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7
FLUID QUEUES, ON=OFF PROCESSES,
AND TELETRAFFIC MODELING
WITH HIGHLY VARIABLE AND
CORRELATED INPUTS
S
IDNEY
R
ESNICK AND
G
ENNADY
S
AMORODNITSKY
Cornell University, School of Operations Research and Industrial Engineering,
Ithaca, NY 14853
7.1 INTRODUCTION
Large teletraf®c data sets exhibiting nonstandard features incompatible with classical
assumptions of short-range dependence and exponentially decreasing tails can now
be explored, for instance, at the ITA Web site www.acm.org=sigcomm=ITA=.
These data sets exhibit the phenomena of heavy-tailed marginal distributions and
long-range dependence. Tails can be so heavy that only in®nite variance models are
possible (e.g., see Willinger et al. [49]), and sometimes, as in ®le size data, even ®rst
moments are in®nite [1]. See also Beran et al. [3], Crovella and Bestavros [12±14],
Leland et al. [33], Resnick [38], Taqqu et al. [48], and Willinger et al. [49]. Other
areas where heavy tails and long-range dependence are crucial properties are ®nance,
insurance, and hydrology [4±7, 16, 17, 24±26, 35, 37].
New features in the teletraf®c data discussed in recent studies suggest several
issues for study and discussion.
 Statistical. How can statistical models be ®t to such data? Finite variance black
box time series modeling has traditionally been dominated by ARMA or Box±
Jenkins models. These models can be adapted to heavy-tailed data and work

very well on simulated data. However, for real nonsimulated data exhibiting
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.
171
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
dependencies, such ARMA models provide unacceptable ®ts and do not
capture the correct dependence structure. For discussion see Davis and Resnick
[15], Resnick [38, 39], Resnick et al. [42], and Resnick and van den Berg [43].
 Probabilistic. What probability models explain observed features in the data
such as long-range dependence and heavy tails.
 Consequences. Dothe new features revealed by current teletraf®c data studies
mean we have to give up Poisson derived models and exponentially bounded
tails and the highly linear models of time series? Various bits of evidence
emphasize the de®ciencies of classical modeling. There are simulation studies
[34] and the experimental queueing analysis of Erramilli, Narayan, and
Willinger [18]. An analytic example [40] shows that for a simple G=M =1
queue, a stationary input with long-range dependence can induce heavy tails
for the waiting time distribution and for the distribution of the number in the
system.
Connections between long-range dependence and heavy tails need to be more
systematically explored but it is clear that in certain circumstances, long-range
dependent (LRD) inputs can cause heavy-tailed outputs and (as we discuss here)
heavy tails can cause long-range dependence. We discuss three models where heavy
tails induce long-range dependence:
1. A single channel on=off source feeding a single server working at constant rate
r > 0. Transmission or on periods have heavy-tailed distributions.
2. A multisource system where a single server working at constant rate r > 0isfed
by J > 1 on=off sources. Transmission periods have heavy-tailed distributions.

3. An in®nite source model feeding a single server working at constant rate
r > 0. At Poisson time points, nodes or sources commence transmitting.
Transmission times have heavy-tailed distributions.
In each of the three cases, our basic descriptor of system performance is the time
for buffer content to reach a critical level. Such a measure of performance is path
based and makes sense without regard to stability of the model, existence of
moments of input variables, or properties of steady-state quantities.
7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL
7.2.1 BasicSetup
We consider ®rst communication between a single source and a single destination
server. The source transmits for random on periods alternating with random off
periods when the source is silent. During the on periods, transmission is at unit rate.
Let fX
on
; X
n
; n ! 1g be i.i.d. nonnegative random variables representing on
periods. The common distribution is F
on
. Similarly, fY
off
; Y
n
; n ! 1g are i.i.d.
172
FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING
nonnegative random variables independent of fX
on
; X
n

; n ! 1g representing off
periods and these have common distribution F
off
. The means are
m
on


I
0

F
on
s ds; m
off


I
0

F
off
s ds;
which are assumed ®nite and the sum of the means is m : m
on
 m
off
. Using these
random variables we generate an alternating renewal sequence characterized as
follows.

