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 λi j . 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.