I'm considering a problem with a travelling salesman - every morning, he's given a list of $N$ people to sell goods to, with $N$ as a Poisson($30$) rv. He goes through each meeting and spends his time meeting with the clients at an exponential rate - $X$ is an exponential with rate parameter $n$ from $N=n$. I'm trying to find the expected amount of time he spends each day doing his job.
I know that
$$E[T] = \sum_{n\ge 0} E[X \mid N=n] P(N=n)$$
However, I'm not sure how to find $ E[X \mid N=n] $...
My first attempt made me believe that it would simply be $n(\frac{1}{\lambda})$, but doing so would lead me to the unfortunate circumstances that $E[T] = 1$, which I know is false.
What should I be doing?