## 3 Traffic Model
### Spatial Traffic Variation  
In order to be able to control the spatial traffic variation over time, we focus on the sets of time periods \(T_{low}\), \(T_{inc}\), \(T_{high}\), and \(T_{dec}\). More precisely, we consider their circular shifting (as explained below), leaving the ratio \(A^{a,b}_{min}/A^{a,b}_{max}\) fixed for all \((a,b)\in V^{2}\). Wide-range realistic backbone networks often spread over more than one time zone. To account for the temporal shift in traffic due to effects arising from this, we introduce a Time Zone Matrix (TZM)  
\[\Theta=\begin{pmatrix}\theta_{0,0}&\cdots&\theta_{0,|V|-1}\\ \vdots&\ddots&\vdots\\ \theta_{|V|-1,0}&\cdots&\theta_{|V|-1,|V|-1}\end{pmatrix}. \tag{2}\]  
The value \(\theta_{a,b}\in\mathbb{Z}\) for \((a,b)\in V^{2}\) determines the number of time periods the demand curve for node pair \(a,b\) has to be shifted temporally due to the fact that \(a\) and \(b\) lie in time zones differing from a reference time zone. The matrix \(\Theta=0\) is used for a network located completely within the reference time zone.  
Figure 1: Temporal traffic model with exemplary relative demand values and even number of elements in the set of time periods \(T\) covering a day.  
In order to introduce controlled changes of spatial traffic distribution, we define a Spatial Variation Matrix (SVM)  
\[\Gamma=\begin{pmatrix}\gamma_{0,0}&\cdots&\gamma_{0,|V|-1}\\ \vdots&\ddots&\vdots\\ \gamma_{|V|-1,0}&\cdots&\gamma_{|V|-1,|V|-1}\end{pmatrix}, \tag{3}\]  
similarly to the TZM, but with values \(\gamma_{a,b}\in[-1,1]\) for \((a,b)\in V^{2}\). Additionally, we introduce a time displacement parameter \(k\in\{0,\ldots,|T|-1\}\). Then for \((a,b)\in V^{2}\) the value  
\[\zeta_{a,b}=\left\lfloor k\cdot\gamma_{a,b}\right\rfloor, \tag{4}\]  
defines the number of time periods each demand curve should be shifted temporally due to spatial traffic variation (the function \(\left\lfloor x\right\rceil\) stands for rounding of \(x\) to the nearest integer value). More precisely (and taking into account both \(\theta_{a,b}\) and \(\zeta_{a,b}\)), the spatially varied traffic demand \(d_{a,b}(t)\) for \(t\in T\) is computed as  
\[d_{a,b}(t)=\tilde{d}_{a,b}(t-\theta_{a,b}-\zeta_{a,b}+m\cdot|T|), \tag{5}\]  
where the period multiplicity \(m\in\mathbb{Z}\) is defined as  
\[m=\left\lfloor\frac{t-\theta_{a,b}-\zeta_{a,b}}{|T|}\right\rfloor. \tag{6}\]  
Figure 2 shows the effect of the time displacement parameter \(k\) on an exemplary 3-node network with demands for three node pairs (with zero TZM) over a day with the time granularity \(\Delta t\) set to \(1\,\mathrm{h}\) for simplicity. Initially, these single demands follow curves as defined in Section 3.1 (discretized), with different values for \(A^{a,b}_{max}\) and \(A^{a,b}_{min}\) (21, 28, 42 and 3, 4, 6 units, respectively), which results in a total traffic curve of the same type (Figure 2a). For spatial variation, the single demand curves are shifted according to the SVM  
\[\Gamma=\begin{pmatrix}0&-0.4127&1\\ 0&0&-0.1415\\ 0&0&0\end{pmatrix}, \tag{7}\]  
so that for higher values of \(k\) the total traffic variation in the network appears as a distorted version of the initial total traffic curve (compare the red solid curve in Figure 2a with the red solid curves in Figure 2b-d). The total demand curve is different for different values of \(k\) and the same \(\Gamma\) despite the fact that the total amount of traffic over a day \(\sum_{t\in T}\sum_{(a,b)\in V^{2}}d_{a,b}(t)\) is equal for all cases.