A Cumulative Shock Model with Random Failure Threshold and a Change Point

Reliability evaluation plays a pivotal role in the field of shock models. In such models, system failure occurs either when the damage inflicted by shocks surpasses their respective failure thresholds or when the time interval between shocks falls below a critical threshold. In the existing literature, the focus has often been solely on the magnitude of shocks, disregarding their sources. However, it is important to acknowledge that this approach may not always be suitable in real-world scenarios. Systems can experience random shocks originating from various sources, each with different probabilities. Different sources of shocks can have varying implications for a system. Therefore, it is more appropriate to consider the sources of shocks when modeling system reliability. Moreover, most research in the field of shock models utilizes fixed failure thresholds for systems. While fixed failure threshold models can provide a fundamental understanding of system reliability and performance, they may not always accurately reflect real-world conditions. Often, the designer and producer of a part or a system have many diverse users of their products. In practice, the critical threshold value can vary appreciably among users. In this case, a probabilistic, rather than a deterministic threshold value is more appropriate. On the other hand, in practical applications, a system may experience a shock with a stronger or weaker impact due to sudden changes in system behavior or environmental conditions. This represents a point in the data where there is a shift in the underlying distribution or generating process. This research has focused on the investigation of the reliability of a system characterized by random failure thresholds and a change point, which is exposed to cumulative shocks emanating from various sources. Our approach employs Phase-type (PH) distribution and its properties for reliability modeling. To demonstrate the efficiency and accuracy of the proposed model, we presented an illustrative example and conducted a comparative analysis with Monte Carlo simulations. It is imperative to note that accounting for real-world conditions, such as random failure thresholds, change points, and multiple shock sources, can significantly impact the reliability assessment. Engineers and designers stand to gain valuable insights from this model, which can aid in enhancing system reliability and safety and reducing costs throughout the system's lifetime.