Question

A pretimed signal controls a four-way intersection with no turning permitted and zero lost time. The eastbound and westbound traffic volumes are 500 and 600 veh/h. respectively, and the two movements share the same effective green and effective red portions of the cycle. The northbound and southbound directions also share cycle times, with volumes of 650 and 300 veh/h. If the saturation flow of castbound and westbound approaches is 2100 veh/h, and the saturation flow of northbound and southbound approaches is 2700 veh/h the cycle length is 90 s, and D/D/I queueing applies, please determine

 a. The effective red and effective green times that must be allocated to each directional combination (north-south, cast-west) to minimize total vehicle delay and compute the total delay per cycle. 

 b. The minimum total vehicle delay 

Answer 1

Sol: Information provided in the question:

Normal traffic volume in Eastbound = 500 vehicle /hour

Normal traffic volume in Westtbound = 600 vehicle /hour

Normal traffic volume in Norttbound = 650 vehicle /hour

Normal traffic volume in Southbound = 300 vehicle /hour

Saturation flow for Eastbound and Westtbound = 2100 vehicle/hour

Saturation flow for Norttbound and Southbound = 2700 vehicle/hour

Cycle Length (C) = 90 second

(a) Now Using Webster Method of optimum cycle length :

Step (1) : First we find Critical Flow Ratio :

Y = Normal Flow / Saturation flow

For Eastbound and Westtbound traffic volume :

Y_Eastbound = Normal Flow / Saturation flow = 500 vehicle /hour / 2100 vehicle/hour = 0.2380

Y_Westbound = Normal Flow / Saturation flow = 600 vehicle /hour / 2100 vehicle/hour = 0.28571

Here Y_Westbound > Y_Eastbound

So Y_Westbound is taken as Critical Flow Ratio :

For Northbound and Southbound traffic volume :

Y_Northbound = Normal Flow / Saturation flow = 650 vehicle /hour / 2700 vehicle/hour = 0.2407

Y_Southbound = Normal Flow / Saturation flow = 300 vehicle /hour / 2700 vehicle/hour =  0.111

Here  Y_Northbound > Y_Southbound

So Y_Northbound is taken as Critical Flow Ratio

Then Total Critical Flow Ratio:

Y_Total =  Y_Westbound  + Y_Northbound = 0.28571 + 0.2407

Y_Total = 0.52641

Step(2) : For Optimum Cycle Length : (C)

C = (1.5L + 5) / (1 - Y_Total )

Here L = 2x(number of phases) + Effective red time

Let Number of Phases = 2

Then  

90 = (1.5(2x2 + Effective red time)  + 5) / (1 - 0.52641)

42.6231 = (1.5(2x2 + Effective red time)  + 5)

37.6231 = 1.5 (4 + Effective red time)

Effective red time (R) = 37.6231/1.5 - 4 = 21.082066 seconds

Step (3) : Effective Green Time :(G)

G = Y_critical / Y_Total ( C - L)

So For Eastbound and Westtbound traffic volume :

G = (0.28571 / 0.52641)( 90 - (2x2 + 21.082066 ))

G_East-West = 0.54275 x 64.917 = 35.23382 second

For Northbound and Southbound traffic volume :

G = Y_critical / Y_Total ( C - L)

G = (0.2407 / 0.52641) ( 90 - (2x2 + 21.082066 ))

G_{North - South} = 0.45724 x 64.917 = 29.683 seconds

Step (5) Now we compute Total Delay per cycle:

Total Delay per cycle =(cycle length /2) (1 - green time /cycle length) /(1 - Normal flow /saturation flow)

For Eastbound and Westtbound traffic volume :

Total Delay per cycle (Average) = (90/2) (1- 35.2338 /90) / ( 1 - 0.28571) = 38.336 second

For Northbound and Southbound traffic volume :

Total Delay per cycle (Average) = (90/2) ( 1 - 29.683 / 90) / ( 1 - 0.2407) = 39.718819 second

(b) Now to determine Minimum Total Cycle Delay :

For Eastbound and Westtbound traffic volume :

First find Total Cycle Delay = Total Delay per cycle (Average) x 2 = 38.336 x 2 = 76.672 seconds

For Eastbound and Westtbound traffic volume :

Total Cycle Delay = Total Delay per cycle (Average) x 2 = 39.71881 x 2 = 79.4376 seconds

So By Comparing above two values for Total Cycle Delay, We can say

Minimum Total Cycle Delay : 76.672 seconds