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.