Dealing with a geometric distribution here's a question we got for homework. Please excuse my poor translation. Anyhow:

A cellphone company has n1 call users and n2 text users. The
  probabilities of a user to be connected at a given moment are p1
  and p2 accordingly (the probabilities are independent). Also, the
  bit-rate for each service is r1 and r2 bit per second. 
If the maximum network capacity is C bits-per-second, what are the
  chances that the system will work at a given moment?

What is the random variable that I should define? 
I need a hint to get me started, because every idea I have seems to be way to complicated. Thanks!
 A: Independence does not seem physically plausible, but we are explicitly asked to assume it.
Let $X_1$ and $X_2$, respectively, be the numbers of call users and text users connected, or at least wishing to connect.  Then $X_1$ and $X_2$ have binomial distribution. (In the real world there may be truncation.) Let  $W=r_1X_1+r_2X_2$.  Then $W$ is the random variable that we want to be $\le C$. 
It would be reasonable next to use the normal approximation. This is because the distributions of the $X_i$ probably can be well approximated by normals, and then $W$ is a linear combination of independent nearly normals.  The  mean and variance of $W$ are not hard to find.  In extreme cases (the $n_i$ large but the $p_i$ small, with $n_ip_i$ smallish) this might not work well, and we might want to look at linear combinations of Poisson random variables.  
The above approach is probably what you need.  But there could be problems if you want to choose $C$ so that the probability of overload is very small.  In that case you would need specialized estimates for extreme tail probabilities, since the normal approximation tends to underestimate probabilities far in the tail.
Comment: The distributions involved in possible analyses do not include the geometric distribution of the title.
