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5
HEAVY LOAD QUEUEING ANALYSIS
WITH LRD ON=OFF SOURCES
F. B
RICHET AND
A. S
IMONIAN
France TeÂleÂcom, CNET, 92794 Issy-Moulineaux CeÂdex 9, France
L. M
ASSOULIE
Â
Microsoft ResearchLtd., Cambridge, CB2 3NH, United Kingdom
D. V
EITCH
Software Engineering ResearchCentre, Carlton, Victoria 3053, Australia
5.1 INTRODUCTION
In this chapter, we evaluate the impact of long-range dependence on a single network
element (multiplexer or router) of a data network, this element being modeled as a
¯uid queueing system. Such a system has constant output rate C and is fed by a
number N of i.i.d. on=off traf®c sources. The case where the number of sources N is
®xed is treated, for instance, in Boxma and Dumas [4]. For N suf®ciently large, C
exceeds the peak rate of an individual source. If the ratio ``source peak rate=total
output rate C'' decreases with N (e.g., is proportional to 1=N ), then we are in the
realm of ``small'' sources. In the meantime, we consider the ``heavy load'' case when
the output rate C is slightly larger than the total mean input rate so that the queue is
almost always nonempty. This ``heavy load'' situation associated with ``small
sources'' in a ¯uid queueing system motivates the derivation of limit theorems for
both input traf®c and queue occupancy processes when the number N increases to
in®nity, as detailed below.
More precisely, represent an on=off source by mutually independent, alternating
silence periods A and activity periods B. When active, the source emits data at

Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger
ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc.
115
Self-Similar Network Traf®c and Performance Evaluation, Edited by Kihong Park and Walter Willinger
Copyright # 2000 by John Wiley & Sons, Inc.
Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X
constant rate, its peak rate, taken as unity. Given EA1=a and EB1=b, the
activity probability of a source is then n  a=a  b and we require that C > N n so
that the queue has a stationary regime. Provided that the probability density of
duration A  B satis®es a simple regularity condition, we ®rst show in this chapter
that, once properly centered and normalized, the aggregate input rate and traf®c
processes produced by the superposition of N i.i.d. on=off sources converge as
N 4I toward continuous Gaussian processes. This is the content of our
``functional central limit'' theorems. In contrast with the classical heavy load
approximation, note that the limit input rate process obtained this way is generally
non-Markovian for arbitrary distributions of on and off durations A and B.
Denote further by V
N
0
the corresponding queue content in stationary conditions.
Our essential aim is to obtain estimates for the limiting distribution of V
N
0
as
N 4I, the capacity C scaling with N as
C  Nn  g

N
p
5:1

for some positive constant g. Equation (5.1) is clearly identi®ed as a heavy load
condition since the queue load N n=C tends to 1 for increasing N (with speed
1=

N
p
. Now, by considering so-called heavy-tailed distributions for on and=or
off periods, the input process becomes long-range dependent (LRD). Based on the
above convergence results for input processes, our analysis then subsequently
shows that the tail of the limiting distribution of the scaled queue content
V
N
0
=

N
p
is not exponential but Weibullian. Speci®cally, de®ning
lim
N4I
PV
N
0
> x

N
p
hx
for any ®xed x ! 0, we have
lim

x4I
1
x
21ÀH
Á log hxÀ
k
2
2
where both constants k and H P
1
2
; 1 depend on the distributions of on and off
durations. Weibullian tails imply ®nite moments of all orders but, nonetheless,
necessitate buffer sizes that grow much faster with load than in the classical
Markovian case since the exponent 21 À H is strictly less than 1 for H >
1
2
.
Such a result reminds one of the Weibullian nature of the distribution of a ¯uid
queue fed by the so-called fractional Brownian motion (FBM), as discussed in
Norros [13] (see Chapter 4 in this volume for a full account). We also show in this
chapter how the limit input processes obtained as N 4Ican easily be related to
the FBM by means of proper time and space scale changes.
Other heavy load queueing analysis with heavy-tailed distributions has been
considered in related contexts such as the M=G=1 discrete queue (see Chapter 6 in
this volume) and the ¯uid queue with M=G=I input rate process (refer to Chapter 9
in this volume and references contained therein). In comparison to the analysis
116
HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON=OFF SOURCES
performed in Chapters 6, 9 and 10, we stress the fact that the limit considered here,

