Detailed Balance and Reversibility

Cancellation along every edge

Birth-death chains balance edge by edge

The Ehrenfest urn and its binomial equilibrium

Random walks on graphs and reversibility

How does detailed balance strengthen the stationary equation pi p equals pi?

In the three machines one repairman chain, the detailed-balance recursion gives relative weights 1, 10, 45 and 60 on states 0 through 3. What is the stationary probability that all three machines work?

What is the stationary distribution of the Ehrenfest chain with three balls, and what is the underlying reason?

For a random walk on a graph, how does the stationary probability of a vertex depend on the graph?

A certain three-state chain has the uniform vector stationary but has an edge with strictly positive probability in one direction and zero in the other. What can be concluded?

A stationary chain with distribution pi and matrix p is run backward in time. What is the transition probability of the reversed chain from i to j?

For the Ehrenfest chain with three balls, compute the stationary probability of state 1, exactly one ball in the left urn, and write the answer as a fraction or a decimal.

In the Ehrenfest chain with three balls, compute the stationary flow from state 1 to state 2, that is pi of 1 times p of 1, 2, and write the answer as a fraction or a decimal.

A knight performs a random walk on a chessboard, where the degrees of all 64 squares sum to 336. Compute the stationary probability of standing on a given corner square, which has degree 2, and write the answer as a fraction.