服务台可修的GI/PH/1排队系统
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摘要
随着当代科学技术的发展,使得经典排队论中的模型在应用中已不足以反映客观事实。复杂计算机通讯网络、信息高速公路、计算机集成制造系统、柔性制造和装配系统等诸多现代领域都提出了大量的排队论问题。近年来,作为经典排队系统的发展和延伸,可修排队系统成为排队理论研究的一个重要领域。这主要是由于这类排队模型与现实的系统更吻合,因此有更广泛的应用前景。
论文较系统的研究了服务台可修的GI/PH/1排队系统,其中忙期服务台寿命和修复时间也是PH变量。这是一个全新的可修排队模型,是已有文献中模型的推广。首先证明系统在稳态下可转化为一个等价的经典GI/PH/1模型,然后运用矩阵几何解理论和矩阵指数方法给出系统的各种稳态指标。此外,对修复后重新服务和累积服务两类不同接续规则,论文给出了统一的处理。
全文共分为四章。第一章简述了经典排队系统和可修排队系统的发展历史和应用状况。第二章介绍了位相型分布的封闭性及更新过程。第三章讨论了服务台可修的GI/PH/1排队系统,将模型转化为一个经典的GI/PH/1模型进行研究,并对两类不同的接续规则给出了统一的处理。第四章使用矩阵几何解理论和矩阵指数方法给出了该可修系统稳态下的排队指标和可靠性指标,并证明了队长和等待时间都是广义PH变量。
With the development of contemporary science and technology, the models made in classical queuing theory are not enough to reflect the objective fact. Many fields, such as the complicated computer communication network, information superhighway, the computer integrated manufacture system (CIMS),flexible manufacturing and assembly system, etc., have all put forward a large number of queuing theory questions. In recent years, as the development and extension of classical queuing system, repairable queuing systems have become the main content of study in queuing theory. This is mainly because that these queuing models arc more identical with realistic systems, so there are more extensive application prospects.
In this thesis, we study the queuing system GI/PH/1 with repairable service station systematically, where the lifetime and the repair time of the service station are both PH random variables. This is a new repairable queuing model, and it’s the promotion of the models in existing literatures. First, we prove that this queuing system can be transformed into the classical queue model GI/PH/1, then, we use the theory of matrix-geometric solutions and the method of matrix-exponential to give several indexes under stationary state. In addition, we give a unified treatment for renew service and cumulative service.
This thesis is divided into four chapters. In the first chapter, we describe the development history and application status of the classical queuing system and the repairable queuing system. In the second chapter, we introduce the closed nature and update process of phase-type distribution. In the third chapter, we discuss the queuing system GI/PH/1 with repairable service station, and transform this model to a classical GI/PH/1 model to research, and give a unified treatment for renew service and cumulative service. In the fourth chapter, we use the theory of matrix-geometric solutions and the method of matrix-exponential to give the queuing indexes and reliability indexes under stationary state, and prove the queue length and the waiting time are both generalized PH variables.
论文较系统的研究了服务台可修的GI/PH/1排队系统,其中忙期服务台寿命和修复时间也是PH变量。这是一个全新的可修排队模型,是已有文献中模型的推广。首先证明系统在稳态下可转化为一个等价的经典GI/PH/1模型,然后运用矩阵几何解理论和矩阵指数方法给出系统的各种稳态指标。此外,对修复后重新服务和累积服务两类不同接续规则,论文给出了统一的处理。
全文共分为四章。第一章简述了经典排队系统和可修排队系统的发展历史和应用状况。第二章介绍了位相型分布的封闭性及更新过程。第三章讨论了服务台可修的GI/PH/1排队系统,将模型转化为一个经典的GI/PH/1模型进行研究,并对两类不同的接续规则给出了统一的处理。第四章使用矩阵几何解理论和矩阵指数方法给出了该可修系统稳态下的排队指标和可靠性指标,并证明了队长和等待时间都是广义PH变量。
With the development of contemporary science and technology, the models made in classical queuing theory are not enough to reflect the objective fact. Many fields, such as the complicated computer communication network, information superhighway, the computer integrated manufacture system (CIMS),flexible manufacturing and assembly system, etc., have all put forward a large number of queuing theory questions. In recent years, as the development and extension of classical queuing system, repairable queuing systems have become the main content of study in queuing theory. This is mainly because that these queuing models arc more identical with realistic systems, so there are more extensive application prospects.
