An amplify-and-forward-based relay network is considered in which an energy-constrained relay is capable of harvesting energy from radio frequency (RF) signal. The source transmits the RF signal that contains both energy and information. The relay first scavenges the RF energy using the time switching (TS) or power-splitting (PS) protocols and then utilizes this energy to forward the signal to the destination. In this paper, new analytical framework has been presented for the outage and ergodic throughputs of the TS and PS protocols considering the Nakagami-m fading channels. Specifically, an exact closed-form expression for the outage throughput has been derived assuming an integer fading parameter. Furthermore, as the exact closed-form expressions for the outage throughput (for arbitrary fading parameters) and ergodic throughput (for both integer and arbitrary fading parameters) are mathematically intractable, new closed-form approximations and bounds have been derived. Finally all the derived expressions have been validated by the Monte Carlo simulations.