How the model works
The simulation follows every car separately. Each driver has their own desired speed, following distance and way of accelerating, drawn at random from published ranges. Two copies of the road receive exactly the same cars; only the share of zipper drivers differs.
How the cars drive
Each car follows the traffic ahead using the Intelligent Driver Model (IDM), a car-following model widely used in research, extended as in its authors’ Human Driver Model: drivers react with a delay of about a second and watch several cars ahead, not just the one in front. Voluntary lane changes follow MOBIL, which weighs a driver’s own gain against the inconvenience caused to others. The zipper rules (who merges, who lets whom in) are ours: no study measures how often drivers let someone in. Treiber, M.; Hennecke, A.; Helbing, D. (2000).Congested traffic states in empirical observations and microscopic simulationsPhysical Review E 62, 1805–1824Peer-reviewed studyread in fullSources Treiber, M.; Kesting, A.; Helbing, D. (2006).Delays, inaccuracies and anticipation in microscopic traffic modelsPhysica A 360(1), 71–88Peer-reviewed studyread in fullRead as arXiv preprint. Drivers use inputs from T′ earlier, extrapolated (§2.1, §2.3), and sum the IDM over the next n_a vehicles, usually 5 (§2.4–2.5).Open source ↗ · Sources Kesting, A.; Treiber, M.; Helbing, D. (2007).General lane-changing model MOBIL for car-following modelsTransportation Research Record 1999, 86–94Peer-reviewed studyread in fullSources
Parameters and sources
| Parameter | Value | Source |
|---|---|---|
| Road | 4.2 km, two lanes; the right lane ends at 3.8 km | — |
| Taper | 165 m (12 ft lane, 45 mph work-zone speed) | Federal Highway Administration (2009).Manual on Uniform Traffic Control Devices2009 ed., Part 6, Chapter 6CTechnical standardread in fullRead by a research sub-agent.Open source ↗ · Sources |
| Lane-ends signs | 1,567, 762 and 305 m before the merge point (start of the taper) | Federal Highway Administration (2009).Manual on Uniform Traffic Control Devices2009 ed., Part 6, Chapter 6CTechnical standardread in fullRead by a research sub-agent.Open source ↗ · Sources |
| Desired speed (v0) | 94–112 km/h (26–31 m/s), varies by driver: a highway posted at 100 km/h (assumption) | — |
| Time gap (T) | 1.1–1.5 s (published calibrations: 1.0–1.6 s) | Kesting, A.; Treiber, M. (2008).Calibrating car-following models by using trajectory data: methodological studyTransportation Research Record 2088, 148–156Peer-reviewed studyread in fullRead by a research sub-agent.SourcesKesting, A.; Treiber, M.; Helbing, D. (2010).Enhanced intelligent driver model to access the impact of driving strategies on traffic capacityPhilosophical Transactions of the Royal Society A 368, 4585–4605Peer-reviewed studyread in fullRead by a research sub-agent.Sources |
| Maximum acceleration (a) | 1.0–1.5 m/s² | Treiber, M.; Kesting, A. (2013).Traffic Flow Dynamics: Data, Models and SimulationSpringerBookread in fullThe errata (Table 11.2) were read; the URL points to the errata.Open source ↗ · SourcesKesting, A.; Treiber, M.; Helbing, D. (2007).General lane-changing model MOBIL for car-following modelsTransportation Research Record 1999, 86–94Peer-reviewed studyread in fullSources |
| Comfortable braking (b) | 1.8 m/s² (published values: 1.5–2.0) | Treiber, M.; Kesting, A. (2013).Traffic Flow Dynamics: Data, Models and SimulationSpringerBookread in fullThe errata (Table 11.2) were read; the URL points to the errata.Open source ↗ · SourcesKesting, A.; Treiber, M.; Helbing, D. (2007).General lane-changing model MOBIL for car-following modelsTransportation Research Record 1999, 86–94Peer-reviewed studyread in fullSources |
