# Competing Failure Modeling for Systems under Classified Random Shocks and Degradation
###### Abstract  
A new model is presented to analyze reliability for systems subjected to competing risks of a typical degradation process and external random shocks. The external random shocks may cause hard failure and instantaneous increases in the degradation process. Random shocks are classified into three types in the new model according to their sizes. Different types will lead to different instantaneous increases in the degradation process. Besides system reliability, the competing failure probabilities are also studied. Competing failure probability indicates the probability that one risk mode fails while the other risk mode does not fail. Competing failure probabilities are helpful in determining which risk mode is more likely to occur during a system's lifetime. It is impractical if both risk modes fail together for systems under competing failure modes, while the system reliability and failure probabilities, which are simply calculated by expression, do not resolve this impractical situation. The modified values of system reliability and competing failure probability are calculated in this paper with the impractical situation excluded. The effectiveness of the presented model is demonstrated by the reliability analysis of the micro-electromechanical system (MEMS), which proved that the probabilities calculated using this method are more practical. In addition, sensitivity analysis is performed for specific parameters. competing risks; degradation failure; random shock failure; classified random shock; MEMS 2023202337490. 22 June 2023202  
There are also many research articles concerning random shocks which use the five models. [PERSON] used the extreme shock model to analyze the reliability of systems exposed to external random shock [2]. [PERSON] applied the extreme shock model, the \(\delta\) shock model, and the run shock model in his paper to manage shock problems [3]. [PERSON] used the extreme shock model, the \(\delta\) shock model, and the m-shock model to solve random shock problems with shifting thresholds [4]. [PERSON] used the extreme shock model to analyze the random shocks with multi-shifting thresholds [5]. There are many models, and copious research has been conducted to target degradation problems. The approaches of the degradation analysis can be divided into three main categories [6]: (1) the general path model, this model uses an appropriate expression with specific parameters to represent the degradation process; (2) the stochastic process model, whereby the Markov process and BM (Brownian motion) are typical stochastic process models to conduct reliability analyses; and (3) the statistical model, this model asks for degradation data to establish the degradation model. Many research papers regarding degradation systems have been conducted. [PERSON] modeled the deterioration of the organic coating layer that protects steel structures from corrosion by three different stochastic processes, Brownian motion, a non-stationary gamma process, and a two-stage hit-and-grow physical process [7]. [PERSON] used the degradation path to model the degradation process and estimate reliability, in which a maximum-likelihood parameter estimation method is provided [8]. [PERSON] modeled the degradation process by Brownian motion with drift based on statistics from an accelerated degradation test of accelerometers [9]. The degradation models are widely applied when dealing with degradation problems such as lifetime estimation and maintenance policy-making in practice. [PERSON] estimated the reliability and storage life of FOG (fiber optic gyros) by Brownian motion with drift [10]. [PERSON] predicted the degradation time of a civil aircraft engine by the stochastic process (Brownian motion) [11]. [PERSON] studied the failure process of a sliding rail due to wearing, in which the degradation process is modeled by Brownian motion [12]. As previously mentioned, wear and tribology are the most typical degradation processes for mechanical systems.