Diffusion approximation in overloaded switching queueing models

Queueing Systems

Volumn 40, Issue 2, 2002, Pages 143-182

  Anisimov, Vladimir V ^a,b  

^a BILKENT UNIVERSITY   (Turkey)

^b TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KYIV   (Ukraine)

Author keywords

Averaging principle; Diffusion approximation; Fluid limit; Markov process; Networks; Queueing systems; Semi Markov process; Switching process

Indexed keywords

DIFFUSION IN LIQUIDS; MARKOV PROCESSES; NETWORKS (CIRCUITS); QUEUEING THEORY; SWITCHING;

AVERAGING PRINCIPLE; DIFFUSION APPROXIMATIONS; FLUID LIMITS; QUEUEING SYSTEM; SEMI MARKOV PROCESS; SWITCHING PROCESS;

QUEUEING NETWORKS;

EID: 0036111741     PISSN: 02570130     EISSN: 15729443     Source Type: Journal    
DOI: 10.1023/A:1014371517599     Document Type: Article

References (50)

1. V.V. Anisimov, Limit theorems for random processes and their applications to discrete summation schemes, Theory Probab. Appl. 20 (1975) 692-694.
2. V.V. Anisimov, Switching processes, Cybernetics 13(4) (1977) 590-595.
3. V.V. Anisimov, Averaging principle for switching processes, Theory Probab. Math. Statist. 46 (1992) 1-10.
4. V.V. Anisimov, Limit theorems for processes with semi-Markov switching and their applications, Random Oper. Stochastic Equations 2(4) (1994) 333-352.
5. V.V. Anisimov, Switching processes: averaging principle, diffusion approximation and applications, Acta Appl. Math. 40 (1995) 95-141.
6. V.V. Anisimov, Asymptotic analysis of switching queueing systems in conditions of low and heavy loading, in: Matrix-Analytic Methods in Stochastic Models, eds. A.S. Alfa and S.R. Chakravarthy, Lecture Notes in Pure and Applied Mathematics, Vol. 183 (Marcel Dekker, New York, 1996) pp. 241-260.
7. V.V. Anisimov, Averaging principle for near-critical branching processes with semi-Markov switching, Theory Probab. Math. Statist. 52 (1996) 13-26.
8. V.V. Anisimov, Averaging methods for transient regimes in overloading retrial queuing systems, Math. Comput. Modelling 30(3/4) (1999) 65-78.
9. V.V. Anisimov, Switching stochastic models and applications in retrial queues, Top 7(2) (1999) 169-186.
10. V.V. Anisimov and A.O. Aliev, Limit theorems for recurrent processes of semi-Markov type, Theory Probab. Math. Statist. 41 (1990) 7-13.
11. V.V. Anisimov and J.R. Artalejo, Analysis of Markov multiserver retrial queues with negative arrivals, Queueing Systems 39 (2001) 157-182.
12. V.V. Anisimov and V.S. Chabanyuk, On applying of Skorokhod reflecting problem at diffusion approximation of queueing networks, Soviet Phys. Dokl. 35(6) (1990) 505-506.
13. V.V. Anisimov and E.A. Lebedev, Stochastic Queueing Networks. Markov Models (Kiev Univ., Kiev, 1992) (in Russian).
14. V.V. Anisimov and A.P. Yurachkovskiy, A limit theorem for stochastic difference schemes with random coefficients, Theory Probab. Math. Statist. 33 (1986) 1-9.
15. G.P. Basharin, P.P. Boebuov and J.A. Kogan, Analysis of Queues in Computing Networks (Nauka, Moscow, 1989) (in Russian).
16. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).
17. M. Bramson, State space collapse with applications to heavy traffic limits for multiclass queueing networks, Queueing Systems 30(1/2) (1998) 89-148.
18. M. Bramson and J.G. Dai, Heavy traffic limits for some queueing networks, Ann. Appl. Probab. 11 (2001) 49-90.
19. N.P. Buslenko, V.V. Kalashnikov and I.N. Kovalenko, Lectures on the Theory of Complex Systems (Sov. Radio, Moscow, 1973) (in Russian).
20. H. Chen and H. Zhang, Diffusion approximations for some multiclass queueing networks under FIFO disciplines, Math. Oper. Res. 25(4) (2000) 679-707.
21. J.G. Dai, W. Dai, A heavy traffic limit theorems for a class of open queueing networks with finite buffers, Queueing Systems 32 (1999) 5-40.