In this thesis, we study the queuing system GI/PH/1 with repairable service station systematically, where the lifetime and the repair time of the service station are both PH random variables. This is a new repairable queuing model, and it’s the promotion of the models in existing literatures. First, we prove that this queuing system can be transformed into the classical queue model GI/PH/1, then, we use the theory of matrix-geometric solutions and the method of matrix-exponential to give several indexes under stationary state. In addition, we give a unified treatment for renew service and cumulative service.
This thesis is divided into four chapters. In the first chapter, we describe the development history and application status of the classical queuing system and the repairable queuing system. In the second chapter, we introduce the closed nature and update process of phase-type distribution. In the third chapter, we discuss the queuing system GI/PH/1 with repairable service station, and transform this model to a classical GI/PH/1 model to research, and give a unified treatment for renew service and cumulative service. In the fourth chapter, we use the theory of matrix-geometric solutions and the method of matrix-exponential to give the queuing indexes and reliability indexes under stationary state, and prove the queue length and the waiting time are both generalized PH variables.
引文
1邓永录.运筹学中的随机模型.广州:中山大学出版社,1996:77-127
2 A. K. Erlang. the Theory of Probabilities and Telephone Conversations. Nyt Tidsskrift Matematik,1909,(20):33-39
3 A. K. Erlang. Solution of Some Problems in the Theory of Probability of Significance in Automatic Telephone Exchanges. Electroteknikeren,1917,(13):5-13
4 Pollaczek. FLosung Eines Geometrischen Wahrsceinlichkeits Problems. Math Z,1932, (35):230-278
5 A. Y. Khinchin. Mathematical Theory of a Stationary Queue (in Russian). Mat. Sbornik,1932,(39):73-84
6 C. Palm. Analysis of the Erlang Traffic Formulae for Busy Signal Arrangements. Ericsson Tech.,1938,(6):39-58
7 L.库珀, U. N.勃哈特, L. J.勒布朗著.运筹学.魏国华,周仲良译.顾基发校.上海:上海科学技术出版社,1978:1-2
8 D. G. Kendall. Stochastic Processes Occurring in the Theory of Queues and Their Analysis by the Method of Imbedded Markov Chains. Ann Math Statist,1953,(24): 338-354
9 P. Moran. A Probability Theory of Dams and Storage Systems: Modifications of the Release Rule. Aust.Appl.Sci.,1955,(7):130-137
10 K. W. Ross, D. K. Tsang, J. Wang. Monte Carlo Summation and Integration Applied to Multi-chain Queuing Networks. J.Assoc.Comput.Mach.,1994,(41):1110-1135
