基于博弈论的排队经济学模型及策略分析
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摘要
在排队系统中,顾客为了使自身的利益最大化而单独行动,不考虑对整个队列的影响,这种实质上是相互作用的自私行为就导致了一个纳什均衡格局的形成,然而,这个均衡从全局角度看很可能并不是最优的,由此目光敏锐的经济学和排队论学者针对这一经济学现象在排队论和博弈论背景和框架下开始了研究工作,进而逐渐形成了一门经济学、博弈论与经典排队论有机融合的交叉学科,即本文将要研究的排队经济学(Economics of Queues)。鉴于近些年博弈论的迅速崛起,此研究方向也被一些学者称之为排队博弈论(Queueing Games)。
以前人提供的关于排队经济学的文献成果为基础,主要研究了若干类型的排队经济学模型并对其进行了纳什均衡策略和社会最优策略分析,其策略主体既包含顾客也包括服务员,即研究了排队系统中的均衡行为,并把其与社会最优行为进行了对比进而找出了差异,并通过大量的数值算例描绘出了系统主要性能指标的变化规律,以此形象地验证了所得结论的合理性和正确性。
首先,对具有两项互补服务的排队经济学模型进行了策略分析。顾客除了接受第一项主要的服务项目获得效益之外,还需要购买某类商品或接受另一项辅助服务。分析了在竞争机制下形成的两类纳什均衡,分析了顾客的均衡行为,给出了两个服务员各自的均衡价格,即均衡定价策略。然后考虑了垄断机制下的纳什均衡,即两项服务都由一个垄断者提供,并发现了垄断者的目标与社会最优目标是一致的,而且垄断价格要低于竞争机制下两个服务员的价格总和。此外,在三种不同的收费策略下,比较了顾客和两个服务员对各个收费策略的好恶倾向。
其次,对具有两类顾客及双相对优先权参数的排队经济学模型进行了策略分析。其中第一个模型指出了在不同的系统参数和优先权分配策略下可以产生不同类型的纳什均衡,并分析了顾客的均衡行为,以及服务员根据具体的系统参数为使利润最大化而采取的优先权分配策略。而第二个模型则是以系统费用最小化为初衷,针对几种常见的费用函数：线性、最大值、二次以及指数型费用函数,研究了相对优先权在减少系统费用方面的作用。相对于绝对优先权,找到了相对优先权能更好地降低整个系统费用的条件,并明确地给出了服务员对两类顾客的具体的最优优先权分配策略。
除了对具有两类顾客的双相对优先权参数模型进行了研究之外,还考虑了具有N类顾客的带有N个相对优先权参数的排队经济学模型,分析了纳什均衡状态下顾客的平均排队时间问题。在系统中所有顾客数给定的情况下,给出了从任意一个顾客服务开始算起,一个给定所属类型的顾客的平均剩余排队时间,同时也给出了任意顾客从到达开始算起到服务开始的无条件平均排队时间。根据服务员给出的具体的优先权分配方案,其结果可以指导各类顾客做出是否排队等候的决策。
再次,对具有启动关闭期的排队经济学模型也进行了策略分析。考虑了三种启动关闭休假策略：可中断启动关闭策略,可跳跃启动关闭策略和非中断启动关闭策略,目的是为了研究顾客的均衡和社会最优决策问题。对于可视情形,得到了顾客的均衡阈值策略,分析了系统的稳态行为。对于不可视情形,得到了顾客的均衡和社会最优混合策略,并导出了使社会福利最优化和垄断利润最大化的服务员定价策略。
随后,对具有不完全服务信息的排队经济学模型也进行了策略分析。基于最大熵原理,在顾客被告知多种部分服务信息的情形下,给定几种常见的服务时间分布,分别比较了不可视排队系统中风险中立顾客和风险规避顾客的均衡策略和社会最优策略。此外,还分析了服务员在均衡状态下针对各种部分服务信息所作出的利润最大化策略。理论和数值分析表明顾客获得的部分信息越多,进入概率就越大,而且服务员的垄断利润也越大。
除了启动关闭期休假策略,经典休假思想也被引入到了排队经济学之中,并对一个具有多重休假的排队经济学模型进行了初步的策略分析,目的是找出顾客的均衡策略以及社会最优策略。两者进行比较,得到了个人利益最优化会导致系统过度拥塞的结论。
最后,对一个离散时间多服务员排队经济学模型也进行了策略分析。主要研究的问题也是顾客的均衡和社会最优策略,即从个人利益最优化和社会福利最优化两个标准出发,顾客将如何在各个服务员前进行分配,定性地说明了服务员的均衡占用率不会超过社会最优占用率,并通过数值算例定量地加以佐证。
Customers in queueing systems act independently in order to maximize their own wel-fare. Yet, each customer's optimal behavior is affected by acts taken by both the system managers and the other customers. The result is an aggregate "equilibrium" pattern of behavior which may not be optimal from the point of view of society as a whole. Similar observations have been known to economists and researchers of queueing theory for a long time so that they began to solve this economic problem based on game and queueing theory. Then it gradually developed into a cross-subject of Economics, Game Theory and Queueing Theory named as Economics of Queues, which will be studied in the dissertation.
Based on the previous literatures on the subject of Economics of Queues, equilibrium and socially optimal strategies are analyzed in several types of models, where customers' and server's strategies are both considered, and system performance measures under Nash equilibrium are compared with those under social optimization. Moreover, the main results obtained are verified by appropriate numerical examples visually.
