带有止步和中途退出的优先权排队系统

研究了一个带有止步和中途退出的优先权排队系统,其中系统中有两类顾客,第一类顾客具有优先权,而且可能中途退出,第二类顾客可能止步和中途退出.首先,建立了系统稳态概率满足的方程组.其次,采用分块矩阵的方法得到了两类顾客的稳态分布,并且得到了系统中两类顾客的的平均队长、平均中途退出率等性能指标.最后,进行了相应的性能分析与比较,为系统的优化设计提供了参考.     

带止步与中途退出策略的M~x/M/1/N单重工作休假排队系统

研究了带有止步和中途退出的M~x/M/1/N单重工作休假排队系统.顾客成批到达,到达后每批中的顾客,或者以概率b决定进入队列等待服务,或者以概率1-b止步(不进入系统).顾客进入系统后可能因为等待的不耐烦而在没有接受服务的情况下离开系统(中途退出).系统中一旦没有顾客,服务员立即进入单重工作休假.首先,利用马尔科夫过程理论建立了系统稳态概率满足的方程组.其次利用矩阵解法求出了稳态概率的矩阵解并得到了系统的平均队长、平均等待队长以及顾客的平均消失概率等性能指标.最后通过数值例子分析了工作休假时的低服务率η和休假率θ这两个参数对系统平均队长的影响.     

带有止步和中途退出的成批到达的M~x/M/R/N同步休假排队系统的性能分析

研究了带有止步和中途退出的Mx/M/R/N同步休假排队系统.顾客成批到达.到达的顾客如果看到服务员正在休假或者全忙,他或者以概率b决定进入队列等待服务,或者以概率1-b止步(不进入系统).系统根据一定的原则以概率nk在未止步的k个顾客中选择n个进入系统.在系统中排队等待服务的顾客可能因为等待的不耐烦而在没有接受服务的情况下离开系统(中途退出).系统中一旦没有顾客,R个服务员立即进行同步多重休假.首先,利用马尔科夫过程理论建立了系统稳态概率满足的方程组.其次,在证明了相关矩阵可逆性的基础上,利用矩阵解法求出了系统稳态概率的明显表达式,并得到了系统的平均队长、平均等待队长及顾客的平均损失率等性能指标.     

带有止步和中途退出的成批到达的Mx/M/1/N多重休假排队系统的性能分析

本文研究了带有止步和中途退出的M^x／M/1／N多重休假排队系统。顾客成批到达，到达后每批中的顾客，或者以概率b决定进入队列等待服务，或者以概率1-b止步（不进入系统）。顾客进入系统后可能因为等待的不耐烦而在没有接受服务的情况下离开系统（中途退出）。系统中一旦没有顾客，服务员立即进行多重休假。首先，利用马尔科夫过程理论建立了系统稳态概率满足的方程组。其次，在利用高等代数相关知识证明了相关矩阵可逆性的基础上，利用矩阵解法求出了稳态概率的矩阵解，并得到了系统的平均队长、平均等待队长以及顾客的平均损失率等性能指标。     

Simulation of a Fast-Food Operation Where Dissatisfied Customers Renege

This paper presents the results of experiments with a simulation model for the operation of a fast-food kiosk. The kiosk was located in an office complex where a large lunch population, consisting mostly of office workers, was served by a number of competing food facilities. Each customer of the kiosk had an initial expectation of speed and quality of service. A customer arriving at the kiosk might decide not to wait, in view of the length of any queue in front of the kiosk and how long he was willing to wait. This is called balking. If the customer decided to join the queue, he observed the speed of the service others ahead of him received while in the queue, which he used to modify his own perception of the time he would have to spend waiting. A comparison of his time value and the expected wait would lead the customer to a decision as to leaving the queue or staying on. A customer who thus leaves is said to have reneged.A terminating simulation of the operation of the kiosk during a lunch period was based on a Bayesian approach. Experiments with the model revealed many interesting and useful relationships to help better understand the effect of the number of servers in the kiosk and of the food quality on the balking and reneging of the customers.     

