Auger decay is an efficient ultrafast relaxation process of core-shell or inner-shell excited atom or molecule. Generally, it occurs in femto-second or even atto-second time domain. Direct measurement of lifetimes of Auger process of single ionized and double ionized inner-shell state of an atom or molecule is an extremely difficult task. In this paper, we have applied the highly correlated complex absorbing potential-equation-of-motion coupled cluster (CAP-EOMCC) approach which is a combination of CAP and EOMCC approach to calculate the lifetime of the states arising from 2p inner-shell ionization of an Ar atom and 3d inner-shell ionization of Kr atom. We have also calculated the lifetime of Ar(2+)(2p(-1)3p(-1)) (1)D, Ar(2+)(2p(-1)3p(-1)) (1)S, and Ar(2+)(2p(-1)3s(-1)) (1)P double ionized states. The predicted results are compared with the other theoretical results as well as experimental results available in the literature.