Monitoring times between events (TBE) is an important aspect of process monitoring in many areas of applications. This is especially true in the context of high-quality processes, where the defect rate is very low, and in this context, control charts to monitor the TBE have been recommended in the literature other than the attribute charts that monitor the proportion of defective items produced. The Shewhart-type t-chart assuming an exponential distribution is one chart available for monitoring the TBE. The t-chart was then generalized to the t r -chart to improve its performance, which is based on the times between the occurrences of r (≥1) events. In these charts, the in-control (IC) parameter of the distribution is assumed known. This is often not the case in practice, and the parameter has to be estimated before process monitoring and control can begin. We propose estimating the parameter from a phase I (reference) sample and study the effects of estimation on the design and performance of the charts. To this end, we focus on the conditional run length distribution so as to incorporate the 'practitioner-to-practitioner' variability (inherent in the estimates), which arises from different reference samples, that leads to different control limits (and hence to different IC average run length [ARL] values) and false alarm rates, which are seen to be far different from their nominal values. It is shown that the required phase I sample size needs to be considerably larger than what has been typically recommended in the literature to expect known parameter performance in phase II. We also find the minimum number of phase I observations that guarantee, with a specified high probability, that the conditional IC ARL will be at least equal to a given small percentage of a nominal IC ARL. Along the same line, a lower prediction bound on the conditional IC ARL is also obtained to ensure that for a given phase I sample, the smallest IC ARL can be attained with a certain (high) probability. Summary and recommendations are given.

In many situations the times between certain events are observed and monitored instead of the number of events themselves, particularly when the events occur rarely. In this case it is common to assume that the times between events follow an exponential distribution. Control charts are one of the main tools of statistical process control and monitoring. Control charts are used in Phase I to assist operating personnel in bringing the process into a state of statistical control. In this paper, Phase I control charts are considered for the observations from an exponential distribution with an unknown mean. A simulation study is carried out to compare the in-control (IC) robustness and out-of-control (OOC) performance of the proposed chart. It is seen that the proposed charts are considerably more IC robust than two competing charts and have comparable OOC properties.

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