Equilibrium threshold strategies in observable queueing systems with setup/closedown times

This paper considers two types of setup/closedown policies: interruptible and insusceptible setup/closedown policies. When all customers are served exhaustively in a system under the interruptible setup/closedown policy, the server shuts down (deactivates) by a closedown time. When the server reactivates since shutdown, he needs a setup time before providing service again. If a customer arrives during a closedown time, the service is immediately started without a setup time. However, in a system under the insusceptible setup/closedown policy, customers arriving in a closedown time can not be served until the following setup time finishes. For the systems with interruptible setup/closedown times, we assume both the fully and almost observable cases, then derive equilibrium threshold strategies for the customers and analyze the stationary behavior of the systems. On the other hand, for the systems with insusceptible setup/closedown times, we only consider the fully observable case. We also illustrate the equilibrium thresholds and the social benefits for systems via numerical experiments. As far as we know, there is no work concerning equilibrium behavior of customers in queueing systems with setup/closedown times.     

Batch arrival queues under vacation policies with server breakdowns and startup/closedown times

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under vacation policies with startup/closedown times, where the vacation time, the startup time, and the closedown time are generally distributed. When all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time. After shutdown, the server operates one of (1) multiple vacation policy and (2) single vacation policy. When the server reactivates since shutdown, he needs a startup time before providing the service. If a customer arrives during a closedown time, the service is immediately started without a startup time. The server may break down according to a Poisson process while working and his repair time has a general distribution. We analyze the system characteristics for the vacation models.     

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.     

基于排队系统的偏好差异性顾客服务策略分析

在服务商为不同偏好顾客提供差异性服务的排队系统中,其收益受到需求规模、顾客类型等因素影响,如何制定策略才能达到服务特定类型顾客并获取最大收益的目的？本文以服务时间与效果正相关的顾客密集型服务为背景,基于M/M/1模型对服务商为两种偏好顾客提供两种服务的情景进行研究,并将从偏好服务所获基准效用相对较低、较高的顾客分别定义为1类、2类顾客。结果表明,①随着潜在顾客规模的增加,最优策略所适用的3种基本策略依次为公平策略、优先服务2类顾客的策略以及仅服务2类顾客的排他性策略;②3种基本策略的适用范围仅与1类顾客的服务选择标准相关;③在优先服务某类顾客的策略中,有效到达速率及服务速度由另一类顾客的特征决定。     

Equilibrium customer strategies in a single server Markovian queue with setup times

We consider a single server Markovian queue with setup times. Whenever this system becomes empty, the server is turned off. Whenever a customer arrives to an empty system, the server begins an exponential setup time to start service again. We assume that arriving customers decide whether to enter the system or balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine customer behavior under various levels of information regarding the system state. Specifically, before making the decision, a customer may or may not know the state of the server and/or the number of present customers. We derive equilibrium strategies for the customers under the various levels of information and analyze the stationary behavior of the system under these strategies. We also illustrate further effects of the information level on the equilibrium behavior via numerical experiments.      

Equilibrium pricing in an M/G/1 retrial queue with reserved idle time and setup time

Power consumption is a ubiquitous and challenging problem in modern society. To save energy, one should turn off an idle device which still consumes about 60% of its peak consumption and switch it on again when some jobs arrive. However, it is not tolerate for delay sensitive applications. Therefore, there is a trade-off between power consumption and delay performance. In this paper we study an M/G/1 retrial queueing system with setup times in which the server keeps idle for a reserved idle time after completion of a service. If there are arrivals during this reserved idle time, these customers can be served immediately. Otherwise, the server will be turned off for saving energy until a new customer comes to activate the server. The setup time follows an exponential distribution. Based on the reward-cost function and the expected payoff, all customers will make decisions on whether to join or balk the system upon arrival. Given these strategic behaviors we study the optimal pricing strategies from the perspective of the server and social planner, respectively. The optimization of the reserved idle time for maximizing the server’s profit is also studied. Finally, numerical experiments are presented to illustrate the impact of system parameters on the customers’ equilibrium behavior and profit maximization solutions.     

