在等候模型中，估計等候時間及延遲機率是一般研究之主要方向。由M/M/s模型開始推廣，目前已有許多學者在對M/G/s等候線提出相關研究，即放寬服務時間服從指數分配的條件的擁塞測量進行估計模型研究。而對於GI/G/s等候線，將到達過程分配條件放寬，使得等候線模型更符合現實狀態，因此在估計擁塞測量上較為困難，也尚未有一個確切的估計模型。故此，至今持續有許多學者在GI/G/s等候線之各項數值及參數估計進行更深入的研究。本研究將探討，目前GI/G/s等候線之平均等候時間估計模型的估計變異，並以Kimura (1986)之系統插值(System Interpolating) 近似模型為主要觀察對象。在其研究中，以QNA (Queueing Network Analyzer)為基礎，轉換參數設定提出各項擴展模型，並以模型期望等候時間與實際平均等候時間之差作為準確率的判斷，但未提及其估計模型之估計變異，且估計量是否為穩定狀態。在模擬實驗中，已有研究證明變異數縮減技術能幫助模擬估計值更加準確且使實驗更有效率，本研究使用其中一個方法：偏差控制變異技術(Biased Control Variates)，在此方法中加入相關性高的近似法對原模型估計作調整，能使得預估計參數的變異改善，並且使估計量能比原估計模型更穩定。因此，本研究預將QNA之平均等候時間近似模型結合控制變異技術，並與QNA後的擴展模型進行精準度比較。

For the queueing system, generally pay close attention on two issues: the waiting time and the queueing length, because these can show the important information for manager that is where is the delay problem. The M/M/s model has been extended for many years. For GI/G/s queue, the distribution conditions for arrival process are relaxed, making the queueing model more realistic, so it is difficult to estimate the congestion measurement, and there is not yet an exact estimation model. As a result, many researcher have continued to conduct more in-depth studies on the numerical and parameter estimates of the GI/G/s queue.
In Kimura’s research, based on the QNA (Queue Network Analyzer), the conversion parameters were set to propose various expansion models, and the difference between the expected waiting time of the model and the actual average waiting time was used as the judgment of the accuracy rate, but he didn’t mention about the variance and the estimated statement is stable or not.
In simulation experiments, studies have shown that variate reduction techniques can help to make simulation estimates more accurate and make experiments more efficient. This study uses one of the methods: Biased Control Variates, and it be used highly approximated pairs of approximations. The adjustment of the original model estimate can improve the variation of the pre-estimation parameter and make the estimator more stable than the original estimation model. Therefore, in this study, the approximation model of the average waiting time of QNA is combined with the control mutation technique, and the accuracy of the extended model after QNA is compared.

參考文獻
[1] Burman, D.Y. and D.R. Smith, “A light traffic limit theorem for multi-server queues”, Math of Operational Research. 8, pp.15-25, 1983.
[2] Cosmetatos, G.P. “On the implementation of Page's approximation for waiting times in general multi-server queues”, Journal of the Operational Research Society. 33, pp.1158-1159, 1982.
[3] Fredericks, A.A. “A class of approximations for the waiting time distribution in a GI/G/1 queueing system”, Bell System Technology Journal. 61, pp.295-325, 1982.
[4] Glynn, P. W., and W. Whitt. “Indirect Estimation via L =λ\W”, Operations Research. 37, pp.82-103, 1989.
[5] Halachmi, B., and W.R. Franta, “A diffusion approximation to the multi-server queue”, Management Science, 24, pp.522-529, 1978.
[6] Hokstad, P. “Approximations for the M/G/m queue”, Operations Research. 26, pp.510-523, 1978.
[7] Hokstad, P. “Pounds for the mean queue length of the M/KE/m queue”, European Journal of Operational Research. 23, pp.108-117, 1986.
[8] Köllerström, J. “Heavy traffic theory for queues with several servers”, Journal of Applied Probability, 11, pp.544-552, 1974.
[9] Kingman, J.F.C. “The single server queue in heavy traffic”, Proceedings of the Cambridge Philosophical Society.7, pp.902-904, 1961.
[10] Kingman, J.F.C. “On queues in heavy traffic”, Journal of the Royal Statistical Society: Series. B 24, pp.383-392, 1962.
[11] Kingman, J.F.C. “The heavy traffic approximation in the theory of queues”, Proceedings of Symposia in Pure Mathematics on Congestion Theory, Chapel Hill, eds. W.L. Smith and R.I. Wilkinson, pp. 137-169, 1964.
[12] Kimura, T. “Heuristic approximations for the mean delay in the GI/G/s queue”, Economic Journal of Hokkaido University, 16, pp.87-98, 1987.
[13] Kimura, T. “Approximations for the waiting time in the GI/G/s queue “, Journal of the Operational Research. Society of Japan, 34, pp.173-186, 1991.
