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Equivalence of ensembles for large vehicle-sharing models

Christine Fricker 1, 2 Danielle Tibi 3
2 DYOGENE - Dynamics of Geometric Networks
Inria de Paris, CNRS - Centre National de la Recherche Scientifique : UMR 8548, DI-ENS - Département d'informatique - ENS Paris
Abstract : For a class of large closed Jackson networks submitted to capacity constraints, asymptotic independence of the nodes in normal traffic phase is proved at stationarity under mild assumptions, using a Local Limit Theorem. The limiting distributions of the queues are explicit. In the Statistical Mechanics terminology, the equivalence of ensembles - canonical and grand canonical - is proved for specific marginals. The framework includes the case of networks with two types of nodes: single server/finite capacity nodes and infinite servers/infinite capacity nodes, that can be taken as basic models for bike-sharing systems. The effect of local saturation is modeled by generalized blocking and rerouting procedures, under which the stationary state is proved to have product-form. The grand canonical approximation can then be used for adjusting the total number of bikes and the capacities of the stations to the expected demand.
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Submitted on : Wednesday, September 23, 2015 - 5:52:04 PM
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Christine Fricker, Danielle Tibi. Equivalence of ensembles for large vehicle-sharing models. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2017, 27 (2), pp.883-916. ⟨10.1214/16-AAP1219⟩. ⟨hal-01203789⟩



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