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.