Abstract: This paper aims to address the trade-off between the nonlinear age of information (AoI) and the energy efficiency (EE) of time-sensitive amplified forwarding (AF) relay-assisted IoT systems in Nakagami-m fading channels. Firstly, an AF relay transmission model is proposed to find the end-to-end approximate signal-to-noise ratio (SNR). Secondly, the end-to-end block error probability of packet transmission is derived by considering the outdated channel state information (CSI) in the Nakagami-m fading channel. Finally, under the simultaneous consideration of non linear energy harvesting (EH) and outdated CSI, the statistical descriptions of the time duration for the sensor to fully charge its battery are derived as well as those  of the time intervals of update packet delivery. As a result, the average AoI is achieved with closed-form expression and the trade-off model of nonlinear AoI to EE is established by simultaneously exploiting outdated CSI and non-linearity of EH circuit. The experimental results show that when the packet length is between 200~250 b and the source transmit power is 35 dBm, the AoI-EE achieves the optimum.

Key words: nonlinear age of information (AoI), nonlinear energy harvesting(EH), outdated channel state information, Nakagami-m ,