11 K. Rosling. Optimal Inventory Policies for Assembly Systems under Random Demands. Operations Reseach,1989,(37):565-579
12 B. Doshi. Queuing Systems with Vacation A Survey. Queuing Systems,1986,1:29-66
13田乃硕.休假排队综述.运筹学杂志,1994,(13):29-32
14徐光辉.随机服务系统理论(排队论)及其在计算机设计中的应用.计算机应用与应用数学,1975,(1):33-43
15 G. H. Hsu, K. Bosch. Finite Dams with Double Level of Release. Z. Operat.Res.,1983, (22):83-106
16 M. S. Ali Khan. Finite Dams with Dams with Inputs Forming a Markov Chain. J. Appl.Prob.,1970,(7):291-303
17陈洋,朱翼隽,陈燕.具有三种状态可修排队系统.江苏大学学报,2005,26(1):41-44
18朱翼隽,童仁群,王晓春,陈佩树.有可选到达服务台可修的M/G/1重试排队系统.江苏大学学报,2007,28(1):85-88
19李艳.有优先权开关寿命连续型两部件冷贮备可修系统可可靠性分析.兰州大学学报,2003,39(2):18-22
20胥秀珍,朱翼隽.空竭服务多级适应性休假Geomx/G(Geom/G)/1可修排队系统分析.运筹与管理,2005,14(1):8-12
21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,1982,(2):113-127
22曹晋华.服务设备可修的机器服务模型分析.数学研究与评论,1985,(5):89-96
23 B. Avi-Itzhak, P. Naor. Some Queuing Problems with the Service Station Subject to Server Breakdown.Opns.Res.,1963,(10):303-320
24 K. Thiruvengadan. Queuing with Breakdown. Opns.Res.,1963,(11):62-71
25 M. F. Neuts, D. M. Lucantoni. A Markov Queue with N Servers Subject to Breakdowns and Repairs. Mgmt.Sci.,1979,(25):849-861
26 B. Sengupta. A Queue with Service Interruptions in an Alternating Random Environment. Opns.Res.,1990,(38):308-318
27史定华.服务台可修的M/G(Ek/H)/1排队系统的统计平衡理论.运筹学杂志,1989,8 (1):65-66
28史定华,李伟.可修排队系统Em/G(M/H)/1的瞬态解.运筹学杂志,1990,9 (2):39-40
29史定华.可修排队系统的GI/G(M/G)/1的可靠性分析.自动化学报,1995,(21):659-666
30史定华.可修排队与休假排队的结构分析.运筹学杂志,1994,13(2):25-28
31史定华.两部件串联可修模型分析.自动化学报,1995,18(1):16-20
32史定华,田乃硕.服务台可修的GI/M(M/PH)/1排队系统.应用数学学报,1995,18(1): 44-50
33唐应辉.服务台可修的M/G/1排队系统的进一步分析.系统工程理论与实践,1996,16(4):45-51
34唐应辉,唐小我,赵玮.单重休假的Mx/G(M/G)/1可修排队系统(Ⅰ)—一些排队指标.系统工程理论与实践,2000,20(1):40-50
35唐应辉,唐小我.排队论—基础与分析技术.北京:科学出版社,2006:231-242
36刘仁彬,唐应辉,骆川义.一种新型的N部件串联可修系统及其可靠性分析.应用数学,2007,20(1):164-170
37唐应辉,赵玮.可修排队系统可靠性指标的分解特性.运筹学学报,2004,8(4):73-84
38魏英源,唐应辉.N个不同部件串联而成的M/G/1可修排队系统.系统工程理论与实践,2004,24(11):106-110
39唐应辉,俞国建,李晓东.修理设备可更换且修理延迟的两同型并联可修系统.工程数学学报,2005,22(1):1-8
40魏瑛源.服务台由N个部件串联的M/G/1(E,MV)可修排队系统的平稳分布.河西学院学报,2006,22(2):6-10
41李泉林.服务台可修的PH/PH(PH/PH)/1排队系统.数理统计与应用概率,1995,10(2): 75-83
42李泉林.服务台可修的M/SM(PH/SM)/1排队系统.应用数学,1996,9(4):422-428
43岳德权,吕胜利,李静铂.一个修理工的M/M/N可修排队.燕山大学学报,2003,(3): 197-202
44吕胜利,岳德权,李静铂.多服务台可修排队的稳态分布存在条件.运筹与管理,2005,14(4):64-69
45吕胜利,李静铂,岳德权.M/M/N可修排队稳态分布存在条件的一种新形式.系统工程理论与实践,2005,(8):79-84
46李江华,王金亭.具有不耐烦顾客的M/M/N可修排队系统.北京交通大学学报,2006,30(3):84-87
47周文慧,邓永录.具有负顾客到达的M/G/1可修排队系统.运筹学学报,2006,10(2): 28-36
48刘娟.有负顾客的M/G/1重试可修排队系统的极限分布.华东交通大学学报,2007,24 (1):165-167
49唐应辉,刘晓云.修理工带休假的单部件可修系统的可靠性分析.自动化学报,2004,30(3):466-470