with many small sources and heavy load queueing regime, enables the derivation of
continuous limit processes together with the Weibullian queueing behavior, as
opposed to the heavy-tailed queueing behavior obtained when ®xed ``source peak
rate=total output rate C'' ratio and a ®nite number of sources are considered.
The rest of the chapter is organized as follows. Section 5.2 contains some
preliminary notation and basic properties related to the on=off source model. In
Section 5.3, we state and prove functional central limit theorems (CLTs) for the
properly scaled input processes as the number of sources tends to in®nity. In Section
5.4, these CLTs are used to derive general limiting upper and lower bounds for the
stationary distribution of the queue content V
N
0
under the heavy load condition
(5.1). Section 5.5 addresses the impact of long-range dependence on these bounds,
demonstrating their Weibullian behavior in this case and relating the present model
to FBM as considered in Chapter 4. Much of the material presented in this chapter
was ®rst published in Brichet et al. [5]. Alternative arguments for convergence
results can also be found in Kurtz [10]
5.2 PRELIMINARY PROPERTIES OF THE ON/OFF SOURCE MODEL
Let fl
t
g
t!0
be the stationary process representing the input rate of a single on=off
source at time t and o
t


t
0

l
s
ds the input it generates in the interval 0; t.We
denote fL
N
t
g and fW
N
t
g the sum of N i.i.d. copies of the processes fl
t
g and fo
t
g,
respectively. For ®xed t; l
t
is a Bernoulli variable with mean n and variance
s
2
 n1 À n; 5:2
while o
t
is a random variable taking values in 0; t, with mean nt and variance
denoted by
Dt
2
 Varo
t
5:3
It is also useful to note from the independence and homogeneity of all input sources

that, given L
N
0
 r; r Pf0; 1; ...; Ng, the expectation of W
N
t
can be expressed as
EW
N
t
jL
N
0
 rN À rm
a
trm
b
t; 5:4
where m
a
t and m
b
t are the ®rst conditional moments
m
a
tEo
t
jl
0
 0; m

b
tEo
t
jl
0
 15:5
associated with o
t
. Finally, we denote by r the covariance function of process fl
t
g
de®ned for t ! 0by
rtEl
0
À nl
t
À nEl
0
l
t
Àn
2
: 5:6
5.2 PRELIMINARY PROPERTIES OF THE ON/OFF SOURCE MODEL
117
Using basic renewal properties of the on=off rate processes, it can easily be shown
that the Laplace transform of covariance function r is given by
r*p
n1 À n
p

1 À
a  b
p
:
1 À a*p1 À b*p
1 À a*pb*p

5:7
for p > 0, where a* (resp. b*) denotes the Laplace transform of the off (resp. on)
duration A (resp. B).
5.3 FUNCTIONAL CENTRAL LIMIT THEOREMS
The aim of this section is to ®nd suf®cient conditions for the rate and input processes
associated with the on=off sources to satisfy functional CLTs. To this end, we ®rst
establish (Theorem 5.3.1) a CLT for renewal processes, using standard results on
weak convergence [3]. The desired results are then obtained in Section 5.3.2.
5.3.1 CLT forRenewal Processes
For any interval I of R, let dI denote the space of right-continuous functions on I
with left limits. In the sequel, real-valued processes on some interval I are
considered as dI-valued random elements. For bounded I ; dI is endowed
with the J
1
-Skorokhod topology. The space dR

, denoted by d for short, is
endowed with the topology of convergence in the J
1
topology on bounded intervals.
Theorem 5.3.1. Let L be a stationary renewal process, withL0; t representing
the number of renewal points within time interval 0; t; t ! 0, and with®nite and
nonzero mean intensity l. Denoting by P the distribution function of the inter-

renewal times, we assume that:
the density P
H
exists and is bounded in some neighborhood of 0: 5:8
Then, given a sequence fL
j
g
j!0
of i.i.d. copies of L, the sequence of processes
fL
N
g
N>0
de®ned by
L
N
t
1

N
p
P
N
j1
L
j
0; tÀlt; t P R

; 5:9
is tight and converges weakly to a limiting Gaussian process with a.s. continuous

paths as N 4I.
In order to prove Theorem 5.3.1, the following results are needed. The ®rst one is
taken from Billingsley [3].
118
HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON=OFF SOURCES
Theorem 5.3.2. [3, Theorem 15.6, p. 128]. Let I 0; T be some bounded
interval of R. For a sequence fG
N
g
N>0
of dI-valued processes to converge
weakly to the dI-valued process G, it is suf®cient that:
 for all t
1
; ...; t
n
P I, the vector G
N
t
1
; ...; G
N
t
n
 converges weakly to
Gt
1
; ...; Gt
n
,asN4I;

 there exists p > 0,q> 1 and a continuous nondecreasing function F suchthat
EjG
N
tÀG
N
t
1
j
p
ÁjG
N
t
2
ÀG
N
tj
p
 Ft
2
ÀFt
1

q
5:10
for all N and 0 t
1
t t
2
T.
The proof of Lemmas 5.3.3 and 5.3.4 below are deferred to Sections 5.7.1 and 5.7.2.

