窗口能力不等的多服务窗M/M/3排队模型及应用研究
|
摘要
陆传赉在文献[1]中提出了窗口能力不等的多服务窗M/M/n排队模型.在该模型中，假定顾客陆续到达，且他们之间的间隔时间服从参数为λ的指数分布.各服务窗对顾客服务的时间分别服从参数是μi的指数分布，其中下标i表示第i个服务窗.顾客到达系统的时间间隔与顾客接受服务的时间是相互独立的.在文献[1]中，作者只对模型中服务窗个数n=2的情形进行了研究，并运用系统状态转移图得到了K氏方程组.通过把正则性与K氏方程组相结合，得到了系统队长的平稳分布及各项指标.
本文最重要的创新之处是推广了文献[1]的模型，将窗口的个数n=2推广到n=3，讨论了窗口能力不等的多服务窗M/M/3排队模型.本文运用系统状态转移图得到了K氏方程组，再通过把正则性与K氏方程组相结合，得到了系统队长的平稳分布及各项指标.同时通过在模型中令μ1=μ2=μ3=μ
，φ1=φ2=φ3=1/3，获得了与多服务窗等待制排队模型M/M/3完全相同的结果.这一方面说明本文结果也是多服务窗等待制M/M/n模型的推广，另一方面也在一定程度上验证了本文结果的正确性.最后通过举例说明了该模型在生活实际中的应用并根据模型结果得到了实例的各项指标.
Lu Chuanlai once put forward the model of multi-service window-M/M/3queuing with different range of capacity in literature[1]. This model assumed that theinterval between the customers’ arrival obeyed parameters forλof exponentialdistribution; the service time of each window for customers obeyed parameters forμiof exponential distribution and the “i” stands for the number of the servicewindow. The interval between customers’ arrival at the system and the service timeare independent. In the literature[1], the author only had a research on the case thatthe number of the service window was two, that is “n=2”, and got the K’s equationsthrough the system state transition diagram. It got the stationary distribution and theindexes of the queuing length through the combination of K’s equations andregularity.
In this paper, the most important innovation is to promote the model of theliterature[1]: the number of the window from2to3, that is “n=2” to “n=3”;discussing the model of the multi-service window-M/M/3queuing. In the paper, ituses the system state transition diagram to get the K’s equations and it gets thestationary distribution and the indexes of the queuing length through the combinationof the K’s equations and regularity. At the same time, it gets the same result as themodel of multi-service window-M/M/3waiting-queuing through settingμ1=μ2=μ3=μ, φ1=φ2=φ3=1/3in the model. On one hand, it indicatesthat it is the promotion of the model of multi-service window-M/M/nwaiting-queuing in the paper; on the other hand, it also checks, to some extent, theveracity of the result in the paper. Finally, it shows the application of the model inreal life through the examples and it gets the indexes of the examples through theresult of the model.
本文最重要的创新之处是推广了文献[1]的模型，将窗口的个数n=2推广到n=3，讨论了窗口能力不等的多服务窗M/M/3排队模型.本文运用系统状态转移图得到了K氏方程组，再通过把正则性与K氏方程组相结合，得到了系统队长的平稳分布及各项指标.同时通过在模型中令μ1=μ2=μ3=μ
，φ1=φ2=φ3=1/3，获得了与多服务窗等待制排队模型M/M/3完全相同的结果.这一方面说明本文结果也是多服务窗等待制M/M/n模型的推广，另一方面也在一定程度上验证了本文结果的正确性.最后通过举例说明了该模型在生活实际中的应用并根据模型结果得到了实例的各项指标.
Lu Chuanlai once put forward the model of multi-service window-M/M/3queuing with different range of capacity in literature[1]. This model assumed that theinterval between the customers’ arrival obeyed parameters forλof exponentialdistribution; the service time of each window for customers obeyed parameters forμiof exponential distribution and the “i” stands for the number of the servicewindow. The interval between customers’ arrival at the system and the service timeare independent. In the literature[1], the author only had a research on the case thatthe number of the service window was two, that is “n=2”, and got the K’s equationsthrough the system state transition diagram. It got the stationary distribution and theindexes of the queuing length through the combination of K’s equations andregularity.