On a Two-Queue Priority System with Impatience and its Application to a Call Center*

We consider an s-server priority system with a protected and an unprotected queue. The arrival rates at the queues and the service rate may depend on the number n of customers being in service or in the protected queue, but the service rate is assumed to be constant for n > s. As soon as any server is idle, a customer from the protected queue will be served according to the FCFS discipline. However, the customers in the protected queue are impatient. If the offered waiting time exceeds a random maximal waiting time I, then the customer leaves the protected queue after time I. If I is less than a given deterministic time, then he leaves the system, else he will be transferred by the system to the unprotected queue. The service of a customer from the unprotected queue will be started if the protected queue is empty and more than a given number of servers become idle. The model is a generalization of the many-server queue with impatient customers. The global balance conditions seem to have no explicit solution. However, the balance conditions for the density of the stationary state process for the subsystem of customers being in service or in the protected queue can be solved. This yields the stability conditions and the probabilities that precisely n customers are in service or in the protected queue. For obtaining performance measures for the unprotected queue, a system approximation based on fitting impatience intensities is constructed. The results are applied to the performance analysis of a call center with an integrated voice-mail-server.     

Questionnaire design improvement and missing item scores estimation for rapid and efficient decision making

In this paper we consider a single-server, cyclic polling system with switch-over times and Poisson arrivals. The service disciplines that are discussed, are exhaustive and gated service. The novel contribution of the present paper is that we consider the reneging of customers at polling instants. In more detail, whenever the server starts or ends a visit to a queue, some of the customers waiting in each queue leave the system before having received service. The probability that a certain customer leaves the queue, depends on the queue in which the customer is waiting, and on the location of the server. We show that this system can be analysed by introducing customer subtypes, depending on their arrival periods, and keeping track of the moment when they abandon the system. In order to determine waiting time distributions, we regard the system as a polling model with varying arrival rates, and apply a generalised version of the distributional form of Little??s law. The marginal queue length distribution can be found by conditioning on the state of the system (position of the server, and whether it is serving or switching).     

Symmetric queues served in cyclic order

Consider a symmetrical system of n queues served in cyclic order by a single server. It is shown that the stationary number of customers in the system is distributed as the sum of three independent random variables, one being the stationary number of customers in a standard M/G/1 queue. This fact is used to establish an upper bound for the mean waiting time for the case where at most k customers are served at each queue per visit by the server. This approach is also used to rederive the mean waiting times for the cases of exhaustive service, gated service, and serve at most one customer at each queue per visit by the server.     

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.     

带有止步和中途退出的M/M/R/N同步多重工作休假排队系统

研究了带有止步和中途退出的M/M/R/N同步多重工作休假排队系统,利用马尔可夫过程理论和矩阵解法求出了含有两个逆阵的系统稳态概率的矩阵解,并得到了系统的平均队长、服务员处在工作休假期的概率以及顾客的平均止步率等性能指标.最后通过数值例子分析了系统的参数对平均队长的影响.     

The MM CPP/GE/c G-Queue: Sojourn Time Distribution

We obtain the sojourn time probability distribution function at equilibrium for a Markov modulated, multi-server, single queue with generalised exponential (GE) service time distribution and compound Poisson arrivals of both positive and negative customers. Such arrival processes can model both burstiness and correlated traffic and are well suited to models of ATM and other telecommunication networks. Negative customers remove (ordinary) customers in the queue and are similarly correlated and bursty. We consider both the cases where negative customers remove positive customers from the front and the end of the queue and, in the latter case, where a customer currently being served can and cannot be killed by a negative customer. These cases can model an unreliable server or load balancing respectively. The results are obtained as Laplace transforms and can be inverted numerically. The MM CPP/GE/c G-Queue therefore holds the promise of being a viable building block for the analysis of queues and queueing networks with bursty, correlated traffic, incorporating load balancing and node-failures, since the equilibrium behaviour of both queue lengths and response times can be determined in a tractable way.     

