Comparative analysis for the N policy M/G/1 queueing system with a removable and unreliable server

In this paper we analyze a single removable and unreliable server in the N policy M/G/1 queueing system in which the server breaks down according to a Poisson process and the repair time obeys an arbitrary distribution. The method of maximum entropy is used to develop the approximate steady-state probability distributions of the queue length in the M/G(G)/1 queueing system, where the second and the third symbols denote service time and repair time distributions, respectively. A study of the derived approximate results, compared to the exact results for the M/M(M)/1, M/E₂(E₃)/1, M/H₂(H₃)/1 and M/D(D)/1 queueing systems, suggest that the maximum entropy principle provides a useful method for solving complex queueing systems. Based on the simulation results, we demonstrate that the N policy M/G(G)/1 queueing model is sufficiently robust to the variations of service time and repair time distributions.     

The Discrete-Time GI/Geo/1 Queue with Multiple Vacations

We study a discrete-time GI/Geo/1 queue with server vacations. In this queueing system, the server takes vacations when the system does not have any waiting customers at a service completion instant or a vacation completion instant. This type of discrete-time queueing model has potential applications in computer or telecommunication network systems. Using matrix-geometric method, we obtain the explicit expressions for the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.     

An Analysis of <Emphasis Type="Italic">M/M/s</Emphasis> Queueing Systems Based on the Maximum Entropy Principle

There have been some interesting attempts to derive the equilibrium probability distribution of possible states of queueing systems by using the maximum entropy principle. In particular, Shore introduced an entropy model for the M/M/∞ queueing system. In this paper an inconsistency in Shore's procedure is indicated and an exact entropy model for the multiple server queueing system is presented.     

关于M/M/1动态排队系统解的半稳定性

本文讨论了两类 M/M/1 动态系统的数学模型 ,利用常微分方程所描述的 M/M/1 系统的结果证明了较复杂的偏微分方程所描述的 M/M/1 系统的一些性质 ,该方法简化了已有结果     

On the poisson output of queueing systems with dependent service times

In this paper a method is stated to conclude that the output of given queueing system is POISSON from the steady-state probabilities and from the behavior of the queueing system at jump epochs, i.e. at epochs when the system state can be changed. The corresponding statement for queueing systems described by Markov processes with denumber-able state space will be generalized to systems which can have arbitrary service time distributions if the steady-state probabilities are insensitive with respect to these distributions.     

A NEW APPROACH FOR THE STUDY OF THE M X /G/1 QUEUE USING RENEWAL ARGUMENTS

The M ^X /G/1 queueing system as well as several of its variants have long ago been studied by considering the embedded discrete-time Markov chain at service completion epochs. Alternatively other approaches have been proposed such as the theory of regenerative processes, the supplementary variable method, properties of the busy period, etc. In this note we study the M ^X /G/1 queue via a simple new method that uses renewal arguments. This approach seems quite powerful and may become fruitful in the investigation of other queueing systems as well.     

Markov Skeleton Processes and Applications to Queueing Systems

In this paper, we apply the backward equations of Markov skeleton processes to qucueing systems. The transient distribution of the waiting time of a GI/G/1 queueing system, the transient distribution of the length of a GI/G/N queueing system and the transient distribution of the length of queueing networks are obtained.     

M/G/1 queue with event-dependent arrival rates

Motivated by experiments on customers’ behavior in service systems, we consider a queueing model with event-dependent arrival rates. Customers’ arrival rates depend on the last event, which may either be a service departure or an arrival. We derive explicitly the performance measures and analyze the impact of the event-dependency. In particular, we show that this queueing model, in which a service completion generates a higher arrival rate than an arrival, performs better than a system in which customers are insensitive to the last event. Moreover, contrary to the M/G/1 queue, we show that the coefficient of variation of the service does not necessarily deteriorate the system performance. Next, we show that this queueing model may be the result of customers’ strategic behavior when only the last event is known. Finally, we investigate the historical admission control problem. We show that, under certain conditions, a deterministic policy with two thresholds may be optimal. This new policy is easy to implement and provides an improvement compared to the classical one-threshold policy.     

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。     

Goal-Seeking Behaviour in Queueing Systems

In recent years authors have come to recognise the adaptive nature of practical queueing systems and have begun to investigate this adaptability through possible correlations in the customer arrival rate and/or service rate. However, many systems may possess higher orders of behaviour than can adequately be described in terms of simple adaptive relationships. This paper considers the implications of assuming a goal-seeking behaviour in queueing systems. The results from a simple model of a goal-seeking system are presented and these are compared with the results from a conventional queueing model. From this basis the defining characteristics of goal-seeking behaviour are derived and their application to practical systems is discussed.     

