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Queueing Systems

, Volume 70, Issue 4, pp 299–337 | Cite as

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

  • Ahmed El Kharroubi
  • Abdelhak Yaacoubi
  • Abdelghani Ben Tahar
  • Kawtar Bichard
Article

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 ℝ d , 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.

Keywords

Reflected Brownian motion Positive recurrence Skorokhod problems Variational problems Queuing networks Large deviations 

Mathematics Subject Classification (2000)

60F10 60J60 60J65 60K25 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ahmed El Kharroubi
    • 1
  • Abdelhak Yaacoubi
    • 1
  • Abdelghani Ben Tahar
    • 1
  • Kawtar Bichard
    • 1
  1. 1.Faculté des Sciences, Ain ChockUniversite Hassan IICasablancaMaroc

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