Simple cellular automaton model for traffic breakdown, highway capacity, and synchronized flow

Boris S. Kerner, Sergey L. Klenov, and Michael Schreckenberg

Phys. Rev. E 84, 046110 – Published 21 October 2011

Abstract

We present a simple cellular automaton (CA) model for two-lane roads explaining the physics of traffic breakdown, highway capacity, and synchronized flow. The model consists of the rules “acceleration,” “deceleration,” “randomization,” and “motion” of the Nagel-Schreckenberg CA model as well as “overacceleration through lane changing to the faster lane,” “comparison of vehicle gap with the synchronization gap,” and “speed adaptation within the synchronization gap” of Kerner's three-phase traffic theory. We show that these few rules of the CA model can appropriately simulate fundamental empirical features of traffic breakdown and highway capacity found in traffic data measured over years in different countries, like characteristics of synchronized flow, the existence of the spontaneous and induced breakdowns at the same bottleneck, and associated probabilistic features of traffic breakdown and highway capacity. Single-vehicle data derived in model simulations show that synchronized flow first occurs and then self-maintains due to a spatiotemporal competition between speed adaptation to a slower speed of the preceding vehicle and passing of this slower vehicle. We find that the application of simple dependences of randomization probability and synchronization gap on driving situation allows us to explain the physics of moving synchronized flow patterns and the pinch effect in synchronized flow as observed in real traffic data.

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Received 22 June 2011

DOI:https://doi.org/10.1103/PhysRevE.84.046110

Authors & Affiliations

Boris S. Kerner¹, Sergey L. Klenov², and Michael Schreckenberg³

¹Daimler AG, GR/PTF, HPC: G021, D-71059 Sindelfingen, Germany
²Moscow Institute of Physics and Technology, Department of Physics, 141700 Dolgoprudny, Moscow Region, Russia
³Physik von Transport und Verkehr, Universität Duisburg-Essen, D-47048 Duisburg, Germany

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Vol. 84, Iss. 4 — October 2011

