Self-Similar through
High-Variability:
Statistical Analysis of Ethernet
LAN Traffic at the Source Level
Walter Willinger, Murad S. Taqqu,
Robert Sherman, Daniel V. Wilson
Bellcore, Boston University
SIGCOMM’95
Outline
Introduction
Self-similarity through high-variability
Ethernet LAN traffic measurements at the source
level
Implications of the Noah Effect in practice
Conclusion
Introduction
Actual traffic exhibits correlations over a
wide range of time scales (i.e. has longrange dependence).
Traditional traffic models focus on a very
limited range of time scales and are thus
short-range dependent in nature.
Introduction
Two problems that cause the resistance
toward self-similar traffic modeling
What is a physical “explanation” for the
observed self-similar nature of measured traffic
from today’s packet networks?
 What is the impact of self-similarity on network
and protocol design and performance analysis?

Introduction
The superposition of many ON/OFF sources
whose ON-periods and OFF-periods exhibit
the Noah Effect produces aggregate network
traffic that features the Joseph Effect.
Noah Effect: high variability or infinite variance
 Joseph Effect: self-similar or long-range
dependent

Self-Similarity through
High-Variability
Idealized ON/OFF model
An ON-period can be followed by an ONperiod and an OFF-period can succeed another
OFF-period.
 The distributions of the ON and OFF times may
vary.

Idealized ON/OFF Model
Reward sequence
{W(l ), l = 0,1,2,…}
{W(l )} is a 0/1-valued discrete time stochastic
process.
 W(l ) = 1 or 0 depends on whether or not there
is a packet at time l.
 {W(l )} consists of a sequence of 1’s (“ONperiods”) and 0’s (“OFF-periods”)

Idealized ON/OFF Model
The lengths of the ON- and OFF-periods are
i.i.d. positive random variables, denoted Uk, k =
1,2,…
 Let Sk = S0 + U2 + … + Uk , k  0 be the
corresponding renewal times.

1
P( S0  u )  ( E (U )) P(U  u  1), u  0,1,2,...
Idealized ON/OFF Model
Suppose there are M i.i.d. sources
The mth source has its own reward sequence
{W(l ), l 0}
 Superposition reward (“packet load”)

*
M ,b
W
( j) 
b ( j 1) M
 W
( m)
l bj 1 m 1
b: non-overlapping time blocks
j: the aggregation block number
(l ), j  0,1,2,...
Idealized ON/OFF Model
Suppose that U has a hyperbolic tail distribution,
P(U  u) ~ cu

as u  , 1    2, (1)
*
{
W
as M and b ,
M ,b } adequately
normalized is fractional Gaussian noise
{GH , (t ), t  0} , which is self-similar with Hurst
parameter ½  H <1
Idealized ON/OFF Model

Property (1) is the infinite variance syndrome or
the Noah Effect.
  2 implies E(U2) = 
  > 1 ensures that E(U) < , and that S0 is not

infinite
Idealized ON/OFF Model
Theorem 1. For large enough source Number M and
Block aggregation size b, the cumulative load
{WM* ,b ( j ), j  0} behaves statistically as
1
H
1/ 2
bM  b M GH , ( j )
2
1
3 
2


where H 
and
. More
4
E
(
U
)
2
(


1
)(
2


)(
3


)
precisely, 2
L blim
L
lim b

M 
H
M
1/ 2
bM 
 *
WM ,b ( j ) 
  GH , ( j )
2 

where Llim means convergence in the sense of the finitedimensional distributions (convergence in law)
Ethernet LAN Traffic
Measurements at the Source Level
Location

Bellcore Morristown Research and Engineering Center
The first set



The busy hour of the August 1989 Ethernet LAN
measurements
About 105 sources, 748 active source-destination pairs
95% of the traffic was internal
The second set



9 day-long measurement period in December 1994
About 3,500 sources, 10,000 active pairs
Measurements are made up entirely of remote traffic
Textured Plots of Packet Arrival Times
Textured Plots of Packet Arrival Times
Checking for the Noah Effect
Complementary distribution plots
log( P(U  u)) ~ log( c)   log( u), as u  
Hill’s estimate

Let U1, U2,…, Un denote the observed ON-(or
OFF-)periods and write U(1)  U(2) …U(n) for
the corresponding order statistics
1
ˆ n  
k
1

(log U ( n1)  log U ( nk ) )  , (3)

i 0

i  k 1
A Robustness Property of the
Noah Effect
As far as the Noah Effect is concerned, it does not
matter how the OFF-periods have been defined.

u
P(U  u | U  t ) ~   , 1    2 (4)
t
The similar investigation of sensitivity of the ONperiod distributions to the choice of threshold value
reveals the same appealing robustness feature of the
Noah Effect.
Self-Similarity and the Noah
Effect: 1989 Traffic Traces
181(out of 748) source-destination pairs generated
more than 93% of all the packets are considered.
The data at the source-destination level are
consistent with


ON/OFF modeling assumption
Noah Effect for the distribution of ON/OFF-periods
-values for the ON- and OFF-periods may be
different.
Self-Similarity and the Noah
Effect: 1994 Traffic Traces
Non-Mbone traffic
300 (out of 10,000) pairs responsible for 83%
of the traffic are considered.
 Self-similarity property of the aggregate packet
stream is mainly due to the relative strong
presence of the Noah Effect in the OFF-periods.

Self-Similarity and the Noah
Effect: 1994 Traffic Traces
Mbone traffic
Only an analysis of the aggregate packet stream
is performed.
 The strong intensity of the Joseph Effect
become obvious only after aggregation levels
beyond 100ms.
 There is no Noah Effect for ON-periods.


Reason: The use of unsophisticated compression
algorithms resulted in packets bursts separated by
comparatively large idle periods.
Traffic Modeling and Generation
Although network traffic is intrinsically
complex, parsimonious modeling is still
possible.

Estimating a single parameter  (intensity of
the Noah Effect) is enough.
Performance and Protocol
Analysis
The queue length distribution
Traditional (Markovian) traffic: decreases
exponentially fast
 Self-similar traffic: decreases much more
slowly

Protocol design should be expected to take
into account knowledge about network
traffic such as the presence or absence of
the Noah Effect.
Conclusion
The presence of the Noah Effect in
measured Ethernet LAN traffic is confirmed.
The superposition of many ON/OFF models
with Noah Effect results in aggregate packet
streams that are consistent with measured
network traffic, and exhibits the self-similar
or fractal properties.