LIMIT BEHA VIOR OF FLUID QUEUES AND NETW ORKShalweb.uc3m.es/esp/personal/personas/bdauria/files/dauriasamorodnitsky05.pdf · LIMIT BEHA VIOR OF FLUID QUEUES AND NETW ORKS BERNARDO

LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS

BERNARDO D�AURIA AND GENNADY SAMORODNITSKY

Abstract� A superposition of a large number of in�nite source Poisson inputs or that of a largenumber of ON�OFF inputs with heavy tails can look like either a Fractional Brownian Motionor a stable L�evy Motion� depending on the magni�cation at which we are looking at the inputprocess �Mikosch et al� �� In this paper we investigate what happens to a queue driven bysuch inputs� Under such conditions� we study the output of a single uid server and the behaviorof a uid queuing network� For network we obtain random �eld limits describing the activity atdi�erent stations� In general� both kinds of stations arise in the same network� the stations ofthe �rst kind experience loads driven by a Fractional Brownian Motion� while the stations of thesecond kind experience loads driven by a stable L�evy motion�

�� Introduction

After the discovery that the internet data tra�c has unusual statistical characteristics such

as self�similarity� long range dependence �LRD� and heavy tails� many new models have been

developed to explain the origin of these new features� interaction between them� and to take

these features into account�

Two of the most popular models view the data tra�c as the superposition of many contributions

due to independent sources of data� They are the so called in�nite source Poisson model �otherwise

known as the M�G�� input �ow model� and the superposition of ONOFF sources�

In theM�G�� input �ow model� the transmissions �sources� start according to a homogeneous

Poisson process� The durations of the transmissions are mutually independent and identically

distributed� and independent of the Poisson process� If the transmission times are heavy tailed

the resulting input process has slowly decaying correlations and� hence� is viewed as having long

range dependence �see e�g� Samorodnitsky ��

In the superposition of ONOFF sources model� the number of sources is constant and each

source switches between ON�periods �transmission� and OFF�periods �silence�� For each source�

�� Mathematics Subject Classi�cation� Primary �K�� secondary �F�� F�� G��Key words and phrases� queue� queuing network� output process� heavy tailed distribution� long range depen�

dence� fractional Brownian Motion� stable L�evy process� weak convergence�D�Auria�s research was supported by the University of Salerno and Cornell University during his visit to Cornell

University� Samorodnitsky�s research was partially supported by NSF grant DMS�� and NSA grant MSPF��G�� at Cornell University�

�

� B� D�AURIA AND G� SAMORODNITSKY

the ON�periods and OFF�periods are assumed to be two sequences of i�i�d� random variables�

mutually independent� and di�erent sources are independent as well� Similarly to the previous

model� when the transmission times andor silence times are heavy tailed the resulting input

process has slowly decaying correlations and is also viewed as having long range dependence �see

Heath et al� �� Slow decay of correlation often comes together with a certain type of scaling�

hence the observed self�similarity of the tra�c�

While in the literature there is� largely� a consensus on the self�similar nature of the aggregate

tra�c� di�erent authors report divergent conclusions about the marginal distributions for the

cumulative tra�c� Indeed� these have been at times reported as light tailed� heavy tailed or

intermediate tailed� some of the more recent references are Smith et al� �� Campos et al�

�� Downey �� and Gong et al� �� One explanation of this phenomenon can be found

in the recent papers of Mikosch et al� �� and Gaigalas and Kaj �� which studied the limit

of a sequence of properly scaled input processes both for the M�G�� and ON�OFF models� It

turns out that the limit depends on relation between the rate at which the transmissions are

initiated �called connection rate in Mikosch et al� �� and the time scale at which the system

is considered� If the connection rate is relatively high� the deviations of the input process from

its average look like a Fractional Brownian motion� while if the connection rate is relatively low�

these deviations look like a stable motion�

In these papers the attention was focused on the input �ow to a single server� We� on the

other hand� are interested in the output �ows� The knowledge of the output process properties

is very useful as the output �ow from one station is usually �a part of� the input process for a

subsequent queue� Following this line of reasoning it is possible to get insights into the behavior

of a whole queuing network�

In this paper we show that the deviations of the output �ow from its average in a single queuing

system behave similarly to those in the input �ow and� hence� satisfy limit results of the kind

given in Mikosch et al� �� We extend then these results to �uid queuing networks� The

results rely on the fact that the stability of the involved queuing systems assures tightness of the

bu�er content processes� As a consequence� we will see how the marginal distributions� together

with the correlation structure� are propagated across the network�

Roughly speaking� one can view the limit theorems we obtain in the following way� It turns

out that� in addition to the linear drift� the �ow of the work through the network looks Gaussian

in some parts of the network and stable in other parts of the network�

LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS �

This fact has direct practical applications� For example� in the situations where our limiting

results are applicable �i�e� in high capacity networks with high average tra�c and moderate

tra�c intensity� with occasional large tasks� which are considered at large time scales�� empirical

�nding of su�ciently di�erent marginal behavior in two separate points of a network may give

information of low coupling �reducibility� of the network� For instance� if we �nd that the properly

normalized cumulative input �ow at one station has approximately a stable distribution and at

another station it has approximately a Gaussian distribution� then the output of the latter station

does not reach the former� Otherwise� according to the Theorems �� and �� below� the �rst

distribution would be Gaussian as well�

From the modeling point of view� the knowledge of the nature of the limit �ows legitimates

the study of queuing networks in simpli�ed assumption of Brownian Motion � or stable L�evy

Motion � based �ows� There is another important point that is worth emphasizing� Note that

we are studying the behavior of random loads that are� generally� di�erent in di�erent parts

of the network and at di�erent instances of time� That is� we are dealing with a stochastic

process whose �time� is� really� two�dimensional� with the network location playing the role of one

