Synthetic-type control charts for time-between-events monitoring

This paper proposes three synthetic-type control charts to monitor the mean time-between-events of a homogenous Poisson process. The first proposed chart combines an Erlang (cumulative time between events, T-r) chart and a conforming run length (CRL) chart, denoted as Synth-T-r chart. The second proposed chart combines an exponential (or T) chart and a group conforming run length (GCRL) chart, denoted as GR-T chart. The third proposed chart combines an Erlang chart and a GCRL chart, denoted as GR-T-r chart. By using a Markov chain approach, the zero- and steady-state average number of observations to signal (ANOS) of the proposed charts are obtained, in order to evaluate the performance of the three charts. The optimal design of the proposed charts is shown in this paper. The proposed charts are superior to the existing T chart, T-r chart, and Synth-T chart. As compared to the EWMA-T chart, the GR-T chart performs better in detecting large shifts, in terms of the zero- and steady-state performances. The zero-state Synth-T-4 and GR-T-r (r = 3 or 4) charts outperform the EWMA-T chart for all shifts, whereas the Synth-T-r (r = 2 or 3) and GR-T-2 charts perform better for moderate to large shifts. For the steady-state process, the Synth-T-r and GR-T-r charts are more efficient than the EWMA-T chart in detecting small to moderate shifts.