Run Length Distribution of CUSUM Control Schemes for Negative Binominal Processes

Abstract

The CUSUM control scheme is an important tool in detecting the quality of goods or services. It is used in many industries for rapid detection of many industrial processes. The average run length (ARL) and distribution of run length and standard deviation of run length (SDRL) are the essential characteristics in constructing a control chart and are also important indexes for comparing performance with other control charts. Until now, no theoretic derivation for the run length and its standard deviation underlying negative binominal distribution for the CUSUM scheme have been developed. In this paper, an analytic method for steady state average run length as well as SDRL and two-sided ARL and SDRL with different control statistics is developed for negative binomial CUSUM (NB CUSUM). To decrease computer memory requirements and computing time, a sparse matrix operation method is also proposed to obtain the run length when conducting the Markov chain method. This analytic method makes the computations of run length for one-sided and two-sided NB CUSUM schemes easier. A real numerical example also shows the analytic results are consistent with the simulation results.