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Stability of Product Form G-Networks

Published online by Cambridge University Press:  27 July 2009

Erol Gelenbe  and
Rolf Schassberger
Erol Gelenbe
Affiliation:
Ecole des Hautes Etudes en InformatiqueUniversité René Descartes 45 rue des Saints-Pères 75006 Paris, France
Rolf Schassberger
Affiliation:
Institut für Mathematische StochastikTechnische Universität Braunschweig Pockelstrasse 14 D 3300 Braunschweig, Germany
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Abstract

We prove necessary and sufficient conditions for the existence and uniqueness of the stationary solution of the queueing networks (G-networks) with negative and positive customers introduced in Gelenbe [3], which have been shown to have product form. First, the existence of the solution of the nonlinear customer flow equations is established using Brouwer's fixed-point theorem; this result is valid for stable and unstable systems, as well as for certain networks that may not have product form. Then, the result is used to establish general stability related to the usual “load factor less than 1” criterion of queueing systems for G-networks with product form.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 3 , July 1992 , pp. 271 - 276
Copyright
Copyright © Cambridge University Press 1992

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References

Baskett, F., Chandy, K.M., Muntz, R.R., & Palacios, F.G. (04 1975). Open, closed and mixed networks of queues with product form. ACM Journal 22(2): 248–260.CrossRefGoogle Scholar
Fourneau, J.M. & Gelenbe, E. (1992). Multiple class G-networks. EHEI Technical Report.CrossRefGoogle Scholar
Gelenbe, E. (1991). Product form queueing networks with positive and negative customers. Journal of Applied Probability 28: 656–663.CrossRefGoogle Scholar
Gelenbe, E., Glynn, P., & Sigman, K. (1991). Queues with negative arrivals. Journal of Applied Probability 28: 245–250.CrossRefGoogle Scholar
Gelenbe, E. & Mitrani, I. (1980). Analysis and synthesis of computer systems. London and New York: Academic Press.Google Scholar
Kemeny, J.G. & Snell, J.L. (1965). Finite Markov chains. Princeton, NJ:Van Nostrand.Google Scholar