30 Consider a graph with nodes 1, 2, , n and the n 2 arcs (i, j), i j, i, j, 1, , n (See Section 3 6 2 for appropriate definitions ) Suppose that a particle moves along this graph as follows Events occur along the arcs (i, j) according to independent Poisson processes with rates ij An event along arc (i, j) causes that arc to become excited If the particle is at node i at the moment that (i, j) becomes excited, it instantaneously moves to node j, i, j 1, , n Let Pj denote the proportion of time that the particle is at node j Show that Pj 1 n Hint Use time reversibility

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Question: 30. Consider a graph with nodes 1, 2, . . . , n and the n 2 arcs (i, j), i = j, i,

30. Consider a graph with nodes 1, 2, . . . , n and the

n 2

arcs (i, j), i = j, i, j,= 1, . . . , n.

(See Section 3.6.2 for appropriate definitions.) Suppose that a particle moves along this graph as follows: Events occur along the arcs (i, j) according to independent Poisson processes with rates λij . An event along arc (i, j) causes that arc to become excited. If the particle is at node i at the moment that (i, j) becomes excited, it instantaneously moves to node j, i, j = 1, . . . , n. Let Pj denote the proportion of time that the particle is at node j. Show that Pj = 1 n

Hint: Use time reversibility.

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