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Asymptotic analysis of a queueing system by a two-dimensional state space

Published online by Cambridge University Press:  14 July 2016

J. P. C. Blanc
J. P. C. Blanc*
Affiliation:
Centre for Mathematics and Computer Science, Amsterdam
*
∗ Present address: Department of Mathematics and Informatics, Delft University of Technology, Julianalaan 132, 2628 BL, Delft, The Netherlands.
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Abstract

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.

Keywords

CONFORMAL MAPPINGRELAXATION TIME

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 4 , December 1984 , pp. 870 - 886
Copyright
Copyright © Applied Probability Trust 1984 

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References

[1] Blanc, J. P. C. (1982) Applications of the Theory of Boundary Value Problems in the Analysis of a Queueing Model with Paired Services. Mathematical Center Tract 153, Amsterdam.Google Scholar
[2] Blanc, J. P. C. (1985) The relaxation time of two queueing systems in series. Stochastic Models. To appear.CrossRefGoogle Scholar
[3] Cohen, J. W. (1982) The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
[4] Cohen, J. W. and Boxma, O. J. (1983) Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam.Google Scholar
[5] Fayolle, G. and Iasnogorodski, R. (1979) Two coupled processors: the reduction to a Riemann–Hilbert problem. Z. Wahrscheinlichkeitsth. 47, 325–351.Google Scholar
[6] Fayolle, G., King, P. J. B. and Mitrani, I. (1982) The solution of certain two-dimensional Markov models. Adv. Appl. Prob. 14, 295–308.Google Scholar
[7] Golusin, G. M. (1957) Geometrische Funktionentheorie. V. E. B. Deutscher Verlag der Wissenschaft, Berlin.Google Scholar
[8] Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.Google Scholar
[9] Kleinrock, L. (1976) Queueing Systems II, Computer Applications. Wiley, New York.Google Scholar
[10] Markushevich, A. I. (1977) Theory of Functions of a Complex Variable, Vol. 3, 2nd edn. Chelsea, New York.Google Scholar
[11] Muschelischwili, N. I. (1965) Singuläre Integralgleichungen. Akademie-Verlag, Berlin.Google Scholar
[12] Nehari, Z. (1952) Conformal Mapping. McGraw-Hill, New York.Google Scholar
[13] Tsuji, M. (1959) Potential Theory in Modern Function Theory. Maruzen, Tokyo.Google Scholar