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    Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/84949


    Title: 封閉式等候網路機率分配之估計與分析
    Estimation of Probability Distributions on Closed Queueing Networks
    Authors: 莊依文
    Contributors: 陸行
    莊依文
    Keywords: 封閉式等候線網路
    穩定機率
    Closed queueing networks
    Stationary probabilities
    Product-forms
    Phase type
    Date: 2001
    Issue Date: 2016-04-15 16:02:53 (UTC+8)
    Abstract: 在這一篇論文裡，我們討論兩個階段的封閉式等候線網路，其中服務時間的機率分配都是Phase type分配。我們猜測服務時間的機率分配和離開時間間隔的機率分配滿足一組聯立方程組。然後，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。最後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。
    In this thesis, we are concerned with the property of a two-stage closed system in which the service times are identically of phase type. We first conjecture that the 　Laplace-Stieltjes Transforms (LST) of service time distributions may satisfy a system of equations. Then 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. Finally, we establish an algorithm to obtain all the stationary probabilities. The algorithm is expected to work well for relatively large customers in the system.
    封面頁
    證明書
    致謝詞
    論文摘要
    目錄
    1 Introduction
    1.1 Background
    1.2 Literature Review
    1.3 Organization of the thesis
    2 The Model
    2.1 Phase type distribution
    2.2 Problem statement and assumptions
    2.3 Preliminaries results
    2.4 Propositions
    3 A two stage closed network with ρ1 ≧ ρ2
    3.1 Transition rate matrix
    3.2 Balance equation
    3.3 Product form solutions
    3.4 Algorithm for the boundary probabilities
    3.5 A summary of the algorithm
    4 A two stage closed network with ρ1 ＜ ρ2
    4.1 Transition rate matrix
    4.2 Balance equation
    4.3 Product form solutions
    4.4 Algorithm for the unboundary probabilities
    4.5 A summary of the algorithm
    5 Examples
    5.1 Example for -/M/1 → /M/1 system
    5.2 Example for -/E2/1 → /E2/1 system
    6 Conclusions and future research
    6.1 Conclusion
    6.2 Future research
    References
    Appendix
    Reference: 1.Bellman R., Introduction to Matrix Analysis (MacGraw-Hill, London) (1960).
    2.Bertsimas D., An exact FCFS waiting time analysis for a class of G/G/s queueing systems. QUESTA, 3,(1988) 305-320.
    3.Bertsimas D., An analytic approach to a general class of G/G/s queueing systems. Operations Research, 38 (1990) 139-155.
    4.Buzen, J.P., Computational algorithms for the closed queueing networks with exponential servers. Commun. ACM, 16, 9(Sept.), (1973) 527-531.
    5.Conway, A.E., and Georganas, N.D., RECAL--A new efficient algorithm for the exact analysis of multiple-chain closed queuing networks ,Journal-of-the-Association-for-Computer-Machinery , 33, 4(Oct.), (1986) 768-791.
    6. Conway, A.E., and Georganas, N.D., Docomposition and arregation by class in closed queueing networks. IEEE Trans. Softw. Eng., 12, 1025-1040, (1986).
    7. Ganesh, A., and Anantharam, V., Stationary tail in probabilities in exponential server tandem queues with renewal arrivals. in Frank P. Kelly and Ruth J. Williams (eds.), Stochastic Networks, The IMA Volumes in Mathematics and Its Applications, 71, (Springer-Verlag, 1995), 367-385.
    8.Fujimoto, K., and Takahashi, Y., Tail behavior of the stationary distributions in two-stage tandem queues---numerical experiment and conjecture. Journal of the Operations Research Society of Japan, 39-4, (1996) 525-540.
    9. Fujimoto, K., Takahashi, Y., and Makimoto, N., Asymptotic Properties of Stationary Distributions in Two-Stage Tandem Queueing Systems. Journal of the Operations Research Society of Japan, 41-1, (1998) 118-141.
    10. Gordon, W.J., and Newell, G.F., Matrix-Geometric Solutions in Stochastic Models (The John Hopkins University Press, 1981).
    11. Golub, G.H., and Van Loan, C.F., Matrix--Computations (The John Hopkins University Press, 1989).
    12. Chao, X., A Queueing Network Model with Catastrophe and Product Form Solution, Operations Research Letters, 18, (1995) 75-79.
    13. Chao, X., Pinedo, M. and Shaw, D., An Assembly Network of Queues with Product Form Solution, Journal of Applied Probability, 33, (1996) 858-869.
    14. Chao, X., Miyazawa, M., Serfozo, R., and Takada. H., Necessary and sufficient conditions for product form queueing networks, Queueing Systems, 28, (1998),377-401.
    15. Chao, X., and Miyazawa. M., On quasi-reversibility and partial balance: An alternative approach to product form results, Operations Research, 46, (1998) 927-933.
    16. Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models (The John Hopkins University Press, 1981).
    17. Neuts, M.F., and Takahashi, Y., Asymptotic behavior of the stationary distributions in the $GI/PH/c$ queue with heterogeneous servers, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 57 (1988) 441-452.
    18. Le Boudec, J.Y., Steady-state probabilities of the PH/PH/1 queue. Queueing Systems, 3 (1988) 73-88.
    19. Luh, H., Matrix product-form solutions of stationary probabilities in tandem queues. Journal of the Operations Research, 42-4 (1999) 436-656.
    20. Reiser, M., and Kobayashi, H., Queueing networks with multiple closed chains, theory and computational algorithms. IBM J. Res. Dev. , 19,(1975) 283-294.
    21. Reiser, M., and Lavenberg, S. S., Mean value analysis of closed multichain queueing networks. Journal-of-the-Association-for-Computer-Machinery , 27, (1980) 313-322.
    22. Seneta, E., Non-negative Matrices and Markov Chains (Springer-Verlag, 1980).
    23. Takahashi, Y., Asymptotic exponentiality of the tail of the waiting-time distribution in a PH/PH/c queue. Advanced Applied Probability, 13 (1981) 619-630.
    Description: 碩士
    國立政治大學
    應用數學系
    Source URI: http://thesis.lib.nccu.edu.tw/record/#A2002001140
    Data Type: thesis
    Appears in Collections:[Department of Mathematical Sciences] Theses

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