Compute mean value of the exponential of product of 2 exponential variables

Question

I have to compute $E[e^{vS}]$, where $v$ and $S$ are two independent RVs exponentially distributed with different parameters, $\lambda_1, \lambda_2$.

According to this question I started computing $Y = vS$, but then it is hard to continue with the integral and $e^Y$

How can I solve it?

EDIT 1

In particular, I have to compute the mean completion time $C$ of processes in a queuing system with priorities: $$E[C] = (E[e^{vS}]-1)*({E[D]+ \frac{1}{v}})$$ so $v$ is an interarrival time, $S$ is a service time and $D$ is the duration of the process with higher priority.

EDIT 2 The paper that I am studying is this, in I am trying to solve $E[C]$ and $E[C^2 ]$ on page 8. In the start pages, there is the therminolgy used

Answer 1

Your mean is$$\int_0^\infty dv\int_0^\infty dS\lambda_1\lambda_2\exp(vS-\lambda_1v-\lambda_2S)=\int_0^\infty dv\int_0^\infty dS\lambda_1\lambda_2\exp((S-\lambda_1)(v-\lambda_2)-\lambda_1\lambda_2).$$The region $S>\lambda_1,\,v>\lambda_2$ makes an infinite contribution $\lambda_1\lambda_2\exp(-\lambda_1\lambda_2)\int_0^\infty da\int_0^\infty db\exp(ab)$ with $a:=S-\lambda_1,\,b:=\lambda_2$, so the final result is $\infty$.

Answer 2

Making a start:$$\mu=\int_{0}^{\infty}\int_{0}^{\infty}e^{xy}\lambda_{1}\lambda_{2}e^{-\lambda_{1}x}e^{-\lambda_{2}y}dxdy=\lambda_{1}\lambda_{2}\int_{0}^{\infty}e^{-\lambda_{2}y}\int_{0}^{\infty}e^{x\left(y-\lambda_{1}\right)}dxdy$$

Now observe that $\int_{0}^{\infty}e^{x\left(y-\lambda_{1}\right)}dx=\infty$ if $y>\lambda_{1}$ and consequently $\mu=\infty$.