Industry: Transportation
Product: ModelRisk
Application: Signal-Timing Optimization at a Signalized Intersection

When a Fixed Cycle Runs Twice the Delay an Adaptive Signal Would: Stochastic Webster Cycle Optimization

A US city ran the same 110-second fixed-cycle, 42-second green-split plan on a critical CBD-arterial intersection for nine years. The engineering team's deterministic Webster calculation, run once at the original 2015 design demand of 1,210 vph, said the plan was close to optimal. By 2024 the demand on the critical approach had drifted to a mean of 1,450 vph with a peak-extended-period mean of 1,711 vph, and the fixed plan was producing a mean per-vehicle delay of 131 seconds on the critical approach — past the LOS F threshold and tipping oversaturated on roughly half of all 15-minute intervals. The deterministic recalculation said "raise the cycle to 120 s." The simulation said the right answer is not a new fixed cycle. It is an adaptive Webster cycle that re-optimises every 15 minutes against the realised demand.

The clearest way to see the gap is to run both control schemes against the same simulated demand and compare the per-vehicle delay distributions. The fixed plan piles up against the LOS-F ceiling — mean 131 s/veh, with 51% of intervals above the 60-second LOS-E threshold. The adaptive Webster cycle pulls the whole distribution left to a mean of 62 s/veh and 25% above LOS E — a 53% cut in mean delay.

The team rebuilt the intersection model in ModelRisk: critical-approach flow is LogNormal, the green split is solved from the Webster formula

C* = (1.5 × L + 5) / (1 − Σy_i), with L = 12 s lost time/cycle

and per-vehicle delay is computed using the Webster delay equation with the actual degree of saturation x, not the design x.

Why the cycle has to be a distribution, not a number

Critical-approach flow on the corridor is LogNormal with mean 1,450 vph and CV 0.18. Saturation flow per lane is the HCM default 1,900 vph, applied to two through-lanes for 3,800 vph total saturation. Combining the critical-approach y = q/s with the other three phases' aggregate y (Triangular 0.32–0.58 reflecting cross-street empirical variability), the joint Y = Σy_i drives the Webster-optimal cycle, which spans an 82-to-180-second band across the day (mean 132 s):

• P10 of Y = 0.72 → optimal cycle 82 s.
• P50 of Y = 0.82 → optimal cycle 129 s.
• P90 of Y = 0.92 → optimal cycle pinned at the 180 s practical maximum.

A single fixed cycle at 110 s lives near the low end of optimal — too short for the heavy intervals, wrong every 15-minute interval. The Webster delay equation makes this worse: average delay is convex in x, so running x at 0.95 instead of 0.85 multiplies delay rather than nudging it up by 12%.

Deterministic said the fix is a longer cycle. Monte Carlo said the fix is adaptive control.

Running the two control schemes against the same simulated demand:

The adaptive scheme cuts mean per-vehicle delay by 53% and lifts realised throughput on the critical approach by 6%. The deterministic "longer cycle" fix would have moved the mean modestly — useful, but it cannot touch the P90 because it is still a single number facing a distribution.

Three demand regimes, three throughput CDFs

The same adaptive controller delivers different throughput distributions across the day, and the simulation makes the dependence explicit:

• Off-peak (1,088 vph mean demand) — mean throughput 1,086 vph (demand-limited). The adaptive cycle drops toward its 60 s floor.
• Baseline (1,450 vph mean demand) — mean throughput 1,444 vph. Cycle averages ~130 s.
• Peak-extended (1,711 vph mean demand) — mean throughput 1,687 vph. The cycle pins at the 180 s practical maximum on the heaviest intervals and the controller starts dropping low-volume permissive phases.

Across the day the adaptive scheme serves demand right up to the saturation ceiling at each regime, where the single fixed 110-s plan leaves throughput on the table at both ends — too long for off-peak, too short for the peak.

What actually moves the per-vehicle delay

The dominant driver is critical-approach mean flow (a 500 vph range moves mean delay ±14 s). Other-phase aggregate demand y (the Triangular 0.32–0.58) is second; it determines how much green the critical phase can claim. Saturation flow per lane is third — calibrating s up by 100 vph from the HCM default to a field-measured 2,000 vph shaved 9 s/veh from the simulated mean delay. Lost time per cycle matters more than expected: an 8-to-18 s range moves delay ±7 s/veh, and that was what justified replacing the 4-second amber/all-red intervals with 3-second intervals on a corridor with adequate stopping-sight distance.

What the model changed

• Adaptive controller spec. The simulation produced the cycle-length distribution (82–180 s, mean 132 s) the procurement team wrote the adaptive-controller specification against; the previous spec asked for "120 ± 10 s," which would have failed on the large share of intervals whose optimal cycle sits outside that band.
• LOS-E exceedance probability at this intersection dropped from 51% under the fixed plan to 25% under the adaptive plan in the first six months of post-deployment data — within a few points of the simulated prediction.
• Throughput recovered. Realised critical-approach throughput rose from 1,340 to 1,416 vph (+6%), matching the simulation. That recovered throughput translates to roughly 76 additional vehicles processed per peak hour without any physical capacity addition.
• Citywide replication budget. Because the simulated benefit is structural (not site-specific to this intersection), the city committed to converting 38 more high-volume signals to adaptive control over three years, with the per-intersection benefit case anchored on the simulated 53% mean-delay reduction.

ModelRisk Functionality Used

• LogNormal critical-approach flow (mean 1,450 vph, CV 0.18) fitted from one year of detector data, producing the volume tail that breaks any fixed-cycle plan.
• Triangular other-phase demand y (0.32 / 0.45 / 0.58) reflecting cross-street empirical variability that the prior model assumed constant.
• Webster-optimal cycle calculation inside the simulation loop, with the cycle re-solved every iteration against the realised demand — producing the 82–180 s adaptive band the controller spec was built on.
• Webster delay equation applied with stochastic x, capturing the convex blow-up that makes "average x" understate the actual mean delay on a corridor near saturation.
• Comparison of fixed-cycle and adaptive-cycle delay distributions, producing the 131 → 62 s/veh mean improvement and the 51% → 25% LOS-E exceedance reduction.
• Three-regime throughput CDF (off-peak / baseline / peak-extended) that anchored the day-of-week and hour-of-day scheduling of the adaptive controller's mode switches.

A fixed-cycle plan optimises for an average rush hour that does not exist. ModelRisk gave the city the distribution that does — and 62 s/veh of average delay is what living inside that distribution looks like.