50程锋,王聚丰.多服务速率多重休假的M/G(M/G)/1可修排队.工程数学学报,2005, 22(2):249-254
51 M. F. Neuts. Matrix-geometric Solutions in Stochastic Models. Baltimore: The Johns Hopkins Univ. Press, 1981,2-12
52 B. Sengupta. Markov Processes whose Steady State Distribution in Matrix-exponential with an Application to the GI / PH / 1 Queue. Adv.Appl.Prob.,1989, (21):159-180
53田乃硕,李泉林.PH分布及其在随机模型中的应用.应用数学与计算数学学报,1995,9(2):1-15
54田乃硕.休假随机服务系统.北京:北京大学出版社,2001:14-27
55 R. Marier, C. O’Cinneide. A Closure Characterization of Phase-type Distributions. Appl.Prob.,1992,29(1):92-103
56 M. F. Neuts. Markov Chains with Applications in Queuing Theory Which Have a Matrix-geometric Invariant Vector. Adv. Appl.Prob.,1978, (10):185-211
2 A. K. Erlang. the Theory of Probabilities and Telephone Conversations. Nyt Tidsskrift Matematik,1909,(20):33-39
3 A. K. Erlang. Solution of Some Problems in the Theory of Probability of Significance in Automatic Telephone Exchanges. Electroteknikeren,1917,(13):5-13
4 Pollaczek. FLosung Eines Geometrischen Wahrsceinlichkeits Problems. Math Z,1932, (35):230-278
5 A. Y. Khinchin. Mathematical Theory of a Stationary Queue (in Russian). Mat. Sbornik,1932,(39):73-84
6 C. Palm. Analysis of the Erlang Traffic Formulae for Busy Signal Arrangements. Ericsson Tech.,1938,(6):39-58
7 L.库珀, U. N.勃哈特, L. J.勒布朗著.运筹学.魏国华,周仲良译.顾基发校.上海:上海科学技术出版社,1978:1-2
8 D. G. Kendall. Stochastic Processes Occurring in the Theory of Queues and Their Analysis by the Method of Imbedded Markov Chains. Ann Math Statist,1953,(24): 338-354
9 P. Moran. A Probability Theory of Dams and Storage Systems: Modifications of the Release Rule. Aust.Appl.Sci.,1955,(7):130-137
10 K. W. Ross, D. K. Tsang, J. Wang. Monte Carlo Summation and Integration Applied to Multi-chain Queuing Networks. J.Assoc.Comput.Mach.,1994,(41):1110-1135
11 K. Rosling. Optimal Inventory Policies for Assembly Systems under Random Demands. Operations Reseach,1989,(37):565-579
12 B. Doshi. Queuing Systems with Vacation A Survey. Queuing Systems,1986,1:29-66
13田乃硕.休假排队综述.运筹学杂志,1994,(13):29-32
14徐光辉.随机服务系统理论(排队论)及其在计算机设计中的应用.计算机应用与应用数学,1975,(1):33-43
15 G. H. Hsu, K. Bosch. Finite Dams with Double Level of Release. Z. Operat.Res.,1983, (22):83-106
16 M. S. Ali Khan. Finite Dams with Dams with Inputs Forming a Markov Chain. J. Appl.Prob.,1970,(7):291-303
17陈洋,朱翼隽,陈燕.具有三种状态可修排队系统.江苏大学学报,2005,26(1):41-44
18朱翼隽,童仁群,王晓春,陈佩树.有可选到达服务台可修的M/G/1重试排队系统.江苏大学学报,2007,28(1):85-88
19李艳.有优先权开关寿命连续型两部件冷贮备可修系统可可靠性分析.兰州大学学报,2003,39(2):18-22
20胥秀珍,朱翼隽.空竭服务多级适应性休假Geomx/G(Geom/G)/1可修排队系统分析.运筹与管理,2005,14(1):8-12
