M/T-SPH/1排队平稳指标分布的尾部衰减特征

讨论M/T-SPH/1排队平稳队长分布和平稳逗留时间分布的尾部衰减特征,其中T-SPH表示可数状态吸收生灭过程吸收时间的分布。在分布PGF和LST的基础上,给出了两个平稳分布衰减规律的完整分析.结果表明,当参数取不同值时,平稳队长与平稳逗留时间的尾部具有三种不同类型的衰减特征.     

对M/T-SPH/1排队平稳指标的进一步研究—数值计算与渐近分析

讨论M/T-SPH/1排队平稳队长分布的数值计算,以及平稳队长和逗留时间分布各阶矩的数值计算及渐近分析.其中T-SPH表示可数状态吸收生灭链吸收时间的分布.在分布PGF和LST的基础上,首先给出了计算平稳队长分布,平稳队长以及逗留时间分布各阶矩的数值结果的递推公式.其次还讨论了平稳队长及平稳逗留时间分布各阶矩的尾部渐近特征.结果表明当参数取不同值时,两个指标尾部具有三种不同类型的衰减方式.最后还用数值例子检验了方法的有效性.     

Analysis of the GI/Geo/1 Queue with Disasters

The occurrence of disasters to a queueing system causes all customers to be removed if any are present. Although there has been much research on continuous-time queues with disasters, the discrete-time Geo/Geo/1 queue with disasters has appeared in the literature only recently. We extend this Geo/Geo/1 queue to the GI/Geo/1 queue. We present the probability generating function of the stationary queue length and sojourn time for the GI/Geo/1 queue. In addition, we convert our results into the Geo/Geo/1 queue and the GI/M/1 queue.     

A GI/Geo/1 queue with negative and positive customers

The arrival of a negative customer to a queueing system causes one positive customer to be removed if any is present. Continuous-time queues with negative and positive customers have been thoroughly investigated over the last two decades. On the other hand, a discrete-time Geo/Geo/1 queue with negative and positive customers appeared only recently in the literature. We extend this Geo/Geo/1 queue to a corresponding GI/Geo/1 queue. We present both the stationary queue length distribution and the sojourn time distribution.     

带有Bernoulli控制策略的M/M/1多重休假排队模型

研究具有Bernoulli控制策略的M/M/1多重休假排队模型: 当系统为空时, 服务台依一定的概率或进入闲期, 或进入普通休假状态, 或进入工作休假状态. 对该模型, 应用拟生灭(QBD)过程和矩阵几何解的方法, 得到了过程平稳队长的具体形式, 在此基础上, 还得到了平稳队长和平稳逗留时间的随机分解结果以及附加队长分布和附加延迟的LST的具体形式. 结果表明, 经典的M/M/1排队, M/M/1多重休假排队, M/M/1多重工作休假排队都是该模型的特殊情形.     

Analysis of BMAP/G/1 Queue with Reservation of Service

Abstract

In this article, we study BMAP/G/1 queue with service time distribution depending on number of processed items. This type of queue models the systems with the possibility of preliminary service. For the considered system, an efficient algorithm for calculating the stationary queue length distribution is proposed, and Laplace–Stieltjes transform of the sojourn time is derived. Little's law is proved. An optimization problem is considered.     

A MAP/G/1 Queue with Negative Customers

In this paper, we consider a MAP/G/1 queue with MAP arrivals of negative customers, where there are two types of service times and two classes of removal rules: the RCA and RCH, as introduced in section 2. We provide an approach for analyzing the system. This approach is based on the classical supplementary variable method, combined with the matrix-analytic method and the censoring technique. By using this approach, we are able to relate the boundary conditions of the system of differential equations to a Markov chain of GI/G/1 type or a Markov renewal process of GI/G/1 type. This leads to a solution of the boundary equations, which is crucial for solving the system of differential equations. We also provide expressions for the distributions of stationary queue length and virtual sojourn time, and the Laplace transform of the busy period. Moreover, we provide an analysis for the asymptotics of the stationary queue length of the MAP/G/1 queues with and without negative customers.     

Numerical Computation and Tail Asymptotic for Stationary Indices of a Discrete-Time Vacation Queue

In this paper we study a Geo/Geo/1 queue with T-IPH vacations, where T-IPH denotes the discrete-time phase type distribution defined on a birth and death process with countably many states. Both the multiple and single vacation strategies are considered. For each case, based on the system of stationary equations and using complex analysis method, we firstly give the probability generating functions (PGFs) of stationary distributions for queue length and sojourn time. Moreover, by analysis the PGFs, recursive and asymptotic formulas for additional queue length and additional delay are also given. Finally, we further give some numerical examples to show the effectiveness of the method.     

