Fig. 5 presents a finite-size scaling plot of ρ$_{n}$ ( t ) ob- tained by simulating the NaSch model with v$_{max}$ = 3 and p = 0, with system sizes L = 10 $^{3}$, 5 × 10 3 and 10 4 and a variety of values of α, β corresponding to both the low and high density phases. The data collapse is excel- lent, providing strong evidence for the ansatz obtained from (1), (3), and (26). As for the case of p = 0 when v$_{max}$ = 1 we note the sharpness of the cusp at t = L/ | v$_{c}$ | , again suggesting that any corrections to the ansatz (1) are very small. Each simulation performed consisted of right) equal to h$_{t}$ . Then the maximum speed this vehicle can safely achieve at the next time step is taken to be v$_{safe}$ = min( v$_{t}$ + 1 , v$_{max}$, h$_{t}$ ), which allows for unit accel- eration provided the speed limit is obeyed and crashes are avoided. Provided v$_{safe}$ > 0, a random deceleration is then applied so that with probability p the new speed is v$_{t}$$_{+1}$ = v$_{safe}$ − 1, otherwise v$_{t}$$_{+1}$ = v$_{safe}$ . Finally, in the bulk of the system, the vehicle hops v$_{t}$$_{+1}$ sites to its right, so that x$_{t}$$_{+1}$ = x$_{t}$ + v$_{t}$$_{+1}$ . All vehicles in the bulk of the system are updated in this way in parallel. The bulk dynamics clearly reduces to parallel-update TASEP when v$_{max}$ = 1. It remains to consider the boundary dynamics. We again wish to apply open boundary conditions, however choosing an appropriate implementation of such bound- ary conditions for the NaSch model is actually surpris- ingly subtle, and has been an active topic of research over recent years [18–23]. In particular, it was argued in [21] that in order to observe the maximum-current phase when v$_{max}$ > 1 one needs to implement the inflow of vehicles into the system in a rather careful manner. Since our interest in the present context is confined to the high and low density phases however, we have chosen to implement the boundary conditions in the following simple way. We augment the system, which has sites 1 ≤ i ≤ L , with two boundary sites; one at i = 0 and another at i = L + 1. With probability α a vehicle with speed v$_{max}$ is inserted on site 0, and we immediately compute v$_{safe}$ for this vehicle. If v$_{safe}$ > 0 we move the vehicle to site v$_{safe}$ otherwise we delete it. The output is performed similarly. With probability 1 − β we insert a vehicle on site L +1, which then acts as a blockage to vehicles exiting the system. If the rightmost vehicle in the system has x$_{t}$ ≥ L − v$_{max}$ we define its new speed to be v$_{safe}$ and attempt to move the vehicle to site x$_{t}$$_{+1}$ = x$_{t}$ + v$_{safe}$ . If x$_{t}$$_{+1}$ > L the vehicle is removed from the system. When v$_{max}$ = 1 the above prescription reduces to the boundary rules for the simple TASEP described in section 1. Finally, we also considered the case of v$_{max}$ = 3 with p = 0 . 25. For v$_{max}$ > 1 and p > 0 we are not aware of any exact predictions for v$_{c}$ , however it seems reasonable to conjecture that v$_{c}$ is independent of β ( α ) in the low (high) density phase. We therefore simulated the NaSch model with v$_{max}$ = 3, p = 0 . 25 and α = 0 . 25 at four different values of β > α , which should then correspond to a single value of v$_{c}$ . By considering a single value of v$_{c}$ we can still use a finite-size scaling plot of ρ$_{n}$ ( t ) to test the conjectures (1) and (3). Fig. 6 provides strong evidence to support their validity at v$_{max}$ > 1 and p > 0. By varying the value of | v$_{c}$ | used to produce the scaling plot of ρ$_{n}$ ( t ) so that the support edge lay at | v$_{c}$ | t/L = 1 we obtained v$_{c}$ ≈ 2 . 65. We remark that, assuming the validity of (1) and (3), this method can be used as a way to obtain approximate values of | v$_{c}$ | when v$_{max}$ > 1 and p > 0. We now describe our simulations of the NaSch model as defined above. To our knowledge, no rigorous results are known for v$_{c}$ when v$_{max}$ > 1. However, for the deter- ministic case ( p = 0) we expect that for any v$_{max}$ , since in the low-density phase the deter- ministic movement of vehicles from left to right should control the dynamics, while in the high-density phase we expect that it is the movement of holes (traveling with speed 1) from right to left which is important. More gen- erally, we expect the form (24) to remain valid, but with an unknown function g , that will in general depend on v$_{max}$ . We have studied the NaSch model in the low and high density phases via Monte Carlo simulation, and found that to a very good approximation the autocorrelation 10 $^{4}$L/ | v$_{c}$ | iterations (with v$_{c}$ given by (26)), with the first 10 $^{3}$L/ v$_{c}$ time-steps discarded. The above simulations were also performed for v$_{max}$ = 5 with identical results. FIG. 5: Color online. Finite-size scaling plot of ρ n ( t ) for p = 0 NaSch with v max = 3 in the high-density and low- density phases, for a variety of choices of α, β and L . The exact value of v c is unknown in this case but here we chosen v c according to (26).  0  0.2  0.4  0.6  0.8  1  0  0.5  1  1.5  2 n (t) |  v$_{c}$  |  t  L = 1000, p = 0,  = 0.25,   = 0.50 = 1000, p = 0,  = 0.50,   = 0.25 = 1000, p = 0,  = 0.50,   = 0.75 = 1000, p = 0,  = 0.75,   = 0.50 = 5000, p = 0,  = 0.25,   = 0.50 = 5000, p = 0,  = 0.50,   = 0.25 = 5000, p = 0,  = 0.50,   = 0.75 = 5000, p = 0,  = 0.75,   = 0.50 = 10000, p = 0,  = 0.25,   = 0.50 = 10000, p = 0,  = 0.50,   = 0.25 = 10000, p = 0,  = 0.50,   = 0.75 = 10000, p = 0,  = 0.75,   = 0.50 6 4.1. Simulations v$_{c}$ = { v$_{max}$, low density phase , − 1 , high density phase , (26) 5. DISCUSSION