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Gibbs Distribution and the Repairman Problem

Author

Listed:
  • Hassan Chetouani

    (Applied Mathematics Laboratory, Université de Technologie de Compiègne, Sorbonne University Alliance, 60203 Compiègne, France)

  • Nikolaos Limnios

    (Applied Mathematics Laboratory, Université de Technologie de Compiègne, Sorbonne University Alliance, 60203 Compiègne, France)

Abstract

In this paper, we obtain weak convergence results for a family of Gibbs measures depending on the parameter θ > 0 in the following form d P θ ( x ) = Z θ exp − H θ ( x ) / θ d Q ( x ) , where we show that the limit distribution is concentrated in the set of the global minima of the limit Gibbs potential. We also give an explicit calculus for the limit distribution. Here, we use the above as an alternative to Lyapunov’s function or to direct methods for stationary probability convergence and apply it to the repairman problem. Finally, we illustrate this method with a numerical example.

Suggested Citation

  • Hassan Chetouani & Nikolaos Limnios, 2023. "Gibbs Distribution and the Repairman Problem," Mathematics, MDPI, vol. 11(19), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4120-:d:1250656
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    References listed on IDEAS

    as
    1. H. Chetouani & V. S. Korolyuk, 2000. "Stationary distribution for repairable systems," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 16(3), pages 179-196, July.
    2. Y.M. Kaniovski & G.C. Pflug, 1997. "Limit Theorems for Stationary Distributions of Birth-and-Death Processes," Working Papers ir97041, International Institute for Applied Systems Analysis.
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