第二种服务可选的M/G/1排队模型的适定性
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摘要
本文分两章.第一章分两节.第一节回顾排队论的历史,第二节中先介绍补充变量方法,然后提出本文所要研究的问题.第二章共分两节.第一节中首先介绍第二种服务可选的M/G/1排队的数学模型,接着引入状态空间、主算子及其定义域,然后将该模型转化成Banach空间中的抽象Cauchy问题.第二节中研究该排队模型的适定性.运用泛函分析中的Hille-Yosida定理,Phillips定理和Fattorini定理证明该模型存在唯一的概率瞬态解.
This paper is divided into two chapters.Chapter 1 is divided into two sections.In Section 1,we introduce briefly the history of queueing theory.In Section 2,we first introduce supplementary variable technique,then we put forward the problem that we study in this thesis.Chapter 2 is split into two sections.In Section 1,first we introduce the M/G/1 queueing model with optional second service,next we convert the model into an abstract Cauchy problem in a Banach space by introducing state space,operators and their domains. In Section 2,we study well-posedness of this queueing model,that is,prove existence and uniqueness of a positive time-dependent solution of the queueing model by using the HilleYosida theorem,the Phillips theorem and the Fattorini theorem in functional analysis.
This paper is divided into two chapters.Chapter 1 is divided into two sections.In Section 1,we introduce briefly the history of queueing theory.In Section 2,we first introduce supplementary variable technique,then we put forward the problem that we study in this thesis.Chapter 2 is split into two sections.In Section 1,first we introduce the M/G/1 queueing model with optional second service,next we convert the model into an abstract Cauchy problem in a Banach space by introducing state space,operators and their domains. In Section 2,we study well-posedness of this queueing model,that is,prove existence and uniqueness of a positive time-dependent solution of the queueing model by using the HilleYosida theorem,the Phillips theorem and the Fattorini theorem in functional analysis.
引文
[1]Kailash C.Madan.An M/G/1 queue with second optional service.Queueing Systems [J],2000,34:37-46.
[2]赵会波.具有可选服务的M/G/1排队模型的适定性[D].乌鲁木齐:新疆大学硕士学位论文.2007.
[3]Xing Xi-min.The resolvent set of M/M/1 queueing model with second optional service .Journal of Xinjiang University(Natural Science Edition)[J].2008,25:403-425.
[4]阿不都外力·吉力力.具有可选服务的M/M/1排队模型研究中出现的几个不等式的证明[D].乌鲁木齐:新疆大学学士学位论文,2008.
[5]Gautam Choudhury.Some aspects of an M/G/1 queueing system with optional second service.Sociedad de Estadistica e Investigaci6n Operativa[J].2003,11:141-150.
[6]A.K.Erlang.Sandsynlighedsregning og telefon samtaler,Nyt Tidskrift for Mathematic Copenhagen,B[C].1909,20:33.
[7]J.H.Dshalalow.Frontiers in Queueing[M].Boca Raton:CRC Press,1973.
[8]T.Engset.The probability theory for computing the number of suitching equipments in automatic telephone exchange.ETZ[J].1918,31:304-305.
[9]A.Y.Khintchine.Mathematical Methods in the Theory of Queueing[M].London:Griffin,1980.
[10]L.Kosten.Stochastic Theory of Service Systems[M].Oxford:Pergamon Press,1973.
[11]J.D.C.Little.A proof for the queueing formula:L=1W.Operations Research[J].1961,9(3):383-387.
[12]越民义.排队论中之一问题-M/M/n.数学学报[J].1959,9:494-502.
[13]吴方.关于排对过程GI/M/n.数学学报[J].1961,11:295-305.
[14]徐光辉.排队过程GI/M/n的瞬时性质.数学学报[J].1965,15:91-120.
[15]曹晋华,程侃.服务台可修的M/G/1排队系统分析.应用数学学报[J].1982,5:113-127.
[16]D.R.Cox.The analysis of non-Markovian stochastic process by the inclusion of supplementary variables.Proceedings of Cambridge-Philosophical Society[J].1955,51:433-441.
[17]D.R.Cox,H.D.Miller.The Theory of Stochastic Processes[M].London:Methuen,1965.
[18]N.K.Jaiswal.Bulk-service queueing problem.Operations Research[J].1960,8:139-143.
[19]M.L.Chaudhry(edited).Queueing Systems[J].1992,10:No 1-4.
