Discrete Time Geo/G/1 Queue with Multiple Adaptive Vacations

This paper treats the discrete time Geometric/G/1 system with vacations. In this system, after serving all customers in the system, the server will take a random maximum number of vacations before returning to the service mode. The stochastic decomposition property of steady-state queue length and waiting time has been proven. The busy period, vacation mode period, and service mode period distributions are also derived. Several common vacation policies are special cases of the vacation policy presented in this study.     

An MX/G/1 queueing system with a setup period and a vacation period

This paper deals with an M^X/G/1 queueing system with a vacation period which comprises an idle period and a random setup period. The server is turned off each time when the system becomes empty. At this point of time the idle period starts. As soon as a customer or a batch of customers arrive, the setup of the service facility begins which is needed before starting each busy period. In this paper we study the steady state behaviour of the queue size distributions at stationary (random) point of time and at departure point of time. One of our findings is that the departure point queue size distribution is the convolution of the distributions of three independent random variables. Also, we drive analytically explicit expressions for the system state probabilities and some performance measures of this queueing system. Finally, we derive the probability generating function of the additional queue size distribution due to the vacation period as the limiting behaviour of the M^X/M/1 type queueing system. This revised version was published online in June 2006 with corrections to the Cover Date.     

The queue-length distribution for Mx/G1 queue with single server vacation

1 IntroductionDuring recent decades many authors studied M/G/l queues with server vacations (seeRefS[1 ~ 6]). They not only studied the stocliastic decomposition properties of the queue lengthand waiting time when the system is in equilibrium, but also studied its transient and equilibrium distributions. Although Baba[7] studied bulk-arrival M"/G/1 with vacation time andShils] studied a kind of M"/G(M/H)/1 queue with repairable service station, they didll't studythe transient and equilibr…     

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。     

GI／G／1和GI／M／1的队长的瞬时分布

在本中给出了GI/G／1和GI／M／1的队长L(t)的瞬时分布的计算方法。     

Operating characteristic analysis on the M/G/1 system with a variant vacation policy and balking

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under a variant vacation policy, where the server leaves for a vacation as soon as the system is empty. The server takes at most J vacations repeatedly until at least one customer is found waiting in the queue when the server returns from a vacation. If the server is busy or on vacation, an arriving batch balks (refuses to join) the system with probability 1 − b. We derive the system size distribution at different points in time, as well as the waiting time distribution in the queue. Finally, important system characteristics are derived along with some numerical illustration.     

M/G/1一般减量服务单重休假排队系统的随机分解

本文讨论了M/G/1型一般减量服务单重休假排队模型,运用结构分析法得到稳态队长和服务时间的随机分解的母函数和拉式变换,并给出稳态分布成立的条件及其概率含义．     

带启动时间的多级适应性休假的M/G/1排队

本研究带启动时间的多级适应性休假的M/G/1间排队。给出稳态队长分布和母函数、等待时间分布和其LST及其随机分解结果，推导出忙期、假期和启动期的母函数。带有启动时间的单重休假和多重休假是本中模型的两个极端情况。     

带启动-关闭期和N策略的多重休假M/G/1排队

研究多重休假带启动-关闭期和N策略的M/G/1排队系统,根据嵌入Markov链的方法推导出状态转移概率矩阵,利用M/G/1型排队系统结构矩阵解析法,得出顾客服务完离去后系统稳态队长分布及其母函数的表达式;从而由经典随机分解原理,给出稳态队长的随机分解结果.此外,利用LST变换处理卷积,得到忙期的母函数及数学期望的表达式;进而得到忙期、启动期和关闭期的母函数及在稳态下服务员处于各状态的概率.最后提出一些数值例子以验证结论.     

An M[X]/G/1 retrial g-queue with single vacation subject to the server breakdown and repair

An M[X]/G/1 retrial G-queue with single vacation and unreliable server is investigated in this paper. Arrivals of positive customers form a compound Poisson process, and positive customers receive service immediately if the server is free upon their arrivals; Otherwise, they may enter a retrial orbit and try their luck after a random time interval. The arrivals of negative customers form a Poisson process. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for a single vacation as soon as the system empties. In this paper, we analyze the ergodical condition of this model. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities.     

