Distributions and Causation Probabilities of Multiple-Run-Rules and Their Applications in System Reliability, Quality Control, and Start-Up Tests

Statistical decision rules have been used successfully in various engineering fields for many years: system reliability, quality control, start-up tests, and switching rules. Many of these decision rules are made of multiple-run-rules with their statistical distributions and characteristics that remain largely unknown. This paper shows that, given multiple-run-rules, there exists a single run-rule defined by a compound pattern which is equivalent to the multiple-run-rules and its waiting time is finite Markov chain imbeddable. Their distributions and causation probabilities can be obtained through the finite Markov chain imbedding (FMCI) technique which provides ways to obtain theoretical and numerical results for many probabilistic engineering systems problems. Two examples of multiple-run-rules, one in system reliability and the other in quality control, are provided to demonstrate the imbedding method and theoretical results.