The Annals of Applied Probability

Equivalence of ensembles for large vehicle-sharing models

Christine Fricker and Danielle Tibi

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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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 2 (2017), 883-916.

Dates
Received: July 2015
First available in Project Euclid: 26 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1495764369

Digital Object Identifier
doi:10.1214/16-AAP1219

Mathematical Reviews number (MathSciNet)
MR3655856

Zentralblatt MATH identifier
1370.60166

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F05: Central limit and other weak theorems

Keywords
Closed Jackson networks finite capacity queues product form distribution equivalence of ensembles asymptotic independence local limit theorem

Citation

Fricker, Christine; Tibi, Danielle. Equivalence of ensembles for large vehicle-sharing models. Ann. Appl. Probab. 27 (2017), no. 2, 883--916. doi:10.1214/16-AAP1219. https://projecteuclid.org/euclid.aoap/1495764369


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