## 3 Model and Preliminaries
### Traffic Information Collection  
In this paper, we proposed a traffic information collection system via VANET for urban areas. In this system, each vehicle was assumed to have a GPS receiver and VANET equipped to report its location. This assumption is reasonable as currently more and more vehicles are equipped with such devices. Considering that there are always traffic lights at the intersection of urban roads, we used vehicles stopped at red lights to collect traffic information by checking the location of all vehicles inside their VANET communication range periodically.  
As shown in Figure 1, when the traffic light turned red for the east-west direction, the first vehicle stopped at the west side was selected as the traffic information collector (TIC). If there were no vehicle stops at the west side, the first vehicle stopped at the east side would be the TIC. In the example of Figure 1, vehicle \(V11\) is the TIC and collects the traffic information during the red light period.  
1. When the traffic light turns to red at \(T0\), \(V11\) broadcasts traffic information collection request.
2. All of the vehicles in the communication range \(R\) of \(V11\) will report their locations to \(V11\) after receiving the request from \(V11\).
3. V11 catalogs the vehicles to four vehicle sets according to the location. They are marked as \(Vset(E,\ T0)\), \(Vset(S,\ T0)\), \(Vset(W,\ T0)\), and \(Vset(N,\ T0)\).
4. After time \(\tau<R/V_{max}\), V11 collects the traffic information again according to Steps 1-3 and obtains \(Vset(E,\ T0+\tau)\), \(Vset(S,\ T0+\tau)\), \(Vset(W,\ T0+\tau)\), \(Vset(N,\ T0+\tau)\). \(V_{max}\) is the maximum allowed velocity. Time \(\tau<R/V_{max}\) will let all vehicles running towards the intersection be detectable at time \(T0+\tau\).  
For example:  
If \(R=150\) m and \(V_{max}=20\)m/s, \(\tau=5\) s can be used as \(5<\frac{150}{20}=7.5\). The maximal length a vehicle can run during \(\tau\) is \(\frac{20}{5}\times 5\) s \(=100\) m \(<150\) m. Then, no vehicle entering the intersection at T0 can run outside the communication range of V11 and be detectable at \(T0+\tau\).
5. The vehicles' set run from road \(A\) to road \(B\) is calculated by formula: \(Vset(A\to B,\ T0+\tau)=Vset(A,\ T0)\cap Vset(B,\ T0+\tau)\).  
For example:  
As shown in Figure 1, the traffic output from \(S\) to \(E\) from \(T0\) to \(T0+\tau\) is:  
\[Vset(S\to E,\ T0+\tau)=Vset(S,\ T0)\cap Vset(E,\ T0+\tau)\] \[=\{V1,V2,V3,V4\}\cap\{V1,V8,V9,V10\}=\{V1\}\]
6. The TIC calculates the traffic output of each road with time interval \(\tau\) until the traffic light for the east-west direction turns green. As the traffic light for the south-north direction turns red, the first vehicle stopped at the north or south side will be selected as the TIC and collect traffic information continuously.  
As a part of the advance travel information system (ATIS), the TIC will send out the collected traffic information via VANET for applications such as traffic prediction. In normal urban traffic conditions, there should always be vehicles stopped at the red light to act as the TIC. If there is no vehicle stopped at the red light to be the TIC, the traffic information of the last time slot will be used. This is acceptable as it normally happens in very low traffic density cases and real-time traffic information is not important.  
With the collected traffic information, the traffic output of each road section \(j\) can be calculated by:  
\[Q^{(h)}{}_{ij}(t)=\sum_{n}\sum_{m}Vset(j\to j_{m},T+n\tau),\]  
where \(T+n\tau\in(t-1,t]\), \(j\) is the \(h\) order neighbor of road \(i\), \(j_{m}\) is the \(h-1\) order neighbor of road \(i\) (\(j_{m}\) is \(i\) when \(h=1\)), \(m\) is the total number of \(j_{m}\) which are downstream \(j\).