  We show optimality, in a well-defined sense, using cumulative sum (CUSUM)
charts for detecting changes in distributions. We consider a setting with
multiple changes between two known distributions. This result advocates the use
of non-restarting CUSUM charts with an upper boundary. Typically, after
signalling, a CUSUM chart is restarted by setting it to some value below the
threshold. A non-restarting CUSUM chart is not reset after signalling; thus is
able to signal continuously. Imposing an upper boundary prevents the CUSUM
chart rising too high, which facilitates detection in our setting. We discuss,
via simulations, how the choice of the upper boundary changes the signals made
by the non-restarting CUSUM charts.