GI／G／1排队系统的队长的瞬时性态（I）

本文用三种方式给出计算队长L(t)的瞬时分布。     

有优先权顾客带启动时间多重休假的M(1)+M(2)/G/1排队系统

研究了带启动时间有顾客优先权多重休假的M(1)+M(2)/G/1排队系统,分别给出了两类顾客的稳态队长的母函数和等待时间分布的LST及其随机分解的结果,推导出忙期、假期和启动期的LST等.     

具有不同到达率的带有启动时间的多级适应性休假Mξ/G/1排队模型

本文研究具有不同到达率的带有启动时间的多级适应性休假M~ξ／G／1排队模型,应用嵌入马尔可夫链方法推导出了稳态队长和等待时间(先到先服务规则)分布,并验证了稳态队长和稳态等待时间具有随机分解性,而且给出了忙期分布．许多关于M~ξ／G／1的排队模型都可以看作是此模型的特例．     

批量到达带启动时间的单重休假M/G/1排队系统

本文研究批量到达带启动时间的单重休假的M／G／1排队系统，给出稳态队长的母函数和等待时间分布的LST及其它们的随机分解结果，推导出忙期、闲期和线期母函数和均值。     

带有Bernoulli反馈的多级适应性休假的Geo/G/1排队系统分析

考虑带有Bernoulli反馈的多级适应性休假的Geo/G/1离散时间排队系统.通过引入服务员忙期和使用一种简洁的分解方法,讨论了队长的瞬时分布,得到了在任意时刻n队长为j的概率关于时刻n的z-变换的递推式,及队长平稳分布的递推式,且证明了稳态队长的随机分解性质.最后,给出了在特殊情形下相应的一些结果和数值计算实例.     

An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule

This paper treats an M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Whenever the system becomes empty at a service completion instant, the server goes for a single working vacation. In the working vacation, a customer is served at a lower speed, and if there are customers in the queue at the instant of a service completion, the server is resumed to a regular busy period with probability p   (i.e., the vacation is interrupted) or continues the vacation with probability 1-p$1-p$. Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by using supplementary variable technique. We also develop a variety of stationary performance measures for this system and give a conditional stochastic decomposition result. Finally, several numerical examples are presented.     

推广的多重休假$M^X/G/1$排队系统

在平稳状态下,Baba利用补充变量方法研究了多重休假的MX/G/1排队,但作者假定了休假时间和服务时间都有概率密度函数.本文考虑推广的多重休假MX/G/1排队,在假定休假时间和服务时间都是一般概率分布函数下,我们研究了队长的瞬态和稳态性质.通过引进"服务员忙期"和使用不同于Baba文中使用的分析技术,我们导出了在任意时刻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.特别地,通过本文可直接获得多重休假的M/G/1与标准的MX/G/1排队系统相应的结果.     

An M/G/1 system with startup server and J additional options for service

We consider an M^[x]/G/1 queueing system with a startup time, where all arriving customers demand first the essential service and some of them may further demand one of other optional services: Type 1, Type 2, … , and Type J service. The service times of the essential service and of the Type i  (i=1,2,…,J)$(i=1\text{,}2\text{,}\dots \text{,}J)$ service are assumed to be random variables with arbitrary distributions. The server is turned off each time when the system is empty. As soon as a customer or a batch of customers arrives, the server immediately performs a startup which is needed before starting each busy period. We derive the steady-state results, including system size distribution at a random epoch and at a departure epoch, the distributions of idle and busy periods, and waiting time distribution in the queue. Some special cases are also presented.     

两类具有N-策略和单重休假的M/G/1排队系统的最优控制策略

本文考虑两类具有N-策略和服务员单重休假的M/G/1排队系统,其中一类是休假不可中断,另一类是休假可中断。利用系统稳态队长的随机分解特性导出稳态队长的概率母函数,并讨论了系统空闲率与附加平均队长对系统一些参数的敏感性。进一步,在建立费用结构的基础上,应用更新报酬过程理论导出了系统长期运行单位时间内所产生的成本期望费用的显示表达式,同时通过数值计算实例确定了使得系统在长期运行单位时间内所产生的成本期望费用最小的控制策略N^*,以及当休假时间为定长T时的二维最优控制策略（N^*,T^*）。