| Gap when stopped (s0) | 2 m | Treiber, M.; Hennecke, A.; Helbing, D. (2000).Congested traffic states in empirical observations and microscopic simulationsPhysical Review E 62, 1805–1824Peer-reviewed studyread in fullSources |
| Car length | 4.5 m (published values: 4–5 m) | Kesting, A.; Treiber, M.; Helbing, D. (2007).General lane-changing model MOBIL for car-following modelsTransportation Research Record 1999, 86–94Peer-reviewed studyread in fullSourcesTreiber, M.; Hennecke, A.; Helbing, D. (2000).Congested traffic states in empirical observations and microscopic simulationsPhysical Review E 62, 1805–1824Peer-reviewed studyread in fullSources |
| Lane changing (MOBIL) | politeness 0.2; threshold 0.2 m/s²; safe braking 4 m/s² | Kesting, A.; Treiber, M.; Helbing, D. (2007).General lane-changing model MOBIL for car-following modelsTransportation Research Record 1999, 86–94Peer-reviewed studyread in fullSources |
| Smallest gap accepted when merging | 0.75 s | Daamen, W.; Loot, M.; Hoogendoorn, S.P. (2010).Empirical analysis of merging behavior at freeway on-rampTransportation Research Record 2188, 108–118Peer-reviewed studyabstract onlySources |
| Anticipation | each driver reacts to the 5 cars ahead, not just the one in front | Treiber, M.; Kesting, A.; Helbing, D. (2006).Delays, inaccuracies and anticipation in microscopic traffic modelsPhysica A 360(1), 71–88Peer-reviewed studyread in fullRead as arXiv preprint. Drivers use inputs from T′ earlier, extrapolated (§2.1, §2.3), and sum the IDM over the next n_a vehicles, usually 5 (§2.4–2.5).Open source ↗ · Sources |
| Reaction time | 0.5–0.9 s, varies by driver: they respond to what they saw that long ago, extrapolated to now (shorter than the 0.9–1.2 s in the source’s examples; see the assumptions) | Treiber, M.; Kesting, A.; Helbing, D. (2006).Delays, inaccuracies and anticipation in microscopic traffic modelsPhysica A 360(1), 71–88Peer-reviewed studyread in fullRead as arXiv preprint. Drivers use inputs from T′ earlier, extrapolated (§2.1, §2.3), and sum the IDM over the next n_a vehicles, usually 5 (§2.4–2.5).Open source ↗ · Sources |
| Relaxation after a merge | time gap halved, back to normal over 20 s | Cohen, S.L. (2004).Application of relaxation procedure for lane changing in microscopic simulation modelsTransportation Research Record 1883, 50–58Peer-reviewed studyabstract onlySources |
| Work zone past the taper | 72 km/h (20 m/s) limit from the start of the taper, which drivers reach by easing off beforehand (0.5 m/s²); time gap +0.5 s (assumptions tuned to work-zone capacity) | Codjoe, J.; Thapa, R.; Osafo, Y.A. (2021).Evaluation of DOTD's Existing Queue Estimation ProceduresLTRC Final Report 642Government research reportread in fullOpen source ↗ · Sources |
| Time step | 0.1 s | Kesting, A.; Treiber, M.; Helbing, D. (2007).General lane-changing model MOBIL for car-following modelsTransportation Research Record 1999, 86–94Peer-reviewed studyread in fullSources |
Assumptions with no field data behind them
- Zipper drivers follow the advice. While traffic near the merge moves faster than 56 km/h (35 mph), they move over early, once past the first sign, matching the open lane’s speed. Once it slows below that (and until it is back above 72 km/h, 45 mph), they stay in their lane, start sorting into alternating order 100 m before the merge point, braking barely at first and comfortably by the taper, and change lanes at the start of the taper. Speed is measured over the last 450 m, as sensor-triggered zipper systems do (they switch on below 35–45 mph).
- Each driver in the continuing lane lets at most one car in at the merge point.
- Early mergers decide to leave the ending lane somewhere between the first and second sign. From then on they slow to the speed of the open lane and line up behind the car they mean to follow at their normal following distance, as a driver looking for a gap does.
