A discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times

The authors discuss a discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times. As soon as the orbit is empty, the server takes a vacation. However, the server is allowed to take a maximum number J of vacations, if the system remains empty after the end of a vacation. If there is at least one customer in the orbit at the end of a vacation, the server begins to serve the new arrivals or the arriving customers from the orbit. For this model, the authors focus on the steady-state analysis for the considered queueing system. Firstly, the authors obtain the generating functions of the number of customers in the orbit and in the system. Then, the authors obtain the closed-form expressions of some performance measures of the system and also give a stochastic decomposition result for the system size. Besides, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided.     

带负顾客和不耐烦顾客的离散时间Geo/G/1重试排队

考虑了一个带负顾客和不耐烦顾客且重试时间为一般分布的离散时间Geo/G/1重试排队系统. 负顾客带走一个正在服务的顾客, 而对重试组中的顾客无影响.正顾客到达系统若遇服务器忙则可能进入重试组也可能离开系统.通过对此排队系统的嵌入马氏链进行分析, 得到了重试组队长和系统队长的概率母函数. 进而得到了一系列重要的排队指标. 此外, 还推导出了系统的稳态存在条件. 以及对无负顾客和不耐烦顾客时的特例进行了分析. 最后通过几个具体的数值实例演示了一些参数对系统关键性能指标的影响.     

Optimal control of an M/G/1 retrial queue with vacations

In this note, we consider an M/G/1 retrial queue with server vacations, when retrial times, service times and vacation times are arbitrary distributed. The distribution of the number of customers in the system in stationary regime is obtained in terms of generating function. Next, we give heavy traffic approximation of such distribution. We show that the system size can be decomposed into two random variables, one of which corresponds to the system size of the ordinary M/G/1 FIFO queue without vacation. Such a stochastic decomposition property is useful for the computation of performance measures of interest. Finally, we solve simple problems of optimal control of vacation and retrial policies.     

A batch arrival retrial queue with starting failures,feedback and admission control

This paper is concerned with the analysis of a feedback M^[X]/G/1 retrial queue with starting failures and general retrial times. In a batch, each individual customer is subject to a control admission policy upon arrival. If the server is idle, one of the customers admitted to the system may start its service and the rest joins the retrial group, whereas all the admitted customers go to the retrial group when the server is unavailable upon arrival. An arriving customer (primary or retrial) must turn-on the server, which takes negligible time. If the server is started successfully (with a certain probability), the customer gets service immediately. Otherwise, the repair for the server commences immediately and the customer must leave for the orbit and make a retrial at a later time. It is assumed that the customers who find the server unavailable are queued in the orbit in accordance with an FCFS discipline and only the customer at the head of the queue is allowed for access to the server. The Markov chain underlying the considered queueing system is studied and the necessary and sufficient condition for the system to be stable is presented. Explicit formulae for the stationary distribution and some performance measures of the system in steady-state are obtained. Finally, some numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.     

A single server retrial queue with general retrial times and two-phase service

An M / G / 1 retrial queue with a first-come-first-served (FCFS) orbit, general retrial time, two-phase service and server breakdown is investigated in this paper. Customers are allowed to balk and renege at particular times. Assume that the customers who find the server busy are queued in the orbit in accordance with an FCFS discipline. All customers demand the first “essential” service, whereas only some of them demand the second “optional” service, and the second service is multioptional. During the service, the server is subject to breakdown and repair. Assume that the retrial time, the service time, and the repair time of the server are all arbitrarily distributed. By using the supplementary variables method, the authors obtain the steady-state solutions for both queueing and reliability measures of interest. This research is supported by the National Natural Science Foundation of China under Grant No. 10871020.     

Analysis of batch arrival queue with randomized vacation policy and an un-reliable server

This paper examines an M^[x]G/1 queueing system with an unreliable server and a delayed repair,in which the server operates a randomized vacation policy with multiple vacations.Whenever the system is empty,the server immediately takes a vacation.If there is at least one customer found waiting in the queue upon returning from a vacation,the server will be immediately activated for service.Otherwise,if no customers are waiting for service at the end of a vacation,the server either remains idle with probability p or leaves for another vacation with probability 1-p.Whenever one or more customers arrive when the server is idle,the server immediately starts providing service for the arrivals.The server may also meet an unpredictable breakdown and the repair may be delayed.For such a system the authors derive the distributions of some important system characteristics,such as the system size distribution at a random epoch and at a departure epoch,the system size distribution at the busy period initiation epoch,and the distribution of the idle period and the busy period.The authors perform a numerical analysis for changes in the system characteristics,along with changes in specific values of the system parameters.A cost effectiveness maximization model is constructed to explain the benefits of such a queueing system.     

