Question

The next four questions are based on the following information. At a car wash station, on average, there are 4 cars coming in for the service every 10 minutes. The average wash time is 2 minutes. The Poisson distribution is appropriate for the arrival rate and service times are exponentially distributed. Please convert all rates into cars per hour and answer the following questions.

1.

The average time a car spent in the waiting line is ___________ hours, and the total time a car spent in this car wash station is _____________ hour. (Please round to two decimal points and include no units.)

2.  

The average number of cars in this car wash station is __________ . (Please round to the closest integer and include no units.)

3.

The probability that there are no cars in this station is __________ . (Please round to one decimal points and include no units.)

4.

The probability that there are exactly two cars in this station is_____________ . (Please round to three decimal points and include no units.)

Answer 1

Arrival rate = λ = 4 cars per 10 mins = 24 cars / hour

Service Rate = μ = 2 mins per car = 30 cars / hour

1. The average time a car spent in the waiting line = λ/μ(μ-λ) =  24/(30*6) = 0.13 hrs = 8 mins

and the total time a car spent in this car wash station = 1/(μ-λ) = 1/6 = 0.167 hrs = 10 mins

2.  The average number of cars in this car wash station = λ/(μ-λ) = 24/6 = 4

3. The probability that there are no cars in this station = P_o = 1 - λ/μ = 1 - 24/30 = 0.2

4. The probability that there are exactly two cars in this station = P₂ = (λ/μ)ⁿP_o = (24/30)²*0.2 = 0.128