A Heavy Traffic Approximation for Workload Processes with Heavy Tailed Service Requirements

A system with heavy tailed service requirements under heavy load having a single server has an equilibrium waiting time distribution which is approximated by the Mittag-Leffler distribution. This fact is understood by a direct analysis of the weak convergence of a sequence of negative drift random walks with heavy right tail and the associated all time maxima of these random walks. This approach complements the recent transform view of Boxma and Cohen (1997).

[1]  Sidney I. Resnick,et al.  Discussion of the Danish Data on Large Fire Insurance Losses , 1997, ASTIN Bulletin.

[2]  J. Geluk,et al.  Regular variation, extensions and Tauberian theorems , 1987 .

[3]  S. Resnick.  Heavy tail modeling and teletraffic data: special invited paper , 1997 .

[4]  Gennady Samorodnitsky,et al.  Activity periods of an infinite server queue and performance of certain heavy tailed fluid queues , 1999, Queueing Syst. Theory Appl..

[5]  Ilkka Norros,et al.  A storage model with self-similar input , 1994, Queueing Syst. Theory Appl..

[6]  J. W. Cohen.  Superimposed renewal processes and storage with gradual input , 1974 .

[7]  Laurens de Haan,et al.  Safety First Portfolio Selection, Extreme Value Theory and Long Run Asset Risks , 1994 .

[8]  Sem C. Borst,et al.  Asymptotic behavior of generalized processor sharing with long-tailed traffic sources , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[9]  Murad S. Taqqu,et al.  On the Self-Similar Nature of Ethernet Traffic , 1993, SIGCOMM.

[10]  Azer Bestavros,et al.  Self-similarity in World Wide Web traffic: evidence and possible causes , 1996, SIGMETRICS '96.

[11]  Laurens de Haan,et al.  On regular variation and its application to the weak convergence of sample extremes , 1973 .

[12]  A. McNeil.  Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory , 1997, ASTIN Bulletin.

[13]  Martin F. Arlitt,et al.  Web server workload characterization: the search for invariants , 1996, SIGMETRICS '96.

[14]  Armand M. Makowski,et al.  On a reduced load equivalence for fluid queues under subexponentiality , 1999, Queueing Syst. Theory Appl..

[15]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[16]  Walter Willinger,et al.  Self-Similarity in High-Speed Packet Traffic: Analysis and Modeling of Ethernet Traffic Measurements , 1995 .

[17]  Onno J. Boxma,et al.  Fluid queues with long-tailed activity period distributions , 1997, Comput. Commun..

[18]  John Frank Charles Kingman,et al.  The single server queue in heavy traffic , 1961, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  WillingerWalter,et al.  Self-similarity through high-variability , 1995 .

[20]  Sidney I. Resnick.  Point processes, regular variation and weak convergence , 1986 .

[21]  D. Applebaum.  Stable non-Gaussian random processes , 1995, The Mathematical Gazette.

[22]  Walter Willinger,et al.  Proof of a fundamental result in self-similar traffic modeling , 1997, CCRV.

[23]  Gennady Samorodnitsky,et al.  Patterns of buffer overflow in a class of queues with long memory in the input stream , 1997 .

[24]  J. Harrison,et al.  Brownian motion and stochastic flow systems , 1986 .

[25]  Predrag R. Jelenkovic,et al.  A Network Multiplexer with Multiple Time Scale and Subexponential Arrivals , 1996 .

[26]  Nick Duffield,et al.  Large deviations and overflow probabilities for the general single-server queue, with applications , 1995 .

[27]  N. Veraverbeke.  Asymptotic behaviour of Wiener-Hopf factors of a random walk , 1977 .

[28]  V. Zolotarev.  The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments , 1964 .

[29]  Sلأren Asmussen,et al.  Applied Probability and Queues , 1989 .

[30]  A. Lazar,et al.  Asymptotic results for multiplexing subexponential on-off processes , 1999, Advances in Applied Probability.

[31]  C. D. Vries,et al.  The Limiting Distribution of Extremal Exchange Rate Returns , 1991 .

[32]  A. Zwart.  A fluid queue with a finite buffer and subexponential input , 2000, Advances in Applied Probability.

[33]  R. Chou,et al.  ARCH modeling in finance: A review of the theory and empirical evidence , 1992 .

[34]  Hansjörg Furrer,et al.  Risk processes perturbed by α-stable Lévy motion , 1998 .

[35]  Walter Willinger,et al.  Long-range dependence in variable-bit-rate video traffic , 1995, IEEE Trans. Commun..

[36]  Gennady Samorodnitsky,et al.  Heavy Tails and Long Range Dependence in On/Off Processes and Associated Fluid Models , 1998, Math. Oper. Res..

[37]  C. G. D. Vries,et al.  On the relation between GARCH and stable processes , 1991 .

[38]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[39]  Takis Konstantopoulos,et al.  Macroscopic models for long-range dependent network traffic , 1998, Queueing Syst. Theory Appl..

[40]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[41]  A. Pakes.  ON THE TAILS OF WAITING-TIME DISTRIBUTIONS , 1975 .

[42]  Gennady Samorodnitsky,et al.  How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails , 1999 .

[43]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

[44]  A. Lazar,et al.  Subexponential asymptotics of a Markov-modulated random walk with queueing applications , 1998, Journal of Applied Probability.

[45]  J. W. Cohen,et al.  Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions , 1999, Queueing Syst. Theory Appl..

[46]  Ward Whitt,et al.  Waiting-time tail probabilities in queues with long-tail service-time distributions , 1994, Queueing Syst. Theory Appl..

[47]  J. Cohen.  SOME RESULTS ON REGULAR VARIATION FOR DISTRIBUTIONS IN QUEUEING AND FLUCTUATION THEORY , 1973 .

[48]  E. Willekens,et al.  Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time , 1992, Queueing Syst. Theory Appl..

[49]  E CrovellaMark,et al.  Self-similarity in World Wide Web traffic , 1996 .

[50]  A. W. Kemp,et al.  Applied Probability and Queues , 1989 .

[51]  W. Whitt.  The continuity of queues , 1974, Advances in Applied Probability.