  Timely status updates are crucial to enabling applications in massive
Internet of Things (IoT). This paper measures the data-freshness performance of
a status update system with an energy harvesting transmitter, considering the
randomness in information generation, transmission and energy harvesting. The
performance is evaluated by a non-linear function of age of information (AoI)
that is defined as the time elapsed since the generation of the most up-to-date
status information at the receiver. The system is formulated as two queues with
status packet generation and energy arrivals both assumed to be Poisson
processes. With negligible service time, both First-Come-First-Served (FCFS)
and Last-Come-First-Served (LCFS) disciplines for arbitrary buffer and battery
capacities are considered, and a method for calculating the average penalty
with non-linear penalty functions is proposed. The average AoI, the average
penalty under exponential penalty function, and AoI's threshold violation
probability are obtained in closed form. When the service time is assumed to
follow exponential distribution, matrix geometric method is used to obtain the
average peak AoI. The results illustrate that under the FCFS discipline, the
status update frequency needs to be carefully chosen according to the service
rate and energy arrival rate in order to minimize the average penalty.