Firstly in the queues with two complementary services, customers have to purchase a product in addition to a service in order to enjoy a benefit. Customers'equilibrium behavior and price strategies of the two servers in the two kinds of Nash equilibria under competition are presented, which are compared with the equilibrium strategies under monopoly. The competition set of prices is higher than that of the monopolist, which is just caused by competition between the two servers. Moreover, the orientations of likes and dislikes of both customers and servers are compared under three types of payment structures.
Secondly, two models with double relative priority parameters are discussed. In the first model, customers'equilibrium behavior as well as server's profit-maximizing strategies based on specific system parameters is analyzed in different types of equilibria. In the other one, the use of relative priority policies for minimizing the queueing system cost is illustrated and the conditions under which relative priorities outperform absolute priorities are presented. Besides the linear and maximum cost functions, much more attention is focused on the quadratic bivariable and exponential cost functions.
Besides the queues with double relative priority parameters, the model with N parame- ters is also considered. First, for a job given its class, the mean remaining queueing times starting from the service commencement of an arbitrary job are obtained. Then the mean queueing times upon arrival are also derived.
Subsequently, queues with three types of setup/closdown policies-interruptible, skip-pable and insusceptible-are considered, respectively. For the observable case, the equilib-rium threshold strategies of customers are derived and the stationary behavior of the sys-tems is analyzed. For the unobservable case, both the equilibrium and socially optimal mixed strategies of customers are found. Furthermore, the appropriate pricing strategy that maximizes both the social welfare and the monopolistic profit is obtained.
Then, two unobservable queues with partial information on service times are analyzed. Given several service time distributions and various partial information, risk-neutral and risk-averse customers'equilibrium and socially optimal strategies are all derived based on maximum entropy principle. Moreover, the profit-maximizing strategies of the server under each type of partial information are also described. The rule is observed that more informa-tion encourages customers to join the queue, which is beneficial for the server's profit.
Besides the setup/closedown vacation policies, queueing model with classical multiple vacations is studied. The equilibrium and socially optimal joining strategies of customers, both in vacation and busy period, have been derived and contrasted. The conclusion is summarized that individual optimization leads to excessive congestion without regulation.
Finally, the allocation problem of customers in a discrete-time multi-server queueing system is discussed and customers'equilibrium and socially optimal strategies are obtained. Comparisons shows that the servers used in equilibrium are no more than those used in the socially optimal outcome, which is also verified by numerical examples.
