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HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON=OFF SOURCES F.. In contrast with the classical heavy loadapproximation, note that the limit input rate process obtained this way is generallynon-

HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON=OFF SOURCES

F BRICHET ANDA SIMONIAN

France TeÂleÂcom, CNET, 92794 Issy-Moulineaux CeÂdex 9, France

More precisely, represent an on=off source by mutually independent, alternatingsilence periods A and activity periods B When active, the source emits data atSelf-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc.

115

Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger

Copyright # 2000 by John Wiley & Sons, Inc Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X

constant rate, its peak rate, taken as unity Given E…A† ˆ 1=a and E…B† ˆ 1=b, theactivity probability of a source is then n ˆ a=…a ‡ b† and we require that C > Nn sothat the queue has a stationary regime Provided that the probability density ofduration A ‡ B satis®es a simple regularity condition, we ®rst show in this chapterthat, once properly centered and normalized, the aggregate input rate and traf®cprocesses produced by the superposition of N i.i.d on=off sources converge as

N " ‡1 toward continuous Gaussian processes This is the content of our

``functional central limit'' theorems In contrast with the classical heavy loadapproximation, note that the limit input rate process obtained this way is generallynon-Markovian for arbitrary distributions of on and off durations A and B.Denote further by V0…N†the corresponding queue content in stationary conditions.Our essential aim is to obtain estimates for the limiting distribution of V0…N† as

N " ‡1, the capacity C scaling with N as

for some positive constant g Equation (5.1) is clearly identi®ed as a heavy loadcondition since the queue load Nn=C tends to 1 for increasing N (with speed1=pN† Now, by considering so-called heavy-tailed distributions for on and=oroff periods, the input process becomes long-range dependent (LRD) Based on theabove convergence results for input processes, our analysis then subsequentlyshows that the tail of the limiting distribution of the scaled queue content

V0…N†=pN is not exponential but Weibullian Speci®cally, de®ning

limN"‡1P…V0…N†> xpN† ˆ h…x†

for any ®xed x  0, we have

limx"‡1

1

x2…1 H†
log h…x† ˆ k22where both constants k and H 2 Š1

2; 1‰ depend on the distributions of on and offdurations Weibullian tails imply ®nite moments of all orders but, nonetheless,necessitate buffer sizes that grow much faster with load than in the classicalMarkovian case since the exponent 2…1 H† is strictly less than 1 for H >1

2.Such a result reminds one of the Weibullian nature of the distribution of a ¯uidqueue fed by the so-called fractional Brownian motion (FBM), as discussed inNorros [13] (see Chapter 4 in this volume for a full account) We also show in thischapter how the limit input processes obtained as N " ‡1 can easily be related tothe FBM by means of proper time and space scale changes

Other heavy load queueing analysis with heavy-tailed distributions has beenconsidered in related contexts such as the M=G=1 discrete queue (see Chapter 6 inthis volume) and the ¯uid queue with M=G=1 input rate process (refer to Chapter 9

in this volume and references contained therein) In comparison to the analysis

performed in Chapters 6, 9 and 10, we stress the fact that the limit considered here,with many small sources and heavy load queueing regime, enables the derivation ofcontinuous limit processes together with the Weibullian queueing behavior, asopposed to the heavy-tailed queueing behavior obtained when ®xed ``source peakrate=total output rate C'' ratio and a ®nite number of sources are considered.The rest of the chapter is organized as follows Section 5.2 contains somepreliminary notation and basic properties related to the on=off source model InSection 5.3, we state and prove functional central limit theorems (CLTs) for theproperly scaled input processes as the number of sources tends to in®nity In Section5.4, these CLTs are used to derive general limiting upper and lower bounds for thestationary distribution of the queue content V0…N† under the heavy load condition(5.1) Section 5.5 addresses the impact of long-range dependence on these bounds,demonstrating their Weibullian behavior in this case and relating the present model

to FBM as considered in Chapter 4 Much of the material presented in this chapterwas ®rst published in Brichet et al [5] Alternative arguments for convergenceresults can also be found in Kurtz [10]

5.2 PRELIMINARY PROPERTIES OF THE ON/OFF SOURCE MODELLet fltgt0 be the stationary process representing the input rate of a single on=offsource at time t and otˆ„0tlsds the input it generates in the interval ‰0; tŠ Wedenote fL…N†t g and fWt…N†g the sum of N i.i.d copies of the processes fltg and fotg,respectively For ®xed t; ltis a Bernoulli variable with mean n and variance

ma…t† ˆ E…otjl0ˆ 0†; mb…t† ˆ E…otjl0ˆ 1† …5:5†associated with ot Finally, we denote by r the covariance function of process fltgde®ned for t  0 by

r…t† ˆ E…l0 n†…lt n† ˆ E…l0lt† n2: …5:6†5.2 PRELIMINARY PROPERTIES OF THE ON/OFF SOURCE MODEL 117

