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78
Is Network Traffic Approximated By Stable Lévy Motion Or Fractional Brownian Motion?
, 1999
"... Cumulative broadband network traffic is often thought to be well modelled by fractional Brownian motion. However, some traffic measurements do not show an agreement with the Gaussian marginal distribution assumption. We show that if connection rates are modest relative to heavy tailed connection le ..."
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Cited by 110 (12 self)
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Cumulative broadband network traffic is often thought to be well modelled by fractional Brownian motion. However, some traffic measurements do not show an agreement with the Gaussian marginal distribution assumption. We show that if connection rates are modest relative to heavy tailed connection length distribution tails, then stable L'evy motion is a sensible approximation to cumulative traffic over a time period. If connection rates are large relative to heavy tailed connection length distribution tails, then FBM is the appropriate approximation. The results are framed as limit theorems for a sequence of cumulative input processes whose connection rates are varying in such a way as to remove or induce long range dependence.
The Effect of Multiple Time Scales and Subexponentiality on the Behavior of a Broadband Network Multiplexer
, 1996
"... The Effect of Multiple Time Scales and Subexponentiality on the Behavior of a Broadband Network Multiplexer Predrag R. Jelenkovi'c The main theme of this dissertation is the evaluation of the capacity of broadband multimedia network multiplexers. This problem calls for the modeling of network ..."
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Cited by 65 (16 self)
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The Effect of Multiple Time Scales and Subexponentiality on the Behavior of a Broadband Network Multiplexer Predrag R. Jelenkovi'c The main theme of this dissertation is the evaluation of the capacity of broadband multimedia network multiplexers. This problem calls for the modeling of network traffic streams and the analysis of a network multiplexer that is loaded with the corresponding models. For modeling we focus on MPEG video traffic streams that are expected to be predominant in the traffic mixture of future multimedia networks. We experimentally demonstrate that real-time MPEG video traffic exhibits multiple time scale characteristics, as well as subexponential first and second order statistics. Then we construct a model of MPEG video that captures both of these characteristics and accurately predicts queueing behavior for a broad range of buffer and capacity sizes. Depending on whether a network multiplexer (loaded with MPEG) is strictly or weakly stable the dominant effect o...
Asymptotic approximation of the move-to-front search cost distribution and least-recently-used caching fault probabilities
, 1999
"... Consider a finite list of items n = 1 � 2 � � � � � N, that are requested according to an i.i.d. process. Each time an item is requested it is moved to the front of the list. The associated search cost C N for accessing an item is equal to its position before being moved. If the request distribu ..."
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Cited by 41 (8 self)
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Consider a finite list of items n = 1 � 2 � � � � � N, that are requested according to an i.i.d. process. Each time an item is requested it is moved to the front of the list. The associated search cost C N for accessing an item is equal to its position before being moved. If the request distribution converges to a proper distribution as N → ∞, then the stationary search cost C N converges in distribution to a limiting search cost C. We show that, when the (limiting) request distribution has a heavy tail (e.g., generalized Zipf’s law), P�R = n � ∼ c/n α as n → ∞, α> 1, then the limiting stationary search cost distribution P�C> n�, or, equivalently, the least-recently used (LRU) caching fault probability, satisfies P�C> n� lim n→ ∞ P�R> n � =
Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows
- Theory Appl
, 2000
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On a Reduced Load Equivalence for Fluid Queues Under Subexponentiality
, 1998
"... We propose a general framework for obtaining asymptotic distributional bounds on the stationary backlog W A1+A2 ;c in a buffer fed by a combined fluid process A 1 + A 2 and drained at a constant rate c. The fluid process A 1 is an (independent) on-off source with average and peak rates ae 1 and r ..."
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Cited by 34 (0 self)
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We propose a general framework for obtaining asymptotic distributional bounds on the stationary backlog W A1+A2 ;c in a buffer fed by a combined fluid process A 1 + A 2 and drained at a constant rate c. The fluid process A 1 is an (independent) on-off source with average and peak rates ae 1 and r 1 , respectively, and with distribution G for the activity periods. The fluid process A 2 of average rate ae 2 is arbitrary but independent of A 1 . These bounds are used to identify subexponential distributions G and fairly general fluid processes A 2 such that the asymptotic equivalence P \Theta W A1+A2 ;c ? x P \Theta W A1 ;c\Gammaae 2 ? x (x ! 1) holds under the stability condition ae 1 + ae 2 ! c and under the non-triviality condition c \Gamma ae 2 ! r 1 . The stationary backlog W A1 ;c\Gammaae 2 in these asymptotics results from feeding source A 1 into a buffer drained at reduced rate c \Gamma ae 2 . This reduced load asymptotic equivalence extends to a larger class o...