以前人提供的关于排队经济学的文献成果为基础,主要研究了若干类型的排队经济学模型并对其进行了纳什均衡策略和社会最优策略分析,其策略主体既包含顾客也包括服务员,即研究了排队系统中的均衡行为,并把其与社会最优行为进行了对比进而找出了差异,并通过大量的数值算例描绘出了系统主要性能指标的变化规律,以此形象地验证了所得结论的合理性和正确性。
首先,对具有两项互补服务的排队经济学模型进行了策略分析。顾客除了接受第一项主要的服务项目获得效益之外,还需要购买某类商品或接受另一项辅助服务。分析了在竞争机制下形成的两类纳什均衡,分析了顾客的均衡行为,给出了两个服务员各自的均衡价格,即均衡定价策略。然后考虑了垄断机制下的纳什均衡,即两项服务都由一个垄断者提供,并发现了垄断者的目标与社会最优目标是一致的,而且垄断价格要低于竞争机制下两个服务员的价格总和。此外,在三种不同的收费策略下,比较了顾客和两个服务员对各个收费策略的好恶倾向。
其次,对具有两类顾客及双相对优先权参数的排队经济学模型进行了策略分析。其中第一个模型指出了在不同的系统参数和优先权分配策略下可以产生不同类型的纳什均衡,并分析了顾客的均衡行为,以及服务员根据具体的系统参数为使利润最大化而采取的优先权分配策略。而第二个模型则是以系统费用最小化为初衷,针对几种常见的费用函数：线性、最大值、二次以及指数型费用函数,研究了相对优先权在减少系统费用方面的作用。相对于绝对优先权,找到了相对优先权能更好地降低整个系统费用的条件,并明确地给出了服务员对两类顾客的具体的最优优先权分配策略。
除了对具有两类顾客的双相对优先权参数模型进行了研究之外,还考虑了具有N类顾客的带有N个相对优先权参数的排队经济学模型,分析了纳什均衡状态下顾客的平均排队时间问题。在系统中所有顾客数给定的情况下,给出了从任意一个顾客服务开始算起,一个给定所属类型的顾客的平均剩余排队时间,同时也给出了任意顾客从到达开始算起到服务开始的无条件平均排队时间。根据服务员给出的具体的优先权分配方案,其结果可以指导各类顾客做出是否排队等候的决策。
再次,对具有启动关闭期的排队经济学模型也进行了策略分析。考虑了三种启动关闭休假策略：可中断启动关闭策略,可跳跃启动关闭策略和非中断启动关闭策略,目的是为了研究顾客的均衡和社会最优决策问题。对于可视情形,得到了顾客的均衡阈值策略,分析了系统的稳态行为。对于不可视情形,得到了顾客的均衡和社会最优混合策略,并导出了使社会福利最优化和垄断利润最大化的服务员定价策略。
随后,对具有不完全服务信息的排队经济学模型也进行了策略分析。基于最大熵原理,在顾客被告知多种部分服务信息的情形下,给定几种常见的服务时间分布,分别比较了不可视排队系统中风险中立顾客和风险规避顾客的均衡策略和社会最优策略。此外,还分析了服务员在均衡状态下针对各种部分服务信息所作出的利润最大化策略。理论和数值分析表明顾客获得的部分信息越多,进入概率就越大,而且服务员的垄断利润也越大。
除了启动关闭期休假策略,经典休假思想也被引入到了排队经济学之中,并对一个具有多重休假的排队经济学模型进行了初步的策略分析,目的是找出顾客的均衡策略以及社会最优策略。两者进行比较,得到了个人利益最优化会导致系统过度拥塞的结论。
最后,对一个离散时间多服务员排队经济学模型也进行了策略分析。主要研究的问题也是顾客的均衡和社会最优策略,即从个人利益最优化和社会福利最优化两个标准出发,顾客将如何在各个服务员前进行分配,定性地说明了服务员的均衡占用率不会超过社会最优占用率,并通过数值算例定量地加以佐证。
Customers in queueing systems act independently in order to maximize their own wel-fare. Yet, each customer's optimal behavior is affected by acts taken by both the system managers and the other customers. The result is an aggregate "equilibrium" pattern of behavior which may not be optimal from the point of view of society as a whole. Similar observations have been known to economists and researchers of queueing theory for a long time so that they began to solve this economic problem based on game and queueing theory. Then it gradually developed into a cross-subject of Economics, Game Theory and Queueing Theory named as Economics of Queues, which will be studied in the dissertation.