[20]C.Knessl,B.J.Matkowsky,Z.Schuss and C.Tier.On the performance of state-dependent single server queues.SIAM Journal of Applied Mathematics[J].1986,46:657-697.
[21]M.L.Chaudhry,U.C.Gupta.Modelling and analysis of M/G~(a,b)/1/N queue-a simple alternative approach.Queueing Systems[J].1999,31:95-100.
[22]艾尼·吾甫尔.M/M/1模型的非负解的存在唯一性.系统工程理论与实践[J].1998,18(10):48-58.
[23]Geni Gupur,Xue-zhi Li,Guang-tian Zhu.Functional Analysis Method in Queueing Theory[M].Hertfordshire,Research Information Ltd,2001.
[24]艾尼·吾甫尔,曹珺.M/M/1排队系统四个指标的渐近性质.数学的实践与认识[J].2003,33:45-48.
[25]Geni Gupur,MIijit Abaif.An eigenvalue of M/M/1 operators and its Application.Acta Analysis Functionalis Applicata[J].1999,1:181-192.
[26]朱广田,艾尼·吾甫尔.M/M/1算子的谱特征.控制理论与应用[J].1999,16(12):55-61.
[27]Geni Gupur,Li Xue-zhi,Zhu Guang-tian.Existence and uniqueness of nonnegative solution of M/G~B/1 queueing model.Computers & Mathematics with Applications[J].2000,39:199-209.
[28]Geni Gupur,Xue-zhi Li,Guang-tian Zhu.The application of C_0-semigroup theory to dynamic queueing systems.Semigroup Forum[J].2001,62:205-216.
[29]Geni Gupur.Semigroup method for M/G/1 queueing system with exceptional service time for the first customer in each busy period.Indian Journal of Mathematics[J].2002,44(2):125-146.
[30]Geni Gupur.Well-posedness of M/G/1 queueing model with single vacations.Computers & Mathematics with Applications[J].2002,44:1041-1056.
[31]Geni Gupur.Asymptotic property of the solution of exhaustive-service M/M/1 queueing model with single vacations.International Journal of Differential Equation and Applications [J].2002,6(1):29-51.
[32]Geni Gupur.An eigenvalue of exhaustive-service M/M/1 queueing model with single vacations.International Journal of Differential Equations and Applications[J].2003,7:211-237.
[33]Geni Gupur.Semigroup method for M/G/1 retrial queue with general retrial times.International Journal of Pure and Applied Mathematics[J].2005,18:405-429.
[34]Zhang Furong,Geni Gupur.The resolvent set of M/M/1 retrial queueing model.International Journal of Pure and Applied Mathematics[J].2005,20(4):449-469.
[35]Jiang Mei,Geni Gupur.An eigenvalue of M/M/1 retrial queueing model with special retrial times.Acta Analysis Functionalis Applicata[J].2007,9:193-203.
[36]Cheng Zhi-song,Geni Gupur.Other eigenvalues of the exhaustive-service M/M/1 queueing model with single vacations.International Journal of Pure and Applied Mathematics [J].2007,41:1171-1186.
[37]张隆,艾尼·吾甫尔.M/M/1算子的另一个特征值.应用泛函分析学报[J].2008,10:81-91.
[38]R.A.Adams.Sobolev Space[M].New York:Academic Press,1975.
[2]赵会波.具有可选服务的M/G/1排队模型的适定性[D].乌鲁木齐:新疆大学硕士学位论文.2007.
[3]Xing Xi-min.The resolvent set of M/M/1 queueing model with second optional service .Journal of Xinjiang University(Natural Science Edition)[J].2008,25:403-425.
[4]阿不都外力·吉力力.具有可选服务的M/M/1排队模型研究中出现的几个不等式的证明[D].乌鲁木齐:新疆大学学士学位论文,2008.
[5]Gautam Choudhury.Some aspects of an M/G/1 queueing system with optional second service.Sociedad de Estadistica e Investigaci6n Operativa[J].2003,11:141-150.
[6]A.K.Erlang.Sandsynlighedsregning og telefon samtaler,Nyt Tidskrift for Mathematic Copenhagen,B[C].1909,20:33.
[7]J.H.Dshalalow.Frontiers in Queueing[M].Boca Raton:CRC Press,1973.
[8]T.Engset.The probability theory for computing the number of suitching equipments in automatic telephone exchange.ETZ[J].1918,31:304-305.
[9]A.Y.Khintchine.Mathematical Methods in the Theory of Queueing[M].London:Griffin,1980.