In this paper, the most important innovation is to promote the model of theliterature[1]: the number of the window from2to3, that is “n=2” to “n=3”;discussing the model of the multi-service window-M/M/3queuing. In the paper, ituses the system state transition diagram to get the K’s equations and it gets thestationary distribution and the indexes of the queuing length through the combinationof the K’s equations and regularity. At the same time, it gets the same result as themodel of multi-service window-M/M/3waiting-queuing through settingμ1=μ2=μ3=μ, φ1=φ2=φ3=1/3in the model. On one hand, it indicatesthat it is the promotion of the model of multi-service window-M/M/nwaiting-queuing in the paper; on the other hand, it also checks, to some extent, theveracity of the result in the paper. Finally, it shows the application of the model inreal life through the examples and it gets the indexes of the examples through theresult of the model.
引文
[1]陆传赉.排队论[M].北京邮电学院出版社,2009.2,28-200.
[2]王梓坤,杨向群.生灭过程与马尔可夫链[M].北京:科学出版社,2005,174-224.
[3]徐光辉.随机服务系统[M].北京:科学出版社,1980,23-150.
[4]侯冬倩，高世泽.服务率可变且窗口能力不等的M/M/n排队模型研究[J].重庆师范大学学报(自然科学版)，2010,27(2):46-48.
[5]C.Palm.Methods of judging the annoyance Caused by congection．Tele No.2(1953)：1-20
[6]S．M．Brodi．On all integro-differential equation for systems withτ waiting(in Ukrainian),Dop．AN URSR6(1959)：571-573.
[7]S．M．Brodi．On a systems problem(in Russian)，Tr．N．Vsesojuznogo sovescanija po teoriivemhamostej Imatematiceskoj statistike．Izd-voANArm．SSR．Erevan(1960)：143—147.
[8]B.W.Gnedenko.I～N Kowalenko,Einfuhrung in die Bedienungstheorie(1sted．in Russian,Nauka,Moscow,1966)．Akademie—Verlag，Berlin，1974,56-58.
[9]E．Robert Stanford．Reneging phenomenon in single channel queues．Mathematics of OperationsResearch,1979，4(2)：162—178.
[10]Moshe Haviv,Ya'acov Ritov．Homogeneous Customers Renege from Invisible Queues atRandom Times under Deteriorating Waiting Conditions．Queuing Systems，2001，56：169-180.
[11]B．Krishna Kumar,J．Raja．On multiserver feedback retrial queues with balking and controlretrial rate．Ann．Oper．Res，2006，141：2ll-232.
[12]Burke PJ. Delays in Single-Server Queues With Batch Input.Opans.Res.23(1975):830-833.
[13]Burke PJ.The Output of a Queueing System.Opns.Res.4(1956):699-704.
[14]Cohen J W.The Single Server Queue,North-Holland,New York,1982.
[15]Feller W.An Introduction to Probability Theory and its Applications,2nd Edition.New York:JohnWiley&Sons,1957.
[16]Cox D R,Smith W L.Queues.Methuen&co.London,1961.
[17]Kleinrock L.Queueing Systems,Volume I:Theory.John Wiley,New York,1975.
[18]Kleinrock L.Queueing Systems,Volume II:Computer Applications.John Wiley,New York,1976.
[19]Cohen J W. The Single Server Queue, Revised edition. North-Holland Publishing Campany,Amsterdam,1982.
[20]孙荣恒.应用概率论.科学出版社[M],1998.
[21]孟玉珂.排队论基础及应用[M].上海:同济大学出版社,1989.
[22]盛骤,谢式千，潘承毅.概率论与数理统计[M].高等教育出版社,2001(12):86-123.
[23]孙荣恒,雷玉浩.关于M/G/1系统等待时间的一个注记[J].重庆大学学报,1999,22(1).
[24]盛友招.排队论及其在计算机通信中的应用[M].北京邮电大学出版社，1998(9):45-72.
[25]孙荣恒.随机过程及其应用[M].北京：清华大学出版社,2004,97-107.
[26]唐应辉，唐小我.排队论基础与分析技术[M].北京：科学出版社,2006,12-20.
[27]孙荣恒,李建平.排队论基础[M].北京:科学出版社,2002,122-125.
