Distribution of $k$-th Maximum Order Statistics of Independent, & Non-Identical SNR random variables in $κ-μ$ fading and its applications in $6G$

Abstract

This paper employs extreme value theory to establish the asymptotic distribution of the $k$-th maximum order statistics of signal to noise ratio (SNR) for a $κ-μ$ fading channel with independent and non-identically distributed (i.n.i.d.) parameters. Since $κ-μ$ encompasses well-known distributions such as Rice, Rayleigh, and Nakagami-m, the order statistics for these are also derived as special cases. We demonstrate the practical significance of our results by showcasing their applicability in several applications, including antenna selection in MIMO systems, backscatter systems, reconfigurable intelligent surfaces, and UAV-assisted relay selection systems. Furthermore, by utilizing the $k$-th maximum order statistics, we derive expressions for outage probability and average throughput of $k$-th maximum order statistics. Comprehensive Monte Carlo simulations are carried out to verify the accuracy of the proposed results.