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.)


1 Answer 1


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.

  • $\begingroup$ 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. $\endgroup$
    – user140832
    May 10, 2014 at 19:32
  • $\begingroup$ @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. $\endgroup$
    – Ritz
    May 12, 2014 at 7:21

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