On the role of Rouché's theorem in queueing analysis

In analytic queueing theory, Rouch'{e's theorem is frequently used, and when it can be applied, leads quickly to tangible results concerning ergodicity and performance analysis. For more complicated models it is sometimes difficult to verify the conditions needed to apply the theorem. The natural question that arises is: Can one dispense with this theorem, in particular when the ergodicity conditions are known? In the present study we consider an $M/G/1$-type queueing problem which can be modelled by $N$ coupled random walks. It is shown that it can be fully analysed without using Rouch'{e's theorem, once it is known that the relevant functional equation has a unique solution with prescribed regularity properties