Empty taxis arrive at the airport pickup area according to a Poisson process at a rate...

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Empty taxis arrive at the airport pickup area according to a Poisson process at a rate of 3 per minute. Passengers arrive at the airport pickup area according to a Poisson process of 2 per minute. The taxis pick up a single passenger at a time if there are any that are waiting. The passengers will only wait for a taxi if there are less than 4 people people waiting ahead of them. (a) You arrive at the pickup area and there is nobody else in line for a taxi. What is the expected time for you to be picked up? (b) Let us model this process with a continuous-time Markov chain. Let the state space be given by {0,1, 2, 3, 4} denoting the number of people waiting at the pickup area. Show that the steady-state probabilities are given by: 81 54 36 24 16 По П II2 II3 II4 211 211 211 211 ' 211 (c) Assume that the process is in its steady-state. If you arrive at the pickup area and you join the queue, what is the expected number of people ahead of you in the line? (d) Assume that the process is in its steady-state. If you arrive at the pickup area and you join the queue, what is your expected waiting time for a taxi? Empty taxis arrive at the airport pickup area according to a Poisson process at a rate of 3 per minute. Passengers arrive at the airport pickup area according to a Poisson process of 2 per minute. The taxis pick up a single passenger at a time if there are any that are waiting. The passengers will only wait for a taxi if there are less than 4 people people waiting ahead of them. (a) You arrive at the pickup area and there is nobody else in line for a taxi. What is the expected time for you to be picked up? (b) Let us model this process with a continuous-time Markov chain. Let the state space be given by {0,1, 2, 3, 4} denoting the number of people waiting at the pickup area. Show that the steady-state probabilities are given by: 81 54 36 24 16 По П II2 II3 II4 211 211 211 211 ' 211 (c) Assume that the process is in its steady-state. If you arrive at the pickup area and you join the queue, what is the expected number of people ahead of you in the line? (d) Assume that the process is in its steady-state. If you arrive at the pickup area and you join the queue, what is your expected waiting time for a taxi?