Probability of the waiting time M/G/1 and G/M/1 queue Task:
Given is
$F(t)=1-\frac{1}{2}e^{-2t}-\frac{1}{2}e^{-\frac{1}{2}t}$ and $\rho=\frac{1}{2}$.
My task is to calculate 
1) $\mathbb{E}[W]$, $\mathbb{P}(W>0)$ and $\mathbb{P}(W>1)$ in an M/G/1 queue with $A_i$ i.i.d. $\sim$ exp and $B_i$ i.i.d. $\sim$ $F(t)$.
2) $\mathbb{E}[W]$, $\mathbb{P}(W>0)$ and $\mathbb{P}(W>1)$ in an G/M/1 queue with $A_i$ i.i.d. $\sim$ $F(t)$ and $B_i$ i.i.d. $\sim$ exp.
W denotes the waiting time.
My ideas so far:
For $\mathbb{E}[W]$ in task 1) I used the formular 
$$
\mathbb{E}[W]=\frac{\rho}{1-\rho} \frac{\mathbb{E}[B^2]}{2 \mathbb{E}[B]}
$$
and calculated the first of second moment of the service time $B$ with the Laplace-Stieljtes Transform.
My result for this is
$$
\tilde{B}(s)=\frac{1}{s+2} + \frac{1}{3s+2}.
$$
which gives 
$\mathbb{E}[W]= \frac{5}{4}.$
Moreover, I know the Laplace Stieltjes Transform for the waiting time. 
Can I use this to calculate $\mathbb{P}(W>0)$ and $\mathbb{P}(W>1)$? If yes, how?
What can I do for the G/M/1 queue?
 A: 1) For a general M/G/1 you only have an expression for the Laplace transform of the waiting time (Pollaczek-Khinchine formula), and so typically there are no explicit expressions for the probability $\mathbb{P}(W>t)$. The exception is the atom at zero which can be computed for example by Little's law:
$$
\mathbb{P}(W>0)=\mathbb{E}\mathbf{1}(W>0)=\lambda \mathbb{E}[B]=\rho.
$$
Other probabilities you can only approximate by numerical inversion of the Laplace Transform.
2) For the G/M/1 it is actually easier because the sojourn time is exponentially distributed (as in the M/M/1 queue). This can be shown by using the fact that the queue length upon arrival is geometric with parameter $\sigma$ that is the solution of
$$
\sigma=\tilde{A}(\mu(1-\sigma)),
$$
where $\tilde{A}(s)=\mathbb{E}[e^{-sA}]$ is the LST of the inter-arrival times. The sojourn time is then given by a geometric sum of exponential random variables with rate $\mu$ and so it is also exponential with rate $\mu(1-\sigma)$. 
The waiting time distribution is then (again, as in the M/M/1 queue) a mixture of an atom with probability $1-\rho$ at zero and an exponential distribution with probability $\rho$. Therefore,
$$
\mathbb{P}(W> t)=e^{-\mu(1-\sigma)t}.
$$ 
*The above assumes that you are familiar with the standard transform analysis of the queues, otherwise some more details are required and I suggest going over the relevant chapters in a queueing theory text book.
