Cycle length and split optimisation: Webster vs HCM practice

1. The problem

Every fixed-time or coordinated signal plan starts with one number: the cycle length. Two traditions are commonly used to estimate it. In the first, Webster’s optimum cycle (Road Research Laboratory, 1958) picks the cycle that approximately minimises total delay. In the second, the critical volume-to-capacity (v/c) relationship found in Highway Capacity Manual (HCM) practice gives the shortest cycle that keeps the intersection’s critical v/c at or below a chosen target.

The two methods use the same inputs but answer different questions, and neither answer is final until pedestrian and vehicle change intervals are checked against the MUTCD. Below: the equations, what the HCM does and does not do, and a hypothetical 4-phase example where the pedestrian minimum sets the practical cycle.

Two clarifications first:

• HCM is an evaluation framework, not an optimiser. FHWA’s Signalized Intersections: Informational Guide (FHWA-HRT-04-091, 2004, written against the HCM 2000 procedure) explains that for HCM operational analysis “a complete signal operating plan, including phasing, cycle length, and green times, must be developed externally”. It also notes that the HCM warns against using its Quick Estimation Method (QEM) for signal timing design [7]. The critical v/c relationship checks or estimates a cycle; it does not prescribe an optimum. The current HCM is the 7th Edition (2022), with Edition 7.1 replacement chapters [6].
• The MUTCD currently in force is the 11th Edition with Revision 1 (December 2025). The Revision 1 Final Rule was published, and took effect, on March 5, 2026; the original 11th Edition (December 2023) became effective on January 18, 2024 [8]. Pedestrian intervals are in Section 4I.06, “Pedestrian Intervals and Signal Phases”, and vehicle change intervals are in Section 4F.17, “Yellow Change and Red Clearance Intervals” [8].

2. Definitions

Symbol	Meaning
v_i	Demand flow of the critical lane group in phase i (veh/h)
s_i	Saturation flow of that lane group (veh/h of green)
y_i = v_i / s_i	Critical flow ratio of phase i
Y = Σ y_i	Sum of critical flow ratios over the critical path
l_i	Lost time of phase i (s)
L = Σ l_i	Total lost time per cycle (s)
C	Cycle length (s)
g_i	Effective green of phase i (s)
X_c	Critical (intersection) v/c ratio
x_i = y_i C / g_i	Degree of saturation of phase i

Critical movement analysis. This step identifies the set of movements that cannot run at the same time [4]. For each phase on the critical path, the lane group with the highest flow ratio is the critical lane group [14]. Y is the sum of those ratios only. Movements that run concurrently with a critical movement do not add to Y.

Lost time. The 2008 FHWA Traffic Signal Timing Manual notes that the HCM defines a default total lost time of 4 s per phase [4]. The University of Idaho NIATT lab manual describes Webster’s L as usually taken to be the sum of the intergreen periods [2]. These two conventions can give different L values, so state which one you use.

3. Method

3.1 Webster’s optimum cycle

Webster’s delay-minimising cycle [1][2]:

C_0 = (1.5 L + 5) / (1 − Y)

Webster’s split rule allocates the available effective green in proportion to the critical flow ratios [2]:

g_i = (C − L) · y_i / Y

With this rule, every critical phase ends up at the same degree of saturation, x_i = Y·C/(C − L).

3.2 Critical v/c and minimum cycle (HCM practice)

The critical v/c of the intersection is

X_c = Y · C / (C − L)

This is the critical v/c relationship as given in HCM-based teaching material [3]. Solving it for C at a chosen target X_c gives the minimum cycle that achieves that target (with L = N·l for N critical phases of equal lost time l):

C_min = L · X_c / (X_c − Y)

The 2008 FHWA Traffic Signal Timing Manual, citing the HCM 2000, describes critical v/c bands as follows [4]:

• below 0.85: operating under capacity
• 0.85–0.95: near capacity
• 0.95–1.00: unstable flow with a wide range of delay
• above 1.00: demand exceeds capacity

C_min is a capacity-driven floor. C_0 is a delay-driven target. Both become meaningless as Y approaches the limiting value: 1 for Webster, X_c for C_min.

3.3 Vehicle change intervals

MUTCD Section 4F.17 requires the durations of the yellow change and red clearance intervals to be “determined using engineering practices”. As Guidance, it gives a yellow change interval of 3 s minimum and 6 s maximum, and a red clearance interval not exceeding 6 s. These values are as reproduced in California’s 2026 MUTCD draft of Section 4F.17, which is based on the federal 11th Edition text [11].

