Joint Distribution of Ages of Information in Networks

Abstract

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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">age vector</i> ) models the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">continuous</i> state of the system. While the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">marginal</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">joint</i> 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.