Statistical physics of the development of Kerner's synchronized-to-free-flow instability at a moving bottleneck in vehicular traffic

With the use of simulations of a stochastic microscopic traffic model in the framework of the three-phase traffic theory, we have revealed the statistical physics of a traffic flow instability with respect to a transition from synchronized flow (S) to free flow (F) (Kerner's $\mathrm{S}\to \mathrm{F}$ instability) at a moving bottleneck (MB) occurring through a slow-moving vehicle in vehicular traffic. We have found that the $\mathrm{S}\to \mathrm{F}$ instability can occur at the MB more frequently than at an on-ramp bottleneck. From a comparison of the occurrence of the $\mathrm{S}\to \mathrm{F}$ instability at the MB and on-ramp bottleneck at the same probability of traffic breakdown and the same flow rate it has been found that, whereas the frequency of the $\mathrm{S}\to \mathrm{F}$ instability at the on-ramp bottleneck barely changes, the larger the velocity of the MB, the more frequently the $\mathrm{S}\to \mathrm{F}$ instability occurs at the MB. Contrarily, when the MB velocity decreases considerably, then rather than the $\mathrm{S}\to \mathrm{F}$ instability, in synchronized flow at the MB the classical traffic flow instability leading to the emergence of wide-moving jams ($\mathrm{S}\to \mathrm{J}$ instability) occurs. It has been found that the physics of the intensification of the $\mathrm{S}\to \mathrm{F}$ instability at the MB with the increase in the MB velocity is associated with the increase in the mean space gap (mean time headway) between vehicles in synchronized flow. For this reason, when the MB velocity increases, there is an MB velocity at which the $\mathrm{S}\to \mathrm{F}$ instability dominates the $\mathrm{S}\to \mathrm{J}$ instability: The MB velocity influences considerably on the competition between the $\mathrm{S}\to \mathrm{F}$ and classical traffic flow instabilities in synchronized flow.

Figure 1

A known empirical example of phase transitions in traffic flow illustrating two traffic flow instabilities of three-phase theory (real measured traffic data of road detectors installed along a three-lane highway) (a), (b) [30] and illustrations of associated hypotheses of three-phase theory (c), (d). (a) Sketch of section of three-lane highway in Germany with three bottlenecks. (b) Speed data measured with road detectors installed along road section in (a); data [30] are presented in space and time with averaging method described in Sec. C.2 of [35]. (c) Hypothesis of three-phase theory about the discontinuous character of over-acceleration probability [27, 28, 30, 31]. (d) Hypothesis of three-phase theory about $\mathrm{F}\to \mathrm{S}\to \mathrm{J}$ phase transitions in traffic flow: 2Z characteristic for phase transitions [28, 30]. F = free flow phase, S = synchronized flow phase, J = wide-moving jam phase. In (b), “$\mathrm{sp}$” = spontaneous $\mathrm{F}\to \mathrm{S}$ transition, “$\mathrm{ind}$” = induced $\mathrm{F}\to \mathrm{S}$ transition [30].

Figure 2

Model of a moving bottleneck MB (a) and an on-ramp bottleneck (b) on a two-lane highway. At the beginning of the main lanes is a constant flow rate $q_{\mathrm{in}}$. The MB is a slow-moving vehicle with constant velocity $v_{\mathrm{MB}}$, which position $x_{\mathrm{MB}}$ changes with time $t$. Behind the MB is the moving merging region with length $L_{\mathrm{M}}$, where the lane changing conditions differ from the rest of the main lanes. Vehicles appear at the beginning of the on ramp according to the constant flow rate $q_{\mathrm{on}}$. Inside the merging region $L_{\mathrm{m}}$ vehicles change from the on ramp onto the main lanes. The end of the merging region and the on ramp is donated with $x_{\mathrm{on}}^{\left(e\right)}$.

Figure 3

Comparison of the probabilities of traffic breakdown at the MB $P_{\mathrm{FS},\mathrm{MB}}^{\left(\mathrm{B}\right)}$ and at the on-ramp bottleneck $P_{\mathrm{FS},\mathrm{on}}^{\left(\mathrm{B}\right)}$ as functions of the flow rate $q_{\mathrm{in}}$ for four examples with different velocities $v_{\mathrm{MB}}$ of the MB and different flow rates $q_{\mathrm{on}}$ of the on-ramp bottleneck. Probabilities of traffic breakdown for MB (blue circles) and on-ramp bottleneck (red crosses). Number of realizations is $N_{\mathrm{r}}=100$ and observation time is $T_{\mathrm{ob}}=30\min$. Based on the discrete points, the fit ${\tilde{P}}_{\mathrm{FS},\mathrm{MB}}^{\left(\mathrm{B}\right)}$ for MB (blue, dashed line) and ${\tilde{P}}_{\mathrm{FS},\mathrm{on}}^{\left(\mathrm{B}\right)}$ for on-ramp (red, dot-dashed line) from (4) and (3) was determined. The four parameters of the two fits were used to determine the pairings $\left(v_{\mathrm{MB}}\right|q_{\mathrm{on}})$ according to condition (5). Model parameters used in simulations are presented in Tables A1– A3 of [32] (Appendix A of this paper); the exception is parameter ${\lambda }_{\mathrm{b}}$ of Table A.3 in [32] that is used in accordance with (A30).