�time component� and the physical time being the second ��time component�� Such stochastic

processes with multivariate time are commonly referred to as random �elds� Our limiting results

preserve this random �eld point of view� Having a random �eld description of the limiting �ows in

a queuing network gives one a state�time description of the loads experienced by the network and�

hence� allows one a better understanding of what may happen in such a network� For example�

it is possible to relate� in principle� what happens in one place in the network in one instance of

time and in another place in the network at a later instance of time� Furthermore� it is possible

to simulate scenarios of behavior of the entire network or its part� For comparison� a limiting

description of only the marginal distributions at individual stations� or of dynamical description

of individual stations separately would not have provided the same insight�

The results of this paper contribute to the existing literature on queuing networks� and a large

body of publications already studies the behavior of networks under various limiting procedures�

Considering such limiting behavior is important both for large or heavily loaded networks� and

because exact distributional results for non�limiting case are available mostly in Markovian cases�

and even then only under limited circumstances �see� for example� Kella �� for tandem�type

networks�� Common limiting procedures considered in the literature include heavy�tra�c anal�

ysis� beginning with Harrison �� and Reiman �� Here one usually studies the limiting

distribution of the queue length process� which� under appropriately light�tailed input� turns out


to be a re�ected Brownian motion in a region� For �discrete� queuing networks with heavy tailed

input similar results but with re�ected fractional Brownian motion in a region were obtained by

Konstantopoulos and Lin �� A very general framework for results of this type for ��uid�

queuing networks� including both light tailed and heavy tailed cases� is in the Section �� of

Whitt �� Another type of limiting procedures involves studying �uid limits of discrete queu�

ing networks �the so�called functional laws of large numbers� and the corresponding deviations

from the limit given by a central limit theorem� Here the limits are taken as the number of nodes

in the system and the arrival rates increase� See Mandelbaum et al� ��

Note that the reason to consider a limiting behavior of a network is to use such results to

approximate the behavior of �nite networks that are nearly extreme in some ways� Fluid Gaussian

limits are obtained in both many of the mentioned publications� and also� in some situations� in

the present paper� This means that queuing networks may have approximately Gaussian behavior

under di�erent �extreme� scenarios� Certain �but not all� features of this approximation carry

over from one situation to another one� For example� under many heavy tra�c �uid limits

a curious phenomenon occurs� that has been called a snapshot principle by Reiman �� a

consequence of scaling the space more heavily than scaling the time� The snapshot principle says

that the workload in the system does not change much on the scale of job processing times and�

hence� information about the workload can be e�ciently transfered through a heavy tra�c system�

Networks considered in the present paper are not under heavy tra�c� but we are still scaling the

space more heavily than the time� and so the snapshot principle holds� On the other hand� both

normalizations and certain other features of the approximation� like the correlation function of

the Gaussian limit may di�er between heavy tra�c situations and the networks considered in this

paper�

This paper is organized as follows� Section formally de�nes the models we are studying in

the case of a single queuing system� Section � contains the results on the tightness of the bu�er

content process and Section � gives limit results for the output process of a single �uid queue�

Sections �� and � extend these results to queuing networks�

� Two models of a fluid queue

In this section we describe two models for a single �uid queue� The queue has an in�nite

bu�er and drains it at constant service rate� say C� The instantaneous input and output �ows


are denoted respectively by i�t� and o�t� while the cumulative input and output processes are

I�t� �

Z t

�i�s�ds and O�t� �

Z t

�o�s�ds�

The bu�er content� that is also the workload in the system� is denoted by W �t�� Unless noted

otherwise we assume thatW �� The workload is related to the cumulative input and output

processes through the relation

O�t� � I�t��W �t��

that expresses the conservation of the �ow�

We now introduce the two di�erent models for the instantaneous input process i�t��

�� M�G�� input �ow� This model is also known as the in�nite source Poisson input model�

Here the input �ow i�t� is given by the sum of the contributions of various sources arriving

according to a Poisson point process and being active for a random duration�

If ftigi�N are the arrival points of a Poisson point process with intensity �� and fXigi�N is a

sequence of i�i�d� �nite mean random variables independent of the Poisson process� then

i�t� �

�Xi��

�fti � t � ti �Xig��

In order to study the limit behavior of the output process we suppose to have a sequence of

�uid queues� indexed by T � and we suppose that the intensity of the arrival Poisson process will

depend on T � i�e� � � ��T �� In particular ��T � is assumed to be an unbounded function of T �

The subscripts as in iT � IT � etc� will show that a process corresponds to a particular system�

We will show that under our limiting procedure� the normalized di�erence between the input

and output processes converges to � �the zero process�� To show that this is true� we may allow

the random variables fXigi�N to have an arbitrary distribution with a �nite mean � � E�Xi��

However� in order to use the known results on convergence of the input process� in the sequel we

suppose that these random variables are appropriately heavy tailed� Speci�cally� denoting by F �x�

their common distribution function� we make the following assumption on its tail �F �x�� F �x�

�F �x� � x��L�x�� x � ��

where L�x� is a slowly varying at in�nity function �Feller ��

The behavior of the normalized version of IT �Tt� as T � � was studied in Mikosch et al�

�� In the next section we prove that the �nite dimensional distributions of OT �Tt�� IT �Tt�

are tight for T � ��

B� D�AURIA AND G� SAMORODNITSKY

�� Superposition of ON�OFF sources� The superposition of ON�OFF sources input model

views the tra�c as the sum of independent contributions ofM �ows� each one being a sequence of

ON and OFF periods� Speci�cally � suppose that fXoni�jg

�i�� fX

oi�j g

�i�� are M independent pairs

of independent sequences of i�i�d� random variables� � � j �M � A generic random variable Xon

is distributed according to some distribution F on�x�� while a generic random variable Xo has a

distribution denoted by F o�x�� In the following �on � E�Xon�� o � E�Xo� are assumed to be

�nite� Denote � � �on � �o and pon � �on

� �

Construct independent sequences of i�i�d� random variables f �Xonj gMj�� and f

�Xoj gMj�� having�

respectively� distributions �F on�x� � ��on

R x��F on�s�ds� �F o�x� � �

�o�

R x��F o�s�ds� independent of

the previously constructed random objects� Following the construction used in Mikosch et al�

�� we de�ne the random variables

T��j � Bj� �Xonj �Xo

j � � ��Bj� �Xoj

with �Bj� being a sequence of i�i�d� Bernoulli random variables� once again independent of

everything else and with PfB � �g � pon� De�ne� further� for n � �� Tn�j � T��j �Pn

i��fXoni�j �

Xoi�j �� then

i�t� �

MXj��

�Bj�ft � �Xon

j g��Xn��

�fTn�j � t � Tn�j �Xonn��jg

��

As with the previous model� we will restrict our attention to the case

�F on�x� � x��on

Lon�x�� x � �� on � ��

and

�F o�x� � x��o�

Lo�x�� x � �� o � ��

with Lon�x�� Lo�x� slowly varying at in�nity functions and � � �on � �o� The assumption

that the ON periods have heavier tails than the OFF periods is made exclusively to con�rm with

the setup in Mikosch et al� �� Similar results can be obtained without this assumption� see

Mikosch and Stegeman ��

Once again� we will suppose to have a sequence of queuing systems indexed by T whose input

process is a superposition of sources� The number of sources M�T � is an unbounded function of

T �

As in the M�G�� case� we will prove that the �nite dimensional distributions of OT �Tt� �

IT �Tt� are tight for T � ��


�� Tightness of the buffer content

In this section we show that the �nite dimension distributions of OT �T �� IT �T �� are tight for

T � �� That is done in the next subsections separately for the two models� The main idea is to

show that the bu�er content under di�erent models remains tight� hence the title of the section�

�� M�G�� input �ow� In the statement of the lemma below the notation Xst� Y means that

P �X � x� � P �Y � x� for all x � R�

Lemma �� Suppose we have a series of M�G��input �uid queuing systems indexed by T � �

with service rate C�T � � ��T � and the arrival rate of the M�G�� input process� ��T � ��

such that � � �� Then� there exists a random variable H such that

WT �t�st� H �T� t � ��

Proof� The bu�er process WT �t� satis�es the Reich formula

WT �t� � sup��s�t

fIT �t�� IT �s�� C�T ��t� s�g ��

Evidently WT �t�st�WT ��

��WT � since IT has stationary increments� Here WT has the station�

ary distribution� a simple regenerative argument shows existence of this distribution thanks to

the fact that �� T �C T �

� �� So in the sequel we consider only the stationary distributions� and we

prove that they are uniformly stochastically bounded in T �

Let us consider the stationary bu�er distribution W�� As the queuing system is stable� it is a

well de�ned random variable�

Consider for each T � � another queuing system� this time not a �uid one� At each time a

new session arrives� it adds to the bu�er an amount of workload equivalent to its duration� Let

�HT �t�� t � �� be the bu�er content process associated with this new queuing system� Evidently

we have that

HT �t� �WT �t� �T� t � ��

and the relation is valid also for the stationary distribution that exists thanks to the fact that

the tra�c intensity for the new system is less then �� Hence in the particular case of T � � we

have W�

st� H��

To complete the proof� let � � T �� Observe that �HT �

��t�� t � �� is the bu�er content

of the system where the Poisson point process of arrivals has intensity ��T � � �� and service

rate ��C�T � � ��T � � �� C�� Hence HT ��

fidi� H�� and so HT

d� H�� which

proves �� with H � H��


Recall that � � OT �T �� IT �T �� WT �T �� for all T� t � �� Since a family of random variables

that is stochastically bounded in absolute value by a �nite random variable is tight� we conclude

that one�dimensional distributions of OT �T ��IT �T �� are tight for T � �� which implies the same

thing for all �nite�dimensional distributions�

�� Superposition of ON�OFF sources�

Lemma �� Let QM be a stationary queuing system whose input process is the superposition of

M independent and identically distributed stationary ON�OFF processes� Let CM � �M be the

service rate of the queuing system QM � with � � �� Then the stationary workload �WM �M � ��

is tight�

Proof� We transform the �uid system QM into a G�G�� queuing system in the following way�

At each instance of activation of an ON period� the amount of work equal to the length of the

ON period is added to the bu�er content� As in the proof of Lemma �� the bu�er content of

the new system cannot be smaller than that of the �uid system� That is� if �WM�t�� t � �� is the

bu�er content process of QM and �W �M�t�� t � �� is the bu�er content process of the new system�

then

WM �t� �W �M �t� t � ��

It is clear that the new system has a unique stationary distribution of the bu�er content� Let us

call it W �M � Then

W �M

st�WM M � ��

Changing time by replacing the bu�er process W �M�� by W �

M ��

CM�� preserves the stationary

distributionW �M � In the time�changed system� the arrival process is the superposition of M i�i�d�

ON�OFF processes with the time dilated by CM � and service rate �� Hence� this is a G�G��

queue� with the arrival rate �� on��o��

� and mean service time �on� We will use the following

Proposition �� VI� of Daley and Vere�Jones ��

Proposition Let �N be a simple point process on R with �nite intensity �� and let �NM denote the

point process obtained by superimposing M independent replicates of �N and dilating the time

scale by a factor M � Then as M �� NM converges weakly to a homogeneous Poisson process

with intensity ��

This implies� in particular� that we are in the framework of the continuity theorem� e�g� Theo�

rem �� of Brandt and Lisek �� with the limiting system being a stable M�G�� queue�


If W �� is the steady state bu�er content of the latter� we have W �

M W �� hence �W

�M �M � ��

is tight and then so is �WM �M � ��

Let now M � M�T � and switch to our usual notation WT �t� � WM T ��t� and WT � WM T ��

As in the proof of Lemma �� the tightness of the stationary workload implies the tightness of

the family WT �t�� T� t � �� Now the argument at the end of the previous subsection establishes

tightness of the �nite�dimensional distributions of OT �T �� IT �T ��

�� Stable motion or Fractional Brownian Motion at the output from a fluid

queue

As a corollary of the results in the previous section� we can extend the convergence results in

