I would assume that customers arrive at the queue according to the poisson process, and then decide whether to enter the queue or leave as per the rules in the question.
I interpret "enter the system" as "join the queue".
The expected time for this will be
E(time until there is a free slot) + E(time for someone to arrive once a slot is free).
Noting that the additional time taken for someone to arrive once a spot is free is independant of the time that the slot became free (memorylessness property of poisson process)
The waiting time of a is with mean
Your post suggests you already understand where comes from.