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

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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.     

延迟N-策略M/G/1排队系统队长的瞬态和稳态分布

研究延迟N-策略M/G/1排队系统,讨论了队长的瞬态和稳态性质.通过引进"服务员忙期"和使用全概率分解技术,导出了在任意时刻t瞬态队长分布的L变换的递推表达式和稳态队长分布的递推表达式,以及平稳队长的随机分解.     

延迟N-策略Geo/G/1排队系统的队长分布及数值计算

考虑延迟N-策略离散时间Geo/G/1排队系统,使用全概率分解技术,从任意初始状态出发,研究了队长的瞬态和稳态性质,导出了在任意时刻n瞬态队长分布的z-变换的递推表达式和稳态队长分布的递推表达式,以及稳态队长的随机分解.最后,通过数值实例, 讨论了稳态队长分布对系统参数的敏感性,并阐述了获得便于计算的稳态队长分布的表达式在系统容量设计中有重要的价值.     

基于多重休假的min(N,V)-策略M/G/1排队系统的队长分布

运用全概率分解技术和拉普拉斯变换工具，研究了基于服务员多重休假的min（N，V）-策略M/G/1排队系统，其中N是预设的休假终止的门限值.讨论了从任意初始状态出发队长的瞬态分布，获得了队长瞬态分布的拉普拉斯变换的递推表达式和稳态队长分布的递推表达式，同时求出了附加队长分布的显示表达式.进一步讨论了当休假时间V分别服从负指数分布和定长分布P{V=T}=1，以及当N=1，N→∞，P{V=0}=1与P{V=∞}=1时的特殊情形.最后，通过数值实例阐述了获得便于计算的稳态队长分布的表达式在系统容量设计中的重要价值.     

延迟D-策略Geo/G/1排队系统的队长分布及容量的优化设计

考虑延迟D-策略离散时间Geo/G/1排队系统, 使用全概率分解技术, 从任意初始状态出发, 研究了队长的瞬态和稳态性质, 推导出了在任意时刻n⁺ 瞬态队长分布的z-变换的递推表达式和稳态队长分布的递推表达式, 并获得稳态队长的随机分解结果, 同时得到了系统在三种任意时刻(n^-, n, n⁺)处稳态队长分布的重要关系. 最后, 通过数值实例, 讨论了稳态队长分布对系统参数的敏感性, 并阐述了获得便于计算的稳态队长分布的表达式在系统容量优化设计中的重要应用价值.     

Queue Size Distribution of Geo/G/1 Queue Under the Min(N,D)-Policy

This paper considers a discrete-time Geo/G/1 queue under the Min(N,D)-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D,whichever occurs first(Min(N,D)-policy).By using renewal process theory and total probability decomposition technique,the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state,and obtain both the recursive expression of the z-transformation of the transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n~+.Meanwhile,the authors obtain the explicit expressions of the additional queue length distribution.Furthermore,the important relations between the steady state queue length distributions at different time epochs n~-,n and n~+ are also reported.Finally,the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution,and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.     

延迟多重休假MX/G/1排队系统的队长分布

考虑延迟多重休假的M^x／G／1排队，在假定延迟时间、休假时间和服务时间都是一般概率分布函数下，研究了队长的瞬态和稳态性质、通过引进“服务员忙期”，导出了在任意时刻t瞬态队长分布的L变换的递推表达式和稳态队长分布的递推表达式，以及平稳队长的随机分解．     

Recursive Solution of Queue Length Distribution for Geo/G/1 Queue with Delayed Min(N,D)-Policy

In this paper we consider a discrete-time Geo/G/1 queue with delayed Min(N, D)-policy.Using renewal process theory, total probability decomposition technique and z-transform, we study the transient and equilibrium properties of the queue length from an arbitrary initial state, and obtain both the recursive expressions of the transient state queue length distribution and the steady state queue length distribution at arbitrary time epoch n~+. Furthermore, we derive the important relations between equilibrium queue length distributions at different time epochs n~-, n and n~+. Finally, we give some numerical examples about capacity decision in queueing systems to demonstrate the application of the analytical results reported in this paper.     

具有多重休假和Min(N,V)-策略控制的Geo/G/1离散时间排队

考虑服务员具有多重休假和系统采用min(N,V)-策略控制的离散时间Geo/G/1排队系统,使用全概率分解技术和更新过程理论,研究了系统在任意时刻n+的瞬态队长分布和稳态队长分布,得到了瞬态队长分布的z-变换表达式和稳态队长分布的递推表达式.进一步,得到了系统在时刻点n,n~-和外部观察时刻点的稳态队长分布.特别地,本文直接获得了一些特殊离散时间排队系统相应的结果.最后,通过数值实例阐述了获得便于计算的稳态队长分布的表达式在系统容量设计中的重要价值.     

