The number of customers that arrive at a service station during a time t is a Poisson
random variable with parameter \beta t . The time required to service each customer is an exponential random variable with parameter a .
Identify the distribution of the number of customers N that arrive during the service time T of a speci ficc customer by:

finding the MGF of N

f(t)=ae^{-at}
N(n)=\frac{\beta^n t^n}{n!}} e^{-\beta t}

I have found the respective moment generating functions which are:

M_f (s)= \frac{a}{a-s} for the exponential distribution and

M_N (s)= e^{\beta t{e^s-\beta t}

Now i have no idea how to combine the results to find the required PDF.
And i cant find anything in my notes about this.