Some new real-time monitoring schemes for Gumbel's bivariate exponential time between the events

Monitoring the vector of times between multiple events is essential in a high-quality process such as healthcare operations. To this end, the multivariate time between events (TBE) process monitoring schemes are regularly used as one of the most straightforward and appealing visual tools. The existing literature on multivariate TBE schemes focuses almost exclusively on using complete information availed in vector-based TBE data, often making delayed monitoring as it requires observing the complete set of time values in a vector-valued observation. To address this issue, we recommend monitoring the minimum time value of vector TBE data to reach decisions faster and more efficiently. We introduce several new real-time exponentially weighted moving average (EWMA) schemes for monitoring Gumbel's bivariate exponential TBE processes. We compare them with existing schemes using fully observed vector-based schemes. A Markov chain method is developed to compute the average time to signal (ATS), and the optimal parameters are found. Finally, three real-life examples are used to illustrate the implementation of the proposed schemes.