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.     

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.     

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.     

Randomized policy of a poisson input queue with <Emphasis Type="Italic">J</Emphasis> vacations

This paper studies the operating characteristics of an M/G/1 queuing system with a randomized control policy and at most J vacations.After all the customers are served in the queue exhaustively, the server immediately takes at most J vacations repeatedly until at least N customers are waiting for service in the queue upon returning from a vacation.If the number of arrivals does not reach N by the end of the J~(th) vacation,the server remains idle in the system until the number of arrivals in the queue re...     

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.     

Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation

This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to Bernoulli vacation policy.It is assumed that the server, after each service completion,begins a process of search in order to find the following customer to be served with a certain probability,or begins a single vacation process with complementary probability. This paper analyzes the Markov chain underlying the queueing system and obtain its ergodicity condition.The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle,busy or on vacation.Finally,the author gives two stochastic decomposition laws,and as an application the author gives bounds for the proximity between the system size distributions of the model and the corresponding model without retrials.     

Strategic Behavior and Optimization in an Unobservable Constant Retrial Queue with Balking and Set-Up Time

An M/M/1 constant retrial queue with balking customers and set-up time is considered.Once the system becomes empty, the server will be turned down to reduce operating costs, and it will be activated only when there is a customers arrives. In this paper, the almost unobservable case is studied, in which the information of the queue length is unavailable, whereas the state of the server can be obtained. Firstly, the steady state solutions are derived and the individual equilibrium strategies are analyzed. In addition, social optimization problems, including cost analysis and social welfare maximization are investigated by using the PSO algorithm. Finally, by appropriate numerical examples, the sensitivity of some main system parameters is shown.     

Many-server queues with customer abandonment: A survey of diffusion and fluid approximations

The performance of a call center is sensitive to customer abandonment.In this survey paper,we focus on G/GI/ n + GI parallel-server queues that serve as a building block to model call center operations.Such a queue has a general arrival process(the G ),independent and identically distributed(iid) service times with a general distribution(the first GI),and iid patience times with a general distribution(the +GI).Following the square-root safety staffing rule,this queue can be operated in the quality- and efficiency-driven(QED) regime,which is characterized by large customer volume,the waiting times being a fraction of the service times,only a small fraction of customers abandoning the system,and high server utilization.Operational efficiency is the central target in a system whose staffing costs dominate other expenses.If a moderate fraction of customer abandonment is allowed,such a system should be operated in an overloaded regime known as the efficiency-driven (ED) regime.We survey recent results on the many-server queues that are operated in the QED and ED regimes.These results include the performance insensitivity to patience time distributions and diffusion and fluid approximate models as practical tools for performance analysis.     

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.     

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.     

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.     

Equilibrium Customer Strategies in Markovian Queues with Setup Times and Partial Failures

This paper considers the equilibrium behavior of customers in a Markovian queue with setup times and partial failures, where the reactivated server must go through a period of setup time to reach the normal working state and a failure can occur at any time during the normal service. When a partial failure occurs, the server continues to serve the customers on spot at a low rate and does not admit a new arrival. Once the system becomes empty, an exponential repair time starts. Assuming that all the customers have the option of joining or balking based on a linear reward-cost structure, the authors analyze the equilibrium strategies of the customers and the average social benefits of the system in the fully observable case and the partially observable case, respectively. And on this basis, the effect of several parameters on customers' strategic behavior is presented by some numerical examples.     

具有强占优先权的不耐烦顾客的M/M/m/k排队模型

首先研究只有一类不耐烦顾客的M/M/m排队模型,其中顾客到达服从相互独立的泊松分布,服务时间服从相互独立的指数分布,到达率与服务率随着系统中的顾客数而发生变化。顾客的耐心等待时间(截止到服务开始前)服从指数分布。在此基础上进一步研究两类顾客到达的M/M/m/k排队系统。其中第一类顾客对于第二类顾客有强占优先权,两类顾客的到达率与服务率随着系统中顾客人数而发生变化。采用矩阵分析的方法得到了两类顾客各自的稳态分布,并有相应的性能分析,为系统的优化设计提供了依据。     

Geom/G_1,G_2(Geom/G)/1/1可修Erlang消失系统的可靠性指标及其计算机仿真分析

研究Bernoulli到达且无等待空间的单服务员离散时间可修Erlang消失排队系统.系统中服务员可向顾客提供两种不同类型的服务,即常规服务和可选二次服务.在系统运行过程中服务设备的故障可以引起系统中顾客的清空.采用一种新型的离散补充变量技术,给出了系统稳态可用度,稳态失效频度,首次故障前平均时间,服务员空闲概率,故障概率,工作概率以及系统稳态损失概率等一系列性能指标.最后通过数值实例和计算机仿真验证了理论分析技术的合理性和有效性.     

