We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The environment is cast in terms of a Jackson/Gordon–Newell network although alternative interpretations are possible. The main tool is the detailed balance equations. The difference compared to earlier works is that the position of the random walk influences the transition intensities of the network environment and vice versa, creating strong correlations. The form of the stationary distribution is closely related to the well-known product formula.

Consider the following self-organizing rule called POS(i): after a book in the jth position of a shelf is borrowed, it is moved up one position if ji, and is moved to the ith position if j > i. This is a family of move-forward rules, with POS(l) being the move-to-front rule and POS(n − 1) being the transposition rule where n is the number of books to be organized. We derive explicitly the stationary distribution under the POS(i) rule and show that its search cost compares favorably with that of move-to-front rule under any book access probabilities p1, p1, ···, pn.

Books are arranged on a shelf. After use, a book is returned to the left hand end of the shelf. Expressions are derived for the probability that, in equilibrium, a given book is in a given position on the shelf, in terms of the frequencies with which the different books are demanded.