# 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  
This section will describe all the concepts involved in evaluating service reliability. These concepts are derived from international standards for conceptual transport data models; specifically, Transmodel [31], a European reference data model for public transport information. The first concept that we took from this conceptual model is that of transport network. According to this model, a transport network is made up of all the entities that represent the places, roads and routes necessary to spatially describe vehicle operations. One such entity is points on the transport network. These points may represent places where various actions are performed; for example: point where passengers board or alight from the vehicle, timetable control points, points of interest, etc. Each of these activities is associated with a classification, and a point may belong to more than one classification. For example, it may be both a point where passengers board and alight and a timetable control point. The points of particular interest for the purposes of our system are those points where passengers board or alight from the vehicle, which will be called, generically, stops. Examples of these points are platforms at stations or bus stops. In formal terms, the transport network may be modelled using a directed graph \(G=(N,A)\), where \(N\) is the set of nodes, in which each node \(p\in N\) represents a point of the transport network. \(A\) is the set of directed arcs, wherein each arc \(a\in A\) represents the section of the route connecting two nodes \(p\), \(k\in N\), where \(p\) is the start node of the arc and \(k\) the end node of the arc. A route on the transport network is a route that runs systematically, starting at a node that is a stop on the route, the start node, and ending at another stop, the end node. The set of routes on the transport network is represented by \(R\) and each route on the network by \(r\), \(r\in R\). A route \(r\) is an ordered sequence of arcs such that \(a\in A\), thus establishing the passing order for each node on the route: \(r=(a_{1},a_{2},...,a_{n})\). The subscript of each arc of the sequence indicates the position in the ordered sequence, such that the start node on the first arc of the sequence, \(a_{1}\), is the start node of the route and the end node of the last arc of the sequence, \(a_{n}\), is the end node of the route. Given a route \(r\), defined by a sequence of \(n\) arcs, the number of nodes involved in the definition of a line will be \(n+1\), of which at least two must be a stop. To evaluate the punctuality of route \(r\), stop nodes are required. This subset of nodes of route \(r\) will be called route \(r\) set of stops, and will be represented by \(P_{r}\); they are an ordered set of stop nodes, in which the order of each node in the set \(P_{r}\) matches the order in which they are visited on route \(r\). A public transport operator's schedule, \(SP\), consists of a set \(SP=\{S_{i}\}\), where each element \(S_{i}\) is called a scheduling unit. Each scheduling unit, \(S_{i}\), is assigned a vehicle resource, a driver resource and a set of operations to be carried out at a scheduled date and time. If the schedule is not met, this means an inefficient use of the allocated resources. The operations included in a \(S_{i}\) scheduling unit may be of different types, but for the purposes of monitoring quality of service, line service operations are of greatest interest. A line service is defined as the completion of route \(r\) at a scheduled date and time. The set of line service operations on route \(r\), \(L_{r}=\{l_{i}\}\), is the set that comprises all line service operations of route \(r\) established in \(SP\); moreover, to represent only the line services on route \(r\) to be carried out during a period of time, \(T\), the notation \(L_{r,T}\) is used.