Energy and delay trade-offs of end-to-end vehicular communications using a hyperfractal urban modelling

We characterise trade-offs between the end-to-end communication delay and the energy in urban vehicular communications with infrastructure assistance. Our study exploits the self-similarity of the location of communication entities in cities by modelling them with a hyperfractal model which characterises the distribution of mobile nodes and relay nodes by a fractal dimension d_F and d_r, both larger than the dimension of the embedded map. We compute theoretical bounds for the end-to-end communication hop count considering two different energy-minimising goals: either total accumulated energy or maximum energy per node. Let δ > 1 be the attenuation factor in the street, we prove that when we aim to a total energy cost of order n^{(1−δ)(1−α)}, the hop count for an end-to-end transmission is of order n^1−α/(d_F⁻¹⁾, with α < 1 is a tunable parameter. This proves that for both goals, the energy decreases as we allow choosing routing paths of higher length. The asymptotic limit of the energy becomes significantly small when the number of nodes becomes asymptotically large. A lower bound on the network throughput capacity with constraints on path energy is also given. We show that our model fits real deployments where open data sets are available. The results are confirmed through simulations using different fractal dimensions in a Matlab simulator.