21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,1982,(2):113-127
22曹晋华.服务设备可修的机器服务模型分析.数学研究与评论,1985,(5):89-96
23 B. Avi-Itzhak, P. Naor. Some Queuing Problems with the Service Station Subject to Server Breakdown.Opns.Res.,1963,(10):303-320
24 K. Thiruvengadan. Queuing with Breakdown. Opns.Res.,1963,(11):62-71
25 M. F. Neuts, D. M. Lucantoni. A Markov Queue with N Servers Subject to Breakdowns and Repairs. Mgmt.Sci.,1979,(25):849-861
26 B. Sengupta. A Queue with Service Interruptions in an Alternating Random Environment. Opns.Res.,1990,(38):308-318
27史定华.服务台可修的M/G(Ek/H)/1排队系统的统计平衡理论.运筹学杂志,1989,8 (1):65-66
28史定华,李伟.可修排队系统Em/G(M/H)/1的瞬态解.运筹学杂志,1990,9 (2):39-40
29史定华.可修排队系统的GI/G(M/G)/1的可靠性分析.自动化学报,1995,(21):659-666
30史定华.可修排队与休假排队的结构分析.运筹学杂志,1994,13(2):25-28
31史定华.两部件串联可修模型分析.自动化学报,1995,18(1):16-20
32史定华,田乃硕.服务台可修的GI/M(M/PH)/1排队系统.应用数学学报,1995,18(1): 44-50
33唐应辉.服务台可修的M/G/1排队系统的进一步分析.系统工程理论与实践,1996,16(4):45-51
34唐应辉,唐小我,赵玮.单重休假的Mx/G(M/G)/1可修排队系统(Ⅰ)—一些排队指标.系统工程理论与实践,2000,20(1):40-50
35唐应辉,唐小我.排队论—基础与分析技术.北京:科学出版社,2006:231-242
36刘仁彬,唐应辉,骆川义.一种新型的N部件串联可修系统及其可靠性分析.应用数学,2007,20(1):164-170
37唐应辉,赵玮.可修排队系统可靠性指标的分解特性.运筹学学报,2004,8(4):73-84
38魏英源,唐应辉.N个不同部件串联而成的M/G/1可修排队系统.系统工程理论与实践,2004,24(11):106-110
39唐应辉,俞国建,李晓东.修理设备可更换且修理延迟的两同型并联可修系统.工程数学学报,2005,22(1):1-8
40魏瑛源.服务台由N个部件串联的M/G/1(E,MV)可修排队系统的平稳分布.河西学院学报,2006,22(2):6-10
41李泉林.服务台可修的PH/PH(PH/PH)/1排队系统.数理统计与应用概率,1995,10(2): 75-83
42李泉林.服务台可修的M/SM(PH/SM)/1排队系统.应用数学,1996,9(4):422-428
43岳德权,吕胜利,李静铂.一个修理工的M/M/N可修排队.燕山大学学报,2003,(3): 197-202
44吕胜利,岳德权,李静铂.多服务台可修排队的稳态分布存在条件.运筹与管理,2005,14(4):64-69
45吕胜利,李静铂,岳德权.M/M/N可修排队稳态分布存在条件的一种新形式.系统工程理论与实践,2005,(8):79-84
46李江华,王金亭.具有不耐烦顾客的M/M/N可修排队系统.北京交通大学学报,2006,30(3):84-87
47周文慧,邓永录.具有负顾客到达的M/G/1可修排队系统.运筹学学报,2006,10(2): 28-36
48刘娟.有负顾客的M/G/1重试可修排队系统的极限分布.华东交通大学学报,2007,24 (1):165-167
49唐应辉,刘晓云.修理工带休假的单部件可修系统的可靠性分析.自动化学报,2004,30(3):466-470
50程锋,王聚丰.多服务速率多重休假的M/G(M/G)/1可修排队.工程数学学报,2005, 22(2):249-254
51 M. F. Neuts. Matrix-geometric Solutions in Stochastic Models. Baltimore: The Johns Hopkins Univ. Press, 1981,2-12
52 B. Sengupta. Markov Processes whose Steady State Distribution in Matrix-exponential with an Application to the GI / PH / 1 Queue. Adv.Appl.Prob.,1989, (21):159-180
53田乃硕,李泉林.PH分布及其在随机模型中的应用.应用数学与计算数学学报,1995,9(2):1-15
54田乃硕.休假随机服务系统.北京:北京大学出版社,2001:14-27
55 R. Marier, C. O’Cinneide. A Closure Characterization of Phase-type Distributions. Appl.Prob.,1992,29(1):92-103
56 M. F. Neuts. Markov Chains with Applications in Queuing Theory Which Have a Matrix-geometric Invariant Vector. Adv. Appl.Prob.,1978, (10):185-211