Figure 4

In (a) we show for an MB the determined parameter $q_{\mathrm{p},\mathrm{MB}}$ from the fit ${\tilde{P}}_{\mathrm{FS},\mathrm{MB}}^{\left(\mathrm{B}\right)}$ in (4) of the probability of traffic breakdown as a function of the velocity $v_{\mathrm{MB}}$. Accordingly (b) shows the determined parameter $q_{\mathrm{p},\mathrm{on}}$ from the fit ${\tilde{P}}_{\mathrm{FS},\mathrm{on}}^{\left(\mathrm{B}\right)}$ in (3) for an on ramp as a function of the on-ramp flow rate $q_{\mathrm{on}}$. Following the condition in (5), pairings of MB and on ramp $\left(v_{\mathrm{MB}}\right|q_{\mathrm{on}})$ were determined. The result is shown in (c). Other model parameters are the same as those explained in Fig. 3.

Figure 5

Simulated traffic phases (F = free flow, S = synchronized flow, J = wide-moving jam) in time and space shown in system coordinate moving at the velocity $v_{\mathrm{MB}}$ for different velocities $v_{\mathrm{MB}}$. The flow rate $q_{\mathrm{in}}$ is chosen so that condition (2) is satisfied. The flow rate $q_{\mathrm{in}}$ is $1314\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (a), $1379\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (b), (c), and $1489\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (d). Other model parameters are the same as those explained in Fig. 3.

Figure 6

Simulated traffic phases (F = free flow, S = synchronized flow, J = wide-moving jam) in time and space shown for the MB in the system coordinate moving at the velocity $v_{\mathrm{MB}}$ for different pairings $\left(v_{\mathrm{MB}}\right|q_{\mathrm{on}})$. Panels (a) and (b) show two comparisons between MB and on ramp for the pairings $\left(v_{\mathrm{MB}}\right|q_{\mathrm{on}})\text{in}$ Figs. 3 and 3. The flow rate $q_{\mathrm{in}}$ is chosen so that condition (2) is satisfied. The flow rate $q_{\mathrm{in}}$ is $1276\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (a) and $1325\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (b). Other model parameters are the same as those explained in Fig. 3.

Figure 7

Simulated traffic phases (F = free flow, S = synchronized flow, J = wide-moving jam) in time and space shown for the MB in the system coordinate moving at the velocity $v_{\mathrm{MB}}$ for different pairings $\left(v_{\mathrm{MB}}\right|q_{\mathrm{on}})$. Panels (a) and (b) show two comparisons between MB and on-ramp for the pairings $\left(v_{\mathrm{MB}}\right|q_{\mathrm{on}})\text{in}$ Figs. 3 and 3. The flow rate $q_{\mathrm{in}}$ is chosen so that condition (2) is satisfied. The flow rate $q_{\mathrm{in}}$ is $1379\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (a), and $1553\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (b). Other model parameters are the same as those explained in Fig. 3.

Figure 8

Emergence of the $\mathrm{S}\to \mathrm{F}$ transition at the MB shown in Fig. 5 through the occurrence of an $\mathrm{S}\to \mathrm{F}$ instability. (a) Single-vehicle trajectories on the left lane around the $\mathrm{S}\to \mathrm{F}$ transition. The bold dashed lines denote the development of the $\mathrm{S}\to \mathrm{F}$ instability. (b) A further zoom-in on the trajectories from (a) with five marked trajectories through bold, black curves. The speed along the trajectories 1 and 2 in dependence of time is shown in (c). Vehicle 1 is moving on the left lane and has reached the downstream boundary of the synchronized flow. Therefore it is able to accelerate from the speed inside synchronized flow to the higher speed inside free flow. Vehicle 2, following vehicle 1, also starts to accelerate. However, a vehicle on the right lane uses the gap between vehicles 1 and 2 to change to the left lane, which forces vehicle 2 to decelerate, creating the speed peak. The color corresponds to the determined traffic phases (F = free flow, S = synchronized flow, J = wide-moving jam). Other model parameters are the same as those explained in Fig. 3.

