Abstract
In analytic queueing theory, Rouche's theorem is frequently used to prove the existence of a certain number of zeros in the domain of regularity of a given function. If the theorem can be applied it leads in a simple way to results concerning the ergodicity condition and the construction of the solution of the functional equation for the generating function of the stationary distribution. Unfortunately, the verification of the conditions needed to apply Rouche's theorem is frequently quite difficult. We prove the theorem which allows to avoid some difficulties arising in applying classical Rouche's theorem to an analysis of queueing models.
Similar content being viewed by others
References
L. Abolnikov and A. Dukhovny, Necessary and sufficient conditions for the ergodicity of Markov chains with transition Δm,n (Δ ′ m,n )-matrix, J. Appl. Probab. Simulation 1 (1987) 13-23.
A.N. Dudin and V.I. Klimenok, Queueing systems with passive servers, J. Appl.Math. Stochastic Anal. 9 (1996) 185-204.
H.R. Gail, S.L. Hantler and B.A. Taylor, Spectral analysis of M/G/1 and G/M/1 type Markov chains, Adv. in Appl. Probab. 22 (1996) 114-165.
F.D. Gakhov, E.I. Zverovich and S.G. Samko, Increment of argument, logarithmic residue and generalized principle of argument, Dokl. National Acad. Sci. USSR 213 (1973) 1233-1236.
F.N. Gouweleeuw, A General Approach to Computing Loss Probabilities in Finite-Buffer Queues (Thesis Publishers, Amsterdam, 1996).
G.P. Klimov, Stochastic Queueing Systems (Nauka, Moscow, 1966) (in Russian).
M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications (Marcel Dekker, New York, 1989).
H. Takagi, Queueing Analysis (North-Holland, Amsterdam, 1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Klimenok, V. On the Modification of Rouche's Theorem for the Queueing Theory Problems. Queueing Systems 38, 431–434 (2001). https://doi.org/10.1023/A:1010999928701
Issue Date:
DOI: https://doi.org/10.1023/A:1010999928701