1. The interarrival distribution is F
on
à F
off
and the mean interarrival time is
m  m
on
 m
off
.
2. The renewal times are
0;
P
n
i1
X
i
 Y
i
; n ! 1

:
Because of the ®niteness of the means, the renewal process has a stationary version:
D; D 
P
n
i1
X
i
 Y

i
; n ! 1

:
where D is a delay random variable satisfying
PD > x

I
x
PX
on
 Y
off
> s
m
ds


I
x
1 À F
on
à F
off
s
m
ds:
However, making the process stationary in this manner has the disadvantage that the
initial delay period D does not decompose into an on and an off period the way
subsequent inter-renewal periods do and the following procedure is preferable for

generating the stationary alternating renewal process. De®ne independent random
variables B; X
0
on
; Y
0
off
, which are assumed independent of fX
on
; X
n
; n ! 1g and
fY
off
; Y
n
; n ! 1g,by
PB  1
m
on
m
 1 À PB  0;
PX
0
on
> x

I
x
1 À F

on
s
m
on
ds : 1 À F
0
on
x;
PY
0
off
> x

I
x
1 À F
off
s
m
off
ds : 1 À F
0
off
x:
7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL
173
The delay random variable D
0
is de®ned by
D

0
 BX
0
on
 Y
off
1 À BY
0
off
:
This delayed renewal sequence
fS
n
; n ! 0g : D
0
; D
0

P
n
i1
X
i
 Y
i
; n ! 1

is a stationary renewal process.
7.2.2 High Variability Induces Long-Range Dependence
Consider the indicator process fZ

t
g, which is 1 iff t is in an on period. Thus, for
t ! D
0
,
Z
t

1; if S
n
t < S
n
 X
n1
; some n
0; if S
n
 X
n1
t < S
n1
; some n

and if 0 t < D
0
we de®ne
Z
t

1; if B  1 and 0 t < X

0
on
;
0; otherwise:
(
A standard renewal argument gives the following result [22].
Proposition 7.2.1. fZ
t
; t ! 0g is strictly stationary and
PZ
t
 1
m
on
m
:
Conditional on Z
t
 1, the subsequent sequence of on=off periods is the same as
seen from time 0 in the stationary process with B  1.
It is easiest to express long-range dependence in terms of slow decay of
covariance functions so we consider the second-order properties of the stationary
process fZ
t
g (See Heath et al. [22].) The basis for the next result is a renewal theory
argument.
Theorem 7.2.2. The covariance function
gsCovZ
t
; Z

ts

174
FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING
of the stationary process fZt; t ! 0g is
gs
m
on
m
m
off
m
À

s
0

F
off
s À uF
0
on
à Udu


m
on
m
m
off

m
À F
0
on
à U Ã1 À F
off
s


m
on
m
off
m
2
À
1
m

s
0
zs À oU dw;
where
U 
P
I
n0
F
on
à F

off

nÃ
and
zt

t
0

F
off
x

F
on
t À x dx
 m
on
F
0
on
Ã1 À F
off
t
 m
off
F
0
off
Ã1 À F

on
t:
How do we analyze the asymptotic behavior of gÁ as a function of s? Note gs is
of the form
gsconst lim
v3I
z à U vÀz à Us
hi
so we need rates of convergence in the key renewal theorem. This can be based on a
theorem of Frenk [19] and is given in Heath et al. [22].
Theorem 7.2.3. Assume that there is an n ! 1 such that F
on
à F
off

nÃ
is non-
singular. Suppose

F
on
tt
Àa
Lt; t 3I;
where 1 < a < 2 and L is slowly varying at in®nity and assume

F
off
to


F
on
t; t 3I:
Then
gt$
m
2
off
a À 1m
3
t
ÀaÀ1
Lt; t 3I:
7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL
175
So gt decreases like a constant times t

F
on
t. Such a slow decay of gt at an
algebraic rate is characteristic of long-range dependence. One way to think about this
result is that, with heavy-tailed on periods, there is a signi®cant probability that a
very long on period can cover both the time points s and t  s, thereby inducing
strong correlation between these two time points.
Taqqu et al. [48] use the long-range dependence of the on=off process and
superimpose many such processes. This superposition is approximately a fractional
Brownian motion, giving one explanation of the observed self-similarity of Ethernet
traf®c. See alsoLeland et al. [33]. Other limiting procedures leading toLe
Â
vy motion

with heavy tails are possible and are also brie¯y discussed in Leland et al. [33]. See
also Konstantopoulos and Lin [32].
7.2.3 Single Channel Fluid Queues with Constant Service Rates
Suppose work enters a communication system according to the on=off process. The
server works off the load at constant rate r assuming there is load to work on.
Here are the formal model ingredients for the single source model.
1. The Input Process
At :

t
0
Z
v
dv:
Since At$tm
on
=m, the long-term input rate is m
on
=m.
2. The Output Process. There is a release rate functionÐthe release rate from the
system when contents are at level x is
rx :
r; if x > 0;
0; if x  0:

