  We establish a simple connection between certain in-control characteristics
of the CUSUM Run Length and their out-of-control counterparts. The connection
is in the form of paired integral (renewal) equations. The derivation exploits
Wald's likelihood ratio identity and the well-known fact that the CUSUM chart
is equivalent to repetitive application of Wald's SPRT. The characteristics
considered include the entire Run Length distribution and all of the
corresponding moments, starting from the zero-state ARL. A particular practical
benefit of our result is that it enables the in- and out-of-control
characteristics of the CUSUM Run Length to be computed concurrently. Moreover,
due to the equivalence of the CUSUM chart to a sequence of SPRTs, the ASN and
OC functions of an SPRT under the null and under the alternative can all be
computed simultaneously as well. This would double up the efficiency of any
numerical method one may choose to devise to carry out the actual computations.
