多服务台排队问题研究及其利润最大化讨论
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摘要
多服务台排队系统作为排队理论研究的一个重要方面,它是以服务台个数多于一个的排队系统作为研究对象,在管理和决策上发挥着重要的作用。排队论作为一个理论,它是概率论和随机过程理论的结合,是运筹学的一个重要分支。本文在已有多服务台排队理论的基础上,主要对以下的几个方面对多服务台排队系统进行研究：
首先,建立了一个更为广泛的多服务台排队模型,在模型中不仅具有止步和中途退出的一般特征,还加入了重试的现象,所谓的重试是指已经接受完服务的顾客再一次进入系统寻求服务的过程,使得模型更加的贴近于实际生活。然后利用矩阵分析的方法得到了模型稳态时的各个目标参量,从而解决了可重试排队且带止步和中途退出的多服务台同步多重休假排队问题。
其次,根据一般排队的特点,建立了服务台个数分别为奇数和偶数的两类服务台之间能够相互帮助的,也就是具有辅助性服务的排队系统。然后利用矩阵拆分的方法得到了系统在稳态时各个状态的稳态概率,进而得到具有辅助性服务的多服务台排队系统的各个目标参量。
再次,利用随机过程理论,建立多服务台排队系统中的系统收入公式。再利用统计的方法得到码头服务系统收入的期望,从而计算出系统的总收入,为下一步求系统的利润最大化奠定基础。
最后运用多服务台排队系统的理论和方法,分析和研究了集装箱码头的运行机制,建立了码头服务系统的数学模型。根据内外贸集装箱吞吐量的实际数据和数学软件编制的程序找到最优的服务台数量,以及适宜的系统容量空间来使得系统总利润达到最大化。
As an important aspect in the research of line, Multi-service-line-system studies the line-system which has more than one sercive desk, and plays an important role in the management and decision-making. The queuing theory, as a combining of Probability Theory and Random process, is an important branch of Operations. Basing on the existing Multi-service queuing theory, this paper researches the multi-service-line-system in several aspects as follows.
Firstly, we established a more extensive Multi-service queuing model, which not only has the general characteristics of stop and dropout, but also added the retry(a process that customers who have already accepted the service again get into the system for service), making this model more closed to the actual life. Then, using matrix analysis methods, we got the steady-state model parameters of each goal, thus solve the queuing problem with retry queue, balking and reneging multi-serviced multiple synchrorous vacations.
Secondly, according to the characteristics of ordinary queuing, we built two kinds of line-system in which the numbers of service desks are respectively odd and even, and could help each other. That is to say, this service-system has auxiliary services. Then, using matrix analysis methods, we got probability of each state in the steady-state, thus we got each goal reference variables of multi-service-line-system with assistance service.
Again, utilizing the stochastic process theory, we established an income-formula of the multi-service-line-system. Then, utilizing the statistical method, we got the income expectation of terminal-service-system and total income of the system, which built the foundation for the profit maximization of the system in the next step.
Finally, using the multi-service-line-system theory and method, we analyzed and studied the operation mechanism of a container terminal, and established a mathematical model of the terminal service system. What's more, using the real datas of throughputs of domestic and forergn trade container and the program developed by mathematical sofrware, we found the optimal service quantity and surtable capacity-space which maximizes the total profits of this system.
首先,建立了一个更为广泛的多服务台排队模型,在模型中不仅具有止步和中途退出的一般特征,还加入了重试的现象,所谓的重试是指已经接受完服务的顾客再一次进入系统寻求服务的过程,使得模型更加的贴近于实际生活。然后利用矩阵分析的方法得到了模型稳态时的各个目标参量,从而解决了可重试排队且带止步和中途退出的多服务台同步多重休假排队问题。
其次,根据一般排队的特点,建立了服务台个数分别为奇数和偶数的两类服务台之间能够相互帮助的,也就是具有辅助性服务的排队系统。然后利用矩阵拆分的方法得到了系统在稳态时各个状态的稳态概率,进而得到具有辅助性服务的多服务台排队系统的各个目标参量。
再次,利用随机过程理论,建立多服务台排队系统中的系统收入公式。再利用统计的方法得到码头服务系统收入的期望,从而计算出系统的总收入,为下一步求系统的利润最大化奠定基础。
最后运用多服务台排队系统的理论和方法,分析和研究了集装箱码头的运行机制,建立了码头服务系统的数学模型。根据内外贸集装箱吞吐量的实际数据和数学软件编制的程序找到最优的服务台数量,以及适宜的系统容量空间来使得系统总利润达到最大化。
As an important aspect in the research of line, Multi-service-line-system studies the line-system which has more than one sercive desk, and plays an important role in the management and decision-making. The queuing theory, as a combining of Probability Theory and Random process, is an important branch of Operations. Basing on the existing Multi-service queuing theory, this paper researches the multi-service-line-system in several aspects as follows.
Firstly, we established a more extensive Multi-service queuing model, which not only has the general characteristics of stop and dropout, but also added the retry(a process that customers who have already accepted the service again get into the system for service), making this model more closed to the actual life. Then, using matrix analysis methods, we got the steady-state model parameters of each goal, thus solve the queuing problem with retry queue, balking and reneging multi-serviced multiple synchrorous vacations.
Secondly, according to the characteristics of ordinary queuing, we built two kinds of line-system in which the numbers of service desks are respectively odd and even, and could help each other. That is to say, this service-system has auxiliary services. Then, using matrix analysis methods, we got probability of each state in the steady-state, thus we got each goal reference variables of multi-service-line-system with assistance service.
