Bias optimality of admission control in a non-stationary repairable queue

We study a non-stationary repairable queue with a single server and multiple customers’ types. The difference between types of customers is defined by the offered rewards. We show that the bias optimal policy has the trunk reservation (threshold) form. Furthermore, under some given conditions, we also prove that the control level of the bias optimal policy is monotone about time.     

On the optimality of a maintenance queueing system

We consider a finite single-server maintenance queue with multiple types of customers. The difference between customers' types is defined by the offered rewards. We show that the optimal admission control policy for maximizing the long-run average reward per unit time has a trunk reservation structure. Meanwhile, if the equipment is off, there exists a threshold of the queue length, above which the optimal repair speed is increasing in the queue length and below which the optimal repair speed is 0.     

On non-equilibria threshold strategies in ticket queues

In many real-life queueing systems, a customer may balk upon arrival at a queueing system, but other customers become aware of it only at the time the balking customer was to start service. Naturally, the balking is an outcome of the queue length, and the decision is based on a threshold. Yet the inspected queue length contains customers who balked. In this work, we consider a Markovian queue with infinite capacity and with customers that are homogeneous with respect to their cost reward functions. We show that that no threshold strategy can be a Nash equilibrium strategy. Furthermore, we show that for any threshold strategy adopted by all, the individual’s best response is a double threshold strategy. That is, join if and only if one of the following is true: (i) the inspected queue length is smaller than one threshold, or (ii) the inspected queue length is larger than a second threshold. Our model is under the assumption that the response time of the server when he finds out that a customer balked is negligible. We also discuss the validity of the result when the response time is not negligible.     

The Infinite Server Queue with Arrivals Generated by a Non-Homogeneous Compound Poisson Process

This paper considers the infinite server queue with arrivals generated by a non-homogeneous compound Poisson process. In such a system, customers arrive in groups of variable size, the arrival epochs of groups being points of a non-homogeneous Poisson process, and they are served without delay. The service times of customers who belong to the same group need not be independent nor identically distributed. Assuming that the system is initially empty, the transient distribution of the queue size and of the counting departure process is obtained. Also the limiting queue size distribution (when it exists) is determined and it is found to be insensitive to the form the service time distribution functions.     

在D-策略控制下服务员单重休假且休假不中断的M/G/1排队系统分析

该文研究在D-策略控制下服务员单重休假且休假不中断的M/G/1排队系统,其中当服务员休假结束归来时,如果系统中等待服务的顾客所需的总服务时间之和不小于事先给定的正数阀值D,服务员就立即开始服务.运用全概率分解技术、更新过程理论和拉普拉斯变换工具,本文在任意初始状态下讨论了队长的瞬态分布,导出了队长瞬态分布的拉普拉斯变换的表达式和稳态队长分布的递推表达式.同时给出了稳态队长的随机分解结构、附加队长分布的显示表达式.进一步借用稳态队长分布{pj,j=0,1,2,?},讨论了系统容量的优化设计,并阐述了稳态队长分布对系统容量优化设计所起的重要作用.最后,在建立费用模型的基础上,导出了系统在长期单位时间内期望费用的显示表达式,并通过数值实例不仅确定了使系统在长期单位时间内的期望费用最小的控制策略D?,而且还得到了当休假时间长度为固定时长T(>0)时系统的联合控制策略(T?,D?).     

The N threshold policy for the GI/M/1 queue

We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.     

Transient analysis of a Markovian queue with deterministic rejection

We analyze the transient behavior of the M/M/1+D queue. Considering an Erlang distribution for customers’ waiting time, we approximate the real system by a Markov chain. We obtain the Laplace Transform of the transient probabilities in the approximated model and the Laplace transform of the main performance measures for the real system. We next analyze the busy period of this queue. One interesting insight is that the busy period of the unstable M/M/s queue has a finite coefficient of variation.     

To wait or not to wait: Strategic behaviors in an observable batch-service queueing system

This paper deals with an observable batch service queueing system in which customers rationally choose whether to form a batch with another customer or not, in addition to deciding whether or not to join the queue. When choosing whether to form a batch, a customer in an incomplete batch decides on an optimal waiting time for the next customer to arrive and share the service fee. When choosing whether to join the queue, customers follow a threshold strategy, which depends on the strategy identified in the former game.     

On equilibrium threshold strategies when choosing between observable and unobservable queues

We consider threshold equilibrium strategies in a two-server memoryless queueing system where customers inspect one queue before deciding which queue to join. We show that such an equilibrium may not exist. A numerical study indicates that if threshold equilibria exist, the effective arrival rate to the unobserved queue is higher when the regime there is last-come first-served rather than first-come first-served.     

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.     

Busy period analysis of the state dependent M/M/1/K queue

In this paper, we study the transient behavior of a state dependent M/M/1/K queue during the busy period. We derive in closed-form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. For two special cases called the threshold policy and the static policy we determine simple expressions for their joint transform.     

A mixed priority retrial queue with negative arrivals,unreliable server and multiple vacations

A retrial queue accepting two types of positive customers and negative arrivals, mixed priorities, unreliable server and multiple vacations is considered. In case of blocking the first type customers can be queued whereas the second type customers leave the system and try their luck again after a random time period. When a first type customer arrives during the service of a second type customer, he either pushes the customer in service in orbit (preemptive) or he joins the queue waiting to be served (non-preemptive). Moreover negative arrivals eliminate the customer in service and cause server’s abnormal breakdown, while in addition normal breakdowns may also occur. In both cases the server is sent immediately for repair. When, upon a service or repair completion, the server finds no first type customers waiting in queue remains idle and activates a timer. If timer expires before an arrival of a positive customer the server departs for multiple vacations. For such a system the stability conditions and the system state probabilities are investigated both in a transient and in a steady state. A stochastic decomposition result is also presented. Interesting applications are also discussed. Numerical results are finally obtained and used to investigate system performance.     

