Question #249026

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.

Expert's answer

Part a

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

Part b

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

Part c

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

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