Find the point of no recovery.
Start at 10 arrivals per minute. Increase demand, then give each setting time. Look for a queue that repeatedly clears versus one that keeps growing. Finally stop new arrivals and watch the backlog drain.
FIELD NOTE 18 / THE FOUR-WAY STOP LAB
After you.
Four roads. A shared space. A thousand small decisions.
Discover how a little more traffic can become a much longer wait.
0:00 model time
The two-second hesitation
WORLD A
The reference
The original arrival stream and stop rules.
Waiting for the first arrivals.
- Waiting now
- 0 cars
- Cleared the crossing
- 0 cars
- Mean wait · cleared cars
- —
- Longest current wait
- 0.0 s
WORLD B
With one small change
The same cars, with a two-second hesitation after one model minute.
Waiting for the first arrivals.
- Waiting now
- 0 cars
- Cleared the crossing
- 0 cars
- Mean wait · cleared cars
- —
- Longest current wait
- 0.0 s
Colour = waiting time01250+seconds
Tap a car to inspect it. White outline = selected car; amber indicators show turns. Up to five waiting cars per road are drawn; every car beyond the view still counts. Queue positions are schematic.
A SMALL INTERRUPTION
B’s next eligible driver will hesitate for two extra seconds after model time 1:00.
Follow a driver.
Waiting for a driver.
A selected car’s path and reason for waiting appear here.
The stop-line order is separate from the time a car joined the back of a queue.
COMPARE THE TWO WORLDS
Where did the waiting go?
- Extra waiting cars in B
- 0
- Extra accumulated wait in B
- 0 car·s
The same arrivals enter both worlds. Only the hesitation changes.
Accumulated wait includes cars still waiting, cars crossing and cars already cleared. A negative difference means less waiting in B. A changed departure order can change which movements overlap; one traffic sample does not establish a general effect.
Does the queue recover?
All four approaches combinedA · referenceB · comparison · dashed
A queue can grow during a burst and disappear later. A sustained upward trend suggests demand is exceeding the output achievable under these conditions. The vertical marker shows the latest extra hesitation.
Arriving. Waiting. Getting through.
Collecting the first minute of output.Arrivals · both worlds · dottedB arrivals · violet dotsA clearedB cleared · dashed
Arrivals − cleared = cars waiting + cars still crossing. A widening gap exposes a backlog, including vehicles outside the picture. Measured output is not automatically the intersection’s capacity.
Four roads. Different experiences.
World A · compare the wait and output on each approach.
| Road | Waiting | Cleared | Mean wait, cleared |
|---|---|---|---|
| North | 0 | 0 | — |
| East | 0 | 0 | — |
| South | 0 | 0 | — |
| West | 0 | 0 | — |
Average waiting time includes only cars that have cleared. Always check the current queues and longest waits too: the cars having the worst experience may still be waiting.
Running. Live controls preserve the cars and clock.
Predict the queue. Change the timing. Follow the consequences.
Model notes & references ↗FOLLOW THE SCIENCE
The wait belongs
to the whole system.
One driver sees a gap, a stop sign and a few neighbours. Across the intersection, those local decisions determine how quickly everyone can move. A queue is a memory of earlier arrivals and departures. Removing a disturbance does not instantly erase it.
This experiment connects traffic engineering with queueing theory and coordination. Four-way-stop research studies arrival distributions, turning movements and the division of demand between approaches because those details change delays and the opportunities for vehicles to move together.
MAKE A PREDICTION FIRST
Four ways to investigate.
Same cars. Different timing.
Both worlds receive the same ordered cars, roads and turns. A spaces them evenly. B groups them into four-car platoons from the same approach. Compare at the ends of complete four-car cycles.
What can two seconds change?
World B pauses its next eligible driver for two extra seconds after model time 1:00. The arrival stream is identical in A and B. Follow the queue difference and accumulated waiting.
Give the disturbance some room.
Repeat the hesitation with only seven arrivals per minute. Does the extra queue disappear? Does its disappearance erase the time people already spent waiting?
A small change can use up the breathing room.
If an illustrative intersection clears 20 cars per minute while 18 arrive, it has only two cars per minute of spare capacity. A backlog of ten extra cars takes about five minutes to remove if those rates stay constant. The two-car difference matters more to recovery than the much larger total flow.
Our simulated output is not fixed at 20. It emerges from the sequence of turns, occupied paths, stop and decision times. A quiet intersection has low measured output because few cars arrive; that does not reveal its maximum capacity.
The same number of cars can tell another story.
Even arrivals give an intersection regular opportunities to clear. A platoon can deliver several cars before the previous ones have left. During the gaps, the intersection may be quiet again. Equal average demand does not imply equal queues or waits.
Tian and colleagues found longer delays and queues under platoon arrivals in their all-way-stop simulation study. The paired timing trial here lets you explore that mechanism with the same cars and routes. Read the research abstract ↗
Some paths can share a moment.
Opposing straight movements use separate lanes. Other combinations cross or merge into the same exit. The model checks the space swept by each vehicle’s body and permits only compatible routes to overlap.
A turning car can occupy the shared space for a different time than a car going straight. Because each approach has a single queue, the next driver’s intended movement also affects opportunities for the cars behind. A fast overall output can still coexist with an uneven distribution of waits.
Two seconds can reach beyond one driver.
The hesitation is added when a driver would otherwise be preparing to depart. Conflicting vehicles retain their yield obligations. Compatible traffic may keep moving. After the hesitation ends, later vehicles can inherit part of the delay.
The queue difference can return to zero even though people have accumulated extra waiting. A changed departure order can also create different opportunities for overlapping paths. Compare complete traces and several traffic samples instead of assuming that every disturbance has the same effect.
