Detection of process change with non-geometric failure time distribution.

Abstract

Detection of process change is an important issue in quality control, system reliability and maintenance. We consider a system that always starts in a good state (G) and after (random variable) time T, the system enters a bad (B) state. The system stays in B until an exogenous action is taken place. Monitoring, which is free and takes place at the end of every period, gives imperfect information about the state of the system. Checking is an action that gives perfect information about the system state. A decision to check or not to check needs to be made at the end of every period. In extending known analyses involving geometric failure time distributions, we consider general failure time and monitoring distributions. Balancing costs associated with false alarms against with detection time, we formulate the single replacement (SR) and infinite horizon (IH) problems in a dynamic programming framework. For these two problems, the optimal checking policy is proved to be a conditional probability threshold rule (CPTR). Algorithms for solving for these optimal thresholds are also provided. Since finding the optimal thresholds requires a great amount of computer memory, we use an approximate policy--a fixed probability threshold rule (FPTR). We then search for the best fixed probability threshold (BFPT) that achieves the minimum total expected cost over all possible fixed thresholds. We show that FPTR is optimal if a plot of the total expected cost vs. the fixed threshold shows a particular pattern. Using FPTR, we study the special cases of a uniform failure time distribution with either Bernoulli or Normal monitoring. For Bernoulli monitoring, we derive efficient algorithms for finding the BFPT for both exponential and uniform failure time distributions. For Normal monitoring, we simulate the total expected cost given a fixed threshold. For both monitoring distributions, numerical results show small errors associated with using FPTR instead of CPTR for both SR and IH problems.