3. The Stability Condition. Input does not overwhelm the release rate. This
necessitates
1 > r >
m
on

m
: 7:1
The restriction that r < 1 results from the normalization that work arrives at
rate 1 during on periods and prevents the contents process from having 0 as an
absorbing state.
4. The contents process fX tg satis®es the storage equation
dX tdAtÀrX t dt:
Note that during an on period, the net input rate is 1 À r since work is inputted at
unit rate but the server works at rate r. During an off period, the release rate is r
176
FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING
(provided there is liquid to release). This means the paths of X Á are sawtoothed
shaped.
7.2.3.1 Regeneration Times Recall that the stationary alternating renewal pro-
cess is
S
n
 D
0

P
n
i1
X
i
 Y
i
; n ! 0

:

Since the contents process is stable, we can de®ne regeneration times
fC
n
g :fS
n
: X S
n
À  0g;
which are times when a dry period ends and input commences. So the standard limit
theorems due to Smith for regenerative processes [45] guarantee limit distributions
exist in discrete and continuous time:
PX S
n
 > x31 À W x; PX t > x31 À V x:
There are also connections with standard random walk theory and Lindley's equation
holds. If we compare X S
n
 with X S
n1
 we get
X S
n1
X S
n
1 À rX
n1
À rY
n1



X S
n
x
n1


;
where
x
n1
1 À rX
n1
À rY
n1
and fx
j
g i.i.d. It is important to distinguish between the random walk with steps fx
n
g
and the random walk with steps fX
n
 Y
n
g.
Assume
1 À F
on
xx
Àa
Lx; a > 1; x 3I:

From standard random walk theory [11, 36] we get
1 À W x$
r
1 À r
1 À r
aÀ1
a À 1m
on
x
ÀaÀ1
Lx
: bx
ÀaÀ1
Lx; x 3I;
7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL
177
sothat the tail of W is heavy and comparable to the integral of the tail of F
on
. The
de®nition of r is
r 
m
on
m
off
1 À r
r
< 1:
As expected from the sawtooth shape of the paths, the tail of V x is heavier because
of a bigger multiplicative factor

1 À V x$ b 
1 À r
aÀ1
ma À 1
!
x
ÀaÀ1
Lx:
See Boxma and Dumas [9] and Heath et al. [22] and the references therein and also
the discussion in Chapter 10. The tails of both W x and V x are of the form
const  x

F
on
x. So the equilibrium content of the system, in either continuous or
discrete time, can get quite large with nontrivial probability. This point will be
reinforced in the discussion of the time it takes for buffer content to reach a critical
level.
7.2.4 Extremes, Level Crossings, and Buffer Over¯ow
The distributions W x and V x are standard queueing quantities and convey some
performance information. Another performance measure, one that is less dependent
on notions of stability and existence of moments, is the time until buffer over¯ow.
We formulate the time to buffer over¯ow as the hitting time of a high level L:
tL : inf ft ! 0 : X t!Lg:
De®ne M t
W
t
s0
X smaxfX s : 0 s tg, and the reason for interest in
the maximum content up to t is that as a process it is the inverse of the tÁ process

since
tLinf fs > 0 : X s!Lginf fs > 0 : M s!Lg : M
2
L;
where M
2
Á is the right continuous inverse of the monotone function MÁ.Ifwe
understand the asymptotic behavior of M Á, then we will understand the asymptotic
behavior of tÁ. To understand the behavior of MÁ we ®rst study the extremes of
fX S
n
g and then ®ll in the behavior between the discrete points fS
n
g. We study the
maxima of the random walk generated by fx
n
g over cycles and then knit cycles
together. Recall
x
n1
1 À rX
n1
À rY
n1
:
178
FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING