Equivalence of ensembles for large vehicle-sharing models. (English) Zbl 1370.60166
Summary: 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.
MSC:
60K25 | Queueing theory (aspects of probability theory) |
60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
60F05 | Central limit and other weak theorems |
90B20 | Traffic problems in operations research |
90B22 | Queues and service in operations research |