Mikosch et al� �� to the output processes�

In this section it is essential that the random variables that de�ne the input processes are

appropriately heavy tailed� Moreover� as stated in Mikosch et al� �� the limit process will

depend on the rate of convergence to in�nity of the quantities ��T � and M�T � as T �� These

is the reason the Growth Conditions are introduced�

In the following b�t� is the quantile function

b�t� �� F

��t� or b�t� �

�� F on

��t��

depending on the model� where for a non�decreasing function F �

�F ��t� � inffx � R � F �x� � tg�

�� M�G�� input �ow�

De�nition ��

Slow Growth Condition �SGC�� limT��b � T �T �

T � ��

Fast Growth Condition �FGC�� limT��b � T �T �

T ��

Using the previous lemma it is easy to prove the following theorem� Our notation for sta�

ble distributions follows that in Samorodnitsky and Taqqu �� Furthermore� the following

standard notation and terminology for weak convergence in the context of stochastic processes

will be used� Let XT �t�� t � �� T � � be a family of stochastic processes and Y �t�� t � � be

another stochastic process� If for every �nite collection of times t�� tk� the family of �the dis�

tributions of� k�dimensional random vectors �XT �t�� XT �tk�� converges weakly as T � �

to �the distribution of� the corresponding random vector �Y �t�� Y �tk�� then we say that

�� B� D�AURIA AND G� SAMORODNITSKY

XT �t�� t � � converges in �nite�dimensional distributions as T � � to Y �t�� t � �� and use the

notation XT ��fidi� Y ��

If both the processes in the family XT �t�� t � �� T � � and the process Y �t�� t � � have

continuous sample functions� then they can be viewed as random vectors in the metric space

C�� In that case� if the random vector XT �t�� t � � converges weakly in C�� as T ��

to the random vector Y �t�� t � �� then we also say that the stochastic process XT �t�� t � �

converges in distribution to the stochastic process Y �t�� t � �� and use the notation XT ��d�

Y �� Convergence in distribution implies convergence in �nite�dimensional distributions� but the

converse statement requires tightness� See Billingsley ��

Theorem �� Suppose we have a series of M�G�� input �uid queuing systems indexed by T �

service rate C�T � � ��T � with � � � and the arrival rate ��T �� and ��T � �� Consider the

output �ow OT �� Depending on which growth condition is veri�ed� the following holds�

SGC � OT T ��T� T � ��b � T �T �

fidi� X��

FGC � OT T ��T� T � ��

�� T �T � �F T ��

d� BH�� H �

��

��

Here X�� is an ��stable L�evy Process with positive jumps� � � ��

h�

��

i� and BH��

is the standard Fractional Brownian Motion�

Proof� Using the equation �� and subtracting the mean values� we obtain

OT �T �� T��T ��

g�T ��

IT �T �� T��T ��

g�T ��WT �T ��

g�T ��

where g�T � is� depending on the growth condition� equal to

SGC � g�T � � b��T �T ��

FGC � g�T � ��T �T � �F �T �

� ��

In any case limT�� g�T � �� hence for every t � �

WT �Tt�

g�T �

T�� in probability��

and� together with the results of Theorem � and Theorem � in Mikosch et al� �� respectively

for the SGC and for the FGC� this completes the proof for the �di convergence�

To prove the weak convergence �or convergence in distribution� in the FGC case� we start with

several observations� First of all� weak convergence in C �� is equivalent to weak convergence

in C ��H� for all H � �� Since the argument for di�erent time lengths is exactly the same� we

will prove weak convergence in C �� Second� weak convergence in C �� is metrizable �e�g�

LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS ��

by Prohorov�s metric� see e�g� Theorem �� in Whitt �� Given a family of a vectors in a

metric space� indexed by a continuous parameter T � �� convergence of these vectors as T �� is

equivalent to convergence� as n�� of all countable subfamily indexed by Tn� n � � with Tn

�� Therefore� we need to prove weak convergence of�OTn�Tn��Tn��Tn��

��Tn�T

�n�F �Tn��

��

in C �� along subsequences Tn �� Since convergence of �nite�dimensional distributions has

already been established� it only remains to prove tightness� and it is established in Lemma ��

below�

�� Superposition of ON�OFF sources� First we adapt the de�nitions to the present case�

Now we set

De�nition ��

Slow Growth Condition �SGC�� limT��b M T �T �

T � ��

Fast Growth Condition �FGC�� limT��b M T �T �

T ��

Even though the setup is di�erent from the previous case� we still have a similar result about

the behaviour of the output from a single queue�

Theorem �� Suppose we have a series of �uid queuing systems indexed by T � fed by the su�

perposition of M�T � i�i�d� ON�OFF processes� service rate C�T � � �M�T �� with � � � and

M�T � �� Consider the output �ow OT �� Depending on which growth condition is veri�ed�

the following holds�

SGC � OT T ��onTM T � ��

b M T �T �

fidi� cX��

FGC � OT T ��onTM T � ��

�M T �T � �F on T ��

d� �BH�� H �

��

��

Here � C� ��

� � C� ��

� ��cos �� c ��o�

�� X�� is an ��stable L�evy Process� ��

� �o��

� and BH�� is a standard Fractional Brownian Motion�

The argument is the same as in the previous case� For the tightness we need Lemma �� below�

�� Tightness� To prove weak convergence in C �� in Theorems �� and �� under FGC

condition� we need to prove tightness of the probability measures induced byOTn T ��E�OTn Tn��

g Tn�

�with the function g�T � appropriate to theM�G�� and ON�OFF cases� for increasing to in�nity

sequences Tn�


Lemma �� Let �Tn� be a sequence increasing to in�nity� If for some function g�T � the proba�

bility measures induced byITn Tn��E�ITn Tn��

g Tn�are tight in �C �� J�� then so are the probability

measures induced byOTn Tn��E�OTn Tn��

g Tn��

Proof� By Theorem �� of Billingsley �� we need to check that for every � � � and any � �

there is � � � and n� � � such that

P

�� sup

��s�t��

t�s�

��WTn�Tnt��WTn�Tns�

g�Tn�

��

A � ��

for all n � n��

We prove �� by dividing the probability in two parts

P

�� sup

��s�t��

t�s�

��WTn�Tnt��WTn�Tns�

g�Tn�

��

A � P

�� sup

��s�t��

t�s�

WTn�Tnt��WTn�Tns�

g�Tn��

A

� P

�� sup

��s�t��

t�s�

WTn�Tns��WTn�Tnt�

g�Tn��

A �

and show that we can make both probabilities in the right hand side arbitrarily small� Fix � � �

and consider the event

��

�� sup

��s�t��

t�s�


g�Tn��

��

For any � �� choose � � s� � t� � �� t� � s� � � such that

WTn�Tnt��WTn�Tns�� g�Tn��

Let

u� � supfr � �s�� t�� WTn�Tnr� � �g �

with the convention that supf�g � s�� Then

�g�Tn� � WTn�Tnt��WTn�Tnu�� ITn�Tnt�� ITn�Tnu�� C�Tn��Tn�t� � u��

� ITn�Tnt�� ITn�Tnu�� E�ITn��Tn�t� � u��

as in the period �u�� t�� the bu�er is always non�empty and� hence� the system constantly drains

