A truck transports goods among $10$ points located on a circular route. These goods are carried only from one point to the next with probability $p$, or to the preceding point with probability $q=1-p$.

- Write the transition probability matrix.
- Find the limiting stationary distribution.
- Write your conclusions about this question.

The transition matrix is the circulant matrix $M = q cdot P + p cdot P^T$, where $P$ is the permutation matrix in the link. Computing the stationary distribution can be done by computing the solution to the system $(M – I)x = 0$.

However, rather than solving this system of equations, we can more easily prove that your guess of the stationary distribution $pi = (1/10,dots,1/10)$ is correct by verifying that $pi M = M$. To see that this holds, note that $pi = frac 1{10} (1,dots,1)$, and that the entries of $(1,dots,1)M$ are the column-sums of $M$. The only non-zero entries of a given column of $M$ are $p$ and $q$, which means that every entry of $(1,dots,1)M$ will be $p+q = 1$, which means that we have

$$

(1,dots,1)M = (1,dots,1)M implies pi M = pi.

$$

So, $pi$ is indeed the stationary distribution.