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34
Reducedload equivalence and induced burstiness in GPS queues with longtailed traffic flows
 Theory Appl
, 2000
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Exact asymptotics for fluid queues fed by multiple heavytailed onoff flows
 Ann. Appl. Probab
"... We consider a fluid queue fed by multiple On–Off flows with heavytailed (regularly varying) On periods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a reduced system. The reduced system consists of a “dominant ” subset of the flows, ..."
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Cited by 22 (10 self)
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We consider a fluid queue fed by multiple On–Off flows with heavytailed (regularly varying) On periods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a reduced system. The reduced system consists of a “dominant ” subset of the flows, with the original service rate subtracted by the mean rate of the other flows. We describe how a dominant set may be determined from a simple knapsack formulation. The dominant set consists of a “minimally critical ” set of On– Off flows with regularly varying On periods. In case the dominant set contains just a single On–Off flow, the exact asymptotics for the reduced system follow from known results. For the case of several On–Off flows, we exploit a powerful intuitive argument to obtain the exact asymptotics. Combined with the reducedload equivalence, the results for the reduced system provide a characterization of the tail of the workload distribution for a wide range of traffic scenarios.
Capacity Regions for Network Multiplexers with HeavyTailed Fluid OnOff Sources
, 2001
"... Consider a network multiplexer with a finite buffer fed by a superposition of independent heterogeneous OnOff sources. An OnOff source consists of a sequence of alternating independent activity and silence periods. During its activity period a source produces fluid with constant rate. For this sys ..."
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Cited by 17 (6 self)
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Consider a network multiplexer with a finite buffer fed by a superposition of independent heterogeneous OnOff sources. An OnOff source consists of a sequence of alternating independent activity and silence periods. During its activity period a source produces fluid with constant rate. For this system, under the assumption that the residual activity periods are intermediately regularly varying, we derive explicit and asymptotically exact formulas for approximating the stationary overflow probability and loss rate. The derived asymptotic formulas, in addition to their analytical tractability, exhibit excellent quantitative accuracy, which is illustrated by a number of simulation experiments. We demonstrate through examples how these results can be used for efficient computing of capacity regions for network switching elements. Furthermore, the results provide important insight into qualitative tradeoffs between the overflow probability, offered traffic load, available capacity, and buffer space. Overall, they provide a new set of tools for designing and provisioning of networks with heavytailed traffic streams. KeywordsNetwork multiplexer, Finite buffer fluid queue, OnOff process, Heavytailed distributions, Subexponential distributions, Longrange dependence I.
Overflow Behavior in Queues with Many LongTailed Inputs
 ADVANCES IN APPLIED PROBABILITY
, 1999
"... We consider a fluid queue fed by a superposition of n homogeneous onoff sources with generally distributed on and offperiods. We scale buffer space B and link rate C by n, such that we get nb and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponenti ..."
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Cited by 16 (7 self)
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We consider a fluid queue fed by a superposition of n homogeneous onoff sources with generally distributed on and offperiods. We scale buffer space B and link rate C by n, such that we get nb and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponentially in the number of sources n; we specifically examine the situation in which also b is large. We explicitly compute asymptotics for the case in which the onperiods have a subexponential distribution, e.g., Pareto, Lognormal, or Weibull. We provide a detailed interpretation of our results. Crucial is the shape of the function v(t) := log P(A* > t) for large t, A* being the residual onperiod. If v(&middot;) is slowly varying (e.g., Pareto, Lognormal), then, during the trajectory to overflow, the input rate will only slightly exceed the link rate. Consequently, the buffer will fill `slowly', and the typical time to overflow will be `more than linear' in the buffer size. In contrast, if v(&middot;) ...
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...
Tail asymptotics for discriminatory processor sharing queues with heavytailed service requirements
 Performance Evaluation
, 2005
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A Reducedload Equivalence for Generalised Processor Sharing Networks with Heavytailed Input Flows
 Queueing Systems
, 2000
"... We consider networks where traffic is served according to the Generalised Processor Sharing (GPS) principle. GPSbased scheduling algorithms are considered important for providing differentiated quality of service in integratedservices networks. We are interested in the workload of a particular flo ..."
