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 針對一個開放式馬可夫等候網路, 若要知道該系統是否穩定,則需知道該系統之延遲時間。這種系統的延遲時間已由C. Humes, Jr., J. Ou 和P. R. Kumar 討論過, 從他們獲得的結論中我們瞭解在開放式馬可夫等候網路中平均人數被函數化。由線性規劃我們可以得到系統平均人數的上界和下界, 其型式為□ ,這裡□ 所代表的是系統的負載,且□ ,而M是極點重數。理論上, M之值可趨近無限大, 但是目前所有文獻中均無設計出M > 2之實例, 因此本論文中, 我們將限制條件轉換成矩陣型式和設計出一些例子使得極點重數M大於先前結果, 之後再改變限制條件並且利用給定的例子去解對應的線性規劃, 我們將發現 ”極點重數M” 之值亦大於先前的結果。
 For an open Markovian queueing network, to know whether the system is stable, one needs to know the delay of the system. The delay of the system was discussed by C. Humes, Jr., J. Ou, and P. R. Kumar, we study the functional dependence of the mean number in the open Markovian queueing networks. The upper and lower bounds of mean number in the system can be obtained by linear programs, and presented as□ , where □ is the nominal load on the system, and 0 □□< 1, and M is the pole multiplicity. In this paper we transform constraints into matrix expression, and create some examples so that the pole multiplicity M is larger than the previous results. Furthermore, we vary constraints and solve the corresponding linear programs associated with the given examples, we will find the “pole multiplicity M” is also larger than the previous results.
 Contents 1 Introduction ……………………………………………………………………1 2 Notations and Definitions ……………………………………………………3 3 Matrix Representation ………………………………………………………..6 4 Applications of the matrix form …………………………………………..16 5 Conclusions ………………………………………………………………….25 References ……………………………………………………………………..26
 Andrew Grace, ''Optimization Toolbox,'' The MathWorks, Inc., 1990-1994.Borse, G. J. ''Numerical Methods with MALAB,'' LehighUniversity, 1995.Edward J. Anderson, and Peter Nash, ''Linear programmingin infinite dimensional spaces,'' John Wiley & Sons, Inc., 1987.Gong, W. B. and J. Q. Hu, ''The Maclaurin series for theGI/G/1 queue,'' Journal of Applied Probability, vol. 29,pp.176-184, 1992.Humes, C. Jr., J. Ou, and P. R. Kumar,''The delay of open Markovian queueing networks: Uniform functional bounds, heavy traffic pole multiplicities, and Stability,''Mathmatics of operations research, vol.22. pp921-pp945, 1997.Kumar, P. R. and Meyn, S. P. ''Stability of queueingnetworks and scheduling policies,'' IEEE Transactions on Automatic Control, vol. 40, pp.251-260, February 1995.Kumar, P. R. and Meyn, Sean ''Duality and linear programsfor stability and performace analysis of queueing Networks and scheduling policies,'' 1995.Kumar, Sunil and Kumar, P. R. ''Performance bounds forqueueing networks and scheduling policies,'' IEEE Transactions on Automatic Control, vol. Ac-39, pp. 1600-1611, August 1994.Lippman, S. ''Applying a new device in the optimization of exponential queueing systems,'' OperationsResearch}, vol. 23, pp. 687-710, 1975.Murty, Katta G. ''Linear programming,'' John Wiley & Sons,Inc., 1983.Romeijn, H. Edwin and Robert L. Smith, ''Shadow pricesin infinite dimensional linear programming,'' Mathmatics ofoperations research, vol. 23, pp239-pp255, 1997.Romeijn, H. Edwin and Robert L. Smith, and James C. Bean''Duality in infinite dimensional linear programming,'' Mathematical programming, vol. 53, pp79-97, 1992.
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