## Fluid Queues, Leaky Buckets, On-Off Processes and Teletraffic Modeling with Highly Variable and Correlated Inputs (1998)

Venue: | in Self-Similar Network Traffic and Performance |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Resnick98fluidqueues,,

author = {Sidney Resnick and Gennady Samorodnitsky},

title = {Fluid Queues, Leaky Buckets, On-Off Processes and Teletraffic Modeling with Highly Variable and Correlated Inputs},

booktitle = {in Self-Similar Network Traffic and Performance},

year = {1998}

}

### OpenURL

### Abstract

INTRODUCTION There now exist several large teletraffic data sets exhibiting non-standard features incompatible with classical assumtions of short range dependence and rapidly decreasing tails. For instance, it is worth exploring the variety of data catalogued at the ITA web site www.acm.org/sigcomm/ITA/. These data sets exhibit the phenomena of heavy-tailed marginal distributions and long range dependence. Tails can be so heavy that only infinite variance models are possible (eg, [43]), and sometimes, as in file size data, even first moments are infinite. See [1]. Heavy tails have been fit to file lengths ([1], [9],[10]) cpu time to complete a job, call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics ([12], [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially consider

### Citations

1930 | On the self-similar nature of Ethernet traffic
- Leland, Taqq, et al.
- 1993
(Show Context)
Citation Context ...tion with the Hurst phenomenon. See [21], [22], [4], [5]. In telecommunications, long range dependence has been found in video conference data ( [3]), packet counts per unit time in ethernet traffic (=-=[27]-=-) and in bytes per unit time in WWW traffic ([9], [10]). i ii FLUID QUEUES, LEAKY BUCKETS Tails of many teletraffic quantities are heavy and there is suspicion that they are getting heavier as WWW use... |

1439 | Generalized autoregressive conditional heteroskedasticity
- Bollerslev
- 1986
(Show Context)
Citation Context ..., call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics ([12], [13], [20], =-=[6]-=-, [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], [4], [5]. In telecommunica... |

1225 | Self-Similarity in World Wide Web Traffic: Evidence and Possible Causes
- Crovella, Bestavros
- 1997
(Show Context)
Citation Context ...so heavy that only infinite variance models are possible (eg, [43]), and sometimes, as in file size data, even first moments are infinite. See [1]. Heavy tails have been fit to file lengths ([1], [9],=-=[10]-=-) cpu time to complete a job, call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and e... |

523 |
ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence
- Bollerslev, Chou
- 1992
(Show Context)
Citation Context ...l holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics ([12], [13], [20], [6], =-=[7]-=-) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], [4], [5]. In telecommunications... |

465 |
Extreme Values, Regular Variation and Point Processes
- Resnick
- 1987
(Show Context)
Citation Context ...with finite right limits.) Theorem 3 Assume fX(t)g stable and 1 \Gamma F on (x) = x \Gammaff L(x); ff ? 1; x !1: Define b(s) = ` 1 1 \Gamma F on '/ (s): Let fY ff (t); t ? 0g be the extremal process (=-=[37]-=-) generated by \Phi ff (x) = expf\Gammax \Gammaff g; x ? 0 so that P [Y ff (t)sx] = \Phi t ff (x): Define S ff (t) = 1 \Gamma r 1=ff Y ff (t): Then in D r (0; 1) \Theta D l (0; 1), as u !1 ` M(u\Delta... |

428 | Web Server Workload Characterization: The Search for Invariants
- Arlitt, Williamson
- 1996
(Show Context)
Citation Context ...al distributions and long range dependence. Tails can be so heavy that only infinite variance models are possible (eg, [43]), and sometimes, as in file size data, even first moments are infinite. See =-=[1]-=-. Heavy tails have been fit to file lengths ([1], [9],[10]) cpu time to complete a job, call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42... |

314 | Experimental queueing analysis with long-range dependent packet traffic
- Erramilli, Narayan, et al.
- 1996
(Show Context)
Citation Context ...cism. However, various bits of evidence point to the deficiencies of classical modeling. There are simulation studies ([29] and the experimental queueing analysis of Erramilli, Narayan and Willinger (=-=[14]-=-) who fed a single server queue with deterministic service times the actual long range dependent traffic from the Bellcore ethernet. They observed a sharp increase in the average delay as a function o... |

264 |
Long-term storage capacity of reservoirs
- Hurst
- 1951
(Show Context)
Citation Context ...e and economics ([12], [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See =-=[21]-=-, [22], [4], [5]. In telecommunications, long range dependence has been found in video conference data ( [3]), packet counts per unit time in ethernet traffic ([27]) and in bytes per unit time in WWW ... |

