An M/G/1 Retrial Queueing System with Two-Phase Service and Preemptive Resume

An M/G/1 retrial queueing system with additional phase of service and possible preemptive resume service discipline is considered. For an arbitrarily distributed retrial time distribution, the necessary and sufficient condition for the system stability is obtained, assuming that only the customer at the head of the orbit has priority access to the server. The steady-state distributions of the server state and the number of customers in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analysed numerically. A general decomposition law for this retrial queueing system is established.     

一类批量服务排队系统的求解

给出一类特殊的批量服务排队系统-公交系统的稳态概率分布的求解过程,并在此基础上给出这类服务系统平均队长的算法,中给出具体例子说明计算过程。     

On the Three Threshold Policy in the Multi-Server Queueing System with Vacations

In this paper, we study an M/M/c queue with a three threshold vacation policy denoted by (e, d, N). With such a policy, the servers keep serving the customers until the number of idle servers reaches d and then e of d servers start taking a vacation together. These e servers keep taking vacations until the number of customers in the system is at least N at a vacation completion instant, then the e servers return to serve the queue again. Using the matrix analytic method, we obtain the stationary performance measures and prove the conditional stochastic decomposition properties for the waiting time and queue length. This model is a generalization of previous multi-server vacation models and offers a useful performance evaluation and system design tool in multi-task server queueing systems.     

Scheduling in a multi-class series of queues with deterministic service times

We consider a problem of scheduling in a multi-class network of single-server queues in series, in which service times at the nodes are constant and equal. Such a model has potential application to automated manufacturing systems or packet-switched communication networks, where a message is divided into packets (or cells) of fixed lengths. The network is a series-type assembly or transfer line, with the exception that there is an additional class of jobs that requires processing only at the first node (class 0). There is a holding cost per unit time that is proportional to the total number of customers in the system. The objective is to minimize the (expected) total discounted holding cost over a finite or an infinite horizon. We show that an optimal policy gives priority to class-0 jobs at node 1 when at least one of a set ofm–1 inequalities on partial sums of the components of the state vector is satisfied. We solve the problem by two methods. The first involves formulating the problem as a (discrete-time) Markov decision process and using induction on the horizon length. The second is a sample-path approach using an interchange argument to establish optimality.The research of this author was supported by the National Science Foundation under Grant No. DDM-8719825. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.     

Analysis of a Discrete-Time Queueing System with Timed Vacations

We consider a discrete-time GI-G-1 queueing system with server vacations. Vacations occur whenever the queue becomes empty or whenever a timer expires. When the timer expires one of the following four actions are considered: the server completes transmission of the present packet before leaving for a vacation, the service immediately leaves for a vacation and the interrupted packet's service is either continued, repeated or resampled and repeated after the vacation. Using a probability generating functions approach, we derive various performance measures such as moments of the buffer contents at various time epochs in equilibrium and moments of the packet delay in equilibrium. By means of an example, we then compare the operation modes under consideration.     

基于排队的库存服务系统最优控制策略

研究一个连续盘点的(s,Q)补货的库存服务系统。基于排队理论建立库存水平状态平衡方程,并推导出库存水平稳态概率分布以及作为库存控制的系统稳态性能指标。以库存成本最小化为目标,构建服务水平约束的库存控制模型。针对模型的非线性约束与整数型变量的特征,采用一种改进的遗传算法(IGA)用于决策变量的寻优。数值实验表明,当目标服务水平大于库存系统内生的服务水平时,实施服务水平约束能够降低库存控制成本。     

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.     

具有负顾客和特殊类顾客的串联排队系统分析

讨论了带有普通类顾客、负顾客和特殊类顾客的M/M/1→M/M/1两级串联排队系统模型,负顾客在一级服务系统中,一对一抵消队尾的普通类顾客(若有),若一级服务系统无普通类顾客,负顾客自动消失,负顾客不进入二级服务系统.特殊类顾客不经过一级服务系统,直接进入二级服务系统等待接受服务.用拟生灭过程和矩阵几何解方法,得到了系统稳态队长的分布,以及系统忙期的分布和顾客逗留时间的分布等相关指标.     

On the optimality of static policy in stochastic open shop

Consider n jobs and two machines. Each job has to be processed on both machines. The order in which it is dome is immaterial. However, the decision maker has to decide in advance which jobs will be processes first on machine 1 (2). We assume that processing times on each machine are identically exponentially distributed random variables. We prove that assigning equal number of jobs to be first processed by machine 1 (2) stochastically minimizes the makespan.     

批量到达的、具有两种服务阶段和服务中断的重试排队系统的适定性

运用Hille-Yosida定理,Phillips定理与Fattorini定理证明批量到达、具有两种服务阶段和服务中断的重试排队系统存在唯一的、非负的、满足概率性质的时间依赖解.     

M/M/m/m防空系统射击效能的排队概率特性

研究了具有消失制的M/M/m/m防空系统的射击效能,利用排队论及随机运筹学的有关知识,在模型的条件与假设下给出了其平稳状态的队长的分布律πk,平均工作的防空武器数E,敌机的突防概率πm,忙期长度等指标.     