Using basic renewal properties of the on=off rate processes, it can easily be shownthat the Laplace transform of covariance function r is given by

5.3 FUNCTIONAL CENTRAL LIMIT THEOREMS

The aim of this section is to ®nd suf®cient conditions for the rate and input processesassociated with the on=off sources to satisfy functional CLTs To this end, we ®rstestablish (Theorem 5.3.1) a CLT for renewal processes, using standard results onweak convergence [3] The desired results are then obtained in Section 5.3.2

5.3.1 CLT forRenewal Processes

For any interval I of R, let d…I† denote the space of right-continuous functions on Iwith left limits In the sequel, real-valued processes on some interval I areconsidered as d…I†-valued random elements For bounded I; d…I† is endowedwith the J1-Skorokhod topology The space d…R‡†, denoted by d for short, isendowed with the topology of convergence in the J1topology on bounded intervals.Theorem 5.3.1 Let L be a stationary renewal process, withL…0; t† representingthe number of renewal points within time interval Š0; tŠ; t  0, and with®nite andnonzero mean intensity l Denoting by P the distribution function of the inter-renewal times, we assume that:

the density P0exists and is bounded in some neighborhood of 0: …5:8†Then, given a sequence fLjgj0 of i.i.d copies of L, the sequence of processes

fL…N†gN>0 de®ned by

L…N†…t† ˆ 1

N

p PNjˆ1…Lj…0; t† lt†; t 2 R‡; …5:9†

is tight and converges weakly to a limiting Gaussian process with a.s continuouspaths as N " ‡1

In order to prove Theorem 5.3.1, the following results are needed The ®rst one istaken from Billingsley [3]

Theorem 5.3.2 [3, Theorem 15.6, p 128] Let I ˆ ‰0; TŠ be some boundedinterval of R For a sequence fG…N†gN>0 of d…I†-valued processes to convergeweakly to the d…I†-valued process G, it is suf®cient that:

 for all t1; ; tn2 I, the vector …G…N†…t1†; ; G…N†…tn†† converges weakly to

…G…t1†; ; G…tn††, as N " ‡1;

 there exists p > 0, q > 1 and a continuous nondecreasing function F suchthatE‰jG…N†…t† G…N†…t1†jp
jG…N†…t2† G…N†…t†jpŠ  ‰F…t2† F…t1†Šq …5:10†for all N and 0  t1 t  t2 T

The proof of Lemmas 5.3.3 and 5.3.4 below are deferred to Sections 5.7.1 and 5.7.2.Lemma 5.3.3 Let L be a stationary renewal process satisfying (5.8) Let a0> 0denote an upper bound of the density P0in some neighborhood of 0 Then for smallenough E > 0, process L can be constructed on the same probability space as ahomogeneous Poisson process M with intensity m ˆ l=…1 lE† _ a0=…1 a0E† sothat with probability one, the paths of M dominate those of L on ‰0; EŠ; that is,

L…t†  M…t†; t 2 ‰0; EŠ:

Lemma 5.3.4 In the setting of Theorem 5.3.1, let

P…N†…t1; t; t2† ˆ E‰jL…N†…t† L…N†…t1†j2
jL…N†…t2† L…N†…t†j2Š …5:11†for 0  t1 t  t2, where the process L…N† is de®ned by Eq (5.9) It holds that

P…N†…t1; t; t2†  2fE…x2Z2† ‡ E…x2†E…Z2† ‡ ‰E…xZ†Š2g; …5:12†where the random pair …x; Z† is distributed as …L…t1; t† l…t t1†, L…t; t2†

Kolmogorov's regularity criterion, that a version of G exists with almost surelycontinuous paths The version is thus a d-valued random element.

