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題名 計算 PH/PH/1/N佇列機率分配之研究
A New Approach to Analyze Stationary Probability Distributions of a PH/PH/1/N Queue
作者 王心怡
貢獻者 陸行
王心怡
日期 2002
上傳時間 9-May-2016 16:39:12 (UTC+8)
摘要 　　在這一篇論文裡， 我們討論開放式有限容量等候系統。其中到達時間和服務時間的機率分配都是 Phase type 分配， 我們發現到達時間和服務時間的機率分配滿足一個方程式。然後根據平衡方程式，我們推導出非邊界狀態的穩定機率可以被表示成 product-form 的線性組合， 而每個 product-form 可以用聯立方程組的根來構成。利用非邊界狀態的穩定機率， 我們可以求出邊界狀態的機率。此外，我們介紹廣義逆矩陣來解出這個複雜的數值問題。再者，我們建立一個求穩定機率的演算過程。利用這個演算方法， 可以簡化求穩定機率的複雜度。最後我們利用這個演算法處理四個例子。
　　In this thesis, we analyze the PH\\PH\\1\\N open queueing system with finite capacity, N. We find two properties which show that the Laplace transforms of interarrival and service times distributions satisfy an equation of a simple form. According to the state balance equations, we present that the stationary probabilities on the unboundary states can be written as a linear combination of product-forms. Each component of these products can be expressed in terms of roots of the system of equations. Instead of solving complicated numerical problem, we introduced the pseudo-inverse to find the solution. Furthermore, we establish an algorithm for solving stationary probabilities and calculating the performance of PH\\PH\\1\\N system which reduces the computational complexity. Finally, we give four examples solved by the algorithm.
謝辭
     Abstract-----i
     中文摘要-----ii
     Contents-----iii
     1 Introduction 1
     　　1.1 The Motivation-----1
     　　1.2 Literature Review-----3
     　　1.3 Organization of the Thesis-----4
     2 Analysis of PH/PH/1/N Systems-----5
     　　2.1 Formulation of the Model as a Continuous-time Markov Chain-----5
     　　　　2.1.1 Interarrival and Service Times-----5
     　　　　2.1.2 Assumptions and Problem Description-----6
     　　　　2.1.3 The State Balance Equations-----6
     　　2.2 Analysis of Equations-----8
     　　　　2.2.1 Separation of Variables Technique-----8
     　　　　2.2.2 Propositions-----10
     3 Steady-state Probabilities of PH/PH/1/N Systems-----15
     　　3.1 The Model with Matrix Forms-----15
     　　　　3.1.1 Phase Type Distribution-----15
     　　　　3.1.2 Transition Rate Matrix-----16
     　　　　3.1.3 Balance Equations-----19
     　　3.2 Product Form Solutions-----19
     　　3.3 The Boundary Probabilities-----20
     4 The Numerical Method-----22
     　　4.1 The System of Linear Equations-----22
     　　4.2 The Least Square Algorithm-----23
     5 The Relevant Information of the System-----25
     　　5.1 The System-size Probability-----25
     　　5.2 A Summary of the Algorithm-----26
     6 Examples-----27
     　　6.1 Examples of M/M/1/7 System-----27
     　　　　6.1.1 Example of Case 1-----27
     　　　　6.1.2 Example of Case 2-----30
     　　6.2 Example of E2/E2/1/4 System-----32
     　　6.3 Example of C2/C2/1/5 System-----35
     7 Conclusions and Future Research-----38
     　　7.1 Conclusions-----38
     　　7.2 Future Research-----39
     References-----40
     Appendix A-----42
     Appendix B-----46
     Appendix C-----47
     Appendix D-----48
參考文獻 [1] Adan, W.A. and Wessels, J. Analyzing Ek/Er/c queues. European Journal of Operational Research 92, 112-124, (1996).
     [2] Bellman R., Introduction to Matrix Analysis, MacGraw-Hill, London, (1960).
     [3] Bertsimas D., An exact FCFS waiting time analysis for a class of G/G/s queueing systems. QUESTA 3,305-320, (1988).
