Phase Transition of Message Propagation Speed in Delay-Tolerant Vehicular Networks

Delay-tolerant network (DTN) architectures have recently been proposed as a means to enable efficient routing of messages in vehicular area networks (VANETs), which are characterized by alternating periods of connectivity and disconnection. Under such architectures, when multihop connectivity is available, messages propagate at the speed of radio over connected vehicles. On the other hand, when vehicles are disconnected, messages are carried by vehicles and propagate at vehicle speed. Our goal in this paper is to analytically determine what gains are achieved by DTN architectures and under which conditions, using the average message propagation speed as the primary metric of interest. We develop an analytical model for a bidirectional linear network of vehicles, as found on highways. We derive both upper and lower bounds on the average message propagation speed by exploiting a connection with the classical pattern-matching problem in probability theory. The bounds reveal an interesting phase transition behavior. Specifically, we find out that, below a certain critical threshold, which is a function of the traffic density in each direction, the average message speed is the same as the average vehicle speed, i.e., DTN architectures provide no gain. On the other hand, we determine another threshold above which the average message speed quickly increases as a function of traffic density and approaches radio speed. Based on the bounds, we also develop an approximation model for the average message propagation speed that we validate through numerical simulations.