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Figure 1
Fundamental empirical features of traffic breakdown. (a) Spontaneous traffic breakdown (speed in space and time) [3, 4, 5]. (b) Probability of spontaneous traffic breakdown at a bottleneck [5]. (c), (d) Induced traffic breakdown [(c) speed in space and time, (d) representation of (c) in space-time plane] [6]. Arrows labeled by “on-ramp” and “off-ramp” show locations of on-ramp and off-ramp bottlenecks, respectively.Reuse & Permissions
Figure 2
Simulations of fundamental empirical features of traffic breakdown shown in Fig. 1 with the KKS CA model (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) made for a two-lane road with on-ramp bottleneck(s). (a) Spontaneous traffic breakdown at an on-ramp bottleneck; time delay of traffic breakdown $T_{FS}^{(B)} = 9$ min; flow rate in free flow upstream of the bottleneck $q_{in} = 1450$ vehicles/(h lane); on-ramp inflow rate $q_{on} = 600$ vehicles/h. (b) Probability of spontaneous traffic breakdown at on-ramp bottleneck $P_{FS}^{(B)}$ as a function of the flow rate downstream of the bottleneck $q_{sum} = q_{in} + 0.5 q_{on}$ [40] at a given flow rate $q_{in} = 1450$ vehicles/(h lane) when on-ramp inflow $q_{on}$ changes; bottleneck location $x_{on} = 15$ km. (c), (d) Induced traffic breakdown: (c) speed in space and time, (d) representation of (c) in space-time plane. Open road boundary conditions are used; the boundary and initial conditions used in simulations are the same as those used in the KKW model [13] (see Sec. 16.2.5 in [8]). For vehicle merging from the on-ramp lane (300 m long) onto two-lane main road, rules (4) are used at $p_{c} = 1$ ; after merging the speed of the merging vehicle is set to $v_{n} = \min (v_{n} + 1, v_{n}^{+})$ ; while moving in the on-ramp lane within merging region, no speed adaptation has been applied; that is, in the rules of vehicle motion we set $G = 0$ . Space cell $δ x = 1.5$ m, time step is 1 s, $d = 5$ (7.5 m), $v_{free} = 25$ (135 km/h), $k = 3$ , $p = 0.05$ , $p_{c} = 0.07$ , $δ = 2$ (10.8 km/h), $g_{c} = 15$ (22.5 m), $L_{a} = 100$ (150 m). In (a), (c), (d), data in left lane are shown. In (b), probability $P_{FS}^{(B)}$ is calculated with the use of 40 different realizations for each given $q_{sum}$ ; $T_{ob} = 30$ min. In (c), (d), locations of on-ramp bottlenecks “on-ramp 1” and “on-ramp 2” are $x_{on} = 12$ and 16 km, respectively; $q_{in} = 1390$ vehicles/(h lane), $q_{on 1}, q_{on 2} = (100, 500)$ vehicles/h; SP at downstream bottleneck “on-ramp 1” occurs through the increase in $q_{on 1}$ to 900 vehicles/h between $t = 5$ and $t = 10$ min.Reuse & Permissions
Figure 3
Vehicle trajectories in the space-time plane within a widening synchronized flow pattern (WSP) shown in Fig. 2a. (a) Overview (trajectories for each third vehicle are shown only). (b) Speed disturbance decays. (c) Speed disturbance grows, leading to traffic breakdown. In (b) and (c) trajectories of all vehicles are shown. The left and right panels correspond to left and right lanes, respectively. The dashed lines in (a), (c) mark the upstream front of synchronized flow separating synchronized flow downstream of the front and free flow upstream.Reuse & Permissions
Figure 4
Microscopic characteristics of synchronized flow within WSP shown in Fig. 2a. (a), (b) Single-vehicle speed and associated time headways as time functions at location $x =$ 12.5 km for free flow (a) and synchronized flow (b). (c), (d) Averaged data [moving averaging of single-vehicle data shown in (a), (b) over 10 vehicles is used] in the space-gap–speed (c) and flow-density (d) planes: F, free flow; S, synchronized flow; $G$ , synchronization space gap; $g_{safe}$ , the safe space gap ( $g_{safe} = v$ ). In (c), (d), a two-dimensional region (hatched region) is related to steady states of synchronized flow used in the KKS CA model. The left and right panels correspond to left and right lanes, respectively. Flow rates within WSP and in free flow upstream of WSP are, respectively, 1250 and 1450 vehicles/(h lane).Reuse & Permissions
Figure 5