240 | Proof of a fundamental result in self-similar traffic modeling
- Taqqu, Willinger, et al.
- 1997
(Show Context)
Citation Context ...[1]. Heavy tails have been fit to file lengths ([1], [9],[10]) cpu time to complete a job, call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], =-=[42]-=-). Other areas where heavy tails abound are finance and economics ([12], [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydro... |

224 |
Introduction to the Theory of Coverage Processes
- Hall
- 1988
(Show Context)
Citation Context ... being no active sources. The length of an activity period is a random variable whose distribution is known by means of a Laplace xiv FLUID QUEUES, LEAKY BUCKETS transform formula as given in [41] or =-=[16]-=-. Based on this, we may derive another interesting quantity. Letsbe the number of sessions in an activity period and define p+ (u) = P [ i=1 X i ? u] to be the probability that at least one of the ses... |

129 |
Self-similarity through high variability: statistical analysis of ethernet lan traffic at the source level
- Willinger, Taqqu, et al.
- 1995
(Show Context)
Citation Context ...cm.org/sigcomm/ITA/. These data sets exhibit the phenomena of heavy-tailed marginal distributions and long range dependence. Tails can be so heavy that only infinite variance models are possible (eg, =-=[43]-=-), and sometimes, as in file size data, even first moments are infinite. See [1]. Heavy tails have been fit to file lengths ([1], [9],[10]) cpu time to complete a job, call holding times, inter-arriva... |

84 |
Heavy tails and long range dependence in on/off processes and associated ßuid models
- Heath, Resnick, et al.
- 1998
(Show Context)
Citation Context ...nst ! Sn +Xn+1 ; some n 0; if Sn + Xn+1st ! Sn+1 ; some n and if 0st ! D (0) we define Z t = ( 1; if B = 1 and 0st ! X (0) on ; 0; otherwise. : A standard renewal argument gives the following result (=-=[18]-=-). Proposition 1 fZ t ; ts0g is strictly stationary and P [Z t = 1] =son : Conditional on Z t = 1, the subsequent sequence of on/off periods is the same as seen from time 0 in the stationary process w... |

81 |
On the tails of waiting-time distributions
- Pakes
- 1975
(Show Context)
Citation Context ...ant to distinguish between the random walk with steps f n g and the random walk with steps fXn + Yn g. Assume 1 \Gamma F on (x) = x \Gammaff L(x); ff ? 1; x !1: From standard random walk theory ([8]; =-=[31]-=-) we get 1 \Gamma W (x)sae 1 \Gamma ae (1 \Gamma r) ff\Gamma1 (ff \Gamma 1) on x \Gamma(ff\Gamma1) L(x) =:bx \Gamma(ff\Gamma1) L(x); x !1; so that the tail of W is heavy and comparable to the integral... |

79 | Explaining world wide web traffic self-similarity
- Crovella, Bestavros
- 1995
(Show Context)
Citation Context ... be so heavy that only infinite variance models are possible (eg, [43]), and sometimes, as in file size data, even first moments are infinite. See [1]. Heavy tails have been fit to file lengths ([1], =-=[9]-=-,[10]) cpu time to complete a job, call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance ... |

79 |
Estimating the Tails of Loss Severity Distributions using Extreme Value Theory
- McNeil
- 1997
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Citation Context ...l times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics ([12], [13], [20], [6], [7]) and insurance analysis (=-=[30]-=-, [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], [4], [5]. In telecommunications, long range dependence has be... |

71 | Asymptotic results for multiplexing subexponential on-off processes
- Jelenković, Lazar
- 1999
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Citation Context ...0] dt so the input rate is random depending on the number of active sources at a given time. To insure stability we assume long term input rate =son ! r: 1.4.1 Activity periods. Following [24], [25], =-=[26]-=-, we define an activity period to be a busy period in the corresponding M/G/1 queue so if at time 0 the initial conditions are that N(0) = 0 and a node turns on, then activity period = infft ? 0 : N(t... |

60 | Heavy Tail Modeling and Teletraffic Data
- Resnick
- 1997
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Citation Context ..., even first moments are infinite. See [1]. Heavy tails have been fit to file lengths ([1], [9],[10]) cpu time to complete a job, call holding times, inter-arrival times between packets in a network (=-=[39]-=-), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics ([12], [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range depende... |

47 |
Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and …rst passage probabilities. The Annals of Applied Probability
- Asmussen
- 1998
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Citation Context ...Gamma r)Xn+1 \Gamma rYn+1 : The first downgoing ladder epoch of f P n i=0si ; ns0g is N = inffn ? 0 : n X i=0sis0g: (1.2) The tail behavior of the maximum in a (discrete) cycle is described next. See =-=[2]-=-, [17]. Proposition 2 For the stable queuing process fX(Sn )g satisfying 1 \Gamma F on (x) = x \Gammaff L(x); ff ? 1; x !1; the maximum over a cycle has a distribution tail asymptotic to the tail of t... |

