This paper analyzes a tractable Markov model of a limit order book (LOB) driven by stochastic fluctuations between supply and demand from long-term investors. The model assumes Poisson arrivals of unit-sized bids and asks with independent prices, no cancellations, and immediate matching of compatible orders. The authors establish the existence of thresholds κ_b and κ_a such that bids below κ_b and asks above κ_a are never executed, while the interval (κ_b, κ_a) is recurrently emptied. Using fluid limits, they prove positive recurrence of binned (discretized) versions of the model, which justifies the derivation of limiting distributions for the highest bid and lowest ask. These distributions satisfy a system of differential equations derived from probability flux balance, and explicit solutions are given for the case of uniform arrival prices. The model is extended to include market orders and high-frequency trading strategies, such as market making, sniping, and mixtures thereof. The authors compute optimal strategies for a single trader and analyze Nash equilibria when multiple high-frequency traders compete. They find that competition among sniping strategies leads to a high rate of order cancellations, a feature observed in real markets. The paper also compares continuous-time trading with frequent batch auctions, showing that batch auctions slightly reduce profits for sniping traders. Overall, the model provides a mathematically rigorous foundation for understanding price formation and strategic trading in limit order books, with implications for market design.