Optimal design for a retrial queueing system with state-dependent service rate

This paper considers a single server retrial queue in which a state-dependent service policy is adopted to control the service rate. Customers arrive in the system according to a Poisson process and the service times and inter-retrial times are all exponentially distributed. If the number of customers in orbit is equal to or less than a certain threshold, the service rate is set in a low value and it also can be switched to a high value once this number exceeds the threshold. The stationary distribution and two performance measures are obtained through the partial generating functions. It is shown that this state-dependent service policy degenerates into a classic retrial queueing system without control policy under some conditions. In order to achieve the social optimal strategies, a new reward-cost function is established and the global numerical solutions, obtained by Canonical Particle Swarm Optimization algorithm, demonstrate that the managers can get more benefits if applying this state-dependent service policy compared with the classic model.     

重试,反馈M/M/s/k排队的呼叫中心性能分析

由于CTI(计算机电话集成)技术的发展,使呼叫中心得到广泛的应用.与呼叫中心实现技术的发展相比,对呼叫中心管理的研究显得有些滞后,而针对呼叫中心排队模型的研究,更是如此.针对呼叫中心服务系统中的重试和反馈问题,考虑一种带重试和反馈的M/M/s/k排队模型.将等待位置和服务台数推广到有限个.在模型求解过程中,尝试采用矩阵迭代的新方法,使求解过程简单明了.然后,采用逼近的方法给出模型的数值解,并得出反馈对系统的影响随系统负荷的增大而快速增大等结论.     

Analysis of a discrete-time GI/Geo/1/N queue with multiple working vacations

This paper analyzes a finite-buffer renewal input single server discrete-time queueing system with multiple working vacations. The server works at a different rate rather than completely stopping working during the multiple working vacations. The service times during a service period, service time during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The analysis of actual waiting-time distribution and some performance measures are carried out. We present some numerical results and discuss special cases of the model.     

一类休假的Geo~X/G/1可修排队的可靠性分析

考虑单重休假、Bernoulli反馈和可变输入率的离散时间Geo~X/G/1可修排队.顾客的批到达速率与服务器的休假有关.刚服务完的顾客以概率1-θ进入队列寻求下次服务.服务器在服务过程中可能故障需修复后再继续工作.借助更新过程理论、z变换和一种分解法,研究了时刻n+位于服务器忙期的条件概率、服务器的瞬态和稳态不可用度以及(0~+,n~+]时间内服务器的平均故障次数和稳态故障频度,揭示了这类离散时间可修排队中服务器可靠性指标的结构,得到了一些特殊可修排队的可靠性结果.最后通过数值实例分析了系统参数对服务器可靠性指标的影响.     

The M/M/1 queue with working vacations and vacation interruptions

In this paper, we study the M/M/1 queue with working vacations and vacation interruptions. The working vacation is introduced recently, during which the server can still provide service on the original ongoing work at a lower rate. Meanwhile, we introduce a new policy：, the server can come back from the vacation to the normal working level once some indices of the system, such as the number of customers, achieve a certain value in the vacation period. The server may come back from the vacation without completing the vacation. Such policy is called vacation interruption. We connect the above mentioned two policies and assume that if there are customers in the system after a service completion during the vacation period, the server will come back to the normal working level. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions and the stochastic decomposition structures for the number of customers and the waiting time and provide some indices of systems.     

The structure of departure process and optimal control strategy <Emphasis Type="Italic">N</Emphasis>* for <Emphasis Type="Italic">Geo/G</Emphasis>/1 discrete-time queue with multiple server vacations and Min(<Emphasis Type="Italic">N,V</Emphasis>)-Policy

This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n⁺ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0⁺, n⁺] from any initial state. Meanwhile, the relationship among departure process, server’s state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures, including the expected length of server busy period, server’s actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N* for minimizing the system cost under a given cost structure.     

