An Algebraic Approach to Practical and Scalable Overlay Network
Monitoring
Yan Chen, David
Bindel, Hanhee Song,
Randy H. Katz
Presented by Mahesh Balakrishnan

 Overlay networks
 Monitoring of end-to-end paths
 The need for a separate Monitoring Service
 Metrics: Latency... Loss Rate?
 The Goal: A Scalable Overlay Loss Rate
Monitoring Service

 Latency-only Schemes
 Clustering:
– Nodes are clustered together, and cluster representative is monitored
– Claim: Inaccurate for congestion detection
 Co-ordinates:
– Cannot give congestion information

 Network Tomography: Determining internal network properties from black-box measurements
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Shavitt, et al.
Algebraic approach
Ozmutlu, et al. Selecting minimal set of paths to cover all links
 General Metric Systems: RON

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Assumptions:
– Access to link composition of paths
– Ability to measure path (but not link) characteristics
From the possible n 2 end-to-end paths, select a basis set of k paths (k << n 2 ) to monitor.
The characteristics of all paths can be inferred from this basis set.
Centralized algorithm: all nodes send measurements to central node.

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Eq 1: 1
 p
1

( 1
 l
1
)( 1
 l
2
)
Represent paths as vectors: v



1


1
0




AD
BD
AC log( 1
 p
1
)
 log( 1
 l
1
)
 log( 1
 l
2
)


1 1 0


 log(


 log( log(
1
1
1


 l l l
1
2
)
3
)
)





B p
1
D
A l2 l1
3
C

Path Matrix Link Rates Path Rates
… x

R s

1
=
G

{ 0 | 1 } r
 s b

R r

1

A
G





1
0
1
1
0
1
Gx
 b
0
1
1




AB
AC
BC k = Number of essential paths
1 < k <= s
G is rank deficient: k < s
B p
1
D l2 l1
3
C

s
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 k = # of essential paths
= rank ( G ) k <= s k
Usually G is rankdeficient: k < s
Select k linearly independent paths to monitor:

…
G x
G
 b
=

One-time QR Decomposition:
O(rk 2 ) time… O(n 4 )!
Inferring other paths: O(k 2 )

 Accuracy
 Scalability: How does k grow w.r.t n ?
 Other concerns:
– centralized solution
– compute time under churn
– storage load

 Star Topology, Strict Hierarchy: s = O (n ), => k
= O( n )
 Clique: Each path (end host pair) contains a unique link, hence k = O( n 2 )
 Hierarchy is good, Dense Connectivity is bad
 Conjecture: k = O( nlogn ) for the internet
 What if only a small % of end nodes are on overlay?

Synthetic Hierarchical Real

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Path Addition: O(k 2 )
Path Removal: O(k 2 ) [Naïve : O(rk 2 )
Node Addition: O(nk 2 )
Node Removal: O(nk 2 )
–
–
Cannot use path removal algorithm directly; path will be replaced using another path involving node
Remove all paths, then look for replacements
 Cubic in n: Churn in large systems?

 End-to-end internet paths are generally stable
 Traceroute
 Topology checked on a daily basis, in presence of drastic loss rate changes
 If path has changed at certain links, other paths with that link are checked as well

Load Balancing/Topology
Measurement Errors
 Paths in G are randomly reordered before basis set is selected
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Untraceable paths/segments are modeled as single links; they always get selected in basis
Router aliases – one physical link presented as several virtual links – all virtual links get similar loss rates

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Three synthetic BRITE topologies: Barabasi-
Albert, Waxman, hierarchical
One ‘real’ router topology (Mercator)
 Methodology:
–
–
Loss Distribution: Good = 0-1%, Bad = 5-10%
Loss Model: Bernoulli, Gilbert
 Simulate loss for selected paths, infer for other paths

 All Configurations under 0.008, 1.18

Synthetic Hierarchical Topology
Real Topology

 3 seconds for 100 nodes, 21 minutes for 500!

Node Addition
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Path Addition: 125 msec
Path Removal: 445 msec
Node Addition: 1.18 sec
Node Removal: 16.9 sec
What about n >> 60?
Network Link Removal
Node Deletion

 51 hosts, each from different organization
 Each node sends a UDP packet to every other host in each trial
 300 trials of 300 msec each
 Receiver counts packets for loss rate
 Traceroute used for topology measurement

Average Abs. Error = 0.0027, Average Error Factor 1.1
Cumulative coverage/FP Cumulative error (Worst Run)

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Sensitivity Analysis done at night, on empty networks
Threshold at 12.8
Mbps
Why do this?

 Algebraic Method for inferring loss rates of all paths from a basis set
 Quite Accurate
 Reasonable load imposed on each node
 But is it really scalable?
 Centralized solution, cubic dependence on n for handling node addition/removal