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Variational problem in the non-negative orthant of ℝ3: reflective faces and boundary influence cones

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Abstract

In this paper we consider the variational problem in the non-negative orthant of ℝ3. The solution of this problem gives the large deviation rate function for the stationary distribution of an SRBM (Semimartingal Reflecting Brownian Motion). Avram, Dai and Hasenbein (Queueing Syst. 37, 259–289, 2001) provided an explicit solution of this problem in the non-negative quadrant. Building on this work, we characterize reflective faces of the non-negative orthant of ℝd, we construct boundary influence cones and we provide an explicit solution of several constrained variational problems in ℝ3. Moreover, we give conditions under which certain spiraling paths to a point on an axis have a cost which is strictly less than the cost of every direct path and path with two pieces.

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Authors and Affiliations

  1. Faculté des Sciences, Ain Chock, Universite Hassan II, B.P. 5366, MAARIF, Casablanca, Maroc

    Ahmed El Kharroubi, Abdelhak Yaacoubi, Abdelghani Ben Tahar & Kawtar Bichard

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  1. Ahmed El Kharroubi
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  2. Abdelhak Yaacoubi
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  3. Abdelghani Ben Tahar
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  4. Kawtar Bichard
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Correspondence to Ahmed El Kharroubi.

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El Kharroubi, A., Yaacoubi, A., Ben Tahar, A. et al. Variational problem in the non-negative orthant of ℝ3: reflective faces and boundary influence cones. Queueing Syst 70, 299–337 (2012). https://doi.org/10.1007/s11134-012-9278-x

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  • DOI: https://doi.org/10.1007/s11134-012-9278-x

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