Instantaneous multihop connectivity of one-dimensional vehicular ad hoc networks with general distributions of communication nodes

Connected and automated vehicle technologies hold great promises for improving the safety, efficiency, and environmental impacts of the transportation sector. In this study we are concerned with multihop connectivity of instantaneous vehicular one-dimensional ad hoc networks (VANETs) formed by connected vehicles along a communication path in a road network with given either vehicle locations or traffic densities, market penetration rates, and transmission ranges. We first define a new random variable for the location of the end node of a communication chain, which is a discrete random variable with given vehicle locations and a mixed random variable with given traffic densities. Then recursive, iterative, or differential equation models of instantaneous multihop connectivity between two communication nodes are derived from the relationships between end node probability mass or density function and connectivity. Assuming a simple communication model, the new models are applicable for general distribution patterns of vehicles and communication nodes, including non-evenly placed vehicles and nonhomogeneous Poisson distributions of nodes. With given vehicle locations, the computational cost for this new model is linear to the number of vehicles; with given traffic densities, we derive a new closed-form connectivity model for homogeneous Poisson distributions of communication nodes and an approximate closed-form model when distribution patterns of communication nodes are given by spatial renewal processes. We then apply the models to evaluate impacts on connectivity of traffic patterns, including shock waves, and road-side stations. The connectivity model could be helpful for designing routing protocols in VANETs and developing their applications in transportation systems.