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61
Fitting Mixtures Of Exponentials To LongTail Distributions To Analyze Network Performance Models
, 1997
"... Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and interva ..."
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Cited by 156 (13 self)
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Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail 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 longtail distributions, being well described by distributions such as the Pareto and Weibull. It is known that longtail distributions can have a dramatic effect upon performance, e.g., longtail servicetime distributions cause longtail waitingtime distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component longtail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a longtail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove tha...
Asymptotic results for multiplexing subexponential onoff processes
 Advances in Applied Probability
, 1998
"... Consider an aggregate arrival process AN obtained by multiplexing N OnOff 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 characteri ..."
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Cited by 71 (20 self)
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Consider an aggregate arrival process AN obtained by multiplexing N OnOff 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 noninteger 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 OnOff 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.
Stochastically Bounded Burstiness for Communication Networks
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1999
"... 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 ..."
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Cited by 67 (4 self)
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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 realtime traffic, like Markov modulated processes and other multiple timescale processes. We expect that this new calculus will be of particular interest in the implementation of services providing statistical guarantees.
Subexponential Asymptotics of a MarkovModulated Random Walk with Queueing Applications
, 1996
"... Let f(Xn; Jn)g be a stationary Markovmodulated 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 distribut ..."
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Cited by 47 (15 self)
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Let f(Xn; Jn)g be a stationary Markovmodulated 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 WienerHopf factor) has longtailed 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
Market Mechanisms for Network Resource Sharing
, 1999
"... 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 marketbased approach is twofold. First, in modern multiservice networks, resources such as bandwidth and buffer space have dif ..."
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Cited by 37 (7 self)
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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 marketbased 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
Analysis on generalized stochastically bounded bursty traffic for communication networks
 in Proc. IEEE Local Computer Networks 2002
, 2002
"... 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 Bursti ..."
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Cited by 33 (9 self)
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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 selfsimilar input processes, such as fractional Brownian motion, gSBB traffic contains nonGaussian selfsimilar input processes, such as αstable selfsimilar 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 workconserving 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.
Reducedload equivalence and induced burstiness in GPS queues with longtailed traffic flows
 Theory Appl
, 2000
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Multiplexing OnOff Sources with Subexponential On Periods: Part I
, 1997
"... Consider an aggregate arrival process A N obtained by multiplexing N OnOff sources with exponential Off periods of rate and subexponential On periods ø on . For this process its activity period I N satisfies P[I N ? t] ¸ (1 + Eø on ) N \Gamma1 P[ø on ? t] as t !1; for all sufficien ..."
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Cited by 23 (6 self)
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Consider an aggregate arrival process A N obtained by multiplexing N OnOff sources with exponential Off periods of rate and subexponential On periods ø on . For this process its activity period I N satisfies P[I N ? t] ¸ (1 + Eø on ) N \Gamma1 P[ø on ? t] as t !1; for all sufficiently small . When N goes to infinity, with N ! , A N approaches an M=G=1 type process, for which the activity period I 1 , or equivalently a busy period of an M=G=1 queue with subexponential service requirement ø on , satisfies P[I 1 ? t] ¸ e Eø on P[ø on ? t] as t !1. For a simple subexponential OnOff fluid flow queue we establish a precise asymptotic relation between the Palm queue distribution and the time average queue distribution. Further, a queueing system in which one OnOff source, whose On period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential sources with aggregate expected rate Ee t , is shown to be asymptotically ...
Modeling Heterogeneous Network Traffic in Wavelet Domain: Part II  NonGaussian Traffic
 IEEE/ACM Transactions on Networking
, 1999
"... 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 nonGaussian distribution or a periodic structure. To include a nonGaussian distributi ..."
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Cited by 20 (1 self)
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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 nonGaussian distribution or a periodic structure. To include a nonGaussian distribution, we first investigate what higherorder statistics are pertinent by exploring a relationship between timescale 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 higherorder statistics as well as localized spatial and temporal dependence of periodic traffic at different time scales, the resulting wavelet models for both nonGaussian and periodic traffic are simple and accurate with the lowest computational complexity attainable. 1 I...
Asymptotic Behavior of Generalized Processor Sharing with LongTailed Traffic Sources
 IN: PROC. INFOCOM 2000 CONFERENCE
, 1999
"... We analyze the asymptotic behavior of longtailed traffic sources under the Generalized Processor Sharing (GPS) discipline. GPSbased scheduling algorithms, such as Weighted Fair Queueing, have emerged as an important mechanism for achieving differentiated qualityofservice in integratedservices n ..."
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Cited by 18 (7 self)
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We analyze the asymptotic behavior of longtailed traffic sources under the Generalized Processor Sharing (GPS) discipline. GPSbased scheduling algorithms, such as Weighted Fair Queueing, have emerged as an important mechanism for achieving differentiated qualityofservice in integratedservices networks. Under certain conditions, we prove that in an asymptotic sense an individual source with longtailed traffic characteristics is effectively served at a constant rate, which may be interpreted as the maximum feasible average rate for that source to be stable. Thus, asymptotically, the source is only affected by the traffic characteristics of the other sources through their average rate. In particular, the source is essentially immune from excessive activity of sources with `heavier'tailed traffic characteristics. This suggests that GPSbased scheduling algorithms provide an effective mechanism for extracting high multiplexing gains, while protecting individual connections.