  A source provides status updates to monitors through a network with state
defined by a continuous-time finite Markov chain. An age of information (AoI)
metric is used to characterize timeliness by the vector of ages tracked by the
monitors. Based on a stochastic hybrid systems (SHS) approach, first order
linear differential equations are derived for the temporal evolution of both
the moments and the moment generating function (MGF) of the age vector
components. It is shown that the existence of a non-negative fixed point for
the first moment is sufficient to guarantee convergence of all higher order
moments as well as a region of convergence for the stationary MGF vector of the
age. The stationary MGF vector is then found for the age on a line network of
preemptive memoryless servers. From this MGF, it is found that the age at a
node is identical in distribution to the sum of independent exponential service
times. This observation is then generalized to linear status sampling networks
in which each node receives samples of the update process at each preceding
node according to a renewal point process. For each node in the line, the age
is shown to be identical in distribution to a sum of independent renewal
process age random variables.
