Question

A toll tunnel has decided to experiment with the use of a debit card for the collection of tolls. Initially, only one lane will be used. Cars are estimated to arrive at this experimental lane at the rate of 600 per hour. It will take exactly 5.0 seconds to verify the debit card.

a. In how much time would you expect the customer to wait in line, pay with the debit card, and leave? (Round your intermediate calculations to 3 decimal places and final answer to 1 decimal place.)

b. How many cars would you expect to see in the system? (Round your intermediate calculations to 3 decimal places and final answer to 2 decimal places.)

Answer 1

Arrival rate, λ = 600 per hour
Service rate, μ = 1 in 5 seconds = (3600/5) per hour = 720 per hour

The service rate is constant. Hence the formulae for M/D/1 queue will be used.

(a)

Average waiting time, W_q = λ/{2μ. (μ - λ)} = 600/(2*720*(720-600)) = 0.003472 hours = 12.5 seconds

Average time in the system, W_s = W_q + Service time = 12.5 + 5 = 17.5 seconds.

(b)

Average number of cars in the system, L_s = W_s * λ = (17.5/3600)*600 = 2.92 cars