Peter's goal in launching Auto-Luxo (AL) is to provide premium customers with the ability to rent luxury vehicles when they need it. Peter must now decide on the size of his fleet of luxury vehicles. Studies show that the duration of the rental of such cars follows an exponential distribution with an average of 3 days. This includes the time needed to clean and prepare the car for the next customer.

Peter expects to receive about 40 requests per week (the exact request per week is random and the interval between rentals is exponentially distributed). He plans to operate the business seven days a week. Also suppose that customers who want to rent a car when one is not available are added to a waiting list on a "first come, first served" basis, until they get their vehicle.

Part 1 [2 points] How many vehicles should Peter have in his fleet if his goal is that on average there are between 2 and 3 customers on the list? In this case, what proportion of customers obtain their vehicle immediately, without going through the list?

Part 2 [2 points] How many vehicles should Peter have in his fleet if his goal is that, on average, the proportion of customers who must go through the list before obtaining a vehicle is 5%? In this case, on average, how many customers are on the waiting list?

 

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Pour le problme 5 Scenario Taux d'arrives Taux de services #de serveurs Q Capacit W Wq L Lq 1 2.33 inf 2 3 3 2.33 4 0.56 0.13 22.39 5.25 0.34 0.01 13.58 0.24 0.49 0.06 19.58 2.44 0.34 0.00 13.46 0.12 0.46 0.03 18.40 1.26 0.33 0.00 13.40 0.06 0.45 0.02 17.82 0.68 0.33 0.00 13.36 0.03 0.44 0.01 17.52 0.37 0.33 0.00 13.35 0.02 0.43 0.01 17.35 0.21 0.33 0.00 13.34 0.01 0.43 0.00 17.26 0.11 0.33 0.00 13.34 0.00 0.43 0.00 17.20 0.06 0.33 0.00 13.33 0.00 0.43 0.00 17.18 0.03 0.33 0.00 13.33 0.00 5 6 7 8 9 10 11 12 13 14 15 16 17 18 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 2.33 3 2.33 2.33 3 2.33 223323 2.33 2.33 3 2.33 3 ~~28~~~~~~~~~~8855 19 19 20 20 21 21 22 22 23 23 24 24 26 26 27 27 EEEEEEEEEE inf inf inf inf inf inf inf inf inf inf inf inf EEEEEE inf inf inf inf inf Util PO PrBalk PrWait 0.90 0.00 0.00 0.57 0.70 0.00 0.00 0.10 0.86 0.00 0.00 0.41 0.67 0.00 0.00 0.06 0.82 0.00 0.00 0.63 0.00 0.00 0.78 0.00 0.61 0.00 0.75 0.00 0.58 0.00 0.71 0.00 0.56 0.00 0.69 0.00 0.53 0.00 0.66 0.00 0.51 0.00 0.63 0.00 0.49 0.00 0.28 0.04 0.00 0.19 0.02 0.13 0.01 0.08 0.01 0.05 0.00 0.03 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Pour le problme 5 Scenario Taux d'arrives Taux de services #de serveurs Q Capacit W Wq L Lq 1 2.33 inf 2 3 3 2.33 4 0.56 0.13 22.39 5.25 0.34 0.01 13.58 0.24 0.49 0.06 19.58 2.44 0.34 0.00 13.46 0.12 0.46 0.03 18.40 1.26 0.33 0.00 13.40 0.06 0.45 0.02 17.82 0.68 0.33 0.00 13.36 0.03 0.44 0.01 17.52 0.37 0.33 0.00 13.35 0.02 0.43 0.01 17.35 0.21 0.33 0.00 13.34 0.01 0.43 0.00 17.26 0.11 0.33 0.00 13.34 0.00 0.43 0.00 17.20 0.06 0.33 0.00 13.33 0.00 0.43 0.00 17.18 0.03 0.33 0.00 13.33 0.00 5 6 7 8 9 10 11 12 13 14 15 16 17 18 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 3 2.33 3 2.33 2.33 3 2.33 223323 2.33 2.33 3 2.33 3 ~~28~~~~~~~~~~8855 19 19 20 20 21 21 22 22 23 23 24 24 26 26 27 27 EEEEEEEEEE inf inf inf inf inf inf inf inf inf inf inf inf EEEEEE inf inf inf inf inf Util PO PrBalk PrWait 0.90 0.00 0.00 0.57 0.70 0.00 0.00 0.10 0.86 0.00 0.00 0.41 0.67 0.00 0.00 0.06 0.82 0.00 0.00 0.63 0.00 0.00 0.78 0.00 0.61 0.00 0.75 0.00 0.58 0.00 0.71 0.00 0.56 0.00 0.69 0.00 0.53 0.00 0.66 0.00 0.51 0.00 0.63 0.00 0.49 0.00 0.28 0.04 0.00 0.19 0.02 0.13 0.01 0.08 0.01 0.05 0.00 0.03 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00