Based on the previous literatures on the subject of Economics of Queues, equilibrium and socially optimal strategies are analyzed in several types of models, where customers' and server's strategies are both considered, and system performance measures under Nash equilibrium are compared with those under social optimization. Moreover, the main results obtained are verified by appropriate numerical examples visually.
Firstly in the queues with two complementary services, customers have to purchase a product in addition to a service in order to enjoy a benefit. Customers'equilibrium behavior and price strategies of the two servers in the two kinds of Nash equilibria under competition are presented, which are compared with the equilibrium strategies under monopoly. The competition set of prices is higher than that of the monopolist, which is just caused by competition between the two servers. Moreover, the orientations of likes and dislikes of both customers and servers are compared under three types of payment structures.
Secondly, two models with double relative priority parameters are discussed. In the first model, customers'equilibrium behavior as well as server's profit-maximizing strategies based on specific system parameters is analyzed in different types of equilibria. In the other one, the use of relative priority policies for minimizing the queueing system cost is illustrated and the conditions under which relative priorities outperform absolute priorities are presented. Besides the linear and maximum cost functions, much more attention is focused on the quadratic bivariable and exponential cost functions.
Besides the queues with double relative priority parameters, the model with N parame- ters is also considered. First, for a job given its class, the mean remaining queueing times starting from the service commencement of an arbitrary job are obtained. Then the mean queueing times upon arrival are also derived.
Subsequently, queues with three types of setup/closdown policies-interruptible, skip-pable and insusceptible-are considered, respectively. For the observable case, the equilib-rium threshold strategies of customers are derived and the stationary behavior of the sys-tems is analyzed. For the unobservable case, both the equilibrium and socially optimal mixed strategies of customers are found. Furthermore, the appropriate pricing strategy that maximizes both the social welfare and the monopolistic profit is obtained.
Then, two unobservable queues with partial information on service times are analyzed. Given several service time distributions and various partial information, risk-neutral and risk-averse customers'equilibrium and socially optimal strategies are all derived based on maximum entropy principle. Moreover, the profit-maximizing strategies of the server under each type of partial information are also described. The rule is observed that more informa-tion encourages customers to join the queue, which is beneficial for the server's profit.
Besides the setup/closedown vacation policies, queueing model with classical multiple vacations is studied. The equilibrium and socially optimal joining strategies of customers, both in vacation and busy period, have been derived and contrasted. The conclusion is summarized that individual optimization leads to excessive congestion without regulation.
Finally, the allocation problem of customers in a discrete-time multi-server queueing system is discussed and customers'equilibrium and socially optimal strategies are obtained. Comparisons shows that the servers used in equilibrium are no more than those used in the socially optimal outcome, which is also verified by numerical examples.
