Some New Results for M/G/1 and GI/G/1 Queues

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An M/G/1 Queueing System with Delay Server Vacations

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Some Reliability Problems Arising in the M~x/G(M/G)/1 Repairable Queueing System with Single Delay Vacation

1　 IntroductionIn Ref.[1 ] some queueing indices of the Mx/ G( M/ G) / 1 repairable queueing system withsingle delay vacation were discussed,and the transient solution,stationary solution andstochastic decomposition of the queue length were obtained.Butwhile the service stationis up and operating,it is subject to breakdowns.Once the service station breaks down,there is a repair operation that bring the broken service station back to a new state andoperation immediately.So it is important to…     

N个不同部件串联而成的M/G/1可修排队系统

考虑服务台由N个不同部件串联而成的M/G/1可修排队系统,其中服务台正常当且仅当N个部件都正常.在假定每个部件的寿命服从指数分布,而故障部件的修理时间服从一般分布下,通过引入服务台的"广义忙期",提出了分析服务台有关可靠性指标的一种新方法,用该方法更加简洁的讨论了服务台的许多感性趣的可靠性指标,得到了一些重要的可靠性结果,推广了已有的结论.     

Optimal control of an M/G/1 retrial queue with vacations

In this note, we consider an M/G/1 retrial queue with server vacations, when retrial times, service times and vacation times are arbitrary distributed. The distribution of the number of customers in the system in stationary regime is obtained in terms of generating function. Next, we give heavy traffic approximation of such distribution. We show that the system size can be decomposed into two random variables, one of which corresponds to the system size of the ordinary M/G/1 FIFO queue without vacation. Such a stochastic decomposition property is useful for the computation of performance measures of interest. Finally, we solve simple problems of optimal control of vacation and retrial policies.     

M/G/1排队系统的性能灵敏度估计与仿真

对具有一般分布的排队系统-M／G／l排队系统给出了一种通过其嵌入Markov链来计算系统势能及性能导数的仿真算法。由于该算法基于分析系统的一条单一样本轨道，故可直接用于系统的控制与优化。最后提供一个数值例子来表明该算法的应用。     

多重假期中以概率p进入的M/G/1可修排队系统

考虑在服务员假期中到达的顾客以概率p(0＜p≤1)进入系统的多重休假M/G/1可修排队系统,运用全概率分解技术和拉普拉斯变换,研究了服务台的下列可靠性指标:1)首次失效前的寿命分布;2)瞬时不可用度和稳态不可用度;3)在(0,t]时间内的平均失效次数.获得了服务台一系列的可靠性结果.     

具有启动失效的MAP/G/1一般重试系统分析

考虑具有启动失效的M AP/G/1一般重试排队系统,即系统中顾客在服务台空闲或服务结束时,以概率θ成功接受服务,而以概率=θ1-θ服务失效,并且服务台开始处于修理期。利用补充变量法和RG分解求得稳态时系统的队长分布以及系统的平均忙期等指标。     

G/M/1排队系统的性能灵敏度分析与仿真

对G／M／1排队系统，通过研究其嵌入Markov链，讨论了系统的稳态性能灵敏度分析问题，导出了系统的稳态分布与其嵌入Markov链的稳态分布之间的关系式，给出了用嵌入Markov链的势能表示的稳态性能灵敏度公式。并提出了通过嵌入Markov链来计算系统性能灵敏度的仿真算法，由于该算法基于系统的一条单一样本轨道，故可直接应用于系统的控制与优化。最后提供了两种具体的G／M／1排队系统的例子来说明该算法的应用。     

受控M/G/1排队系统的性能优化及迭代算法

通过嵌入Markov链的方法,讨论了受控M/G/1排队系统,在无限水平平均代价准则下的最优平稳策略问题。定义了平均Poisson方程,导出了平均代价模型在紧致行动集上的最优性方程。最后给出了一个求解最优平稳策略的迭代算法,并给出了一个仿真实例来说明该算法的应用。     

一类休假的Geo~X/G/1可修排队的可靠性分析

考虑单重休假、Bernoulli反馈和可变输入率的离散时间Geo~X/G/1可修排队.顾客的批到达速率与服务器的休假有关.刚服务完的顾客以概率1-θ进入队列寻求下次服务.服务器在服务过程中可能故障需修复后再继续工作.借助更新过程理论、z变换和一种分解法,研究了时刻n+位于服务器忙期的条件概率、服务器的瞬态和稳态不可用度以及(0~+,n~+]时间内服务器的平均故障次数和稳态故障频度,揭示了这类离散时间可修排队中服务器可靠性指标的结构,得到了一些特殊可修排队的可靠性结果.最后通过数值实例分析了系统参数对服务器可靠性指标的影响.     

服务台可修的M／G／1排队系统的进一步分析

在文［１］的基础上，本文进一步讨论了如下问题：１）在广义服务时间和系统忙期内服务台的失效次数；２）在（０，ｔ］内服务台平均失效次数的渐近展；３）在广义服务时间和系统忙期内服务台总的失效时间，并得到一系列新结果.     