On the Counting Process for a Class of Markovian Arrival Processes with an Application to a Queueing System

In this paper, we consider the counting process for a class of Markovian arrival processes (MAPs). We assume that the representing matrices in such MAPs are expanded in terms of matrix representations of the standard generators in the Lie algebra of the special linear group. The primary purpose of this paper is to construct an explicit solution of the time-dependent distribution and factorial moments of the number of arrival events in (0,t] of the counting process for this class of MAPs. Our construction relies on the Baker–Hausdorff lemma and the specific structure of the representing matrices. To investigate the efficiency of CPU usage with the explicit solution, we have conducted numerical experiments on computing the time-dependent distribution of the counting process through the explicit solution and uniformization-based method. We show that the CPU times required to compute the time-dependent distribution of the number of arrival events in (0,t] through the explicit solution have little sensitivity to t, while the consumption of CPU times with the uniformization-based method becomes greater as t increases. For illustrative purposes, we present a system performance analysis of a queueing system for possible use in automatic call distribution (ACD) systems. As an application of the explicit solution, we use it to express the waiting time distribution of the queueing system. Some numerical examples are also given with comparisons to computer simulations.AMS subject classification: 60K20, 60K25, 68M20This revised version was published online in June 2005 with corrected coverdate     

Min(N,V)--策略休假的M/G/1排队系统分析

在一个M／G／1休假排队系统中，同时考虑N-策略和多重休假策略，休假终止准则为任一个条件满足，我们称其为Min（N，V）-策略。本文给出了在此策略下的排队系统的稳态队长、忙期分布等基本指标。首次使用条件等待时间方法得到稳态等待时间的LST（Laplace-Stieltjes transform），同时还列举了一个应用的实例。最后指出本文模型是几个已研究模型的推广。     

On the Exact Analysis of a Discrete-Time Queueing System with Autoregressive Inputs

In this paper, we provide an exact analysis of a discrete-time queueing system driven by a discrete autoregressive model of order 1 (DAR(1)) characterized by an arbitrary marginal batch size distribution and a correlation coefficient. Closed-form expressions for the probability generating function and mean queue length are derived. It is shown that the system performance is quite sensitive to the correlation of the arrival process. In addition, a comparison with traditional Markovian processes shows that arrival processes of DAR(1) type exhibit larger queue length as compared with the traditional Markovian processes when the marginal densities and correlation coefficients are matched.     

On the output process of an M/M/1 queue with randomly varying system parameters

In this paper an analysis of the output process from an M/M/1 queue where the arrival and service rates vary randomly is presented. The results include expressions for the mean, variance and distribution of the interdeparture interval, the joint density function of two successive interdeparture intervals and their correlation. An interesting feature of the results is that the moments of the interdeparture time are expressed in terms of the expected times to first and second departures from an arbitrary point in time.     

On the order of tandem queues

We consider the optimal order of servers in a tandem queueing system withm stages, an unlimited supply of customers in front of the first stage, and a service buffer of size 1 but no intermediate storage buffers between the first and second stages. Service times depend on the servers but not the customers, and the blocking mechanism at the first two stages is manufacturing blocking. Using a new characterization of reversed hazard rate order, we show that if the service times for two servers are comparable in the reversed hazard rate sense, then the departure process is stochastically earlier if the slower server is first and the faster server is second than if the reverse is true. This strengthens earlier results that considered individual departure times marginally. We show similar results for the last two stages and for other blocking mechanisms. We also show that although individual departure times for a system with servers in a given order are stochastically identical to those when the order of servers is reversed, this reversibility property does not hold for the entire departure process.     

Convergence of a stochastic approximation algorithm for the GI/G/1 queue using infinitesimal perturbation analysis

Discrete-event systems to which the technique of infinitesimal perturbation analysis (IPA) is applicable are natural candidates for optimization via a Robbins-Monro type stochastic approximation algorithm. We establish a simple framework for single-run optimization of systems with regenerative structure. The main idea is to convert the original problem into one in which unbiased estimators can be derived from strongly consistent IPA gradient estimators. Standard stochastic approximation results can then be applied. In particular, we consider the GI/G/1 queue, for which IPA gives strongly consistent estimators for the derivative of the mean system time. Convergence (w.p.1) proofs for the problem of minimizing the mean system time with respect to a scalar service time parameter are presented.     

On the equivalence of three types of blocking in non-Markovian tandem queues

Definitions of blocking types for tandem queueing systems are given where each stage consists of multiple servers. Some of them are shown to be identical with respect to service initiation epochs at each stage. Such identical relationships are proved using sample path correspondence, and are therefore valid irrespective of the arrival process and the service time distributions.     

On the Analysis of the Virtual Waiting Time in Open Queueing Networks

The model of an open queueing network in heavy traffic has been developed. These models are mathematical models of computer networks in heavy traffic. A limit theorem has been presented for the virtual waiting time of a customer in heavy traffic in open queueing networks. Finally, we present an application of the theorem—a reliability model from computer network practice.     

Scheduling networks of queues: Heavy traffic analysis of a simple open network

We consider a queueing network with two single-server stations and two types of customers. Customers of type A require service only at station 1 and customers of type B require service first at station 1 and then at station 2. Each server has a different general service time distribution, and each customer type has a different general interarrival time distribution. The problem is to find a dynamic sequencing policy at station 1 that minimizes the long-run average expected number of customers in the system.The scheduling problem is approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved, and the solution is interpreted in terms of the queueing system in order to obtain an effective sequencing policy. Also, a pathwise lower bound (for any sequencing policy) is obtained for the total number of customers in the network. We show via simulation that the relative difference between the performance of the proposed policy and the pathwise lower bound becomes small as the load on the network is increased toward the heavy traffic limit.