引文
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56 L. Kleinrock. Queueing Systems, Vol.2:Computer Applications. New York:John Wiley & Sons, 1976
57 K. Lin, S. Ross. Admission control with incomplete information of a queueing system. Operations Research,2003,51(4):645-654
58 U. Sumita, Y. Masuda, S. Yamakawa. Optimal internal pricing and capacity planning for service facility with finite buffer. European Journal of Operational Research,2001,128:192-205
59 C. Bell, S. Stidham. Individual versus social optimization in allocation of customers to alternative servers. Management Science,1983,29:831-839
60 R. Bradford. Incentive compatible pricing and routing policies for multi-server queues. European Journal of Operational Research,1996,89:226-236
61 H. Lee, M. Cohen. Multiagent customer allocation in a stochastic service system. Management Science,1985,31
62 J. Cohen, F. Kelly. A paradox of congestion in a queueing network. Journal of Applied Probability, 1990,27:730-734
63 B. Calvert, W. Solomon, I. Ziedins. Braess's paradox in a queueing network with state dependent routing. Journal of Applied Probability,1997,34:134-154
64 R. Larson. Perspectives on queues:social justice and the psychology of queueing. Operations Research,1987,35:895-909
65 K. Balachandran. Purchasing priorities in queues. Management Science,1972,18:319-326
66 K. Balachandran, B. Srinidhi. A stable cost application scheme for service center usage. Journal of Accounting, Auditing and Finance,1988,15:87-99
67 K. Balachandran, J. Lukens. Stable pricing in service systems. Zeitschrift fur Operations Research, 1976,20:189-201
68 I. Adiri, U. Yechiali. Optimal priority purchasing and pricing decisions in nonmonpoly and monopoly queues. Operations Research,1974,22:1051-1066
69 R. Hassin, M. Haviv. Equilibrium threshold strategies:the case of queues with priorities. Operations Research,1997,45:966-973
70 H. Alperstein. Optimal pricing for the service facility offering a set of priority prices. Management Science,1988,34:666-671
71 M. Agastya. personal communication,2001
72 A. Krueger. The political economy of the rent-seeking society. American Economic Review,1974, 64:291-303
73 G. Tullock. The welfare cost of traffics, monopolies, and theft. Western Economic Journal,1967, 5:224-232
74 R. Hassin, J. Puerto, F. Fernandez. The use of relative priorities in optimizing the performance of a queueing system. European Journal of Operational Research,2009,193:476-483
75 Y. Hayel, B. Tuffin. Pricing for Heterogeneous Services at a Discriminatory Processor Sharing queue. IRISA,2004
76 K. Rege, B. Sengupta. Queue Length Distribution for the Discriminatory Processor Sharing Queue. Operations Research,1996,44(4):653-657
77 D. Mitra, A. Weiss. A Closed Network with a Discriminatory Processor Sharing Server. Proceed-ings of ACM Sigmetrics and Performance,1989
78 G. Fayolle, I. Mitrani, R. Iasnogorodski. Sharing a processor among many job classes. Journal of the Association for Computing Machinery,1980,27:519-532
79 M. Haviv, J. Van der Wal. Waiting time in queues with relative priorities. Operations Research Letters,2007,35(5):591-594
80 R. Hassin, H. Haviv. Equilibrium of Customer Behavior in Queueing Systems:To Queue or not to Queue. Kluwer,2003
81 M. Haviv, J. Van der Wal. Equilibrium strategies for processor sharing and queues with relative priorities. Probability in the Engineering and Informational Sciences,1997,11:403-412
82 M. Marchand. Priority pricing. Management Science,1974,26:1131-1140
83 S. Ghanem. Computing central optimization by a pricing priority policy. IBM Systems Journal, 1975,14:272-292
84 R. Dolan. Incentive mechanisms for priority queueing problems. Bell Journal of Economics,1978, 9:421-436
85 T. Groves, M. Loeb. Incentives and public inputs. Journal of Public Economics,1975,4:211-226
86 A. Ha. Incentive-compatible pricing for a service facility with joint production and congestion externalities. Management Science,1998,44(12):1623-1636
