I have problems to derive the following:
"A gas station offers two services. For each service customers arrive according to a Poisson process. On average 20 customers per hour for service 1 and 5 customers for service 2. The service times are exponential, both with a mean service time of two minutes.
Determine the distribution of the total number of customers in the system."
Using minutes as time units, I get $\rho_1=\lambda_1 \cdot \mathbb{E}[B]= \frac{1}{3} \cdot 2 = \frac{2}{3}$ and $\rho_2=\lambda_2 \cdot \mathbb{E}[B]= \frac{1}{12} \cdot 2 = \frac{1}{6}$. So the distribution of the number of customers looking at the different services individually is $$\mathbb{P}(L_1=n)=(1-\rho_1)(\rho_1)^n=\frac{1}{3}\Big(\frac{2}{3}\Big)^n$$ and $$\mathbb{P}(L_2=n)=(1-\rho_2)(\rho_2)^n=\frac{5}{6}\Big(\frac{1}{6}\Big)^n .$$
For the distibution of the total number of customers, I thought about $$\mathbb{P}(L=n)=\sum_{k=0}^{n}\mathbb{P}(L_1=k,L_2=n-k).$$
From there, I do not really know how to compute is further. Am I on the right track? Does anybody has a hint how to compute it?