Lemma 5.3.3. Let L be a stationary renewal process satisfying (5.8). Let a
0
> 0
denote an upper bound of the density P
H
in some neighborhood of 0. Then for small
enough E > 0, process L can be constructed on the same probability space as a
homogeneous Poisson process M with intensity m  l=1 À lEa
0
=1 À a
0
E so
that with probability one, the paths of M dominate those of L on 0; E; that is,
Lt Mt; t P0; E:
Lemma 5.3.4. In the setting of Theorem 5.3.1, let
P
N
t
1
; t; t
2
EjL
N
tÀL
N
t
1
j
2
ÁjL

N
t
2
ÀL
N
tj
2
5:11
for 0 t
1
t t
2
, where the process L
N
is de®ned by Eq. (5.9). It holds that
P
N
t
1
; t; t
2
 2fEx
2
Z
2
Ex
2
EZ
2
ExZ

2
g; 5:12
where the random pair x; Z is distributed as Lt
1
; tÀlt À t
1
,Lt; t
2
À
lt
2
À t.
Moreover, for E > 0 as in Lemma 5.3.3, there exists a constant d > 0 suchthat
t
2
À t
1
E A P
N
t
1
; t; t
2
 dt
2
À t
1

2
: 5:13

Proof of Theorem 5.3.1. The plan is to apply Theorem 5.3.2. Note ®rst that, by
Lemma 5.3.3, the variance of L0; t is ®nite for all t > 0. Thus, for any ®nite
collection t
1
; ...; t
n
, the ®nite-dimensional vector L
N
t
1
; ...; L
N
t
n
 converges
weakly to a multidimensional Gaussian distribution, as a consequence of the
classical CLT. These ®nite-dimensional Gaussian distributions satisfy Kolmogorov's
consistency criterion, hence the existence of some Gaussian process G with the
corresponding ®nite-dimensional distributions. Lemma 5.3.3 provides bounds on
the quantities EjGtÀGsj
2
or order jt À sj, which enables us to deduce, by
5.3 FUNCTIONAL CENTRAL LIMIT THEOREMS
119
Kolmogorov's regularity criterion, that a version of G exists with almost surely
continuous paths. The version is thus a d-valued random element.
We now show that for any interval I 0; T, condition (5.10) of Theorem 5.3.2
is met. Let E be ®xed as in Lemma 5.3.3. Let t
1
t t

2
lie in I .Ift
2
À t
1
E,
Condition (5.10) is met with p  q  2 and Fx

d
p
x, in view of (5.13). It
remains to ®nd a suitable bound on P
N
t
1
; t; t
2
 when t
2
À t
1
> E. Using the basic
inequality 2jabj a
2
 b
2
in conjunction with the Cauchy±Schwarz inequality,
(5.12) entails that
P
N

t
1
; t; t
2
 3Ex
4
 EZ
4
:
This bound is in turn less than
3fELt
1
; t
4
 ELt; t
2

4
lt À t
1

4
lt
2
À t
4
g:
In order to bound the means in the last display expression, note that, by Lemma
5.3.3, both Lt
1

; t and Lt; t
2
 are less than the sum of dT=Ee random variables, each
of which is less than a Poisson random variable with mean mE. Let m
4
denote the
fourth moment of such a Poisson random variable. In view of the straightforward
inequality 
P
k
i1
a
i

n
k
nÀ1
P
k
i1
ja
i
j
n
, each of the means in the preceding bound is
less than dT =EeÂdT =Ee
4
m
4
dT=Ee

5
m
4
. One thus obtains
P
N
t
1
; t; t
2
 6fdT=Ee
5
m
4
lT
4
g:
Denoting the latter upper bound by K, t
2
À t
1
! E then entails that
P
N
t
1
; t; t
2

K

E
2
t
2
À t
1

2
:
Thus condition (5.10) of Theorem 5.3.2 is met, with p  q  2 and Fx

d k=E
2

p
x. j
5.3.2 CLT forRate and Input Processes
The propositions to follow state the functional CLT for the two sequences of
processes fL
N
t
g and fW
N
t
g introduced in Section 5.2.
Proposition 5.3.5. Let A  B denote the total duration of two successive on and off
periods of rate process fl
t
g and assume that the distribution of A  B has a bounded
density in the neighborhood of 0. The sequence of processes Y

