Age of Gossip in Ring Networks With Non-Poisson Updates

We consider a network consisting of $n$ nodes connected in a ring formation and a source that generates updates according to a renewal process and disseminates them to the ring network according to a Poisson process. The nodes in the network gossip with each other according to a push-based gossiping protocol, and disseminate version updates. Gossip between two neighbors happens at the arrivals of renewal processes with finite mean and variance. All renewal processes and Poisson processes in the network are independent but not identically distributed. We consider both uni-directional ring networks and bi-directional ring networks. We use version age of information to quantify the freshness of information at each node. Prior work has used the stochastic hybrid systems (SHS) approach or a first passage percolation (FPP) approach to analyze ring networks with edges following identical Poisson processes. In this work, we use a sample-path backtracking approach to characterize the probabilistic scaling of the version age of information of an arbitrary node in the gossip network, where each edge follows an independent but not identically distributed renewal process. We show that the version age of information of any node in the network is stochastically equivalent to $\sqrt{n}$ at any time instant after the node has received its first update from the source.