Relay technology has recently gained great interest in millimeter wave (60 GHz or above) radio frequencies as a promising transmission technique improving the quality of service, providing high data rate, and extending the coverage area without additional transmit power in deeply shadowed fading environments. The performance of relay-based systems considerably depends on which relay selection protocols (RSPs) are used. These RSPs are typically using the channel side information (CSI). Specifically, the relay terminal (RT) is chosen among all available RTs by a central entity (CE) which receives all RTs' CSI via feedback channels. However, in the millimeter wave radio frequencies, the rate of the CSI variation is much higher than that of the CSI variation in 6 GHz frequencies under the same mobility conditions, which evidently results in a serious problem causing that the CSI at the CE is inaccurate for the RSP since the feedback channels have a backhaul / transmission delay. However and fortunately, the shadowing side information (SSI) varies very slowly in comparison to the rate of the CSI variation. In this context, we propose in this paper a partial-RSP in dual-hop amplify-and-forward relaying system, which utilize only the SSI of the RTs instead of their CSI. Then for the performance analysis, we obtain an exact average unified performance (AUP) of the proposed SSI-based partial-RSP for a variety shadowed fading environments. In particular, we offer a generic AUP expression whose special cases include the average bit error probability (ABEP) analysis for binary modulation schemes, the ergodic capacity analysis and the moments-generating function (MGF)-based characterization. The correctness of our newly theoretical results is validated with some selected numerical examples in an extended generalized-K fading environment.