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Until recently [33 35, 42] traf c has almost exclusively been modeled by exponential distributions In the last few years it has been shown that this is insuf cient and that heavy-tailed distributions are more appropriate and can be used to explain selfsimilar behavior or long-range dependent behavior Given a random variable X with distribution function F x P X x , its distribution function F x is called heavy tailed if its tail 1 F x P X > x decreases subexponentially for large values of x This means that if a random variable represents waiting time, then for a random variable with a heavy-tailed distribution the longer the already accumulated waiting time is the lower in the likelihood of the next arrival within the following time interval The models we consider are based on the exponential, the Weibull, the Pareto, or the lognormal probability distributions.

Past work has concentrated on the exponential, the lognormal, and the Pareto distribution, but not the Weibull distribution Often, if the conditions of ``strict randomness'' of the exponential distribution are not satis ed, the Weibull distribution is a suitable alternative [23] The Weibull distribution is a generalization if the exponential distribution in the sense that a variable x has a Weibull distribution of y x=a c has an exponential distribution with probability density function p y e y As the value of c decreases the probability of longer as well as shorter values increases, and the burstiness of the traf c increases The de nitions and the maximum likelihood estimators for all the probability distributions can be found in Tables 153 and 154 For more details see Johnson and Kotz [23] Note that all but the exponential distributions are twoparameter distributions and can have heavy tails.

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53 Chang, I Fear of SARS, fear of strangers New York Times, May 21, 2003, Section A, Column 2, p 31 54 Person, B; Sy, F; Holton, K; Govert, B; Liang, A National Center for Infectious Diseases/SARS Community Outreach Team Fear and stigma: the epidemic within the SARS outbreak Emerging Infect Dis 2004, 10, 358 363 55 Levine, S Sick cruise ships Cleaning vessels US News World Rep Dec 16, 2002, 133, 50 56 Gunn, R A; Terranova, W A; Greenberg, H B; Yashuk, J; Gary, G W; Wells, J G; Taylor, P R; Feldman, R A Norwalk virus gastroenteritis aboard a cruise ship: an outbreak on ve consecutive cruises Am J Epidemiol 1980, 112, 820 827 57 McKinlay, M A; Pevear, D C Treatment of the picornavirus common cold by inhibitors of viral uncoating and attachment.

19.4 CLASSIFICATION OF IMMUNE-MEDIATED INJURY (HYPERSENSITIVITY)

For the analysis of the traf c traces described above we follow the approach of using goodness-of- t measures suggested by Paxson [34, 35] for the analysis of wide-area TCP connections To judge if the t of one model is better than the other.

Under certain circumstances immune responses can produce tissue damage. These deleterious reactions are collectively known as hypersensitivity or allergy. Hypersensitivity reactions have been divided into four types (originally proposed by Gell and Coombs) based on mechanism (Table 19.5). In all cases the adverse effects of hypersensitivity develop in two stages: (1) Induction (sensitization) requires a suf cient or cumulative exposure dose of the sensitizing agent to induce immune responses that cause no obvious symptoms. (2) Elicitation occurs in sensitized individuals upon subsequent exposure to the antigen and results in adverse antigen-speci c responses that include in ammation. Type I hypersensitivity (sometimes referred to as atopy) is mediated by an antigenspeci c cytophilic antibody (usually IgE) that binds to mast cells and basophils. On subsequent exposure, the allergen binds to these cell-bound antibodies and cross-links IgE molecules, causing the release of mediators such as histamine and slow-reacting substance of anaphylaxis (SRS-A). These mediators cause vasodilation and leakage of uid into the tissues, plus sensory nerve stimulation (leading to itching, sneezing, and cough). Type I is also called immediate-type hypersensitivity because reactions occur within minutes after exposure of a previously sensitized individual to the offending

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