N
de®ned by
Y
N
t

L
N
t
À N n

N
p
; t ! 0; 5:14
then converges weakly in the space d toward a zero-mean stationary Gaussian
process Y withalmost surely continuous paths.
120
HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON=OFF SOURCES
Proof. Write ®rst Eq. (5.14) as
Y
N
t

1

N
p
P
N
j1

l
j;t
À n; 5:15
where the fl
j;t
g are i.i.d. copies of the process fl
t
g. In view of the ordinary CLT, the
®nite-dimensional distributions of the process Y
N
converge weakly to limiting
Gaussian distributions. Weak convergence of the sequence fY
N
g
N>0
will then hold
provided it is tight. Since L
N
is the superposition of N i.i.d. on=off rate processes l
j
with peak rate 1, Eq. (5.15) entails that
Y
N
t
 Y
N
0

1


N
p
P
N
j1
fL

j
0; tÀL
À
j
0; tg 5:16
for t ! 0, where L

j
0; t (resp. L
À
j
0; t is the number of upward (resp. downward)
jumps of process l
j
over time interval 0; t. Each point process L

j
(resp. L
À
j
)isa
stationary renewal process with intensity l, where 1=l  1=a  1=b is the mean cycle
period of each process l

j
. We therefore deduce from Eq. (5.16) that
Y
N
t
 Y
N
0
 L
N

tÀL
N
À
t; 5:17
where
L
N
Æ
t
1

N
p
P
N
j1
fL
Æ
j

0; tÀltg:
In view of decomposition (5.17), in order to prove the tightness of the sequence
fY
N
g, it is suf®cient to show that each sequence fY
N
0
g, fL
N

g, and fL
N
À
g is tight.
The sequence fY
N
0
g converges in distribution and is therefore tight. Tightness of
both sequences L
N

and L
N
À
follows from Theorem 5.3.1. Since the weak limits of
both sequences have a.s. continuous paths, it results that the weak limit of Y
N
also
has a.s. continuous paths. Finally, stationarity of each process Y
N

ensures
stationarity of the limiting Gaussian process. j
The following result on the convergence of the normalized input process can be
deduced as a simple corollary of the previous proposition.
Proposition 5.3.6. In the framework of Proposition 5.3.5, the sequence of
processes fO
N
t
g de®ned by
O
N
t

W
N
t
À N nt

N
p
; t ! 0; 5:18
5.3 FUNCTIONAL CENTRAL LIMIT THEOREMS
121
converges weakly toward a zero-mean, continuous Gaussian process fO
t
g with
stationary increments and O
0
 0. In addition, this process is such that
lim

t3I
O
t
t
 0 a:s:
Proof. Note that
O
N
t


t
0
Y
N
u
du
with Y
N
introduced in Eq. (5.14). The mapping j, which associates to some
f P d0; T the integrated function j f : t U3

t
0
fu du, is continuous on the
subset C0; T of d0; T consisting of continuous functions (indeed, this holds
since the trace of the J
1
-Skorokhod topology on C0; T coincides with the
topology of uniform convergence; see Billingsley [3]). As Y

N
converges weakly
to a limiting process Y with a.s. continuous paths, O
N
 jY
N
 converges weakly
to jY. Process Y being Gaussian and centered, the limit process O is clearly
centered, continuous, and Gaussian with O
0
 0. The stationarity of increments of O
is readily deduced from the stationarity of process Y . In view of such stationarity, the
ergodic theorem can be used to yield the existence of an almost sure limit for O
t
=t as
t 3I. This limit is necessarily a Gaussian random variable, and we will be done
if its variance is equal to zero. This variance is equal to (see Eq. (5.32))
lim
t3I
2
t
2

t
0
t À uru du;
where ru is as de®ned in Eq. (5.6). It can be shown that this limit equals zero
provided ru tends to zero as u 3I. Recall that the process fl
t
g is regenerative,

with cycle lengths distributed as A  B. Under the present assumptions, this cycle
length distribution has a density in the neighborhood of zero, and hence is nonlattice.
This ensures (e.g., see Asmussen [1]) that l
t
converges weakly to its stationary
distribution, irrespective of its initial value, as t 3I. Writing
rtnPl
t
 1jl
0
 1Àn
2
;
we conclude that rt indeed goes to zero as t 3I. j
5.4 QUEUE CONTENT DISTRIBUTION IN HEAVY LOAD CONDITION
Using the above convergence results for both the normalized rate process and the
input process created by the superposition of a large number of on=off sources, we
can now assert general limit bounds for the distribution of the normalized queue
122
HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON=OFF SOURCES
content V
N
0
=