     [4] Bertsimas D., An analytic approach to a general class of G/G/s queueing systems. Operations Research 38 , 139-155, (1990).
     [5] Chao, X., Pinedo, M. and Shaw, D., An Assembly Network of Queues with Product Form Solution. Journal of Applied Probability, 33, 858-869, (1996).
     [6] Chao, X., Miyazawa, M., Serfozo, R., and Takada. H., Necessary and sufficient conditions for product form queueing networks. Queueing Systems, Vol 28, 377-401, (1998).
     [7] Golub, G.H., and Van Loan, C.F., Matrix-Computations, The John Hopkins University Press, (1989).
     [8] Hille, E. Analytic Function Theory, vo 1, 252-256, (1962).
     [9] Le Boudec, J.Y., Steady-state probabilities of the PH/PH/1 queue. Queueing Systems 3,73-88, (1988).
     [10] Luh, H. Matrix product-form solutions of stationary probabilities in tandem queues. Journal of the Operations Research 42-4, 436-656, (1999).
     [11] Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press, (1981).
     [12] Neuts, M.F. and Takahashi, Y. Asymptotic behavior of the stationary distributions in the GI/PH/C queue with heterogeneous servers. Z. Wahrschem-lichkeitstheorie verw. Gebiete, 57, 441-452, (1988).
     [13] Noble, B. and Daniel, J.W. Applied Linear Algebra, Prentice-Hall International Editions, (1988).
     [14] Pollaczek, F. Theorie Analytique des Problemes Stochastiques Relatifs a un Groupe de Lignes Telephoniques Avec Dispositif d`Attente, Gauthier, Paris, (1961).
     [15] Seneta, E. Non-negative Matrices and Markov Chains, Springer-Verlag, (1980).
     [16] Smit, J.H.A. The Queue GI/M/s with Customers of Different Types or the Queue of GI/Hm/s, Advanced Applied Probability15, 392-419, (1983).
     [17] Smit, J.H.A. A Numerical Solution for the Multi-Server Queue with Hyper-Exponential Service Times. Operations Research Letters. 2, 217-224, (1983).
     [18] Takahashi, Y. Asymptotic exponentiality of the tail of the waiting-time distribution in a PH/PH/c queue. Advanced Applied Probability 13, 619-630, (1981).
     [19] Winston, W.L. Operations Research, Duxbury Press, (1994).
     [20] Zhang, Z.H. Linear Algebar, Wen Sheng Publishing House, (1991).
     [21] Zhuang, Y.W. Estimation of Probability Distributions on Closed Queueing Networks, The National Chengchi University Press, (2001).
描述 碩士
國立政治大學
應用數學系
89751004
資料來源 http://thesis.lib.nccu.edu.tw/record/#A2010000178
資料類型 thesis
dc.contributor.advisor 陸行zh_TW
dc.contributor.author (Authors) 王心怡zh_TW
dc.creator (作者) 王心怡zh_TW
dc.date (日期) 2002en_US
dc.date.accessioned 9-May-2016 16:39:12 (UTC+8)-
dc.date.available 9-May-2016 16:39:12 (UTC+8)-
dc.date.issued (上傳時間) 9-May-2016 16:39:12 (UTC+8)-
dc.identifier (Other Identifiers) A2010000178en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/95617-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 89751004zh_TW
dc.description.abstract (摘要) 　　在這一篇論文裡， 我們討論開放式有限容量等候系統。其中到達時間和服務時間的機率分配都是 Phase type 分配， 我們發現到達時間和服務時間的機率分配滿足一個方程式。然後根據平衡方程式，我們推導出非邊界狀態的穩定機率可以被表示成 product-form 的線性組合， 而每個 product-form 可以用聯立方程組的根來構成。利用非邊界狀態的穩定機率， 我們可以求出邊界狀態的機率。此外，我們介紹廣義逆矩陣來解出這個複雜的數值問題。再者，我們建立一個求穩定機率的演算過程。利用這個演算方法， 可以簡化求穩定機率的複雜度。最後我們利用這個演算法處理四個例子。zh_TW
dc.description.abstract (摘要) 　　In this thesis, we analyze the PH\\PH\\1\\N open queueing system with finite capacity, N. We find two properties which show that the Laplace transforms of interarrival and service times distributions satisfy an equation of a simple form. According to the state balance equations, we present that the stationary probabilities on the unboundary states can be written as a linear combination of product-forms. Each component of these products can be expressed in terms of roots of the system of equations. Instead of solving complicated numerical problem, we introduced the pseudo-inverse to find the solution. Furthermore, we establish an algorithm for solving stationary probabilities and calculating the performance of PH\\PH\\1\\N system which reduces the computational complexity. Finally, we give four examples solved by the algorithm.en_US
