A street noodle vendor (one server) in Singapore can service an average of 20 customers per hour. Given an average arrival rate of 12 customers per hour, use the Poisson distribution to calculate the probability that the vendor can handle the demand.
-Calculate the mean waiting time in the queue.
A)4.5 minutes
B)2.5 minutes
C)1.5 minutes
D)3.5 minutes
E)5 minutes

Question

Quizplus · Accepted Answer

The mean arrival rate is 12 customers per hour, and the mean service rate is 20 customers per hour. The probability that the vendor can handle the demand is given by the Poisson distribution:P(X = k) = (e^-λ * λ^k) / k!where λ is the mean arrival rate.In this case, λ = 12, so the probability that the vendor can handle the demand is:P(X = 0) = (e^-12 * 12^0) / 0! = 0.0002P(X = 1) = (e^-12 * 12^1) / 1! = 0.0024P(X = 2) = (e^-12 * 12^2) / 2! = 0.0144P(X = 3) = (e^-12 * 12^3) / 3! = 0.0576P(X = 4) = (e^-12 * 12^4) / 4! = 0.1728P(X = 5) = (e^-12 * 12^5) / 5! = 0.3456P(X = 6) = (e^-12 * 12^6) / 6! = 0.4147P(X = 7) = (e^-12 * 12^7) / 7! = 0.3456P(X = 8) = (e^-12 * 12^8) / 8! = 0.2304P(X = 9) = (e^-12 * 12^9) / 9! = 0.1209P(X = 10) = (e^-12 * 12^10) / 10! = 0.0504Therefore, the probability that the vendor can handle the demand is 0.0002 + 0.0024 + 0.0144 + 0.0576 + 0.1728 + 0.3456 + 0.4147 + 0.3456 + 0.2304 + 0.1209 + 0.0504 = 1.7544.The mean waiting time in the queue is given by:W = λ / (μ - λ)where μ is the mean service rate.In this case, μ = 20, so the mean waiting time in the queue is:W = 12 / (20 - 12) = 4.5 minutesTherefore, the correct answer is A.

A Street Noodle Vendor (One Server) in Singapore Can Service