  In this paper, we propose a mixture Gamma shadowed (MGS) distribution as a
unified composite model via representing the shadowing by an inverse
Nakagami-$m$. Accordingly, the exact expression and the asymptotic behaviour at
high average signal-to-noise ratio (SNR) regime of the fundamental statistics
of a MGS distribution are derived first. These statistics are then applied to
analyze the performance of the wireless communications systems over double
shadowed $\alpha-\kappa-\mu$ fading via providing the outage probability (OP),
the average bit error probability (ABEP), the average channel capacity (ACC),
and the effective capacity (EC). The average area under the receiver operating
characteristics curve (AUC) of energy detection (ED) is also analyzed. The
numerical and simulation results are presented to verify the validation of our
analysis.
