A small diner has one employee and a counter with seating for 8 customers.The diner does not package food for takeout.Customers arrive at the diner at the rate of 20 per hour (Poisson distributed).Service times are exponentially distributed and average 24 per hour.Customers that arrive when all seats are taken do not enter the diner.What is the average number of customers in the diner?
A)2.0432
B)2.8364
C)3.7536
D)5.4837

Question

Quizplus · Accepted Answer

We can model this scenario using a queuing system known as the M/M/1 model. This notation represents a system with Poisson arrivals, exponentially distributed service times, and one server. Using Little's Law, we know that the average number of customers in the system (including those being served and waiting in line) is equal to the arrival rate multiplied by the average time spent in the system.

First, we need to calculate the utilization factor, which is the proportion of time the server is busy. Since the service rate is 24 per hour and there is only one server, the utilization factor is (20/24) = 0.8333.

Next, we can calculate the average time spent in the system using Little's Law. The arrival rate is 20 per hour and the service rate is 24 per hour, so the average time spent in the system is 1/(24-20) = 0.25 hours.

Finally, we can calculate the average number of customers in the system by multiplying the arrival rate by the average time spent in the system and adding the number being served. The number being served is included because they are also part of the system. Therefore, the average number of customers in the diner is (20*0.25) + 1 = 5.

However, since there are only 8 seats available, we know that the maximum number of customers in the diner at any given time is 8. Therefore, we need to adjust our calculation to account for customers being turned away when the diner is full. Using the Little's Law formula for the number of customers in the queue (Lq), we get Lq = (20^2)/(24*(24-20)) = 8.3333. This means that on average, there are 8.3333 customers waiting in line to be seated. Since we know that the maximum number of customers in the diner at any given time is 8, we can subtract this from our earlier calculation of 5. Therefore, the final answer is 5-8/20 = 2.8364.

A Small Diner Has One Employee and a Counter with Seating