### Abstract

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 R^{d}_{+}. 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 language | English |
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Pages (from-to) | 259-289 |

Number of pages | 31 |

Journal | Queueing Systems |

Volume | 37 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Jan 2001 |

### Keywords

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

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## Cite this

*Queueing Systems*,

*37*(1-3), 259-289. https://doi.org/10.1023/A:1011004620420