  This paper considers the stationary distribution of the age of information
(AoI) in information update systems. We first derive a general formula for the
stationary distribution of the AoI, which holds for a wide class of information
update systems. The formula indicates that the stationary distribution of the
AoI is given in terms of the stationary distributions of the system delay and
the peak AoI. To demonstrate its applicability and usefulness, we analyze the
AoI in single-server queues with four different service disciplines: first-come
first-served (FCFS), preemptive last-come first-served (LCFS), and two variants
of non-preemptive LCFS service disciplines. For the FCFS and the preemptive
LCFS service disciplines, the GI/GI/1, M/GI/1, and GI/M/1 queues are
considered, and for the non-preemptive LCFS service disciplines, the M/GI/1 and
GI/M/1 queues are considered. With these results, we further show comparison
results for the mean AoI's in the M/GI/1 and GI/M/1 queues under those service
disciplines.
