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

• Published in: Queueing systems Vol. 70; no. 4; pp. 299 - 337
• Main Authors: El Kharroubi, Ahmed, Yaacoubi, Abdelhak, Ben Tahar, Abdelghani, Bichard, Kawtar
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.04.2012
• Subjects: Business and Management; Computer Communication Networks; Control; Operations Research/Decision Theory; Probability Theory and Stochastic Processes; Supply Chain Management; Systems Theory; Reflected Brownian motion; Positive recurrence; Queuing networks; 60J65; Skorokhod problems; 60F10; 60J60; Large deviations; 60K25; Variational problems
• ISSN: 0257-0130; 1572-9443
• DOI: 10.1007/s11134-012-9278-x

In this paper we consider the variational problem in the non-negative orthant of ℝ 3 . 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 ℝ 3 . 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.