Reconfigurable intelligent surface (RIS) is a promising solution to support a large volume of data traffic and massive connectivity for future wireless mobile networks. Especially, RIS can mitigate the drawbacks in massive multiple-input multiple-output (MIMO) systems, such as the blockage caused by obstacles and the signal processing overhead by constructively and passively reflecting the incident wave toward the destination. In this paper, we provide an asymptotic analysis of the distribution of sum rate (SR) in RIS-aided massive MIMO systems. Using the asymptotic distribution of SR, the achievable scheduling gain and the optimal number of users are determined. In addition, we examined the channel hardening effect and outage probability through the achievable scheduling gain, and the optimal number of users is utilized to develop a low-complexity scheduling algorithm. Simulation results reveal that the SR obtained from our analysis closely aligns with the actual SR. The results also show that the channel hardening effect can vanish with many users thereby achieving the multiuser diversity gain, and an RIS-aided system is more reliable than a conventional massive MIMO system in terms of the outage probability. Furthermore, the proposed scheduling algorithm is shown to reduce computational complexity compared to the conventional scheduling algorithm.