Estimation of the degradation factor of a censored geometrical process

Aim. The article examines the behaviour of renewable objects that are complex systems and generate temporally unhomogeneous failure flows. The objects’ dependability is described with a geometrical processes model. The mathematical model of such processes allows considering both the ageing and renewal of a system. In the first case the failure flow rate increases with time. That corresponds with the period of ageing, when the failure rate progressively grows and the system fails more and more frequently. In the second case, the failures that show high rate at the beginning of operation become rare with time. In technical literature, this stage of operation is called the burn-in period. Normal renewal process is a special case of the geometric process model. In real operation conditions not all operation times end with a failure. Situations arise when as part of preventive maintenance a shortcoming is identified in an observed object, that gets replaced as the result. Or, for a number of reasons, a procedure is required, for which the object is removed from service and also replaced with an identical one. The object that was removed from service is repaired, modernized or simply stored. Another situation of unfinished operation occurs when the observation of an object is interrupted. More precisely, the object continues operating at the time the observation stops. For example, it may be known that at the current time the object is in operation. Both of the described situations classify the operation time as right censored. The task is to estimate the parameters of the mathematical model of geometric process using the known complete and right censored operation times that are presumably governed by the geometric process model. For complete operation times, this task was solved for various distributions [11-16]. As it is known, taking into consideration censored data increases the estimation quality. In this paper the estimation task is solved subject to the use of complete and right censored data. Additionally, the article aims to provide an analytical justification of increased estimation quality in cases when censoring is taken into account, as well as a practical verification of the developed method with real data.

Methods. The maximum likelihood method is used for evaluation of the parameters of the geometrical process model. The likelihood function takes into consideration right censored data. The resulting system of equations is solved by means of the Newton-Raphson method.

Conclusions. The article introduces formulas for evaluation of model parameters according to the maximum likelihood method on the assumption of various distribution laws of the time to first failure. The resulting formulas enable the estimation of the parameters of the geometrical process model involving uncertainty in the form of right censoring. Analytical evidence is produced on increased accuracy of estimation in cases when right censored data is taken into consideration. Parameter estimation was performed based on real operational data of an element of the Bilibino NPP protection control system. 

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