Probability question using no-memory property of exponential distribution 
A customer must be served first by server 1, then by server 2, and
  finally by server 3. The amount of time required for service by server
  i is an exponential random variable with rate µi , for i = 1, 2, 3.
  Suppose you enter the system when it contains a single customer who is
  being served by server 3.

Find the expected amount of time that you spend in the system. (Whenever you encounter a busy server, you must wait for the service in progress to end before you can enter service.)
Attempt:
$1/\mu_1+1/\mu_2+1\mu_3 \frac{\mu_1}{\mu_1+\mu_2} \frac{\mu_2}{\mu_2+\mu_3)}1/\mu_3$

I'm not sure if this is correct, but you can do this with a conditioning argument: $E(T_1+T_2+T_3)+E(S|S>T_1+T_2)P(S>T_1+T_2)+E(S|S<T_1+T_2)P(S<T_1+T_2)=\frac{1}{\mu_1}+\frac{1}{\mu_2}+\frac{1}{\mu_3}+\frac{\mu_1}{\mu_1+\mu_3}\frac{\mu_2}{\mu_2+\mu_3}\frac{1}{\mu_3}$, since $E(S|S<T_1+T_2)=0$.