47 | Subexponential Asymptotics of a Markov-Modulated Random Walk with Queueing Applications
- Jelenkov́ıc, Lazar
- 1998
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Citation Context ...ma r1 [X(t)?0] dt so the input rate is random depending on the number of active sources at a given time. To insure stability we assume long term input rate =son ! r: 1.4.1 Activity periods. Following =-=[24]-=-, [25], [26], we define an activity period to be a busy period in the corresponding M/G/1 queue so if at time 0 the initial conditions are that N(0) = 0 and a node turns on, then activity period = inf... |

44 |
Extreme values in the GI/G/1 queue
- Iglehart
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Citation Context ...eto. The histogram shows that 85% of the time, the contents process crosses level L due to a very long on period and not due to gradual buildup. 1.2.5 Contrast with exponential tails. From results of =-=[23]-=- we may contrast the heavy tailed case with the case of exponentially decreasing tails. Suppose: For some fl ? 0: E(e fl 1 ) = 1 (so that F off helps determine the growth rate fl) and for this value o... |

42 | Limit Theory for Bilinear Processes with Heavy-Tailed Noise, Annals of Applied Probability 6
- Davis, Resnick
- 1996
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Citation Context ...nsimulated data exhibiting dependencies, such ARMA models provide unacceptable fits and apparently do not come near to capturing the correct dependence structure. For discussion see [33], [39], [40], =-=[11]-=-, [36]. ffl Probabilistic. What probability models explain observed features in the data? Long range dependence and heavy tails are striking properties. What models qualitatively explain these propert... |

39 |
Variable-bit-rate video traffic and long-range dependence. Unpublished manuscript
- Beran, Sherman, et al.
- 1992
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Citation Context ...dence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], [4], [5]. In telecommunications, long range dependence has been found in video conference data ( =-=[3]-=-), packet counts per unit time in ethernet traffic ([27]) and in bytes per unit time in WWW traffic ([9], [10]). i ii FLUID QUEUES, LEAKY BUCKETS Tails of many teletraffic quantities are heavy and the... |

35 | periods of an infinite server queue and performance of certain heavy tailed fluid queues. Queueing Systems 33
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- 1999
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Citation Context ...er than u. Then based on the Takacs formula we may derive the following: p+ (u) = F on (u)sF on (u) R u 0 e \Gamma R t 0sF on (x)dx dt + e \Gamma R u 0sF on (x)dx e F on (u) u !1: This allows us (see =-=[35]-=-) to generalize a result of Jelenkovic and Lazar ([26]). Theorem 5 Let I be the change in the buffer content during the activity period. Assume that the stability condition holds, and that 0 ! r ! 1 +... |

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- 1997
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Citation Context ... r)Xn+1 \Gamma rYn+1 : The first downgoing ladder epoch of f P n i=0si ; ns0g is N = inffn ? 0 : n X i=0sis0g: (1.2) The tail behavior of the maximum in a (discrete) cycle is described next. See [2], =-=[17]-=-. Proposition 2 For the stable queuing process fX(Sn )g satisfying 1 \Gamma F on (x) = x \Gammaff L(x); ff ? 1; x !1; the maximum over a cycle has a distribution tail asymptotic to the tail of the on ... |

28 |
The impact of autocorrelation on queueing systems
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- 1993
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Citation Context ...ising from the evolving nature of network traffic arouses some engineering skepticism. However, various bits of evidence point to the deficiencies of classical modeling. There are simulation studies (=-=[29]-=- and the experimental queueing analysis of Erramilli, Narayan and Willinger ([14]) who fed a single server queue with deterministic service times the actual long range dependent traffic from the Bellc... |

25 |
Methods of using long-term storage in reservoirs
- Hurst
- 1956
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Citation Context ...economics ([12], [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], =-=[22]-=-, [4], [5]. In telecommunications, long range dependence has been found in video conference data ( [3]), packet counts per unit time in ethernet traffic ([27]) and in bytes per unit time in WWW traffi... |

23 | Multiplexing on-off sources with subexponential on periods: Part II
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- 1997
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Citation Context ...[X(t)?0] dt so the input rate is random depending on the number of active sources at a given time. To insure stability we assume long term input rate =son ! r: 1.4.1 Activity periods. Following [24], =-=[25]-=-, [26], we define an activity period to be a busy period in the corresponding M/G/1 queue so if at time 0 the initial conditions are that N(0) = 0 and a node turns on, then activity period = infft ? 0... |

