The Annals of Applied Probability

Equivalence of ensembles for large vehicle-sharing models

Christine Fricker and Danielle Tibi

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • [1] Armendáriz, I. and Loulakis, M. (2009). Thermodynamic limit for the invariant measures in supercritical zero range processes. Probab. Theory Related Fields 145 175–188.
  • [2] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [3] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin. Corrected reprint of the 2nd (1998) ed.
  • [4] Dobrushin, R. L. and Tirozzi, B. (1977). The central limit theorem and the problem of equivalence of ensembles. Comm. Math. Phys. 54 173–192.
  • [5] Economou, A. and Fakinos, D. (1998). Product form stationary distributions for queueing networks with blocking and rerouting. Queueing Systems Theory Appl. 30 251–260.
  • [6] Fayolle, G. and Lasgouttes, J.-M. (1996). Asymptotics and scalings for large product-form networks via the central limit theorem. Markov Process. Related Fields 2 317–348.
  • [7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II. 2nd ed. Wiley, New York.
  • [8] Fricker, C. and Gast, N. (2016). Incentives and redistribution in bike-sharing systems with stations of finite capacity. EURO Journal on Transportation and Logistics 5 261–291.
  • [9] Fricker, C., Gast, N. and Mohamed, H. (2012). Mean field analysis for inhomogeneous bike sharing systems. In 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA’12). Discrete Math. Theor. Comput. Sci. Proc. AQ 365–376. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
  • [10] George, D. K. and Xia, C. H. (2010). Asymptotic analysis of closed queueing networks and its implications to achievable service levels. SIGMETRICS Performance Evaluation Review 38 3–5.
  • [11] Gnedenko, B. V. (1948). On a local limit theorem of the theory of probability. Uspehi Matem. Nauk (N. S.) 3 187–194.
  • [12] Großkinsky, S., Schütz, G. M. and Spohn, H. (2003). Condensation in the zero range process: Stationary and dynamical properties. J. Stat. Phys. 113 389–410.
  • [13] Khinchin, A. I. (1949). Mathematical Foundations of Statistical Mechanics. Dover Publications, Inc., New York. Translated by G. Gamow.
  • [14] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin.
  • [15] Malyshev, V. A. and Yakovlev, A. V. (1996). Condensation in large closed Jackson networks. Ann. Appl. Probab. 6 92–115.
  • [16] Serfozo, R. (1999). Introduction to Stochastic Networks. Applications of Mathematics (New York) 44. Springer, New York.