Figure 9

Velocity along sections of the single vehicular trajectories 2–5 marked in Fig. 8. The speed (a) in dependence of time and (b) in dependence of relative location. The speed peak from Fig. 8 is the beginning of a speed wave of local speed increase within synchronized flow. While propagating further upstream, the amplitude and spatiotemporal extent of the speed wave increases, leading to an area of free flow inside synchronized flow. The color corresponds to the determined traffic phases (F = free flow, S = synchronized flow, J = wide-moving jam). Other model parameters are the same as those explained in Fig. 3.

Figure 10

Determined probability $P_{\mathrm{SF}}$ of an $\mathrm{S}\to \mathrm{F}$ instability for MB against the velocity $v_{\mathrm{MB}}$ in (a) and for on ramp against the flow rate $q_{\mathrm{on}}$ in (b). A direct comparison between MB and on ramp can be made by plotting $P_{\mathrm{SF}}$ against the flow rate $q_{\mathrm{in}}$ on the main lanes, shown in (c). The vertical lines mark the examples from Figs. 6 and 7. Line 1 corresponds to $(0\mathrm{km}/\mathrm{h}|1222\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, 2 to $(16\mathrm{km}/\mathrm{h}|1183\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, 3 to $(34\mathrm{km}/\mathrm{h}|1102\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, and 4 to $(50\mathrm{km}/\mathrm{h}|844\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$. The parameters $q_{\mathrm{in}}, v_{\mathrm{MB}}$ and $q_{\mathrm{on}}$ satisfy condition (2) (Fig. 4). The observation time after traffic breakdown is $T_{\mathrm{ob}}^{\left(\mathrm{cong}\right)}=60\min$. Other model parameters are the same as those explained in Fig. 3.

Figure 11

Distribution of the space gap $g^{\left(\mathrm{S}\right)}$ in synchronized flow before an instability for the pairings $\left(v_{\mathrm{MB}}\right|q_{\mathrm{on}})\text{shown}$ in Figs. 6 and 7. The histograms include all $N_{\mathrm{r}}^{\left(\mathrm{B}\right)}$ realizations of the simulation, not only the ones shown in Figs. 6 and 7. The flow $q_{\mathrm{in}}$ on the main lanes is $1276\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (a), $1325\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (b), $1379\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (c), and $1553\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ (d). For a better visualization the $x$ axis is limited to $50\mathrm{m}$. The blue, dashed line is the mean ${\overline{g}}^{\left(\mathrm{S}\right)}$ and the red, dotted line the median of the space gap $g^{\left(\mathrm{S}\right)}$. Other model parameters are the same as those explained in Fig. 3.

Figure 12

The determined mean space gap ${\overline{g}}^{\left(\mathrm{S}\right)}$ in synchronized flow before an instability as a function of the velocity $v_{\mathrm{MB}}$ for an MB in (a) and of the flow rate $q_{\mathrm{on}}$ for an on ramp in (b). A direct comparison between MB and on ramp can be made by plotting ${\overline{g}}^{\left(\mathrm{S}\right)}$ against the flow rate $q_{\mathrm{in}}$ on the main lanes, shown in (c). The vertical lines mark the examples from Figs. 6 and 7. Line 1 corresponds to $(0\mathrm{km}/\mathrm{h}|1222\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, 2 to $(16\mathrm{km}/\mathrm{h}|1183\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, 3 to $(34\mathrm{km}/\mathrm{h}|1102\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, and 4 to $(50\mathrm{km}/\mathrm{h}|844\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$. Other model parameters are the same as those explained in Fig. 3.

Figure 13

Determined probability $P_{\mathrm{SJ}}$ of an $\mathrm{S}\to \mathrm{J}$ instability for MB against the velocity $v_{\mathrm{MB}}$ (a) and for on-ramp bottleneck against the flow rate $q_{\mathrm{on}}$ (b). A direct comparison between MB and on-ramp bottleneck can be made by plotting $P_{\mathrm{SJ}}$ against the flow rate $q_{\mathrm{in}}$ on the main lanes (c). The vertical lines mark the examples from Figs. 6 and 7. Line 1 corresponds to $(0\mathrm{km}/\mathrm{h}|1222\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, 2 to $(16\mathrm{km}/\mathrm{h}|1183\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, 3 to $(34\mathrm{km}/\mathrm{h}|1102\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$, and 4 to $(50\mathrm{km}/\mathrm{h}|844\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right))$. The parameters $q_{\mathrm{in}}, v_{\mathrm{MB}}$ and $q_{\mathrm{on}}$ satisfy condition (2) (Fig. 4). The observation time after traffic breakdown is $T_{\mathrm{ob}}^{\left(\mathrm{cong}\right)}=60\min$. Other model parameters are the same as those explained in Fig. 3.

