具有可选服务的M／M／1排队模型的一个特征值及其应用

证明0是具有可选服务的M／M／1排队模型的主算子及其共轭算子的几何重数为1的特征值,由此推出该模型的时间依赖解强收敛于该模型的稳态解．     

具有工作休假及休假中止的$M/M/1$排队模型的主算子的点谱

本文考虑具有工作休假及休假中止的$M/M/1$排队模型的主算子的点谱. 证明该模型主算子在左半轴有不可数无穷多个特征值. 此结果描述了主算子的点谱. 然后证明该主算子生成的$C_0$-半群的本质增长界为0,由此推出该$C_0$-半群不是紧算子、它的本质谱半径等于1. 此外,这些结果蕴含该模型的时间依赖解不可能指数收敛于其稳态解.     

Time-dependent analysis for a queue modeled by an infinite system of partial differential equations

By studying the spectrum of the underlying operator corresponding to the exhaustive-service M/G/1 queueing model with single vacations we prove that the time-dependent solution of the model strongly converges to its steady-state solution.     

Spectrum of the Operator Corresponding to the M/M/1 Queueing Model with Vacations and Multiple Phases of Operation and Application

We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation. Then by using this result we prove that the essential growth bound of the C₀-semigroup generated by the operator is 0, the C₀-semigroup is not compact, not eventually compact, even not quasi-compact. Moreover, we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution. In addition, we obtain the spectral radius and essential spectral radius of the C₀-semigroup. Lastly, we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum.     

关于M/M/n排队模型的动态解及稳定性

文章讨论动态 M/M/n排队模型 ,运用算子半群理论证明了该模型动态正解的存在唯一性 .并进一步表明零点是系统的一个本征值 ,相应的本征函数为系统的一个定态正解 ,系统的动态正解强稳定到定态解     

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.     

可变输入率M/M/n模型的动态解及其渐近稳定性

本文运用有界线性算子半群理论讨论了可变输入率M/M/n排队模型,证明模型主算子生成C0半群,并运用一定的技巧证明动态解渐近稳定到其定态解.     

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.     

M/G/1重试排队系统的渐近稳定性

用算子半群理论研究了带有重试排队的M/G/1系统.通过解算子方程和预解方程,证明了0是系统算子的本征值,且为虚轴上唯一的谱点.从而得出了当时间趋于无穷时系统时间依赖解收敛于稳态解的结论.     

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.     

服务中断的M/M/1重试排队模型的稳态解

研究服务中断的M/M/1重试排队模型的稳态解,证明当α+μ>λ时0不足该模型主算子的特征值.由此推出该模型不存在稳态解.     

具有可变到达率的多重休假Geo~(λ_1,λ_2)/G/1排队分析

本文考虑顾客到达与服务员休假相关的多重休假离散时间排队系统,用更新过程及u-变换分析了系统的队长性质.分别得到系统在三种时点(n~-,n~+,n)处的队长分布的递推解,进而揭示了在不同到达率条件下系统队长分布不再具有随机分解特性,得到了系统在四种时点(n~-,n~+,n,离去时点D_n)处稳态队长分布的重要关系(不同于连续时间排队系统).     

常微分方程形式的M/M/1排队模型的一个注

讨论动态Ｍ／Ｍ／１排队模型，运用半群理论证明了该模型存在唯一的正解，并研究了相应算子的谱特征．     

Departure Processes of BMAP/G/1 Queues

A unified approach is applied to analyze the departure processes of finite/infinite BMAP/G/1 queueing systems for both vacationless and vacation arrangements via characterizing the moments, the z-transform of the scaled autocovariance function of interdeparture times C _P (z), and lag n (n1) covariance of interdeparture times. From a structural point of view, knowing departure process helps one to understand the impact of service mechanisms on arrivals. Through numerical experiments, we investigate and discuss how the departure statistics are affected by service and vacation distributions as well as the system capacity. From a practical perspective, output process analysis serves to bridge the nodal performance and connectionwise performance. Our results can be then used to facilitate connection- or networkwise performance analysis in the current high-speed networks.     

Derivation of the N-step interdeparture time distribution in GI/G/1 queueing systems

The departure process of a queueing system has been studied since the 1960s. Due to its inherent complexity, closed form solutions for the distribution of the departure process are nearly intractable. In this paper, we derive a closed form expression for the distribution of interdeparture time in a GI/G/1 queueing model. Without loss of generality, we consider an embedded Markov chain in a general K_M/G/1 queueing system, in which the interarrival time distribution is Coxian and service time distribution is general. Closed form solutions of the equilibrium distribution are derived for this model and the Laplace–Stieltjes transform (LST) of the distribution of interdeparture times is presented. An algorithmic computing procedure is given and numerical examples are provided to illustrate the results. With the analysis presented, we provide a novel analytic tool for studying the departure process in a general queueing model.     

Maximum entropy analysis to the N policy M/G/1 queueing system with a removable server

We study a single removable server in an M/G/1 queueing system operating under the N policy in steady-state. The server may be turned on at arrival epochs or off at departure epochs. Using the maximum entropy principle with several well-known constraints, we develop the approximate formulae for the probability distributions of the number of customers and the expected waiting time in the queue. We perform a comparative analysis between the approximate results with exact analytic results for three different service time distributions, exponential, 2-stage Erlang, and 2-stage hyper-exponential. The maximum entropy approximation approach is accurate enough for practical purposes. We demonstrate, through the maximum entropy principle results, that the N policy M/G/1 queueing system is sufficiently robust to the variations of service time distribution functions.     

M／M／1算子的另一个特征值

证明2√λμ-λ-μ是偏微分方程形式的M／M／1排队模型主算子的几何重数为1的特征值．     

Analysis of the MAP/PH/1/K queue with service control

We consider a finite capacity queue with Markovian arrivals, in which the service rates are controlled by two pre-determined thresholds, M and N. The service rate is increased when the buffer size exceeds N and then brought back to normal service rate when the buffer size drops to M. The normal and fast service times are both assumed to be of phase type with representations (β, S), and β θS), respectively, where θ>1. For this queueing model, steady state analysis is performed. The server duration in normal as well as fast periods is shown to be of phase type. The departure process is modelled as a MAP and the parameter matrices of the MAP are identified. Efficient algorithms for computing system performance measures are presented. We also discuss an optimization problem and present an efficient algorithm for arriving at an optimal solution. Some numerical examples are discussed.     

修理工可单重休假的带有一个冷贮备部件的Gaver并联系统的时间依赖解的渐进行为

研究修理工可单重休假的带有一个冷贮备部件的Gaver并联系统的时间依赖解.运用C₀-半群理论与算子理论研究该模型相应算子的谱的特征,获得了该系统的时间依赖解强收敛于该系统的稳态解.