In a spirit akin to the sandpile model of self- organized criticality, we present a simple statistical model of the cellular-automaton type which simulates the role of an asperity in the dynamics of a one-dimensional fault. This model produces an earthquake spectrum similar to the characteristic-earthquake behaviour of some seismic faults. This model, that has no parameter, is amenable to an algebraic description as a Markov Chain. This possibility illuminates some important results, obtained by Monte Carlo simulations, such as the earthquake size-frequency relation and the recurrence time of the characteristic earthquake.