M/M/1抢占优先权排队平稳指标的尾部分析

讨论M/M/1抢占优先权排队模型,该模型可以用一个具有可数位相的拟生灭(QBD)过程来描述.对该过程,我们得到平稳状态时低优先权顾客数分布的概率母函数,结果表明它不是一个有理函数.在此基础上,进一步指出,对该过程,低优先权顾客的平稳队长和平稳逗留时间分别具有几何衰减和指数衰减的特性.     

带启动时间的Geom/Geom/1多重工作休假排队

本文中研究了一个带有启动时间的Geom/Geom/1多重工作休假排队模型。服务台在休假期间,不停止服务,而是以较低的服务率为顾客提供服务。运用拟生灭过程和矩阵几何解的方法,给出了该模型的稳态队长分布,并求出了平均队长以及顾客的平均逗留时间。     

有负顾客到达的M/G/1休假可修排队系统

采用补充变量法和母函数的方法研究了有负顾客到达的M/G/1休假可修排队系统,其中负顾客的抵消规则是带走正在接受服务的正顾客并使得服务器处于修理状态.休假策略是空竭服务多重休假.文中给出了系统存在稳态的充要条件,系统稳态队长分布的概率母函数及系统可靠度的L变换.     

推广的单重休假M~x/G/1排队系统

研究了服务前需要重新调整机器的单重休假Mx/G/1排队系统,在LS变换和L变换下得到了服务员忙期中队长的瞬态分布和队长稳态分布的概率母函数.     

推广的多级适应性休假M~x/G/1排队系统

用LS变换和L变换研究了推广的多级适应性休假Mx/G/1排队系统中队长的瞬态分布,并且进一步用LS变换的终值定理和洛比达法则得到了队长平稳分布的概率母函数,这个结果可以应用到很多模型.     

输入率可变、反馈的、且有负顾客的优先排队模型

针对实际应用中存在输入率可变、因服务出差错而导致顾客需要重新排队接受服务以及不同的顾客类需要不同的服务质量等现状,建立了输入率可变、有反馈及负顾客的、服务时间服从一般分布优先排队模型.得出了"强占优先"与"非强占优先"两种服务规则下,系统中每一类顾客的队长、等待时间、逗留时间的平稳分布均存在,并求出了每一类顾客的队长、等待时间、逗留时间及他们的L-S变换,忙期等指标,最后还指出了模型在应用中的注意事项及要进一步解决的问题.     

基于Geo/Geo/1(E,SV)排队系统的均衡止步策略

基于单重休假Geo/Geo/1排队系统,研究顾客的均衡止步策略,首次将休假服务机制引入到离散时间排队经济学模型中. 顾客基于“收入--支出”结构,自主决定去留. 利用拟生灭过程理论,运用差分方程求解技巧,对系统进行了稳态分析,得到了顾客的平均逗留时间;进而构造适当的函数,给出了寻找均衡止步策略的具体方法并证明之;而后分析了在均衡策略下, 系统的稳态行为和社会收益;最后通过数值实验讨论了系统参数对均衡行为的影响.     

具有负顾客和特殊类顾客的串联排队系统分析

讨论了带有普通类顾客、负顾客和特殊类顾客的M/M/1→M/M/1两级串联排队系统模型,负顾客在一级服务系统中,一对一抵消队尾的普通类顾客(若有),若一级服务系统无普通类顾客,负顾客自动消失,负顾客不进入二级服务系统.特殊类顾客不经过一级服务系统,直接进入二级服务系统等待接受服务.用拟生灭过程和矩阵几何解方法,得到了系统稳态队长的分布,以及系统忙期的分布和顾客逗留时间的分布等相关指标.     

N策略带启动时间的Geom/Geom/1工作休假排队

考虑N策略带启动时间的Geom/Geom/1工作休假排队,服务员在休假期间并未完全停止工作而是以较低的速率为顾客服务.运用拟生灭链和矩阵几何解方法,给出了该模型的稳态队长的分布和等待时间的概率母函数,并证明了队长和等待时间的条件随机分解结构.     

M/M/1排队系统四个指标的渐近性质

应用 C0 -半群理论研究 M/M/1排队系统中四个指标 :系统中顾客的平均等待时间 ,顾客的平均逗留时间 ,顾客总数和等待服务的顾客总数的渐近性质 ,得到这四个指标的渐近稳定性结果 .     

The conditional sojourn time distribution in the GI/M/1 processor-sharing queue in heavy traffic

We treat the GI/M/1 queue with a processor-sharing server, in the heavy traffic case. Using perturbation methods, we construct asymptotic expansions for the conditional sojourn time distribution of a tagged customer conditioned on the tagged customer's service time. The resulting approximation is simple in form and involves only the first three moments of the interarrival time distribution.     

