  We study a general setting of status updating systems in which a set of
source nodes provide status updates about some physical process(es) to a set of
monitors. The freshness of information available at each monitor is quantified
in terms of the Age of Information (AoI), and the vector of AoI processes at
the monitors (or equivalently the age vector) models the continuous state of
the system. While the marginal distributional properties of each AoI process
have been studied for a variety of settings using the stochastic hybrid system
(SHS) approach, we lack a counterpart of this approach to systematically study
their joint distributional properties. Developing such a framework is the main
contribution of this paper. In particular, we model the discrete state of the
system as a finite-state continuous-time Markov chain, and describe the coupled
evolution of the continuous and discrete states of the system by a piecewise
linear SHS with linear reset maps. Using the notion of tensors, we first derive
first-order linear differential equations for the temporal evolution of both
the joint moments and the joint moment generating function (MGF) for an
arbitrary set of age processes. We then characterize the conditions under which
the derived differential equations are asymptotically stable. The generality of
our framework is demonstrated by recovering several existing results as special
cases. Finally, we apply our framework to derive closed-form expressions of the
stationary joint MGF in a multi-source updating system under non-preemptive and
source-agnostic/source-aware preemptive in service queueing disciplines.
