A Stochastic Hybrid Systems Approach to the Joint Distribution of Ages of Information in Networks

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 destination nodes (or 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 system's continuous and discrete states by a piecewise linear SHS with linear reset maps. We start our analysis by deriving first-order linear differential equations for the temporal evolution of both the joint moments and the joint moment generating function (MGF) of all possible pairwise combinations formed by the age vector components. We then derive the conditions under which the derived differential equations are asymptotically stable. Finally, we apply our framework to characterize the stationary joint MGF in a multi-source updating system under several queueing disciplines including non-preemptive and source-agnostic/source-aware preemptive in service queueing disciplines.