Four Protected Left-Turn Phases on a 90-Second Cycle
A four-leg signalized intersection of a four-lane major street (two through lanes plus a dedicated left-turn lane on every approach, 12-ft lanes) and a two-lane minor street runs a 90-s cycle with four protected left-turn phases. PM peak-hour volumes are: eastbound 1,380 vph (180 left), westbound 1,240 (150 left), northbound 620 (120 left from a dedicated left lane), southbound 540 (90 left from a dedicated left lane). PHF is 0.92. Saturation flow is 1,900 pc/h/ln for through lanes and 1,700 pc/h/ln for protected lefts. Field observations: major-street through queues fail to clear in two or three cycles during the PM peak, while the northbound and southbound left phases routinely serve only two or three vehicles on a 12-s green. Pedestrians cross the 68-ft major street at 3.5 ft/s; the eastbound left-turn bay is 150 ft long.
1. The agency wants to retain all four protected left-turn phases because 'protected lefts are safer.' Evaluate that position quantitatively using critical lane volumes and recommend a phase plan.
Read the worked answer
Method. Use the critical-lane-volume concept: for each phase take the heaviest competing movement, express it as v/s, and sum across phases to get Y.
Current four-phase plan
- P1 E/W protected left: 180/1,700 = 0.106
- P2 E/W through: max(1,200/2, 1,090/2) = 600/1,900 = 0.316
- P3 N/S protected left: 120/1,700 = 0.071
- P4 N/S through: max(500, 450)/1,900 = 0.263
- Y = 0.756; lost time L = 4 phases x 4 s = 16 s
- Webster: C = (1.5(16)+5)/(1-0.756) = 29/0.244 = 119 s
Three-phase alternative (protected E/W lefts; N/S lefts permissive from dedicated lanes with flashing yellow arrow): Y = 0.106 + 0.316 + 0.263 = 0.685; L = 12 s.
Judgment. Dropping the two minor-street protected phases removes 0.071 of Y and 4 s of lost time, cutting the required cycle from about 119 s to about 73 s. The northbound left (120 vph) faces only 450 vph of opposing through traffic that arrives as a platoon and clears in roughly 7 s, so permissive capacity in a dedicated lane is several hundred vph - more than adequate. The safety argument does not hold when the alternative to a protected phase is a green that is too short to clear the through queues: spillback and cycle failures create rear-end and blocking crashes. Recommend three phases, monitoring the N/S lefts for two or three cycles before locking in.
2. Compute a defensible cycle length for the recommended phasing and check the resulting degree of saturation on the critical through movement and on the eastbound left.
Read the worked answer
Cycle length. Y = 0.685, L = 12 s: C = (1.5(12) + 5)/(1 - 0.685) = 23/0.315 = 73 s -> round to 75 s (the next 5-s step, which also gives a workable pedestrian interval).
Green splits. Total effective green = 75 - 12 = 63 s.
- E/W through: g = 63 x (0.316/0.685) = 29.0 s; capacity = 1,900 x 29.0/75 = 735 pc/h/ln
- EB through demand = 1,200/2 = 600 pc/h/ln -> v/c = 0.82
- EB protected left: g = 63 x (0.106/0.685) = 9.7 s; capacity = 1,700 x 9.7/75 = 220 vph -> v/c = 180/220 = 0.82
- N/S through: g = 63 x (0.263/0.685) = 24.2 s; capacity = 1,900 x 24.2/75 = 613 vph vs 500 -> v/c = 0.82
Pedestrian check. 68 ft at 3.5 ft/s = 19.4 s walk-plus-clearance, well under the 29-s major-street green.
Judgment. A v/c of 0.82 at the design hour is a deliberate choice: it is high enough to keep the cycle short and delay low, but leaves roughly 18 percent reserve. Do not chase a lower v/c by lengthening the cycle - that is exactly the failure mode being corrected. Confirm with field counts after implementation rather than relying on the PHF-adjusted estimate.
3. Using the timing developed above, check whether the 150-ft eastbound left-turn bay is adequate, and state what you would recommend given that the bay cannot be lengthened without moving a retaining wall but the intersection is being rebuilt next year.
Read the worked answer
Storage demand. Left turns arrive throughout the ~65 s the phase is red plus the green:
- EB lefts arriving during red = 180 x (75 - 9.7)/3,600 = 3.3 vehicles average
- With Poisson variation, the 95th-percentile queue is roughly 1.6 to 1.8 times the mean -> about 5.5 to 6 vehicles
- At 25 ft per queued vehicle, required storage = 140 to 150 ft before any residual queue carries over.
Result. The 150-ft bay is exactly at - not above - the 95th-percentile demand. Any cycle failure on the through movement, a stalled vehicle, or a shift in the volume mix will spill left-turners into the through lane, which is the mechanism that produced the original queue failures.
Recommendation. Since the wall precludes lengthening, reallocate: give the EB left phase about 12 s instead of 9.7 s (capacity rises to ~270 vph, v/c 0.67, and the mean arrival-during-red queue shrinks because the red interval shortens), and confirm that the through movement still holds v/c below about 0.90. If the rebuild allows any realignment at all, target 200 ft of bay; otherwise add upstream lane-use signing and consider a protected-permissive display so left-turners can discharge on the through green when gaps permit. Verify with a field queue count before finalizing the plans.