## 5 Graphical Analysis and Empirical Analysis  
A graphical analysis for speed selection of outflow control is conducted as well as an empirical solution similar to analysis in [33; 34]. Suppose that LP 1 and LP 2 are running the ad hoc simulation (Figure 2) with a rollback threshold (\(\mu\) = 200 veh/hr/ln). LP 1 has estimated the average flow rate on Link B until 7:20 AM simulation time would be 120 veh/hr/ln and a sudden flow increase would occur at 7:20 AM simulation time to 600 veh/hr/ln (Figure 3). Further suppose LP 2 uses 120 veh/hr/ln as an initial input flow rate on Link B until it receives updated information regarding the flow change. Even though LP 1 sends a higher flow rate of 7:21 AM simulation time traffic state to the server on Link B, the 4 min flow rate average (240 veh/hr/ln) does not warrant a rollback immediately in the server, since the difference between 240 veh/hr/ln (the 4 min flow rate average by LP 1) and 120 veh/hr/ln (the current input rate of LP 2) is smaller than the given threshold (\(\mu\)  
Figure 2: Two logical process examples. 200 veh/hr/ln). However, LP 1's simulation advances one more simulation minute and LP 1 sends a much higher average flow rate 360 veh/hr/ln at 7:22 AM simulation time. The server compares the difference between 360 veh/hr/ln and 120 veh/hr/ln (the current input rate of LP 2) and sends a rollback message to LP 2, since the difference is greater than the given threshold. Similarly, the difference of 7:23 AM simulation time traffic states (difference between 480 veh/hr/ln by LP 1 and 360 veh/hr/ln, the new input rate for LP 2) is not large enough to force a rollback. One more simulation minute later, LP 2 needs to alter its input flow rate again when the 4 min flow rate average at 7:24 AM simulation time traffic state is greater than 360 veh/hr/ln (the new input rate for LP 2) by more than the given threshold (\(\mu\): 200 veh/hr/ln). A rollback is processed whenever the difference between estimates is greater than the given threshold. This implies the system is dependent on the size of threshold. For example, if the size of threshold becomes smaller, then the system would have more rollbacks, which implies more computational overheads although generally higher agreements between LP estimates across the network. On the other hand, a larger threshold is expected to reduce the computational overheads although may result in higher discrepancies between the _LPs_. Figure 4 demonstrates cumulative number of vehicles served on Link B. A(t) represents the cumulative arrivals on Link B in LP 1 and D(t) corresponds to Link B cumulative departure in LP 2 (i.e., cumulative number of entering vehicles on LP 2). As seen in Figure 3, (1) LP 1 sends its traffic estimates to the server and the arrival flow rate from upstream LP 1 is constant as a(t) from simulation time 7:00 AM to 7:20 AM, (2) downstream LP 2 continues its simulation with the departure flow rate equal to d(t), (3) the estimated arrival flow rate from upstream LP 1 begins to increase after 7:20 AM simulation time and the information is sent to the server and saved in the Space-Time Memory, (4) when the arrival rate of 7:22 AM simulation time is sent to the server, the difference between a(t) and d(t) is greater than the given threshold, which prompts the first rollback by the server, (5) the server invalidates traffic states of simulation time 7:22 AM and after (gray dotted line) provided by LP 2 in its Space-Time Memory, (6) the server also sends the new arrival rate information to LP 2, and (7) after receiving the rollback message with the updated flow rate, LP 2 rolls back to simulation state of 7:22 AM simulation time and renews its simulation. Similarly, LP 2 processes another rollback at 7:24 AM. Figure 3: Flow rate diagram for ad hoc distributed simulation. A drawback of the current threshold method may also be seen in this analysis.