Forgetting about the painless means/parameters confusion, I feel like this is almost correct (I just changed a notation), except that I think you count one too many: the time spent by the previous customer at server2 is included in your last term (it only matters if you're served before he exits), so you don't have to take it in account.now for the waiting time, I basically reasoned it as follows:
Assume that there is only one customer and he's at server 2, leaving server 1 empty.
so his expected time in the system will be (where I get the probability from http://www.mathhelpforum.com/math-he...tribution.html)
using the above result would yield:
where the last term is your wait time given that you finished with server 1 before the customer finished with server 2.
I would have expressed it differently, like: the waiting time is the sum of
- the waiting time before going to server 1, which equals (the serving time at server 1 + the possible waiting time to go to server 2)
- the possible waiting time to go to server 2.
Your computation heavily uses the "memoryless property" of exponential variables. Depending on what you're supposed to justify, additional explanation may be good, notably about the time spent at waiting that the other customer, if any, exits the system.