Markov Chains & Long-Run Behavior · Challenge problem

A buffer holds at most $2$ jobs. Each time step, independently, a new job arrives with probability $1/2$, and, if the buffer was non-empty at the start of the step, one job finishes with probability $1/2$. A job arriving when the buffer is full and no job finishes in that step is lost; if a job finishes and one arrives in the same step, the count is unchanged. In the long run, what fraction of time steps start with a full buffer? Express your answer as a common fraction.