Asymptotic Behavior of a Multiplexer Fed by a Long-Range Dependent Process
- JOURNAL OF APPLIED PROBABILITY
, 1997
"... In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called "M/G/1 input process" or "Cox input process". Asymptotic lower bounds are obtained for any distributio ..."
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Cited by 33 (4 self)
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In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called "M/G/1 input process" or "Cox input process". Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g., G corresponding to either the Pareto or lognormal distribution and c \Gamma ae ! 1, where ae is the arrival rate to the buffer.
Large Deviation Analysis of Subexponential Waiting Times in a Processor Sharing Queue
, 2001
"... We investigate the distribution of the waiting time V in a stable M/G/1 processor sharing queue with trac intensity < 1. When the distribution of a customer service request B belongs to a large class of subexponential distributions with tails heavier than e , it is shown that P[V > x] = ..."
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Cited by 26 (5 self)
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We investigate the distribution of the waiting time V in a stable M/G/1 processor sharing queue with trac intensity < 1. When the distribution of a customer service request B belongs to a large class of subexponential distributions with tails heavier than e , it is shown that P[V > x] = P[B > (1 )x](1 + o(1)) as x !1: Furthermore, we demonstrate that the preceding relationship does not hold if the service distribution has a lighter tail than e .
Multiplexing On-Off Sources with Subexponential On Periods: Part II
, 1997
"... 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 ∞ ..."
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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 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.
Subexponential loss rates in a GI/GI/1 queue with applications
- QUEUEING SYSTEMS 33
, 1999
"... Consider a single server queue with i.i.d. arrival and service processes, {A, An, n � 0} and {C, Cn, n � 0}, respectively, and a finite buffer B. The queue content process {Q B n, n � 0} is recursively defined as Q B n+1 = min((Q B n + An+1 − Cn+1) +, B), q + = max(0, q). When E(A − C) < 0, and A ..."
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Cited by 23 (4 self)
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Consider a single server queue with i.i.d. arrival and service processes, {A, An, n � 0} and {C, Cn, n � 0}, respectively, and a finite buffer B. The queue content process {Q B n, n � 0} is recursively defined as Q B n+1 = min((Q B n + An+1 − Cn+1) +, B), q + = max(0, q). When E(A − C) < 0, and A has a subexponential distribution, we show that the stationary expected loss rate for this queue E(Q B n + An+1 − Cn+1 − B) + has the following explicit asymptotic characterization: E(Q B n + An+1 − Cn+1 − B) + ∼ E(A − B) + as B →∞, independently of the server process Cn. For a fluid queue with capacity c, M/G/ ∞ arrival process At, characterized by intermediately regularly varying on periods τ on, which arrive with Poisson rate Λ, the average loss rate λ B loss satisfies λ B loss ∼ Λ E(τ on η − B) + as B →∞, where η = r + ρ − c, ρ = EAt <c; r (c � r) is the rate at which the fluid is arriving during an on period. Accuracy of the above asymptotic relations is verified with extensive numerical and simulation experiments. These explicit formulas have potential application in designing communication networks that will carry traffic with long-tailed characteristics, e.g., Internet data services.
The Asymptotic Workload Behavior of Two Coupled Queues
- QUEUEING SYSTEMS
, 2002
"... We consider a system of two coupled queues, Q 1 and Q 2 . When both queues are backlogged, they are each served at unit rate. However, when one queue empties, the service rate at the other queue increases. Thus, the two queues are coupled through the mechanism for dynamically sharing surplus serv ..."
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Cited by 22 (5 self)
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We consider a system of two coupled queues, Q 1 and Q 2 . When both queues are backlogged, they are each served at unit rate. However, when one queue empties, the service rate at the other queue increases. Thus, the two queues are coupled through the mechanism for dynamically sharing surplus service capacity. We derive the
