A generalised δ-shock model with two types of shocks. Issue 4 (2nd October 2017)

• Record Type: Journal Article
• Authors: Wang, Guan Jun; Peng, Rui
• Is Part Of: International journal of systems science. Volume 4:Issue 4(2017)
• Page Start: 372
• Page End: 383
• Publication Date: 2017-10-02
• Subjects: Generalised δ-shock model -- NBU -- asymptotic exponentiality -- Markov renewal process -- limiting distribution -- L-S transform; Operations research -- Periodicals; Logistics -- Periodicals; System analysis -- Periodicals; 003
• DOI: 10.1080/23302674.2016.1206986
• Languages: English
• ISSNs: 2330-2674

Abstract

ABSTRACT: In the δ-shock model, a system fails if the interval time between two consecutive shocks is less than a fixed threshold δ, which can be seen as the system recovery time from a shock. In this paper, we generalise the δ-shock model by assuming that the system is subject to two types of shocks, and the system recovery time for a type i shock is δ i, i = 1, 2. If a shock occurs while the system still has not recovered from the consequence of the previous shock, the system fails. The shock to the system is governed by a homogeneous Poisson process with intensity λ, and each shock is of type 1 with probability p, type 2 with probability q = 1 − p . The explicit expression of the reliability function for the lifetime T of the generalised δ-shock model is derived. Both the new better than used property and the asymptotic exponentiality of the lifetime T of the generalised δ-shock model are proven. Moreover, by assuming that the system can be repaired at failure and work again, the limiting distribution of the system state is obtained. Finally, some numerical results are also provided to illustrate the proposed model.