limit beha vior of fluid queues and netw...
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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
� B� D�AURIA AND G� SAMORODNITSKY
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
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS �
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 � ��
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS �
�� 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��
� B� D�AURIA AND G� SAMORODNITSKY
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�
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS �
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�
�� B� D�AURIA AND G� SAMORODNITSKY
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�
WTn�Tnt��WTn�Tns�
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�� �
�� �� �����������
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS ��
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 � ������ �
for all n � n��
Therefore� for each � � � and � �� there are �� � � and n� � � such that
P
�� sup
��s�t��
t�s��
WTn�Tnt��WTn�Tns�
g�Tn�� �
A �
������
for all n � n��
It remains to prove that� for every � � � and � �� there is �� � � and n� � � such that
P
�� sup
��s�t��
t�s��
WTn�Tns��WTn�Tnt�
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����
WTn�Tns��WTn�Tnt�
g�Tn�� �
��
for all n � n� and any � � s � �� Let us �x any � � ��� ��� For � � s � � we have
�
�P
�sup
s�t� s����
WTn�Tns��WTn�Tnt�
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������
�� B� D�AURIA AND G� SAMORODNITSKY
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
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS ��
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�
� B� D�AURIA AND G� SAMORODNITSKY
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 ����
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS ��
��� 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
�� B� D�AURIA AND G� SAMORODNITSKY
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
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS ��
�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�
�� B� D�AURIA AND G� SAMORODNITSKY
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 � �� � ��
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS ��
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���� � ��
�� B� D�AURIA AND G� SAMORODNITSKY
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��
LIMIT BEHAVIOR OF FLUID QUEUES AND NETWORKS ��
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
�� B� D�AURIA AND G� SAMORODNITSKY
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�
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� B� D�AURIA AND G� SAMORODNITSKY
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