فایل ورد کامل رفتار امتناعی توازن در سیستم صف بندی Geo/Geo/1 با چندین بیکاری

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بخشی از مقاله انگلیسیعنوان انگلیسی:Equilibrium balking behavior in the Geo=Geo=1 queueing system with multiple vacations~~en~~

Abstract

This paper studies the equilibrium behavior of customers in the queueing system under multiple vacation policy. The server leaves for repeated vacations as soon as the system becomes empty. Customers decide for themselves whether to join or to balk, which is more sensible than the classical viewpoint in queueing theory. Equilibrium customer behavior is considered under four cases: fully observable, almost observable, almost unobservable and fully unobservable, which cover all the levels of information. Based on the reward-cost structure, we obtain the equilibrium balking strategies in all cases. Furthermore, the stationary system behavior is analyzed and a variety of performance measures are developed under the corresponding strategies. Finally, we present several numerical experiments that demonstrate the effect of the information level as well as several parameters on the equilibrium behavior and social benefit. The research results not only offer the customers optimal strategies but also provide the managers with a good reference to discuss the pricing problem in the queueing system.

۱ Introduction

Due to the widely applications for management in service system and electronic commerce, there exists an emerging tendency to study customers’ behavior in queueing models. In these models, customers are allowed to make decisions as to whether to join or to balk, buy priority or not etc., which is more sensible to describe queueing models. Traditionally, queueing systems are divided into the observable model and the unobservable model regarding whether the information of the queue length is available to customers or not prior to their actions. The observable queueing system was first analyzed by Naor [1], who studied equilibrium and social optimal strategies in an M=M=1 queue with a simple linear reward-cost structure. Afterward, Naor’s model and results had been extended in several literatures, see e.g. [2–۴]. Chen and Frank [5] generalized Naor’s model assuming that both the customers and the server maximize their expected discounted utility using a common discount rate. Erlichman and Hassin [6] discussed a priority queue in which customers have the option of overtaking some or all of the customers. On the other hand, Edelson and Hildebrand [7] presented the pioneering literature on the unobservable queue in which the properties of the basic unobservable M=M=1 queue were discovered. Littlechild [8] extended the model of Edelson and Hildebrand assuming that customers have different service values. Chen and Frank [9] discussed the robustness of the main result of Edelson and Hildebrand, that a profit maximizer chooses a socially optimal admission fee, when the assumption of a linear utility function is removed. Balachandran [10] considered an unobservable M=G=1 model with a fixed cost of running the service facility. Subsequently, several authors had investigated the equilibrium strategies in various unobservable models incorporating many diverse characteristics. The fundamental results on this subject in both the observable and unobservable queueing systems can be found in the comprehensive monograph of Hassin and Haviv [11].

Discrete-time queueing systems with vacations have been widely studied in the past because of their extensively use in digital communication and telecommunication networks. An excellent and complete study on discrete-time vacation models had been presented by Takagi [12]. Zhang and Tian [13] presented the detailed analysis on the Geo=G=1 queue with multiple adaptive vacations and further in [14] Tian and Zhang dealt with a GI=Geo=1 queue with multiple vacations and exhaustive service. Recently, Samanta et al. [15] and Tang et al. [16] did research on the discrete-time GeoX=G=1 vacation queue with different characteristics.

As for the research on the equilibrium customer behavior in vacation queue models, the first was presented by Burnetas and Economou [17], who explored both the observable and unobservable cases in a single server Markovian queue with set up times. Then, Economou and Kanta [18] analyzed the equilibrium balking strategies in the observable single-server queue with breakdowns and repairs. Recently, Sun et al. [19] considered the equilibrium behavior of customers in an observable M=M=1 queue under interruptible and insusceptible setup/closedown policies. Economou et al. [20] extended the analysis done for the almost and fully unobservable queues in [17] to the non-Markovian case. Liu et al. [21] studied the equilibrium threshold strategies in observable queues under single vacation policy. However, there was no work concerning the equilibrium balking behavior in the discrete-time queues with multiple vacations.

In the present paper we analyze the equilibrium balking strategies in the discrete-time Geo=Geo=1 queue with multiple vacations. To the authors’ knowledge, this is the first time that the multiple vacation policy is introduced into the economics of queues. The customers’ dilemma is whether to join the system or balk. They make decisions based on a nature reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We explore various cases with regard to the level of information available to customers upon arrival. More specifically, at his arrival epoch a customer may or may not know the number of customers present and/or the state of the server. Therefore, four combinations emerge, ranging from full to no information. In each of the four cases we discuss customer equilibrium strategies, analyze the stationary behavior of the corresponding system and derive the equilibrium social benefit for all customers. Furthermore, we present several numerical experiments to explore the effect of the information level as well as several parameters on the equilibrium behavior and the social benefit.

This paper is organized as follows. In Section 2, we give the model description and the reward-cost structure. Section 3 discusses the observable queue in which customers observe the length of the queue. We distinguish two subcases depending on the additional information, or lack thereof, of the server state. We determine equilibrium threshold strategies and analyze the resulting stationary system behavior. Then Section 4 studies the unobservable queue where the queue length is not available to customers. We derive the corresponding mixed equilibrium strategies and investigate the stationary behavior for the almost and fully unobservable models. In Section 5, we illustrate the effect of the information level on the equilibrium behavior and the social benefit via analytical and numerical comparisons. Finally, in Section 6, we give a necessary conclusion.

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