Moderate deviations for queues in critical loading

Queueing Systems

Volumn 31, Issue 3-4, 1999, Pages 359-392

  Puhalskii, Anatolii A ^a,b  

^a INSTITUTE FOR INFORMATION TRANSMISSION PROBLEMS   (Russian Federation)

^b UNIVERSITY OF COLORADO   (United States)

Author keywords

Heavy traffic; Large deviation principle; Queues

Indexed keywords

DIFFUSION APPROXIMATIONS; HEAVY TRAFFICS; LARGE DEVIATION PRINCIPLE; LOGARITHMIC ASYMPTOTICS; MODERATE-DEVIATIONS; OPEN QUEUEING NETWORKS; QUEUES; SINGLE SERVER QUEUE;

QUEUEING THEORY;

EID: 0033243599     PISSN: 02570130     EISSN: 15729443     Source Type: Journal    
DOI: 10.1023/a:1019174601483     Document Type: Article

References (49)

1. J. Abate, G.L. Choudhury and W. Whitt, Waiting-time tail probabilities in queues with long-tail service-time distributions, Queueing Systems 16 (1994) 311-338.
2. J. Abate, G.L. Choudhury and W. Whitt, Exponential approximations for tail probabilities in queues, I: Waiting times, and II: Sojourn time and workload, Oper. Res. 43 (1995) 885-901 and 44 (1996) 758-763.
3. J. Abate, G.L. Choudhury and W. Whitt, Exponential approximations for tail probabilities in queues, I: Waiting times, and II: Sojourn time and workload, Oper. Res. 43 (1995) 885-901 and 44 (1996) 758-763.
4. V. Anantharam, How large delays build up in a GI/G/1 queue, Queueing Systems 5 (1989) 345-367.
5. S. Asmussen, Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/GI/1 queue, Adv. in Appl. Probab. 14 (1982) 143-170.
6. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).
7. A.A. Borovkov, Stochastic Processes in Queueing Theory (Nauka, Moscow, 1972) (in Russian, English translation: Springer, Berlin, 1976).
8. C.-S. Chang, Sample path large deviations and intree networks, Queueing Systems 20 (1995) 7-36.
9. C.-S. Chang, D.D. Yao and T. Zajic, Moderate deviations for queues with long-range dependent input, in: Stochastic Networks, Lecture Notes in Statistics, Vol. 117 (Springer, New York, 1996) pp. 275-298.
10. H. Chen and A. Mandelbaum, Leontief systems, RBV's and RBM's, in: Proc. Imperial College Workshop on Applied Stochastic Processes, eds. M.H.A. Davis and R.J. Elliot (Gordon and Breach, London, 1991).
11. H. Chen and W. Whitt, Diffusion approximations for open queueing networks with service interruptions, Queueing Systems 13 (1993) 335-359.
12. A. Dembo and O. Zeitouni, Large Deviations, Techniques and Applications (Jones and Bartlett, 1993).
13. J.D. Deuschel and D.W. Stroock, Large Deviations (Academic Press, New York, 1989).
14. R.L. Dobrushin and E.A. Pechersky, Large deviations for tandem queueing systems, J. Appl. Math. Stochastic Anal. 7 (1995) 301-330.
15. N.G. Duffield and N. O'Connell, Large deviations and overflow probabilities for the general single server queue, with applications, Math. Proc. Cambridge Philos. Soc. 118 (1995) 363-374.
16. P. Dupuis and R.S. Ellis, The large deviation principle for a general class of queueing systems I, Trans. Amer. Math. Soc. 347 (1995) 2689-2751.
17. P. Dupuis and H. Ishii, On Lipshitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics 35 (1991) 31-62.
18. P. Dupuis, H. Ishii and H.M. Soner, A viscosity solution approach to the asymptotic analysis of queueing systems, Ann. Probab. 18 (1990) 226-255.
19. P. Dupuis and K. Ramanan, Convex duality and the Skorokhod problem, preprint (1998).
20. A. Ganesh and V. Anantharam, Stationary tail probabilities in exponential server tandems with renewal arrivals, Queueing Systems 22 (1996) 203-247.
21. P.W. Glynn, Large deviations for the infinite server queue in heavy traffic, Stochastic Networks, eds. F.P. Kelly and R.J. Williams, IMA Volumes in Mathematics and its Applications, Vol. 71 (Springer, New York, 1995) pp. 387-393.
