# System Proposal for Mass Transit Service Quality Control Based on GPS Data
## 3 Preliminary
### Conceptual and Formal Model of the Public Transport Network and Activity
For each line service, \(l_{i}\), passing times through each of the nodes of set \(P_{r}\) are scheduled, and that will be the information advertised to transport users. On line service \(l_{i}\) on route \(r\) with \(n\) stops, the timetabled passing times of said line service is an ordered cumulative sequence of times specifying what time, \(t\), the vehicle is scheduled to pass each of the stops on the route, i.e., \(TP_{li}=\{l_{1},l_{2},...,l_{n}\}\) where \(l_{1}\) indicates at what time the vehicle is scheduled to start the line service, \(t_{2}\) the arrival time at the second stop, and so on until the last time, \(t_{n}\), which indicates the scheduled time at which the vehicle will reach the last stop on the route. The passing times of line service \(l_{i}\) on route \(r\) vary and depend on various factors, some of which are related to transportation issues and some of which are not. More specifically, when schedulingpassing times at the various stops on a line service, it is necessary to consider the time required to complete each of the arcs that define the route and the time needed by passengers to board or alight from the vehicle at each stop on the route. The time spent completing the different arcs of the route depends on the vehicle speed, which in turn depends on variable factors such as weather conditions when the line service is running or the traffic status, which in turn is a factor dependent on the time of day the route is operated or the day type, e.g., working day, weekend or holiday, and even on the time of year. To draw this paragraph on conceptual models to a close, we would like to mention that they are implemented through a database that contains all the entities and the relationships between the entities representing the transport network, the deployed resources and the records reflecting all the facets of public transport operations. This database is called the transport database (TDB). As mentioned in the previous section on related studies, there are different criteria for assessing adherence to the timetabled passing times on the line services included in \(SP\). To decide on what criteria to use, the type of line service must be taken into account. For line services scheduled by frequency, typical of urban transport, the criterion used is the regularity with which the service is provided. However, for line services scheduled by timetable, typical of intercity transport, the criterion used is punctuality. As was stated in the previous section, to evaluate the punctuality of the line service, the OTP and RTV metrics are used. RTV is calculated using a basic metric called Arrival Delay (AD) obtained for each stop, \(p\), of a line service on route \(r\), \(AD_{li,p}\). AD is defined as the difference between the observed passing time through stop \(p\), \(OT_{li,p}\), and the scheduled passing time through that stop, \(ST_{li,p}\), when operating line service \(l_{i}\). Considering that the passing times for each stop are measured with respect to the time at which the line service began, this metric is calculated using the following expression:  
\[AD_{li,p}=OT_{li,p}-ST_{li,p}. \tag{1}\]  
The RTV metric is calculated using the following expression:  
\[RTV_{li}=n^{-1}\times\sum_{p=0}^{n}\frac{\left|OT_{li,p}-ST_{li,p}\right|}{OT_{ li,p}} \tag{2}\]  
As stated in the previous section, for line services operating by frequency, the criterion used is regularity. To evaluate regularity, the HV and EWT metrics are used. To calculate these metrics, first the Headway Ratio (HR) parameter has to be calculated using the following expression:  
\[HR_{li,k}^{p}=\left(\frac{H_{li,k}^{p}}{f_{li,k}^{p}}\right)\times 100 \tag{3}\]  
where \(f_{li,k}^{p}\) is thescheduled frequency between two line services \(l_{i}\), \(l_{i+1}\) at stop \(p\) on route \(r\) and \(H_{li,k}^{p}\)the frequency observed between both line services at stop \(p\) on route \(r\). The factor 100 represents a perfect match between observed and scheduled time.