6.262.PS9 - degree of vertex k d(k is the number of edges...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 – Discrete Stochastic Processes Problem Set # 9 Issued: April 16, 2010 Due: April 23, 2010 Reading: Sections 6.1 – 6.3 1) Exercise 5.10 (Use figure 5.5 rather than 5.4. Assume the system is positive recurrent for parts a) and c) only. Recall from pp. 130 – 133 that Little’s theorem holds quite generally for G/G/m queues. 2) Random Walk on a Graph A graph G is a collection of n vertices ( i.e., nodes ) connected by l edges (i.e., links ). A graph is said to be connected if there is a path from every vertex to every other vertex in the graph. A self-loop is an edge that begins and ends at the same vertex. Two vertices are said to be neighbors if they are connected by an edge. The
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Unformatted text preview: degree of vertex k, d(k), is the number of edges that terminate at vertex k. Four examples of connected graphs with no self-loops are shown below. Let G be a finite connected graph with no self-loops. A random walk on G starts at a given vertex, and at each time step it moves to one of its neighbors. All its neighbors are chosen with equal probability. a) Explain how this random walk can be represented as a finite Markov chain. Determine the transition probabilities P ij in terms of the interconnectivity of the graph. b) Find the steady-state probabilities π ur for this random walk. Is the Markov chain reversible? 3) Problem 6.7 from the course notes. 4) Problem 6.11 from the course notes. 5) Problem 6.16 from the course notes....
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6.262.PS9 - degree of vertex k d(k is the number of edges...

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