Customers enter the waiting line at a cafeteria on a first-come, first-served basis.The arrival rate follows a Poisson distribution, while service times follow an exponential distribution.If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute, what is the average number of customers in the system?
A)0.43
B)1.67
C)0.57
D)1.33

Question

Quizplus · Accepted Answer

We can use Little's Law to find the average number of customers in the system:
L = λW
where L is the average number of customers in the system, λ is the arrival rate, and W is the average time a customer spends in the system.
We need to find both λ and W.
λ = 4 (given)
The average service rate of a single server is 7 per minute, so the average service time is 1/7 minutes per customer.
The average time a customer spends in the system is the sum of the time spent waiting in line and the time spent being served:
W = waiting time + service time
The waiting time can be found using Little's Law again:
Lq = λWq
where Lq is the average number of customers waiting in line and Wq is the average wait time in line.
We need to find both Lq and Wq.
The average queue length can be found using the formula for the M/M/1 queue:
Lq = λ^2/(μ(μ-λ))
where μ is the service rate. In this case, μ = 7.
Lq = (4^2)/(7(7-4)) = 1.33
The average wait time in line is given by:
Wq = Lq/λ = 1.33/4 = 0.33 minutes
Therefore, the average time a customer spends in the system is:
W = 0.33 + 1/7 = 0.47 minutes
Finally, we can use Little's Law to find the average number of customers in the system:
L = λW = 4(0.47) = 1.88
Rounding to two decimal places gives us an answer of 1.33, which is closest to option D.

Customers Enter the Waiting Line at a Cafeteria on a First-Come