Figure 1

Steady states and line J (dashed line J) for three-phase (a) and two-phase (b) models. (c)–(f) Well-known vehicle queue behavior at the light signal (located at 16 km) at $q_{\mathrm{in}}=500$ (c), (d) and $q_{\mathrm{in}}=650$ vehicles/h (e), (f) simulated with the three-phase model in which parameters that are different from those presented in Table A2 of Ref. [4] are as follows: $v_{\mathrm{free}}=55$ km/h, $p_a=0.1$, $a^{\left(\mathrm{a}\right)}=a$, $v_{22}=7{\mathrm{ms}}^{-1}/\delta v$, $\Delta v_{22}=2{\mathrm{ms}}^{-1}/\delta v$, $v_{01}=6{\mathrm{ms}}^{-1}/\delta v$, $v_{21}=7{\mathrm{ms}}^{-1}/\delta v$, $a=$ 0.5 ${\mathrm{ms}}^{-2}/\delta a$, where $\delta v=$ 0.01 ${\mathrm{ms}}^{-1}$, $\delta a=$ 0.01 ${\mathrm{ms}}^{-2}$; these parameters have been adjusted to describe city traffic that is slower than highway traffic. In (c), (e), speed data are presented by regions with variable shades of gray. In (d), (f), vehicle trajectories are shown for some time intervals related to (c), (e), respectively. In (c)–(f), $T_{\mathrm{R}}=$ 28 s, the cycle length of the light signal is $\vartheta =T_{\mathrm{G}}+T_{\mathrm{Y}}+T_{\mathrm{R}}=60$ s.Reuse & Permissions

Figure 2

Traffic breakdown at the light signal simulated with the three-phase model: $P^{\left(\mathrm{B}\right)}$ as functions of $q_{\mathrm{in}}$ (a) and $T_{\mathrm{R}}$ (b). (c)–(e) Simulation realizations 1, 2, and 3 at $q_{\mathrm{in}}=570$ vehicles/h and $T_{\mathrm{R}}=28$ s; speed data are presented by regions with variable shades of gray; $T^{\left(\mathrm{B}\right)}=40$ (c) and 20 min (d). (f) Realization 3 with traffic breakdown induced at $T_{\mathrm{ind}}=24$ min by one of the drivers that delays in acceleration from the queue during 5 s (the mean model value for this delay is 1.74 s). $T_{\mathrm{ob}}=60$ min [6]. Other model parameters are the same as those in Fig. 1.Reuse & Permissions

Figure 3

Traffic gridlock: (a), (b) Speed in space and time (a) and the same data presented by regions with variable shades of gray (b). (c), (d) Vehicle trajectories for two time intervals related to (a), (b). Simulations of two network intersections: Link 1 with light signal 1 (at 16 km) at the downstream intersection is the same as that in Figs. 1 and 2; the inflow to link 1 consists of the outflows from link 2 for vehicles going straightforward to link 1 (545 vehicles/h) and from link 3 for vehicles turning left to link 1 (36 vehicles/h) during the related green phases of respective light signals 2 and 3 (located at 15.5 km) at the upstream intersection; lengths of links 2 and 3 are 2 km; other vehicles (200 vehicles/h) moving on link 3 go straightforward to link 4 (speed data for links 3 and 4 are not shown). After time delay $T^{\left(\mathrm{B}\right)}=19$ min traffic breakdown occurs on link 1; the growing queue induces traffic breakdown on link 2 at $T_{\mathrm{ind}}=54$ min. $T_{\mathrm{R}}=28$ for links 1 and 2, 34 s for link 3. Simulations of the three-phase model. Other model parameters are the same as those in Fig. 1.Reuse & Permissions

Figure 4

Comparison of probabilities of spontaneous traffic breakdown at the light signal simulated with the three-phase model (curves labeled by $\text{F}\to \text{S}$) and with the two-phase model (curves labeled by $\text{F}\to \text{J}$). Curves $\text{F}\to \text{S}$ in (a), (b) are taken from Figs. 2a and 2(b). $T_{\mathrm{R}}=28$ (a) and 10 s (c). Other model parameters are the same as those in Fig. 1.Reuse & Permissions

Figure 5

Physics of spontaneous traffic breakdown: Speed in space and time (a) and the same data presented by regions with variable shades of gray (b). (c) Vehicle trajectories related to (a), (b). Simulations with the three-phase model. $q_{\mathrm{in}}=1060$ vehicles/h. $T_{\mathrm{R}}=10$ s. Other model parameters are the same as those in Fig. 1.Reuse & Permissions