Modeling vehicular traffic networks. Part I

We propose three models for the traffic of vehicles within a network formed by sites (cities, car-rental agencies, parking lots, etc.) and connected by two-way arteries (roads, highways), that allow forecasting the vehicular flux in a sequence of n consecutive steps, or units of time. An essential approach consists in using, as an a priori information, previous observations and measurements. The formal tools used in our analysis consists in: (1) associating a digraph to the network where the edges correspond to arteries and the vertices with loops represent the sites. (2) From a distribution of vehicles within the network, we construct a matrix from which we derive a stochastic matrix (SM). This matrix becomes the generator of the evolution of the traffic flow. And (3), we use the Perron-Frobenius theory for a formal analysis. We investigate three models: (a) a closed network with conserved number of vehicles; (b) to this network we add an influx and an outflux of vehicles to picture an open system. And (c), we construct a nonlinear model whose formal structure exhibits the existence of several (L) stationary states for the distribution of vehicles at each site, that alternate cyclically with time. Each state represents the traffic for L different moments. These models are hybridized and compared numerically to the effective vehicular traffic in a sector of the city of Tigre, localized in the province of Buenos Aires, Argentina. The empirical data and the traffic modelization are presented in a following paper, referred as Part II.