Performance analysis approximation in a queueing system of type M/G/1

In this work, we apply the strong stability method to obtain an estimate for the proximity of the performance measures in the M/G/1 queueing system to the same performance measures in the M/M/1 system under the assumption that the distributions of the service time are close and the arrival flows coincide. In addition to the proof of the stability fact for the perturbed M/M/1 queueing system, we obtain the inequalities of the stability. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results, the STR-STAB algorithm which we execute for a determined queueing system: M/Coxian − 2/1. The accuracy of the approach is evaluated by comparison with simulation results.     

Analysis of an SPP/G/1 system with multiple vacations and E-limited service discipline

Many researchers have studied variants of queueing systems with vacations. Most of them have dealt with M/G/1 systems and have explicitly analyzed some of their performance measures, such as queue length, waiting time, and so on. Recently, studies on queueing systems whose arrival processes are not Poissonian have appeared. We consider a single server queueing system with multiple vacations and E-limited service discipline, where messages arrive to the system according to a switched Poisson process. First, we consider the joint probability density functions of the queue length and the elapsed service time or the elapsed vacation time. We derive the equations for these pdf's, which include a finite number of unknown values. Using Rouché's theorem, we determine the values from boundary conditions. Finally, we derive the transform of the stationary queue length distribution explicitly.     

M/G/1排队系统的性能灵敏度分析

非Markov型排除系统经常被用来作为某些实际工程问题（如通讯网络）的研究模型,对于一般的M／G／1排队系统,本文通过研究其嵌入Markov链,讨论了系统的稳态性能灵敏度分析问题,并给出用嵌入Markov链的势能表示的稳态性能灵敏度公式,由于嵌入Markov链要比描述其系统状态的半Markov过程简单得多,故本文的结果对M／G／1排队系统的性能灵敏度仿真计算及系统的优化,都将带来极大的方便。     

Analyses of subensembles ofM/G/1 queueing sequences

Although many queueing processes of various principles have extensively been investigated, little attention has been paid to the sampling aspect of the theory, by which the nature of sample sequences of finite or infinite length can be examined with respect to some given ensemble of queueing sequences. In this paper we wish to identify classes of sample sequences of an M/G/1 model and investigate several hitherto unknown properties of queueing phenomenon of a given particular service system over a finite or infinite length of time. The method to be used is an extension of both the method of imbedded Markow chains, cf. D. G. Kendall [4], and semi-Markovian processes, Smith [9], Lévy [5], Pyke[7,8], Fabens [2], Neuts [6], etc.     

VIRTUAL WAITING TIMES IN PRIORITY-M/G/1 QUEUES WITH VACARIONS

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具有相型服务分布的排队系统的性能灵敏度分析

具有非线数服务分布的排队网络已被广泛应用于许多领域,如通讯网络和管理系统。本文借助于无穷小说矩阵摄动方法,研究了M/PH/1排队系统的稳态性能灵敏度分析问题,给出了性能灵敏度公式,并表明了稳态性能灵敏度很容易通过系统势能进行计算。同时,给出一种计算势能及性能导数的算法。这个算法可直接用于系统的控制与优化,因为它基于分析系统的一条单一样本轨道。最后提供一个数值例子来表明这个算法的应用。     

多级适应性休假$M^X/G/1$排队系统的队长分布

考虑多级适应性休假的MX/G/1排队系统.采用一种较简单的分析方法,讨论了队长分布的瞬态和稳态性质,得到了队长瞬态分布的拉普拉斯变换的递推表达式和稳态分布的递推表达式,以及稳态队长的随机分解,并给出了服务台闲期、服务台忙循环期的分布函数.另外,从讨论中直接导出了一些特殊排队模型的相应指标.     

A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity

We study a single removable server in an infinite and a finite queueing systems with Poisson arrivals and general distribution service times. The server may be turned on at arrival epochs or off at service completion epochs. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service time, to obtain the steady state probability distribution of the number of customers in a finite system. The method is illustrated analytically for three different service time distributions: exponential, 3-stage Erlang, and deterministic. Cost models for infinite and finite queueing systems are respectively developed to determine the optimal operating policy at minimum cost.     

带反馈的多重休假M~X/G/1排队

研究批量到达带反馈的多重休假M/G/1排队.建立休假,反馈,和成批到达的多类型相结合的排队模型.采用了嵌入马尔可夫链的方法研究了该排队系统,推导出稳态队长分布的母函数及其随机分解结果,给出忙期的LST和全假期的均值.最后考虑了批量等于1的特殊情况.