Transcript SNMP - Simple Network Measurements Please!

Internet Measurement Conference 2003
27-29 Of October, 2003
Miami, Florida, USA
http://www.icir.org/vern/imc-2003/
Date for student travel grant applications: Sept 5th
AT&T Labs - Research
An Information-Theoretic
Approach to Traffic Matrix
Estimation
Yin Zhang, Matthew Roughan, Carsten Lund – AT&T Research
David Donoho – Stanford
Shannon Lab
AT&T Labs - Research
Problem
Have link traffic measurements
Want to know demands from source to destination
B
C
A

.
TM  
.
 .
AT&T Labs - Research
x A, B
.
.
.
x A ,C
.
.
.




Example App: reliability analysis
Under a link failure, routes change
want to find an traffic invariant
B
C
A
AT&T Labs - Research
Approach
Principle *
“Don’t try to estimate something
if you don’t have any information about it”
 Maximum Entropy
Entropy is a measure of uncertainty
More information = less entropy
To include measurements, maximize entropy subject to the
constraints imposed by the data
Impose the fewest assumptions on the results
 Instantiation: Maximize “relative entropy”
Minimum Mutual Information
AT&T Labs - Research
Mathematical Formalism
Only measure traffic at links
1
Traffic y1
link 1
router
link 2
link 3
3
AT&T Labs - Research
2
Mathematical Formalism
1
Traffic y1
Traffic matrix element x1
route 1
route 3
2
router
route 2
3
Problem: Estimate traffic matrix (x’s) from the link measurements (y’s)
AT&T Labs - Research
Mathematical Formalism
y1  x1  x3
1
route 1
route 3
2
router
route 2
3
Problem: Estimate traffic matrix (x’s) from the link measurements (y’s)
AT&T Labs - Research
Mathematical Formalism
y1  x1  x3
1
route 1
route 3
route 2
3
2
router
 y1   1 0 1  x1 
  
 
 y2    1 1 0  x2 
 y   0 1 1  x 
 3 
 3 
Problem: Estimate traffic matrix (x’s) from the link measurements (y’s)
AT&T Labs - Research
Mathematical Formalism
y1  x1  x3
1
route 1
route 3
2
router
Routing matrix
route 2
3
y = Ax
For non-trivial network
UNDERCONSTRAINED
AT&T Labs - Research
Regularization
 Want a solution that satisfies constraints: y = Ax
Many more unknowns than measurement: O(N2) vs O(N)
Underconstrained system
Many solutions satisfy the equations
Must somehow choose the “best” solution
 Such (ill-posed linear inverse) problems occur in
Medical imaging: e.g CAT scans
Seismology
Astronomy
 Statistical intuition => Regularization
Penalty function J(x)
solution:
arg min
x
 y  Ax
2
 2 J x 
AT&T Labs - Research

