Cusum-schemes with variable sampling intervals and sample sizes

CUSUM-schemes with variable sampling intervals and sample sizes are introduced and investigated for situations where a production process switches at an unknown time from an in-control state to an out-of-control state. Suitable performance criteria are derived to compare CUSUM-schemes with this additional feature. The gain from this feature may be substantial. Without seriously affecting the run length properties under the out-of-control state it is possible to simultaneously reduce the average number of sampled items per time unit (25% to 50%) and to increase the average run length under the in-control state (40% to 50%). Furthermore it is shown that one may restrict to simple schemes that have only two different sample sizes and equally spaced tim-iintervals between the observations.