Asymptotic independence and a network traffic model

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.

[1]  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 .

[2]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[3]  S. Resnick,et al.  Smoothing the Hill estimator , 1997 .

[4]  Sidney I. Resnick,et al.  Heavy Tail Modelling and Teletraffic Data , 1995 .

[5]  Walter Willinger,et al.  Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level , 1997, TNET.

[6]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[7]  V. Paxson,et al.  WHERE MATHEMATICS MEETS THE INTERNET , 1998 .

[8]  Ward Whitt,et al.  An Introduction to Stochastic-Process Limits and their Application to Queues , 2002 .

[9]  Sidney I. Resnick,et al.  Weak Convergence of high-speed network traffic models , 2000 .

[10]  L. Breiman,et al.  On Some Limit Theorems Similar to the Arc-Sin Law , 1965 .

[11]  C. Stărică,et al.  Multivariate extremes for models with constant conditional correlations , 1999 .

[12]  S. Resnick,et al.  Is network traffic approximated by stable Levy motion or fractional Brownian motion , 2002 .

[13]  A. Skorokhod.  Limit Theorems for Stochastic Processes , 1956 .

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

[15]  Sidney I. Resnick,et al.  Self-similar communication models and very heavy tails , 2000 .

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

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

[18]  Walter Willinger,et al.  Experimental queueing analysis with long-range dependent packet traffic , 1996, TNET.

[19]  Ward Whitt,et al.  LIMITS FOR CUMULATIVE INPUT PROCESSES TO QUEUES , 2000, Probability in the Engineering and Informational Sciences.

[20]  Murad S. Taqqu,et al.  Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards , 2000 .

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

[22]  Bojan Basrak,et al.  The sample autocorrelation function of non-linear time series , 2000 .

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

[24]  Walter Willinger,et al.  Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level , 1997, TNET.

[25]  Ward Whitt,et al.  Some Useful Functions for Functional Limit Theorems , 1980, Math. Oper. Res..

[26]  Laurens de Haan,et al.  Sea and Wind: Multivariate Extremes at Work , 1998 .

[27]  Gennady Samorodnitsky,et al.  A Heavy Traffic Approximation for Workload Processes with Heavy Tailed Service Requirements , 2000 .

[28]  S. Resnick,et al.  Empirical Testing Of The Infinite Source Poisson Data Traffic Model , 2003 .

[29]  Vladas Pipiras,et al.  The limit of a renewal reward process with heavy-tailed rewards is not a linear fractional stable motion , 2000 .

[30]  S. Resnick,et al.  A network traffic model with random transmission rate , 2002 .

[31]  Mark Crovella,et al.  Characteristics of WWW Client-based Traces , 1995 .