411 solve the previous problem in the concrete case 3

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Unformatted text preview: A 1 . 150 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS If Xt is irreducible and has stationary distribution ⇡ then pt (i, j ) ! ⇡ (j ) as t ! 1 Detailed balance condition. A su cient condition to be stationary is that ⇡ (i)q (i, j ) = ⇡ (j )q (j, i) There may not be a stationary distribution with this property, but there is one if we have a birth and death chain: i.e., the state space is {0, 1, . . . r}, where r may be 1, and we have q (i, j ) = 0 when |i j | > 1. In this case we have ⇡ (n) = ··· 0 · ⇡ (0) µn · · · µ1 n1 Queues provide a number of interesting examples of birth and death chains. 4.8 Exercises 4.1. A salesman flies around between Atlanta, Boston, and Chicago as follows. A B C A 4 3 5 B 2 4 0 C 2 1 5 (a) Find the limiting fraction of time she spends in each city. (b) What is her average number of trips each year from Boston to Atlanta? 4.2. A small computer store has room to display up to 3 computers for sale. Customers come at times of a Poisson process with rate 2 per week to buy a computer and will buy one if at least 1 is available. When the store has only...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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