Answer to Question #226455 in Excel for Ernest

Question

A. Drivers using the Accra-Tema motorway have to stop at the tollbooth to pay their tolls. Motorists arrive at the tollbooth at a rate of 200 per 8-hour-day. The single tollbooth attendant can serve, on average, 220 motorists per 8-hour-day.

a. What is the average minutes’ time a motorist waits in the queue to receive service at the toll?

b. What is the average number of motorists in the queuing system?

Management of the motorway likes to have its operators working 90% of the time. What must the arrival rate be for the tollbooth attendant to be as busy as management would like?

Answer 1

Motorists arrive in time unit:

$λ = 200/8/60 = 0.4167min^{-1}$

Motorists served in time unit:

$µ = 220/8/60 = 0.4583min^{-1}$

1) The average minutes’ time a motorist waits in the queue to receive service at the toll:

$\frac{λ}{µ*(µ-λ)} = 21.8min$

2) The average number of motorists in the queuing system:

$\frac{λ^{2}}{µ*(µ-λ)} = 9$

3) The arrival rate be for the tollbooth attendant to be busy 90% of the time:

attendant to be busy 90% of the time = probability that attendant is busy $p = \frac{λ_2}{µ} = 0.9$

$λ_2 = 0.9µ \implies$ arrival rate = 220*0.9 = 198 motorists per 8-hour-day