Figure 14

Determined probability of an $\mathrm{S}\to \mathrm{F}$ instability $P_{\mathrm{SF}}$, of an $\mathrm{S}\to \mathrm{J}$ instability $P_{\mathrm{SJ}}$, and of uninterrupted synchronized flow $P_{\mathrm{S}}$ for the MB against the velocity $v_{\mathrm{MB}}$. Characteristic parameters are $v_{\mathrm{MB}}^{\left(\min \right)}=25\mathrm{km}/\mathrm{h}, v_{\mathrm{MB}}^{\left(\mathrm{S}\right)}=33\mathrm{km}/\mathrm{h}, v_{\mathrm{MB}}^{\left(\mathrm{J}\right)}=37\mathrm{km}/\mathrm{h}$, and $v_{\mathrm{MB}}^{\left(\max \right)}=39\mathrm{km}/\mathrm{h}$. The parameters $q_{\mathrm{in}}, v_{\mathrm{MB}}$, and $q_{\mathrm{on}}$ satisfy condition (2) (Fig. 4). The observation time after traffic breakdown is $T_{\mathrm{ob}}^{\left(\mathrm{cong}\right)}=60\min$. Other model parameters are the same as those explained in Fig. 3.

Figure 15

Determined probability of an $\mathrm{S}\to \mathrm{F}$ instability $P_{\mathrm{SF}}$, of an $\mathrm{S}\to \mathrm{J}$ instability $P_{\mathrm{SJ}}$ and of uninterrupted, synchronized flow $P_{\mathrm{S}}$ for the MB and on-ramp bottleneck against the mean space gap ${\overline{g}}^{\left(\mathrm{S}\right)}$. Characteristic parameters are $g_{\min }=15.7\mathrm{m}, g_{\mathrm{S}}=20.3\mathrm{m}, g_{\mathrm{J}}=25.1\mathrm{m}$, and $g_{\max }=26.5\mathrm{m}$. The parameters $q_{\mathrm{in}}, v_{\mathrm{MB}}$, and $q_{\mathrm{on}}$ satisfy condition (2) (Fig. 4). The observation time after traffic breakdown is $T_{\mathrm{ob}}^{\left(\mathrm{cong}\right)}=60\min$. Other model parameters are the same as those explained in Fig. 3.

Figure 16

Simulated traffic phases (F = free flow, S = synchronized flow, J = wide-moving jam) in time and space shown in system coordinate moving at the velocity $v_{\mathrm{MB}}$ for different realizations of identical simulation parameters. In (a) the synchronized flow is interrupted by an $\mathrm{S}\to \mathrm{F}$ instability, in (b) by an $\mathrm{S}\to \mathrm{J}$ instability, and in (c) there is no instability. The velocity of the MB is $v_{\mathrm{MB}}=36\mathrm{km}/\mathrm{h}$, which corresponds to mean space gap ${\overline{g}}^{\left(\mathrm{S}\right)}=24.0\mathrm{m}$ and the maximum of probability $P_{\mathrm{S}}$ (Figs. 14 and 15). Probabilities are $P_{\mathrm{SF}}=0.69, P_{\mathrm{SJ}}=0.07$, and $P_{\mathrm{S}}=0.24$. The flow rate $q_{\mathrm{in}}$ is $1398\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$. Other model parameters are the same as those explained in Fig. 3.

Figure 17

Simulations of the concave growth of traffic oscillations of Jiang et al. with the Kerner-Klenov model. Panels (a) and (b) show a continuation from Fig. 6 of the simulated traffic phases (F = free flow, S = synchronized flow, J = wide-moving jam) in time and space for the MB in system coordinates moving at the velocity $v_{\mathrm{MB}}$ on the left main lane. The parallelogram-shaped region with a width of $60\min$ marks the data used for the analyses in (c). By dividing this region in cells of length $10\mathrm{m}$ and averaging inside these cells and over the time, we calculated in (c) the mean speed $\overline{v}$ and the corresponding standard deviation ${\sigma }_v$ that exhibits the concave growth. This method was used in [100] to show concave growth in speed of Jiang et al. in experimental data. The standard deviation ${\sigma }_v$ is shown only upstream of the merging region $L_{\mathrm{M}}=300\mathrm{m}$ [Fig. 2] to focus on the concave growth. In (a) and (b), the moving jams (traffic oscillations) emerge through the classical traffic instability ($\mathrm{S}\to \mathrm{J}$ instability) that occur in synchronized flow at the MB with probability $P_{\mathrm{SJ}}=1$, whereas the Kerner's $\mathrm{S}\to \mathrm{F}$ instability does not occur (probability $P_{\mathrm{SF}}=0$) for $v_{\mathrm{MB}}=0\mathrm{km}/\mathrm{h}$ and $v_{\mathrm{MB}}=16\mathrm{km}/\mathrm{h}$ (Fig. 14). The flow rate $q_{\mathrm{in}}$ is $1276\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ in (a), $1325\mathrm{vehicles}/\left(\mathrm{h}\mathrm{lane}\right)$ in (b). Other model parameters are the same as those explained in Fig. 3.