Comment on “Analytical investigation of the open boundary conditions in the Nagel-Schreckenberg model”

In their recent paper [Phys. Rev. E 79, 031115 (2009)], Ning Jia and Shoufeng Ma used some Markov chain arguments for the analytical description of inflow in the deterministic Nagel-Schreckenberg model with open boundaries. In this context, they considered two different mechanisms of injecting vehicles: the standard injection rule and a popular expanded injection rule. While the results for the first one seem to be correct, simulations show that the inflow formula in case of the expanded injection rule yields only approximate results. Therefore, this comment provides the exact formula also in this case and explains the shortcoming in the derivation of Jia and Ma.

Figure 1

(Color online) Comparison between simulation results $(\times )$ and inflow formula (4) [solid line (blue)] in case of $v_{\text{max}}=2$ for the expanded injection rule.

Figure 2

Difference concerning inflow behavior in case of an empty and nonempty minisystem ($v_{\text{max}}=2$, system after injection phase, new vehicles indicated by circles).

Figure 3

Transition graph of the Markov chain representing the complete inflow process (based on [9]).

Figure 4

(Color online) Comparison between simulation results $(\times )$ and inflow formulas (4) [middle line (blue)] and (8) [upper line (red)] in case of $v_{\text{max}}=1$ for the expanded injection rule. The lower line (green) shows the difference between both formulas.

Figure 5

(Color online) Comparison between simulation results $(\times )$ and inflow formulas (4) [middle line (blue)] and (8) [upper line (red)] in case of $v_{\text{max}}=2$ for the expanded injection rule. The lower line (green) shows the difference between both formulas.

Figure 6

(Color online) Comparison between inflow formulas (4) [middle line (blue)] and (8) [upper line (red)] in case of $v_{\text{max}}=5$ for the expanded injection rule. The lower line (green) shows the difference between both formulas.