Equivalence of ensembles for large vehicle-sharing models

TL;DR

This paper proves the asymptotic independence and ensemble equivalence in large vehicle-sharing network models, providing explicit queue distributions and a product-form stationary state for systems with capacity constraints.

Contribution

It establishes the equivalence of canonical and grand canonical ensembles for large vehicle-sharing networks with capacity constraints, including models with mixed node types.

Findings

Asymptotic independence of nodes in large networks is proven.

Explicit limiting distributions of queues are derived.

The stationary state has a product-form under generalized blocking and rerouting.

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.