Again, utilizing the stochastic process theory, we established an income-formula of the multi-service-line-system. Then, utilizing the statistical method, we got the income expectation of terminal-service-system and total income of the system, which built the foundation for the profit maximization of the system in the next step.
Finally, using the multi-service-line-system theory and method, we analyzed and studied the operation mechanism of a container terminal, and established a mathematical model of the terminal service system. What's more, using the real datas of throughputs of domestic and forergn trade container and the program developed by mathematical sofrware, we found the optimal service quantity and surtable capacity-space which maximizes the total profits of this system.
引文
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[2]赵文.排队论(随即服务系统理论)概述.天津商学院学报,1995.
[3]冼世靖,毛利.使用随机过程解决排队论问题.应用科学,2008.
[4]Ji-Hong Li,Nai-Shuo Tian,Zhan-You Ma. Performance analysis of GI/M/1 queue with working vacations and vacation interruption. Applied Mathematical Modelling 32 (2008) 2715-2730.
[5]Kyung C. Chae, Dae E. Lim,Won S. Yang. The GI/M/1 queue and the GI/Geo/1 queue both with singleworking vacation. Performance Evaluation 66 (2009) 356-367.
[6]Raik Stolletz. Approximation of the non-stationary M(t)/ M(t)/ c(t)-queue using stationary queueing models:The stationary backlog-carryover approach, European Journal of Operational Research 190 (2008) 478-493.
[7]R.O.Al-Seedy,A.A.El-Sherbiny,S.A.El-Shehawy,S.I.Ammar,Transient solution of the M/M/ c queue with balking and reneging.Computers and Mathematics with Applications 57 (2009) 1280_1285.
[8]Y.Levy,U.Yechiali,An M/M/c queue with servers Vacations,INFOR14(1976)153-163.
[9]N.Tian,Q.Li,J.Cao,Conditional stochastic decomposition in M/M/c queue with server vacations,StochasticMod.15(2)(1999)367-377.
[10]夏茂辉,田乃硕,同步N-策略多重休假M／M／c排队.运筹学学报,1997.
[11]侯玉梅,田乃硕,M／M／c休假排队系统—综述.运筹学学报,2000.
[12]成国庆,李玲,唐应辉,部分服务台同步多重休假的M／M／c／k排队及其优化.应用数学与计算数学学报,2007.
[13]陈杰,朱翼隽,同步N-策略多重休假的M／M／c／WV排队.成都信息工程学院学报,2008.
[14]王 楠,王金亭,高晋芳,异步服务的M／M／2重试排队算法,2007.
[15]田乃硕,高作峰,张忠君.异步休假的M／M／c排队的稳态理论.应用数学学报,2001年.
[16]Zhe George Zhang,Naishuo Tian,Analysis on queueing systems with synchronous vacation so partial servers, Performance Evaluation 52 (2003) 269-282.
[17]刘洺辛,马占友等,部分服务台异步N-策略多重休假M／M／c排队,2006.
[18]田乃硕,徐秀丽,马振友,韦才敏,部分服务台同步多重休假M／M／c排队.运筹学学报,2001.
[19]刘洺辛,马占友,徐秀丽,田乃硕,部分服务台异步N-策略多重休假M／M／c排队.燕山大学学报,2006.
[20]田乃硕,徐秀丽,拟生灭过程与部分服务台休假的M／M／c排队.燕山大学学报,2000.
[21]Yang Woo Shin,Taek Sik Choo, M/M/s queue with impatiet customers and retrials, Applied Mathematical Modelling 33 (2009) 2596-2606.
[22]Patrick Wuchner,Janos Sztri,Hermann de Mee,Finite-source M /M/s retrial queue with search for balking and impatient customers from theorbit,2009.
[23]Yang Woo Shin, Convergence of the stationary distributions of M/M/ s/K retrial queue as K tends to infinity, European Journal of Operational Research 189 (2008) 1104-1117.
[24]Abou ElA ta M O Hariri A M A,The M/M/c/N queue with balking and reneging. Computers and Operations Research,1992,19 (8):713—716.
[25]朱翼隽,朱仁祥,基于重试、不耐烦M／M／s／K+M排队的呼叫中心性能分析.江苏大学学报,2004.
[26]岳德权,孙妍平,带有止步和中途退出的M／M／c／N部分服务员同步多重休假排队系统的等待时间.系统工程理论与实践,2008.
[27]孙妍平,岳德权,带有止步和中途退出的M／M／R／N部分服务员同步单重休假排队系统的性能分析.海南大学学报自然科学版,2008.
[28]孙妍平,岳德权,带有止步和中途退出的M／M／S／N同步多重休假排队系统的性能分析.系统工程理论与实践,2007.
[29]鲍媛媛,朱翼隽,基于负顾客M／M／s／k+M重试、反馈排队的呼叫中心性能分析.成都信息工程学院学报,2008.
[30]朱翼隽,朱仁祥,方基奎,基于重试、反馈M／M／S／k排队的呼叫中心性能分析.系统工程学报,2006.
[31]Patrick Wuchner, Janos Sztrik, Hermann de Meer, Finite-source M/M/ c retrial queue with search for balking and impatient customers from the orbit.2009.
[32]厉莉,朱翼隽,待反馈优先的M／M／c排队在通信网络中的应用.成都信息工程学院学报,2006年.
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