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Barrera, J., & Fontbona, J. (2010). The Limiting Move-To-Front Search-Cost In Law Of Large Numbers Asymptotic Regimes. Ann. Appl. Probab., 20(2), 722–752.
Abstract: We explicitly compute the limiting transient distribution of the search-cost in the move-to-front Markov chain when the number of objects tends to infinity, for general families of deterministic or random request rates. Our techniques are based on a “law of large numbers for random partitions,” a scaling limit that allows us to exactly compute limiting expectation of empirical functionals of the request probabilities of objects. In particular, we show that the limiting search-cost can be split at an explicit deterministic threshold into one random variable in equilibrium, and a second one related to the initial ordering of the list. Our results ensure the stability of the limiting search-cost under general perturbations of the request probabilities. We provide the description of the limiting transient behavior in several examples where only the stationary regime is known, and discuss the range of validity of our scaling limit.
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Villena, M. J., & Araneda, A. A. (2017). Dynamics and stability in retail competition. Math. Comput. Simul., 134, 37–53.
Abstract: Retail competition today can be described by three main features: (i) oligopolistic competition, (ii) multi-store settings, and (iii) the presence of large economies of scale. In these markets, firms usually apply a centralized decisions making process in order to take full advantage of economies of scales, e.g. retail distribution centers. In this paper, we model and analyze the stability and chaos of retail competition considering all these issues. In particular, a dynamic multi-market Cournot Nash equilibrium with global economies and diseconomies of scale model is developed. We confirm the non-intuitive hypothesis that retail multi-store competition is more unstable than traditional small business that cover the same demand. The main sources of stability are the scale parameter, the number of markets, and the number of firms. (C) 2016 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
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