Control charts for monitoring the time between events can be applied in various areas. In this study, we focus on the exponential control chart and consider the phase II problem (when process parameters are known) as well as the phase I problem (when process parameters are unknown). An exponential chart designed with the conventional approach has the disadvantage that the Average Run Length (ARL) value may increase when the process deviates from the nominal state. An ARL-unbiased design approach is therefore proposed for both phase II and phase I exponential charts. A sequential sampling scheme is adopted for the phase I exponential chart. The proposed ARL-unbiased design approach has several advantages over the conventional one, as it provides a self-starting feature and can significantly improve the ARL performance. Specific guidelines are suggested regarding the time to stop updating the estimates of parameters and control limits based on the actual false alarm rate. The phase I exponential chart can be calibrated to a constant in-control ARL value for each successive event accumulated to date. Simulated and real data examples are given to demonstrate the use and efficiency of the proposed design approach.
In this paper, control charts for monitoring exponentially distributed time between events (TBE) are studied. In particular, a Gamma chart which monitors the time until the rth event is proposed and investigated. A new method based on a random-shift model for calculating the out-of-control average time to signal (ATS) of the Gamma chart is developed. It is shown to be much more accurate than the conventional method based on a fixed-shift model through comparing with Monte Carlo simulation. A comparison is also made among the exponential, the Gamma and the exponential CUSUM charts, which shows that the Gamma chart is more sensitive than the exponential chart and the performance of a Gamma chart with r ¼ 4 is comparable with that of an exponential CUSUM optimally designed. However, the advantage of the Gamma chart is the ease involved in the design, evaluation and implementation. The use of the Gamma chart is illustrated with two real and one simulated examples.
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