dc.description.abstract (摘要) 謝辭
     Abstract-----i
     中文摘要-----ii
     Contents-----iii
     1 Introduction 1
     　　1.1 The Motivation-----1
     　　1.2 Literature Review-----3
     　　1.3 Organization of the Thesis-----4
     2 Analysis of PH/PH/1/N Systems-----5
     　　2.1 Formulation of the Model as a Continuous-time Markov Chain-----5
     　　　　2.1.1 Interarrival and Service Times-----5
     　　　　2.1.2 Assumptions and Problem Description-----6
     　　　　2.1.3 The State Balance Equations-----6
     　　2.2 Analysis of Equations-----8
     　　　　2.2.1 Separation of Variables Technique-----8
     　　　　2.2.2 Propositions-----10
     3 Steady-state Probabilities of PH/PH/1/N Systems-----15
     　　3.1 The Model with Matrix Forms-----15
     　　　　3.1.1 Phase Type Distribution-----15
     　　　　3.1.2 Transition Rate Matrix-----16
     　　　　3.1.3 Balance Equations-----19
     　　3.2 Product Form Solutions-----19
     　　3.3 The Boundary Probabilities-----20
     4 The Numerical Method-----22
     　　4.1 The System of Linear Equations-----22
     　　4.2 The Least Square Algorithm-----23
     5 The Relevant Information of the System-----25
     　　5.1 The System-size Probability-----25
     　　5.2 A Summary of the Algorithm-----26
     6 Examples-----27
     　　6.1 Examples of M/M/1/7 System-----27
     　　　　6.1.1 Example of Case 1-----27
     　　　　6.1.2 Example of Case 2-----30
     　　6.2 Example of E2/E2/1/4 System-----32
     　　6.3 Example of C2/C2/1/5 System-----35
     7 Conclusions and Future Research-----38
     　　7.1 Conclusions-----38
     　　7.2 Future Research-----39
     References-----40
     Appendix A-----42
     Appendix B-----46
     Appendix C-----47
     Appendix D-----48		-
dc.description.tableofcontents 謝辭
     Abstract-----i
     中文摘要-----ii
     Contents-----iii
     1 Introduction 1
     　　1.1 The Motivation-----1
     　　1.2 Literature Review-----3
     　　1.3 Organization of the Thesis-----4
     2 Analysis of PH/PH/1/N Systems-----5
     　　2.1 Formulation of the Model as a Continuous-time Markov Chain-----5
     　　　　2.1.1 Interarrival and Service Times-----5
     　　　　2.1.2 Assumptions and Problem Description-----6
     　　　　2.1.3 The State Balance Equations-----6
     　　2.2 Analysis of Equations-----8
     　　　　2.2.1 Separation of Variables Technique-----8
     　　　　2.2.2 Propositions-----10
     3 Steady-state Probabilities of PH/PH/1/N Systems-----15
     　　3.1 The Model with Matrix Forms-----15
     　　　　3.1.1 Phase Type Distribution-----15
     　　　　3.1.2 Transition Rate Matrix-----16
     　　　　3.1.3 Balance Equations-----19
     　　3.2 Product Form Solutions-----19
     　　3.3 The Boundary Probabilities-----20
     4 The Numerical Method-----22
     　　4.1 The System of Linear Equations-----22
     　　4.2 The Least Square Algorithm-----23
     5 The Relevant Information of the System-----25
     　　5.1 The System-size Probability-----25
     　　5.2 A Summary of the Algorithm-----26
     6 Examples-----27
     　　6.1 Examples of M/M/1/7 System-----27
     　　　　6.1.1 Example of Case 1-----27
     　　　　6.1.2 Example of Case 2-----30
     　　6.2 Example of E2/E2/1/4 System-----32
     　　6.3 Example of C2/C2/1/5 System-----35
     7 Conclusions and Future Research-----38
     　　7.1 Conclusions-----38
     　　7.2 Future Research-----39
     References-----40
     Appendix A-----42
     Appendix B-----46
     Appendix C-----47
     Appendix D-----48		zh_TW
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#A2010000178en_US
dc.title (題名) 計算 PH/PH/1/N佇列機率分配之研究zh_TW
dc.title (題名) A New Approach to Analyze Stationary Probability Distributions of a PH/PH/1/N Queueen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Adan, W.A. and Wessels, J. Analyzing Ek/Er/c queues. European Journal of Operational Research 92, 112-124, (1996).
