## 5 Graphical Analysis and Empirical Analysis
A rollback occurs where the slope difference of two curves, i.e., the difference between a(t) and d(t), is greater than the given threshold, since the rollback comparison is based on the point flow rate difference (i.e., absolute flow difference at a time instance, not cumulative difference) in the proposed model. For example, 100 veh/hr/ln arrival rate and 150 veh/hr/ln departure rate with 100 veh/hr/ln threshold does not warrant a rollback in the proposed model, even though 50% more vehicles (50 vehicles) would be generated over an hour. While difference in cumulative vehicle counts would be a good potential measure to detect changes in traffic conditions, it would require additional system measurements, such as counting the number of vehicles entering and exiting the network. Furthermore, the proposed system is associated with numerous rollbacks across the network during the simulation time period. Therefore, the impact of system overhead would need to be considered in collecting cumulative vehicle counts. As discussed earlier, downstream traffic information is transmitted to upstream _LPs_ in congested traffic conditions. To accomplish this transmission, the outflow rate on the exiting link of the upstream _LPs_ is controlled by changing speed of vehicles on the link. This is required as the simulation model (VISSIM(r)) has no direct means to throttle the flow rate on an unconstrained link. A question concerning selection of the speed to apply in order to meter the same number of vehicles with downstream _LPs_ is addressed in this section. Suppose the current 4 min average flow rate which is stored in the Space-Time Memory is calculated as an average of Flow A and Flow B in Figure 5. To process the average flow rate--Flow D (which is \(\frac{(FlowA+FlowB)}{2}\), i.e., the average of Flow A and Flow B) in upstream LP, new speed needs to be applied on Link C of the upstream LP in Figure 2. If the slope of \(\overline{OC}\) (average of slope between \(\overline{OA}\) and \(\overline{OB}\); that is average speed of \(V_{A}\) and \(V_{B}\)) is applied as a new speed, the actual flow rate which the upstream LP processes would be the flow rate of C (Flow C), which is higher than the flow rate of D. This can explain why constant speed can produce more traffic volume. For example, more traffic can be processed during the constant 50 km/hr for two hour period than 100 km/hr for one hour and 0 km/hr for one hour. Based on the flow and density relationship, the speed to produce \(\frac{(FlowA+FlowB)}{2}\)  
Figure 4: Cumulative number of vehicle diagram for ad hoc distributed simulation. on Link C in LP 1 would be \(V_{D}\) (a slope of \(\overline{OD}\)), instead of \(V_{C}\) (a slope of \(\overline{OC}\)). Therefore, the average speed of downstream LP should not be applied directly onto the link in upstream _LPs_ to reproduce the same traffic flow rate. Instead, lower speed than the average speed of downstream LP needs to be exercised. After recognizing the average speed may not meter the intended number of vehicles, applying speed selection is investigated based on empirical analysis from VISSIM(r) model. Figure 6 depicts speed flow diagram obtained from VISSIM(r) test runs after an initialization period [35]. In each test run the desired speed of each vehicle is altered from 48 km/hr to a lower speed at simulation time 600 s with at total simulation time of 1200 s. The lower speeds simulated where 0.5 km/hr to 15 km/hr in 0.5 km/hr increment and 16 km/hr to 30 km/hr in 1 km/hr increment. Five runs were completed for each speed for a total 245 runs. Each dot in Figure 6 represents the average of flow rate and speed during a one minute interval between 660 s and 1200 s excluding transition period data between 600 s and 660 s. Figure 6 reveals that the data display a close agreement with flow and speed relationship. In the proposed model, the first 120 s estimates are excluded in aggregation right after a new speed is applied to exclude the transition period data.