具有插队和止步行为的M/M/1/m+1排队系统的等待时间分析

本文研究具有插队和止步行为的M/M/1/m+1排队系统中顾客的等待时间问题.首先,将顾客分为两类,一类顾客到达系统在队尾排队等待服务,称为常规顾客;另一类顾客总是尽可能的靠近队首插队,称为插队顾客.利用负指数分布、Laplace-Stieltjes变换、全概率公式,本文给出处于等待队列位置n的顾客、常规顾客、插队顾客的等待时间的表达式,并在此基础上,给出了相关指标随系统参数的变化情况.     

一类带负顾客和反馈的M/G/1休假排队系统

将负顾客和反馈相结合研究了一类带负顾客和反馈M/G/1的休假排队系统,正顾客服务完后以概率1-θ反馈到队尾等待下次服务,以概率θ(0θ≤1)离开系统。负顾客抵消正在接受服务的正顾客,休假策略为单重休假。给出了它们稳态存在的充分必要条件,利用补充变量法和状态转移分析模型,得到了系统主要排队指标和稳态队长概率母函数及概率母函数的随机分解结果。     

具有插队和止步行为的M/M/c 排队系统

研究了具有插队和止步行为的M/M/c排队系统. 将到达顾客分为常规顾客和插队顾客, 常规顾客在队尾排队等待服务, 插队顾客总是尽可能的靠近队首插队等待服务. 插队行为由到达顾客的插队概率和队列中等待顾客对插队行为的容忍来描述. 利用负指数分布的性质、Laplace-Stieltjes变换和全概率公式, 给出了处于等待队列位置n的顾客、任意一个常规顾客和任意一个插队顾客的等待时间的表达式. 在此基础上, 讨论了系统相关指标随系统参数的变化情况.     

可选服务的M/M/1多重休假排队系统驱动的流模型

在M/M/1多重休假排队驱动系统的基础上引入可选服务,探讨一类体现第二次服务可选的M/M/1多重休假驱动系统的流排队.构建净输入率结构,同时结合拟生灭过程方法获得驱动系统的平稳分布,利用Laplace变换(LT)方法得到流模型稳态库存量的Laplace-Stieltjes(LST),空库概率及均值的表达式.最后,通过数值分析验证系统性能指标的变动规律.     

GI~X/M~b/1/N排队系统

本文主要研究了顾客一般独立批到达、指数批服务、缓冲器容量有限的单个服务器的排队系统，本文首先使用补充变量和嵌入马氏链的方法，在部分拒绝和全部拒绝情形下，得到系统排队队长的稳态分布，进而得到相应的性能指标，如系统的平均排队长、平均等待时间、损失概率等．其次对等待时间进行了分析．     

附有选择性服务与无等待能力的M/G/1排队系统的系统主算子的性质

研究了附有选择性服务与无等待能力的M/G/1排队系统.运用C_0半群理论,通过服务率均值的观念,对系统主算子的谱上界进行了估值,并得到谱上界即为各服务率均值的最小者的相反数.然后运用了共尾的概念及相关的理论,得到了系统主算子的谱上界与系统主算子产生的半群的增长界相等,从而得到其增长界也是各服务率均值的最小者的相反数.     

分析M/M/1多重工作休假排队的一种新方法

用一种新方法对经典的M/M/1工作休假排队系统建立模型.对该模型,用无限位相GI/M/1型Markov过程和矩阵解析方法进行分析,不但得到了所讨论排队模型平稳队长分布的具体结果,还给出了平稳状态时服务台具体位于第几次工作休假的概率.这些关于服务台状态更为精确的描述是该排队系统的新结果.     

关于GI/G/1/∞排队系统的平稳等待时间分布的局部尾等价式

关于GI/G/1/∞排队系统的平稳等待时间分布W,已有许多经典的结果描述了其尾分布[AKW-](x)=1-W(x)的等价极限情况.该文结合一些保险与金融领域重要的风险变量,研究了关于分布W的各种局部尾等价式问题.     

非零服务期M/G/1闸门服务休假排队系统的随机分解

讨论了非零服务期M/G/1闸门服务排队系统的随机分解.得到稳态队长和等待时间母函数(PGF)及拉氏变换(LST).并且可根据其随机分解的结构特征,为非空竭服务休假排队系统的排队指标的控制提供直接依据.     

服务员单重休假的N-策略M/G/1排队系统

研究了同时考虑单重休假和N-策略两种休假策略的排队系统,其休假准则为任一个条件满足.我们给出了此排队系统的稳态队长,忙期分布等基本指标,并得到稳态等待时间的LST(Laplace-Stieltjes Trans-form)。     

Tail asymptotics for the queue length in an M/G/1 retrial queue

In this paper, we study the tail behavior of the stationary queue length of an M/G/1 retrial queue. We show that the subexponential tail of the stationary queue length of an M/G/1 retrial queue is determined by that of the corresponding M/G/1 queue, and hence the stationary queue length in an M/G/1 retrial queue is subexponential if the stationary queue length in the corresponding M/G/1 queue is subexponential. Our results for subexponential tails also apply to regularly varying tails, and we provide the regularly varying tail asymptotics for the stationary queue length of the M/G/1 retrial queue. AMS subject classifications: 60J25, 60K25     

A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule

We consider an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system , including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.     

