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


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.


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

Mathematics Subject Classification (2000)

60F10 60J60 60J65 60K25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avram, F., Dai, J.G., Hasenbein, J.J.: Explicit solutions for variational problems in the quadrant. Queueing Syst. 37, 259–289 (2001) CrossRefGoogle Scholar
  2. 2.
    Bazaraa, M., Sherali, H., Shetty, C.: Non-Linear Programming, Theory and Algorithms, 3rd edn. John Wiley & Sons, New York (2006) CrossRefGoogle Scholar
  3. 3.
    Bernard, A., El Kharroubi, A.: Régulations deterministes et stochastiques dans le premier orthant de ℝn. Stoch. Stoch. Rep. 34, 149–167 (1991) Google Scholar
  4. 4.
    Bramson, M., Dai, J.G., Harrison, J.M.: Positive recurrence of reflecting Brownian motion in three dimensions. Ann. Appl. Probab. 20, 753–783 (2010) CrossRefGoogle Scholar
  5. 5.
    Budhiraja, A., Dupuis, P.: Simple necessary and sufficient conditions for the stability of constrained processes. SIAM J. Appl. Math. 59(5), 1686–1700 (1999) CrossRefGoogle Scholar
  6. 6.
    Dupuis, P., Williams, R.J.: Lyapunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22, 680–702 (1994) CrossRefGoogle Scholar
  7. 7.
    Dupuis, P., Ramanan, K.: A time-reversed representation for the tail probabilities of stationary reflected Brownian motion. Stoch. Process. Appl. 98(2), 253–288 (2002) CrossRefGoogle Scholar
  8. 8.
    El Kharroubi, A., Ben Tahar, A., Yaacoubi, A.: Sur la récurrence positive du mouvement Brownien réfléchi dans l’orthant positif de ℝn. Stoch. Stoch. Rep. 68, 229–253 (2000) Google Scholar
  9. 9.
    El Kharroubi, A., Ben Tahar, A., Yaacoubi, A.: On the stability of the linear Skorokhod problem in an orthant. Math. Methods Oper. Res. 56, 243–258 (2002) CrossRefGoogle Scholar
  10. 10.
    Harrison, J.M., Hasenbein, J.J.: Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution. Queueing Syst. 61, 113–138 (2009) CrossRefGoogle Scholar
  11. 11.
    Harrison, J.M., Williams, R.J.: Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115 (1987) CrossRefGoogle Scholar
  12. 12.
    Hobson, D.G., Rogers, L.C.G.: Recurrence and transience of reflecting Brownian motion in the quadrant. Math. Proc. Camb. Philos. Soc. 113, 387–399 (1993) CrossRefGoogle Scholar
  13. 13.
    Majewski, K.: Large deviation of the steady-state distribution of reflected processes with applications to queueing systems. Queueing Syst. 29, 351–381 (1998) CrossRefGoogle Scholar
  14. 14.
    Murty, K.G.: On the number of solutions to the complementarity problem and spanning properties of complementary cones. Linear Algebra Appl. 5, 65–108 (1972) CrossRefGoogle Scholar
  15. 15.
    Samelson, H., Thrall, R.M., Besler, O.: A partition theorem for Euclidean n-space. Proc. Am. Math. Soc. 9, 805–807 (1958) Google Scholar
  16. 16.
    Taylor, L.M., Williams, R.J.: Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Theory Relat. Fields 96, 283–317 (1993) CrossRefGoogle Scholar
  17. 17.
    Williams, R.J.: Semimartingale reflecting Brownian motions in the orthant. In: Kelly, F.P., Williams, R.J. (eds.) Stochastic Networks. The IMA Volumes in Mathematics and its Applications, vol. 71, pp. 125–137. Springer, New York (1995) Google Scholar

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

Personalised recommendations