Scaling Limits for Cumulative Input Processes

References

• Alsmeyer G. On generalized renewal measures and certain first passage times. Ann. Probab. (1992) 20:1229–1247Crossref, Google Scholar
• Baccelli F., Brémaud P.Elements of Queueing Theory (1994) (Springer, Berlin, Germany) . Palm Martingale Calculus and Stochastic RecurrencesGoogle Scholar
• Billingsley P.Convergence of Probability Measures (1968) (Wiley, New York) Google Scholar
• Bingham N. H., Goldie C. M., Teugels J. L.Regular Variation (1987) (Cambridge University Press, Cambridge, UK) Crossref, Google Scholar
• Crovella M., Bestavros A. Self-similarity in world wide web traffic: Evidence and possible causes. Performance Eval. Rev. (1996) 24:160–169Crossref, Google Scholar
• Crovella M., Bestavros A., Taqqu M. S., Adler R., Epstein R. Heavy-tailed probability distributions in the world wide web. A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions (1999) (Birkhauser, Boston, MA) 3–26Google Scholar
• Daley D., Vere-Jones D.An Introduction to the Theory of Point Processes (2003) 1(Springer, Berlin, New York) Google Scholar
• Embrechts P., Maejima M.Selfsimilar Processes (2002) (Princeton University Press, Princeton, NJ) Google Scholar
• Embrechts P., Goldie C., Veraverbeke N. Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie verw. Geb. (1979) 49:335–347Crossref, Google Scholar
• Embrechts P., Klüppelberg C., Mikosch T.Modelling Extremal Events for Insurance and Finance (1997) (Springer, Berlin, New York) Crossref, Google Scholar
• Faÿ G., González-Arévalo B., Mikosch T., Samorodnitsky G. Modeling teletraffic arrivals by a Poisson cluster process. Queueing Systems Theory Appl. (2006) 54:121–140Crossref, Google Scholar
• Gaigalas R., Kaj I. Convergence of scaled renewal processes and a packet arrival model. Bernoulli (2003) 9:671–703Crossref, Google Scholar
• Guerin C. A., Nyberg H., Perrin O., Resnick S. I., Rootzén H., Stărică C. Empirical testing of the infinite source Poisson data traffic model. Stochastic Models (2003) 19:151–200Crossref, Google Scholar
• Gut A.Stopped Random Walks: Limit Theorems and Applications (1988) (Springer, Berlin, New York) Crossref, Google Scholar
• Heath D., Resnick S. I., Samorodnitsky G. Heavy tails and long range dependence in ON/OFF processes and associated fluid models. Math. Oper. Res. (1998) 23:145–165Link, Google Scholar
• Hernandes-Campos H., Marron S., Samorodnitsky G., Smith F. D. Variable heavy tailed durations in Internet traffic. Performance Eval. J. (2004) 58:261–284Crossref, Google Scholar
• Hernandes-Campos H., Jeffay K., Park C., Marron S., Resnick S. I. Extremal dependence: Internet traffic applications. Stochastic Models (2005) 21:1–35Crossref, Google Scholar
• Hohn N., Veitch D. Inverting sampled traffic. ACM/SIGCOMM Internet Measurement Conf. (2003) Miami, FL:222–233Crossref, Google Scholar
• Hohn N., Veitch D., Abry P. Cluster processes: A natural language for network traffic. IEEE Trans. Signal Process (2003) 51:2229–2244Crossref, Google Scholar
• Jacod J., Shiryaev A. N.Limit Theorems for Stochastic Processes (1987) (Springer, Berlin, New York) Crossref, Google Scholar
• Kallenberg O.Foundations of Modern Probability (2001) 2nd ed.(Springer, Berlin, New York) Google Scholar
• Karr A. F.Point Processes and Their Statistical Inference (1986) (Marcel Dekker, Basel, New York) Google Scholar
• Konstantopoulos T., Lin S. J. Macroscopic models for long-range dependent network traffic. Queueing Systems Theory Appl. (1998) 28:215–243Crossref, Google Scholar
• Leland W. E., Taqqu M. S., Willinger W., Wilson D. V. On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking (1994) 2:1–15Crossref, Google Scholar
• Levy J. B., Taqqu M. S. Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards. Bernoulli (2000) 6:23–44Crossref, Google Scholar
• Maulik K., Resnick S. I. Small and large time scale analysis of a network traffic model. Queueing Systems Theory Appl. (2003) 43:221–250Crossref, Google Scholar
• Maulik K., Resnick S. I., Rootzéen H. Asymptotic independence and a network traffic model. J. Appl. Probab. (2002) 39:671–699Crossref, Google Scholar
• Mikosch T., Resnick S. I., Rootzén H., Stegeman A. Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Probab. (2002) 12:23–68Crossref, Google Scholar
• Nualart D.The Malliavin Calculus and Related Topics (1995) (Springer, Berlin, New York) Crossref, Google Scholar
• Petrov V. V.Sums of Independent Random Variables (1975) (Springer, Berlin, New York) Crossref, Google Scholar
• Pipiras V., Taqqu M. S. The limit of a renewal reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli (2000) 6:607–614Crossref, Google Scholar
• Pipiras V., Taqqu M. S., Levy J. B. Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed. Bernoulli (2004) 10:121–163Crossref, Google Scholar
• Resnick S. I.Extreme Values, Regular Variation, and Point Processes (1987) (Springer, New York) Crossref, Google Scholar
• Samorodnitsky G., Taqqu M. S.Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance (1994) (Chapman & Hall, New York) Google Scholar
• Stegeman A. Extremal behavior of heavy-tailed on-periods in a superposition of on/off processes. Adv. Appl. Probab. (2002) 34:179–204Crossref, Google Scholar
• Taqqu M. S. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie verw. Geb. (1975) 31:287–302Crossref, Google Scholar
• Taqqu M. S., Willinger W., Sherman R. Proof of a fundamental result in self-similar traffic modeling. Comput. Comm. Rev. (1997) 27:5–23Crossref, Google Scholar
• Willinger W., Taqqu M. S., Sherman R., Wilson D. Self-similarity through high variability: Statistical analysis of Ethernet LAN traffic at the source level. Comput. Comm. Rev. (1995) 25:100–113Proc. ACM/SIGCOMM'95, Cambridge, MACrossref, Google Scholar