Traffic measurements from communication networks have shown that many quantities characterizing network performance have long-tail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and intervals between connection requests in Internet traffic all have been found to have long-tail distributions, being well described by distributions such as the Pareto and Weibull. It is known that long-tail distributions can have a dramatic effect upon performance, e.g., long-tail service-time distributions cause long-tail waiting-time distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component long-tail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a long-tail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove tha...

Consider an aggregate arrival process AN obtained by multiplexing N On-Off processes with exponential Off periods of rate λ and subexponential On periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/ ∞ arrival process A ∞ t and capacity c. When On periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable QP t observed at the beginning of the arrival process activity periods P[Q P t +ρ−c> x] ∼ Λr P[τ c−ρ x/(r+ρ−c) on> u]du x → ∞, where ρ = EA ∞ t < c; r (c ≤ r) is the rate at which the fluid is arriving during an On period. The asymptotic (time average) queuedistributionlower boundis obtained undermoregeneral assumptions on On periods than regular variation. In addition, we analyze a queueing system in which one On-Off process, whose On period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate Eet. This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value Eet.

We develop a network calculus for processes whose burstiness is stochastically bounded by general decreasing functions. This calculus enables us to prove the stability of feedforward networks and obtain statistical upper bounds on interesting performance measures such as delay, at each buffer in the network. Our bounding methodology is useful for a large class of input processes, including important processes exhibiting "subexponentially bounded burstiness" such as fractional Brownian motion. Moreover, it generalizes previous approaches and provides much better bounds for common models of real-time traffic, like Markov modulated processes and other multiple time-scale processes. We expect that this new calculus will be of particular interest in the implementation of services providing statistical guarantees.

Let f(Xn; Jn)g be a stationary Markov-modulated random walk on R\Theta E (E finite), defined by its probability transition matrix measure F = fF ij g; F ij (B) = P[X 1 2 B; J 1 = jjJ 0 = i]; B 2 B(R); i; j 2 E. If F ij ([x; 1))=(1 \Gamma H(x)) ! W ij 2 [0; 1), as x! 1, for some longtailed distribution function H, then the ascending ladder heights matrix distribution G+ (x) (right Wiener-Hopf factor) has long-tailed asymptotics. If EXn! 0, at least one W ij? 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by P \Theta sup n0 Sn? x

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The theme of this thesis is the design and analysis of decentralized and distributed market mechanisms for resource sharing in multiservice networks. The motivation for a market-based approach is twofold. First, in modern multiservice networks, resources such as bandwidth and buffer space have different value to different users, and these valuations cannot, in general, be accurately known in advance as users compete against each other for the resources. Second, the network resources themselves are distributed, and often, not subject to any single authority. We present

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We introduce the concept of generalized Stochastically Bounded Burstiness (gSBB) for Internet traffic, the tail distribution of whose burstiness can be bounded by a decreasing function in a function class with little restrictions. This new concept extends the concept of Stochastically Bounded Burstiness (SBB) introduced by previous researchers to a much larger extent, —while the SBB model can apply to Gaussian self-similar input processes, such as fractional Brownian motion, gSBB traffic contains non-Gaussian self-similar input processes, such as α-stable self-similar processes which are not SBB in general. We develop a network calculus for gSBB traffic. We characterize gSBB traffic by the distribution of its queue size. We explore property of sums of gSBB traffic and relation of input and output processes. We apply this calculus to a work-conserving system shared by a number of gSBB sources, to analyze the behavior of output traffic for each source and to estimate the probabilistic bounds for delays. We expect that this new calculus will be of particular interest in the implementation of services with statistical qualitative guarantees. 1.

We consider an aggregate arrival process A N obtained by multiplexing N On-Off sources with exponential Off periods of rate λ and generally distributed On periods τ on. When N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. For a fluid queue with the limiting M/G/ ∞ arrivals A ∞ t, regularly varying On periods with noninteger exponent, and capacity c, we obtain a precise asymptotic behavior of the queue length random variable QP t observed at the beginning of the arrival process activity periods P[Q P t> x] ∼ Λ r+ρ−c P[τ c−ρ on> u]du x → ∞, x/(r+ρ−c) where ρ = EA ∞ t < c; r (c ≤ r) is the rate at which the fluid is arriving during an On period. (In particular, when P[τ on> x] ∼ x −α,1 < α < 2, the above formula applies to the so-called long-range dependent On-Off sources.) Based on this asymptotic result and the results from a companion paper we suggest a computationally efficient approximation for the case of finitely many long-tailed On-Off sources. The accuracy of this approximation is verified with extensive simulation experiments.

Following our work described in Part I of this paper that modeled various correlation structures of Gaussian traffic in wavelet domain, we extend our previous models to heterogeneous network traffic with either a non-Gaussian distribution or a periodic structure. To include a non-Gaussian distribution, we first investigate what higher-order statistics are pertinent by exploring a relationship between time-scale analysis of wavelets and cumulative traffic. We then develop a novel algorithm in the wavelet domain to capture the important statistics. By utilizing local properties of wavelet basis in both space and time, we further extend such wavelet models to periodic MPEG traffic. As wavelets provide a natural fit to higher-order statistics as well as localized spatial and temporal dependence of periodic traffic at different time scales, the resulting wavelet models for both non-Gaussian and periodic traffic are simple and accurate with the lowest computational complexity attainable. 1 I...