Switch place of 2 infinite summations In our course about queuing theory sometimes this rule is used:  
$$\sum_{n=0}^\infty \sum_{k=n+1}^\infty n\cdot k = \sum_{k=1}^\infty \sum_{n=0}^{k-1} n\cdot k$$
Another example (but without infinity):
$$\sum_{n=0}^m n \sum_{i=0}^{n-1} i = \sum_{i=0}^{m-1} i \sum_{n=i+1}^{m} n$$ for a given m.
These are just 2 examples but similar operations like these are used a few times and I can't seem to figure out what the general rule is. I also couldn't find it on the wikipedia page about summation (http://en.wikipedia.org/wiki/Summation).
Could someone explain/prove the rule or point me to a link where I can find information about it?

It's more of a process than a rule, but here's how I would do it. Take your first sum $\sum_{n=0}^\infty \sum_{k=n+1}^\infty n\cdot k$, and look through first what values of $k$ does appear at all. Since the lowest value of $k$ anywhere in the sum is $1$, and $k$ is unbounded above, the outer summation has to be $\sum_{k=1}^\infty$.
Now we look at a given value of $k$, and see what values of $n$ permits that $k$ to appear in the original sum. We see that this happens when $k\geq n+1$ which means that $n\leq k-1$, so $k-1$ has to be the upper limit of the inner sum. The value $n=0$ is permitted by any value of $k$, and it is the lowest value of $n$ appearing, and therefore has to be the lower limit of the sum. The inner sum is therefore $\sum_{n=0}^{k-1}$.

I figured an illustration by picture might come in handy as well. Assume we have some function $f(n, k)$ (in your case $f(n, k) = nk$), and we want to find the sum
$$
\sum_{n=0}^\infty \sum_{k=n+1}^\infty f(n, k)
$$
Then the picture below is a picture of (a small part of) the first quadrant of the $nk$-plane (including the axes). The brown dots represents the points whose function value you are interested in:

Now, this picture is taken out of its original context, but works fine, assuming we have $n$ along the $x$-axis, and $k$ aling the $y$-axis. Then we see that if we want $k$ to be the first index of summation, we sww that it goes from $1$ to $\infty$, and that for each $k$, $n$ goes from $0$ to $k-1$. This way of drawing a picture of the terms you're interested in always works, also with finite sums. It's also a big help if you're ever going to do the same for integration.