�ow at the rate C�Tn� � E�ITn��

Since t� � u� � �� we conclude that

��

�� sup

��s�t��

t�s�

ITn�Tnt�� ITn�Tns�� E�ITn��Tn�t� s��

g�Tn��

��


Since Mikosch et al� �� established tightness for IT T ��E�IT T ��g T � � we can use Theorem �� of

Billingsley �� to conclude that there is a �� and n� � � such that

P � ��


Therefore� for each � � � and � �� there are �� and n� � � such that

P

�� sup

��s�t��

t�s��


g�Tn��

A �

��


It remains to prove that� for every � � � and � �� there is �� and n� � � such that

P

�� sup

��s�t��

t�s��


g�Tn��

A �

��

since then �� will follow by choosing n� � max�n�� n�� and � � min�� The argument in

Theorem �� in Billingsley �� shows that it is enough to prove that for any � � � and � �

there is � � � � � and n� � � such that

�

�P

�sup

s�t� s��


g�Tn��

��

for all n � n� and any � � s � �� Let us �x any � � �� For � � s � � we have

�

�P

�sup

s�t� s��


g�Tn��

��

�P �WTn�Tns� � �g�T ��

for n large enough� sinceWTn Tns�g Tn�

� in probability�

Therefore� �� follows� and so we are done�

�� The fluid network model

In the following parts of the paper we use the results of the previous sections to describe the

limiting behavior of a network of �uid queues� We start with de�ning our �uid network�

The network consists ofN �uid queuing systems� say fQigi� in the sequel referred to as stations�

each one with constant service rate Ci� The output from station Qi is routed to the other stations

in a deterministic way according to the routing rates fpijgj� i�e� if oi�t� is the instantaneous

outgoing �ow from the station Qi� the station Qj receives instantaneous input of pijoi�t��

The rates satisfy the obvious conditions

pij � �� i� j � N and

NXj��

pij � �� i � N��


Hence the matrix P � fpijg� called routing matrix� is a substochastic matrix�

The portion of output �ow from the station Qi to the outside of the network is given by

pi� � ��PN

j�� pij� We remark that the case pii � � is not excluded and it refers to the presence

of a loop around station Qi�

Similarly to the previous sections� we use the notation

Ii�t� �

Z t

�

ii�s�ds and Oi�t� �

Z t

�

oi�s�ds�

to denote the cumulative input and output processes at station Qi� while the bu�er content is

denoted by Wi�t��

Let �ii�t� and �Ii�t� be respectively the amount of instantaneous and cumulative input to the

queuing station Qi from the outside of the network�

The cumulative input to the queuing station Qi can be expressed for each t � � as

Ii�t� � �Ii�t� �

NXj��

pjiOj�t�

and� using the fact that

Oj�t� � Ij�t��Wj�t�� j � N�

we obtain the following system of equations

Ii�t��NXj��

pjiIj�t� � �Ii�t��NXj��

pjiWj�t�� i � N��

De�ning the row vectors �I�t� � �I��t�� IN�t��I�t� � ��I��t�� IN�t�� and �W �t� � �W��t��

WN �t�� this system can be expressed in vector form as

�I�t��I � P � � ��I�t�� W �t�P�

Suppose that for every i � �� N � p k�i� � � for some k � � �i�e� no �ow is destined to stay

in the system forever�� Then the matrix �I � P � is invertible �Feller �� XV�� Denoting

H � �I � P �� the solution of the system of equations �� can be written in the form

�I�t� � ��I�t�H � �W �t�PH��

�� M�G�� input processes� In this case� each external input process �Ii� results from a

M�G�� input process�

We denote by �i the intensity rates of the Poisson point processes� and by F i� and �i the

distribution functions and the means of the activity periods� respectively� For the limit results


we derive in the sequel we assume regular variation

�F i��x� � x��iLi�x�� x � �� i � ��

where Li�x� is a slowly varying at in�nity function� Note that we are not assuming that �i� i �

�� N are all equal� As in the case of a single station� the assumption of the regular variation

is not required for the bu�er content tightness established below�

The tra�c intensities at di�erent stations �i are given by

�i ��

Ci

NXj��

hjiEh�Ij��

i��

Ci

NXj��

hji�j�j��

where H � �hij� is as above�

Suppose we have a sequence of such queuing networks� indexed by T � in which some components

of the arrival intensity vector ��T � increase to in�nity� In this case we characterize the limit

behavior of the normalized �ows circulating in the network�

As in the previous sections� we keep constant the values of the tra�c intensities by varying the

service rates of the queuing stations� i�e�

Ci�T � ��

�i

NXj��

hji�j�T ��j��

�� Superposition of ON�OFF sources� In this case� the external input processes ��Ii� � �

i � N�� result from a superposition of i�i�d� stationary Mi ON�OFF sources�

We denote by �oni and �oi the average ON and OFF times for the external input at the station

i� and use the notation �i � �oni � �oi and poni ��oni�i� Furthermore� the limit results we will

prove �but not the bu�er content tightness� require the regular variation assumptions on the ON

times and OFF times distribution functions

�F oni �x� � x��

oni Loni �x�� x � �� oni � ��

and

�Fio�x� � x��

o�i Loi �x�� x � �� oi � ��

with Loni �x�� Loi �x� slowly varying at in�nity functions and �i � �oni � �oi � Once again� the

indices �i� i � � � � � � N do not have to be all equal� Note that� as before� the assumption that

the ON periods have heavier tails than the OFF periods is only made to con�rm the existing

literature� and that similar results can be obtained without this assumption�


In the same way as before� the tra�c intensities are given by

�i ��

Ci

NXj��

hjiEh�Ij��

i��

Ci

NXj��

hjiMjponj ��

where H � �hij� is as above�

Once again� we consider a limiting procedure in which we have a sequence of such queuing

networks� indexed by T � with some components of the source�number vector �M�T � increasing to