17 | How system performance is affected by the interplay of averages in a ¯uid queue with long range dependence induced by heavy tails
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Citation Context ...Note the complexity allows only analysis of expected times and that only the order is obtained. For functions f; g we write fsg to mean 0 ! lim inf x!1 f(x) g(x)slim sup x!1 f(x) g(x) ! 1: Theorem 6 (=-=[19]-=-) Suppose the session length distribution F on is heavy tailed 1 \Gamma F on (x) = x \Gammaff L(x); ff ? 1; and r \Gammason ? 0 is non-integer. SUMMARY xvii Then E(fl)sfl 1\Gammak ` 1 1 \Gamma F on (f... |

13 | Performance decay in a single server exponential queueing model with long range dependence
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11 | Discussion of the Danish Data on Large Fire Insurance Losses
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- 1998
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4 |
de Vries, "Safety First Portfolio Selection, Extreme Value Theory and Long Run Asset Risks
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- 1994
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Citation Context ... to complete a job, call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics (=-=[12]-=-, [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], [4], [5].... |

4 |
On Banach Algebras, Renewal Measures and Regenerative Processes
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Citation Context ...f fl(\Delta) as a function of s? Note fl(s) is of the form fl(s) = const h lim v!1 zsU(v) \Gamma zsU(s) i so we need rates of convergence in the key renewal theorem. This can be based on a theorem of =-=[15]-=- and is given in [18]. Theorem 2 Assume that there is an ns1 such that (F onsF off ) n is nonsingular. SupposesF on (t) = t \Gammaff L(t); t !1; where 1 ! ff ! 2 and L is slowly varying at infinity an... |

3 |
de Vries. The limiting distribution of extremal exchange rate returns
- Hols, G
- 1991
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Citation Context ... a job, call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics ([12], [13], =-=[20]-=-, [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], [4], [5]. In telecomm... |

3 |
How misleading can sample acf's of stable ma's be? (Very!). Technical report 1209; available at www.orie.cornell.edu/trlist/trlist.html; to appear: Annals of Applied Probability
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- 1998
(Show Context)
Citation Context ...eal nonsimulated data exhibiting dependencies, such ARMA models provide unacceptable fits and apparently do not come near to capturing the correct dependence structure. For discussion see [33], [39], =-=[40]-=-, [11], [36]. ffl Probabilistic. What probability models explain observed features in the data? Long range dependence and heavy tails are striking properties. What models qualitatively explain these p... |

2 |
On the tail of the stationary waiting time distribution and limit theorems for the M/G/1 queue
- Cohen
- 1972
(Show Context)
Citation Context ...mportant to distinguish between the random walk with steps f n g and the random walk with steps fXn + Yn g. Assume 1 \Gamma F on (x) = x \Gammaff L(x); ff ? 1; x !1: From standard random walk theory (=-=[8]-=-; [31]) we get 1 \Gamma W (x)sae 1 \Gamma ae (1 \Gamma r) ff\Gamma1 (ff \Gamma 1) on x \Gamma(ff\Gamma1) L(x) =:bx \Gamma(ff\Gamma1) L(x); x !1; so that the tail of W is heavy and comparable to the in... |

1 |
Schemes exhibiting Hurst behavior
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- 1988
(Show Context)
Citation Context ...ics ([12], [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], =-=[4]-=-, [5]. In telecommunications, long range dependence has been found in video conference data ( [3]), packet counts per unit time in ethernet traffic ([27]) and in bytes per unit time in WWW traffic ([9... |

1 |
On the expected range and expected adjusted range of partial sums of exchangeable random variables
- Boes, Cruz
- 1973
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Citation Context ...[12], [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], [4], =-=[5]-=-. In telecommunications, long range dependence has been found in video conference data ( [3]), packet counts per unit time in ethernet traffic ([27]) and in bytes per unit time in WWW traffic ([9], [1... |

1 |
de Vries. On the relation between Garch and stable processes
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- 1991
(Show Context)
Citation Context ...mplete a job, call holding times, inter-arrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics ([12], =-=[13]-=-, [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially considered in hydrology in connection with the Hurst phenomenon. See [21], [22], [4], [5]. In te... |

1 |
non-linearities can ruin the heavy tailed modeler's day
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- 1997
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1 |
An Introduction to Queueing Theory
- Tak'acs
- 1962
(Show Context)
Citation Context ...on there being no active sources. The length of an activity period is a random variable whose distribution is known by means of a Laplace xiv FLUID QUEUES, LEAKY BUCKETS transform formula as given in =-=[41]-=- or [16]. Based on this, we may derive another interesting quantity. Letsbe the number of sessions in an activity period and define p+ (u) = P [ i=1 X i ? u] to be the probability that at least one of... |