A suburban specialty restaurant has developed a single drive-thru window.Customers order,pay,and pick up their food at the same window.Arrivals follow a Poisson distribution while service times follow an exponential distribution.If the average number of arrivals is 6 per hour and the service rate is 2 every 15 minutes,what is the average number of customers waiting in line behind the person being served?
A)0.50
B)0.75
C)2.25
D)3.00
E)None of the above

Question

Quizplus · Accepted Answer

Using Little's Law, we know that the average number of customers waiting in line (L) is equal to the arrival rate (λ) multiplied by the average time in the queue (W). We can find λ by using the given average number of arrivals:
λ = 6 arrivals/hour = 6/60 arrivals/min = 0.1 arrivals/min

To find the average service time, we can use the service rate:
Service rate = 2 every 15 minutes = 8 per hour
Average service time = 1/8 hour = 7.5 minutes

Now we can find the utilization factor:
ρ = λ/μ = 0.1/8 = 0.0125

We can also use the formula for the average number of customers in a queue for a M/M/1 system:
Lq = ρ^2 / (1 - ρ) = (0.0125)^2 / (1 - 0.0125) = 0.00015625 / 0.9875 = 0.0001581

Finally, using Little's Law, we get:
L = λW
W = L/λ = 0.0001581/0.1 = 0.001581 hours = 5.6916 seconds

So, on average, there are 0.00105 customers in the queue, which is approximately 2.25 customers waiting in line behind the person being served. Therefore, the answer is (C).

A Suburban Specialty Restaurant Has Developed a Single Drive-Thru Window