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

The $$\delta $$ -shock model is one of the basic shock models which has a wide range applications in reliability, finance and related fields. In existing literature, it assumed that recovery time system from damage induced by constant as well shocks magnitude. However, technical systems gradually deteriorate with time, takes more to recover this damage, whereas larger magnitude also results same effect. Therefore, paper, we introduce general when depends on both arrival times magnitudes shocks. Moreover, consider flexible process, namely, Poisson generalized gamma process. It includes homogeneous non-homogeneous Pólya process particular cases. For defined survival model, derive relationships for function mean lifetime study some relevant stochastic properties. As an application, example corresponding optimal replacement policy discussed.