Bobby is considering building a car wash with 2 washing stations that can be modeled as an M/M/2/10 queuing system. He predicts that cars will arrive at a rate of λ=10 cars/hr. On average it takes 15 minutes to wash a car (so μ=4 cars/hr). For each car wash Bobby charges
$20.

(a) Draw the rate transition diagram with the numerical values for the arrival and service rates
(b) Calculate the following quantities
i. Expected time customers will have to wait inline
ii. The probability of having n = 0,1,2,...,10 customers
iii. The effective arrival rate
iv. How many cars per hour are unable to enter the car wash? What is the lost revenue
(per hour)?
(c) Bobby is now considering building another washing station for $100,000 (so now it’s an M/M/3/10 system). Calculate the following quantities
i. Expected time customers will have to wait in line
ii. The effective arrival rate
iii. How many cars per hour are unable to enter the car wash? What is the lost revenue (per hour)?
iv. How much revenue per hour is added with the new washing station (find the differ-ence between b-iv and c-iii)? If the car-wash is open 12 hours per day, how many days will it take to pay-off the extra construction costs?