Customers arrive at a music store at an average of 1 per minute (Poisson arrivals). The service rate is 15 customers per hour (exponential service times). What is the minimum number of servers needed to keep the waiting time in the system under 5 minutes?
A) 4
B) 5
C) 6
D) 7

Question

Quizplus · Accepted Answer

First, we need to calculate the utilization factor of the system. 
Utilization factor = Arrival rate * Service time = 1/minute * 1/15 hour/customer = 1/15

Using the formula for the waiting time in the M/M/s queueing system, Wq:

Wq = (p^s / s!(s-p)^2) * (1/(1-p/s))^2 * rho/(1-rho)

where p is the utilization factor, s is the number of servers, and rho is the traffic intensity (rho=p/s)

We want to find the minimum number of servers s that makes Wq<5 minutes.

Substituting the values we get:

Wq = (1/5^s*s!(1/14))^2 * (s/14*(s-p))^2 * 1/14(1-1/14)

Taking derivative we can find where the ratio is minimum, which is at s=5.

Therefore, the minimum number of servers needed to keep the waiting time in the system under 5 minutes is 5.

Customers Arrive at a Music Store at an Average of 1