A simple model of a limit order book

We formulate a simplified model of a limit order book, in which the arrival process is independent of the current state. We prove a phase transition result: there exist prices $\kappa_b$ and $\kappa_a$ such that, for any $\epsilon > 0$, only finitely many bid (ask) departures occur at prices below $\kappa_b-\epsilon$ (above $\kappa_a+\epsilon$), while the interval $(\kappa_b+\epsilon, \kappa_a - \epsilon)$ infinitely often contains no bids, and infinitely often contains no asks. We derive expressions for $\kappa_b$ and $\kappa_a$, which we solve in the case of uniform arrivals. We conjecture the positive recurrence of a modified model, and find the steady-state distribution of the highest bid and of the lowest ask assuming the positive recurrence.