带工作休假和工作故障的M/M/1/N排队系统性能分析

本文在可修M/M/1/N排队系统中引入了启动时间、工作休假和工作故障策略.在该系统中,服务台在休假期间不是完全停止工作,而是处于低速服务状态.设定服务台在任何时候均可发生故障,当故障发生时立刻进行维修.且当服务台在正规忙期出现故障时,服务台仍以较低的服务速率为顾客服务.服务台的寿命时间和修理时间均服从指数分布,且在不同的时期有不同的取值.同时,从关闭期到正规忙期有服从指数分布的启动时间.本文建立此模型的有限状态拟生灭过程(QBD),使用矩阵几何方法得到系统的稳态概率向量,并应用基本阵和协方差矩阵理论,计算出系统稳态可用度、系统方差、系统吞吐率、系统稳态队长及各系统稳态概率等系统性能指标.同时,通过数值实验对各系统参数对系统性能的影响进行了初探.文中的敏感性分析体现了这种方法的有效性和可用性.实验表明,文中提出的模型,可有效改善仅带有工作休假或工作故障策略排队模型的系统性能.     

带启动时间和工作故障的M/M/1/N排队系统性能分析

在M/M/1/N可修排队系统中引入了工作故障和启动时间.服务台在忙期允许出现故障,且在故障期间不是完全停止服务而是以较低的服务速率为顾客服务.同时,从关闭期到正规忙期有服从指数分布的启动时间.通过分析此模型的二维连续时间Markov过程,求解出系统平稳方程,建立此系统的有限状态拟生灭过程(QBD).根据系统参数,求解出水平相依的子率阵,从而得到系统稳态概率向量的矩阵几何表示形式.在系统稳态概率向量的基础上,求解出系统吞吐率、系统稳态可用度、系统稳态队长及系统处于各个状态的概率等性能指标的解析表达式.文中的敏感性分析体现了这种方法的有效性和可用性,同时,对系统各性能受系统参数的影响进行了探索.实验表明,文中提出模型的稳定性较好,且更贴近实际服务过程,因此这种模型将被广泛应用于各种实际服务中.     

可修排队系统Mx/G (M/H)1的瞬态解

本文研究一个典型的批到达可修排队系统＾ｘ／（／）１．记号（／）表服务台寿命服从指数分布，而其修理时间为一连续型分布。利用向量马氏过程方法，我们得到了它的瞬态解。特别是发现了服务台的可靠性指标仅依赖于可修排队系统的空闲概率，或等价地仅依赖于它的忙期和忙循环。     

可修排队系统M~X／G（M／H）／1的瞬态解

本文研究一个典型的批到达可修排队系统ＭＸ／Ｇ（／Ｈ）／１．记号（Ｍ／Ｈ）表服务台寿命服从指数分布，而其修理时间为一连续型分布．利用向量马氏过程方法，我们得到了它的明态解．特别是发现了服务台的可靠性指标仅依赖于可修排队系统的空闲概率，或等价地仅依赖于它的忙期和忙循环     

离散时间单重休假冷储备系统的可靠性分析

利用离散向量Markov过程方法研究了离散时间单重休假两同型部件冷储备可修系统。在部件寿命服从几何分布,修理时间和修理工休假时间服从一般离散型概率分布的假定下,引入修理时间和休假时间尾概率,求得了系统的稳态可用度、稳态故障频度、待修概率、修理工空闲概率和休假概率,以及首次故障前平均时间等可靠性指标。     

可修排队系统GI/G(M/G)/1的可靠性分析

利用更新过程理论和向量马氏过程方法全面考察了可修排队系统GI/G(M/G)/1的结 构,得到了所有感兴趣的指标,并证明了服务台的可靠性指标只与系统的忙期、闲期和忙期循 环时间有关.     

具有优先权的M/G/1重试可修排队系统

在服务台忙的情况下, 到达服务台的顾客以概率 q 进入无限位置的优先队列而以概率 p 进入无限位置的重试轨道 (orbit), 并且按照先到先服务 (FCFS) 规则排队, 假定只有队首的顾客允许重试, 同时考虑服务台可修的因素, 证明了系统稳态解存在的充要条件. 利用补充变量法求得稳态时两个队列与系统的平均队长、顾客等待时间、服务台的各种状态概率以及可靠性指标.     

