Transportation Engineering & Urban Mobility
Webster Traffic Signal Cycle & Green Split Formula
Minimize intersection delay using F. V. Webster’s (1958) optimal cycle length C_opt = (1.5 * L + 5) / (1 - Y), critical flow-ratio sum Y = SUM(q_i / s_i), and proportional effective green splits.
Balance startup lost time against red-light queue wait with F. V. Webster’s (1958) classic formula!
Short traffic-signal cycles waste a huge fraction of every minute on yellow/all-red clearance and driver startup perception-reaction Lost Time L (sec/cycle), starving intersection capacity; overly long cycles force cross-street platoons to sit at empty red lights! F. V. Webster (1958) at the UK Road Research Laboratory proved that for N conflicting signal phases with critical arrival flows q_i (veh/h) and saturation discharge rates s_i (veh/h of green), Total Critical Flow Ratio Y = SUM(q_i / s_i) must satisfy Y < 1 for stability, Minimum Non-Saturating Cycle Length is C_min = L / (1 - Y), Delay-Minimizing Optimal Cycle Length is C_opt = (1.5 * L + 5) / (1 - Y) seconds, and Phase i Effective Green Time is allocated equisaturatingly via g_i = (y_i / Y) * (C_opt - L)!
Verify on a 3-phase arterial junction with Total_Lost_Time L = 12.0 sec (G2) and Phase Critical Flows / Saturation Rates: Phase_1_Main_NS (q_1 = 630, s_1 = 1,800 -> y_1 = 0.35), Phase_2_Cross_EW (q_2 = 450, s_2 = 1,800 -> y_2 = 0.25), and Phase_3_Protected_Left (q_3 = 160, s_3 = 1,600 -> y_3 = 0.10): Critical Flow Ratio Sum is Y = 0.35 + 0.25 + 0.10 = 0.7000 (< 1.00), Minimum Stable Cycle is 12 / 0.30 = 40.00 sec, Webster Optimal Cycle is (1.5 * 12 + 5) / (1 - 0.70) = 23 / 0.30 = 76.67 sec, Total Usable Effective Green is 76.67 - 12 = 64.67 sec, and Main_NS Effective Green Split is (0.35 / 0.70) * 64.6667 = 32.33 sec!
| Signal Timing Metric (L = 12 s, y = [0.35, 0.25, 0.10]) | Formula | Verified Output |
|---|---|---|
| Intersection Critical Flow Sum (Y) | SUM(q_i / s_i) | 0.7000 (70.0% Saturation) |
| Minimum Non-Oversaturated Cycle (C_min) | L / (1 - Y) = 12 / 0.30 | 40.00 sec |
| Webster Delay-Minimizing Cycle (C_opt) | (1.5 * L + 5) / (1 - Y) | 76.67 sec (23 / 0.30) |
| Phase 1 Effective Green Split (g_1) | (y_1 / Y) * (C_opt - L) | 32.33 sec (50.0% of Green) |
Webster (1958) Delay-Minimizing Optimal Cycle Length Formula
Guard oversaturated intersections Y >= 1 (where queues grow unboundedly under fixed-time control and require lane expansion, turn prohibitions, or ramp metering)!
=LET(l_sc, G2, y_sm, SUM(D2:D4), IF(OR(l_sc <= 0, MIN(B2:B4) < 0, MIN(C2:C4) <= 0, y_sm <= 0), "Invalid Signal Input", IF(y_sm >= 1, "Oversaturated Junction (Y >= 1)", ROUND((1.5 * l_sc + 5) / (1 - y_sm), 2))))Equisaturation Phase-1 Effective Green Time g_1 Formula
Evaluate C_opt unrounded inside LET before multiplying by y_1 / Y so phase green splits sum to C_opt - L without rounding drift!
=LET(l_sc, G2, y_1, D2, y_sm, SUM(D2:D4), IF(OR(l_sc <= 0, y_sm <= 0, y_sm >= 1), "Oversaturated Junction (Y >= 1)", LET(c_op, (1.5 * l_sc + 5) / (1 - y_sm), ROUND((y_1 / y_sm) * (c_op - l_sc), 2))))Try the example
Use the original values or change them to test your understanding.