## 4 System Modelling
### Traffic Flow and Travel Delay  
In DIFTOS, time is represented as a set of successive time slots, \(T=\{t_{1},t_{2},\ldots,t_{n}\}\), we have used the flow speed on the road link within the time slots to indicate the traffic condition of that road, \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Description** \\ \hline \(G=(I,\Omega,V)\) & Road network graph that represents the city map \\ I & Set of all vertices in the road network graph \\ \(\Omega\) & Set of all edges in the road network graph \\ \(V\) & Set of all vehicles in the city \\ \(r_{i,j}\) & The road segment from intersection \(i\) to intersection \(j\) \\ \(T=\{t_{1},t_{2},\ldots,t_{n}\}\) & Set of successive time slots \\ \(x_{i,j}^{t}\) & The mean of the speed of all vehicles driving within the traffic flow on the road \\ \(\delta_{ij}^{t}(v_{i})\) & Segment \(r_{i,j}\) within the time slot \(t\) \\ \(\delta_{ij}^{t}(v_{i})\) & The velocity of the vehicle \(v_{i}\) in road segment \(r_{i,j}\) during time slot \(t\) \\ \(d_{i,j}\) & Travel delay required to cross road segment \(r_{i,j}\) \\ \(l_{i,j}\) & The length of the road segment \(r_{i,j}\) \\ \(\omega_{j}\) & The waiting delay at intersection \(j\) \\ \(C_{i,j}\) & The capacity of the road segment \(r_{i,j}\) \\ \(R_{i,j}(t)\) & The number of available positions in the road segment \(r_{i,j}\) during time slot \(t\). \\ \(w_{ij}(t)\) & The weight of road segment \(r_{i,j}\) during time slot \(t\) \\ \(P=\{r_{s_{r}},\ldots,r_{d}\}\) & The path yielded by connecting the road segments from \(r_{s}\) to \(r_{d}\) \\ \(\delta_{\Omega,T}\) & Road reservation matrix of the road set \(\Omega\) during the period \(T\) \\ \(w_{ij}^{\theta}(t)\) & The quota limit of the road \(r_{i,j}\) during time slot \(t\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations summary. where the traffic condition of a road segment \(r_{i,j}\) in a given time slot \(t\) is denoted as \(x_{i,j}^{t}\) and is defined as the mean of the speed of all vehicles driving within the traffic flow on that road segment within the time slot \(t\)[23]. \[x_{i,j}^{t}=E\Big{(}\vartheta_{i,j}^{t}(v_{i})\Big{)} \tag{1}\]  
where \(\vartheta_{i,j}^{t}(v_{i})\) is the speed of the vehicle \(v_{i}\) in road \(r_{i,j}\) during time slot \(t\). To estimate the mean speed of traffic flow, the average of all vehicles speed on the road segment during the time slot \(t\) is calculated. The average is calculated over different speeds of all vehicles. In this context, we have made the assumption that during each time slot, the traffic condition on a segment is uniform. The traffic statistics of the mean speed for each road segment is calculated through the vehicles' travel experience reports; it is also estimated using the vehicular traffic measurement at loop detectors in network roads [24]. Let \(d_{i,j}\) denote the link delay time, which is the rounded number of time slots required for a vehicle to cross the road segment \(r_{i,j}\) that has the length \(l_{i,j}\) and the traffic condition \(x_{i,j}^{t}\), note that \(d_{i,j}\) also includes intersection waiting delay \(\omega_{j}\) that vehicles take to exit intersection \(j\) before crossing to the next road segment \(r_{j,k}\). Thus, the link travel delay for road segment \(r_{i,j}\) is computed in (2). \[d_{i,j,the}=\frac{l_{i,j}}{x_{i,j}^{t}}+\omega_{j} \tag{2}\]  
During each time slot \(t_{x}\), each road segment \(r_{i,j}\) is associated with a weight, denoted as \(w_{i,j}(x)\), that changes dynamically in each time slot according to the road segments reservations, and it can be computed using (3):  
\[w_{i,j}(x)=C_{i,j}-R_{i,j}(x) \tag{3}\]  
where: \(R_{i,j}(x)\): is the number of reserved positions in the road segment \(r_{i,j}\) during time slot \(x\).