A generalized δ-shock model in the discrete-time framework

Document Type : Original Scientific Paper

Authors

Department of Statistics‎, ‎Razi University‎, ‎Kermanshah‎, ‎Iran

Abstract

According to the δ-shock model‎, ‎a system subjected to random shocks fails when the inter-arrival time between two successive shocks is less than a critical threshold δ‎. ‎Several generalizations of the δ-shock model have been proposed to study shock-exposed systems in different scenarios‎. ‎In this paper‎, ‎we examine a recently proposed generalization of the δ-shock model within a discrete-time framework‎, ‎where inter-arrival times follow a geometric distribution‎. ‎We derive the probability generating function‎, ‎the mean time to failure‎, ‎and the variance of the system lifetime‎. ‎Furthermore‎, we present ‎an illustrative example related to crop insurance is presented to demonstrate the practical relevance and applicability of the proposed discrete-time model.

Keywords

Main Subjects

References

Bladt, M. and Nielsen, B.F. (2017). Matrix-Exponential Distributions in Applied Probability. New York: Springer.
Eryilmaz, S. (2013). On the lifetime behavior of a discrete time shock model. Journal of Computational and Applied Mathematics, 237(1):384–388.
Farhadian, R. and Jafari, H. (2025). Some new approaches to δ-shock modeling. Applied Mathematical Modelling, 137:115705.
Grimmett, G. and Welsh, D. (2014). Probability: An Introduction. Second edition, Oxford University Press.
Li, Z.H., Chan, L.Y. and Yuan, Z.X. (1999). Failure time distribution under a δ-shock model and its application to economic design of system. International Journal of Reliability, Quality and Safety Engineering, 6(03):237–247.
Lorvand, H., Nematollahi, A. and Poursaeed, M.H. (2020). Life distribution properties of a new δ-shock model. Communications in Statistics-Theory and Methods, 49(12):3010–3025.
Lorvand, H. and Eryilmaz, S. (2024). A new extended δ-shock model with the consideration of shock magnitude. Applied Stochastic Models in Business and Industry, 40(5):1355–1364.
Parvardeh, A. and Balakrishnan, N. (2015). On mixed δ-shock models. Statistics & Probability Letters, 102:51–60.
Poursaeed, M.H. (2019). A generalization of δ-shock model. Journal of Advanced in Mathematical Modeling, 9(1):58–70.
Poursaeed, M.H. (2020). On δ-shock model in a multi-state system. Communications in Statistics-Theory and Methods, 49(7):1761–1767.
Tuncel, A. and Eryilmaz, S. (2018). System reliability under δ-shock model. Communications in Statistics-Theory and Methods, 47(19):4872–4880.
Wang, G.J. and Peng, R. (2017). A generalized δ-shock model with two types of shocks. International Journal of Systems Science: Operations & Logistics, 4(4):372–383.
Journal of Statistical Modelling: Theory and Applications
Volume 6, Issue 2
July 2025
Pages 127-135

Files

History

  • Receive Date: 25 October 2025
  • Revise Date: 03 August 2026
  • Accept Date: 12 August 2026

Share

How to cite

Statistics