The engineering practice most often used is the kinematic equation. ITE’s Recommended Practice Guidelines for Determining Traffic Signal Change and Clearance Intervals (RP-040B, 2020) bases its method on the “extended kinematic equation” [12]. The basic form, given in the 2008 FHWA Traffic Signal Timing Manual as a single change-period expression (Eq. 5-2, credited there to Kell and Fullerton) whose first term is the yellow and second term the red clearance, is [4]:

Y = t + 1.47 V / (2(a + 32.2 g)) R = (W + L_v) / (1.47 V)

Here t is the perception-reaction time (s), V the approach speed (mph), a the deceleration rate (ft/s²), g the grade (decimal, uphill positive), W the intersection width (ft), and L_v the vehicle length (ft). The manual states that “t = 1.0 s, a = 10 ft/s², and L_v = 20 ft are often cited” [4]. Arizona DOT’s guideline uses the same default values [13]. For design, use RP-040B itself.

3.4 Pedestrian intervals (MUTCD 11th Edition, Section 4I.06)

The Section 4I.06 values below were confirmed from California’s 2026 MUTCD, which is based on the federal 11th Edition, and from MoDOT’s Engineering Policy Guide 902.9 [9][10]:

• Walk interval: should be at least 7 s (Guidance). It may be as short as 4 s where pedestrian volumes and characteristics do not require 7 s (Option).
• Pedestrian clearance time: should let a pedestrian who left the curb at the end of the walk interval travel at 3.5 ft/s to at least the far side of the traveled way (Guidance). Where slower pedestrians routinely use the crossing, a walking speed below 3.5 ft/s should be considered. Where extended push-button presses are provided, up to 4 ft/s may be used.
• Walk plus clearance check: walk interval plus pedestrian clearance time should let a pedestrian starting at the detector travel to the far side at 3 ft/s. If no detector is present, the start point is 6 ft behind the curb face or pavement edge. Any extra time needed goes into the walk interval.
• Buffer interval (Standard): a steady UPRAISED HAND shall be displayed for at least 2 s before any conflicting vehicular movement is released, and the buffer shall not begin later than the start of the red clearance interval, if used. The pedestrian change interval plus the buffer interval shall not be less than the calculated pedestrian clearance time.

The 2 s minimum is the 11th Edition value, reproduced without California amendment in [9]; MoDOT’s EPG 902.9 still carries the 3 s minimum of the 2009 MUTCD (Section 4E.06) [10][15]. Revision 1 touched Section 4I.06 only at Paragraph 19 (conflicting turning volumes may be left or right turns) and made no change to Section 4F.17; the walk, clearance and buffer provisions are unchanged [16]. Read the value from the edition your agency has adopted.

4. Worked example (hypothetical)

All values are illustrative; the Python script reproduces every number.

Layout. A 4-phase, fixed-time intersection with protected lefts on both streets:

Phase	Critical lane group	v (veh/h)	s (veh/h)	y
1	EB/WB left	210	1,750	0.120
2	EB/WB through+right	1,260	3,500	0.360
3	NB/SB left	150	1,750	0.086
4	NB/SB through+right	640	3,400	0.188

Y = 0.754. Lost time is 4 s per phase (L = 16 s); lost time is not the intergreen, which is computed separately below.

Webster. C_0 = (1.5·16 + 5)/(1 − 0.754) = 29/0.246 = 117.9 s.

Minimum cycle at X_c = 0.90. C_min = 16·0.90/(0.90 − 0.754) = 98.6 s, which rounds up to 100 s in practice.

Splits by y-ratio (split = g + l):

Cycle	g1	g2	g3	g4	Split 4	X_c
100 s	13.4	40.1	9.5	21.0	25.0	0.898
120 s	16.6	49.7	11.8	26.0	30.0	0.870

Phase 4 change intervals. Approach speed is 35 mph (51.45 ft/s), on a level grade, with t = 1.0 s, a = 10 ft/s², W = 90 ft and L_v = 20 ft:

• Y = 1.0 + 51.45/20 = 3.57 s, used as 3.6 s. This is within the 3–6 s Guidance range.
• R = 110/51.45 = 2.14 s, used as 2.1 s.

