ps2_2017_solutions.pdf - MS&E 221 Ramesh Johari Problem Set...

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MS&E 221 Problem Set 2 Ramesh Johari Due: February 9, 2017, 12:00 PM , on Gradescope Problem 1. (a) The process is a Markov chain on states 0 , 1 , ..., N : (b) If p > 0 and q > 0 , then the chain is irreducible and all states will be recurrent. (c) Let h i = P ( at least one customer is blocked before the queue empties when the current state is i ) . Obviously h 0 = 0 . Use first transition analysis to obtain: h i = qh i - 1 + (1 - p - q ) h i + ph i +1 , for i = 1 , ..., N - 1 , and h N = qh N - 1 + (1 - p - q ) h N + p. Now we have a problem in the birth and death chain’s form, and the solution is h i = 8 < : i N +1 , if p = q 1 - ( p q ) i 1 - ( p q ) N +1 , if p 6 = q (d) Let’s express the probability that exactly three customers are blocked before the queue re- turns to being completely empty as P . We need the following events to occur: Starting from empty, our first move must be to state 1 (Prob = p ) From state 1, we must be blocked before returning to state 0 (Prob = h 1 ) After we are blocked for the first time, the chain will restart from state N due to the Strong Markov Property. Thus, from state N , we must be blocked 2 more times (Prob = h 3 N ) 1
Now, after we have been blocked for the 3rd time, the queue must return to empty before the next blocking (Prob = 1 - h N ) Thus, P will be the product of the probabilities listed above.
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