[28]左宏明,于显威.等待制排队模型在教务员岗位数量确定中的应用[J].沈阳农业大学学报,2005-04,36(2):214-217.,
[29]高世泽.概率统计引论[M].重庆大学出版社，2000（7）51-58.
[30]周炯槃.通信网理论基础[M].北京:人民邮电出版社,1991.
[31]王承恕.通信网基础[M].北京:人民邮电出版社,1999.
[2]王梓坤,杨向群.生灭过程与马尔可夫链[M].北京:科学出版社,2005,174-224.
[3]徐光辉.随机服务系统[M].北京:科学出版社,1980,23-150.
[4]侯冬倩，高世泽.服务率可变且窗口能力不等的M/M/n排队模型研究[J].重庆师范大学学报(自然科学版)，2010,27(2):46-48.
[5]C.Palm.Methods of judging the annoyance Caused by congection．Tele No.2(1953)：1-20
[6]S．M．Brodi．On all integro-differential equation for systems withτ waiting(in Ukrainian),Dop．AN URSR6(1959)：571-573.
[7]S．M．Brodi．On a systems problem(in Russian)，Tr．N．Vsesojuznogo sovescanija po teoriivemhamostej Imatematiceskoj statistike．Izd-voANArm．SSR．Erevan(1960)：143—147.
[8]B.W.Gnedenko.I～N Kowalenko,Einfuhrung in die Bedienungstheorie(1sted．in Russian,Nauka,Moscow,1966)．Akademie—Verlag，Berlin，1974,56-58.
[9]E．Robert Stanford．Reneging phenomenon in single channel queues．Mathematics of OperationsResearch,1979，4(2)：162—178.
[10]Moshe Haviv,Ya'acov Ritov．Homogeneous Customers Renege from Invisible Queues atRandom Times under Deteriorating Waiting Conditions．Queuing Systems，2001，56：169-180.
[11]B．Krishna Kumar,J．Raja．On multiserver feedback retrial queues with balking and controlretrial rate．Ann．Oper．Res，2006，141：2ll-232.
[12]Burke PJ. Delays in Single-Server Queues With Batch Input.Opans.Res.23(1975):830-833.
[13]Burke PJ.The Output of a Queueing System.Opns.Res.4(1956):699-704.
[14]Cohen J W.The Single Server Queue,North-Holland,New York,1982.
[15]Feller W.An Introduction to Probability Theory and its Applications,2nd Edition.New York:JohnWiley&Sons,1957.
[16]Cox D R,Smith W L.Queues.Methuen&co.London,1961.
[17]Kleinrock L.Queueing Systems,Volume I:Theory.John Wiley,New York,1975.
[18]Kleinrock L.Queueing Systems,Volume II:Computer Applications.John Wiley,New York,1976.
[19]Cohen J W. The Single Server Queue, Revised edition. North-Holland Publishing Campany,Amsterdam,1982.
[20]孙荣恒.应用概率论.科学出版社[M],1998.
[21]孟玉珂.排队论基础及应用[M].上海:同济大学出版社,1989.
[22]盛骤,谢式千，潘承毅.概率论与数理统计[M].高等教育出版社,2001(12):86-123.
[23]孙荣恒,雷玉浩.关于M/G/1系统等待时间的一个注记[J].重庆大学学报,1999,22(1).
[24]盛友招.排队论及其在计算机通信中的应用[M].北京邮电大学出版社，1998(9):45-72.
[25]孙荣恒.随机过程及其应用[M].北京：清华大学出版社,2004,97-107.
[26]唐应辉，唐小我.排队论基础与分析技术[M].北京：科学出版社,2006,12-20.
[27]孙荣恒,李建平.排队论基础[M].北京:科学出版社,2002,122-125.
[28]左宏明,于显威.等待制排队模型在教务员岗位数量确定中的应用[J].沈阳农业大学学报,2005-04,36(2):214-217.,
[29]高世泽.概率统计引论[M].重庆大学出版社，2000（7）51-58.
[30]周炯槃.通信网理论基础[M].北京:人民邮电出版社,1991.
[31]王承恕.通信网基础[M].北京:人民邮电出版社,1999.