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23
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 78 (18 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.
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 56 (14 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
Multiplexing OnOff Sources with Subexponential On Periods: Part II
, 1997
"... We consider an aggregate arrival process A N obtained by multiplexing N OnOff 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 ∞ ..."
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Cited by 25 (6 self)
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We consider an aggregate arrival process A N obtained by multiplexing N OnOff 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 socalled longrange dependent OnOff 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 longtailed OnOff sources. The accuracy of this approximation is verified with extensive simulation experiments.
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 22 (9 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.
Traffic Source Modeling
, 1999
"... Designing and planning networks is often done by simulating the inuence of various traffic types. This simulation approach depends on reliable and realistic traffic models that are capable of covering first and secondorder statistics of the observed network traffic. In this report, an overview ove ..."
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Cited by 21 (5 self)
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Designing and planning networks is often done by simulating the inuence of various traffic types. This simulation approach depends on reliable and realistic traffic models that are capable of covering first and secondorder statistics of the observed network traffic. In this report, an overview over stateoftheart models for the simulation of network traffic will be given.
Induced Burstiness in Generalized Processor Sharing Queues with LongTailed Traffic Flows
, 2000
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Asymptotic Loss Probability in a Finite Buffer Fluid Queue with Heterogeneous HeavyTailed OnOff Processes
, 2000
"... Consider a fluid queue with a finite buffer B and capacity c fed by a superposition of N independent OnOff processes. An OnOff process consists of a sequence of alternating independent activity and silence periods. Successive activity, as well as silence, periods are identically distributed. The p ..."
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Cited by 14 (5 self)
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Consider a fluid queue with a finite buffer B and capacity c fed by a superposition of N independent OnOff processes. An OnOff process consists of a sequence of alternating independent activity and silence periods. Successive activity, as well as silence, periods are identically distributed. The process is active with probability p on and during its activity period produces fluid with constant rate r. For this queueing system, under the assumption that the residual activity periods are intermediately regularly varying, we derive explicit and asymptotically exact formulas for approximating the stationary loss probability and loss rate. In the case of homogeneous sources with residual activity periods equal in distribution to on r , the queue overflow probability is asymptotically, as B !1, equal to P[Q B = B] = ` N k 0 ' p k 0 on P on r ? B k 0 (r \Gamma ae) +N ae \Gamma c k 0 (1 + o(1)); where ae = rp on , N ae ! c and k 0 is the smallest integer greater than (c...
Finite buffer queue with generalized processor sharing and heavytailed input processes
 Computer Networks
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Network Multiplexer with Generalized Processor Sharing and Heavytailed OnOff Flows
 In: Teletraffic Engineering in the Internet Era, Proc. ITC17
, 2001
"... this paper we focus on the latter ..."
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RealTime Generation of Fractal ATM Traffic: Model, Algorithm, and Implementation
 Algorithm, and Implementation, CTR, Tecnical report, CU/CTR/TR
, 1996
"... We design and implement a fractal traffic generator module as a part of the realtime traffic generation and monitoring system built at Columbia University. Fast generation of diverse types of synthetic fractal traffic has been considered difficult to achieve with traditional fractal models. This fra ..."
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Cited by 5 (0 self)
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We design and implement a fractal traffic generator module as a part of the realtime traffic generation and monitoring system built at Columbia University. Fast generation of diverse types of synthetic fractal traffic has been considered difficult to achieve with traditional fractal models. This fractal module employs a fractal traffic model based on the superposition of i.i.d. fractal renewal point processes (SupFRP), a generation algorithm, and several approximation techniques used to meet the realtime constraint. This module can represent a traffic source, realtime or not, exhibiting fractal behavior (such as longrange dependence) over a wide range of time scales, with tunable statistics such as mean, variance, and the Hurst parameter. The capability of the fractal module is fully analyzed by simulations and experiments over a broad range of model parameters. Numerical results show that (i) this generator produces fractal ATM cell streams at very high bitrates (10  25 Mbps) in real time;...