Multi-RIS Communication Systems: Asymptotic Analysis of Best RIS Selection for i.n.i.d. Random Variables Using Extreme Value Theory

We investigate the performance of multiple reconfigurable intelligent surfaces (multi-RIS) systems where the RIS link with the highest signal-to-noise ratio (SNR) is selected at the destination. In practice, all the RISs will not have the same number of reflecting elements. Hence, selecting the RIS link with the highest SNR will involve characterizing the distribution of the maximum of independent, non-identically distributed (i.n.i.d.) SNR random variables (RVs). Using extreme value theory, we derive the asymptotic distribution of the normalized maximum of i.n.i.d. non-central chi-square distributed SNR RVs with one degree of freedom and then extend the results fork-th order statistics. Using these asymptotic results, the outage capacity and average throughput expressions are derived for the multi-RIS system. The results for independent and identically distributed SNR RVs are then derived as a special case of i.n.i.d. RVs. All the derivations are validated through extensive Monte Carlo simulations, and their utility is discussed.