Stochastic

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Unformatted text preview: Calls come in to the dispatcher at times of a Poisson process with rate 2 per hour. Suppose that each requires an exponential amount of time with mean 20 minutes, and that callers will hang up if they hear there are no cabs available. (a) What is the probability all three cabs are busy when a call comes in? (b) In the long run, on the average how many customers are served per hour? 4.23. Detailed balance for three state chains. Consider a chain with state space {1, 2, 3} in which q (i, j ) > 0 if i 6= j and suppose that there is a stationary distribution that satisfies the detailed balance condition. (a) Let ⇡ (1) = c. Use the detailed balance condition between 1 and 2 to find ⇡ (2) and between 2 and 3 to find ⇡ (3). (b) What conditions on the rates must be satisfied for there to be detailed balance between 1 and 3? 4.24. Kolmogorov cycle condition. Consider an irreducible Markov chain with state space S . We say that the cycle condition is satisfied if given a cycle of states x0 , x1 , . . . , xn = x0 with q (x...
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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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