�� To keep the tra�c intensities constant� the service rates are assumed to be given by

Ci�T � ��

�i

NXj��

hjiMj�T �ponj ��

�� Tightness of the buffer content for the stations in a network

In this section we establish the workload tightness for a queuing network in a manner similar

to that used for a single queue in Section ��

�� M�G�� input processes�

Theorem �� Suppose we have a series of �uid queuing networks� indexed by T � consisting of

N stations fQigi and with routing matrix P such that� for each � � i � N � p k�i� � � for some

k � �� The external input to each station Qi is a M�G�� process with intensity �i�T �� The

service rates satisfy the relation �� and so the tra�c intensities �i � � are kept constant�

Then the stationary workload �Wi�T � T � �� at each station � � i � N is tight�

Proof� For every station i and t � �� Oi�t� � Ii�t�� Therefore

Ii�t� � �Ii�t� �

NXj��

pjiIj�t�� i � N� t � ��

Since the matrix H � �I � P �� P�

n�� Pn has nonnegative entries� we conclude that

Ii�t� �NXj��

hji �Ij�t�� i � N� t � ��

Notice that if every �Ii�� is an M�G�� input process� then so isPN

j�� hji�Ij�� Hence our

statement follows from Lemma ��


�� Superposition of ON�OFF sources�

Theorem �� Suppose we have a series of �uid queuing networks� indexed by T � consisting of

N stations fQigi and with routing matrix P such that� for each � � i � N � p k�i� � � for some

k � �� The external input to each station Qi is a superposition of Mi�T � stationary ON�OFF

processes� The service rates satisfy the relation �� and so the tra�c intensities �i � � are

kept constant�

Then the stationary workload �Wi�T � T � �� at each station � � i � N is tight�

Proof� Using the same argument as for the M�G�� case we have

Ii�t� �NXj��

hji�Ij�t� � � i � N� t � ��

Letting � �Wi�t�� t � �� be the bu�er content process of a queuing system fed by the process in the

right side of �� and with service rate Ci� we have that

�Wi�t� � sup��s�t

��

NXj��

hji

��Ij�t�� Ij�s�

�� Ci�t� s�

��

�NXj��

hji sup��s�t

n�Ij�t�� Ij�s�� iE

h�Ij��

i�t� s�

o�

NXj��

hji �Wji�t��

where ��i �CiPN

j�� hjiE��Ij ��

� � and where � �Wji�t�� t � �� is the bu�er occupancy process of a �uid

queue fed by the input process �Ij� with service rate ��iE��Ij�� Hence the tightness follows from

Lemma ��

�� Random field limits for the flows in fluid networks

This section brings together the results on single queues and on tightness of the workload for in

the network case� considered previously� and establishes a random �eld limit for a �uid queuing

network� Since the language describing what happens in the M�G�� model is a bit di�erent in

the ON�OFF model� we start with introducing some unifying additional notation� First of all� let

fi�T � �

��

�i�T �� for the M�G�� model

Mi�T �� for the ON�OFF model��

i � �� n be the rate function �appropriate to the input model� for the external input into the

network entering through the station i� The quantile functions corresponding to this external


input are denoted by

bi�t� �� F i�

��t� or bi�t� �

�� Fi

on��t��

once again depending on the model�

It turns out� as will be seen from the limit results below� that parts of the network have a

Fractional Brownian motion�like behavior while some other parts of the network have a L�evy

stable�like behavior� More explicitly� it is precisely those parts of the network that are reached

from the nodes the external input to which is under the Fast Growth Condition will have a

Fractional Brownian motion�like behavior�

Therefore� everything depends both on the regime of the external inputs and on the connections

within the network� We start with classifying the nodes depending on the growth condition of

the external input to that node� Denote N � f�� Ng� and let

Ns �

�i � N � lim

T��

bi�fi�T �T �

T� �

�

be the set of nodes the external input to which is under the Slow Growth Condition �SGC��

Correspondingly� let Nf � N n Ns be the set of nodes the external input to which is under the

Fast Growth Condition �FGC�� Recall that we always assume that the external input to each

node is either under the Fast or the Slow Growth Condition� We know from Theorems �� and

�� that� in order to obtain a non�degenerate �nite limit� one has to use di�erent normalizations

under FGC and SGC� It is� therefore� not surprising that in order to use a uniform notation for

the proper normalization corresponding to the external input at each node� we have to de�ne

di�T � �

�bi��i�T �T � if i � Ns��i�T �T

� �F i��T �� if i � Nf

��

when the input is the M�G�� model� and

di�T � �

�bi�Mi�T �T � if i � Ns�Mi�T �T

� �Fion�T �

� �� if i � Nf

��

when the input is the superposition of ON�OFF models� An important observation is that

limT��

di�T �

T�

�� if i � Ns

� if i � Nf��

Next� we need to introduce the notation describing the connections between the nodes in the

network� First of all� for i � �� N let

Ri �nj � N � there exist n � �� s�t� p

n�ji � �

o


�with the usual convention p ��ij � � if i � j and � otherwise� be the set of the nodes the external

input to which reaches station i� Furthermore� let

S � fi � �� N � Ri Nf � �g

be the part of the network that is only reachable from the nodes with slow growth external inputs�

and its complement�

B � fi � �� N � Ri Nf �� g

be the part of the network that is reachable from some nodes with fast growth external inputs�

For i � S let

Rsi � Ri Ns

be the set of the slow growth external inputs that reach the station i� and for i � B let

Rfi � Ri Nf

be the set of the fast growth external inputs that reach the station i�

Among all the nodes the external input to which reaches a given station� only the ones with

the largest order of magnitude of deviations from the mean will contribute to the limiting random

