Probability, Markov Chains, Queues, and Simulation : The Mathematical Basis of Performance Modeling by Stewart William J

Author:Stewart, William J.
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2009-11-15T00:00:00+00:00

10. 8 Exercises

Exercise 10.1.1 State Gerschgorin’s theorem and use it to prove that the eigenvalues of a stochastic matrix cannot exceed 1 in magnitude.

Exercise 10.1.2 Show that if λ is an eigenvalue of P and x its associated left-hand eigenvector, then 1 − α(1 − λ) is an eigenvalue of P(α) = I − α(I − P) and has the same left-hand eigenvector x, where (i.e., the real line with zero deleted). Consider the matrix

obtained from the previous procedure. What range of the parameter α results in the matrix P being stochastic? Compute the stationary distribution of P.

Exercise 10.1.3 Look up the Perron-Frobenius theorem from one of the standard texts. Use this theorem to show that the transition probability matrix of an irreducible Markov chain possesses a simple unit eigenvalue.

Exercise 10.1.4 Part of the Perron-Frobenius theorem provides information concerning the eigenvalues of transition probability matrices derived from periodic Markov chains. State this part of the Perron-Frobenius theorem and use it to compute all the eigenvalues of the following two stochastic matrices given that the first has an eigenvalue equal to 0 and the second has an eigenvalue equal to −0.4:

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