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.

for (a)

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.