N-策略工作休假M/M/1排队系统中的顾客行为研究

考虑N-策略M/M/1排队,休假期间服务员并未停止工作而是以较低的速率为顾客服务.系统的决策主体是顾客,基于"收益-成本"结构,利用马尔可夫过程理论,采取均值分析的方法,以顾客追求利益最大化为出发点,分析了全可见和几乎可见两种情况下的顾客行为.通过求解平衡方程,得到几乎可见情况下系统的稳态概率,进而求得几乎可见状态下顾客的期望逗留时间.构建均衡社会收益函数,并通过数值模拟,分析系统的各个参数对社会均衡收益的影响.     

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.     

在一个二阶排队网络中对两类不同顾客的模糊控制

研究一个二阶的串联排队网络 ,其中第一阶有一个服务员 ,第二阶有两个并联服务员 ,每个服务员都有其自己的具有无限存储空间的排队队列 .在第一阶的存储空间中包含有足够多的两类顾客 .两类不同的顾客需要首先在第一阶的服务员上接受服务然后按顺序先后进入第二阶分别排在和两类顾客相对应的服务员前 .不管什么时侯只要第一阶的服务员服务完一个顾客 ,系统就得到一个收益 ,与此同时 ,每个在第二阶排队中的顾客都需要付一笔占位费 .系统的控制目标是根据系统的状态动态地选择在第一阶中下一个被服务的顾客的种类 ,以保证系统在一个无限长的时间里平均收益最大 .这个控制模型可以应用在生产和通讯网络中 .在这篇论文里 ,一个模糊控制的新方法被提出用以解决这个复杂问题 .计算机模拟的结果显示这种新方法是非常有效并极富研究潜力的 .     

A comparative study of the inventory system with service facility and postponed demands

In this article, we present a continuous review （s,S） inventory system with a service facility consisting of finite buffer （capacity N ） and a single server. The customers arrive according to a Poisson process. The individual customer＇s unit demand is satisfied after a random time of service, which is assumed to be exponential. When the inventory level drops to s＇an order for Q（= S-s） items is placed. The lead time of reorder is assumed to be exponential distribution. An arriving customer, who finds the buffer is full, enters into the pool of infinite size or leaves the system according to a Bernolli trial. At the time of service completion, if the buffer size drops to a preassigned level L （1 〈 L 〈 N） or below and the inventory level is above s, we select the customers from the pool according to two different policy ： in first policy, with probability p （0 〈 p 〈 1） we select the customer from the head of the pool and we place the customer at the end of the buffer; in the second policy, with p （0 〈 p 〈 1） the customer from the pool is transferred to the buffer for immediate service and after completion of his service we provide service to the customer who is in the buffer with probability one. If at a service completion epoch the buffer turns out to be empty, there is at least one customer in the pool and the inventory level is positive, then the one ahead of all waiting in the pool gets transferred to the buffer, and his service starts immediately. The joint probability distribution of the number of customers in the pool, number of customers in the buffer and the inventory level is obtained in the steady-state case. Various stationary system performance measures are computed and total expected cost rate is calculated. A comparative result of two models is illustrate numerically.     

SINGLE SERVER QUEUES WITH A BATCH MARKOVIAN ARRIVAL PROCESS AND BULK RENEWAL OR NON-RENEWAL SERVICE

We first consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process(BMAP).The server serves customers in batches of maximum size 'b' with a minimum threshold size 'a'.The service time of each batch follows general distribution independent of each other as well as the arrival process.The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch.Next we obtain queue-length distributions at various other epochs such as,pre-arrival,arbitrary and pre-service using relations with post-departure epoch.Later we also obtain the system-length distributions at post-departure and arbitrary epochs using queue-length distribution at post-departure epoch.Some important performance measures,like mean queue-lengths and mean waiting times have been obtained.Total expected cost function per unit time is also derived to determine the locally optimal values of a and b.Secondly,we perform similar analysis for the corresponding infinite-buffer single server queue where arrivals occur according to a BMAP and service process in this case follows a non-renewal one,namely,Markovian service process(MSP).