Scenario B.1
A single ticket taker can tear tickets and direct movie patrons to their seats at a rate of 90 per hour.Customers arrive every minute for assistance and always wait,regardless of how long the line gets.Arrivals are governed by the Poisson distribution and service is governed by the exponential distribution.
-Use the information in Scenario B.1.What is the average number of customers in line?
A)0.33
B)0.66
C)1.33
D)2.00

Question

Quizplus · Accepted Answer

To find the average number of customers in line, we need to use Little's Law, which states that the average number of customers in the system (in line or being served) is equal to the arrival rate multiplied by the average time they spend in the system.

First, let's find the arrival rate. The problem tells us that customers arrive according to the Poisson distribution. Since they arrive every minute, the arrival rate is also λ=1.

Next, we need to find the average time a customer spends in the system. This includes the time waiting in line and the time being served. We can calculate this as the sum of the average time in line (waiting for service) and the average service time (time to tear tickets and direct patrons to their seats).

The average time in line is equal to the average number of customers in line divided by the arrival rate. Let L be the average number of customers in line:

L/λ = L/1 = average time in line

We don't know L yet, but we can use Little's Law to find it:

L = λ * W
where W is the average time a customer spends in the system.

The average service time is the reciprocal of the service rate, which is given as 90 customers per hour. So the average service time is 1/90 hours per customer.

W = average time in line + average service time
W = L/1 + 1/90
W = L + 1/90

Now let's substitute W into Little's Law:

L = λ * W
L = 1 * (L + 1/90)
L = L + 1/90
L - L = 1/90
0 = 1/90

This is a problem! We've ended up with an equation that makes no sense. The issue is that our system is unstable: the arrival rate is greater than the service rate, so the queue will continue to grow without bound. This is why we cannot find a reasonable answer for the average number of customers in line.

Scenario B.1 A Single Ticket Taker Can Tear Tickets and Direct Movie