87 A. Ha. Optimal incentive-compatible pricing for processor sharing queues with customer-chosen service requirements. Technical report, Yale School of Management, NewHaven, CT,1999
88 L. Kleinrock. Optimal Bribing for Queue Positionsn. Operations Research,1967,15:304-318
89 R. Hassin. Decentralized regulation of a queue. Management Science,1995,41:163-173
90 F. Lui. An equilibrium queueing model of bribery. Journal of Political Economy,1985,93:760-781
91 A. Glazer, R. Hassin. Stable priority purchasing in queues. Operations Research Letters,1986, 4:285-288
92 P. Jehiel, B. Moldovanu. Efficient Design with Interdependent Valuations. Econometrica,2001, 69:1237-1259
93 P. Dasgupta, E. Maskin. Efficient Auctions. Quarterly Journal of Economics,2000,115:341-388
94 T. Kittsteiner, B. Moldovanu. Auction-based Queue Disciplines. Working paper, University of Bonn,2003
95 T. Kittsteiner, B. Moldovanu. Priority Auctions and Queue Disciplines that Depend on Processing Time. Management Science,2005,51(2):236-248
96 G. Myrdal. Asian Drama:An Inquiry into the Poverty of Nations. New York:Pantheon,1968
97 L. Li, Y. Lee. Pricing and delivery-time performance in a competitive environment. Management Science,1994,40:633-646
98 C. Davidson. Equilibrium in servicing industries:an economic application of queueing theory. Journal of Business,1988,61:347-367
99 I. Luski. On partial equilibrium in a queueing system with two servers. The Review of Economic Studies,1976,43:519-525
100 D. Levhari, I. Luski. Duopoly pricing and waiting lines. European Economic Review,1978,11:17-35
101 G. Cachon, P. Harker. Service competition, outsourcing and co-production in a queueing game. Working paper, University of Pennsylvania, Philadelphia, PA.,1999
102 D. Reitman. Endogenous quality differentiation in congested markets. The Journal of Industrial Economics,1991,39
103 M. Armony, M. Haviv. Price and delay competition between two service providers. European Journal of Operational Research,2003,147(1):32-50
104 H. Chen, Y Wan. Price competition of make-to-order firms. HE Transactions,2003,35(9):817-832
105 C. Loch. Pricing in markets sensitive to delay. Ph.D. Dissertation, Stanford University,1991
106 N. Edelson. Congestion tolls under monopoly. American Economic Review,1971,61:873-882
107 H. Chen, M. Frank. Monopoly Pricing When Customers Queue. IEE Transactions,2004,36:1-13
108 A. Veltman, R. Hassin. Equilibrium in queueing systems with complementary products. Queueing Systems,2005,50:325-342
109 M. Chaudhry, J. Templeton. A first course in bulk queues. John Wiley & Sons,1983
110 R. Hassin, M. Haviv. Who should be given priority in a queue? Operations Research Letters,2006, 34
111 B. Kolman, R. Beck. Elementary Linear Programming with Applications (Second Edition). Elsevier Inc.,1995
112 D. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont
113 C.E. Shannon. A Mathematical Theory of Communication. The Bell System Technical Journal, 1948,27:379-423,623-656
114 E.T. Jaynes. Information theory and statistical mechanics. Physical Review,1957,106:620-630
115 J.N. Kapur, H.K. Kesavan. Entropy optimization principles with applications. Academic Press, Boston,1992
116 D. Koutsoyiannis. The scaling properties in the distribution of hydrological variables as a result of the maximum entropy principle (solicited). European Geosciences Union, Vienna,2005
117 H. Takagi. Queueing Analysis, Vol.1 Vacation and priority systems.1991,1-203
118 N. Tian, G. Zhang. Vacation Queueing Models-Theory and Applications. New York:Springer-Verlag,2006
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43 H. Chen, M. Frank. Monopoly pricing when customers queue. Working Paper, Faculty of Com-merce and Business Administration, University of British Columbia,1994
44 K. Balachandran. Incentive and regulation in queues. Lecture Notes in Economics and Mathemat-ical Systems,1991,370:162-176
