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On the role of Rouché’s theorem in queueing analysis. (English) Zbl 0879.60097
Summary: In analytic queueing theory, Rouché’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? 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 analyzed without using Rouché’s theorem, once it is known that the relevant functional equation has a unique solution with prescribed regularity properties.

MSC:
60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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