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.