This paper considers a status update communication system consisting of a source-destination link with timeliness requirements. First, we study the properties of a sample path of the age of information (AoI) process at the destination. Under the assumption of ergodicity, we obtain a general formula of the stationary distribution of the AoI. We relate this result to a discrete time queueing system and provide a general expression of the generating function of AoI in relation with the system time and the peak age of information (PAoI). Furthermore, we consider the first-come-first-served (FCFS) Geo/Geo/1 queue and we obtain closed-form expressions of the generating functions and the stationary distributions of the AoI and the PAoI. We built upon these results to provide a methodology for analyzing general non-linear age functions for this type of systems.