22. J.G. Dai and T. Kurtz, A multiclass station with Markovian feedback in heavy traffic, Math. Oper. Res. 20 (1995) 721-742.
23. S.N. Ethier and T.G. Kurtz, Markov Processes, Characterization and Convergence (Wiley, New York, 1986).
24. I.I. Ežov and A.V. Skorokhod, Markov processes which are homogeneous in the second component, Theory Probab. Appl. 14 (1969) 679-692.
25. G.I. Falin and J.G.C. Templeton, Retrial Queues (Chapman and Hall, London, 1997).
26. E. Gelenbe and G. Pujolle, Introduction to Queueing Networks (Wiley, Chichester, 1998).
27. I.I. Gikhman and A.V. Skorokhod, Theory of Random Processes III (Springer, Berlin, 1978).
28. J.M. Harrison, Balanced fluid models of multiclass queueing network: A heavy traffic conjecture, in: Stochastic Networks, eds. F. Kelly and R. Williams, IMA Volumes in Mathematics and Its Applications, Vol. 71 (Springer, New York, 1995) pp. 1-20.
29. J.M. Harrison and R.J. Williams, A multiclass closed queueing network with unconventional heavy traffic behavior, Ann. Appl. Probab. 6(1) (1996) 1-47.
30. R. Hersh, Random evolutions: survey of results and problems, Rocky Mountain J. Math. 4(3) (1974) 443-475.
31. D.L. Iglehart, Limit diffusion approximation for the many server queue and the repairman problem, J. Appl. Probab. 2 (1965) 429-441.
32. R. Kertz, Random evolutions with underlying semi-Markov processes, Publ. Res. Inst. Math. Sci. 14 (1978) 589-614.
33. V.S. Korolyuk and A.V. Swishchuk, Random Evolutions (Kluwer Academic, Dordrecht, 1994).
34. E.V. Krichagina, R.S. Liptser and A.A. Puhalsky, Diffusion approximation for the system with arrival process depending on queue and arbitrary service distribution, Theory Probab. Appl. 33 (1988) 124-135.
35. T. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal. 12 (1973) 55-67.
36. R.S. Liptser and A.N. Shiryaev, Theory of Martingales (Kluwer Academic, Dordrecht, 1989).
37. A. Mandelbaum and W. Masey, Strong approximation for time-dependent queues, Math. Oper. Res. 20(1) (1995) 33-64.
38. A. Mandelbaum, W. Masey and M.I. Reiman, Strong approximation for Markovian service networks, Queueing Systems 30(1/2) (1998) 149-202.
39. A. Mandelbaum and G. Pats, State-dependent stochastic networks. Part I: Approximations and applications with continuous diffusion limits, Ann. Appl. Probab. 8(2) (1998) 569-646.
40. M. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications (Marcel Dekker, New York/Basel, 1989).
41. G. Papanicolaou and R. Hersh, Some limit theorems for stochastic equations and applications, Indiana Univ. Math. J. 21 (1972) 815-840.
42. M. Pinsky, Random Evolutions, Lecture Notes in Mathematics, Vol. 451 (Springer, Berlin, 1975) pp. 89-100.
43. M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9(3) (1984) 441-458.
44. M.I. Reiman, A multiclass feedback queue in heavy traffic, Adv. in Appl. Probab. 20 (1988) 179-207.
45. A.V. Skorokhod, Limit theorems for random processes, Theory Probab. Appl. 1 (1956) 289-319.
46. A.V. Skorokhod, Stochastic equations for diffusion processes with boundaries. I, II, Theory Probab. Appl. 6 (1961) 287-298; 7 (1962) 5-25.
47. A.V. Skorokhod, Stochastic equations for diffusion processes with boundaries. I, II, Theory Probab. Appl. 6 (1961) 287-298; 7 (1962) 5-25.
48. W. Whitt, On the heavy- traffic limit theorem for GI/G/∞ queues, Adv. in Appl. Probab. 14 (1982) 171-190.
49. R.J. Williams, On the approximation of queueing networks in heavy traffic, in: Stochastic Networks. Theory and Applications, eds. F.P. Kelly, S. Zachary and I. Zieding (Oxford Univ. Press, Oxford, 1996) pp. 35-56.
50. R.J. Williams, Diffusion approximation for open multiclass queueing networks: Sufficient conditions involving state space collapse, Queueing Systems 30(1/2) (1998) 27-88.