Minimal number of phone connections to ensure a sufficient number of direct lines A set of telephone connections should be installed in order to connect the cities $A$ and $B$. City $A$ has $2000$ phones.
Assuming that each user in city $A$ needs a telephone connection to city $B$ for $2$ minutes on average during the $10$ working hours and that these calls are completely random, what is the minimal number $M$ of telephone connections to city $B$ in order to ensure, that not more than $1$% of the calls from city $A$ to city $B$ do not get a direct line?
I am supposed to assume a Gaussian distribution.
All I found is, that the mean of the Gaussian should be $2000\times\frac{2}{60}=6.6667$ connections, based on the given time intervals. - $6\frac{2}{3}$ people are expected to have a call at any given moment in  the specified time interval.
How do I proceed from here? - How do I obtain a variance in order to get the 99% confidence interval, necessary to solve the problem.
Furthermore, how do I deal with the, in reality, impossible situations where I have less than 0 or more than 2000 calls at any given moment? Do I just renormalize the clamped section so the integral over it gives 1 or do I ignore those technically impossible cases, or how to deal with them?
 A: In that case, I think the best you can do is to assume a standard deviation of 1 (as that is the 'default').  However it is unclear whether you should assume that is the standard deviation of the calls (in which case the units are clearly minutes) or the standard deviation of total time of connection between City A and City B.  
If it is the former, then there is a practical problem because a non-trivial amount of the distribution would include calls with negative time.  Ignoring the problem of stochastic time selection and assuming the distribution of call volume is uniform (a slightly different assumption than call start times are uniform), then one could calculate the standard error of the mean $\sigma = \frac{1}{\sqrt{2000}} = 0.02236068$.  Given that the standard error of the mean is the same thing as the standard deviation of the mean, we then select the 99% point of the Gaussian, a call duration of 2.052019 minutes leaving 68.40062 hours of connection between City A and City B.
If the later, then one assumes the standard deviation would be expressed in hours, but it might also be expressed in minutes.  In either case, the solution would be trivial once you made the assumption about the standard deviation.

EDIT
Ah, that makes a big difference! Your source material includes the (English) text "The sum of the probabilities for occupied lines, which is higher than the number of lines, should be less than 1%. Approximate the probability function by a Gaussian distribution".  So you aren't directly meant to assume a Gaussian distribution, you are meant to assume a Gaussian distribution as an approximation of the binomial distribution whuber specified!
If you cf. this resource you will see that you can estimate the standard deviation of the number of connections at any given time with the formula $\sigma = \sqrt{np(1-p)}$.  
As @whuber observed, the probability of any given connection being used at any time is $p= \frac{2}{60*10} = 0.003333333$,.  We know the $n$ of connections is $2000$.  Thus, by substitution we estimate $\sigma = 2.577682$, in R sqrt(2000*(2/(10*60))*(1-(2/(10*60)))).  This value matches well a simulation of $\sigma$ in R, sd(rbinom(200000,2000,2/(60*10))).
Combined with your calculated mean number of connections, it is just a matter of identifying the 99th percentile of the specified distribution.  In R, qnorm(.99,6.6667,sqrt(2000*(2/(10*60))*(1-(2/(10*60)))))$ = 12.66329$.  This matches well a simulation in R using rbinom, quantile(rbinom(200000,2000,2/(60*10)),.99) (although this later result is 13 because random binomial draws are discrete values not proportions).
A: You would want to consider a Poisson distribution for the number of calls that are being currently made. You can then either take the tabulated value of the cdf (i.e., use the appropriate statistical software to compute the 99th percentile), or use the asymptotic Gaussian approximation to the Poisson.
