I'm forgotten most of this. I used M/M/1 Theory to refresh my memory. It seems fine although some of the characters are funky. I'm not sure what you have as a given. Also, I am assuming the is the arrival rate and is the service rate.
This is just an idea.
Well if you know that the
where N is the number of people in the queue when the person in question arrives. Breaking up the probabilities based on how many are in the queue makes good sense to me since it ought to be easier to figure out the rest. Then
Now you just have to figure out and then do the sum. Hint: think Poisson!
Also, I'm not sure why the problem was broken in two pieces. It seems to me that if the bottom expression becomes your top expression.