Modeling the Expected Data Collection Time for Vehicular Networks Using Random Walks on a Torus

We are interested in modeling the total expected data collection time for a set of cars (or other mobile nodes) to aggregate the information that they are each carrying to a single vehicle. Data collection processes between cars - or other mobile nodes - can be modeled as random walks of particles on a graph, and in this model, data is transferred between data-possessing cars once they encounter each other on the same node of the graph. We analyze this process in the special case in which this graph is a square grid torus graph. Recursion occurs in the system because once an encounter between multiple cars that possess data occurs, the system becomes analogous to simpler systems with the same graph and with fewer cars that possess data. We make approximations for the total data collection time using this recursive property.