How is total lost time L = 19 s obtained?

2 s per phase for two phases plus 15 s pedestrian all-red

What is Webster's optimum cycle time here?

C = (1.5L + 5)/(1 - Y) = 86.1 s, adopted as 90 s

Asked 2/20
2073 Shrawan · 8 marks

Webster signal: roads H and F, 3 s and 4 s headways

Short answer

Saturation flows are 1200 and 900 PCU/h, giving Y=0.611. With assumed 2 s amber/lost time per phase and 15 s all-red, L=19 s, so optimum cycle 86.1 s, adopt 90 s. Effective green 71 s split as 38.7 and 32.3 s; displayed greens 39 and 32 s plus 2 s ambers.

The average normal flow of traffic on cross roads H and F during design period are 400 and 250 PCU per hour, the saturated headway on these roads are estimated as 3 secs and 4 secs respectively. The all red time required for pedestrian crossing is 15 secs. Design two phase traffic signal by Webster's method.

Saturation flow from the saturation headway: road H $s = 3600/3 = 1200$ PCU/h; road F $s = 3600/4 = 900$ PCU/h.

Step 1: Flow ratios

Flow ratio $y = q/s$ for each approach; the larger value in a phase is the critical one.

Phase	Approach	q	s	y = q/s
Phase 1	Road H	400	1200	0.333
Phase 2	Road F	250	900	0.278

Y = 0.333 + 0.278 = 0.611

Step 2: Lost time

The lost time and amber are not given. An amber of 2 s per phase is assumed and the lost time per phase is taken as 2 s (equal to the amber), plus the all-red period for pedestrians $R$ = 15 s per cycle.

L = (0.0+2.0) + (0.0+2.0) + 15 = 19.0\ \text{s}

Step 3: Optimum cycle time (Webster)

C_0 = \frac{1.5L + 5}{1 - Y} = \frac{1.5(19.0) + 5}{1 - 0.611} = 86.1\ \text{s}

Adopt C = 90 s.

Step 4: Green times

Total effective green $= C - L = 90 - 19.0 = 71.0$ s, divided in proportion to $y$ :

g_{1} = \frac{y}{Y}(C-L) = \frac{0.333}{0.611}(71.0) = 38.7\ \text{s}

g_{2} = \frac{y}{Y}(C-L) = \frac{0.278}{0.611}(71.0) = 32.3\ \text{s}

Displayed green $G = g +$ start-up loss (so that $\sum G + \sum I + R = C$ ):

Phase	Effective g (s)	Start-up loss (s)	Green G (s)	Amber (s)
Phase 1	38.7	0.0	39	2
Phase 2	32.3	0.0	32	2

Check: 39 + 2 + 32 + 2 + 15 = 90 s.

Phase diagram

Cycle = 90 s
Road H:      [ GREEN 39 ][A 2][          RED 49           ]
Road F:      [     RED 41     ][ GREEN 32 ][A 2][ ALL-RED 15 ]
Pedestrians: [          DON'T WALK 75           ][ WALK 15 ]

✓ Answer checked by Claude Opus 5.5 (high effort)

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