Barber College is the best deal in town! For only $5, you can get a shampoo and cut by any one of four untrained stylists completed in 7.5 minutes on average. Perhaps this is why customers show up on average two minutes apart (both arrival times and service times are exponentially distributed). What percentage of the time are the four stylists working with customers?
A)94%
B)90%
C)84%
D)80%

Question

Quizplus · Accepted Answer

We can use the M/M/c queuing model to solve this problem. Here, c = 4 (number of servers) and ρ is the utilization of the servers, which is equal to λ/μc, where λ is the arrival rate and μ is the service rate of each server.
From the problem, we know that the arrival rate is 0.5 customers per minute (i.e. 2 minutes per customer) and the average service time is 7.5 minutes per customer. Therefore, μ = 1/7.5 = 0.1333 customers per minute.
Using the formula ρ = λ/μc, we get ρ = (0.5/0.1333*4) = 0.938.
Now, we can use the formula P(c busy) = ((c*ρ)^c / c!) / (Σ((c*ρ)^k / k!), k=0 to c), which gives us the probability that all servers are busy. P(c busy) equals the probability that the four stylists are working with customers.
Substituting the values, we get P(4 busy) = ((4*0.938)^4 / 4!) / (Σ((4*0.938)^k / k!), k=0 to 4) = 0.94 or 94%.
Therefore, the answer is (A) 94%.

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