Question

 At the exit of a toll gate with a single booth, vehicle arrive at random at a rate of 20 vehicles per minute. The service has an average rate of 22 vehicles per minute. Estimate the following:

(a) Average length of queue formed at the toll gate.

(b) Average waiting time of vehicles.

(c) Average time vehicles spent in the system.

Answer 1

Part a

$Q= \frac{\rho^2}{1- \rho}\\ Q= \frac{(\frac{20}{22})^2}{1- \frac{20}{22}}\\ Q=9.08 vehicles$

Part b

$t= (\frac{1}{t_1}+\frac{1}{t_2})*60\\ t= (\frac{1}{22}+\frac{1}{20})*60\\ t=5.73 sec$

Part c

$t= \frac{2- \rho}{2 \mu (1- \rho)}\\ t= \frac{2- \frac{20}{22}}{2*22 (1-\frac{20}{22})}\\ t=0.27 min$