带启动时间的N-策略M/G/1排队系统的队长

从任意初始状态出发, 直接研究了系统队长的瞬态分布和稳态分布.通过引进的“服务员忙期", 使用全概率分解技术和拉普拉斯变换,导出了在任意时刻t队长的瞬态分布的拉普拉斯变换的表达式,进一步获得了有重要应用价值的稳态分布的具体的递推式子,以及稳态队长的随机分解结果. 特别地,还直接获得了一些特殊排队系统的更实用的稳态队长分布的递推表达式.     

第二类故障期间以概率p进入的M/G/1可修排队

假设服务台在忙期和闲期内都可能发生故障,且具有不同的故障率,并且在闲期的故障状态期间到达顾客以概率p进入系统.使用全概率分解技术和利用拉普拉斯变换工具,研究了服务台的瞬态不可用度、稳态不可用度、(0,t]时间内的平均故障次数和稳态故障频度,获得服务台一些重要的可靠性结果,并且分别讨论了当p取值为0和取值为1时的特殊情况.     

Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation

This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to Bernoulli vacation policy.It is assumed that the server, after each service completion,begins a process of search in order to find the following customer to be served with a certain probability,or begins a single vacation process with complementary probability. This paper analyzes the Markov chain underlying the queueing system and obtain its ergodicity condition.The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle,busy or on vacation.Finally,the author gives two stochastic decomposition laws,and as an application the author gives bounds for the proximity between the system size distributions of the model and the corresponding model without retrials.     

带负顾客和不耐烦顾客的离散时间Geo/G/1重试排队

考虑了一个带负顾客和不耐烦顾客且重试时间为一般分布的离散时间Geo/G/1重试排队系统. 负顾客带走一个正在服务的顾客, 而对重试组中的顾客无影响.正顾客到达系统若遇服务器忙则可能进入重试组也可能离开系统.通过对此排队系统的嵌入马氏链进行分析, 得到了重试组队长和系统队长的概率母函数. 进而得到了一系列重要的排队指标. 此外, 还推导出了系统的稳态存在条件. 以及对无负顾客和不耐烦顾客时的特例进行了分析. 最后通过几个具体的数值实例演示了一些参数对系统关键性能指标的影响.     

Transient Queue Size Distribution Solution of Geom / G / 1 Queue with Feedback-A Recursive Method

This paper considers the Geom / G / 1 queueing model with feedback according to a late arrival system with delayed access (LASDA). Using recursive method, this paper studies the transient property of the queue size from the initial state N(0⁺) = i. Some new results about the recursive expression of the transient queue size distribution at any epoch n ⁺ and the recursive formulae of the equilibrium distribution are obtained. Furthermore, the recursive formulae of the equilibrium queue size distribution at epoch n ⁻, and n are obtained, too. The important relations between stationary queue size distributions at different epochs are discovered (being different from the relations given in M / G / 1 queueing system). The model discussed in this paper can be widely applied in all kinds of communications and computer network. This research is supported by the National Natural Science Foundation of China under Grant No. 70871084, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 200806360001, and the Scientific Research Fund of Southwestern University of Finance and Economics.     

Bernoulli反馈排队的N策略Geom／G／1排队系统的队长分布

考虑具有Bernoulli反馈排队和N策略体假的离散时间Geom/G/1排队系统,采用一种直观、简洁的全概率分解方法以及U-变换技术,研究了该排队系统队长的瞬态性质,得到队长瞬态分布的U-变换形式的递推表达式;进一步导出队长稳态分布的递推表达以及平稳队长分布的随机分解;最后,计算出稳态队长分布,并且考察了其统计性质.     

基于Geom/G/1休假排队的顾客止步策略研究

将决策行为引入排队模型中, 以顾客追求利益最大化为出发点, 研究空竭服务、多重休假Geom/G/1排队模型中顾客的优化止步策略. 在不可见排队的前提下, 基于"收益-成本"结构, 采取均值分析的方法, 构建顾客个人和顾客总体的收益函数, 进而, 在不同参数范围内, 分析并确定出顾客均衡策略和社会最优策略. 最后, 通过数值模拟完善结论.     

The Queue Length Distribution for M/G/1 Queue with Delay Single Server Vacation

1　IntroductionDuringrecentseveraldecadesmanyauthorsstudiedM/G/1queueswithdifferentservervacationregimes(seeRefs.[1～6]).Theynotonlystudiedthestochasticdecompositionpropertiesofthequeuelengthandwaitingtimewhenthesystemisinequilibrium,butalsostudiedthetransientandequilibriumdistributionsofthequeuelength.InRef.[6]theau-thorsstudiedM/G/1queuewithdelaymultipleservervacationsinwhichtheserverhasthreestates:vacation-preparation,vacationandbusyperiod.InthispaperwefirstdiscusstheM/G/1queuewithdelay…