Automatic Parking Problem Solving

Dhifallah

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Joined
May 11, 2019
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I have a mathematical problem hope if you can help in solve it.
Consider that we have an Automatic Parking Garage contains 900 units , each unit can contains just one car.
The Garage open 24 hours per day , 7 days per week (24*7).
The working day will divide into three parts

1- (8:00 am - 3:00 Pm) ( the rent cost in this level is 1.2 $ per hour)
2- ( 3 Pm- 10 Pm) ( the rent cost in this level is 0.9 $ per hour)
3- ( 10 Pm- 8 am) ( the rent cost in this level is 2.2 $ Overnight)

The calculation well be for the first two cases like ( if the care parking for less on hour the pay will pe for one hour at least and if it is above one hour and less two hours the pay will be for two hours and so on), For the last case if the care parking for any time the pay well be like if it is stay the whole night .

When any new car want to parking the process needs 5 min, also when another car want to go out the garage it needs 5 min, so there is a wasting time.

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Now if we want to calculate the income for this garage in case the garage will
1- full capacity in 30 Days ( with consideration the waiting time when cars go in and out)
2-1/2 Capacity in 30 Days ( with consideration the waiting time when cars go in and out)
3- 1/4 Capacity in 30 Days ( with consideration the waiting time when cars go in and out)
4- 1/8 Capacity in 30 Days ( with consideration the waiting time when cars go in and out) .

How we can do it ?
 
Can you define full capacity as it is not obvious? How does full capacity affect the waiting time? As soon as some leaves a spot then someone else come into the spot immediately OR does it mean that exactly 900 cars are always in the garage.

Here is the big problem. Here are two cases and even forget about waiting times. Note that if we can compute the revenue for one day then we can simply multiply it my 30 to get the revenue for 30 days
Case 1: 900 cars stay for 24 hours
Case 2: 900 cars stay for 15 minute (forget waiting time)and then another 900 cars enter. Suppose this happens all day long.
Clearly case 2 will make more money than case 1.

This above argument is the same for 1/2 capacity, 1/4 capacity and 1/8 capacity.
 
Thank you for your reply Jomo, Is there is any mathematical way to figure out most the scenarios and the probability to calculate the revenue in different cases.
 
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