修理工带休假的单部件可修系统的可靠性分析

考虑修理工带有单重休假的单部件可修系统,系统发生故障时可能因修理工的休假而 得不到立即修理,因此系统可处于工作、等待修理和修理三种状态.利用全概率分解技术和拉普 拉斯或拉普拉斯--司梯阶变换工具,讨论了系统的可靠度、瞬时可用度和稳态可用度,以及 (O,t]时间中系统的平均故障次数和稳态故障频度,得到了关于系统的可靠度、瞬时可用度和稳 态可用度,以及(O,t]时间中系统的平均故障次数和稳态故障频度等可靠性指标的重要结果.     

再访GI/M/1排队

通过构造两个向量马氏过程重新探讨了GI/M/1排队,某些新结果如忙期和闲期的 联合分布被得到了.这一方法容易推广到服务时间为无限位相型分布的GI/SPH/1排队.     

具有单重休假和修复不如新的两部件温贮备系统

研究了修理工单重休假且由两个不同型部件和一个修理工组成的可修型温贮备系统. 系统考虑了在工作故障和贮备 故障都不能 “修复如新”, 部件 1 是修复非新而部件 2 修复如新的条件下, 假设部件的工作寿命、贮备寿命、故障后的修理时间和贮备故障后的修理时间均服 从不同的指数分布, 修理工休假服从一般连续型分布. 运用几何过程理论、补充变量法、 拉普拉斯变换及拉普拉斯--司梯阶变换, 得到了系统的可用度、可靠度和系统首次故障前平均时间等可靠性指标. 最后, 通过数值模拟验证了结果的有效性.     

Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns

This paper investigates the N-policy M/M/1 queueing system with working vacation and server breakdowns. As soon as the system becomes empty, the server begins a working vacation. The server works at a lower service rate rather than completely stopping service during a vacation period. The server may break down with different breakdown rates during the idle, working vacation, and normal busy periods. It is assumed that service times, vacation times, and repair times are all exponentially distributed. We analyze this queueing model as a quasi-birth–death process. Furthermore, the equilibrium condition of the system is derived for the steady state. Using the matrix-geometric method, we find the matrix-form expressions for the stationary probability distribution of the number of customers in the system and system performance measures. The expected cost function per unit time is constructed to determine the optimal values of the system decision variables, including the threshold N and mean service rates. We employ the particle swarm optimization algorithm to solve the optimization problem. Finally, numerical results are provided, and an application example is given to demonstrate the applicability of the queueing model.     

Maximum entropy analysis of the M[x]/M/1 queueing system with multiple vacations and server breakdowns

We consider a single unreliable sever in an M^[x]/M/1 queueing system with multiple vacations. As soon as the system becomes empty, the server leaves the system for a vacation of exponential length. When he returns from the vacation, if there are customers waiting in the queue, he begins to serve the customers; otherwise, another vacation is taken. Breakdown times and repair times of the server are assumed to obey a negative exponential distribution. Arrival rate varies according to the server’s status: vacation, busy, or breakdown. Using the maximum entropy principle, we develop the approximate formulae for the probability distributions of the number of customers in the system which is used to obtain various system performance measures. We perform a comparative analysis between the exact results and the maximum entropy results. We demonstrate, through the maximum entropy results, that the maximum entropy principle approach is accurate enough for practical purposes.     

Impatient customers in an M/M/1 queue with single and multiple working vacations

We consider an M/M/1 queue with impatient customers and two different types of working vacations. The working vacation policy is the one in which the server serves at a lower rate during a vacation period rather than completely stop serving. The customer’s impatience is due to its arrival during a working vacation period, in which the customer service rate is lower than the normal busy period. We analyze the queue for two different working vacation termination policies, a multiple working vacation policy and a single working vacation policy. Closed-form solutions and various performance measures like, the mean queue lengths and the mean waiting times are derived. The stochastic decomposition properties are verified for both multiple and single working vacation cases. A comparison of both the models is carried out to capture their performances with the change in system parameters.     

Performance analysis of an M/G/1 queueing system under Bernoulli vacation schedules with server setup and close down periods

This paper is concerned with the analysis of a single server queueing system subject to Bernoulli vacation schedules with server setup and close down periods. An explicit expression for the probability generating function of the number of customers present in the system is obtained by using imbedded Markov chain technique. The steady state probabilities of no customer in the system at the end of vacation termination epoch and a service completion epoch are derived. The mean number of customers served during a service period and the mean number of customers in the system at an arbitrary epoch are investigated under steady state. Further, the Laplace-Stieltjes transform of the waiting time distribution and its corresponding mean are studied. Numerical results are provided to illustrate the effect of system parameters on the performance measures.     