Pedestrian crossing concurrent with Phase 4 (crossing the 72 ft major street):

• Pedestrian clearance time = 72/3.5 = 20.6 s. With a 7 s walk interval, walk + clearance = 27.6 s.
• The 3 ft/s check from 6 ft behind the curb gives (72 + 6)/3.0 = 26.0 s. That is below 27.6 s, so no time needs to be added to the walk interval.
• Minimum Phase 4 split under a conservative convention: walk + pedestrian clearance must end by the end of yellow, which leaves the red clearance to carry the buffer. That gives 7 + 20.6 + 2.1 = 29.7 s, rounded up to 30 s; the 2.1 s red clearance covers the 2 s buffer minimum. The minimum Phase 4 effective green is therefore 26 s. Agencies credit the buffer and red clearance differently; use yours.

Result. At the 100 s capacity floor, the y-ratio split gives Phase 4 only 25.0 s, which is short of the 30 s pedestrian minimum. Fixing Phase 4 at 30 s and re-allocating the remaining green by y-ratio gives:

C (s)	Max phase x (Phases 1–3)	Phase 4 x
100	0.975	0.724
105	0.943	0.760
110	0.915	0.796
115	0.891	0.833
120	0.870	0.869

Once the pedestrian constraint is respected, the shortest cycle that keeps every critical phase at or below 0.90 is 115 s. At 120 s, close to Webster’s 117.9 s, the y-based Phase 4 green (25.97 s) is effectively equal to the pedestrian minimum, so the pedestrian constraint stops binding. A practical plan is C = 120 s with integer splits of 21 / 53 / 16 / 30 s, which sum to 120. The resulting phase degrees of saturation are 0.847, 0.882, 0.857 and 0.869.

Lesson: C_min is valid only if its phase lengths are feasible. Here the pedestrian minimum pushed the feasible cycle about 16 s above it, towards Webster’s value.

import math
ph=[(210,1750),(1260,3500),(150,1750),(640,3400)]
y=[v/s for v,s in ph]; Y=sum(y); l=4.0; L=4*l
C0=(1.5*L+5)/(1-Y); Cmin=L*0.90/(0.90-Y)          # 117.9, 98.6
V=1.47*35; Yc=1.0+V/(2*10); R=(90+20)/V             # 3.57, 2.14
walk=7; PCT=72/3.5; chk=(72+6)/3.0                  # 20.57, 26.0
gmin4=math.ceil(walk+PCT+round(R,1))-l              # 26 s
for C in range(90,135,5):
    g=[(C-L)*yi/Y for yi in y]
    if g[3]<gmin4:
        rest=C-L-gmin4; ys=Y-y[3]
        g=[rest*y[0]/ys,rest*y[1]/ys,rest*y[2]/ys,gmin4]
    print(C,[round(yi*C/gi,3) for yi,gi in zip(y,g)])

5. Limitations

• Webster’s model assumptions. The equation comes from a delay model for an isolated, fixed-time, undersaturated signal with random arrivals. It does not describe actuated control (phase lengths vary cycle by cycle) or oversaturation (queues carry over; Y near or above 1).
• Coordination. In a coordinated system the cycle is a system variable chosen for progression; isolated C_0 and C_min are inputs, not the answer [4].
• Saturation flow and lost time. Results are very sensitive to s and L. In the example, raising Y by 0.03 (0.754 to 0.784) moves C_0 from 117.9 s to 134.2 s. Measure saturation flows where possible.
• Lane utilisation, shared lanes and permitted lefts. These need the full HCM lane-group procedure. A single y per phase is a simplification.
• Pedestrian convention. How the buffer interval and the red clearance count toward pedestrian clearance changes the minimum split by a few seconds. Document the convention used.
• HCM role. The HCM evaluates a timing plan you supply (capacity, v/c, delay, LOS). Optimisation tools or engineering judgement must produce that plan [7].

6. Practitioner checklist

1. Confirm the governing editions: HCM 7th (with 7.1 updates, if adopted), the MUTCD 11th Edition with Revision 1 or your state’s edition, and any agency timing guideline.
2. Define the phasing and identify the critical path and critical lane groups.
3. Use measured saturation flows where possible, and state the lost-time convention.
4. Compute Y, C_0 and C_min at the agency’s target X_c.
5. Compute yellow and red clearance with the agency method (e.g., ITE RP-040B) and check them against Section 4F.17.
6. Compute the walk interval, the pedestrian clearance at 3.5 ft/s (or slower where warranted), and the 3 ft/s total check. Confirm the 2 s buffer minimum (Section 4I.06) against the edition your agency has adopted.
7. Convert pedestrian minimums into minimum splits and re-allocate the remaining green.
8. Round to a practical cycle and integer splits, then recompute every x_i.
9. For coordinated or actuated operation, carry these values into the system or controller model rather than using them directly.
10. Record every input and assumption so a reviewer can reproduce the result.