The Structure of Departure Process and Optimal Control Strategy N~* for Geo/G/1 Discrete-Time Queue with Multiple Server Vacations and Min(N,V)-Policy

This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n~+ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0~+, n~+] from any initial state. Meanwhile, the relationship among departure process, server's state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures,including the expected length of server busy period, server's actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N*for minimizing the system cost under a given cost structure.     

Analysis of discrete-time queues with batch renewal input and multiple vacations

This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed.Service and vacation times are mutually independent and geometrically distributed.The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch.The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system.Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival,arbitrary and outside observer’s observation epochs for partial-batch rejection policy.The blocking probability of the first-,an arbitrary-and the last-job in a batch have been discussed.The analysis of actual waiting-time distributions measured in slots of the first-,an arbitrary- and the last-job in an accepted batch,and other performance measures along with some numerical results have also been investigated.     

Performance Analysis of a Discrete-Time Queue with Working Breakdowns and Searching for the Optimum Service Rate in Working Breakdown Period

This paper deals with a discrete-time Geo/Geo/1 queueing system with working breakdowns in which customers arrive at the system in variable input rates according to the states of the server. The server may be subject to breakdowns at random when it is in operation. As soon as the server fails, a repair process immediately begins. During the repair period, the defective server still provides service for the waiting customers at a lower service rate rather than completely stopping service.We analyze the stability condition for the considered system. Using the probability generating function technique, we obtain the probability generating function of the steady-state queue size distribution.Also, various important performance measures are derived explicitly. Furthermore, some numerical results are provided to carry out the sensitivity analysis so as to illustrate the effect of different parameters on the system performance measures. Finally, an operating cost function is formulated to model a computer system and the parabolic method is employed to numerically find the optimum service rate in working breakdown period.     

ON THE TRANSIENT DEPARTURE PROCESS OF M^x/G/1 QUEUEING SYSTEM WITH SINGLE SERVER VACATION

This paper studies the transient departure process of M^x/G/1 queueing system with single server vacation. We present a simple probability decomposition method to derive the expected number of departures occurring in finite time interval from any initial state and the asymptotic expansion of the expected number. Especially, we derive some more practical results for some special cases.     

负顾客、带启动期和备用服务员的M/M/1休假排队系统

考虑一类有正、负顾客, 带启动期和有备用服务员的M/M/1休假排队系统. 负顾客一对一抵消队尾的正顾客(若有), 若系统中无正顾客, 到达的负顾客自动消失, 负顾客不接受服务.系统中两个服务员, 其中一个在岗工作时另外一个备用.上岗服务员若因为某种原因休假, 备用服务员立即替换上岗.当系统变空时, 系统关闭.用拟生灭过程和矩阵几何解方法, 得到了稳态队长的分布, 此外, 证明了稳态条件下队长的条件随机分解并得到了附加队长的分布. 最后, 通过两个数值例子说明该模型可以较好的模拟一些实际问题.     

On the single server retrial queue with priority subscribers and server breakdowns

The author concerned the reliability evaluation as well as queueing analysis of M1, M2/G1, G2/1 retrial queues with two different types of primary customers arriving according to independent Poisson flows. In the case of blocking, the first type of customers can be queued whereas the second type of customers must leave the service area but return after some random period of time to try their luck again. The author assumes that the server is unreliable and it has a service-type dependent, exponentially distributed life time as well as a service-type dependent, generally distributed repair time. The necessary and sufficient condition for the system to be stable is investigated. Using a supplementary variable method, the author obtains a steady-state solution for queueing measures, and the transient as well as the steady-state solutions for reliability measures of interest.     

The Queue Length Distribution for M/G/1 Queue with Delay Single Server Vacation

1　IntroductionDuringrecentseveraldecadesmanyauthorsstudiedM/G/1queueswithdifferentservervacationregimes(seeRefs.[1～6]).Theynotonlystudiedthestochasticdecompositionpropertiesofthequeuelengthandwaitingtimewhenthesystemisinequilibrium,butalsostudiedthetransientandequilibriumdistributionsofthequeuelength.InRef.[6]theau-thorsstudiedM/G/1queuewithdelaymultipleservervacationsinwhichtheserverhasthreestates:vacation-preparation,vacationandbusyperiod.InthispaperwefirstdiscusstheM/G/1queuewithdelay…