Abstract
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? In the present study we consider an M/G/1-type queueing problem which can be modelled byN coupled random walks. It is shown that it can be fully analysed without using Rouché's theorem, once it is known that the relevant functional equation has a unique solution with prescribed regularity properties.
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References
D.G. Down and O.J. Boxma, A polling model with threshold switching, in:Proceedings of the Nordic Teletraffic Seminar NTS-12, eds. I. Norros and J. Virtamo (Espoo, Finland, 1995).
H.R. Gail, S.L. Hantler and B.A. Taylor, Spectral analysis of M/G/1 and G/M/1 type Markov chains,Stochastic Models 10 (1994) 1–43.
H.R. Gail, S.L. Hantler and B.A. Taylor, Spectral analysis of M/G/1 type Markov chains,Advances in Applied Probability 28 (1996) 114–165.
D.-S. Lee and B. Sengupta, Queueing analysis of a threshold based priority scheme for ATM networks,IEEE/ACM Transactions on Networking 1 (1993) 709–717.
T. Lindvall,Lectures on the Coupling Method (Wiley, New York, 1992).
M. Marcus and H. Minc,A Survey of Matrix Theory and Matrix Inequalities (Allyn and Bacon, Boston, 1964).
S.P. Meyn and R.L. Tweedie,Markov Chains and Stochastic Stability (Springer-Verlag, London, 1993).
I. Mitrani and D. Mitra, A spectral expansion method for random walks on semi-infinite strips, in:Iterative Methods in Linear Algebra, eds. R. Beauwens and P. de Groen (Elsevier, 1992).
M.F. Neuts, Queues solvable without Rouché's theorem,Operations Research 27 (1979) 767–781.
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Cohen, J.W., Down, D.G. On the role of Rouché's theorem in queueing analysis. Queueing Syst 23, 281–291 (1996). https://doi.org/10.1007/BF01206561
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DOI: https://doi.org/10.1007/BF01206561