A student observes the elevator usage of a building with a single elevator. This elevator will only allow people to go from the main floor to the 10th of the building. Once passengers are placed on the 10th floor, the elevator will then return back to the main floor, regardless of whether there are people waiting on the main floor or not. The entire trip from the main floor, up to the 10th floor, and back down takes exactly 5 minutes. If there are no people waiting, the elevator will idle on the main floor. If at least 1 person is present, the elevator will take those waiting up to the 10th floor. However, the elevator can only take at most 6 people. If the requested number of users ever exceeds 6, the first 6 people to arrive will be allowed on and the remaining must take the stairs (they do not wait for the elevator to return).
The student notices that people arrive according to a Poisson process with rate ? = 1 per minute.
a) If the elevator just left with 3 people, what is the probability that it will have to leave immediately again after returning from the 10th floor?
b) What is the probability that an elevator must take exactly 4 people?
c) If an elevator has been idle for 30 seconds, what is the expected time until it must make
another trip?
d) A new system is set up where the elevator will only leave immediately if there are 2 or more
people waiting. If there is only a single person, the elevator will wait 1 extra minute before leaving. Given that a passenger waits while an elevator is idle, what is the probability that the extra length of their wait is less than 30 seconds?
e) Assume that the managers decide that instead of waiting for 2 passengers, they prefer 3 passengers to be present before an elevator leaves. However, they also want to ensure that if an elevator must wait, only 50% of the times will it leave before reaching the required 3 people. How long must they set their waiting time for this to happen?