N
p
when N 4I and heavy load condition (5.1) is ful®lled. To
derive such limits, ®rst recall [2] that the stationary queue content V
N

0
can be
de®ned as the supremum of the transient process fW
N
t
À Ctg, namely,
V
N
0
 sup
t!0
W
N
t
À Ct: 5:19
Second, given a ¯uid queue with output rate g, it has been shown [16] that, provided
the input process fO
t
g
t!0
is such that (1) O
t


t
0
Y
u
du for all t, where the rate
process fY

t
g
t!0
is stationary, and (2) O
t
À gt tends to ÀI almost surely as t 4I,
then, for any x ! 0, the corresponding stationary distribution of the queue content
U
0
veri®es the bounds
sup
t>0
PO
t
> x  gt PU
0
> x;
PU
0
> x

g
ÀI
g À yPY
0
P dy

I
0
f

y
t; x  gt dt;
5:20
where f
y
t;: denotes the probability density of O
t
with given Y
0
 y. In order to
state the next proposition, let F denote the distribution function of a standard normal
variable and introduce the notation
mtm
b
tÀm
a
t;
Dt
2
 Dt
2
À mt
2
s
2
;
St; x
Dt
s Á Dt
g Àx  gt

mts
2
Dt
2

;
Rt; x
s Á Dt
Dt
St; xFSt; x 
1

2p
p
e
ÀSt;x
2
=2

5:21
where s, Dt, m
a
t, and m
b
t are de®ned by Eqs. (5.2), (5.3), and (5.5),
respectively.
Proposition 5.4.1. Assume that heavy load condition (5.1) holds for some positive
constant g. Then the distribution of V
N
0

=

N
p
converges weakly as N 4I.
Furthermore, de®ning
hx lim
N4I
PV
N
0
> x

N
p

for any x ! 0, the bounds
qx hx Qx
5.4 QUEUE CONTENT DISTRIBUTION IN HEAVY LOAD CONDITION
123
hold with
qxsup
t>0

F
x  gt
Dt

; 5:22
where


F  1 À F and
Qx

I
0
exp À
x  gt
2
2Dt
2
!
Rt; x
Dt

2p
p
dt 5:23
where Rt; x is given by Eq. (5.21).
Proof. First, using the scaling condition (5.1) for C together with notation
(5.18), relation (5.19) for V
N
0
reads
V
N
0

N
p

 sup
t>0
O
N
t
À gt:
Let f denote the map d P d U3 fdsup
t>0
d
t
À gt. From the weak conver-
gence of processes O
N
toward process O, one can conclude that
V
N
0
=

N
p
 fO
N
 converges weakly to U
0
 fO provided the distribution of
O puts no mass on the set of trajectories d P d for which f is discontinuous. The
fact that O has a.s. continuous paths and is such that O
t
=t 3 0 a.s., as implied by

Proposition 5.3.6, is suf®cient to conclude that this is so (the detailed argument is
left to the reader). It therefore follows that, for ®xed x ! 0, PV
N
0
> x

N
p
3
hx as N 4I, where
hxP sup
t>0
O
t
À gt > x

: 5:24
The limit distribution of V
N
0
=

N
p
is therefore identical to that of the stationary
distribution of a queue fed by input O
t
g and with output rate g > 0.
Second, process fO
t

g verifying the conditions recalled above at the beginning of
this section, the lower bound hx!qx can readily be derived, where qx
denotes the left-hand side of lower bound (5.20). Each variable O
t
being Gaussian
and centered with variance Dt
2
for ®xed t > 0, expression (5.22) follows.
Third, we further obtain the upper bound hx Qx for upper bound (5.24),
where Qx denotes the right-hand side of upper bound (5.20). To compute this
upper bound, we note that both variables Y
0
and O
t
are Gaussian so that f
y
t;Á is a
Gaussian density. Its mean and variance, by use of standard relations [15, pp. 302,
305] for Gaussian correlation and with notation (5.2) and (5.3), are given by
m
y
tCovY
0
; O
t
Â
y
VarY
0


 EY
0
O
t
Â
y
s
2
5:25
124
HEAVY LOAD QUEUEING ANALYSIS WITH LRD ON=OFF SOURCES