Optimal admission and pricing control problem with deterministic service times and sideline profit

We consider a discrete-time admission control problem in a company operating in service industries with two classes of customers. For the first class of customers, the company then (1) has an option to accept or reject him/her (admission control), or (2) decides on an offering price (pricing control). The second-class (sideline) customers are only served if no first-class customers are in the system, and this yields the sideline profit. In this paper, we discuss both admission control and pricing control problems within an identical framework, and we examine the properties of the optimal policies to maximize the total expected present discounted net profits. We show that when the sideline profit is large, the optimal policies may not be monotone in the number of first-class customers in the system.     

Optimal server scheduling in nonpreemptive finite-population queueing systems

We consider a finite-population queueing system with heterogeneous classes of customers and a single server. For the case of nonpreemptive service, we fully characterize the structure of the server's optimal service policy that minimizes the total average customer waiting costs. We show that the optimal service policy may never serve some classes of customers. For those classes that are served, we show that the optimal service policy is a simple static priority policy. We also derive sufficient conditions that determine the optimal priority sequence.     

Two thresholds of a batch arrival queueing system under modified T vacation policy with startup and closedown

This paper studies the operating characteristics of the variant of an M^[x]/G/1 vacation queue with startup and closedown times. After all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time, and then takes at most J vacations of constant time length T repeatedly until at least one customer is found waiting in the queue upon returning from a vacation. If at least one customer is present in the system when the server returns from a vacation, then the server reactivates and requires a startup time before providing the service. On the other hand, if no customers arrive by the end of the J th vacation, the server remains dormant in the system until at least one customer arrives. We will call the vacation policy modified T vacation policy. We derive the steady‐state probability distribution of the system size and the queue waiting time. Other system characteristics are also investigated. The long‐run average cost function per unit time is developed to determine the suitable thresholds of T and J that yield a minimum cost. Copyright © 2007 John Wiley & Sons, Ltd.     

Admission control strategies for tandem Markovian loss systems

Motivated by communication networks, we study an admission control problem for a Markovian loss system comprised of two finite capacity service stations in tandem. Customers arrive to station 1 according to a Poisson process, and a gatekeeper, who has complete knowledge of the number of customers at both stations, decides whether to accept or reject each arriving customer. If a customer is rejected, a rejection cost is incurred. If an admitted customer finds that station 2 is full at the time of his service completion at station 1, he leaves the system and a loss cost is incurred. The goal is to find easy-to-implement policies that minimize long-run average cost per unit time. We formulate two intuitive, extremal policies and provide analytical results on their performances. We also present necessary and/or sufficient conditions under which each of these policies is optimal. Next, we show that for some states of the system it is always optimal to admit new arrivals. We also fully characterize the optimal policy when the capacity of each station is two and discuss some characteristics of optimal policies in general. Finally, we design heuristic admission control policies using these insights. Numerical experiments indicate that these heuristic policies yield near-optimal long-run average cost performance.     

不同价格水平下排队策略研究

在M/M/1排队中引入了不同的服务价格,基于"收益-成本"结构,以顾客和企业均追求利益最大化为出发点,在两种不可见情形下,研究了顾客均衡策略行为和企业最优服务定价决策,通过数值模拟,描述了休假期服务价格对顾客均衡策略的影响,以及几乎不可见情况下休假期服务价格对企业收益的作用和完全不可见情况下休假期服务价格随潜在到达率的变化情况,以及当企业获得最大收益时,正常工作期和休假期服务价格的关系.     

Containment of socially optimal policies in multiple-facility Markovian queueing systems

We consider a Markovian queueing system with N heterogeneous service facilities, each of which has multiple servers available, linear holding costs, a fixed value of service and a first-come-first-serve queue discipline. Customers arriving in the system can be either rejected or sent to one of the N facilities. Two different types of control policies are considered, which we refer to as ‘selfishly optimal’ and ‘socially optimal’. We prove the equivalence of two different Markov Decision Process formulations, and then show that classical M/M/1 queue results from the early literature on behavioural queueing theory can be generalized to multiple dimensions in an elegant way. In particular, the state space of the continuous-time Markov process induced by a socially optimal policy is contained within that of the selfishly optimal policy. We also show that this result holds when customers are divided into an arbitrary number of heterogeneous classes, provided that the service rates remain non-discriminatory.     

