## 4 Numerical Example #2: Grid Network  
In this section, the proposed day-to-day dynamic process model of route choice and signal control is further analyzed through numerical tests carried out on the ten-link network shown in Figure 9. Three of the network nodes are assumed to represent intersections controlled by traffic signals operating on a two-phase plan with a fixed cycle of 90 s. Each of the six routes connecting the origin to the destination traverses at least one signal-controlled intersection. The functional characteristics of the ten network links are specified in Table 1. Note that the capacities of links 3-8 are determined endogenously as the product of the respective saturation flows and green splits, while those of the remaining links (1, 2, 9, and 10) are fixed exogenously. Figure 9: Grid network with three signal-controlled intersections. Travel times of unsignalized links are modeled by means of the classical BPR function [38] as follows:  
\[T=T_{0}\Big{(}1+0.15\,x^{4}\Big{)} \tag{73}\]  
where \(T_{0}\) represents travel time under free-flow conditions and \(x\) is the link's flow-to-capacity ratio. On the other hand, travel times of signalized links are computed by adding to their free-flow value a delay term representing the effect of signal control under congestion. To this end, we employ a \"sheared\" delay function, which extends the well-known Webster formula [39] to cover flow-to-capacity ratios greater than unity, thus accounting for the effect of temporary oversaturation on overall vehicular delay. Among the several functions of this type documented in the literature, we adopt in this study the expression commonly known as the \"Canadian formula\" [40]  
\[\overline{d}=\frac{c(1-G)^{2}}{2[1-G\cdot\min(x,1)]}+900\tau\bigg{[}x-1+\sqrt {(x-1)^{2}+\frac{4x}{\tau QG}}\bigg{]} \tag{74}\]  
where for a given signal-controlled link  
\(\overline{d}\) is the average overall vehicular delay (s);  
\(c\) is the signal cycle length (s);  
\(G\) is the green split;  
\(x\) is the flow-to-capacity ratio;  
\(\tau\) is the duration of the arrival overflow (h);  
\(Q\) is the saturation flow (veh./h). Note that the first term of (74) coincides with the average uniform delay of [PERSON]'s formula and provides an estimate of the delay induced by the cyclic nature of the traffic signal operation. The second component of (74), instead, differs from [PERSON]'s incremental term in that it represents the delay caused by sustained oversaturation conditions in addition to the effect of \"instantaneous\" overflows arising from the stochastic nature of vehicular arrivals. In the numerical example described in this section, expression (74) is applied to the signal-controlled links of the test network setting \(c=90\) s, \(\tau=0.25\) h, and the saturation flow values shown in Table 1. Unlike these parameters, which are fixed exogenously, \(G\) and \(x\) are endogenous variables. Since the analysis presented in the following focuses on the relationship between the values of \(\gamma\) ensuring fixed-point stability and the route choice parameter \(\theta\), the values of \(\alpha\) and \(\beta\) were kept fixed in the numerical tests carried out on the grid network. According to [PERSON] [41], plausible values for these parameters lie in the ranges 0.4-0.6 and 0.4-0.8, respectively; based on this indication, the midpoints of such intervals (\(\alpha=0.5;\ \beta=0.6\)) are assumed as representative values in our analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Link** & **Free-Flow Travel Time** & **Saturation Flow** & **Capacity** \\  & **(min.)** & **(veh./h)** & **(veh./h)** \\ \hline
1 & 5 & – & 1500 \\
2 & 5 & – & 1500 \\
3 & 5 & 1400 & – \\
4 & 12 & 2000 & – \\
5 & 5 & 1600 & – \\
6 & 5 & 1400 & – \\
7 & 12 & 2000 & – \\
8 & 5 & 1500 & – \\
9 & 5 & – & 1500 \\
10 & 5 & – & 1500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test network data. It is apparent in Figures 10-12 that the range of \(\gamma\) values ensuring fixed-point stability tends to shrink progressively as \(\theta\) grows.