- Upstream, a driver lets an early merger in once its nose is half a car length ahead (courtesy), then no one else for 60 s.
- An early merger who still hasn’t found a gap after 40 s, in traffic under 14 km/h, pushes in: the driver alongside has to let them in, even if they just let someone else in, or they take a gap of at least 1 m if the car behind can stop for it with hard braking (up to 6 m/s²). With typical drivers, these forced merges are 9% of all merges.
- The ending lane is usable only to the middle of the taper (1.8 m of width left). Within 25 m of that point, or after 20 s stuck at the taper, a driver takes such a gap too, so nothing can lock up.
- A driver who has stopped to let someone in accepts them through any gap that clears their bumper (0.5 m).
- Near the merge point, an early merger in the open lane who sees someone trying to get in beside them or just ahead closes up on the car in front (half their usual time gap, 1 m when stopped). They let that car in only after it has waited 6 s. They close up without the usual reaction delay, since they are watching the car ahead closely: with the delay, gaps this short ended in collisions.
- Before they decide to move over, early-type drivers use whichever lane is faster, including the ending lane: about 153 of them hop into it per 20 minutes and merge back further on. Without that, the queue would be 38% shorter with zipper merging instead of 33% (both from the same 3 draws).
- After a normal merge, both drivers accept a shorter gap that returns to normal over 20 s; after a forced merge, the car behind brakes to restore its gap.
- Past the taper, the work zone caps throughput. We tuned that cap (speed and extra time gap) so queued throughput lands in the range measured in the field. It does more than set the level: how much extra room drivers leave here also decides which merging style passes more cars (see the sensitivity table).
- Reaction times are shorter than the 0.9–1.2 s in the anticipation paper’s examples: longer ones made cars collide here, most likely because these drivers keep shorter gaps. Lining up, letting someone in and stopping at the end of the lane happen without that delay, since drivers see them coming.
- 100% zipper means perfect compliance: everyone stays in their lane and takes turns. On real roads, even with zipper signs, many drivers still merge early (38 to 49% use the ending lane, not 50%).
- Queue: 20 m stretches moving under 20 km/h, with gaps of up to 40 m bridged, measured from the start of the taper to the last car of the queue.
Calibration
We compare the model with ranges measured in the field. “Typical drivers” means most drivers merge early and about one in ten tries to zipper. Heavy traffic: 2,400 vehicles an hour for 20 minutes, 5 different random draws. Results generated on 2026-10-31.
“Fit”: we tuned parameters until the model matched, so it isn’t evidence. “Test”: nothing was tuned to reach it. “Consistency”: follows largely from our own rules; shown for transparency, not as evidence.
| Check | Kind | Model | Field | Accepted | Result |
|---|---|---|---|---|---|
| Queue discharge at a 2→1 closure (cars only) Krammes & Lopez 1994; LTRC 484; Vaughan et al. 2026; HCM6 via Codjoe et al. 2021 (lit. §4.1) | Fit | 1,449 / 1,415 veh/h | 1,300–1,700 veh/h | 1,300–1,750 veh/h | PASS |
| Throughput difference, 100% zipper vs typical drivers Beacher et al. 2004; Radwan et al. 2009; LTRC 484: no significant difference (lit. §1.2) | Fit | −2% | −5–5% | −10–10% | PASS |
| Ending-lane share near the taper, typical drivers, at 1,800 / 2,000 / 2,400 / 3,200 veh/h, against the McCoy–Pesti relation for the same queue length McCoy & Pesti relation used by NCDOT, in Vaughan et al. 2026 (lit. §4.5); accepted within ±3 points at every traffic level | Test | 12 / 13 / 12 / 12% | 14 / 12 / 10 / 5% | ±3 points | MISS |
