Two finite-difference methods for solving MAP(t)/PH(t)/1/K queueing models

In this paper two solution methods to the MAP(t)/PH(t)/1/K queueing model are introduced, one based on the Backwards Euler Method and the other on the Uniformization Method. Both methods use finite-differencing with a discretized, adaptive time-mesh to obtain time-dependent values for the entire state probability vector. From this vector, most performance parameters such as expected waiting time and expected number in the system can be computed. Also presented is a technique to compute the entire waiting (sojourn) time distribution as a function of transient time. With these two solution methods one can examine any transient associated with the MAP(t)/PH(t)/1/K model including time-varying arrival and/or service patterns. Four test cases are used to demonstrate the effectiveness of these methods. Results from these cases indicate that both methods provide fast and accurate solutions to a wide range of transient scenarios. This revised version was published online in June 2006 with corrections to the Cover Date.     

Almost sure comparison of birth and death processes with application to M/M/s queueing systems

Given conditions, which concern the infinitesimal parameters of two birth and death processes, the processes are constructed on the same probability space such that one process is almost surely larger than the other. Application is made to M/M/s queueing systems. Stochastic comparisons of queue length and virtual waiting time in two M/M/s systems are obtained.     

排队系统中的泰勒展开方法

综述了排队系统中的泰勒展开方法。它由Gong和Hu在1990s首次提出,并在最近几年里有了一些新的发展。首先,通过GI/GI/1队列的简单例子介绍其基本原理;其次,展示如何应用该方法分析相关性队列和离去过程;然后,阐述如何基于该方法发展排队网络近似的高阶矩方法;最后,讨论未来的几个可能研究方向。     

Statistical analysis of queueing systems

This paper provides an overview of the literature on statistical analysis of queueing systems. Topics discussed include: model identification, estimation, hypothesis testing and other related aspects. Not all of these statistical problems are covered in books on queueing theory or stochastic processes. The bibliography is not exhaustive, but comprehensive enough to provide sources from the literature.     

Lattice path counting andM/M/c queueing systems

We apply the lattice path counting method to the analysis of the transientM/M/c queueing system. A closed-form solution is obtained for the probability of exactlyi arrivals andj departures within a time interval of lengtht in anM/M/c queueing system that is empty at the initial time. The derivation of the probability is based on the counting of paths from the origin to(i,j) on thexy-plane, that have exactly r_d x-steps whose depth from the liney=x isd (d=0,1,...,c–1). The closed-form solution has an expression useful for numerical calculation.     

Transient flows in queueing systems via closure approximations

We present a method for obtaining approximations to the distribution of flow times of customers in arbitrary queueing systems. We first propose approximations for uni-variate and multi-variate distributions of non-negative random variables. Then using a closure approximation, we show that the distribution of flow time can be calculated recursively. Computational results for the single server, multi-server and tandem queues are encouraging, with less than 5%average error in the mean flow time in most cases. The average error in the variance of flow times is found to be less than 10% for the more regular distributions.     

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.     

服务台可修的GI／M（M／PH）／1排队系统

本文首次讨论一个到达间隔为一般分布的可修排队系统。假定服务时间、忙期服务台寿命都服从指疏分布，修复时间是ＰＨ变量。首先证明该系统可转化为一个经典的ＧＩ／￣ＰＨ／１排队模型，然后给出系统在稳态下的各种排队论指标和可靠性指标。     

相依修理的可修排队系统 MAP/PH(M/PH)/2

系统地研究了两个不同并行服务台的可修排队系统MAP/PH(M/PH)/2,其中两个不同的服务台拥有一个修理工.若其中一台处于修理状态,则另一台失效后就处于待修状态.利用拟生灭过程理论,我们首先讨论了两个服务台的广义服务时间的相依性,然后给出了系统的稳态可用度和稳态故障度,最后得到了系统首次失效前的时间分布及其均值.     

M/M/c休假排队系统稳态分布的数值计算

从数值计算角度研究M/M/c休假排队系统稳定状态的概率分布.采用GMRES方法求解概率分布向量所满足的大型线性方程,构造了一个循环预处理算子加速GMRES方法的收敛.数值实例验证了该算法的优越性.     

A time-dependent approximation for the queue M/M(1, s)/c

A time-dependent approximation for the mean number of customersin the queue for the queueing system M/M(1, s)/c is presented,using coordinate transformation. Comparisons between the approximationand an equivalent simulation model are presented, and agreement is shown to be good. The appropriateness of utilizing themodel as an upper bound for queues with smaller coefficientsof variation is discussed. Possible applications of the modelare presented.     

