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Starting at time t=0, cars arrive at a car wash according to a Poisson process with rate of cars/hour. At any given moment, the car wash is either...
Starting at time t=0, cars arrive at a car wash according to a Poisson process with rate of λ cars/hour. At any given moment, the car wash is either free or occupied. The car wash is initially free at time t=0. If a car arrives at the car wash when it is free, the car is serviced immediately. Service lasts for 1/4 hours, during which the car wash is occupied. If a car arrives at the car wash when it is occupied, the car is denied service and it leaves the car wash.
Whenever necessary, write down your answers using the standard notation. Type fact(n) for n!.
Write down the PMF pN(k) of N, the number of cars arriving at the car wash between times 0 and 3, in terms of λ and k.
For k=0,1,...,
A) pN(k)=
B) Find the probability that a car is accepted for service given that it arrives at time t=1/6. Write down your answer in terms of λ, using the standard notation.
C) Find the PDF fT2(t) of T2, the time until the second serviced car leaves the car wash.
For t≥0.5,
fT2(t)=
D) Find the probability that exactly one car is accepted for service between t=0 and t=1.
