Age of information and energy efficiency of amplify-and-forward relay-assisted Internet of Things with nonlinear energy harvesting and imperfect channel state information over Nakagami-m$$ m $$ fading channels

The emergence of the fifth generation of communication networks (5G) has paved the way for the fast development of Internet of Things (IoT) as well as many real-time and mission-critical applications. Because the status update required to be as timely as possible, a new metric termed as age of information (AoI) was born with the ability of capturing the timeliness. However, in practical implementation the sensor is energy constrained and transmits very low-power signals, which yield high AoI so that the received status update can be stale or even outdated. To overcome this problem, this work proposes an IoT system model by integrating energy harvesting (EH), amplify-and-forward (AF) relay cooperation, and short packet communication (SPC) techniques. In the proposed scheme, the relay that is connected to fixed power grid, transfers energy radio frequency signals to charge the sensor and assists forwarding status update by using time switching scheme. The employment of SPC overcomes the freshness loss caused by relay cooperation and EH. Then, this work investigates the AoI and energy efficiency of the proposed scheme under finite block length communications over Nakagami-m$$ m $$ fading channels by specially considering channel state information (CSI) and nonlinear EH model. Under these practical constraints, the tractable AF relay transmission model and the end-to-end approximated signal-to-noise ratio are found. The end-to-end block error probability of finite block length transmission is derived, too. Third, under the simultaneous consideration of nonlinear EH and outdated CSI, the statistical descriptions of the time duration for the sensor fully charging its battery are derived as well as the ones of the time intervals of update packet delivery. As a result, the average AoI is achieved with closed-form expression and the tradeoff model of AoI to energy efficiency is established by simultaneously exploiting outdated CSIs and nonlinearity of EH circuit. The presented numerical analysis exploits the impact of the outdated CSIs and nonlinear EH model and gives the insights of such model.