統計分佈中，Erlang分佈和指數分佈為伽馬分佈的特例，然而由於指數分佈管制圖只須累積發生一個事件就開始進行監控，穩健性可能較差，因此本研究將指數分佈延伸到累積r個事件的Erlang分佈，並且利用兩個統計模型間的資訊落差(Kullback-Leibler information)來建構建立一個Erlang分佈管制圖，其中假設r已知，僅對事件間隔時間的變化進行監控，也就是對到達率進行監控，而實務上則可以應用於監控機台故障間隔時間和診所病患就診的間隔時間等。本研究所建構的管制圖不需要參數設定，然而過往的管制圖例如指數加權移動平均管制圖和累積和管制圖都要進行參數設定，若參數沒有設定在最佳區間時可能會造成管制圖監控績效不佳。此外，由於本研究只需要利用型一誤差的相對應關係便可以得到管制界線，相比指數加權移動平均管制圖和累積和管制圖可以較容易得到管制界線，因此本研究所建構的管制圖在操作上較為容易並且更加方便使用於實務。
本研究所建構的Erlang KLI管制圖會藉由每一期觀察值的加入重新計算檢定統計量以觀察事件間隔時間是否發生變化，並且分別探討當到達率上升或下降時管制圖的監控績效，也會包含同時監控到達率參數λ的上升和下降。本研究所建構的Erlang KLI管制圖與指數KLI管制圖、伽馬EWMA管制圖和伽馬DEWMA管制圖比較後，發現不管是在單邊還是雙邊監控上本研究所建構的管制圖都具有整體或特定位移較佳的優勢，並且管制圖會隨著r越大監控小位移越敏感。最終導入實際案例進行分析，分析中也觀察到Erlang KLI管制圖的監控績效優於指數KLI管制圖、伽馬EWMA管制圖和伽馬DEWMA管制圖。

This study builds an Erlang control chart for monitoring time between events, and extends the exponential distribution to the Erlang distribution with accumulated r events. This study also uses the information difference between the two statistical models (Kullback-Leibler information) to construct a control chart where r is assumed to be known and only the change in event interval times are monitored. The Erlang KLI control chart can be applied to monitor the interval times between machine failures and the interval times between patients visit in clinics, and it does not require parameter(s) setting. However, previous control charts such as EWMA control chart and CUSUM control chart require parameter(s) setting. When the parameter(s) does not set in optimal range, it may result in poor monitoring performance. In addition, the KLI control chart is easier to obtain the control limit than the EWMA control chart and the CUSUM control chart, so it is easier to operate and more convenient to use in practice.

The Erlang KLI control chart constructed in this study compared with the exponential KLI control chart, the gamma EWMA control chart and the gamma DEWMA control chart, it was found that the Erlang KLI control chart has the advantage of monitoring the overall or specific shift. The control chart also becomes more sensitive to small shift as the r increases.

留恩賜，用於監控間隔時間之指數Kullback-Leibler資訊管制圖，國立成功大學
工業與資訊管理研究所碩士論文，民國一百一十年六月。
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