Traffic can jam on an open road with no accident or bottleneck, and the jam cuts throughput. A smooth car-following law and a four-rule automaton both reproduce it, and in both the jam moves backwards at a speed set by how quickly drivers pull away.
In 2008 Sugiyama and coauthors put 22 cars on a circular track 230 metres round and asked the drivers to cruise at about 30 km/h. Three minutes in, a cluster of five stopped cars was drifting backwards around the circle at roughly 20 km/h, while cars outside it moved at about 40 km/h. The track had no obstacle.
Figure 1. Sugiyama's track in the car-following model introduced below, at 10 times real speed: each line is one car's position over the last two minutes (one on narrow screens). To cruise at 30 km/h at this spacing the model's drivers need a time headway of 0.41 s, and each makes small random acceleration errors (seeded). Violet dots mark full stops; the dashed line has the measured jam speed as its slope. At \(a = 1.6\) m/s² the first car stops three to four minutes in, the jam moves back near 18 km/h and free cars reach about 45 km/h, as in the experiment, but the jam holds about 12 cars, not 5. Lower \(a\) gives a slower, longer jam; from about 2 m/s² no car stops within ten minutes.
Each driver reacts only to the car ahead, like a boid in the flocking essay. Two models reproduce the jam: in the Intelligent Driver Model uniform flow is a solution but a small wobble grows, and in the Nagel-Schreckenberg automaton drivers slow down at random. In both the jam travels backwards and costs throughput.
The Intelligent Driver Model (Treiber, Hennecke and Helbing, 2000) sets each car's acceleration from its speed \(v\), the bumper-to-bumper gap \(s\) to the car ahead, and the rate \(\Delta v\) at which that gap is closing:
The first term pulls speed toward \(v_0\); the second brakes when the gap is shorter than the desired gap \(s^*\), which grows with speed through the headway \(T\) and when closing on a slower car. Outside Figure 1, \(v_0 = 30\) m/s, \(T = 1.5\) s, \(s_0 = 2\) m, \(b = 1.5\) m/s² and cars are 5 m long, leaving the maximum acceleration \(a\), how briskly drivers respond. Figure 2 starts evenly spaced cars at the balance speed, one of them 1 m/s slow.
Figure 2. A 1 km ring at about 30 times real speed: cars coloured by speed (click one to brake it) and a space-time diagram, a row every 2 s, time running down. At the defaults the 1 m/s wobble first shrinks, then grows, and cars in the jam stop about 25 simulated minutes in. The readout cross-correlates the speed field 40 s apart once speeds differ by more than 4 m/s.
The red bands tilt left: cars drive into the stopped region from behind and leave from its front, so it moves backwards. The cars it holds thin out the rest of the ring, where speeds rise past the original equilibrium.
Whether the wobble grows can be read off the partial derivatives of the acceleration \(f(s, v, \Delta v)\) at the uniform state. Each car passes its leader's speed fluctuations on with some gain, and uniform flow is stable when the long-wavelength gain stays at or below one, which for any law of this form means (Wilson, 2008)
The left side is the reaction to the gap; the right is damping from \(f_v\) and \(f_{\Delta v}\), both negative. In the IDM \(f_s\) and \(f_v\) scale like \(a\) but \(f_{\Delta v}\) only like \(\sqrt a\), so brisk drivers are stable and sluggish ones are not.
Figure 3. The shaded tongue is where the criterion fails (120 × 80 grid). Each dot is its own ring run 15 simulated minutes from uniform flow with one car 1 m/s slow, filled if the wobble grew; near the edge, especially at the tip, growth can be too slow to show. The ring marks Figure 2's settings: click to move it, or focus the chart and use the arrow keys (Shift for bigger steps).
Flow is stable at low density for any \(a\). Above that the unstable band in Figure 3 widens as drivers get more sluggish: below about \(a = 0.9\) m/s² it runs to bumper-to-bumper, and above about 1.26 m/s² it vanishes. Treiber, Hennecke and Helbing's own freeway value, \(a = 0.73\) m/s², sits well inside it with this page's other settings.
Figure 4. Each dot is its own 1 km ring, started uniform with one car braked to a stop and run 25 simulated minutes; its wave speed comes from cross-correlating the last 10 minutes of the speed field 40 s apart. Filled: cars in the jam stop fully. Grey: \(a = 0.5\) and \(0.8\); the slider adds a teal series, its unstable band along the top. Forward-drifting patterns at the band's low edge are left out. Click a dot (up to 110 cars/km) to load that ring into Figure 2.
Where the jam stops cars fully, its speed barely depends on density (Figure 4): about −14.5 km/h at \(a = 0.8\) and −10.3 km/h at \(a = 0.5\). The front recedes by one car each time a car pulls away. If each departure takes \(\tau\) and a stopped car occupies \(\ell\), it moves back at
with no reference to the traffic arriving. At \(a = 0.8\) stopped cars sit 6.9 m apart, so each waits about 1.7 s; brisker drivers leave sooner, which is why Figure 1's jam speeds up with \(a\). On German motorways wide jams have kept their shape for about 50 minutes over 13 km (Kerner and Rehborn, 1996), and stop-and-go waves travel upstream at about 15 km/h (Treiber, Hennecke and Helbing, 2000), a speed set by how people pull away and queue.
Without the slow car, uniform IDM flow would run forever. Nagel and Schreckenberg (1992) introduced a model that needs no disturbance: a ring of \(L\) cells holding at most one car each, integer speeds from 0 to \(v_{\max}\), and four rules applied to every car in parallel each tick, where \(d\) is the number of empty cells ahead:
With 7.5 m cells and 1 s ticks, \(v_{\max} = 5\) is 135 km/h. Rule 3 is the only randomness; Figure 5 runs the model.