Microscopic motion of vehicles through WSP shown in Fig. 2 (a). (a) Vehicle trajectories in the left lane (dashed line shows the upstream front of WSP). (b), (c) Single-vehicle speed (b) and acceleration (deceleration) (c) for one of the vehicles [solid curve 1 in (a)] propagating through the WSP.Reuse & Permissions
Figure 6
Microscopic characteristics of localized SP (LSP). (a) Vehicle speed in space and time in the left (left) and right (right) lanes for $q_{in} =$ 1340 vehicles/(h lane), $q_{on} =$ 570 vehicles/h. (b) Vehicle trajectories in the left lane (dashed line shows the upstream front of LSP). (c), (d) Single-vehicle speed (c) and acceleration (deceleration) (d) for one of the vehicles [solid curve 1 in (b)] propagating through the LSP.Reuse & Permissions
Figure 7
Random time delay of spontaneous traffic breakdown $T_{FS}^{(B)}$ in two different simulation realizations: $T_{FS}^{(B)} =$ 15 (a), 5 min (b). Other parameters are the same as those in Fig. 2a. To calculate different realizations, different initial conditions for random model fluctuations are chosen in simulations.Reuse & Permissions
Figure 8
Explanation of infinite number of highway capacities: (a) Boundaries $F_{th}^{(B)}$ and $F_{S}^{(B)}$ for respectively minimum capacities $q_{th}^{(B)}$ and maximum capacities $q_{\max}^{(B, free)}$ . (b), (c) $Z$ characteristic for traffic breakdown (b) and probability of breakdown (c) found in 40 simulation realizations (runs) as functions of the flow rate downstream of the bottleneck $q_{sum}$ ; $T_{ob} =$ 30 min; (b), (c) are associated with boundaries $F_{th}^{(B)}$ and $F_{S}^{(B)}$ in (a) for $q_{in} =$ 1450 vehicles/(h lane) shown by a dashed horizontal line in (a): Intersections of this line with boundaries $F_{th}^{(B)}$ and $F_{S}^{(B)}$ give, respectively, the minimum capacity $q_{th}^{(B)}$ and maximum capacity $q_{\max}^{(B, free)}$ shown in (b), (c). In (b), F, average speed in free flow; S, synchronized flow associated with the averaging of synchronized flow speeds measured at a virtual detector (200 m upstream of location $x_{on} =$ 15 km) in different SPs at each given $q_{sum}$ . In (c), the same data as in Fig. 2b are shown.Reuse & Permissions
Figure 9
SPs at small on-ramp inflow rates (a)–(d) and a diagram of congested patterns in KKS model (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), (14) (e) [41]. In (a), we show random distributed short-time living regions of synchronized flow that spontaneously occur at bottleneck in KKS CA model (2)–(12) at $p = const$ in (10). In (b)–(d), we present WSP (b), MSP (c), and a sequence of MSPs (d) under application of (14) in (10). In (a)–(d), $(q_{in}, q_{on})$ are (1600, 400) (a), (b), (1600, 50) (c), and (1600, 100) (d) [vehicles/(h lane), vehicles/h]. $p_{2} = 0.3$ , $p_{3} = 0.05$ . Other parameters are the same as those in Fig. 2a.Reuse & Permissions
Figure 10
Microscopic characteristics of MSPs shown in Fig. 9 (d). (a) Vehicle trajectories in the left lane (dashed lines show the fronts of MSP). (b), (c) Single-vehicle speed (b) and acceleration (deceleration) (c) for one of the vehicles [solid curve 1 in (a)] propagating through the MSP.Reuse & Permissions
Figure 11
Steady states of KKS model (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), (15), (16) in the space-gap–speed (a) and flow-density (b) planes: F, free flow; S, synchronized flow; J, line $J$ ; two-dimensional region (hatched region) is related to steady states of synchronized flow used in the KKS CA model.Reuse & Permissions
Figure 12
General patterns. (a), (b) Speed in left lane in space and time within WSP (a) and WSP (b) in the KKS model (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) when (15) and (16) are not applied. (c), (b) Transformation of WSP (a) into GP (c) and WSP (b) into GP (d) under application of (15) and (16) in (2)–(12). $q_{in} =$ 1430 (a), (c), 1385 (b), (d) vehicles/(h lane), $q_{on} =$ 1000 (a)–(d) vehicles/h. $p_{0} = 0.6$ , $p_{1} = 0.05$ , $v_{pinch} = 8$ (43.2 km/h), $k_{1} = 3$ , $k_{2} = 2$ . Other parameters are the same as those in Fig. 2a.Reuse & Permissions
Figure 13