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.     

On M/G/1 system under NT policies with breakdowns,startup and closedown

This paper studies the vacation policies of an M/G/1 queueing system with server breakdowns, startup and closedown times, in which the length of the vacation period is controlled either by the number of arrivals during the vacation period, or by a timer. After all the customers are served in the queue exhaustively, the server is shutdown (deactivates) by a closedown time. At the end of the shutdown time, the server immediately takes a vacation and operates two different policies: (i) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or the waiting time of the leading customer reaches T units; and (ii) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or T time units have elapsed since the end of the closedown time. If the timer expires or the number of arrivals exceeds the threshold N, then the server reactivates and requires a startup time before providing the service until the system is empty. If some customers arrive during this closedown time, the service is immediately started without leaving for a vacation and without a startup time. We analyze the system characteristics for each scheme.     

Equilibrium analysis of the observable queues with balking and delayed repairs

The equilibrium threshold balking strategies are investigated for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Upon arriving, the customers decide whether to join or balk the queue based on observation of the queue length and status of the server, along with the consideration of waiting cost and the reward after finishing their service. By using Markov chain approach and system cost analysis, we obtain the stationary distribution of queue size of the queueing systems and provide algorithms in order to identify the equilibrium strategies for the fully and partially observable models. Finally, the equilibrium threshold balking strategies and the equilibrium social benefit for all customers are derived for the fully and partially observable system respectively, both with server breakdowns and delayed repairs.     

Distributional Form of Little's Law for FIFO Queues with Multiple Markovian Arrival Streams and Its Application to Queues with Vacations

This paper considers stationary queues with multiple arrival streams governed by an irreducible Markov chain. In a very general setting, we first show an invariance relationship between the time-average joint queue length distribution and the customer-average joint queue length distribution at departures. Based on this invariance relationship, we provide a distributional form of Little's law for FIFO queues with simple arrivals (i.e., the superposed arrival process has the orderliness property). Note that this law relates the time-average joint queue length distribution with the stationary sojourn time distributions of customers from respective arrival streams. As an application of the law, we consider two variants of FIFO queues with vacations, where the service time distribution of customers from each arrival stream is assumed to be general and service time distributions of customers may be different for different arrival streams. For each queue, the stationary waiting time distribution of customers from each arrival stream is first examined, and then applying the Little's law, we obtain an equation which the probability generating function of the joint queue length distribution satisfies. Further, based on this equation, we provide a way to construct a numerically feasible recursion to compute the joint queue length distribution.     

Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions

This paper considers the queue length distribution in a class of FIFO single-server queues with (possibly correlated) multiple arrival streams, where the service time distribution of customers may be different for different streams. It is widely recognized that the queue length distribution in a FIFO queue with multiple non-Poissonian arrival streams having different service time distributions is very hard to analyze, since we have to keep track of the complete order of customers in the queue to describe the queue length dynamics. In this paper, we provide an alternative way to solve the problem for a class of such queues, where arrival streams are governed by a finite-state Markov chain. We characterize the joint probability generating function of the stationary queue length distribution, by considering the joint distribution of the number of customers arriving from each stream during the stationary attained waiting time. Further we provide recursion formulas to compute the stationary joint queue length distribution and the stationary distribution representing from which stream each customer in the queue arrived.     

Implementation strategies of a contract-based MRI examination reservation process for stroke patients

Timely imaging examinations are critical for stroke patients due to the potential life threat. We have proposed a contract-based Magnetic Resonance Imaging (MRI) reservation process [1] in order to reduce their waiting time for MRI examinations. Contracted time slots (CTS) are especially reserved for Neural Vascular Department (NVD) treating stroke patients. Patients either wait in a CTS queue for such time slots or are directed to Regular Time Slot (RTS) reservation. This strategy creates “unlucky” patients having to wait for lengthy RTS reservation. This paper proposes and analyzes other contract implementation strategies called RTS reservation strategies. These strategies reserve RTS for NVD but do not direct patients to regular reservations. Patients all wait in the same queue and are served by either CTS or RTS on a FIFO (First In First Out) basis. We prove that RTS reservation strategies are able to reduce the unused time slots and patient waiting time. Extensive numerical results are presented to show the benefits of RTS reservation and to compare various RTS reservation strategies.     

Optimizing contracted resource capacity with two advance cancelation modes

Critical resources are often shared among different classes of customers. Capacity reservation allows each class of customers to better manage priorities of its customers but might lead to unused capacity. Unused capacity can be avoided or reduced by advance cancelation. This paper addresses the service capacity reservation for a given class of customers. The reservation process is characterized by: contracted time slots (CTS) reserved for the class of customers, requests for lengthy regular time slots (RTS) and two advance cancelation modes to cancel CTS one-period or two-period before. The optimal control under a given contract is formulated as an average cost Markov Decision Process (MDP) in order to minimize customer waiting times, unused CTS and CTS cancelation. Structural properties of optimal control policies are established via the corresponding discounted cost MDP problem. Numerical results show that two-period advance CTS cancelation can significantly improve the contract-based solution.     

Analysis of the MAP/G/1/N queue with multiple vacations

This paper deals with a batch service queue and multiple vacations. The system consists of a single server and a waiting room of finite capacity. Arrival of customers follows a Markovian arrival process (MAP). The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in batches of maximum size ‘b’ with a minimum threshold value ‘a’. We obtain the queue length distributions at various epochs along with some key performance measures. Finally, some numerical results have been presented.