Explicit solutions for variational problems in the quadrant

F. Avram; J. G. Dai; J. J. Hasenbein

doi:10.1023/A:1011004620420

Explicit solutions for variational problems in the quadrant

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

Research output: Contribution to journal › Article › peer-review

27 Citations (Scopus)

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
Pages (from-to)	259-289
Number of pages	31
Journal	Queueing Systems
Volume	37
Issue number	1-3
DOIs	https://doi.org/10.1023/A:1011004620420
Publication status	Published - Jan 2001

Keywords

Large deviations
Queueing networks
Reflecting brownian motions
Skorohod problems
Variational problems

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title = "Explicit solutions for variational problems in the quadrant",

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 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.",

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T1 - Explicit solutions for variational problems in the quadrant

AU - Avram, F.

AU - Dai, J. G.

AU - Hasenbein, J. J.

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N2 - 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.

AB - 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.

KW - Large deviations

KW - Queueing networks

KW - Reflecting brownian motions

KW - Skorohod problems

KW - Variational problems

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DO - 10.1023/A:1011004620420

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VL - 37

SP - 259

EP - 289

JO - Queueing Systems

JF - Queueing Systems

IS - 1-3

ER -

Explicit solutions for variational problems in the quadrant

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