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# Traffic Signal Design | Webster's Formula for Optimum Cycle Length

Webster's method is a rational approach for designing traffic signals. It is simple and is based on the formulae given by Webster. Webster's method is an important topic from the GATE CE exam perspective. Therefore, in this blog, we will cover the method in detail along with examples and practice problems.

### Technical Terms

Before getting into the Webster method, it is important to understand some of the technical terms related to traffic signals as discussed further.

#### Cycle Lenght

Traffic signal works on the time-sharing principle. Cycle length is the time taken to complete one full cycle of the signal at an intersection. For instance, it is the time taken for a signal to go from red, yellow, green, and then come back to the red signal.

#### Green and Red Interval

The green and red interval is the amount of time for the green and red signals respectively.

#### Change Interval

Change interval is the amount of time for the yellow signal. The yellow time is also called the amber time. Amber time can be calculated using simple formula as explained below.

In the above case, as the vehicle can see the amber signal from its stopping sight distance, so, the vehicle can slow down and come to halt before the signal turns red.

But in the above case, the signal is amber only after the vehicle crosses its stopping sight distance. Therefore, sufficient time must be provided so that the vehicle can pass the signal without any disturbance.

The distance needed to be traveled by the vehicle is the summation of the stopping sight distance (SSD) of the vehicle, length of the carriageway (W), and length of the vehicle (L). Now, the amber time is found as,

**Amber time = (SSD + W + L) / v,**

where,

v is the velocity of the vehicle (or design speed)

#### Clearance Interval

Clearance interval is the amount of time for pedestrians to cross and extra time for vehicles to clear the intersection.

#### Phase

Phase is the number of paths crossing at an intersection. For example, in a four-armed intersection, the number of phases is also four. It is also given as the summation of the green interval, change, and clearance interval.

#### Lost Time

In a traffic signal, once the signal is green, the vehicle that is first in queue will take some time to react to the signal and start moving. The second vehicle will take slightly lesser time than the first vehicle and so on. This time will decrease and will eventually reach a constant time called the headway.

The extra time in excess of the headway taken by the vehicles upfront the queue is called lost time. Each phase will have the lost time and needs to be factored in to calculate the optimum cycle length.

#### Saturation Flow (s)

Saturation flow is the highest amount of vehicular flow that is possible. It is given as the inverse of headway. If the headway is in seconds, then the saturation flow is given as,

Saturation flow = 3600/headway in vehicles per hour.

#### Observed Volume (v)

Observed volume is the actual observed volume of traffic flow that is happening at the intersection. It is also represented as vehicles per unit time.

#### Critical Flow Ratio

The critical flow ratio at a phase is the ratio between the observed volume of flow to the saturation flow occurring at all the phases of an intersection. It is given as,

*Critical flow ratio at ith phase = observed volume / saturation flow = v/s at ith phase*

### Optimum Cycle Lenght By Webster Method

Using the above parameters, Webster created a simple formula to calculate the optimum cycle length for an intersection. The optimum cycle length is also taken as the total cycle time for a signal system. Webster's formula is given as,

*Optimum cycle length (Co) = (1.5*L + 5) / (1 - y), *

where,

L - total lost time including all red time,

*L = (n * Lost time at a phase) + All red time*

n - number of phases

All red time is usually taken as zero

Lost time at a phase is usually taken as 2 seconds

y - is the summation of the critical flow ratio at all the phases

#### Green Time by Webster Method

Green time for a road 'a' by Webster's method is given as,

*Ga = (ya/y) * (Co - L),*

where,

ya - critical flow ratio for road 'a'

y - summation of all critical flow ratio

Co - Optimum cycle length

L - lost time including all red time

### Example Problem

**Question: **The normal flow of traffic on crossroads A and B are 400 and 250 vehicles per hour respectively. The saturation of flow for roads A and B are estimated as 1250 and 1000 vehicles per hour respectively. The all-red time for pedestrians to cross is 12 seconds. Design a two-phase traffic system by Webster's method.

**Solution: **

By Webster method, Optimum cycle length (Co) = (1.5*L + 5) / (1 - y),

n = 2

Lost time = 2 s (usually taken)

All red = 12 s (given)

L = (n * Lost time at a phase) + All red time

L = (2*2) + 12 = 16

Road A:

Observed volume (va) = 400

Saturation flow (sa) = 1250

Road B:

Observed volume (vb) = 250

Saturation flow (sb) = 1000

Critical flow ratio on road A (ya) = va/sa = 400/1250 = 0.32

Critical flow ratio on road B (yb)= vb/sb = 200/1000 = 0.25

y = ya + yb = 0.32 + 0.25 = 0.57

Now, Co = (1.5*16 + 5) / (1 - 0.57) = 67.44

*Therefore, the optimum cycle length is 67.44 s*

### Practice Problem

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