We now show that for any interval I ˆ ‰0; TŠ, condition (5.10) of Theorem 5.3.2

is met Let E be ®xed as in Lemma 5.3.3 Let t1 t  t2 lie in I If t2 t1 E,Condition (5.10) is met with p ˆ q ˆ 2 and F…x† ˆpdx, in view of (5.13) Itremains to ®nd a suitable bound on P…N†…t1; t; t2† when t2 t1> E Using the basicinequality 2jabj  a2‡ b2 in conjunction with the Cauchy±Schwarz inequality,(5.12) entails that

iˆ1jaijn, each of the means in the preceding bound isless than dT=Ee  dT=Ee4m4ˆ dT=Ee5m4 One thus obtains

P…N†…t1; t; t2†  6fdT=Ee5m4‡ …lT†4g:

Denoting the latter upper bound by K, t2 t1 E then entails that

P…N†…t1; t; t2† EK2…t2 t1†2:Thus condition (5.10) of Theorem 5.3.2 is met, with p ˆ q ˆ 2 and F…x† ˆ

d _ …k=E2†

p

5.3.2 CLT forRate and Input Processes

The propositions to follow state the functional CLT for the two sequences ofprocesses fL…N†t g and fWt…N†g introduced in Section 5.2

Proposition 5.3.5 Let A ‡ B denote the total duration of two successive on and offperiods of rate process fltg and assume that the distribution of A ‡ B has a boundeddensity in the neighborhood of 0 The sequence of processes Y…N† de®ned by

Proof Write ®rst Eq (5.14) as

Yt…N†ˆ 1

N

p PN

where the flj;tg are i.i.d copies of the process fltg In view of the ordinary CLT, the

®nite-dimensional distributions of the process Y…N† converge weakly to limitingGaussian distributions Weak convergence of the sequence fY…N†gN>0will then holdprovided it is tight Since L…N†is the superposition of N i.i.d on=off rate processes ljwith peak rate 1, Eq (5.15) entails that

Yt…N†ˆ Y0…N†‡ 1

N

p PNjˆ1fL‡

j …0; t† Lj…0; t†g …5:16†for t  0, where L‡

j …0; t† (resp Lj …0; t†† is the number of upward (resp downward)jumps of process ljover time interval ‰0; tŠ Each point process L‡

j (resp Lj) is astationary renewal process with intensity l, where 1=l ˆ 1=a ‡ 1=b is the mean cycleperiod of each process lj We therefore deduce from Eq (5.16) that

Yt…N†ˆ Y0…N†‡ L…N†‡ …t† L…N†…t†; …5:17†where

L…N† …t† ˆ 1

N

p PNjˆ1fL

j …0; t† ltg:

In view of decomposition (5.17), in order to prove the tightness of the sequence

fY…N†g, it is suf®cient to show that each sequence fY0…N†g, fL…N†‡ g, and fL…N†g is tight.The sequence fY0…N†g converges in distribution and is therefore tight Tightness ofboth sequences L…N†‡ and L…N† follows from Theorem 5.3.1 Since the weak limits ofboth sequences have a.s continuous paths, it results that the weak limit of Y…N†alsohas a.s continuous paths Finally, stationarity of each process Y…N† ensures

The following result on the convergence of the normalized input process can bededuced as a simple corollary of the previous proposition

Proposition 5.3.6 In the framework of Proposition 5.3.5, the sequence ofprocesses fO…N†t g de®ned by

converges weakly toward a zero-mean, continuous Gaussian process fOtg withstationary increments and O0ˆ 0 In addition, this process is such that

limt!‡1

with Y…N† introduced in Eq (5.14) The mapping j, which associates to some

f 2 d…‰0; TŠ† the integrated function j… f †: t 7 !„0tf …u† du, is continuous on thesubset C…‰0; TŠ† of d…‰0; TŠ† consisting of continuous functions (indeed, this holdssince the trace of the J1-Skorokhod topology on C…‰0; TŠ† coincides with thetopology of uniform convergence; see Billingsley [3]) As Y…N† converges weakly

to a limiting process Y with a.s continuous paths, O…N†ˆ j…Y…N†† converges weakly

to j…Y† Process Y being Gaussian and centered, the limit process O is clearlycentered, continuous, and Gaussian with O0ˆ 0 The stationarity of increments of O

is readily deduced from the stationarity of process Y In view of such stationarity, theergodic theorem can be used to yield the existence of an almost sure limit for Ot=t as

t ! ‡1 This limit is necessarily a Gaussian random variable, and we will be done

if its variance is equal to zero This variance is equal to (see Eq (5.32))

limt!‡1

r…t† ˆ nP…ltˆ 1jl0ˆ 1† n2;

we conclude that r…t† indeed goes to zero as t ! ‡1 j5.4 QUEUE CONTENT DISTRIBUTION IN HEAVY LOAD CONDITIONUsing the above convergence results for both the normalized rate process and theinput process created by the superposition of a large number of on=off sources, wecan now assert general limit bounds for the distribution of the normalized queue

content V0…N†=pN when N " ‡1 and heavy load condition (5.1) is ful®lled Toderive such limits, ®rst recall [2] that the stationary queue content V0…N† can bede®ned as the supremum of the transient process fWt…N† Ctg, namely,

