Signal timing: roads P 14 m and Q 10.5 m

Isolated two-phase signal with pedestrian indications for road P (14 m wide) and road Q (10.5 m wide).

Design basis (IRC:93-1985, Guidelines on Design and Installation of Road Traffic Signals, with Webster's method): two-phase signal, Phase 1 = road P moves, Phase 2 = road Q moves. Pedestrians cross a road while the traffic on the cross road has green. Assumed (not given): saturation flow 1800 PCU/h per lane, start-up lost time 2 s per phase, pedestrian walking speed 1.2 m/s and initial walk (pedestrian start-up) time 5 s.

Step 1: Amber (change) time

Amber $I = t_r + \dfrac{v}{2a}$ with reaction time $t_r = 1$ s and comfortable deceleration $a = 3\ \text{m/s}^2$, kept within the IRC:93 range of 2-4 s:

$$\begin{aligned} I_P &= 1 + \frac{50/3.6}{2(3)} = 3.31 \to 4\ \text{s}\\ I_Q &= 1 + \frac{35/3.6}{2(3)} = 2.62 \to 3\ \text{s} \end{aligned}$$

Step 2: Pedestrian crossing time

Time to cross = initial walk + width / walking speed:

$$\begin{aligned} t_{p,P} &= 5 + \frac{14}{1.2} = 16.7\ \text{s}\\ t_{p,Q} &= 5 + \frac{10.5}{1.2} = 13.8\ \text{s} \end{aligned}$$

Pedestrians crossing road P walk during Phase 2, so Phase 2 green must be at least 16.7 s. Pedestrians crossing road Q walk during Phase 1, so Phase 1 green must be at least 13.8 s.

Step 3: Webster cycle time from traffic

$$\begin{aligned} y_P &= \frac{210}{1800} = 0.117, \quad y_Q = \frac{120}{1800} = 0.067, \quad Y = 0.183\\ L &= (2+4) + (2+3) = 11\ \text{s}\\ C_0 &= \frac{1.5L+5}{1-Y} = \frac{1.5(11)+5}{1-0.183} = 26.3\ \text{s} \end{aligned}$$ $$\begin{aligned} g_P &= \frac{0.117}{0.183}(15.3) = 9.8\ \text{s}, \quad G_P = g_P + 2 = 11.8\ \text{s}\\ g_Q &= \frac{0.067}{0.183}(15.3) = 5.6\ \text{s}, \quad G_Q = g_Q + 2 = 7.6\ \text{s} \end{aligned}$$

Step 4: Check against pedestrian needs

Adopted cycle length $C = 14 + 4 + 17 + 3 = 38$ s.

Final timings

Pedestrian signals: pedestrians crossing road Q get WALK for 5 s then flashing (clearance) for 9 s during Phase 1 (total 14 s of 14 s green); pedestrians crossing road P get WALK for 5 s then flashing for 12 s during Phase 2 (total 17 s of 17 s green). All pedestrian indications show DON'T WALK (red man) while their cross traffic moves.

Phase diagram

Cycle = 38 s
Road P: [GREEN 14][A 4][  RED 20  ]
Road Q: [ RED 18 ][GREEN 17][A 3]
Ped crossing Q: [WALK+FLASH 14][DON'T WALK 24]
Ped crossing P: [DON'T WALK 18][WALK+FLASH 17][DW]