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.     

Queue size distribution of Geo/G/1 queue under the Min(<Emphasis Type="Italic">N,D</Emphasis>)-policy

This paper considers a discrete-time Geo/G/1 queue under the Min(N,D)-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D, whichever occurs first (Min(N,D)-policy). By using renewal process theory and total probability decomposition technique, the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursive expression of the z-transformation of the transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n ⁺. Meanwhile, the authors obtain the explicit expressions of the additional queue length distribution. Furthermore, the important relations between the steady state queue length distributions at different time epochs n ^-, n and n ⁺ are also reported. Finally, the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution, and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.     

具有温储备失效的M/G/1可修排队系统

本文把“服务台在系统闲期中可能温储备失效”引入到M/G/1可修排队系统中,考虑了具有温储备失效特征的M/G/1可修排队系统.使用全概率分解技术和利用拉普拉斯变换工具,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,进一步获得了队长的稳态分布的递推式,同时,给出了稳态队长和稳态等待时间的随机分解结果. 最后通过数值计算实例讨论了平均附加队长随温储备失效参数和修复参数的变化情况.     

The M/PH/1 queue with working vacations and vacation interruption

We study an M/PH/1 queue with phase type working vacation and vacation interruption where the vacation time follows a phase type distribution. The server serves the customers at a lower rate in a vacation period. The server comes back to the regular busy period at a service completion without completing the vacation. Such policy is called vacation interruption. In terms of quasi birth and death process and matrix-geometric solution method, we obtain the stationary queue length distribution. Moreover we obtain the conditional stochastic decomposition structures of queue length and waiting time when the service time distribution in the regular busy period is exponential.     

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.     

N-策略与Min(N,V)-策略的M/G/1/∞排队系统等待时间的随机分解结构

在一些关于N-策略休假的M/G/1/∞排队模型研究中,由于顾客的等待时间与该顾客到达时刻以后的输入间隔时间不再独立,因此对顾客的稳态等待时间分布的讨论较为困难,更多是集中在系统的稳态队长和附加队长的讨论上,很少有文献讨论顾客的稳态等待时间及其随机分解.本文首先考虑经典N-策略休假的M/G/1/∞排队系统,讨论了顾客的稳态等待时间分布,给出了顾客的稳态等待时间的随机分解结果和顾客的附加延迟时间分布的显式表达式,同时,指出了已有结果的错误.其次,我们考虑在多重休假和单重休假下具有Min(N,V)-策略控制的M/G/1/∞排队系统,给出了顾客的稳态等待时间的随机分解结果,获得了顾客的平均稳态等待时间和平均附加延迟时间表达式.特别地,通过本文可直接获得一些特殊排队系统的相应结果.     

分析M／G／1排队系统队长分布的方法注记

利用系统忙期和ＬＳ变换，本文获得从任意初始状态出发的Ｍ／Ｇ／１排队系统队长瞬态解的ＬＳ变换表达式以及平稳队长分布的表达式，并在Ｍ／Ｍ／１排队系统中验证了其正确性．值得注意的是本文提出的方法简洁、直观．     

M/M/N可修排队稳态分布存在条件的一种新形式

利用母函数方法研究有一个修理工且服务台忙时与闲时故障率不同的M/M/N可修排队,得到系统的一些可靠性指标和系统稳态分布存在条件的一种新形式.     

Strategic Behavior and Optimization in an Unobservable Constant Retrial Queue with Balking and Set-Up Time

An M/M/1 constant retrial queue with balking customers and set-up time is considered.Once the system becomes empty, the server will be turned down to reduce operating costs, and it will be activated only when there is a customers arrives. In this paper, the almost unobservable case is studied, in which the information of the queue length is unavailable, whereas the state of the server can be obtained. Firstly, the steady state solutions are derived and the individual equilibrium strategies are analyzed. In addition, social optimization problems, including cost analysis and social welfare maximization are investigated by using the PSO algorithm. Finally, by appropriate numerical examples, the sensitivity of some main system parameters is shown.     

基于M/M/1/SV排队的均衡门限策略

考虑单重休假M/M/1排队, 在部分可视的前提下, 研究顾客的均衡门限策略, 首次将单重休假机制引入到连续时间排队经济学模型中. 系统的决策主体是顾客, 突破了以往只注重服务机构单方面行为的局限. 基于“收入-支出”结构, 利用马尔可夫过程理论, 通过求解差分方程, 分析了系统的稳态行为, 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 均衡纯门限策略, 均衡混合门限策略的具体方法并证明之; 而后在不同的策略下, 得出了系统的稳态分布和均衡社会收益; 最后, 通过数值实验分析了均衡行为的各指标对系统参数的敏感性. 研究结果为顾客决策提供了优化建议, 同时为管理者研究系统中的定价问题提供了理论参考.