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13
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 18 (9 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 15 (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.
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 13 (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...
Loss Rates for Lévy Processes with Two Reflecting Barriers
, 2005
"... Let {Xt} be a Lévy process which is reflected at 0 and K> 0. The reflected process {V K t} is constructed as V K t = V K 0 + Xt + L0 t − LK t where {L0 t} and {LK t} are the local times at 0 and K, respectively. We consider the loss rate ℓK, defined by ℓK = EπK LK1, where EπK is the expectation unde ..."
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Cited by 6 (0 self)
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Let {Xt} be a Lévy process which is reflected at 0 and K> 0. The reflected process {V K t} is constructed as V K t = V K 0 + Xt + L0 t − LK t where {L0 t} and {LK t} are the local times at 0 and K, respectively. We consider the loss rate ℓK, defined by ℓK = EπK LK1, where EπK is the expectation under the stationary measure πK. The main result of the paper is the identification of ℓK in terms of πK and the characteristic triplet of {Xt}. We also derive asymptotics of ℓK as K → ∞ when EX1 < 0 and the Lévy measure of {Xt} is lighttailed.
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 flows, w ..."
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Cited by 5 (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.
Heavytraffic limits for loss proportions in singleserver queues
 Queueing Syst
, 2004
"... Abstract. We establish heavytraffic stochasticprocess limits for the queuelength and overflow stochastic processes in the standard singleserver queue with finite waiting room (G/G/1/K). We show that, under regularity conditions, the content and overflow processes in related singleserver models ..."
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Cited by 5 (0 self)
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Abstract. We establish heavytraffic stochasticprocess limits for the queuelength and overflow stochastic processes in the standard singleserver queue with finite waiting room (G/G/1/K). We show that, under regularity conditions, the content and overflow processes in related singleserver models with finite waiting room, such as the finite dam, satisfy the same heavytraffic stochasticprocess limits. As a consequence, we obtain heavytraffic limits for the proportion of customers or input lost over an initial interval. Except for an interchange of the order of two limits, we thus obtain heavytraffic limits for the steadystate loss proportions. We justify the interchange of limits in M/GI/1/K and GI/M/1/K special cases of the standard GI/GI/1/K model by directly establishing local heavytraffic limits for the steadystate blocking probabilities.
Finite buffer queue with generalized processor sharing and heavytailed input processes
 Computer Networks
"... www.elsevier.com/locate/comnet ..."
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 ..."
On The Asymptotic Relationship Between The Overflow Probability And The Loss Ratio
, 1999
"... In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate ..."
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Cited by 4 (3 self)
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In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate c and a fluid input. We provide asymptotic upper and lower bounds on the difference between log PfQ ? xg and log PL (x) under different conditions. The conditions for the upper bound are simple and are satisfied by a very large class of input processes. The conditions on the lower bound are more complex but we show that various classes of processes such as Markov modulated and ARMA type Gaussian input processes satisfy them. KEYWORDS: OVERFLOW PROBABILITY; LOSS RATIO; ASYMPTOTIC RELATIONSHIP AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60G15 SECONDARY 60G70;60K25 1.
Loss Probability for a Finite Buffer Multiplexer with the M/G/ ∞ Input Process
, 2004
"... Abstract. This paper studies the impact of longrangedependent (LRD) traffic on the performance of a network multiplexer. The network multiplexer is characterized by a multiplexing queue with a finite buffer and an M/G/ ∞ input process. Our analysis expresses the loss probability bounds using a sim ..."
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Cited by 1 (0 self)
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Abstract. This paper studies the impact of longrangedependent (LRD) traffic on the performance of a network multiplexer. The network multiplexer is characterized by a multiplexing queue with a finite buffer and an M/G/ ∞ input process. Our analysis expresses the loss probability bounds using a simple relationship between loss probability and buffer full probability. Our analysis also derives an exact expression for the buffer full probability and consequently calculates the loss probability bounds with excellent precision. Through numerical calculations and simulation examples, we show that existing asymptotic analyses lack the precision for calculating the loss probability over realistic ranges of buffer capacity values. We also show that existing asymptotic analyses may significantly overestimate the loss probability and that designing networks using our analysis achieves efficient resource utilization.