Sojourn time asymptotics in the M/G/1 processor sharing queue

We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer, iff the sojourn time distribution is regularly varying of index -ν. This result is derived from a new expression for the Laplace–Stieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sojourn Times in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">PH</Emphasis>/1 Processor Sharing Queue

We give in this paper an algorithm to compute the sojourn time distribution in the processor sharing, single server queue with Poisson arrivals and phase type distributed service times. In a first step, we establish the differential system governing the conditional sojourn times probability distributions in this queue, given the number of customers in the different phases of the PH distribution at the arrival instant of a customer. This differential system is then solved by using a uniformization procedure and an exponential of matrix. The proposed algorithm precisely consists of computing this exponential with a controlled accuracy. This algorithm is then used in practical cases to investigate the impact of the variability of service times on sojourn times and the validity of the so-called reduced service rate (RSR) approximation, when service times in the different phases are highly dissymmetrical. For two-stage PH distributions, we give conjectures on the limiting behavior in terms of an M/M/1 PS queue and provide numerical illustrative examples.This revised version was published online in June 2005 with corrected coverdate     

Sojourn time analysis of a two-phase queueing system with exhaustive batch-service and its vacation model

We consider an M/G/1-type, two-phase queueing system, in which the two phases in series are attended alternatively and exhaustively by a moving single-server according to a batch-service in the first phase and an individual service in the second phase. We show that the two-phase queueing system reduces to a new type of single-vacation model with non-exhaustive service. Using a double transform for the joint distribution of the queue length in each phase and the remaining service time, we derive Laplace-Stieltjes transforms for the sojourn time in each phase and the total sojourn time in the system. Furthermore, we provide the moment formula of sojourn times and numerical examples of an approximate density function of the total sojourn time.     

Sojourn time distribution in a MAP/M/1 processor-sharing queue

This paper considers the sojourn time distribution in a processor-sharing queue with a Markovian arrival process and exponential service times. We show a recursive formula to compute the complementary distribution of the sojourn time in steady state. The formula is simple and numerically feasible, and enables us to control the absolute error in numerical results. Further, we discuss the impact of the arrival process on the sojourn time distribution through some numerical examples.     

Batch arrival single-server queue with variable service speed and setup time

In this paper, we consider an M\({}^X\)/M/1/SET-VARI queue which has batch arrivals, variable service speed and setup time. Our model is motivated by power-aware servers in data centers where dynamic scaling techniques are used. The service speed of the server is proportional to the number of jobs in the system. The contribution of our paper is threefold. First, we obtain the necessary and sufficient condition for the stability of the system. Second, we derive an expression for the probability generating function of the number of jobs in the system. Third, our main contribution is the derivation of the Laplace–Stieltjes transform (LST) of the sojourn time distribution, which is obtained in series form involving infinite-dimensional matrices. In this model, since the service speed varies upon arrivals and departures of jobs, the sojourn time of a tagged job is affected by the batches that arrive after it. This makes the derivation of the LST of the sojourn time complex and challenging. In addition, we present some numerical examples to show the trade-off between the mean sojourn time (response time) and the energy consumption. Using the numerical inverse Laplace–Stieltjes transform, we also obtain the sojourn time distribution, which can be used for setting the service-level agreement in data centers.     

Performance Analysis of the GI/M/1 Queue with Single Working Vacation and Vacations

In this paper, we consider a new class of the GI/M/1 queue with single working vacation and vacations. When the system become empty at the end of each regular service period, the server first enters a working vacation during which the server continues to serve the possible arriving customers with a slower rate, after that, the server may resume to the regular service rate if there are customers left in the system, or enter a vacation during which the server stops the service completely if the system is empty. Using matrix geometric solution method, we derive the stationary distribution of the system size at arrival epochs. The stochastic decompositions of system size and conditional system size given that the server is in the regular service period are also obtained. Moreover, using the method of semi-Markov process (SMP), we gain the stationary distribution of system size at arbitrary epochs. We acquire the waiting time and sojourn time of an arbitrary customer by the first-passage time analysis. Furthermore, we analyze the busy period by the theory of limiting theorem of alternative renewal process. Finally, some numerical results are presented.     

Sojourn time asymptotics in processor-sharing queues

Over the past few decades, the Processor-Sharing (PS) discipline has attracted a great deal of attention in the queueing literature. While the PS paradigm emerged in the sixties as an idealization of round-robin scheduling in time-shared computer systems, it has recently captured renewed interest as a useful concept for modeling the flow-level performance of bandwidth-sharing protocols in communication networks. In contrast to the simple geometric queue length distribution, the sojourn time lacks such a nice closed-form characterization, even for exponential service requirements. In case of heavy-tailed service requirements however, there exists a simple asymptotic equivalence between the sojourn time and the service requirement distribution, which is commonly referred to as a reduced service rate approximation. In the present survey paper, we give an overview of several methods that have been developed to obtain such an asymptotic equivalence under various distributional assumptions. We outline the differences and similarities between the various approaches, discuss some connections, and present necessary and sufficient conditions for an asymptotic equivalence to hold. We also consider the generalization of the reduced service rate approximation to several extensions of the M/G/1 PS queue. In addition, we identify a relationship between the reduced service rate approximation and a queue length distribution with a geometrically decaying tail, and extend it to so-called bandwidth-sharing networks. The state-of-the-art with regard to sojourn time asymptotics in PS queues with light-tailed service requirements is also briefly described. Last, we reflect on some possible avenues for further research. AMS Subject Classification 60K25 (primary), 60F10, 68M20, 90B18, 90B22 (secondary).     

对M/T-SPH/1排队平稳指标的进一步研究—数值计算与渐近分析

讨论M/T-SPH/1排队平稳队长分布的数值计算,以及平稳队长和逗留时间分布各阶矩的数值计算及渐近分析.其中T-SPH表示可数状态吸收生灭链吸收时间的分布.在分布PGF和LST的基础上,首先给出了计算平稳队长分布,平稳队长以及逗留时间分布各阶矩的数值结果的递推公式.其次还讨论了平稳队长及平稳逗留时间分布各阶矩的尾部渐近...     

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??We obtain the strong approximation of the sojourn time progress for a two-stage tandem queue in heavy traffic, that is, the traffic intensity $\rho_1=\rho_2=1$. The sojourn time is the period from a customer's arrival to her departure, and the strong approximation is a function of Brownian motion.