## 3 Traffic Multi-Maps Generation Based on Disjoint Path Flows
### Traffic Model  
The topological traffic representation of the urban network \(\Theta\) is described by a graph \(\Theta=G(N,E)\) formed by \(N\) nodes \(\{\eta_{n}\}\) and \(E\) edges (links) \(\{e_{e}\}\) that connect them. The edge \(e_{e}\) connects the nodes \(\eta_{i}\) and \(\eta_{j}\), \(e_{e}=(\eta_{i},\eta_{j})\), with a weight \(\omega_{e}\). Edge weight is the intrinsic minimum cost of traversing it. It depends on physical and logical edge attributes, such as edge length \(l_{e}\), edge max speed \(s_{e}\), and edge tolls \(\tau_{e}\). Edge weight is calculated considering the edge free-flow travel time \(t_{e}^{FF}(x_{e})\) (1) with no tolls where the vehicle assumes that it is the only user of the link [5]. \[\omega_{e}=t_{e}^{FF}(x_{e})=\frac{l_{e}}{s_{e}} \tag{1}\]  
Traffic demand is composed of \(V\) vehicles \([v_{a}]\) that make trips between origin and destination nodes (O/D). The \(D\) unique O/D pairs in the demand form traffic commodities. Traffic flow \(x^{d}\) is formed by the vehicles that belong to the same commodity d, that is, moving from the same source node \(\eta_{i}\) to the destination node \(\eta_{j}\). Sources and destinations may refer to physical network nodes, traffic area zones, or traffic centroids. Total traffic flow (demand) in the network is \(X=\{x^{d}\},\forall d\in D\). Traffic flow \(x^{d}\) may be routed through any of the \(P^{d}\) feasible paths in the network connecting the source node \(\eta_{i}\) to the destination node \(\eta_{j}\). Any path \(p_{i}^{d}\in P^{d}\) is an ordered sequence of connected edges with no loops. Traffic routing selects either statically or dynamically, some of these paths \(\{p_{i}^{d}\}\) and allocate part of flow \(x_{i}^{d}\in x^{d}\) to each path \(p_{i}^{d}\), so that \(\sum\{x_{i}^{d}\}=x^{d}\). From the edge perspective, the traffic flow \(x_{e}\) that traverses an edge \(e_{e}\) is formed by the traffic contribution of all the paths that traverse this edge (2). \[x_{e}=\sum\{x_{i}^{d}\}_{\forall e_{e}\in\{p_{i}^{d}\}} \tag{2}\]  
The travel time \(t_{e}(x_{e})\) to traverse edge \(e_{e}\) depends highly on this traversing flow and is expressed by a volume-delay function \(\Psi\) (VDF). The American Bureau of Public Roads (BPR) [64] defined VDF models to reflect the impact of edge occupancy being (3) the most widely used, where \(W_{e}\) is the link capacity [11] and \(\alpha\) and \(\beta\) are predefined constants (typical values of \(\alpha=0.15\) and \(\beta=4.0\)). This BPR link-cost model is widely used as a basis for traffic assignment. \[t_{e}(x_{e})=\omega_{e}(1+\alpha\frac{x_{e}}{W_{e}})^{\beta} \tag{3}\]  
Cost \(c_{e}\) of an edge includes both the travel time and the eventual tolls \(\tau_{e}\) that apply to the edge (4). Also, the free-flow cost of the edge is considered \(c_{e}^{FF}\). \[c_{e}=t_{e}(x_{e})+\tau_{e},\forall e_{e}\in p_{i}^{d} \tag{4}\]  
\[c_{e}^{FF}=\omega_{e}+\tau_{e},\forall e_{e}\in p_{i}^{d} \tag{5}\]  
Travel time \(t_{i}^{d}\) over path \(p_{i}^{d}\) is formed by the aggregation of the travel time at all the traversed edges (6):  
\[t_{i}^{d}=\sum t_{e}(x_{e}),\forall e_{e}\in p_{i}^{d} \tag{6}\]  
The generalized cost model of a path \(c_{i}^{d}\) considers both the travel time and the possible tolls associated with the edges that are included (7) as described in [25]. \[c_{i}^{d}=\sum c_{e}=\sum t_{e}(x_{e})+\sum\tau_{e},\forall e_{e}\in p_{i}^{d} \tag{7}\]  
We refer to the free-flow cost as \(ffc_{i}^{d}\) (8). \[ffc_{i}^{d}=\sum\omega_{e}+\sum\tau_{e},\forall e_{e}\in p_{i}^{d} \tag{8}\]Route calculation is a complex process executed by the routing agents, which may be located either in the vehicle or in a remote ITS framework. Each routing agent \(v_{a}\) calculates the best route to use for a trip of a commodity in terms of minimal route cost. This calculus is made for every trip and depends on the routing algorithm \(\mathbb{R}\), the origin and destination nodes (commodity d); the network map that it uses \(\mu_{a}\); and the current traffic status and load in the network.