# Analyzing Demand with Respect to Offer of Mobility
## 4 Model and Analyzer
### Paper Aims and Structure  
We considered two parameters that affect drop-offs and pick-ups at a transfer stop \(s_{tra}\): stop density and vehicle line frequency which are formal defined as follows. **Stop density. ** Since the number of stops near a stop \(s\) increases, then the probability that \(s\) is a transfer stop \(s_{tra}\) is higher since more commuters can select \(s\) to transfer according with the next trip by moving to a close stop in multi-stage trips. For example, Figure 5 shows the differences about the stop density in stops A and B.  
Stops/stations may have different mobility modalities (e.g., bus, tram, train), in a modality set \(M\), and may have similar motivations as transfer. For example, a train station is typically a strong point of start in the city to go to work/study, and to go back, compared to a bus/tram stop, which in some cases are city transfer. Thus, we defined a modality weight vector \(W_{md}[1,\dots,|M|]\), where stops/stations with higher modality weight have a higher potential for being a transfer \(s_{tra}\). For example, if \(W_{md}=[0.9,0.05,0.05]\) and \(W_{md}[train]=0.9\) indicates that, a train station has a higher potential to be a transfer one, compared to a stop (\(W_{md}[bus]=0.05\)). Given a circle \(c(s,r)\) with stop \(s\) as its center and the related radius \(r\), we compute the probability that \(s\), in a time interval in an interval vector \(H_{t\in\{mug\_{eff}\}}\), is a transfer one by choosing other stops in \(c(s,r)\), as formally described in the following definition. Figure 4: In the left, an example of single-stage trip having a beginning \(s_{bgg}\) and final \(s_{fin}\) stop. In the right, an example of multi-stage trip having transfer stops \(s_{tra}\) also. Figure 5: Density of stops near two sample stops **A** and **B**. **Definition 1**. : _Let \(P_{tra}^{(stp)}(s,t,k)\) denote the probability that stop \(s\), in interval time \(H(t\in\{mng,aft\})[k]\), is a transfer one by choosing other stops in circle \(c(s,r)\) in interval time \(H(t)[k]\). It is then calculated as in (4)._  
\[P_{tra}^{(stp)}(s,t,k)=e^{(-\frac{1}{s_{stp}(s,t,k)})} \tag{4}\]  
_It is noted that \(\alpha_{stp}(s,t,k)\) is the stop level of stop \(s\) which indicates the normalized weighted sum of the number of stops in circle \(c(s,r)\) in time interval \(H(t)[k]\) and calculated as in (5)._  
\[\alpha_{stp}(s,t,k)=\frac{u_{tra}^{(stp)}(s,t,k)-u_{tra}^{(stp)}(min)}{u_{tra}^ {(stp)}(max)-u_{tra}^{(stp)}(min)} \tag{5}\]  
_where \(u_{tra}^{(stp)}(s,t,k)\) is the stop weighted sum of stop \(s\) in \(H(t)[k]\) and calculated as in (6)._  
\[u_{tra}^{(stp)}(s,t,k)=\sum_{m\in M}^{|M|}N_{stp}(m,t,k).W_{stp}[m] \tag{6}\]  
\(N_{stp}(m,t,k)\) _is the number of stops with transfer modality \(meM\) in circle \(c(s,r)\) with at least one passing vehicle trip in time interval \(H(t)[k]\). Furthermore, \(u_{tra}^{(stp)}(min)\) and \(u_{tra}^{(stp)}(max)\) respectively denote the minimum and maximum stop weighted sums, considering all stops in studied area \(A\)._  
**Vehicle line frequency. ** During a time interval, a higher number of lines passing near a stop/station indicates that, compared to other stops, it is more likely to be used by the commuters as a transfer \(s_{tra}\). For example, _Indipendenza XXVII Aprile_ stop (operated by ATAF) as one of the most crowded stops in the city of Florence with more than _nine_ (or 18 in both directions) lines, is highly used as a transfer stop. Formally, we have:  
**Definition 2**. : _Let \(P_{tra}^{(lin)}(s,t,k)\) denote the probability that \(s\) is a transfer stop by choosing other lines of \(s\) in interval time \(H(t)[k]\).