Figure 5. A 200-cell ring (1.5 km) from randomly placed cars (seeded; Reset draws the next seed): the road, and one row per tick below it. At \(\rho = 0.15\) and \(p = 0.25\) jams appear and dissolve, all drifting left; at \(p = 0\) the same density flows freely once the start sorts itself out. The readout averages the last 100 ticks.
A free car averages \(v_{\max} - p\): rule 3 costs it a cell with probability \(p\) and rule 1 restores it. At \(p = 0\) every car keeps \(v_{\max}\) while \(\rho < 1/(v_{\max} + 1)\), above that the flow is \(J = 1 - \rho\), and jams are frozen leftovers of the start.
The front car of a jam waits \(1/(1 - p)\) ticks on average before rule 3 lets it leave, so the front ought to recede at \(1 - p\) cells per tick. Figure 6 measures it.
Figure 6. Dots: the speed of the stopped cars' pattern on a 2,000-cell ring at \(\rho = 0.35\), cross-correlated 50 ticks apart (seeded). Dashed: \(1 - p\). Line: the slope of the chord from a solid jam (density 1, no flow) to the traffic leaving it, \(J_{\text{out}} / (1 - \rho_{\text{out}})\), with \(J_{\text{out}}\) counted 400 cells past the front of a 1,000-car block and \(\rho_{\text{out}} = J_{\text{out}}/(v_{\max} - p)\). The vertical line follows Figure 5's \(p\).
It recedes more slowly, because cars that have just left still dawdle and the ones behind catch up and stop again. The speed follows the outflow instead. A jam is a boundary between stopped cars and the thinner stream leaving its front, and because cars are conserved the boundary moves at the flow difference over the density difference. At \(p = 0.25\) the outflow is 0.51 cars per tick, the chord gives −0.58 cells per tick and the pattern moves at −0.59, or −16 km/h.
Figure 7. Flow (cars per hour past a point) against density. IDM (teal): uniform equilibrium (dashed) and Figure 4's rings at \(a = 0.8\), averaged over their last 8 minutes, with the unstable band below. The violet jam line starts at the jam density (stopped cars 6.9 m apart) with Figure 4's median wave speed as slope; nothing is fitted. Nagel-Schreckenberg (orange, 7.5 m cells, 1 s ticks): the exact \(p = 0\) diagram and 400-cell rings at \(p = 0.25\) (seeded).
Inside the unstable band the IDM rings carry about 10 to 15 percent less than uniform flow would (less near the edges), and they lie on the jam line (Figure 7). A ring holding a jam and a free stretch mixes the two states, and as density changes the mix slides along the chord between them, so the congested branch's slope is the jam's speed. In the automaton, dawdling slows free cars and seeds jams whose outflow is 0.51 cars per tick, against 0.83 at \(p = 0\).
Real motorways show hysteresis: dense free flow can persist until something breaks it, and a jam's outflow is lower than the flow before it. The basic automaton has none: from evenly spaced cars or from one solid block it settles to the same flow at every density (within 0.006 on a 2,000-cell ring from \(\rho = 0.10\) to 0.18). Barlovic and coauthors (1998) got hysteresis by letting stopped cars dawdle with a higher probability \(p_0\) than moving ones.
Figure 8. The slow-to-start rule with Barlovic and coauthors' \(p = 1/64\) and \(v_{\max} = 5\) on a 1,000-cell ring (seeded). From 30 evenly spaced cars, each step runs 400 ticks and plots the mean flow, then adds five cars mid-way into the largest gap, up to 300, then removes five at random per step back to 30. Grey: free flow \(\rho(v_{\max} - p)\) and their phase-separated branch \((1 - p_0)(1 - \rho)\), one compact jam beside free cars. The slider's lowest setting, \(p_0 = p\), is the plain automaton.
In Figure 8, with \(p_0 = 0.75\), adding cars carries the flow up the free line to about 2,900 cars per hour at 22 cars/km, where cars can no longer keep five empty cells ahead; then a jam forms and the flow drops to about 720. Removing cars, the jam survives down to about 7 cars/km, near Barlovic and coauthors' estimate \(1/(1 + (v_{\max} - p)/(1 - p_0))\), 6.4 cars/km. The lower line's slope, \(-(1 - p_0)\) cells per tick, is the jam's speed: moving cars rarely dawdle here, so the jam stays compact and the simple front argument holds.
Y. Sugiyama et al., "Traffic jams without bottlenecks: experimental evidence for the physical mechanism of the formation of a jam," New Journal of Physics 10, 033001 (2008).
M. Treiber, A. Hennecke, D. Helbing, "Congested traffic states in empirical observations and microscopic simulations," Physical Review E 62, 1805 (2000).
R. E. Wilson, "Mechanisms for spatio-temporal pattern formation in highway traffic models," Philosophical Transactions of the Royal Society A 366, 2017 (2008).
B. S. Kerner, H. Rehborn, "Experimental features and characteristics of traffic jams," Physical Review E 53, R1297 (1996).
K. Nagel, M. Schreckenberg, "A cellular automaton model for freeway traffic," Journal de Physique I 2, 2221 (1992).
R. Barlovic, L. Santen, A. Schadschneider, M. Schreckenberg, "Metastable states in cellular automata for traffic flow," European Physical Journal B 5, 793 (1998).