[14] Kimura, T. “Interpolation approximations for the mean waiting time in a multi-server queue”, Journal of the Operational Research Society of Japan, 35, pp.77-92, 1992.
[15] Kimura, T. “A two-moment approximation for the mean waiting time in the GI/G/s queue“, Management Science. 32, pp.751-763, 1986.
[16] Kramer,W. and M. Langenbach-Belz, “Approximate formulae for the delay in the queueing system GI/G/1”, Proceedings of the 8th International Teletraffic Congress, Melbourne, pp.235.1-8, 1976.
[17] Lavenberg, S. S., and P. D.Welch, “A Perspective on the Use of Control Variables to Increase the Efficiency of Monte Carlo Simulations”, Management Science. 27, pp.322-335, 1981.
[18] Meyn, S. P. and S. Andradottir, K. J. Healy, D. H. Withers, B. L. Nelson, “Efficient simulation of multiclass queueing networks”, 1997 Winter Simulation Conference, pp.216–223, 1997.
[19] Nozari, A., S. F. Arnold and C. D. Pegden, “Control Variates for Multipopulation Simulation Experiments”, IIE Transactions. 16, pp.159-169, 1984.
[20] Nelson, B. L. “A Perspective on Variance Reduction in Dynamic Simulation Experiments”, Communications in Statistics. 16, pp.385-426, 1987a.
[21] Nelson, B. L. “On Control Variate Estimators”, Computers & Operations Research. 14, pp.218-225, 1987b.
[22] Nelson, B. L., B. W., Schmeiser, M. R. Taaffe, and J. Wang, “Approximation-assisted point estimation”, Operations Research Letters. 20, pp.109-118, 1997.
[23] Nelson, B. L.” Batch Size Effects on the Efficiency of Control Variates in Simulation” European Journal of Operational Research. 43, pp.184-196, 1989.
[24] Porta Nova, A.M.O. and J.R. Wilson, "Using Control Variates to Estimate Multiresponse Simulation Metamodels," 1986 Winter Simulation Conference Proceedings (J. Wilson, J. Henrikson, S. Roberts, eds.), pp. 326-334, 1986.
[25] Page, E. Queueing Theory in OR (Butterworth, 1972).
[26] Page, E. “Tables of waiting times for M/M/n, M/D/n and D/M/n and their use to give approximate waiting times in more general queues”, Journal of the Operational Research Society. 33, pp.453-473, 1982.
[27] Schmeiser, B. W. and M. R. Taaffe, “Time-dependent queueing network approximations simulation external control variates”, Operations Research Letters. 16, pp.1–9, 1994.
[28] Schmeiser, B.W., M.R. Taaffe, and J. Wang, “Biased control-variate estimation”, IIE Transaction. 33, pp.219-228, 2001.
[29] Shore, H. “Simple approximations for the GI/G/c queue-- I: The steady-state probabilities”, Journal of the Operational Research Society. 39, pp.279-284, 1988.
[30] Shore, H. “Simple approximations for the GI/G/c queue -- II: The moments, the inverse distribution function and the loss function of the number in the system and of the queue delay”, Journal of the Operational Research Society. 39, pp.381-391, 1988.
[31] Shanthikumar, J.G. and J.A., Buzacott, “On the approximations to the single server queue”, International Journal of Production Research. 18, pp.761-773, 1980.
[32] Seelen, L.P. , H.C. Tijms, and M.H. van Hoorn, “Tables for Multi-Server Queues”, North-Holland, 1985
[33] Tew, J. D. and J. R. Wilson, “Estimating Simulation Metamodels Using Integrated Variance Reduction Techniques”, Technical Report Safety Management System 89- 16, School of Industrial Engineering. Purdue University, West Lafayette, Indian, 1989
[34] Tijums, H.C. “Stochastic Modelling and Analysis: A Computational Approach”, John Wiley & Sons, New York, 1986.
[35] Venkatraman, S. and J. R. Wilson, “The Efficiency of Control Variates in Multiresponse Simulation”, Operations Research Letters, pp.37-42, 1986.
[36] Wilson. J. R. “Variance Reduction Techniques for Digital Simulation”, American Journal of Mathematical and Management Sciences, pp.277-312, 1984.
[37] Wu, J.-S. and W.C. Chan, “Maximum entropy analysis of multi-server queueing systems”, Journal of the Operational Research Society. 40, pp.815-825, 1989.
[38] Whitt, W. “The queueing network analyzer”, Bell System Technology Journal. 62, pp.2779-2815, 1983.
[39] Whitt, W. “Approximations for the GI/G/m queue”, preprint, 1985.
[40] Yao, D.D. “Refining the diffusion approximation for the M/G/m queue”, Operations Research. 33, pp. 1266-1277, 1985.