For an independent review of a study like this, see peer review or get in touch.

Sources

1. Webster, F.V. Traffic Signal Settings. Road Research Technical Paper No. 39, Road Research Laboratory / HMSO, London, 1958 (as commonly cited; original not opened).
2. University of Idaho, NIATT, Traffic Signal Timing Design Lab Manual, “Cycle Length Determination” https://www.webpages.uidaho.edu/niatt_labmanual/Chapters/signaltimingdesign/theoryandconcepts/CycleLengthDetermination.htm ; “Green Split Calculations” https://www.webpages.uidaho.edu/niatt_labmanual/Chapters/signaltimingdesign/theoryandconcepts/GreenSplitCalculations.htm
3. NPTEL, Module 7: Traffic Signal Design. https://archive.nptel.ac.in/content/storage2/courses/105101008/574_SignalLos/point6/point.html
4. FHWA, Traffic Signal Timing Manual, FHWA-HOP-08-024, June 2008 (archived; superseded by [5]). Chapter 3: https://ops.fhwa.dot.gov/publications/fhwahop08024/chapter3.htm ; Chapter 5: https://ops.fhwa.dot.gov/publications/fhwahop08024/chapter5.htm
5. Signal Timing Manual, Second Edition. NCHRP Report 812, Transportation Research Board / National Academies Press, 2015. https://www.nationalacademies.org/publications/22097 A third edition is in preparation (NCHRP Project 03-149, scheduled for completion in late 2026; not yet published as of November 2026): https://trid.trb.org/View/1957089
6. Transportation Research Board. Highway Capacity Manual 7th Edition: A Guide for Multimodal Mobility Analysis. National Academies Press, 2022 (Edition 7.1 replacement chapters available). https://www.nationalacademies.org/publications/26432
7. FHWA, Signalized Intersections: Informational Guide, FHWA-HRT-04-091, August 2004, Chapter 7. https://www.fhwa.dot.gov/publications/research/safety/04091/07.cfm
8. FHWA, Manual on Uniform Traffic Control Devices, 11th Edition (December 2023) and 11th Edition with Revision 1 (December 2025). https://mutcd.fhwa.dot.gov/ ; https://mutcd.fhwa.dot.gov/kno_11th_Editionr1.htm ; table of contents: https://mutcd-dr.fhwa.dot.gov/pdfs/11th_Edition/11thcovtoc.pdf
9. Caltrans, California MUTCD 2026 Edition, Chapter 4I (identified as FHWA’s MUTCD 2023 Edition as amended for use in California; the 4I.06 buffer paragraph carries no California amendment). https://dot.ca.gov/-/media/dot-media/programs/safety-programs/documents/ca-mutcd/2026/2026-camutcd-2026-4i-a11y.pdf
10. Missouri DOT, Engineering Policy Guide 902.9, Pedestrian Control Features (MUTCD Chapter 4I). https://epg.modot.org/index.php?title=902.9_Pedestrian_Control_Features_%28MUTCD_Chapter_4I%29
11. Caltrans, California MUTCD 2026 Edition (Draft), Chapter 4F text, CTCDC Agenda Item 25-10, October 2025. https://dot.ca.gov/-/media/dot-media/programs/safety-programs/documents/ca-mutcd/nmutcd/part4/20251028-ctcdc-mtg-ai-25-10-att-11-camutcd-2026-4f-text-a11y.pdf
12. ITE, Guidelines for Determining Traffic Signal Change and Clearance Intervals, Recommended Practice RP-040B, 2020. https://ite.org/technical-resources/topics/traffic-engineering/traffic-signal-change-and-clearance-intervals ; summary: https://trid.trb.org/view/1692694
13. Arizona DOT, Traffic Guidelines and Processes 621, Signal Phase Change Intervals. https://azdot.gov/sites/default/files/2019/05/tgp0621-2018-01.pdf
14. Wei, H., Zheng, G., Gayah, V., Li, Z. “A Survey on Traffic Signal Control Methods”, arXiv:1904.08117v3, 2020. https://arxiv.org/pdf/1904.08117
15. FHWA, MUTCD 2009 Edition, Section 4E.06 Pedestrian Intervals and Signal Phases. https://mutcd.fhwa.dot.gov/HTM/2009/part4/part4e.htm
16. FHWA, Supplemental Summary of Changes, MUTCD 11th Edition Revision 1 (docket FHWA-2024-0074-0003), copy hosted by WisDOT: https://wisconsindot.gov/dtsdManuals/traffic-ops/manuals-and-standards/wmutcd/fhwa-rev1.pdf