Scheduling service in tandem queues attended by a single server

Consider a tandem queue model with a single server who can switch instantaneously from one queue to another. Customers arrive according to a Poisson process with rate λ . The amount of service required by each customer at the i^th queue is an exponentially distributed random variable with rate μ_i. Whenever two or more customers are in the system, the decision as to which customer should be served first depends on the optimzation criterion. In this system all server allocation policies in the finite set of work conserving deterministic policies have the same expected first passage times (makespan) to empty the system of customers from any initial state. However, a unique policy maximizes the first passage probability of empty-ing the system before the number of customers exceeds K, for any value of K, and it stochastically minimizes (he number of customers in the system at any time t > 0 . This policy always assigns the server to the non empty queue closest to the exit     

Analysis of queueing systems with customer interjections

In this paper we study queueing systems with customer interjections. Customers are distinguished into normal customers and interjecting customers. All customers join a single queue waiting for service. A normal customer joins the queue at the end and an interjecting customer tries to cut in the queue. The waiting times of normal customers and interjecting customers are studied. Two parameters are introduced to describe the interjection behavior: the percentage of customers interjecting and the tolerance level of interjection by individual customers. The relationship between the two parameters and the mean and variance of waiting times is characterized analytically and numerically.     

Single-Server Queues with Markov-Modulated Arrivals and Service Speed

This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is assumed to be general, which may differ with different classes. Further, the service speed depends on the state of the underlying Markov chain. We first show that given such a queue, we can construct the corresponding new queue with constant service speed by means of a change of time scale, and the time-average quantities of interest in the original queue are given in terms of those in the new queue. Next we characterize the joint distribution of the length of a busy period and the number of customers served during the busy period in the original queue. Finally, assuming the FIFO service discipline, we derive the Laplace–Stieltjes transform of the actual waiting time distribution in the original queue.     

Mixed gated/exhaustive service in a polling model with priorities

In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers.     

On the M/G/1 queue with feedback and optional server vacations based on a single vacation policy

We study a single server queue with batch arrivals and general (arbitrary) service time distribution. The server provides service to customers, one by one, on a first come, first served basis. Just after completion of his service, a customer may leave the system or may opt to repeat his service, in which case this customer rejoins the queue. Further, just after completion of a customer's service the server may take a vacation of random length or may opt to continue staying in the system to serve the next customer. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers and the average waiting time in the queue. Some special cases of interest are discussed and some known results have been derived. A numerical illustration is provided.     

Economies of scale,competitiveness, and trade patterns within the European community: Clarendon Press,Oxford, 1983, 193 pages, £20.00

We consider two queues in series with input to each queue, which can be controlled by accepting or rejecting arriving customers. The objective is to maximize the discounted or average expected net benefit over a finite or infinite horizon, where net benefit is composed of (random) rewards for entering customers minus holding costs assessed against the customers at each queue. Provided that it costs more to hold a customer at the first queue than at the second, we show that an optimal policy is monotonic in the following senses: Adding a customer to either queue makes it less likely that we will accept a new customer into either queue; moreover moving a customer from the first queue to the second makes it more (less) likely that we will accept a new customer into the first (second) queue. Our model has policy implications for flow control in communication systems, industrial job shops, and traffic-flow systems. We comment on the relation between the control policies implied by our model and those proposed in the communicationa literature.     

Stationary tail asymptotics of a tandem queue with feedback

Motivated by applications in manufacturing systems and computer networks, in this paper, we consider a tandem queue with feedback. In this model, the i.i.d. interarrival times and the i.i.d. service times are both exponential and independent. Upon completion of a service at the second station, the customer either leaves the system with probability p or goes back, together with all customers currently waiting in the second queue, to the first queue with probability 1−p. For any fixed number of customers in one queue (either queue 1 or queue 2), using newly developed methods we study properties of the exactly geometric tail asymptotics as the number of customers in the other queue increases to infinity. We hope that this work can serve as a demonstration of how to deal with a block generating function of GI/M/1 type, and an illustration of how the boundary behaviour can affect the tail decay rate.     

Strategic behavior in queues with the effect of the number of customers behind

In this paper, we investigate the strategic behavior in queues by considering the effect of the number of customers behind. The equilibrium joining strategy of customers is obtained and its implications for the service system are examined. We find that the complete queue transparency (i.e., disclosing the real-time system information) can have positive effect on customers, which might encourage more customers to join. Further, the follow-the-crowd (FTC) behavior can be observed, which results in multiple equilibria. By comparing the customer welfare under two different information levels, we demonstrate that, somewhat surprisingly, the queue transparency does not necessarily hurt the customer welfare, and a higher customer welfare can be obtained in the transparent case than that in opaque case when the demand volume is large.