4
QUEUEING BEHAVIOR UNDER
FRACTIONAL BROWNIAN TRAFFIC
I
LKKA
N
ORROS
VTT Information Technology, Espoo, Finland
4.1 INTRODUCTION
This chapter gives an overview of some properties of the storage occupancy process
in a buffer fed with ``fractional Brownian traf®c,'' a Gaussian self-similar process.
This model, called here ``fractional Brownian storage,'' is the logically simplest
long-range-dependent (LRD) storage system having strictly self-similar input varia-
tion. The impact of the self-similarity parameter H can be very clearly illustrated
with this model. Even in this case, all the known explicitly calculable formulas for
quantities like the storage occupancy distribution are only limit results, for example,
large deviation asymptotics. Scaling formulas, on the other hand, hold exactly for
this model.
The simplicity is won at the price that the input model is not meaningful at
smallest time scales, where half of the ``traf®c'' is negative. The model can be
justi®ed by rigorous limit theorems, but it should be emphasized that this involves
not only a central limit theorem (CLT) argument for Gaussianity but also a heavy
traf®c limitÐsee Chapter 5. From a less rigorous, practical viewpoint one can say
that fractional Brownian storage gives usable results when, at time scales relevant for
queueing phenomena, the traf®c consists of independent streams such that a large
number of them are simultaneously active, and second-order self-similarity (see
Chapter 1) holds.
Chapter 7 describes many features of storage processes with ®nitely aggregated
on=off input traf®c, which differ qualitatively from those of fractional Brownian
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.

2
.
Regrettably, almost all methodological cornerstones of Brownian motion are based
on independence and thus unavailable when H T
1
2
: independent increments,
Markov property, margingale property, renewal times.
Besides self-similarity, the general methods available for fractional Brownian
storage come from the literature on Gaussian processes. In particular, the beautiful
theory of their large deviations in path space turns out to be well suited for the needs
of performance analysis.
This chapter is structured as follows. The de®nitions are given in Section 4.2.
Some basic scaling formulas are derived in Section 4.3. Results based on large
deviations in path space are presented in Section 4.4. Finally, some other approaches
are outlined in Section 4.5.
4.2 INPUT, OUTPUT, AND STORAGE PROCESSES
We consider in continuous time an unlimited ¯uid storage that is fed by fractional
Brownian traf®c, de®ned below, and emptied at constant service rate c.
4.2.1 The InputProcess, ``Fractional Brownian Traf®c''
The ¯uid input in time interval s; t is denoted by As; t and it has the form
As; tmt À ssZ
t
À Z
s
; s; t P ÀI;I; s t;
102
QUEUEING BEHAVIOR UNDER FRACTIONAL BROWNIAN TRAFFIC
where m and s are nonnegative parameters, m < c, and the process Z
t

2
 ma;
where a, the index of dispersion at unit time, has sometimes been called ``peaked-
ness.'' The point in using ma instead of s
2
is that varying m can now be interpreted
as varying the number of traf®c sources alone, without changing their characteristics.
The parameter H characterizes dependence in the input process. For H P
1
2
; 1,
all the random variables As; t with s < t are strictly positively correlated. For
H 
1
2
, the input process is a Brownian motion, and the storage model is a classical
diffusion approximation for a queueing process. For H P0;
1
2
, inputs on disjoint
intervals are negatively correlated. It is possible that this case has no natural
applications in teletraf®c contexts, but including it comes usually for free, so we
do not exclude it.
We also write
A
t
 A0; t for t ! 0; A
t
ÀAt; 0 for t 0: 4:2
Then As; tA

A0; t=t  m with
probability 1. Since we have assumed m < c, it follows that V
0
is a.s. ®nite. Note that
V is nonnegative, although the input process has (regrettably!) negative increments
also.
The ruggedness (nondifferentiability) of the fractional Brownian path implies a
paradoxical property of V : the storage is almost always nonempty. Indeed, it can be
shown that the supremum in Eq. (4.3) is positive with probability one, and by
stationarity, the positivity must also hold for almost every time point in almost every
realization of the process. The set of times t with V
t
 0 is uncountable a.s., with
almost every point being an accumulation point, so that between any two distinct
busy periods there are a.s. in®nitely many tiny busy periods. This is, of course, an
anomaly of the continuous-time model only, it has no counterpart in the teletraf®c
reality being modeled. Note, on the other hand, that it is a natural feature of a heavy
traf®c limit process (cf. Chapter 5), and that the case H 
1
2
is no exception here.
4.2.3 The Output Process
It is natural to de®ne the output within an interval s; t as
Us; tAs; tÀV
t
 V
s
; U
t
 U0; t for t ! 0: 4:4

parameters m, s
2
, and c. Then
V
m;s
2
;c
t

tPR

d
c À m
a*
V
0;1;1
a
Ã
t

tPR
; where a* 
c À m
s

1=1ÀH
:
Proof. For any a > 0, we have by Eq. (4.1) that
V
t

À1
at À as;
where the similarity in distribution holds for the whole processes, not just for a
single t. Now, choose a  a* by requiring that
sa
ÀH
c À ma
À1
: j
In particular, Proposition 4.3.1 has the following consequences.
Corollary 4.3.2. The storage occupancy distribution obeys the scaling law
PV
m;s
2
;c
> xPV
0;1;1
>
a*
c À m
x

 PV
0;1;1
>
c À m
H=1ÀH
s
1=1ÀH
x