  We consider the age of information in G/G/1/1 systems under two service
discipline models. In the first model, if a new update arrives when the service
is busy, it is blocked; in the second model, a new update preempts the current
update in service. For the blocking model, we first derive an exact age
expression, then we propose two simple to calculate upper bounds for the
average age. The first upper bound assumes the interarrival times to have
log-concave distribution. The second upper bound assumes both the interarrivals
and service times to have log-concave distribution. Both upper bounds are tight
in the case of M/M/1/1 systems. We show that deterministic interarrivals and
service times are optimum for the blocking service model. In addition, using
the age expression for G/G/1/1 systems, we calculate average age expressions
for special cases, i.e., M/G/1/1 and G/M/1/1 systems. Next, for the preemption
in service model, we first derive an exact average age expression for G/G/1/1
systems. Then, we propose a simple to calculate upper bound for the average
age. In addition, similar to blocking discipline, using the age expression for
G/G/1/1 systems, we calculate average age expressions for special cases, i.e.,
M/G/1/1 and G/M/1/1 systems. Average age for G/M/1/1 can be written as a
summation of two terms, the first of which depends only on the first and second
moments of interarrival times and the second of which depends only on the
service rate. In other words, interarrival and service times are decoupled. We
show that deterministic interarrivals are optimum for G/M/1/1 systems. On the
other hand, we observe for non-exponential service times that the optimal
distribution of interarrival times depends on the relative values of the mean
interarrival time and the mean service time.
