it is shown that a continuous-timeM/G/1 retrial queue can be approximated by a discrete-timeGeo/G/1 retrial queue by dividing the time into small intervals of equal length and the approximation approaches the exact when the length of the interval tends to zero. This relationship allows us to apply the recursive formulas derived in this paper to compute the approximate steady-state queue size distribution of the continuous-timeM/G/1 retrial queue and the regularM/G/1 queue.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0046415.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0105828.     

Analysis of the adaptive MMAP[K]/PH[K]/1 queue: A multi-type queue with adaptive arrivals and general impatience

In this paper we introduce the adaptive MMAP[K] arrival process and analyze the adaptive MMAP[K]/PH[K]/1 queue. In such a queueing system, customers of K different types with Markovian inter-arrival times and possibly correlated customer types, are fed to a single server queue that makes use of r thresholds. Service times are phase-type and depend on the type of customer in service. Type k customers are accepted with some probability a_i,k if the current workload is between threshold i − 1 and i. The manner in which the arrival process changes its state after generating a type k customer also depends on whether the customer is accepted or rejected.     

An Exact Solution for the State Probabilities of the Multi-Class,Multi-Server Queue with Preemptive Priorities

We consider a multi-class, multi-server queueing system with preemptive priorities. We distinguish two groups of priority classes that consist of multiple customer types, each having their own arrival and service rate. We assume Poisson arrival processes and exponentially distributed service times. We derive an exact method to estimate the steady state probabilities. Because we need iterations to calculate the steady state probabilities, the only error arises from choosing a finite number of matrix iterations. Based on these probabilities, we can derive approximations for a wide range of relevant performance characteristics, such as the moments of the number of customers of a certain type in the system en the expected postponement time for each customer class. We illustrate our method with some numerical examples. Numerical results show that in most cases we need only a moderate number of matrix iterations (∼20) to obtain an error less than 1% when estimating key performance characteristics.This revised version was published online in June 2005 with corrected coverdate     

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0.     

带负顾客和强占优先权的不耐烦信元排队

考虑带有负顾客的两类信元的强占优先权M/M/1排队系统.两类信元及负顾客的到达过程均为泊松过程.两类信元到达后分别在各自有限的缓冲器内排队,第一类信元较第二类信元有强占优先权,同时第一类信元是不耐烦的.负顾客一对一抵消队尾的第一类信元(若有),若系统中无第一类信元,到达的负顾客就自动消失.负顾客不接受服务.采用矩阵分析的方法得到了两类信元各自的稳态分布,并作了相应的性能分析.     

带负顾客和N-策略的Geo^{lambda_1, lambda_2/Geo/1(MWV)排队系统分析及最优控制策略N*

分析了一个带有负顾客、N-策略控制的Geo/Geo/1多重工作休假排队系统, 其中正顾客在工作休假及正规忙期以不同的到达率进入系统. 利用拟生灭过程和矩阵几何解方法, 给出了该模型的稳态队长分布及平均队长, 以及系统分别处于假期和忙期的概率. 同时, 对该系统的忙期进行了分析, 并讨论了稳态队长分布在系统容量的优化设计中的应用. 最后, 在给定的费用结构下, 用数值计算例子确定了使系统长期单位时间内期望费用最小的最优控制策 N*.     

Performance analysis of a GI-Geo-1 buffer with a preemptive resume priority scheduling discipline

In this paper, we analyze a discrete-time GI-Geo-1 preemptive resume priority queue. We consider two classes of packets which have to be served, where one class has preemptive resume priority over the other. We show that the use of generating functions is beneficial for analyzing the system contents and packet delay of both classes. Moments and (approximate) tail probabilities of system contents and packet delay are calculated. The influence of the priority scheduling is shown by some numerical examples.     

Asymptotics for M/G/1 low-priority waiting-time tail probabilities

We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptive-resume disciplines. We show that the low-priority steady-state waiting-time can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waiting-time distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have non-exponential asymptotics. This phenomenon even occurs when both service-time distributions are exponential. When non-exponential asymptotics holds, the asymptotic form tends to be determined by the non-exponential asymptotics for the high-priority busy-period distribution. We obtain asymptotic expansions for the low-priority waiting-time distribution by obtaining an asymptotic expansion for the busy-period transform from Kendall's functional equation. We identify the boundary between the exponential and non-exponential asymptotic regions. For the special cases of an exponential high-priority service-time distribution and of common general service-time distributions, we obtain convenient explicit forms for the low-priority waiting-time transform. We also establish asymptotic results for cases with long-tail service-time distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the non-exponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waiting-time transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = (p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.     

The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption

Consider a GI/M/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start-up period, after the start-up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0 ? p ? 1) or continue the vacation with probability 1 − p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time.     

Optimal (d, c) vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs

This paper considers a finite buffer M/M/c queueing system in which servers are unreliable and follow a (d, c) vacation policy. With such a policy, at a service completion instant, if the number of customers is reduced to c − d (c > d), the d idle servers together take a vacation (or leave for a random amount of time doing other secondary job). When these d servers return from a vacation and if still no more than c − d customers are in the system, they will leave for another vacation and so on, until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. This study is motivated by the fact that some practical production and inventory systems or call centers can be modeled as this finite-buffer Markovian queue with unreliable servers and (d, c) vacation policy. Using the Markovian process model, we obtain the stationary distribution of the number of customers in the system numerically. Some cost relationships among several related systems are used to develop a finite search algorithm for the optimal policy (d, c) which maximizes the long-term average profit. Numerical results are presented to illustrate the usefulness of such a algorithm for examining the effects of system parameters on the optimal policy and its associated average profit.