A small diner has one employee and a counter with seating for 8 customers.The diner does not package food for take out.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 probability that the diner is full and an arriving customer does not enter?
A)0.8333
B)0.1667
C)0.2067
D)0.0481

Question

Quizplus · Accepted Answer

The arrival rate is λ = 20 customers per hour and the service rate is μ = 24 customers per hour. The utilization factor ρ = λ/μ = (20/24) = 0.8333.

The probability that a customer has to wait for a seat is given by the formula:

P(wait for seat) = (1 - ρ) ρ^n / (1 - ρ^(n+1))

where n is the number of seats. In this case, n = 8. Plugging in the values, we get:

P(wait for seat) = (1 - 0.8333) (0.8333)^8 / (1 - 0.8333^9)
                = 0.0479

The probability that an arriving customer does not enter the diner is the same as the probability that there are already 8 customers in the diner, which is:

P(diner full) = ρ^n (1 - ρ) / (1 - ρ^(n+1))
              = (0.8333)^8 (1 - 0.8333) / (1 - 0.8333^9)
              = 0.0481

Therefore, the answer is D.

A Small Diner Has One Employee and a Counter with Seating