Subexponential asymptotics of a Markov-modulated random walk with queueing applications

Journal of Applied Probability

Volumn 35, Issue 2, 1998, Pages 325-347

  Jelenković, Predrag R ^a   Lazar, Aurel A ^b  

^a LUCENT TECHNOLOGIES   (United States)

^b Och Spine at New York Presbyterian Hospitals   (United States)

Author keywords

Fluid flow queue; Markov modulated random walk; Non Cram r type conditions; Subexponential dependency; Subexponential distributions; Supremum distribution; Weiner Hopf factorization

Indexed keywords

EID: 0032378453     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0021900200014984     Document Type: Article

References (30)

1. ABATE, J., CHOUDHURY, G.L. AND WHITT, W. (1994). Waiting-time tail probabilities in queues with long-tail service-time distributions. Queueing Systems. 16, 311-338.
2. ARNDT, K. (1980). Asymptotic properties of the distributions of the supremum of random walk on a Markov chain. Theory Prob. Appl. 25, 253-267.
3. ASMUSSEN, S. (1989). Aspects of matrix Wiener-Hopf factorization in applied probability. Math. Scientist 14, 101-116.
4. ASMUSSEN, S. (1991). Ladder heights and the Markov-modulated M/G/1 queue. Stoch. Proc. Appl. 37, 313-326.
5. ASMUSSEN, S., HENRIKSEN, L.F. AND KLÜPPELBERG, C. (1994). Large claims approximations for risk processes in a Markovian environment. Stoch. Proc. Appl. 54, 29-43.
6. ATHREYA, K.B. AND NEY, P.E. (1972). Branching Processes. Springer.
7. BACCELLI, F. AND BREMAUD, P. (1994). Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrence. Springer.
8. BILLINGSLEY, P. (1986). Probability and Measure. John Wiley & Sons.
9. BINGHAM, N.H., GOLDIE, C.M. AND TEUGELS, J.L. (1987). Regular Variation. Cambridge University Press, Cambridge.
10. BOROVKOV, A.A. (1976). Stochastic Processes in Queueing Theory. Springer.
11. CHISTAKOV, V.P. (1964). A theorem on sums of independent positive random variables and its application to branching random processes. Theor. Prob. Appl. 9, 640-648.
12. CHOW, Y.S. AND TEICHER, H. (1988). Probability Theory. Springer.
13. CHUNG, K.L. (1960). Markov Chains with Stationary Transition Probabilities. Springer, New York, p. 52.
14. CINLAR, E. (1975). Introduction to Stochastic Processes. Prentice-Hall.
15. CLINE, D.B.H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Rel. Fields 72, 529-557.
16. CLINE, D.B.H. (1987). Convolution of distributions with exponential and subexponential tails. J. Austral. Math. Soc. (Series A) 43, 347-365.
17. COHEN, J.W. (1973). Some results on regular variation for distributions in queueing and fluctuation theory. J. Appl. Prob.10, 343-353.
18. ELWALID, A., HEYMAN, D., LAKSHMAN, T.V., MITRA, D. AND WEISS, A. (1995). Fundamental bounds and approximations for atm multiplexers with applications to video teleconferencing. IEEE Journal on Selected Areas in Communications 13, 1004-1016.
19. EMBRECHTS, P., GOLDIE, C.M. AND VERAVERBEKE, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335-347.
20. FELLER, W. (1971). An Introduction to Probability Theory and its Application, Volume II. Wiley, New York.
21. GARETT, M.W. AND WILLINGER, W. (1994). Analysis, modeling and generation of self-similar VBR video traffic. SIGCOMM'94 269-280.
22. GLYNN, P.V. AND WHITT, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. In Studies in Applied Probability. Papers in honour of Lajos Takaćs, eds. J. Galambos and J. Gani. (J. Appl. Prob. Special Volume 31A.) Alden Press Ltd, Oxford, pp. 131-156.
23. JELENKOVIĆ, P.R., LAZAR, A.A. AND SEMRET, N. (1996). Multiple time scales and subexponentiality in MPEG video streams. International IFIP-IEEE Conference on Broadband Communications.
24. KARAMATA, J. (1930). Sur un mode de croissance régulière des fonctions. Mathematica (Cluj) 4, 38-53.
25. KLÜPPELBERG, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132-141.
26. KLÜPPELBERG, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Rel. Fields 82, 259.
27. LOYNES, R.M. (1968). The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497-520.
28. PAKES, A.G. (1975). On the tails of waiting-time distribution. J. Appl. Prob. 12, 555-564.
29. VERAVERBEKE, N. (1977). Asymptotic behavior of Wiener-Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27-37.
30. WILLEKENS, E. AND TEUGELS, J.L. (1992). Asymptotic expansion for waiting time probabilities in an M/G/1 queue with long-tailed service time. Queueing Systems 10, 295-312.