Count what arrives, what leaves, and what remains.
N(t) = A(t) − D(t) = Q(t) + C(t)
Change in backlog ≈ arrivals − departures
Change in backlog ≈ arrivals − departures
A is the cumulative number arriving, D the number that have cleared, Q the number waiting and C the number still crossing. All queues are counted even if they extend beyond the illustration. This balance is checked directly in the simulation.
Waiting has an area.
Accumulated waiting = ∫ Q(t) dt
If five cars each wait for two seconds, they accumulate ten car-seconds. The area under the queue curve counts time spent waiting by every vehicle, including those still in the queue. It lets us compare a brief large queue with a smaller one that lasts much longer.
Little’s law connects a stable queue to a typical wait.
Average number waiting = arrival rate × average waiting time
Lq = λ Wq
Lq = λ Wq
This long-run relation requires a consistent system boundary and a stable, well-behaved flow. Use compatible units, such as vehicles per second and seconds. A short, growing rush-hour queue or a mean computed only from the cars that have already escaped a backlog is not a reliable steady-state check.
The model, mathematics & references
A teaching model with explicit rules
This is an isolated, right-hand-traffic intersection with four single-lane approaches, unlimited queue storage and free exits. It includes straight, left and right turns. It omits pedestrians, cyclists, lane changes, downstream blockage, collisions and driver negotiation. Queue lengths and waits are calculated; spacing and approach motion in the picture are schematic. No measured city traffic or crash-risk predictions are used.
Stopping, priority and departure
A car arriving at an empty available stop line begins its stop. After a departure, the next car takes 2.2 seconds to move up. Every front car then has an illustrative 0.8-second stop interval. The earliest stop-line arrival has priority over conflicting later arrivals. Ties within the same 0.05-second model step use the vehicle on the right, and opposing left turns yield to conflicting straight or right turns. If these relations cannot choose a unique order, a rotating tie-break resolves the model’s ambiguity. This deterministic tie-break is a simulation convention, not a driving instruction.
A driver with a clear permitted path takes 0.6 seconds to prepare to leave. A requested hesitation adds two seconds to that interval for the next eligible driver in the chosen world. The car continues to hold priority over conflicting later cars. These durations illustrate coordination and queueing; they are not legal stopping requirements or measured reaction times. See SGI’s right-of-way guidance for the real-world context.
Crossing geometry and compatible movements
Cars are 4.5 m long and 1.8 m wide. Lane centres are 4 m apart. A right turn follows a 5 m radius and a left turn a 9 m radius. Each route begins with the car centre 7 m from the intersection centre; its end clears the central crossing. Cars accelerate at 2.2 m/s² up to 5.5 m/s straight, 4.5 m/s left or 3.8 m/s right. A 0.3-second clearance interval follows.
A route reserves its entire swept body envelope until clearance. Envelopes include a 0.25 m margin per side and are sampled at intervals no larger than 0.2 m. Routes that share an entrance or exit, or whose swept rectangles overlap, conflict. Compatible paths may operate at the same time. Holding an entire route is conservative: real drivers can accept gaps after an individual conflict point clears. This model is not a Highway Capacity Manual capacity calculation.
Arrivals and fair comparisons
The demand control is the average across all four roads. North and south share their chosen fraction equally in expectation; east and west share the remainder. Turning marks use 20% right, the selected left fraction and the remaining straight fraction. Reproducible pseudorandom marks assign individual cars. With random arrivals, exponential gaps in integrated demand give a Poisson stream; finite counts fluctuate around the selected average.
Even timing places one arrival per unit of integrated demand. Burst timing places four arrivals near the start of each four-car demand cycle, followed by a gap. The controlled timing trial assigns each four-car group to one approach and reuses the same car identities, roads and turns in A and B. Completed four-car cycles contain equal arrivals; intermediate counts can differ because timing is the tested variable. Live demand changes alter future timing; routes already assigned to either world are retained in both.
In the hesitation trials, both worlds receive the identical arrival events and routes. B’s next eligible driver receives the automatic two-second hesitation after t = 60 s. Adding a manual hesitation cancels the upcoming automatic one. Live sliders preserve the cars, clock and history. The explicit Restart buttons begin a fresh controlled run. Changing arrival mode starts a new future schedule without removing existing traffic.
Time, statistics and limits
The model advances in fixed 0.05-second steps. Arrival, grant and clearance events are resolved on that clock. Paths use analytic constant-acceleration/cruise motion; queue advancement is a prescribed service interval. Counts obey arrivals = queued + crossing + cleared. Accumulated wait integrates the queue count over each time interval. Mean waiting time uses cleared vehicles; the longest current wait exposes people who have not yet contributed to that mean. Output is the number clearing during the previous 60 seconds, displayed only after a full minute.
Charts sample every 0.5 seconds and retain the full observation with up to about 3,600 points, halving temporal sampling density when needed. After 30 model minutes the experiment holds; Continue adds another 30 minutes without resetting. Playback speed changes viewing time only. Hidden tabs freeze, reduced-motion preferences start paused, and Pause/Play resumes from the same state.
Sources and further reading
- Tian et al. (2001), Simulation-Based Study of Traffic Operational Characteristics at All-Way-Stop-Controlled Intersections · arrival distributions, approach balance, queues and delay.
- Wu (2000), Determination of Capacity at All-Way Stop-Controlled Intersections · conflicts between movements and empirically calibrated capacity methods.
- Salter & Ismail (1991), Simulation of Two- and Four-Way Stop Control · arrival patterns, driver assumptions and queue performance.
- Little (1961), A Proof for the Queuing Formula: L = λW · the long-run relation between queue size, arrival rate and time in the system.
- SGI, Saskatchewan Driver’s Handbook: right of way · current driving guidance.