Orders resting at the bid

On a thin venue, the number of orders queued at the best bid is always between 0 and 4. Each tick, with probability $1/3$ a new order joins the queue (unless it already holds 4), with probability $1/2$ one order leaves because it fills or is cancelled (unless the queue is empty), and otherwise nothing changes.

In the long run, what fraction of ticks does the queue spend empty, and what is its average length? Starting from an empty queue, how many ticks does it take on average until the queue is next seen empty? A tick on which it simply stays empty counts as a return.