A Discrete-Time Single-Server Queueing System Under Multiple Vacations and Setup-Closedown Times

Abstract

This article concerns a Geo/G/1/∞ queueing system under multiple vacations and setup-closedown times. Specifically, the operation of the system is as follows. After each departure leaving an empty system, the server is deactivated during a closedown time. At the end of each closedown time, if at least a customer is present in the system, the server begins the service of the customers (is reactivated) without setup; however, if the system is completely empty, the server takes a vacation. At the end of each vacation, if there is at least a customer in the system, the server requires a startup time (is reactivated) before beginning the service of the customers; nevertheless, if there are not customers waiting in the system, the server takes another vacation. By applying the supplementary variable technique, the joint generating function of the server state and the system length together with the main performance measures are derived. We also study the length of the different busy periods of the server. The stationary distributions of the time spent waiting in the queue and in the system under the FCFS discipline are analysed too. Finally, a cost model with some numerical results is presented.     

An optimal age maintenance for an M/G/1 queueing system

This paper discusses an optimal age maintenance scheme for a queueing system. Customers arrive at the system according to a Poisson process. They form a single queue and are served by a server with general service distribution. The system fails after a random time and corrective maintenance is performed at the failure. A preventive maintenance is also performed if the system is empty at age T where ‘age’ refers to the elapsed time since the previous maintenance was completed. If the system is not empty at age T, the system is used until it fails. At the failure, the customers in the system are lost and the arriving customers during the maintenance are also lost. By renewal theory, we study the optimal value of T which minimizes the average number of lost customers over an infinite time horizon.     

Performance of service policies in a specialized service system with parallel servers

This paper considers a scheduling problem occurring in a specialized service system with parallel servers. In the system, customers are divided into the “ordinary” and “special” categories according to their service needs. Ordinary customers can be served by any server, while special customers can be served only by the flexible servers. We assume that the service time for any ordinary customer is the same and all special customers have another common service time. We analyze three classes of service policies used in practice, namely, policies with priority, policies without priority and mixed policies. The worst-case performance ratios are obtained for all of these service policies.     

Optimal admission and pricing control problems in service industries with multiple servers and sideline profit

This paper deals with the problem of selecting profitable customer orders sequentially arriving at a company operating in service industries with multiple servers in which two classes of services are provided. The first class of service is designed to meet the particular needs of customers; and the company (1) makes a decision on whether to accept or to reject the order for this service (admission control) and (2) decides a price of the order and offers it to an arriving customer (pricing control). The second class of service is provided as a sideline, which prevents servers from being idle when the number of customer orders for the first class is less than the number of servers. This yields the sideline profit. A cost is paid to search for customer orders, which is called the search cost. In the context of search cost, the company has an option whether to conduct the search or not. In this paper, we discuss both admission control and pricing control problems within an identical framework as well as examine the structure of the optimal policies to maximize the total expected net profit gained over an infinite planning horizon. We show that when the sideline profit is large, the optimal policies may not be monotone in the number of customer orders in the system. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal static pricing for a service facility with holding costs

We study a service facility modelled as a single-server queueing system with Poisson arrivals and limited or unlimited buffer size. In systems with unlimited buffer size, the service times have general distributions, whereas in finite buffered systems service times are exponentially distributed. Arriving customers enter if there is room in the facility and if they are willing to pay the posted price. The same price is charged to all customers at all times (static pricing). The service provider is charged a holding cost proportional to the time that the customers spend in the system. We demonstrate that there is a unique optimal price that maximizes the long-run average profit per unit time. We also investigate how optimal prices vary as system parameters change. Finally, we consider buffer size as an additional decision variable and show that there is an optimal buffer size level that maximizes profit.     

Comparison of two randomized policy M/G/1 queues with second optional service, server breakdown and startup

The problem addressed in this paper is to compare the minimum cost of the two randomized control policies in the M/G/1 queueing system with an unreliable server, a second optional service, and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service until the system becomes empty. After all customers are served in the queue, the server immediately takes a vacation and the system operates the (T, p)-policy or (p, N)-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of (T, p) and (p, N), respectively. In addition, we obtain the explicit closed form of the joint optimal solutions for those two policies. Based on the minimal cost, we show that the optimal (p, N)-policy indeed outperforms the optimal (T, p)-policy. Numerical examples are also presented for illustrative purposes.