Diagram of congested patterns in KKS model (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), (14), (16), (17) (a) and vehicle speeds in MSP (b) [point A in (a)], in WSP (c) [point B in (a)], in LSP (d) [point C in (a)], in GP (e) [point D in (a)]. In (b)–(e), $(q_{in}, q_{on}) =$ (1610, 50) (b), (1610, 300) (c), (1125, 500) (d), (1290, 500) (e) [vehicles/(h lane), vehicles/h]. $p_{0}^{(2)} = 0.6$ , $p_{1}^{(2)} = 0.3$ , $p_{3} = 0.05$ . Other parameters are the same as those in Fig. 12.Reuse & Permissions
Figure 14
Simulations of MSP propagation on homogeneous road with KKS model (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), (14), (16), (17). (a) Speed in space and time; $v_{cr, FS} \approx 14$ (75.6 km/h), $v_{dis} = 7$ (37.8 km/h), $q_{in} = 1500$ vehicles/(h lane). $T_{dis} = 3$ min. (b)–(d) Single-vehicle speeds at locations 22 km (b) and 2 km (c) and time headways (d) within MSP (a) at location 2 km. Left and right panels are related to left and right lanes, respectively. Other parameters are the same as those in Fig. 13.Reuse & Permissions
Figure 15
Simulations of wide moving jam propagation on homogeneous road with KKS model (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), (14), (16), (17). (a) Speed in space and time; $v_{cr, FJ} \approx 2$ (10.8 km/h), $v_{dis} = 0$ , $q_{in} = 1500$ vehicles/(h lane). $T_{dis} = 3$ min. (b), (c) Single-vehicle speeds (b) and time headways (c) within the jam (a) at location 19.5 km. Left and right panels are related to left and right lanes, respectively. Other parameters are the same as those in Fig. 13.Reuse & Permissions
Figure 16
Robustness of KKS CA model to change in driver behavioral characteristics (a), (b) and in bottleneck type (c). Influence of driver behavior on boundaries $F_{S}^{(B)}$ (curves 1, 3, 5) for maximum capacities and on threshold boundaries $F_{th}^{(B)}$ (curves 2, 4, 6) for minimum capacities. (a) Free flow of 100 $%$ careful drivers for which $g_{c} =$ 20 (30 m) (solid curves 3 and 4). (b) Free flow consisting of 50 $%$ of drivers with usual dynamics and 50 $%$ of more aggressive drivers for which $g_{c} = 10$ (15 m) and $p_{c} = 0.1$ (solid curves 5 and 6); dashed curves (1 and 2) in (a), (b) taken from Fig. 8(a) are associated with usual driver dynamics (Fig. 2). (c) Traffic breakdown occurring with time delay $T_{FS}^{(B)} = 20$ min at bottleneck caused by speed-limit with the resulting development of WPS (speed in space and time in the left and right lanes); $v_{\lim} = 12$ (64.8 km/h), $q_{in} = 1385$ vehicles/(h lane); the beginning of speed limit region of bottleneck located at $x_{B} = 15$ km is labeled by $B$ . Simulations of KKS CA model (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12).Reuse & Permissions
Figure 17
Microscopic characteristics of synchronized flow in the Kerner-Klenov model [35, 36] with model parameters of Ref. [37]. (a) Speed in space and time in the left (left) and right lanes (right) within WSP at $q_{in} = 2120$ vehicles/(h lane), $q_{on} = 400$ vehicles/h. (b) Vehicle trajectories in the left lane (trajectories for each third vehicle are shown only; dashed line shows the upstream front of WSP). (c), (d) Single-vehicle speed (c) and acceleration (deceleration) (d) for one of the vehicles [solid curve 1 in (b)] propagating through the WSP.Reuse & Permissions
Figure 18
Microscopic characteristic of GP shown in Fig. 12c. (a), (d) Vehicle trajectories in the left lane for two fragments within the GP. (b), (c), (e), (f) Single-vehicle speeds (b), (e) and acceleration (deceleration) (c), (f) for, respectively, two vehicles labeled by solid curves 1 and 2 in (a), (d) propagating through the GP.Reuse & Permissions
Figure 19
Microscopic characteristic of GP in the Kerner-Klenov model [35, 36] with model parameters of Ref. [37]: (a) Speed in space and time in the left (left) and right lanes (right) within GP at $q_{in} = 2120$ vehicles/(h lane), $q_{on} = 700$ vehicles/h. (b), (e) Vehicle trajectories in the left lane for two fragments within the GP (trajectories for each third vehicle are shown only). (c), (d), (f), (g) Single-vehicle speeds (c), (f) and acceleration (deceleration) (d), (g) for, respectively, two vehicles labeled by solid curves 1 and 2 in (b), (e) propagating through the GP.Reuse & Permissions

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