How does this relate to other methods?
 Previous methods are just particular cases of J(x)
 Tomogravity (Zhang, Roughan, Greenberg and Duffield)
J(x) is a weighted quadratic distance from a gravity model
 A very natural alternative
Start from a penalty function that satisfies the
“maximum entropy” principle
Minimum Mutual Information
AT&T Labs - Research
Minimum Mutual Information (MMI)
 Mutual Information I(S,D)
Information gained about Source from Destination
I(S,D) = -relative entropy with respect to independent S and D
I(S,D) = 0
S and D are independent
equivalent
p(D|S) = p(D)
gravity model
 Natural application of principle *
Assume independence in the absence of other information
Aggregates have similar behavior to network overall
 When we get additional information (e.g. y = Ax)
Maximize entropy
J(x) = I(S,D)
 Minimize I(S,D)
AT&T Labs - Research
(subject to constraints)
MMI in practice
 In general there aren’t enough constraints
 Constraints give a subspace of possible solutions
y = Ax
AT&T Labs - Research
MMI in practice
 Independence gives us a starting point
independent solution
y = Ax
AT&T Labs - Research
MMI in practice
 Find a solution which
Satisfies the constraint
Is closest to the independent solution
solution
Distance measure is the Kullback-Lieber divergence
AT&T Labs - Research
Is that it?
 Not quite that simple
Need to do some networking specific things
e.g. conditional independence to model hot-potato routing
 Can be solved using standard optimization toolkits
Taking advantage of sparseness of routing matrix A
 Back to tomogravity
Conditional independence = generalized gravity model
Quadratic distance function is a first order approximation
to the Kullback-Leibler divergence
Tomogravity is a first-order approximation to MMI
AT&T Labs - Research
Results – Single example
 ±20% bounds for larger flows
 Average error ~11%
 Fast (< 5 seconds)
 Scales:
O(100) nodes
AT&T Labs - Research
More results
Large errors are in small flows
>80% of demands have <20% error
tomogravity
method
simple
approximation
AT&T Labs - Research
Other experiments
 Sensitivity
Very insensitive to lambda
Simple approximations work well
 Robustness
Missing data
Erroneous link data
Erroneous routing data
 Dependence on network topology
Via Rocketfuel network topologies
 Additional information
Netflow
Local traffic matrices
AT&T Labs - Research
Dependence on Topology
star (20 nodes)
relative errors (%)
30
25
20
15
10
random
geographic
Linear (geographic)
5
0
clique
0
1
2
3
4
5
6
7
8
9
unknowns per measurement
AT&T Labs - Research
10 11
Additional information – Netflow
AT&T Labs - Research
Local traffic matrix (George Varghese)
0%
1%
5%
10%
AT&T Labs - Research
for reference
previous case
Conclusion
 We have a good estimation method
Robust, fast, and scales to required size
Accuracy depends on ratio of unknowns to measurements
Derived from principle
 Approach gives some insight into other methods
Why they work – regularization
Should provide better idea of the way forward
 Additional insights about the network and traffic
Traffic and network are connected
 Implemented
Used in AT&T’s NA backbone
Accurate enough in practice
AT&T Labs - Research
Results
 Methodology
Use netflow based partial (~80%) traffic matrix
Simulate SNMP measurements using routing sim, and
y = Ax
Compare estimates, and true traffic matrix
 Advantage
Realistic network, routing, and traffic
Comparison is direct, we know errors are due to algorithm
not errors in the data
Can do controlled experiments (e.g. introduce known errors)
 Data
One hour traffic matrices (don’t need fine grained data)
506 data sets, comprising the majority of June 2002
Includes all times of day, and days of week
AT&T Labs - Research
Robustness (input errors)
AT&T Labs - Research
Robustness (missing data)
AT&T Labs - Research
Point-to-multipoint
We don’t see whole Internet – What if an edge link fails?
Point-to-point traffic matrix isn’t invariant
AT&T Labs - Research
Point-to-multipoint
 Included in this approach
 Implicit in results above
 Explicit results worse
 Ambiguity in demands in
increased
 More demands use exactly the
same sets of routes
 use in applications is better
Link failure analysis
Point-to-multipoint
Point-to-point
AT&T Labs - Research
Independent model
AT&T Labs - Research
Conditional independence
 Internet routing is asymmetric
 A provider can control exit points for traffic going to
peer networks
peer links
access links
AT&T Labs - Research
Conditional independence
 Internet routing is asymmetric
 A provider can control exit points for traffic going to
peer networks
 Have much less control of where traffic enters
peer links
access links
AT&T Labs - Research
Conditional independence
AT&T Labs - Research
Minimum Mutual Information (MMI)
 Mutual Information I(S,D)=0
Information gained about S from D
I ( S , D)   p( S  s, D  d ) log
s ,d
p ( S  s, D  d )
p( S  s ) p( D  d )
I(S,D) = relative entropy with respect to independence
Can also be given by Kullback-Leibler information divergence
 Why this model
In the absence of information, let’s assume no information
Minimal assumption about the traffic
Large aggregates tend to behave like overall network?
AT&T Labs - Research
Dependence on Topology
Unknowns per
Network
PoPs Links
measurement
Relative Errors (%)
Geographic
Random
Exodus*
17
58
4.69
12.6
20.0
Sprint*
19
100
3.42
8.0
18.9
Abovenet*
11
48
2.29
3.8
11.7
Star
N 2(N-1)
N/2=10
24.0
24.0
Clique
N N(N-1)
1
0.2
0.2
3.54-3.97
10.6
AT&T
-
-
* These are not the actual networks, but only estimates made by Rocketfuel
AT&T Labs - Research
Bayesian (e.g. Tebaldi and West)
J(x) = -log(x), where (x) is the prior model
MLE (e.g. Vardi, Cao et al, …)
In their thinking the prior model generates extra constraints
Equally, can be modeled as a (complicated) penalty function
• Uses deviations from higher order moments predicted by model
AT&T Labs - Research
Acknowledgements
 Local traffic matrix measurements
George Varghese
 PDSCO optimization toolkit for Matlab
Michael Saunders
 Data collection
Fred True, Joel Gottlieb
 Tomogravity
Albert Greenberg and Nick Duffield
AT&T Labs - Research