| Ending-lane share near the taper with zipper signing: model at 75% and 100% compliance Savolainen et al. 2024; Vaughan et al. 2026; Beacher et al. 2004 (lit. §4.5) · can’t fail: 100% gives 50% by construction. It does show the model needs about 75% compliance or more to reach the field range | Consistency | 38.7 / 49.9% | 38–49% | between the two | PASS |
| Queue length reduction, 100% zipper vs typical drivers Measured 8–35%, 50% theoretical ceiling: Kang et al. 2006; MnDOT 2004; McCoy & Pesti 2001 (lit. §1.2) | Fit | 32% | 8–35% | 8–50% | PASS |
| Speed at which the back of the queue moves upstream Bertini & Leal 2005 (lit. §4.7); depends on demand, so the band is wide | Test | 5.7 / 4.8 km/h | 4.8–6.4 km/h | 3–10 km/h | PASS |
| Capacity drop after the queue forms Cassidy & Bertini 1999; Bertini & Leal 2005; HCM6 13.4%; Vaughan et al. 2026 9.8% (lit. §3) | Test | 3.2 / 11.9% | 7–15% | 4–20% | MISS |
| Median time gap left to the new follower, merges with a moving follower (forced included) Daamen et al. 2010: smallest accepted gaps 0.75–1.0 s (lit. §4.5) · excludes merges in front of a stopped car (no time gap defined): 14% with typical drivers, 0% with everyone zippering | Consistency | 1.89 / 1.62 s | 0.75–1 s | 0.75–3 s | PASS |
By share of zipper drivers
| Zipper drivers | Queue after 20 min | Throughput | Ending lane used | Forced merges | Hard braking / 100 merges |
|---|---|---|---|---|---|
| 0% | 2,147 m (2,075–2,315) | 1,530 cars/h (1,520–1,540) | 0% | 2% (1–3%) | 0.6 |
| 10% | 2,115 m (1,975–2,315) | 1,450 cars/h (1,450–1,450) | 12% | 9% (6–11%) | 2.6 |
| 25% | 1,815 m (1,675–2,035) | 1,380 cars/h (1,290–1,460) | 26% | 13% (10–20%) | 2.1 |
| 50% | 1,339 m (1,195–1,515) | 1,460 cars/h (1,370–1,510) | 32% | 8% (5–14%) | 1.9 |
| 75% | 1,367 m (1,215–1,535) | 1,460 cars/h (1,420–1,480) | 39% | 11% (8–14%) | 2.5 |
| 100% | 1,431 m (1,355–1,595) | 1,420 cars/h (1,400–1,430) | 50% | 2% (1–3%) | 0.5 |
Mean of 5 random draws; lowest and highest in brackets. A merge counts as forced by our own rules: a driver takes a smaller gap than drivers normally accept, or pushes in after waiting 40 s at a crawl; with typical drivers most come from the latter rule (see the sensitivity table). Hard braking counts merges after which the car behind loses more than 11 km/h within a second; that can’t happen at a crawl, so it stays rare in a queue. Throughput falls a little, unevenly, as more drivers zipper, but that difference depends on our assumptions (see below).
Over time and by traffic
The queue reduction depends on when you measure it and on traffic. Means of 5 draws over time, 3 by traffic level.
| After | Queue, typical drivers | Queue, everyone zippering | Reduction |
|---|---|---|---|
| 10 min | 1,227 m | 627 m | 49% |
| 20 min | 2,115 m | 1,431 m | 32% |
| 30 min | 2,891 m | 2,283 m | 21% |
| Traffic | Queue reduction after 20 min | Throughput, zipper vs typical |
|---|---|---|
| 1,800 cars/h | 53% | −2.1% |
| 2,000 cars/h | 46% | −1.9% |
| 2,400 cars/h | 33% | −2.4% |
| 3,200 cars/h | 23% | −1.9% |
How much the results depend on the assumptions
One assumption changed at a time, 3 draws each, 2,400 vehicles an hour for 20 minutes. The queue reduction depends most on where early mergers move over; the direction of the throughput difference depends on the work-zone gap and on blocking. A negative reduction means the zipper queue was longer. The last column compares the share of typical drivers still in the ending lane near the taper with the McCoy–Pesti field relation for the same queue length: assumptions that put it far off are ruled out by field counts.