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53 Y. Masuda, S. Whang. Dynamic pricing for network service:equilibrium and stability. Management Science,1999,45:857-869
54 H. Haviv. Stable strategies for processor sharing systems. European Journal of Operational Re-search,1991,52:103-106
55 S. Ross. Stochastic Processes. New York:John Wiley & Sons,1983
56 L. Kleinrock. Queueing Systems, Vol.2:Computer Applications. New York:John Wiley & Sons, 1976
57 K. Lin, S. Ross. Admission control with incomplete information of a queueing system. Operations Research,2003,51(4):645-654
58 U. Sumita, Y. Masuda, S. Yamakawa. Optimal internal pricing and capacity planning for service facility with finite buffer. European Journal of Operational Research,2001,128:192-205
59 C. Bell, S. Stidham. Individual versus social optimization in allocation of customers to alternative servers. Management Science,1983,29:831-839
60 R. Bradford. Incentive compatible pricing and routing policies for multi-server queues. European Journal of Operational Research,1996,89:226-236
61 H. Lee, M. Cohen. Multiagent customer allocation in a stochastic service system. Management Science,1985,31
62 J. Cohen, F. Kelly. A paradox of congestion in a queueing network. Journal of Applied Probability, 1990,27:730-734
63 B. Calvert, W. Solomon, I. Ziedins. Braess's paradox in a queueing network with state dependent routing. Journal of Applied Probability,1997,34:134-154
64 R. Larson. Perspectives on queues:social justice and the psychology of queueing. Operations Research,1987,35:895-909
65 K. Balachandran. Purchasing priorities in queues. Management Science,1972,18:319-326
66 K. Balachandran, B. Srinidhi. A stable cost application scheme for service center usage. Journal of Accounting, Auditing and Finance,1988,15:87-99
67 K. Balachandran, J. Lukens. Stable pricing in service systems. Zeitschrift fur Operations Research, 1976,20:189-201
68 I. Adiri, U. Yechiali. Optimal priority purchasing and pricing decisions in nonmonpoly and monopoly queues. Operations Research,1974,22:1051-1066
69 R. Hassin, M. Haviv. Equilibrium threshold strategies:the case of queues with priorities. Operations Research,1997,45:966-973
70 H. Alperstein. Optimal pricing for the service facility offering a set of priority prices. Management Science,1988,34:666-671
71 M. Agastya. personal communication,2001
72 A. Krueger. The political economy of the rent-seeking society. American Economic Review,1974, 64:291-303
73 G. Tullock. The welfare cost of traffics, monopolies, and theft. Western Economic Journal,1967, 5:224-232
74 R. Hassin, J. Puerto, F. Fernandez. The use of relative priorities in optimizing the performance of a queueing system. European Journal of Operational Research,2009,193:476-483
75 Y. Hayel, B. Tuffin. Pricing for Heterogeneous Services at a Discriminatory Processor Sharing queue. IRISA,2004
76 K. Rege, B. Sengupta. Queue Length Distribution for the Discriminatory Processor Sharing Queue. Operations Research,1996,44(4):653-657
77 D. Mitra, A. Weiss. A Closed Network with a Discriminatory Processor Sharing Server. Proceed-ings of ACM Sigmetrics and Performance,1989
78 G. Fayolle, I. Mitrani, R. Iasnogorodski. Sharing a processor among many job classes. Journal of the Association for Computing Machinery,1980,27:519-532
79 M. Haviv, J. Van der Wal. Waiting time in queues with relative priorities. Operations Research Letters,2007,35(5):591-594
80 R. Hassin, H. Haviv. Equilibrium of Customer Behavior in Queueing Systems:To Queue or not to Queue. Kluwer,2003
81 M. Haviv, J. Van der Wal. Equilibrium strategies for processor sharing and queues with relative priorities. Probability in the Engineering and Informational Sciences,1997,11:403-412
82 M. Marchand. Priority pricing. Management Science,1974,26:1131-1140
83 S. Ghanem. Computing central optimization by a pricing priority policy. IBM Systems Journal, 1975,14:272-292
