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 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Phasetype 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.
 略
 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 theclosed queueing networks with exponential servers. Commun.ACM, 16, 9(Sept.), (1973) 527-531.5.Conway, A.E., and Georganas, N.D., RECAL--A newefficient algorithm for the exact analysis of multiple-chainclosed 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 Mathematicsand 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., andTakada. H., Necessary and sufficient conditions for product formqueueing 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 inStochastic Models (The John Hopkins University Press, 1981).17. Neuts, M.F., and Takahashi, Y.\, Asymptoticbehavior 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/1queue. 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 networkswith multiple closed chains, theory and computational algorithms. IBM J. Res. Dev. , 19,(1975) 283-294.21. Reiser, M., and Lavenberg, S. S., Mean valueanalysis 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.
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