Question

Cars arrive at a parking lot according to a Poisson process with rate \lambda . There are only four parking spaces, and any car that arrives when all the parking spaces are occupied is lost. The parking duration of a car is exponentially distributed with mean 1/\mu . Let pk(t) denote the probability that k cars are parked in the lot at time t, k = 0, 1, . . . , 4. (a) Give the differential equation governing pk(t). (b What are the steady state values of these probabilities? (c) What is the mean first passage time to state 4 given that the process started in state 1?

          Cars arrive at a parking lot according to a Poisson process with rate \lambda . There are only four
parking spaces, and any car that arrives when all the parking spaces are occupied is lost.
The parking duration of a car is exponentially distributed with mean 1/\mu . Let pk(t) denote the
probability that k cars are parked in the lot at time t, k = 0, 1, . . . , 4.
(a) Give the differential equation governing pk(t).
(b What are the steady state values of these probabilities?
(c) What is the mean first passage time to state 4 given that the process started in state 1?

Added by Victoria R.

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