84 R. Dolan. Incentive mechanisms for priority queueing problems. Bell Journal of Economics,1978, 9:421-436
85 T. Groves, M. Loeb. Incentives and public inputs. Journal of Public Economics,1975,4:211-226
86 A. Ha. Incentive-compatible pricing for a service facility with joint production and congestion externalities. Management Science,1998,44(12):1623-1636
87 A. Ha. Optimal incentive-compatible pricing for processor sharing queues with customer-chosen service requirements. Technical report, Yale School of Management, NewHaven, CT,1999
88 L. Kleinrock. Optimal Bribing for Queue Positionsn. Operations Research,1967,15:304-318
89 R. Hassin. Decentralized regulation of a queue. Management Science,1995,41:163-173
90 F. Lui. An equilibrium queueing model of bribery. Journal of Political Economy,1985,93:760-781
91 A. Glazer, R. Hassin. Stable priority purchasing in queues. Operations Research Letters,1986, 4:285-288
92 P. Jehiel, B. Moldovanu. Efficient Design with Interdependent Valuations. Econometrica,2001, 69:1237-1259
93 P. Dasgupta, E. Maskin. Efficient Auctions. Quarterly Journal of Economics,2000,115:341-388
94 T. Kittsteiner, B. Moldovanu. Auction-based Queue Disciplines. Working paper, University of Bonn,2003
95 T. Kittsteiner, B. Moldovanu. Priority Auctions and Queue Disciplines that Depend on Processing Time. Management Science,2005,51(2):236-248
96 G. Myrdal. Asian Drama:An Inquiry into the Poverty of Nations. New York:Pantheon,1968
97 L. Li, Y. Lee. Pricing and delivery-time performance in a competitive environment. Management Science,1994,40:633-646
98 C. Davidson. Equilibrium in servicing industries:an economic application of queueing theory. Journal of Business,1988,61:347-367
99 I. Luski. On partial equilibrium in a queueing system with two servers. The Review of Economic Studies,1976,43:519-525
100 D. Levhari, I. Luski. Duopoly pricing and waiting lines. European Economic Review,1978,11:17-35
101 G. Cachon, P. Harker. Service competition, outsourcing and co-production in a queueing game. Working paper, University of Pennsylvania, Philadelphia, PA.,1999
102 D. Reitman. Endogenous quality differentiation in congested markets. The Journal of Industrial Economics,1991,39
103 M. Armony, M. Haviv. Price and delay competition between two service providers. European Journal of Operational Research,2003,147(1):32-50
104 H. Chen, Y Wan. Price competition of make-to-order firms. HE Transactions,2003,35(9):817-832
105 C. Loch. Pricing in markets sensitive to delay. Ph.D. Dissertation, Stanford University,1991
106 N. Edelson. Congestion tolls under monopoly. American Economic Review,1971,61:873-882
107 H. Chen, M. Frank. Monopoly Pricing When Customers Queue. IEE Transactions,2004,36:1-13
108 A. Veltman, R. Hassin. Equilibrium in queueing systems with complementary products. Queueing Systems,2005,50:325-342
109 M. Chaudhry, J. Templeton. A first course in bulk queues. John Wiley & Sons,1983
110 R. Hassin, M. Haviv. Who should be given priority in a queue? Operations Research Letters,2006, 34
111 B. Kolman, R. Beck. Elementary Linear Programming with Applications (Second Edition). Elsevier Inc.,1995
112 D. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont
113 C.E. Shannon. A Mathematical Theory of Communication. The Bell System Technical Journal, 1948,27:379-423,623-656
114 E.T. Jaynes. Information theory and statistical mechanics. Physical Review,1957,106:620-630
115 J.N. Kapur, H.K. Kesavan. Entropy optimization principles with applications. Academic Press, Boston,1992
116 D. Koutsoyiannis. The scaling properties in the distribution of hydrological variables as a result of the maximum entropy principle (solicited). European Geosciences Union, Vienna,2005
117 H. Takagi. Queueing Analysis, Vol.1 Vacation and priority systems.1991,1-203
118 N. Tian, G. Zhang. Vacation Queueing Models-Theory and Applications. New York:Springer-Verlag,2006