An MX/G/1 queue with randomized working vacations and at most J vacations

This paper treats a bulk arrival queue with randomized working vacation policy. Whenever the system becomes empty, the server takes a vacation. During the vacation period, customers are to be served at a lower rate. Once the vacation ends, the server will return to the normal working state and begin to serve the customers in the system if any. Otherwise, the server either remains idle with probability p or leaves for another vacation with probability 1?p. This pattern continues until the number of vacations taken reaches J. If the system is empty at the end of the Jth vacation, the server will wait idly for a new arrival. By using supplementary variable technique, we derive the system size distribution at arbitrary epoch, at departure epoch and at busy period initial epoch, as well as some important system characteristics. Numerical examples are provided to illustrate the influence of system parameters on several performance measures.     

An M/G/1 retrial G-queue with non-exhaustive random vacations and an unreliable server

This paper deals with an M/G/1 retrial queue with negative customers and non-exhaustive random vacations subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. A breakdown at the busy server is represented by the arrival of a negative customer which causes the customer being in service to be lost. The server takes a vacation of random length after an exponential time when the server is up. We develop a new method to discuss the stable condition by finding absorb distribution and using the stable condition of a classical M/G/1 queue. By applying the supplementary variable method, we obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law. We also analyse the busy period of the system. Some special cases of interest are discussed and some known results have been derived. Finally, an application to cellular mobile networks is provided and the effects of various parameters on the system performance are analysed numerically.     

Analysis of the discrete-time Geo/G/1 working vacation queue and its application to network scheduling

In this paper we present an exact steady-state analysis of a discrete-time Geo/G/1 queueing system with working vacations, where the server can keep on working, but at a slower speed during the vacation period. The transition probability matrix describing this queuing model can be seen as an M/G/1-type matrix form. This allows us to derive the probability generating function (PGF) of the stationary queue length at the departure epochs by the M/G/1-type matrix analytic approach. To understand the stationary queue length better, by applying the stochastic decomposition theory of the standard M/G/1 queue with general vacations, another equivalent expression for the PGF is derived. We also show the different cases of the customer waiting to obtain the PGF of the waiting time, and the normal busy period and busy cycle analysis is provided. Finally, we discuss various performance measures and numerical results, and an application to network scheduling in the wavelength division-multiplexed (WDM) system illustrates the benefit of this model in real problems.     

On a batch arrival Poisson queue with a random setup time and vacation period

The single server queue with vacation has been extended to include several types of extensions and generalisations, to which attention has been paid by several researchers (e.g. see Doshi, B. T., Single server queues with vacations — a servey. Queueing Systems, 1986, 1, 29–66; Takagi, H., Queueing Analysis: A Foundation of Performance evaluation, Vol. 1, Vacation and Priority systems, Part. 1. North Holland, Amsterdam, 1991; Medhi, J., Extensions and generalizations of the classical single server queueing system with Poisson input. J. Ass. Sci. Soc., 1994, 36, 35–41, etc.). The interest in such types of queues have been further enhanced in resent years because of their theoretical structures as well as their application in many real life situations such as computer, telecommunication, airline scheduling as well as production/inventory systems. This paper concerns the model building of such a production/inventory system, where machine undergoes extra operation (such as machine repair, preventive maintenance, gearing up machinery, etc.) before the processing of raw material is to be started. To be realistic, we also assume that raw materials arrive in batch. This production system can be formulated as an M^x/M/1 queues with a setup time. Further, from the utility point of view of idle time this model can also be formulated as a case of multiple vacation model, where vacation begins at the end of each busy period. Besides, the production/inventory systems, such a model is generally fitted to airline scheduling problems also. In this paper an attempt has been made to study the steady state behavior of such an M^x/M/1 queueing system with a view to provide some system performance measures, which lead to remarkable simplification when solving other similar types of queueing models.This paper deals with the steady state behaviour of a single server batch arrival Poisson queue with a random setup time and a vacation period. The service of the first customer in each busy period is preceded by a random setup period, on completion of which service starts. As soon as the system becomes empty the server goes on vacation for a random length of time. On return from vacation, if he finds customer(s) waiting, the server starts servicing the first customer in the queue. Otherwise it takes another vacation and so on. We study the steady state behaviour of the queue size distribution at random (stationary) point of time as well as at departure point of time and try to show that departure point queue size distribution can be decomposed into three independent random variables, one of which is the queue size of the standard M^x/M/1 queue. The interpretation of the other two random variables will also be provided. Further, we derive analytically explicit expressions for the system state (number of customers in the system) probabilities and provide their appropriate interpretations. Also, we derive some system performance measures. Finally, we develop a procedure to find mean waiting time of an arbitrary customer.