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September 19th, 2016, 05:56 AM   #1
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Markov chains

Hi, can someone help me with part (c) of this question? Thank you very much. Below is the full question.

In this problem, we model a queue using a Markov chain. (the queue might represent,
for example, customers waiting to buy something at a shop, a web server attempting to
respond to page requests, or students waiting during office hours.)

Suppose that a certain queue may contain 0, 1, or 2 items.
It is not possible for it to contain 3 or more items. At each time step, one of two things can happen:
(i)with probability 1/3, one item is removed from the queue, if there is an item to remove
(and otherwise nothing happens),
or (ii) with probability 2/3, one item is added to the queue, if there is enough space (i.e. if
there are not already 2 items in the queue) (and otherwise nothing happens).

(a) Formulate a finite Markov chain that describes this system.

(b)If the queue starts off empty, after 4 time steps what is the probability that it is again
empty?


(c) In the long term, what is the average rate at which items are removed from the queue?
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September 19th, 2016, 08:47 AM   #2
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What have you got so far?
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September 19th, 2016, 05:30 PM   #3
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I think that I have to calculate the steady state vector for c and I have done so but I am not sure how to continue from there.
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