I have the following problem: A group of n processors is arranged in an ordered list. When a job arrives, the first processor in line attempts it; if it is unsuccessful, then the next in line tries it; if it too is unsuccessful, then the next in line tries it, and so on. When the job is successfully processed or after all processors have been unsuccessful, the job leaves the system. At this point we are allowed to reorder the processors, and a new job appears. Suppose that we use the one-closer reordering rule, which moves the processor that was successful one closer to the front of the line by interchanging its position with the one in front of it. If all processors were unsuccessful (or if the processor in the first position was successful), then the ordering remains the same. Suppose that each time processor i attempts a job then, independently of anything else, it is successful with probability $p_i$. Define an appropriate Markov chain to analyze this model. Show that it is time reversible. Find the long-run probabilities.
The solution from the solution manual is the following: Consider states $x = (x_1, . . . , x_{i−1}, x_i , x_{i+1}, . . . , x_n)$ and $x^1 = (x_1, . . . , x_{i−1}, x_{i+1}, x_i, . . . , x_n)$. Note that each time $j$ is moved to the left we multiply by $q_j/p_j$ and each time it moves to the right we multiply by $(q_j/p_j)^{−1}$. Since $x_j$, which is initially in position $j$, is to have a net move of $j − x_j$ positions to the left (so it will end up in position $j − ( j − x_j) = x_j$) it follows from the above that $\pi(x) = C\prod_j (q_{x_j}/p_{x_j})^{j−x_j}$. The value of $C$, which is equal to $\pi(1, 2, . . . , n)$, can be obtained by summing over all states $x$ and equating to 1. Since the solution given by the above value of $\pi(x)$ satisfies the time reversibility equations it follows that the chain is time reversible and these are the limiting probabilities.
I was able to obtain the following relation that needs to be proven to demonstrate time reversibility: $\pi(x)=(q_{x_{j+1}}/p_{x_{j+1}})(q_{x_j}/p_{x_j})^{-1}\pi(x^1)$, but I don't understand the above explanation why it is actually true. How can I find $\pi(x)$?