cars get parked in a lot with infinite capacity according to a rate poisson process and indepen

1a. The number of cars parked at time t, N(t), is the difference between the number of cars that

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Cars get parked in a lot (with infinite capacity) according to a λ-rate Poisson process, and independently stay parked for a random duration. The parking time duration of a car follows a common distribution X, with cdf F(x) = P(X ≤ x). Let N(t) be the number of cars parked at time t. 1a. What is the distribution of N(t)? 1b. Assuming a car arrival rate of 1 per minute, and X (in min) ∼ Gamma(3, 1) (as defined in class), what is the expected number of cars parked after 1 hour? 2. In the long run (t → ∞), what is the expected number of cars, as a function of λ and the moments of X?

          Cars get parked in a lot (with infinite capacity) according to a λ-rate Poisson process, and independently stay parked for a random duration. The parking time duration of a car follows a common distribution X, with cdf F(x) = P(X ≤ x). Let N(t) be the number of cars parked at time t.

1a. What is the distribution of N(t)?
1b. Assuming a car arrival rate of 1 per minute, and X (in min) ∼ Gamma(3, 1) (as defined in class), what is the expected number of cars parked after 1 hour?

2. In the long run (t → ∞), what is the expected number of cars, as a function of λ and the moments of X?

Added by Katie H.

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