Explicit solutions for variational problems in the quadrant

F. Avram, J. G. Dai, J. J. Hasenbein

Research output: Contribution to journalArticlepeer-review

27 Citations (Scopus)


We study a variational problem (VP) that is related to semimartingale reflecting Brownian motions (SRBMs). Specifically, this VP appears in the large deviations analysis of the stationary distribution of SRBMs in the d-dimensional orthant Rd+. When d = 2, we provide an explicit analytical solution to the VP. This solution gives an appealing characterization of the optimal path to a given point in the quadrant and also provides an explicit expression for the optimal value of the VP. For each boundary of the quadrant, we construct a "cone of boundary influence", which determines the nature of optimal paths in different regions of the quadrant. In addition to providing a complete solution in the 2-dimensional case, our analysis provides several results which may be used in analyzing the VP in higher dimensions and more general state spaces.

Original languageEnglish
Pages (from-to)259-289
Number of pages31
JournalQueueing Systems
Issue number1-3
Publication statusPublished - Jan 2001


  • Large deviations
  • Queueing networks
  • Reflecting brownian motions
  • Skorohod problems
  • Variational problems


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