How to find the Laplace transform? Jobs arrive to a computer facility according to a Poisson process with rate $\lambda$ jobs / hour. Each job requires a service time $X$ which is uniformly distributed between $0$ and $T$ hours independently of all other jobs. 
Let $Y$ denote the service time for all jobs which arrive during a one hour period. How should I find the Laplace transform of $Y$?

The general form of a Poisson process (or a Levy process) can be defined as; the number of events in time interval $(t, t + T]$ follows a Poisson distribution with associated parameter λT. This relation is given as
\begin{equation}
P[(N(t+T)-N(t))=k] = \frac{(\lambda T)^k e^{- \lambda T}}{k!}
\end{equation}
Where $N(t + T) − N(t) = k$ is the number of events in time interval $(t, t + T]$. It will be obvious to you that $\lambda$ is the rate parameter. Assume in the simplest case for some $T$ where $k=1$ then
\begin{equation}
f(\lambda) = \lambda e^{-\lambda T}
\end{equation}
Then taking Laplace transforms yields
\begin{equation}
\widehat{f(t)} = \frac{\lambda}{\lambda + s}
\end{equation}
I'll leave it to you to fill in the more specific details, I've only dropped the textbook conclusions from a Poisson process and the Laplace transform of such a process.