V0…N†ˆ sup

Second, given a ¯uid queue with output rate g, it has been shown [16] that, providedthe input process fOtgt0 is such that (1) Otˆ„0tYudu for all t, where the rateprocess fYtgt0 is stationary, and (2) Ot gt tends to 1 almost surely as t " ‡1,then, for any x  0, the corresponding stationary distribution of the queue content

U0 veri®es the bounds

supt>0P…Ot> x ‡ gt†  P…U0> x†;

h…x† ˆ lim

N"‡1P…V0…N†> xpN†for any x  0, the bounds

q…x†  h…x†  Q…x†

hold with

q…x† ˆ supt>0 F x ‡ gt D…t† ; …5:22†where F ˆ 1 F and

Proof First, using the scaling condition (5.1) for C together with notation(5.18), relation (5.19) for V0…N† reads

V0…N†

N

p ˆ supt>0…O…N†t gt†:

Let f denote the map d 2 d 7 ! f…d† ˆ supt>0…dt gt† From the weak gence of processes O…N† toward process O, one can conclude that

conver-V0…N†=pN ˆ f…O…N†† converges weakly to U0ˆ f…O† provided the distribution of

O puts no mass on the set of trajectories d 2 d for which f is discontinuous Thefact that O has a.s continuous paths and is such that Ot=t ! 0 a.s., as implied byProposition 5.3.6, is suf®cient to conclude that this is so (the detailed argument isleft to the reader) It therefore follows that, for ®xed x  0, P…V0…N†> xpN† !

my…t† ˆ Cov…Y0; Ot† Var…Yy

0†ˆ E…Y0Ot† 

y

E…Y0Ot† ˆ s2m…t† by deconditioning with respect to the centered variable Y0 Itconsequently follows from the above discussion and relations (5.25) and (5.26) that

my…t† ˆ m…t†y; D…t†2ˆ D…t†2 m…t†2s2: …5:27†Using Eq (5.27) in upper bound (5.20), we then obtain

5.5 IMPACT OF LONG-RANGE DEPENDENCE

We now wish to introduce long-range dependence into the source model and deriveexplicit estimates of bounds q…x† and Q…x† of Proposition 5.4.1 for such a case Asshown below, a natural way to do this is to assume a heavy tail for the distribution ofeither the silence period, the activity period, or both In the rest of this section, we

®rst specify the LRD behavior of the rate process associated with heavy-taileddistributions for on=off durations, then provide the estimates for bounds q…x† and

Q…x† in such a case We conclude this section by providing a weak convergencetheorem for the limiting input process fOtg obtained in Section 5.3, which, oncerescaled in space and time, converges toward the so-called fractional Brownianmotion

5.5.1 LRD Properties

Recall [6] that a stationary process fltg is said to exhibit short- (resp long-) rangedependence if its correlation function r de®ned in Eq (5.6) is such that the integral

„‡1

0 r…t† dt is convergent (resp divergent) If the distributions of on and off periods

A and B are Coxian, that is, their Laplace transforms a* and b* are rationalfunctions, then standard results for Laplace transform inversion applied to expres-sion (5.7) entail that the correlation function t 7 ! r…t† tends exponentially fast to 0

as t " ‡1, thus corresponding to short-term dependence for process fltg Now, thesituation appears totally different in the case when the distribution of either offduration A or on duration B is heavy tailed A distribution function F is here de®ned

for some y > 0 As a consequence of de®nition (5.28), the Laplace transform

f *: p > 0 7 !„0‡1e pt dF…t† of a heavy-tailed distribution F with ®nite mean m andpower r 2 Š1; 2‰ veri®es

f *…p† ˆ 1 mp ‡h
G…2 r†

r 1 pr‡ o…pr† …5:29†for small positive p, where G is Euler's function (see Section 5.7.3) For 1 < r < 2,such an expression for the Laplace transform near 0 in fact characterizes the heavy-tailed behavior of the corresponding distribution F (in the case when r ˆ 2, Eq.(5.29) is identical to a classical Taylor expansion, corresponding to a regulardistribution having a ®nite variance) We can then show the following

Proposition 5.5.1 Assume that the Laplace transforms a* and b* of the off and onperiods have expansions

a*… p† ˆ 1 p

a‡ arpr‡ o… pr†;

b*… p† ˆ 1 pb‡ bsps‡ o… ps†; …5:30†

respectively, withpositive coef®cients ar and bs and where powers r; s 2Š1; 2Š aresuchthat r ‡ s 6ˆ 4 The covariance function of rate fltg then veri®es

is easy to express its variance D2…t† de®ned in Eq (5.3) as

r0 being expressed as in Proposition 5.5.1