[10]L.Kosten.Stochastic Theory of Service Systems[M].Oxford:Pergamon Press,1973.
[11]J.D.C.Little.A proof for the queueing formula:L=1W.Operations Research[J].1961,9(3):383-387.
[12]越民义.排队论中之一问题-M/M/n.数学学报[J].1959,9:494-502.
[13]吴方.关于排对过程GI/M/n.数学学报[J].1961,11:295-305.
[14]徐光辉.排队过程GI/M/n的瞬时性质.数学学报[J].1965,15:91-120.
[15]曹晋华,程侃.服务台可修的M/G/1排队系统分析.应用数学学报[J].1982,5:113-127.
[16]D.R.Cox.The analysis of non-Markovian stochastic process by the inclusion of supplementary variables.Proceedings of Cambridge-Philosophical Society[J].1955,51:433-441.
[17]D.R.Cox,H.D.Miller.The Theory of Stochastic Processes[M].London:Methuen,1965.
[18]N.K.Jaiswal.Bulk-service queueing problem.Operations Research[J].1960,8:139-143.
[19]M.L.Chaudhry(edited).Queueing Systems[J].1992,10:No 1-4.
[20]C.Knessl,B.J.Matkowsky,Z.Schuss and C.Tier.On the performance of state-dependent single server queues.SIAM Journal of Applied Mathematics[J].1986,46:657-697.
[21]M.L.Chaudhry,U.C.Gupta.Modelling and analysis of M/G~(a,b)/1/N queue-a simple alternative approach.Queueing Systems[J].1999,31:95-100.
[22]艾尼·吾甫尔.M/M/1模型的非负解的存在唯一性.系统工程理论与实践[J].1998,18(10):48-58.
[23]Geni Gupur,Xue-zhi Li,Guang-tian Zhu.Functional Analysis Method in Queueing Theory[M].Hertfordshire,Research Information Ltd,2001.
[24]艾尼·吾甫尔,曹珺.M/M/1排队系统四个指标的渐近性质.数学的实践与认识[J].2003,33:45-48.
[25]Geni Gupur,MIijit Abaif.An eigenvalue of M/M/1 operators and its Application.Acta Analysis Functionalis Applicata[J].1999,1:181-192.
[26]朱广田,艾尼·吾甫尔.M/M/1算子的谱特征.控制理论与应用[J].1999,16(12):55-61.
[27]Geni Gupur,Li Xue-zhi,Zhu Guang-tian.Existence and uniqueness of nonnegative solution of M/G~B/1 queueing model.Computers & Mathematics with Applications[J].2000,39:199-209.
[28]Geni Gupur,Xue-zhi Li,Guang-tian Zhu.The application of C_0-semigroup theory to dynamic queueing systems.Semigroup Forum[J].2001,62:205-216.
[29]Geni Gupur.Semigroup method for M/G/1 queueing system with exceptional service time for the first customer in each busy period.Indian Journal of Mathematics[J].2002,44(2):125-146.
[30]Geni Gupur.Well-posedness of M/G/1 queueing model with single vacations.Computers & Mathematics with Applications[J].2002,44:1041-1056.
[31]Geni Gupur.Asymptotic property of the solution of exhaustive-service M/M/1 queueing model with single vacations.International Journal of Differential Equation and Applications [J].2002,6(1):29-51.
[32]Geni Gupur.An eigenvalue of exhaustive-service M/M/1 queueing model with single vacations.International Journal of Differential Equations and Applications[J].2003,7:211-237.
[33]Geni Gupur.Semigroup method for M/G/1 retrial queue with general retrial times.International Journal of Pure and Applied Mathematics[J].2005,18:405-429.
[34]Zhang Furong,Geni Gupur.The resolvent set of M/M/1 retrial queueing model.International Journal of Pure and Applied Mathematics[J].2005,20(4):449-469.
[35]Jiang Mei,Geni Gupur.An eigenvalue of M/M/1 retrial queueing model with special retrial times.Acta Analysis Functionalis Applicata[J].2007,9:193-203.
[36]Cheng Zhi-song,Geni Gupur.Other eigenvalues of the exhaustive-service M/M/1 queueing model with single vacations.International Journal of Pure and Applied Mathematics [J].2007,41:1171-1186.
[37]张隆,艾尼·吾甫尔.M/M/1算子的另一个特征值.应用泛函分析学报[J].2008,10:81-91.
[38]R.A.Adams.Sobolev Space[M].New York:Academic Press,1975.