PH/PH/1/N反馈排队系统的逗留时间

具有反馈依赖于队长的PH／PH／1／N排队系统的队长和忙期的研究已在文[1]中解决，本文主要解决本系统的逗留时间的研究．     

Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N  -policy GI/Geo/1$\mathit{GI}/\mathit{Geo}/1$ queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.     

Sojourn time distributions in the queue defined by a general QBD process

We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.     

带启动时间的Geom/Geom/1多重工作休假排队

本文中研究了一个带有启动时间的Geom/Geom/1多重工作休假排队模型。服务台在休假期间,不停止服务,而是以较低的服务率为顾客提供服务。运用拟生灭过程和矩阵几何解的方法,给出了该模型的稳态队长分布,并求出了平均队长以及顾客的平均逗留时间。     

离散时间排队MAP/PH/3

本文研究具有马尔可夫到达过程的离散时间排队MAP/PH/3,系统中有三个服务台,每个服务台对顾客的服务时间均服从位相型分布。运用矩阵几何解的理论,我们给出了系统平稳的充要条件和系统的稳态队长分布。同时我们也给出了到达顾客所见队长分布和平均等待时间。     

基于Geo/Geo/1(E,SV)排队系统的均衡止步策略

基于单重休假Geo/Geo/1排队系统,研究顾客的均衡止步策略,首次将休假服务机制引入到离散时间排队经济学模型中. 顾客基于“收入--支出”结构,自主决定去留. 利用拟生灭过程理论,运用差分方程求解技巧,对系统进行了稳态分析,得到了顾客的平均逗留时间;进而构造适当的函数,给出了寻找均衡止步策略的具体方法并证明之;而后分析了在均衡策略下, 系统的稳态行为和社会收益;最后通过数值实验讨论了系统参数对均衡行为的影响.     

A network of priority queues in heavy traffic: One bottleneck station

In this paper we consider an open queueing network having multiple classes, priorities, and general service time distributions. In the case where there is a single bottleneck station we conjecture that normalized queue length and sojourn time processes converge, in the heavy traffic limit, to one-dimensional reflected Brownian motion, and present expressions for its drift and variance. The conjecture is motivated by known heavy traffic limit theorems for some special cases of the general model, and some conjectured “Heavy Traffic Principles” derived from them. Using the known stationary distribution of one-dimensional reflected Brownian motion, we present expressions for the heavy traffic limit of stationary queue length and sojourn time distributions and moments. For systems with Markov routing we are able to explicitly calculate the limits.     

A G/M/1 retrial queue with constant retrial rate

In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented.     

Palm calculus for a process with a stationary random measure and its applications to fluid queues

We consider a process associated with a stationary random measure, which may have infinitely many jumps in a finite interval. Such a process is a generalization of a process with a stationary embedded point process, and is applicable to fluid queues. Here, fluid queue means that customers are modeled as a continuous flow. Such models naturally arise in the study of high speed digital communication networks. We first derive the rate conservation law (RCL) for them, and then introduce a process indexed by the level of the accumulated input. This indexed process can be viewed as a continuous version of a customer characteristic of an ordinary queue, e.g., of the sojourn time. It is shown that the indexed process is stationary under a certain kind of Palm probability measure, called detailed Palm. By using this result, we consider the sojourn time processes in fluid queues. We derive the continuous version of Little's formula in our framework. We give a distributional relationship between the buffer content and the sojourn time in a fluid queue with a constant release rate.     

Sojourn time analysis of a two-phase queueing system with exhaustive batch-service and its vacation model

We consider an M/G/1-type, two-phase queueing system, in which the two phases in series are attended alternatively and exhaustively by a moving single-server according to a batch-service in the first phase and an individual service in the second phase. We show that the two-phase queueing system reduces to a new type of single-vacation model with non-exhaustive service. Using a double transform for the joint distribution of the queue length in each phase and the remaining service time, we derive Laplace-Stieltjes transforms for the sojourn time in each phase and the total sojourn time in the system. Furthermore, we provide the moment formula of sojourn times and numerical examples of an approximate density function of the total sojourn time.     

Geo/G/1 queues with disasters and general repair times

This paper discusses discrete-time single server Geo/G/1 queues that are subject to failure due to a disaster arrival. Upon a disaster arrival, all present customers leave the system. At a failure epoch, the server is turned off and the repair period immediately begins. The repair times are commonly distributed random variables. We derive the probability generating functions of the queue length distribution and the FCFS sojourn time distribution. Finally, some numerical examples are given.     

A stochastic network with mobile users in heavy traffic

We consider a stochastic network with mobile users in a heavy traffic regime. We derive the scaling limit of the multidimensional queue length process and prove a form of spatial state space collapse. The proof exploits a recent result by Lambert and Simatos (preprint, 2012), which provides a general principle to establish scaling limits of regenerative processes based on the convergence of their excursions. We also prove weak convergence of the sequences of stationary joint queue length distributions and stationary sojourn times.