Question

Cars arrive at a highway toll booth according to a Poisson distribution with a mean of 90 cars per hour. The service rate for passing through the booth is also Poisson distributed with an average 100 cars per hour. The drivers complain of the long waiting time, and authorities are willing to increase the average passing rate by installing automatic toll collecting device. However, before they incur the expenses, they would like you to calculate the average time in system. It will be closest to:

Answer 1

Using queuing theory, the average time a car spends in the system at the toll booth is calculated as 6 minutes when arrival rate is 1.5 cars/minute, and service rate is 1.6667 cars/minute.

Step-by-step explanation:

To determine the average time a car spends in the system at the toll booth, we need to use queuing theory fundamentals. We're given that cars arrive at the booth with a mean of 90 cars per hour, and the service rate is 100 cars per hour. These rates can be converted to minutes for easier calculation.

Arrival rate (λ): 90 cars/hour = 1.5 cars/minute.
Service rate (μ): 100 cars/hour = 1.6667 cars/minute.

The average time a car spends in the system (both waiting in line and being serviced) can be given by W = 1 / (μ - λ) using the formula for the average time in system in an M/M/1 queue, which assumes exponential service times.

Thus,
W = 1 / (1.6667 - 1.5) = 1 / 0.1667 = 6 minutes.

This means the average time a car spends in the system at this toll booth is 6 minutes.

Answer 2

Answer: Average time in the system = 0.1

Step-by-step explanation:

Given that :

Mean arrival rate ( λ) = 90 cars per Hour

Service rate for passing (mean Numner of people passed per time) (u) = 100 cars per hour

Average time in system (Ws) is given by;

Ws = 1 / (u - λ)

Ws = 1 / ( 100 - 90)

Ws = 1 / 10

Ws = 0.1

Hence, the average time in the system is 0.1