Tail asymptotics of the queue size distribution in the M/M/m retrial queue

We consider an M/M/m retrial queue and investigate the tail asymptotics for the joint distribution of the queue size and the number of busy servers in the steady state. The stationary queue size distribution with the number of busy servers being fixed is asymptotically given by a geometric function multiplied by a power function. The decay rate of the geometric function is the offered load and independent of the number of busy servers, whereas the exponent of the power function depends on the number of busy servers. Numerical examples are presented to illustrate the result.     

Regularly varying tail of the waiting time distribution in M/G/1 retrial queue

We consider an M/G/1 retrial queue where the service time distribution has a regularly varying tail with index −β, β>1. The waiting time distribution is shown to have a regularly varying tail with index 1−β, and the pre-factor is determined explicitly. The result is obtained by comparing the waiting time in the M/G/1 retrial queue with the waiting time in the ordinary M/G/1 queue with random order service policy.     

A G/M/1 retrial queue with constant retrial rate

In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented.     

On the steady-state queue size distribution of the discrete-timeGeo/G/1 queue with repeated customers

In this paper, we study the steady-state queue size distribution of the discrete-timeGeo/G/1 retrial queue. We derive analytic formulas for the probability generating function of the number of customers in the system in steady-state. It is shown that the stochastic decomposition law holds for theGeo/G/1 retrial queue. Recursive formulas for the steady-state probabilities are developed. Computations based on these recursive formulas are numerically stable because the recursions involve only nonnegative terms. Since the regularGeo/G/1 queue is a special case of theGeo/G/1 retrial queue, the recursive formulas can also be used to compute the steady-state queue size distribution of the regularGeo/G/1 queue. Furthermore, it is shown that a continuous-timeM/G/1 retrial queue can be approximated by a discrete-timeGeo/G/1 retrial queue by dividing the time into small intervals of equal length and the approximation approaches the exact when the length of the interval tends to zero. This relationship allows us to apply the recursive formulas derived in this paper to compute the approximate steady-state queue size distribution of the continuous-timeM/G/1 retrial queue and the regularM/G/1 queue.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0046415.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0105828.     

Multi-server retrial queue with negative customers and disasters

We consider a multi-server retrial queue with waiting places in service area and four types of arrivals, positive customers, disasters and two types of negative customers, one for deleting customers in orbit and the other for deleting customers in service area. The four types of arrivals occur according to a Markovian arrival process with marked transitions (MMAP) which may induce the dependence among the arrival processes of the four types. We derive a necessary and sufficient condition for the system to be positive recurrent by comparing sample paths of auxiliary systems whose stability conditions can be obtained. We use a generalized truncated system that is obtained by modifying the retrial rates for an approximation of stationary queue length distribution and show the convergence of approximation to the original model. An algorithmic solution for the stationary queue length distribution and some numerical results are presented.      

A single-server batch arrival queue with returning customers

We consider a new class of batch arrival retrial queues. By contrast to standard batch arrival retrial queues we assume if a batch of primary customers arrives into the system and the server is free then one of the customers starts to be served and the others join the queue and then are served according to some discipline. With the help of Lyapunov functions we have obtained a necessary and sufficient condition for ergodicity of embedded Markov chain and the joint distribution of the number of customers in the queue and the number of customers in the orbit in steady state. We also have suggested an approximate method of analysis based on the corresponding model with losses.     

A unified algorithm for computing the stationary queue length distributions inM(k)/G/1/N and GI/M(k)/1/N queues

An iterative algorithm is developed for computing numerically the stationary queue length distributions in M/G/1/N queues with arbitrary state-dependent arrivals, or simply M(k)/G/1/N queues. The only input requirement is the Laplace-Stieltjes transform of the service time distribution.In addition, the algorithm can also be used to obtain the stationary queue length distributions in GI/M/1/N queues with state-dependent services, orGI/M(k)/1/N, after establishing a relationship between the stationary queue length distributions inGI/M(k)/1/N and M(k)/G/1/N+1 queues.Finally, we elaborate on some of the well studied special cases, such asM/G/1/N queues,M/G/1/N queues with distinct arrival rates (which includes the machine interference problems), andGI/M/C/N queues. The discussions lead to a simplified algorithm for each of the three cases.     

Tail asymptotics for the fundamental period in the MAP/G/1 queue

This paper studies the tail behavior of the fundamental period in the MAP/G/1 queue. We prove that if the service time distribution has a regularly varying tail, then the fundamental period distribution in the MAP/G/1 queue has also regularly varying tail, and vice versa, by finding an explicit expression for the asymptotics of the tail of the fundamental period in terms of the tail of the service time distribution. Our main result with the matrix analytic proof is a natural extension of the result in (de Meyer and Teugels, J. Appl. Probab. 17: 802–813, 1980) on the M/G/1 queue where techniques rely heavily on analytic expressions of relevant functions. I.-S. Wee’s research was supported by the Korea Research Foundation Grant KRF 2003-070-00008.