�eld� Our notation for the slow and fast inputs of this kind are

Rsi��

�j � Rs

i � limT��

dj�T �

dk�T �� for all k � Rs

i

�for i � S and

Rfi��

�j � Rf

i � limT��

dj�T �

dk�T �� for all k � Rf

i

�for i � B accordingly� Once again� we assume that all the relevant normalizing constants are

comparable� i�e� the limits in the de�nition of the sets above exist� These external inputs are

the ones that reach node i with largest individual deviations and they contribute together to the

deviations from the mean that is computed by

gi�T � �

� Pj�Rs

i��dj�T � if i � SP

j�Rfi��

dj�T � if i � B��

It follows from �� that the nodes with the slow condition external input do not contribute to

the overall order of magnitude of gi for i � B� At any rate� we assume that for every i � �� N

the overall order of magnitude increases to in�nity�

gi�T �� as T ��

That is� every node in the network is reached by at least one external input whose intensity grows

to in�nity�


For every external input reaching the node i� its contribution to the limiting random �eld will�

naturally� be weighted by the proportion of its contribution to the overall order of magnitude

gi�T � above� We denote these weights by

ri�j� � limT��

dj�T �

gi�T �for j � Rs

i�� if i � S and for j � Rfi�� if i � B��

Once again� all the limits are assumed to exist�

Example To illustrate the notions introduced so far as well as the results of the following two

theorems describing the limiting behavior of the �ow� we will use the following simple network�

consisting of � stations� with the arrows denoting external inputs and possible routes of the �ow

through the network�

1

32

We assume for simplicity that the distributions Fi of the session lengths in the M�G�� case�

or the distributions Fion in the ON�OFF case are pure Pareto distribution with mean �i and tail

exponent �i� That is�

�F i��x��corr� �Fi

on�x��

��i � �

�i

��i��ii x

��i

for x � �i��i

�i� i � �� This corresponds to

bi�t� ��i � �

�i�it

��i for t � �� i � ��

Let the routing matrix P and the corresponding matrix H � �I � P �� be given by

P �

��

A � H �

��

A �

With a view towards M�G�� input model let us take the intensity rates to be

�i�T � � T i � �i � �� i � ��


The nature of the limiting random �eld depends on the relationships between �i� i � �� and

�i� i � ��

For the sake of an example� let us take

� � ��

This gives us for the sets of fast and slow input nodes

Ns � f�g � Nf � f�� g �

In particular� for all T large enough�

di�T � �

��i � �

�i

��i��

��i��i T i��i�� i � �� and d��T � �

��

�� T

��

Furthermore� we see that

Ri � f�� g for i � �� R� � f�g �

We see immediately that the �stable� and �Brownian� parts of the network are

S � f�g � B � f�� g �

and so

Rs� � f�g and Rf

i � f�� g for i � ��

The structure of the sets Rsi�� and R

fi�� depends on further relations between the parameters

�in addition to �� Let us assume that

��

Then we have

Rfi�� f�� g for i � �� and Rs

�� f�g �

We conclude that

gi�T � �

��

��

��

��

��

��

��

��

�T �� i � ��

and

g��T � ��

�� T

��

Now the ratios in �� are given by Then we have

ri�j� �

��j��j

��j��j��j�

��

��

��

��

for i � �� and j � ��

and

r��


We will come back to this example a bit later� once we state the next theorem that describes

the limiting behavior of the �ow in the network fed by M�G�� external inputs� It follows from

the fact established above in the same way as Theorem �� for a single queue� In both this

theorem and the next one we use the convention Rsi�� if i � B and Rf

i�� if i � S�

Theorem �� Suppose we have a series of �uid queuing networks� indexed by T � The external

input to each station Qi is an M�G�� process with intensity �i�T �� Let Ii�T �� denote the

cumulative input process to station i in the model corresponding to scale T � Under the above

conditions �Ii�T �T ��

PNj�� hji�jT�j�T ��

gi�T �� i � �� N

��

fidi�

�B� Xj�Rs

i��

hjiri�j�X�j �� X

j�Rfi��

hjiri�j�jBHj �� i � �� N

CA

where �j ��

��j

h�j

��j� �

�j

i� Hj �

��j� � and the stable L�evy processes and Fractional Brownian

motions in the right hand side of �� are independent�

Furthermore�

�Ii�T �T ��

PNj�� hji�jT�j�T ��

gfi�T �� i � B

�d�

�B� Xj�Rf

i��

hjiri�j�jBHj�� i � B

CA ��

Example �continued� Let us return to the example of a ��station network� and take� for

concreteness�

��

which satis�es the assumptions on the parameters we have imposed�

In that case� Theorem �� says that the deviation of the input to the �rst station� I��T �T ��

from its mean ��T ��T �� T �� when normalized by ��T �� looks like� ��B��

��B�� the deviation of the input to the second station under the same scaling� I��T �T �� from

its mean ��T �� T � � ��T �� looks like� ��B�� B�� and the deviation of the

input to the third station� I��T �T �� from its mean ��T �� looks like� X�� when normalized

by �� T �� all �looks like� are in the sense of convergence of �nite�dimensional distributions��


Note that the deviations in the �rst and the second stations �look like� �di�erently� weighted

combinations of independent Fractional Brownian motions with parameters �� and �� respec�

tively� and in the third station the deviation from the mean �looks like� ��stable motion with

� � ��

In this case we have B � f�� g and for the �rst two stations we have a full weak convergence�

As mentioned in the introduction� in applications the di�erent shape of the marginal distribu�

tions can give insights on the couplings of paths andor nodes of a network� In this example from

the limit processes one can rightly deduce that the �ow of nodes � and does not reach node ��

Finally� the next theorem establishes a similar result for a network fed ON�OFF external inputs�

Like the previous theorem� it follows from the previously established results in the same way as

Theorem �� for a single queue�

Theorem �� Suppose we have a series of �uid queuing networks� indexed by T � The external

input to each station Qi is a superposition of Mi�T � i�i�d� stationary ON�OFF processes� Let

Ii�T �� denote the cumulative input process to station i in the model corresponding to scale T �

Under the above conditions�Ii�T �T ��

PNj�� hji�

��j �onj TMj�T ��

gsi �T � � gfi�T �� i � �� N

��

fidi�

�B� Xj�Rs

i��

hjiri�j�cjX�j ��j �� X

j�Rfi��

hjiri�j��jBHj�� i � �� N

CA �

Here j � C� �

�j�j � C� is as in Theorem �� cj �

�o�j

��jj

and Hj ��j� � Furthermore� the stable

L�evy processes and Fractional Brownian motions in the right hand side of �� are independent�

Finally��Ii�T �T ��

PNj�� hji�

��j �onj TMj�T ��

gfi�T �� i � B

�d�

�B� Xj�Rf

i��

hjiri�j��jBHj�� i � B

CA ��

We will leave to the reader to see how the statement of this theorem will look on the example

of the ��station network considered above� and how� by changing some of the parameters� one

can obtain di�erent nature of the limit at di�erent stations�

Remark Notice that we have obtained in Theorems �� and �� random �eld descriptions of

the limiting behavior of the network� The limiting random �elds are indexed both by time and by

station� The conclusions are even stronger for the part of the network subject to only Gaussian


limiting �uctuations �denoted by B in both theorems�� Here we have full weak convergence to

the limiting random �eld in the space of continuous functions with the values in the space whose

dimension is the cardinality of B�

This should allow one to try to predict certain patterns of behavior of such a network� In

particular� simulation of the limiting network is possible� Additionally� usage of continuous

mapping theorem in the �Gaussian� part of the network B should allow one to get other qualitative

results� in a manner similar to what has been done� say� in Konstantopoulos and Lin �� and

Whitt �� in the heavy tra�c scenario� We intend to pursue such studies in the future�

Acknowledgments

We would like to thanks L� Leskel!a of University of Helsinki for the interesting discussions and

his clever observations about our work�

References

P� Billingsley �� Convergence of Probability Measures� Wiley� New York�

F� Brandt and B� Lisek �� Stationary Stochastic Models� Wiley� New York�

F� Campos� J� Marron� G� Samorodnitsky and F� Smith �� Variable heavy tailed

durations in internet tra�c� Part I� Understanding heavy tails� In Proceedings of the Tenth

IEEE�ACM International Symposium on Modeling� Analysis and Simulation of Computer and

Telecommunication Systems �IEEE�ACM MASCOTS � ��

D� Daley and D� Vere�Jones �� An Introduction to the Theory of Point Processes�

Springer Series in Statistics� Springer�Verlag� New York�

A� Downey �� The structural cause of �le size distributions� Wellesley College Tech� Report

CSD�TR�� Available at http��rocky�wellesley�edu�downey��lesize�

W� Feller �� An Introduction to Probability Theory and its Applications� volume �� Wiley�

New York� �rd edition�

W� Feller �� An Introduction to Probability Theory and its Applications� volume � Wiley�

New York� nd edition�

R� Gaigalas and I� Kaj �� Convergence of scaled renewal processes and a packet arrival

model� Bernoulli ��"��

W� Gong� Y� Liu� V� Misra and D� Towsley �� On the tails of web �le size distributions�

In Proceedings of ��th Allerton Conference on Communication� Control� and Computing�


J� Harrison �� The di�usion approximation for tandem queues in heavy tra�c� Advances

in Aplied Probability ��" ��

D� Heath� S� Resnick and G� Samorodnitsky �� Heavy tails and long range dependence

in ono� processes and associated �uid models� Mathematics of Operations Research ��"

��

O� Kella �� Makron�modulated feedforward �uid networks� Queueing Systems ��"��

T� Konstantopoulos and S��J� Lin �� Fractional Brownian approximations of queueing

networks� In Stochastic Networks� P� Glasserman� K� Sigman and D� Yao� editors� volume ��

of Lecture Notes in Statistics� Springer� New York� pp� ��"��

A� Mandelbaum� W� Massey and M� Reiman �� Strong approximation for Markovian

service networks� Queueing Systems �� "��

T� Mikosch� S� Resnick� H� Rootzen and A� Stegeman �� Is network tra�c approx�

imated by stable L�evy motion or Fractional Brownian motion# Annals of Applied Probability

��"��

T� Mikosch and A� Stegeman �� The interplay between heavy tails and rates in self�

similar network tra�c� Technical report� Department of Mathematics� University of Groningen�

M� Reiman �� Queuing Networks in Heavy Tra�c� Ph�D� thesis� Department of Operations

Research� Stanford University�

M� Reiman �� The heavy tra�c di�usion approximation for sojourn times in Jackson

networks� In Applied Probability�Computer Science� the Interface� II � R� Disney and T� Ott�

editors� Birkhauser� Boston� pp� �� "��

G� Samorodnitsky �� Long Range Dependence� Heavy Tails and Rare Events� MaPhySto�

Centre for Mathematical Physics and Stochastics� Aarhus� Lecture Notes�

G� Samorodnitsky andM� Taqqu �� Stable Non�Gaussian Random Processes� Chapman

and Hall� New York�

F� D� Smith� F� Hernandez� K� Jeffay and D� Ott �� What TCPIP Protocol Headers

Can Tell Us About the Web� In Proceedings of ACM SIGMETRICS � ��Performance � � �

Cambridge MA� pp� ��"��

W� Whitt �� Stochastic�Process Limits� An Introduction to Stochastic�Process Limits and

Their Applications to Queues� Springer� New York�


Dipartimento di Ingegneria dell� Informazione e Matematica Applicata� University of Salerno�Via Ponte Don Melillo �� Fisciano �SA� Italy

E�mail address� bdauria�unisa�it� dauria�diima�unisa�it

School of Operations Research and Industrial Engineering� Cornell University� Ithaca� NY��

E�mail address� gennady�orie�cornell�edu

Documents

LIMIT BEHA VIOR OF FLUID QUEUES AND NETW ORKShalweb.uc3m.es/esp/personal/personas/bdauria/files/dauriasamorodnitsky05.pdf · LIMIT BEHA VIOR OF FLUID QUEUES AND NETW ORKS BERNARDO