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?
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.