Uniqueness and convergence of solutions to average run length integral equations for cumulative sum and other control charts

The Average Run Length (ARL) is the most frequently used performance measure for selecting and designing CUmulative SUM (CUSUM) as well as other control charts. The derivation of a closed form expression for the ARL of a CUSUM chart, in terms of its design parameters, is intractable. The ARL is often computed as a numerical solution to an integral equation. In this paper, we establish that the ARL is the unique solution to the integral equation under some weak regularity conditions. We prove that numerical algorithms based on Gauss–Legendre and Simpson quadratures provide accurate approximations to the ARL. We develop the proof by establishing the convergence of numerical approximations to the unique solution of the integral equation. Finally, we demonstrate that the integral equation approach and the theory established in this paper easily extend to other control schemes. In particular, we consider an example with a Shewhart chart with correlated observations from an AR(1) process.