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The equilibrium states of large networks of Erlang queues

Part of: Stochastic processes

Published online by Cambridge University Press:  15 July 2020

Davit Martirosyan  and
Philippe Robert Open the ORCID record for Philippe Robert [Opens in a new window]
Davit Martirosyan*
Affiliation:
INRIA
Philippe Robert*
Affiliation:
INRIA
*
*Postal address: INRIA Paris, 2 rue Simone Iff, F-75012 Paris, France. Email: Martirosyan.Davit@gmail.com
**Postal address: INRIA Paris, 2 rue Simone Iff, F-75012 Paris, France. Email: Philippe.Robert@inria.fr, http://team.inria.fr/rap/robert
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Abstract

The equilibrium properties of allocation algorithms for networks with a large number of nodes with finite capacity are investigated. Every node receives a flow of requests. When a request arrives at a saturated node, i.e. a node whose capacity is fully utilized, an allocation algorithm may attempt to reallocate the request to a non-saturated node. For the algorithms considered, the reallocation comes at a price: either extra capacity is required in the system, or the processing time of a reallocated request is increased. The paper analyzes the properties of the equilibrium points of the associated asymptotic dynamical system when the number of nodes gets large. At this occasion the classical model of Gibbens, Hunt, and Kelly (1990) in this domain is revisited. The absence of known Lyapunov functions for the corresponding dynamical system significantly complicates the analysis. Several techniques are used. Analytic and scaling methods are used to identify the equilibrium points. We identify the subset of parameters for which the limiting stochastic model of these networks has multiple equilibrium points. Probabilistic approaches are used to prove the stability of some of them. A criterion of exponential stability with the spectral gap of the associated linear operator of equilibrium points is also obtained.

Keywords

Multiple equilibrium pointsscaled Markov processes

MSC classification

Primary: 60G55: Point processes
Secondary: 92D20: Protein sequences, DNA sequences
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 617 - 654
Copyright
© Applied Probability Trust 2020

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