  The idea behind the recently introduced "age of information" performance
measure of a networked message processing system is that it indicates our
knowledge regarding the "freshness" of the most recent piece of information
that can be used as a criterion for real-time control. In this foundational
paper, we examine two such measures, one that has been extensively studied in
the recent literature and a new one that could be more relevant from the point
of view of the processor. Considering these measures as stochastic processes in
a stationary environment (defined by the arrival processes, message processing
times and admission controls in bufferless systems), we characterize their
distributions using the Palm inversion formula. Under renewal assumptions we
derive explicit solutions for their Laplace transforms and show some
interesting decomposition properties. Previous work has mostly focused on
computation of expectations in very particular cases. We argue that using
bufferless or very small buffer systems is best and support this by simulation.
We also pose some open problems including assessment of enqueueing policies
that may be better in cases where one wishes to minimize more general
functionals of the age of information measures.
