# Rouché theorem in queuing theory

I was looking for the uses of Rouché's theorem, and I came across queuing theory. An article stated that it is a workhorse theorem in this field, but as much as I tried to find some examples on the ways it can be used I still could not.

Could someone show some examples or recommend me some articles / webpages where I can see how this theorem is used for calculating the probability generating function? (A not too complicated example would be nice.)

The following article considers the application of Rouché's theorem in queueing theory.

Adan, van Leeuwaarden and Winands. On the application of Rouché's theorem in queueing theory

For something I can really help you with if you have questions, see the proof of Lemma 4.5 in the following paper.

Selen, Adan and van Leeuwaarden. Product-form solutions for a class of structured multi-dimenensional Markov processes

After some digging I will be able to provide more examples, but maybe this will already be enough to give you some insight.

• Thank you for the answer Ritz, but I have already read the first paper. I would like to see an example, where the PGF is calculated with the help of Rouché's theorem. I see, I was not clear enough with my question, I'll edit it accordingly. May 10, 2014 at 19:32
• @user140832 In my (limited) knowledge, Rouché's theorem is applied to prove that a certain polynomial has a certain number of roots in a bounded region $K$ with a continuous boundary $\delta K$. So, the theorem is not used to actually compute these roots (and thus does not actually compute the PGF), it only shows that there are a certain number of roots in a bounded region. Usually this region is the unit disc. As an example, see Lemma 4.5 in the second paper I referenced. One proves that a polynomial has a certain number of roots in the unit disc, but does not compute these roots.
– Ritz
May 12, 2014 at 7:21