Fundamentals of Transportation/Traffic Signals/Solution

 Solution: With the statements in the problem, we know:
 * Green Time = 30 seconds
 * Red Time = 70 seconds
 * Cycle Length = 100 seconds
 * Arrival Rate = 500 veh/hr (0.138 veh/sec)
 * Departure Rate = 3000 veh/hr (0.833 veh/sec)

Traffic intensity, $$\rho$$, is the first value to calculate.

$$ \rho = \frac = \frac = 0.167\,\!$$

Time to queue clearance after the start of effective green:

$$ t_c {\rm{ }} = {\rm{ }}\frac = \frac = 14.03\ s \,\!$$

Proportion of the cycle with a queue:

$$ P_q {\rm{ }} = {\rm{ }}\frac{C}{\rm{ }} = {\rm{ }}\frac{100}{\rm{ }} = 0.84\,\!$$

Proportion of vehicles stopped:

$$ P_s = \frac = \frac = 0.84 \,\!$$

Maximum number of vehicles in the queue:

$$Q_{\max } {\rm{ }} = {\rm{ }}\lambda r = {\rm{ }}0.138(70) = 9.66 \,\! $$

Total vehicle delay per cycle:

$$ D_t {\rm{ }} = {\rm{ }}\frac{\rm{ }} = {\rm{ }}\frac{\rm{ }} = 406\ veh-s \,\!$$

Average delay per vehicle:

$$ d_{avg} {\rm{ }} = \frac = \frac = 29.41\ s\,\!$$

Maximum delay of any vehicle:

$$ d_{\max } {\rm{ }} = {\rm{ }}r = {\rm{ }} 70\ s \,\!$$

Thus, the solution can be determined:


 * Proportion of the cycle with a queue = 0.84
 * Maximum number of vehicles in the queue = 9.66
 * Total Delay = 406 veh-sec
 * Average Delay = 29.41 sec
 * Maximum Delay = 70 sec

/Solutions