逆問題在模擬學上一直是學術探討的主要領域之一。逆問題是將已知觀測結果轉換為未知系統訊息的方法，而本文的重點在於研究模型重建。本研究以逆問題的角度分析SAN網路圖，並使用Goeva et al.(2014)提出的最大熵模型來分析SAN網路圖。網路圖中的每個活動(activity)皆服從指數分配。我們以最大化熵來作為本次實驗中的目標式，而基本的機率限制做為本次的限制式。一般最大熵模型在馬可夫鏈上的應用是直接引用上述目標與限制式，再經由拉格朗日乘數法取得顧客數的穩態機率。但本次實驗則是透過對偶的轉換與迭代分析來逐漸逼近顧客服務率的原始機率分配。使用此方法最大的特色在於可以用較少的限制條件讓觀測者得到顧客服務率。
本研究的目的是希望透過這種逆推法，藉由已知的數據推論網路圖中任意一個活動所依循的分配。在模擬實驗中，我們由數值結果可以發現，以Goeva et al.(2014)提出的演算法所獲得的最佳化解是能有效應用於SAN網路圖。且透過不同的節點與路徑設計，在缺少部分數據下的預測結果依然是準確的。經由敏感度分析，討論了在不同參數設計下各活動預測結果上的差異，並在最後的模型設計中將均方誤差控制在0.002以下。

Inverse problems have always been one of the main areas of simulation. The inverse problem is a method of converting known observations into unknown system information, and the focus of this article is on model reconstruction. This study analyzes the SAN from the perspective of an inverse problem, and uses the maximum entropy model proposed by Goeva et al. (2014) to analyze the SAN network diagram. Each activity in SAN is subject to exponential allocation. We use maximum entropy as the objective function in this experiment, and the basic probability limit is used as the constraints. The difference from the application of the general maximum entropy model on the Markov chain is to directly quote the above objective function and constraints, and then obtain the steady-state probability of the number of customers through the Lagrange multiplier method.But this experiment is to gradually approach the original probability distribution of customer service rate through dual conversion and iterative analysis.The biggest feature of using this method is that the observer can get the customer service rate with fewer restrictions.
The purpose of this study is to hope to infer the distribution that any activity in SAN follows from the known data through this inverse method. In the simulation experiment, we can find from the numerical results that the optimal solution obtained by the algorithm proposed by Goeva et al. (2014) can be effectively applied to the SAN. And through different node and path designs, the prediction results in the absence of some data are still accurate. Through sensitivity analysis, the differences in the prediction results of various activities under different parameter designs are discussed, and the mean square error is controlled below 0.002 in the final model design.

[1]　Azgomi, M. A. and Movaghar, A., “Hierarchical Stochastic Activity Networks: formal Definitions and Behaviour”, International Journal of Simulation, Systems, Science and Technology, 6(1-2), 56-66, 2005.
[2]　Barron, A. R., and Sheu, C. H., “Approximation of Density Functions by Sequences of Exponential Families”, The Annals of Statistics, 1347-1369, 1991.
[3]　Bingham, N. H. and Pitts, S. M., “Non-parametric Estimation for the M/G/∞ Queue”, Annals of the Institute of Statistical Mathematics, 51(1), 71-97, 1999.
[4]　Basawa, I. V., Bhat, U. N. and Zhou, J., “Parameter Estimation Using Partial Information With Applications To Queueing And Related Models”, Statistics & Probability Letters, 78(12), 1375-1383, 2008.
[5]　Csiszár, I., “Sanov property, generalized I-projection and a conditional limit theorem”, The Annals of Probability, 768-793, 1984.
[6]　Csiszár, I., “Why Least Squares and Maximum Entropy? An Axiomatic Approach to Inference for Linear Inverse Problems”, The annals of statistics, 19(4), 2032-2066, 1991.
[7]　Fearnhead, P., “Filtering Recursions for Calculating Likelihoods for Queues Based on Inter-departure Time Data”, Statistics and Computing, 14(3), 261-266, 2004.
[8]　Ghosh, S. and Lam, H., “A Stochastic Optimization Approach to Assessing Model Uncertainty”, Working paper, 2014.
[9]　Ghosh, S. and Lam, H., “Mirror descent stochastic approximation for computing worst-case stochastic input models”, In 2015 Winter Simulation Conference, pp. 425-436, 2015.
[10]　Goeva, A., Lam, H., and Zhang, B., “Reconstructing Input Models via Simulation Optimization”, In Proceedings of the Winter Simulation Conference 2014, pp. 698-709, 2014.
[11]　Hadamard, J., Lectures on Cauchy's problem in linear partial differential equations, Yale University Press, 1923.
[12]　Heidelberger, P. and Lavenberg, S. S., “Computer performance evaluation methodology”, IEEE Transactions on Computers, (12), 1195-1220, 1984.
[13]　Hansen, P. C. and O’Leary, D. P., “The use of the L-curve in the regularization of discrete ill-posed problems”, SIAM journal on scientific computing, 14(6), 1487-1503, 1993.
[14]　Kaipio, J. and Somersalo, E., Statistical and Computational Inverse problems (Vol. 160), Springer Science and Business Media, 2006.
[15]　Kennedy, M. C. and O’Hagan, A., “Bayesian calibration of computer models”, Journal of the Royal Statistical Society: Series B, 63(3), 425-464, 2001.
[16]　Nakayama, M. K., “Using sectioning to construct confidence intervals for quantiles when applying importance sampling”, In Proceedings of the 2012 Winter Simulation Conference, pp. 1-12, 2012.
[17]　Santner, T. J., Williams, B. J. and Notz, W. I., The design and analysis of computer experiments (Vol. 1), New York: Springer, 2003.
[18]　Sanders, W. H. and Meyer, J. F., “Reduced base model construction methods for stochastic activity networks”, IEEE Journal on Selected Areas in Communications, 9(1), 25-36, 1991.
[19]　Stuart, A. M., “Inverse problems: A Bayesian perspective”, Acta numerica, 19, 451, 2010.
[20]　Tuo, R. and Wu, J., “A Theoretical Framework for Calibration in Computer Models: Parametrization, Estimation and Convergence Properties”, SIAM/ASA Journal on Uncertainty Quantification, 4(1), 767-795, 2016.
[21]　Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation, Society for Industrial and Applied Mathematics, 2005.
[22]　Van Campenhout, J. and Cover, T. M., “Maximum entropy and conditional probability”, IEEE Transactions on Information Theory, 27(4), 483-489, 1981.
[23]　Yuan, J., Ng, S. H. and Tsui, K. L., “Calibration of stochastic computer models using stochastic approximation methods”, IEEE Transactions on Automation Science and Engineering, 10(1), 171-186, 2012.