Large deviations of the steady-state distribution of reflected processes with applications to queueing systems

Queueing Systems

Volumn 29, Issue 2-4, 1998, Pages 351-381

  Majewski, Kurt ^a  

^a SIEMENS AG   (Germany)

Author keywords

Generalized processor sharing; Heavy traffic approximation; Large deviations; Minimizing path; Path space minimization problem; Skorohod map

Indexed keywords

APPROXIMATION ALGORITHMS; PROBABILITY DISTRIBUTIONS;

GENERALIZED PROCESSOR SHARING; HEAVY-TRAFFIC APPROXIMATION; LARGE DEVIATIONS; MINIMIZING PATH; PATH SPACE;

QUEUEING NETWORKS;

EID: 0032339351     PISSN: 02570130     EISSN: 15729443     Source Type: Journal    
DOI: 10.1023/a:1019148501057     Document Type: Article

References (31)

1. M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming, 2nd ed. (Wiley, New York, 1993).
2. C. Berge and A. Ghouila-Houri, Programming, Games and Transportation Networks (Wiley, New York, 1965).
3. n, Stochastics Stochastic Rep. 34 (1991) 149-167.
4. D. Bertsimas, I.C. Paschalidis and J.N. Tsitsiklis, On the large deviations behaviour of acyclic networks of G/G/1 queues, Working paper LIDS-P-2278, Massachusetts Institute of Technology (December 1994).
5. D. Bertsimas, I.C. Paschalidis and J.N. Tsitsiklis, Large deviations analysis of the generalized processor sharing policy, Technical Report MNS-97-108, Systems Group, College of Engineering, Boston University (June 1997).
6. A. Budhiraja and P. Dupuis, Simple necessary and sufficient conditions for the stability of constrained processes, Technical Report LCDS #97-16, Lefschetz Center for Dynamical Systems and Center for Control Sciences, Division of Applied Mathematics, Brown University, Providence, RI (November 1997).
7. C.-S. Chang, Sample path large deviations and intree networks, Queueing Systems 20 (1995) 7-36.
8. R.W. Cottle, J.-S. Pang and R.E. Stone, The Linear Complementarity Problem (Academic Press, New York, 1992).
9. J.G. Dai and J.M. Harrison, Reflected Brownian motion in an orthant: Numerical methods for steady-state analysis, Ann. Appl. Probab. 2(1) (1992) 65-86.
10. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications (Jones and Bartlett, London, 1993).
11. P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastic Rep. 35 (1991) 31-62.
12. P. Dupuis and K. Ramanan, Convex duality and the Skorokhod problem, Technical Report LCDS 97-1, Lefschetz Center for Dynamical Systems and Center for Control Sciences, Division of Applied Mathematics, Brown University, Providence, RI (1997).
13. P. Dupuis and K. Ramanan, A Skorokhod problem formulation and large deviation analysis of a processor sharing model, Queueing Systems 28 (1998) 109-124.
14. J.M. Harrison and V. Nguyen, Brownian models of multiclass queueing networks: Current status and open problems, Queueing Systems 13 (1993) 5-40.
15. J.M. Harrison and M.I. Reiman, Reflected Brownian motion on an orthant, Ann. Probab. 9(2) (1981) 302-308.
16. J.M. Harrison and R.J. Williams, Brownian models of feedforward queueing networks: Quasire-versibility and product form solutions, Ann. Appl. Probab. 2(2) (1992) 263-293.
17. K. Majewski, Large deviations of feedforward queueing networks, Ph.D. thesis, Ludwig-Maximilians-Universität München (May 1996).
18. K. Majewski, Large deviations of stationary reflected Brownian motions, in: Stochastic Networks: Theory and Applications, eds. F.P. Kelly, S. Zachary and I. Ziedins, Lecture Note Series of Royal Statistical Society (Oxford Science Publications, Oxford, 1996) pp. 105-118.
19. K. Majewski, Solving variational problems associated with large deviations of reflected Brownian motions, Working paper (December 1996).
20. K. Majewski, Heavy traffic approximations of large deviations of feedforward queueing networks, Queueing Systems 28 (1998) 125-155.
21. S.G. Nash and A. Sofer, Linear and Nonlinear Programming (McGraw-Hill, New York, 1996).
22. G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization (Wiley, New York, 1988).
23. N. O'Connell, Large deviations for queue lengths at a multi-buffered resource, Technical Report HPL-BRIMS-96-0010, BRIMS, Hewlett-Packard, Bristol (1996).
24. A.K. Parekh and R.G. Gallager, A generalized processor sharing approach to flow control in integrated services networks: The single node case, IEEE/ACM Trans. Network. 1(3) (1993) 345-357.
25. W.P. Peterson, A heavy traffic limit theorem for networks of queues with multiple customer types, Math. Oper. Res. 16(1) (1991) 90-118.
26. A.V. Skorohod, Stochastic equations for diffusion processes with boundaries, SIAM Theory Probab. Appl. 6 (1961) 287-298.
27. H.P. Williams, Model Solving in Mathematical Programming (Wiley, New York, 1993).
28. R.J. Williams, Semimartingale reflecting Brownian motions in the orthant, in: Stochastic Networks, eds. F.P. Kelly and R.J. Williams (Springer, Berlin, 1995) pp. 125-137.
29. Z.-L. Zhang, Large deviations and the generalized processor sharing scheduling for a two-queue system, Queueing Systems 26 (1997) 229-254.
30. Z.-L. Zhang, Large deviations and the generalized processor sharing scheduling for a multiple-queue system, Queueing Systems 28 (1998) 349-376.
31. Z.-L. Zhang, D. Towsley and J. Kurose, Statistical analysis of the generalized processor sharing scheduling discipline, IEEE J. Select. Areas Commun. 13(6) (1995) 1071-1080.