Age Distribution in Arbitrary Preemptive Memoryless Networks

We study the probability distribution of age of information (AoI) in arbitrary networks with memoryless service times. A source node generates packets following a Poisson process, and then the packets are forwarded across the network in such a way that newer updates preempt older ones. This model is equivalent to gossip networks that were recently studied by Yates, and for which he obtained a recursive formula allowing the computation for the average AoI. In this paper, we obtain a very simple characterization of the stationary distribution of AoI at every node in the network. This allows for the computation of the average of an arbitrary function of the age, such as the age-violation probabilities. Furthermore, we show how our simple characterization can yield substantial reductions in the computation time of average AoIs in some structured networks. Finally, we describe how it can yield faster and more accurate Monte Carlo simulations estimating the average AoI, or the average of an arbitrary function of the age.