Evaluation of the System Lifetime Using a New Mixed δ-Shock Model

Jafari, Habib; Farhadian, Reza; Lorvand, Hamed

doi:10.22034/jirss.2026.2048842.1088

Evaluation of the System Lifetime Using a New Mixed δ-Shock Model

Articles in Press

Document Type : Original Article

Authors

Habib Jafari ¹

Reza Farhadian ²

Hamed Lorvand ³

¹ Department of Statistics, Faculty of Science, Razi University, Kermanshah

² Razi University

³ Department of Mathematical Sciences, Isfahan University of Technology, Isfahan

10.22034/jirss.2026.2048842.1088

Abstract

Reliability assessment of complex systems requires accurate modeling of failure mechanisms induced by random external shocks. Classical shock models often focus either on shock magnitudes or on the inter-arrival times between successive shocks, whereas many real-world systems are affected by the joint impact of these two factors. This paper introduces a mixed shock model for systems whose failure is governed by both shock magnitudes and inter-arrival times. In the proposed model, system failure occurs either when the inter-arrival time falls within the interval [α, δ], where 0 ≤ α < δ, or when the magnitude of a single shock exceeds a critical threshold γ. The probability distribution of the system’s stopping time is derived, and the reliability properties of the system’s lifetime are investigated. Numerical examples are provided to illustrate the theoretical findings.

Keywords

Mixed δ-shock model

Inter-arrival time

Magnitude of shock

Reliability

Subjects