A queue at a bank starts with no one in it. Each minute, there is a...

 

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A queue at a bank starts with no one in it. Each minute, there is a 10% chance someone leaves the queue, a 20% chance that someone joins the queue, and a 70% chance that no one leaves or joins. Furthermore, if the queue already has 6 people in it, then any arrivals simply walk away rather than waiting in line. (a) (5 points) How many stationary distributions are there for the Markov chain modeling the queue? (b) (5 points) Compute the stationary distribution(s). (c) (5 points) In the long term, on average, what proportion of the time is the queue full? Does this depend on the initial distribution? A queue at a bank starts with no one in it. Each minute, there is a 10% chance someone leaves the queue, a 20% chance that someone joins the queue, and a 70% chance that no one leaves or joins. Furthermore, if the queue already has 6 people in it, then any arrivals simply walk away rather than waiting in line. (a) (5 points) How many stationary distributions are there for the Markov chain modeling the queue? (b) (5 points) Compute the stationary distribution(s). (c) (5 points) In the long term, on average, what proportion of the time is the queue full? Does this depend on the initial distribution?