22. P.W. Glynn and W. Whitt, Logarithmic asymptotics for steady-state tail probabilities in a single-server queue, J. Appl. Probab. 31A (1994) 131-156.
23. J.M. Harrison and M.I. Reiman, Reflected Brownian motion on an orthant, Ann. Probab. 9 (1981) 302-308.
24. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, I and II, Adv. in Appl. Probab. 2 (1970) 150-177 and 355-369.
25. I.A. Ignatyuk, V. Malyshev and V.V. Scherbakov, Boundary effects in large deviation problems, Russian Math. Surveys 49 (1994) 41-99.
26. J.F.C. Kingman, On queues in heavy traffic, J. Roy. Statist. Soc. B 24 (1962) 383-392.
27. T. Lindvall, Weak convergence of probability measures and random functions in the function space D[0, ∞], J. Appl. Probab. 10 (1973) 109-121.
28. R.Sh. Liptser, Large deviations for a simple closed queueing model, Queueing Systems 14 (1993) 1-31.
29. R.Sh. Liptser and A.N. Shiryaev, Theory of Martingales (Nauka, Moscow, 1986) (in Russian, English translation: Kluwer, Dordrecht, 1989).
30. K. Majewski, Heavy traffic approximations of large deviations of feedforward queueing networks, Queueing Systems 28 (1998) 125-155.
31. A. Mandelbaum, The dynamic complementarity problem, unpublished manuscript (1989).
32. N. O'Connell, Large deviations for departures from a shared buffer, J. Appl. Probab. 34 (1997) 753-766.
33. N. O'Connell, Large deviations for queue lengths at a multi-buffered resource, J. Appl. Probab. 35 (1998) 240-245.
34. Yu.V. Prohorov, Transient phenomena in queueing processes, Liet. Mat. Rink. 3 (1963) 199-206 (in Russian).
35. A. Puhalskii, On functional principle of large deviations, in: New Trends in Probability and Statistics, Vol. 1, eds. V. Sazonov and T. Shervashidze (VSP/Mokslas, Vilnius, 1991) pp. 198-218.
36. A. Puhalskii, Large deviations of semimartingales via convergence of the predictable characteristics, Stochastics 49 (1994) 27-85.
37. A. Puhalskii, Large deviation analysis of the single server queue, Queueing Systems 21 (1995) 5-66; Erratum: Queueing Systems 23 (1996) 337.
38. A. Puhalskii, Large deviations of semimartingales: a maxingale problem approach, I. Limits as solutions to a maxingale problem, Stochastics 61 (1997) 141-243.
39. A. Puhalskii and W. Whitt, Functional large deviation principles for first passage time processes, Ann. Appl. Probab. 7 (1997) 362-381.
40. A. Puhalskii and W. Whitt, Functional large deviation principles for waiting and departure processes, Probab. Engrg. Inform. Sci. 12 (1998) 479-507.
41. M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9 (1984) 441-458.
42. J.S. Sadowsky, The probability of large queue lengths and waiting times in a heterogeneous multiserver queue, part II: Positive recurrence and logarithmic limits, Adv. in Appl. Probab. 27 (1995) 567-583.
43. J.S. Sadowsky and W. Szpankowski, The probability of large queue lengths and waiting times in a heterogeneous multiserver queue, part I: Tight limits, Adv. in Appl. Probab. 27 (1995) 532-566.
44. A. Shwartz and A. Weiss, Large Deviations for Performance Analysis (Chapman and Hall, London, 1995).
45. A.V. Skorohod, Limit theorems for stochastic processes, Theory Probab. Appl. 1 (1956) 261-290.
46. P. Tsoucas, Rare events in series of queues, J. Appl. Probab. 29 (1992) 168-175.
47. S.R.S. Varadhan, Large Deviations and Applications (SIAM, Philadelphia, PA, 1984).
48. A.D. Wentzell, Limit Theorems on Large Deviations for Markov Random Processes (Nauka, Moscow, 1986) (in Russian, English translation: Reidel, Dordrecht, 1989).
49. W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 1 (1980) 67-85.