On history-dependent mixed shock models

Abstract

In this paper, we consider a history-dependent mixed shock model which is a combination of the history-dependent extreme shock model and the history-dependent $\delta$ -shock model. We assume that shocks occur according to the generalized Pólya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Pólya process as the particular cases. For the defined survival model, we derive the corresponding survival function, the mean lifetime and the failure rate. Further, we study the asymptotic and monotonicity properties of the failure rate. Finally, some applications of the proposed model have also been included with relevant numerical examples.

Shock models based on renewal processes with matrix Mittag-Leffler distributed inter-arrival times

“…Furthermore, various mixed shock models, as a combination of two or more basic shock models, were introduced in the literature. For instance, the extreme shock model with the cumulative shock model (Cha and Finkelstein [9]), the extreme shock model with the run shock model (see Eryimaz and Tekin [13]), the extreme shock model with the δ-shock model (Parvardeh and Balakrishnan [38], Wang and Zhang [46], Goyal et al [20]), the cumulative shock model with the run shock model (see Mallor et al [32]), the cumulative shock model with the δ-shock model (Parvardeh and Balakrishnan [38]), the run shock model with the δ-shock model (see, e.g., Eryilmaz [11]).…”

Reliability Analysis and Optimal Replacement Policy for Systems with Generalized Pólya Censored δ Shock Model

“…Cha et al [30] derived and analyzed the corresponding survival and failure rate functions of the extreme shock model under the GPP. Goyal et al [31] studied the survival function and correlation properties of a history-dependent mixed shock model under the GPP. Goyal et al [32] investigated the correlation properties of the time-dependent δ shock model under the GPP and studied the optimal replacement strategy of the established model and the associated random properties.…”

On the general $$\delta $$-shock model

“…Lastly, in the classical δ-shock model, a system fails if the time lag between two successive shocks is less than a predefined threshold value δ (see Li, Chan and Yuan [22], Li, Huang and Wang [23], Li and Kong [24], and references therein). Besides, there are various mixed shock models that are combinations of two or more shock models (see Cha and Finkelstein [4], Finkelstein [13], Eryilmaz and Tekin [11], Wang and Zhang [39], Parvardeh and Balakrishnan [31], Mallor et al [29], Eryilmaz [8], Goyal et al [17], to name a few). Further, by considering different interrelations among model variables (e.g., shock's magnitude, shock's arrival time, intershock time, damage caused by a shock, deterioration process, etc.…”