Wireless powered communication networks (WPCNs) have attracted much research interest in fifth generation (5G) wireless networks. With the help of WPCN, the reliability and battery life of wireless low-power devices can be enhanced. In this paper, we investigate throughput and ergodic capacity in WPCN-assisted amplify-and-forward relaying systems considering two transmission modes, including delay-tolerant and delay-limited. More importantly, we propose energy harvesting protocol so-called hybrid time switching-based and power splitting-based relaying (HTPSR) protocol in order to achieve optimal throughput. In particular, both time switching-based and power splitting-based coefficients in this scheme are considered. Unlike most previous works, we further focus on the impact of outdated channel state information (CSI) in WPCN. In order to evaluate information processing efficiency, the performance can be substantially improved by optimizing harvesting time and power coefficients of the received signal at the relay node for energy and information extraction. Thanks to Monte Carlo simulations, it is confirmed that the system performance is more sensitive to CSI estimation error, noise variance and signal-to-noise ratio which enable us in reasonable computations of HTPSR to obtain QoS requirement.