| Change | Queue reduction after 20 min | Throughput, zipper vs typical | Forced merges, typical / zipper | Ending lane used, typical (field) |
|---|---|---|---|---|
| As published | 33% | −2.4% | 9% / 2% | 12% (10%) |
| No extra following gap in the work zone (0 s instead of 0.5 s) | 30% | −15.6% | 7% / 2% | 12% (11%) |
| Twice the extra following gap in the work zone (1 s) | 33% | +1.7% | 11% / 2% | 12% (10%) |
| Early mergers don’t close up to keep others out | 35% | +1.2% | 8% / 2% | 12% (10%) |
| Early mergers all move over at the first sign | 41% | −3% | 6% / 2% | 12% (9%) |
| Early mergers move over between the 2nd and 3rd sign | 14% | −4% | 6% / 2% | 12% (12%) |
| Early mergers wait for the last 300 m | −5% | +0.1% | 11% / 2% | 41% (13%) |
| No reaction time | 33% | −5.4% | 7% / 3% | 11% (10%) |
| No pushing in after 40 s of waiting | 33% | −1.3% | 6% / 2% | 12% (10%) |
| Early-type drivers never move into the ending lane | 38% | −1.5% | 9% / 2% | 12% (10%) |
Known limitations
- We chose the reaction-time range, and kept anticipation two-way, partly because together they kept the throughput difference and the queue reduction inside the field ranges (a slower range pushed the queue reduction to 40%). That makes both comparisons fits, not tests.
- Capacity drop: in the field, a closure passes 7 to 15% fewer cars once a queue forms. Our model gets about that with everyone zippering (11.9%), where taking turns at the taper causes stop-and-go, but much less with typical drivers (3.2%): once drivers have moved over, a single lane drains smoothly. On real roads the drop with typical drivers is probably larger, so the model may flatter early merging once a queue has formed.
- Results depend on following distances. In a similar simulation study, changing a single headway parameter flipped which merging style passed more cars. Our time gaps come from published calibrations, but the throughput difference between the two styles (a few percent) is the size of shift that parameter alone can cause. Ge, Q.; Menendez, M. (2013).A simulation study for the static early merge and late merge controls at freeway work zones13th Swiss Transport Research Conference (STRC), AsconaConference paperread in fullOpen source ↗ · Sources
- Where merges happen: a video study of a conventional work zone in congestion found 49% of filmed merges at the taper and 29% in the last 900 ft (275 m) before it. In the model, merges within that distance are more bunched at the taper (about 9 in 10). Matching the field figure would leave a quarter to a third of cars in the ending lane near the taper, far above the 5–18% counted elsewhere, so we didn’t; the model probably has too few merges in the last few hundred metres. Datta, T.K.; Hartner, C.; Grillo, L.F. (2007).Evaluation of the Dynamic Late Lane Merge System at Freeway Construction Work ZonesMichigan DOT Report RC-1500, Wayne State UniversityGovernment research reportread in fullOpen source ↗ · Sources
- Drivers’ misjudgement of gaps and speeds, the third part of the Human Driver Model, is left out: adding it lowered the capacity drop instead of raising it. Treiber, M.; Kesting, A.; Helbing, D. (2006).Delays, inaccuracies and anticipation in microscopic traffic modelsPhysica A 360(1), 71–88Peer-reviewed studyread in fullRead as arXiv preprint. Drivers use inputs from T′ earlier, extrapolated (§2.1, §2.3), and sum the IDM over the next n_a vehicles, usually 5 (§2.4–2.5).Open source ↗ · Sources
- Cars only: no trucks, although studies disagree about their effect.
- One layout (two lanes to one, 165 m taper), one approach speed, no driver mistakes.
- Lane changes are instantaneous in the model; the drawing spreads them over about three seconds.
- The model is not calibrated to one particular work zone: it is compared with ranges drawn from several studies.
How we test it
Automated tests check, across 24 combinations of conditions (15 minutes each), that no car ever overlaps another, that none stands still for more than three minutes, and that results are reproducible. Two longer runs (two hours of rising traffic) check that nothing locks up. The calibration checks above also run on every change. The same engine runs in your browser.