Analysis of the GI/Geo/1 Queue with Disasters

The occurrence of disasters to a queueing system causes all customers to be removed if any are present. Although there has been much research on continuous-time queues with disasters, the discrete-time Geo/Geo/1 queue with disasters has appeared in the literature only recently. We extend this Geo/Geo/1 queue to the GI/Geo/1 queue. We present the probability generating function of the stationary queue length and sojourn time for the GI/Geo/1 queue. In addition, we convert our results into the Geo/Geo/1 queue and the GI/M/1 queue.     

Asymptotic behavior of the loss probability for an M/G/1/N queue with vacations

In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N-(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model.     

A simple eigenvalue method for low-order D-BMAP/G/1 queues

This paper is aimed at those engineers and practitioners who need a simple and understandable non-matrix-analytic procedure to compute the performance measures of the discrete-time BMAP/G/1 queueing system when the order of parameter matrices is very low. We develop a set of system equations and derive the vector generating function of the queue length. Starting from the generating function, we propose a eigenvalue approach that can be implemented by those who have basic knowledge on M/G/1 queues and eigenvalue algebra.     

Analyzing the discrete-timeG (G)/Geo/1 queue using complex contour integration

In this paper, a discrete-time single-server queueing system with an infinite waiting room, referred to as theG ^(G)/Geo/1 model, i.e., a system with general interarrival-time distribution, general arrival bulk-size distribution and geometrical service times, is studied. A method of analysis based on integration along contours in the complex plane is presented. Using this technique, analytical expressions are obtained for the probability generating functions of the system contents at various observation epochs and of the delay and waiting time of an arbitrary customer, assuming a first-come-first-served queueing discipline, under the single restriction that the probability generating function for the interarrival-time distribution be rational. Furthermore, treating several special cases we rediscover a number of well-known results, such as Hunter's result for theG/Geo/1 model. Finally, as an illustration of the generality of the analysis, it is applied to the derivation of the waiting time and the delay of the more generalG ^(G)/G/1 model and the system contents of a multi-server buffer-system with independent arrivals and random output interruptions.Both authors wish to thank the Belgian National Fund for Scientific Research (NFWO) for support of this work.     

GI／G／1单重休假随机服务系统的马氏骨架方法

文献[1]引入一类具有广泛应用前景的随机过程-Markov骨架过程。借助Markov骨架过程的方法研究GI/G/1单重休假服务系统队长，及t时刻到达顾客等待时间的瞬时概率分布。     

GI(1)+GI(2)+…+GI(N)/M/1排队模型

文献[1]引入了一类具有广泛应用前景的随机过程--Markov骨架过程.本文借助这类随机过程的方法研究了GI(1)+GI(2)+…+GI(N)/M/1排队模型,求出了此模型到达过程、等待时间及队长的概率分布.     

启动时间的GI／G／1排队系统的非平衡理论

本文利用侯振挺等提出的马尔可夫骨架过程理论讨论了启动时间的GI/G/I排队系统，得到了此系统到达过程，队长，及等待时间的概率分布/     

A GI/Geo/1 queue with negative and positive customers

The arrival of a negative customer to a queueing system causes one positive customer to be removed if any is present. Continuous-time queues with negative and positive customers have been thoroughly investigated over the last two decades. On the other hand, a discrete-time Geo/Geo/1 queue with negative and positive customers appeared only recently in the literature. We extend this Geo/Geo/1 queue to a corresponding GI/Geo/1 queue. We present both the stationary queue length distribution and the sojourn time distribution.     

A semidefinite optimization approach to the steady-state analysis of queueing systems

Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilities are not necessarily exponential (as in most ‘traditional’ queueing systems). Further, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations.     

Analysis of the M/Gi/1 →/M/1 queueing model

An M/GI/1 queueing system is in series with a unit with negative exponential service times and infinite waiting room capacity. We determine a closed form expression for the generating function of the joint queue length distribution in steady state. This result is obtained via the solution of a new type of functional equation in two variables.     

Alternative Numerical Solutions of Stationary Queueing-Time Distributions in Discrete-Time Queues: <Emphasis Type="Italic">GI/G/</Emphasis>1

This paper gives closed-form expressions, in terms of the roots of certain equations, for the distribution of the waiting time in queue, W_q, in the steady-state for the discrete-time queue GI/G/1. Essentially, this is done by finding roots of the denominator of the probability generating function of W _q and then resolving the generating function into partial fractions. Numerical examples are given showing the use of the required roots, even when there is a large number of them. The method discussed in this paper avoids spectrum factorization and uses both closed- and non-closed forms of interarrival- and service-time distributions. Approximations for the tail probabilities in terms of one or three roots taken in ascending order of magnitude are also discussed. The exact computational results that can be obtained from the methods of this study should prove useful to both practitioners and queueing theorists dealing with bounds, inequalities, approximations, and simulation results.     

单重休假闸门服务的Geom/G/1离散时间排队系统

文章研究了单重休假的Geom/G/1闸门服务系统,推导出稳态下系统队长的母函数,FCFS规则下的等待时间的母函数,使用离散时间队长和剩余工作量的分解性质,求出剩余工作量的母函数,最后给出服务周期的性能指标的母函数,及系统处在各种状态的概率.     

对M/T-SPH/1排队平稳队长的分析

研究了M/T-SPH/l排队模型,利用拟生灭过程和算子几何解的方法给出了平稳队长分布的概率母函数.在此基础上,指出该分布不是一个离散PH分布,但在一定条件下却是一个几何尾部分布.     

A factorization property for BMAP/G/1 vacation queues under variable service speed

This paper proposes a simple factorization principle that can be used efficiently and effectively to derive the vector generating function of the queue length at an arbitrary time of the BMAP/G/1/ queueing systems under variable service speed. We first prove the factorization property. Then we provide moments formula. Lastly we present some applications of the factorization principle.     

A discrete-time Geo/G/1 retrial queue with preemptive resume and collisions

We consider a discrete-time Geo/G/1 retrial queue with preemptive resume, collisions of customers and general retrial times. We analyze the Markov chain underlying the considered queueing system and derive its ergodicity condition. Using generating function technique, the system state distribution as well as the orbit size and the system size distributions are studied. Some interesting and important performance measures are obtained. Besides, the stochastic decomposition property is investigated. Finally, some numerical examples are provided.     

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。     

第二种服务可选的M/G/1排队模型的适定性

运用Hille-Yosida定理,Phillips定理与Fattorini定理证明第二种服务可选的M/G/1排队模型存在唯一的概率瞬态解.     

非强占优先权下的M/M/1排队

分析带有两个优先权的非强占M/M/1系统的性能,用补充变量法构造向量马尔可夫过程对此排队系统的状态转移方程进行分析,得到两类顾客在非强占优先权的队长联合分布的母函数,进一步讨论,得出了服务台被两类顾客占有和闲置的概率以及两类信元各自的平均队长.     

服务员强制休假的M/G/1排队模型的适定性

运用Hille-Yosida定理,Phillips定理与Fattorini定理证明服务员强制休假的M/G/1排队模型存在唯一的概率瞬态解.     

A discrete-time Geo/G/1 retrial queue with preferred and impatient customers

This paper is concerned with a discrete-time Geo/G/1 retrial queue with preferred, impatient customers and general retrial times. We analyze the Markov chain underlying the considered queueing system and derive its ergodicity condition. The system state distribution as well as the orbit size and the system size distributions are obtained in terms of their generating functions. These generating functions yield exact expressions for different performance measures. Besides, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of priority and impatience on several performance characteristics of the system.