Steady-state analysis of M/M/c/c-type retrial queueing systems with constant retrial rate

The paper deals with a research of bivariate Markov process \(\{X(t), t\ge 0\}\) whose state space is a lattice semistrip \(S(X)=\{0,1,{\ldots },c\} \times Z_{+}\). The process \(\{X(t), t\ge 0\}\) describes the service policy of a multi-server retrial queue in which the rate of repeated flow does not depend on the number of sources of retrial calls. In this class of queues, a vector–matrix representation of steady-state distribution was obtained. This representation allows to write down the stationary probabilities through the model parameters in closed form and to propose the closed formulas of its main performance measures. The investigative techniques use an approximation of the initial model by means of the truncated one and the direct passage to the limit.     

IPP+M/M／C排队的遍历性(英文)

The IP P＋M/M/c queueing system has been extensively used in the modern communication system.The existence and uniqueness of stationary distribution of the queue length L（t）for IP P＋M/M/1 queue has been proved in[10].In this paper,we shall give the su？cient and necessary conditions of l-ergodicity,geometric ergodicity,and prove that they are neither uniformly polynomial ergodicity nor strong ergodicity.     

A characterisation of (max,+)-linear queueing systems

The (max,+)-algebra has been successfully applied to many areas of queueing theory, like stability analysis and ergodic theory. These results are mainly based on two ingredients: (1) a (max,+)-linear model of the time dynamic of the system under consideration, and (2) the time-invariance of the structure of the (max,+)-model. Unfortunately, (max,+)-linearity is a purely algebraic concept and it is by no means immediate if a queueing network admits a (max,+)-linear representation satisfying (1) and (2). In this paper we derive the condition a queueing network must meet if it is to have a (max,+)-linear representation. In particular, we study (max,+)-linear systems with time-invariant transition structures. For this class of systems, we find a surprisingly simple necessary and sufficient condition for (max,+)-linearity, based on the flow of customers through the network. This revised version was published online in June 2006 with corrections to the Cover Date.     

Characteristics of queueing systems observed at events and the connection between stochastic intensity and palm probability

This article explores the link between the concepts of stochastic intensity and Palm probability and gives a new proof and useful extensions to the so-called PASTA property of queueing theory.     

具有优先权的Lu-Kumar排队网络的扩散逼近

对Lu-Kumar排队网络来说,标准的额定负荷条件,即每个工作站的工作强度ρ<1,并不足以保证该排队网络的稳定性,特别是在具有优先权的服务规则下.论文在讨论了Lu-Kumar排队网络稳定性相关结果的基础上,研究了Lu-Kumar排队网络在具有优先权的服务规则下的扩散逼近.证明了当每个工作站的额定负荷ρ→1时,Lu-Kumar排队网络对具有优先权的服务规则的所有优先级别来说,扩散逼近定理均成立.     

Generalized M/G/C/C state dependent queueing models and pedestrian traffic flows

The generality and usefulness ofM/G/C/C state dependent queueing models for modelling pedestrian traffic flows is explored in this paper. We demonstrate that the departure process and the reversed process of these generalizedM/G/C/C queues is a Poisson process and that the limiting distribution of the number of customers in the queue depends onG only through its mean. Consequently, the models developed in this paper are useful not only for the analysis of pedestrian traffic flows, but also for the design of the physical systems accommodating these flows. We demonstrate how theM/G/C/C state dependent model is incorporated into the modelling of large scale facilities where the blocking probabilities in the links of the network can be controlled. Finally, extensions of this work to queueing network applications where blocking cannot be controlled are also presented, and we examine an approximation technique based on the expansion method for incorporating theseM/G/C/C queues in series, merge, and splitting topologies of these networks.     

A sample path analysis of the M t /M t /c queue

The exact transient distribution of the queue length of the M _t /M _t /1 single server queue with timedependent Poisson arrival rate and timedependent exponential service rate was recently obtained by Zhang and Coyle [63] in terms of a solution to a Volterra equation. Their method involved the use of generating functions and complex analysis. In this paper, we present an approach that ties the computation of these transient distributions directly to the random sample path behavior of the M _t /M _t /1 queue. We show the versatility of this method by applying it to the M _t /M _t /c multiserver queue, and indicating how it can be applied to queues with timedependent phase arrivals or timedependent phase service.     

PH/M/c可修排队系统

研究了一个修理工和c个服务台的可修排队系统.假设顾客的到达过程为PH更新过程,服务台在忙时与闲时具有不同的故障率.顾客的服务时间、服务台的寿命以及服务台的修理时间均服从指数分布.通过建立系统的拟生灭过程,得到了系统稳态分布存在的充要条件.利用矩阵几何解方法,给出了系统的稳态队长.在此基础上,得到了系统的某些排队论和可靠性指标.