need detail solutions!!!!! A student observes the elevator usage of a building w
ID: 3128330 • Letter: N
Question
need detail solutions!!!!!
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 10^th of the building. Once passengers are placed on the 10^th 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 10^th 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 10^th 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 poison process with rate X = 1 per minute. If the elevator just left with 3 people, what is the probability that it will have to leave immediately again after returning from the 10^th floor? What is the probability that an elevator must take exactly 4 people? If an elevator has been idle for 30 seconds, what is the expected time until it must make another trip? 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?Explanation / Answer
(a) a poisson process is memoryless
=>
P(X>=1) = 1 - P(X=0) = 1 - ( e^-5*(5)^0/0!) = 1 - e^-5 =0.993262
(b)
P(X=4) = e^-5 * 5^(4) / 4! = 0.17546737
(c) again memoryless
it will make trip as soon as 1 person comes
and arrival rate is 1 per minute
=> 1 minute
(d) time between two arrivals of a poisson process is given by Exponential distr.
f(t) = lambda * e^(-lambda*t)
lambda = rate = 1 per minute
P(t <0.5 min)
= (integral 0 to 0.5) (1*e^(-1*t)) dt
= e^0 - e^-0.5
= 1 - e^-0.5
= 0.39347
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