     [2] Bellman R., Introduction to Matrix Analysis, MacGraw-Hill, London, (1960).
     [3] Bertsimas D., An exact FCFS waiting time analysis for a class of G/G/s queueing systems. QUESTA 3,305-320, (1988).
     [4] Bertsimas D., An analytic approach to a general class of G/G/s queueing systems. Operations Research 38 , 139-155, (1990).
     [5] Chao, X., Pinedo, M. and Shaw, D., An Assembly Network of Queues with Product Form Solution. Journal of Applied Probability, 33, 858-869, (1996).
     [6] Chao, X., Miyazawa, M., Serfozo, R., and Takada. H., Necessary and sufficient conditions for product form queueing networks. Queueing Systems, Vol 28, 377-401, (1998).
     [7] Golub, G.H., and Van Loan, C.F., Matrix-Computations, The John Hopkins University Press, (1989).
     [8] Hille, E. Analytic Function Theory, vo 1, 252-256, (1962).
     [9] Le Boudec, J.Y., Steady-state probabilities of the PH/PH/1 queue. Queueing Systems 3,73-88, (1988).
     [10] Luh, H. Matrix product-form solutions of stationary probabilities in tandem queues. Journal of the Operations Research 42-4, 436-656, (1999).
     [11] Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press, (1981).
     [12] Neuts, M.F. and Takahashi, Y. Asymptotic behavior of the stationary distributions in the GI/PH/C queue with heterogeneous servers. Z. Wahrschem-lichkeitstheorie verw. Gebiete, 57, 441-452, (1988).
     [13] Noble, B. and Daniel, J.W. Applied Linear Algebra, Prentice-Hall International Editions, (1988).
     [14] Pollaczek, F. Theorie Analytique des Problemes Stochastiques Relatifs a un Groupe de Lignes Telephoniques Avec Dispositif d`Attente, Gauthier, Paris, (1961).
     [15] Seneta, E. Non-negative Matrices and Markov Chains, Springer-Verlag, (1980).
     [16] Smit, J.H.A. The Queue GI/M/s with Customers of Different Types or the Queue of GI/Hm/s, Advanced Applied Probability15, 392-419, (1983).
     [17] Smit, J.H.A. A Numerical Solution for the Multi-Server Queue with Hyper-Exponential Service Times. Operations Research Letters. 2, 217-224, (1983).
     [18] Takahashi, Y. Asymptotic exponentiality of the tail of the waiting-time distribution in a PH/PH/c queue. Advanced Applied Probability 13, 619-630, (1981).
     [19] Winston, W.L. Operations Research, Duxbury Press, (1994).
     [20] Zhang, Z.H. Linear Algebar, Wen Sheng Publishing House, (1991).
     [21] Zhuang, Y.W. Estimation of Probability Distributions on Closed Queueing Networks, The National Chengchi University Press, (2001).		zh_TW