Machine Monitoring Using Probability Thresholds and System Operating Characteristics

1. Introduction We are concerned with detecting a change in the underlying condition of a process, when the available observations are related only probabilistically to this condition. This situation has long been of concern to statisticians, engineers, economists, epidemiologists, etc. In this paper we address the specific context of monitoring a discrete-part production machine, with the objective of effectively determining when to shut the machine down for maintenance or replacement. Applications to other areas such as quality control, health, military surveillance, or economic analysis should be readily apparent. Before presenting any details, we list some definitions and initial assumptions, in order to clarify the general setting that governs our analysis: a) There is an underlying time interval that characterizes the operation of the machine, most often the " part production cycle time ". All times and intervals are subsequently measured in units of this time interval. b) The machine can be in only one of two conditions: " good " or " bad " (denoted G and B, respectively). By the G condition we mean the machine can operate in such a way that it is " in control " or " normal " or otherwise able to produce acceptably; by the B condition we mean it is " out of control " , " failed " or only able to produce bad parts (scrap). c) The machine starts in G but at some random variable operating time T (called the failure time) goes to B. This is called a " failure event " , or more simply, a " failure ". d) Observations of " signals " which are probabilistically related to the machine's condition are made at fixed, predetermined times. e) Immediately following any observation one of two possible actions can be made: " do nothing " or " take an action consistent with believing the machine is in B ". The latter investigative action is called a check. f) When a check is made, production is stopped and the condition of the machine becomes known with certainty. A check that finds the machine in G, called a false alarm, returns the machine to operation (in G) after an interval of length g. A check that finds the machine in B, called a true alarm, re-sets it to " as-new " condition – or, equivalently, replaces it by a new (identical) machine – after an interval of length …