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Cited by 11 (6 self)
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We consider networks where traffic is served according to the Generalised Processor Sharing (GPS) principle. GPSbased scheduling algorithms are considered important for providing differentiated quality of service in integratedservices networks. We are interested in the workload of a particular flow i at the bottleneck node on its path. Flow i is assumed to have longtailed traffic characteristics. We distinguish between two traffic scenarios, (i) flow i generates instantaneous traffic bursts and (ii) flow i generates traffic according to an on/off process. In addition, we consider two configurations of feedforward networks. First we focus on the situation where other flows join the path of flow i. Then we extend the model by adding flows which can branch off at any node, with cross traffic as a special case. We prove that under certain conditions the tail behaviour of the workload distribution of flow i is equivalent to that in a twonode tandem network where flow i is served in is...
Steady State Distribution Of The Buffer Content For M/G/infinity Input Fluid Queues
, 1999
"... . We consider a fluid queue with ON periods arriving according to a Poisson process and having a longtailed distribution. This queue has long range dependence, and we compute the asymptotic behavior of the steady state distribution of the buffer content. The tail of this distribution is much heavi ..."
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Cited by 11 (1 self)
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. We consider a fluid queue with ON periods arriving according to a Poisson process and having a longtailed distribution. This queue has long range dependence, and we compute the asymptotic behavior of the steady state distribution of the buffer content. The tail of this distribution is much heavier than the tail of the buffer content distribution of a queue which does not possess long range dependence and which has light tailed ON periods and the same traffic intensity. 1. Introduction and preliminaries We consider a model of a network server (multiplexer) defined as follows. Sessions arrive to the server according to a Poisson process with rate ? 0. Each session lasts a random length of time with distribution F that has a finite mean . The lengths of different sessions are independent of each other and of the Poisson arrival process. A session generates work or traffic or fluid at unit rate, commonly measured in some units of network traffic, e.g. packets; the work that cannot be ...
Asymptotic behavior of Generalized Processor Sharing queues under subexponential hypothesis
, 2001
"... Abstract: We analyze the behavior of Generalized Processor Sharing (GPS) queues with heavytailed service times. We compute the exact tail asymptotics of the stationary workload of an individual class and give new conditions for reducedload equivalence and induced burstiness to hold. We also show t ..."
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Cited by 9 (0 self)
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Abstract: We analyze the behavior of Generalized Processor Sharing (GPS) queues with heavytailed service times. We compute the exact tail asymptotics of the stationary workload of an individual class and give new conditions for reducedload equivalence and induced burstiness to hold. We also show that both phenomena can occur simultaneously. Our proofs rely on the single big event theorem and new fluid limits obtained for the GPS system that can be of interest by themselves.
Exact Queueing Asymptotics for Multiple HeavyTailed OnOff Flows
, 2001
"... We consider a fluid queue fed by multiple OnOff flows with heavytailed (regularly varying) Onperiods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a reduced system. The reduced system consists of a `dominant' subset of the flo ..."
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Cited by 8 (0 self)
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We consider a fluid queue fed by multiple OnOff flows with heavytailed (regularly varying) Onperiods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a reduced system. The reduced system consists of a `dominant' subset of the flows, with the original service rate subtracted by the mean rate of the other flows. We describe how a dominant set may be determined from a simple knapsack formulation. We exploit a powerful intuitive argument to obtain the exact asymptotics for the reduced system. Combined with the reducedload equivalence, the results for the reduced system provide an asymptotic characterization of the buffer behavior. 2000 Mathematics Subject Classification: 60K25 (primary), 60F10, 90B18, 90B22 (secondary). Keywords and Phrases: fluid models, heavytailed distributions, knapsack problem, large deviations, queueing theory, reducedload equivalence. I.