This paper introduces a tractable Markovian model for the dynamics of a limit order book, focusing on the best bid and ask queues. The model assumes that limit orders, market orders, and cancellations arrive as independent Poisson processes, and that the spread is fixed at one tick. The state of the order book is captured by the sizes of the bid and ask queues, which evolve as a two-dimensional birth-death process. When a queue is depleted, the price moves by one tick, and new queue sizes are drawn from a stationary distribution. The model is analytically tractable, allowing the derivation of closed-form expressions for key quantities such as the conditional distribution of the time until the next price change, the probability of an upward price move given the current queue sizes, and the autocorrelation structure of price changes. The authors also study the diffusion limit of the price process, showing that at longer time scales the price behaves like a Brownian motion with volatility expressed in terms of the arrival rates of orders and cancellations. The model's predictions are validated against empirical data from stocks such as Citigroup, General Electric, and General Motors, showing good agreement with observed transition frequencies and volatility patterns. This work provides a structural link between order flow and price dynamics, offering insights for high-frequency trading and risk management.