Exponential type charts are useful tools to monitor the time between events in high-quality processes with a low defect rate. Most studies on exponential charts are designed with the average run length (ARL) metric. The only use of ARL in the design of control charts is sometimes criticized because the shape of the run length (RL) distribution of control charts changes with the shift size. In fact, the RL distribution of the exponential exponentially weighted moving average (EWMA) chart is skewed, especially when the process is in-control. Hence, the median run length (MRL) serves as a more meaningful indicator. Moreover, in some situations, the shift size in the process is unknown in advance. Under this case, the expected median run length (EMRL) can be used as the metric. In this paper, the RL properties of both the upper- and lower-sided exponential EWMA charts are studied through a Markov chain approach. Two optimal design procedures are developed for one-sided exponential EWMA charts based on the MRL and EMRL, respectively. The choices of reflecting boundaries for one-sided exponential EWMA charts are discussed through many numerical studies. The MRL and EMRL performances of the one-sided exponential EWMA charts are investigated.

指数类型图是有用的工具，可以在缺陷率低的高质量过程中监视事件之间的时间。指数图表上的大多数研究都是使用平均游程长度（ARL）指标设计的。有时会批评在控制图的设计中仅使用ARL，这是因为控制图的游程长度（RL）分布的形状随班次大小而变化。实际上，指数加权加权移动平均值（EWMA）图的RL分布是偏斜的，尤其是在过程处于受控状态时。因此，中间游程长度（MRL）作为更有意义的指标。而且，在某些情况下，该过程中的移位大小是预先未知的。在这种情况下，可以将预期的平均行程长度（EMRL）用作度量。在本文中，通过马尔可夫链方法研究了上下两侧指数EWMA图的RL特性。针对分别基于MRL和EMRL的单面指数EWMA图，开发了两种最佳设计程序。通过许多数值研究，讨论了单面指数EWMA图的反射边界的选择。研究了单面指数EWMA图的MRL和EMRL性能。