Link Failure Monitoring
Using Network Coding
Hamed Firooz
Sumit Roy, Linda Bai
firooz,sroy,lyb3@u.washington
.edu
Outline

Network Tomography


Introduction (Network Monitoring)
Approaches:




Deterministic vs. Stochastic
Active vs Passive
Challenges: Overhead, Identifiability
Network Coding



Applications to network monitoring: new method
Optimization : speed/complexity tradeoffs
OPNET Implementation
Fundamentals of Networking Lab(FunLab)
Network Tomography

Networks: set of nodes, links modeled as
graph G(V,E)

Network monitoring


Node or
network
Involves collection of network performance
statistics (link delay, link loss or failure
status)
Important for QoS guarantees (media
streaming, interactive video applications) 5
1
2
8
3
4
7
6
A Logical Network
Challenges

Choice of appropriate measurement
technique and algorithmics
Fundamentals of Networking Lab(FunLab)
G(V,E)
Measurement Methods

Node-oriented: These
methods are based on
cooperation among network
nodes, e.g. ping or traceroute
 Using Ping, round trip delay to
every node can be measured.
 Uses Internet control message
protocol (ICMP) packets


Many routers do NOT
respond to these packets
Many service providers do not
own the entire network
Fundamentals of Networking Lab(FunLab)
D l1
l1
R
D l2
l2
R
R
Measurement Methods

Edge-oriented: Access is
S
available to nodes at the edge
only (and not to any in the
interior)


Does not require exchanging
special control messages between
interior nodes
Inverse problem: estimate link
level status from end-2-end (path
level) measurements
Fundamentals of Networking Lab(FunLab)
S
Network(?)
S
S
Measurement Methods

Active (sending probe packets)
- Adds overhead to normal data traffic by
introducing new control packets


Passive (insitu traffic analysis)
- No overhead; temporal and spatial
dependence might bias measurement
Our method: edge-oriented, active network
tomography
 Given a network, and a limited number of
end hosts, when can we infer failure
status of the links?
Fundamentals of Networking Lab(FunLab)
Network
?
End-to-End Probing
•
End1
link1
router1
link2
Probes are inserted into a
data stream, and end-to-end
properties on that route
measured.
• Probes are exchanged
between end nodes using
routing matrix of the graph
link3
End2
Routing matrix A
End3
Fundamentals of Networking Lab(FunLab)
link 1
link 2
link 3
End 1  End 2
1
1
0
End 1  End 3
1
0
1
End 2  End 3
0
1
1
End-to-End Probes


Routing matrix relates link
attribute to route attribute
For some parameters like
delay or path loss, this
relation is linear under some
assumptions
 D End 1 End 2   1
 D End 1 End 3    1

 
 D End 2  End 3   0
1
0
1
0   D l1 


1  D l2 

1   D l 3 
Fundamentals of Networking Lab(FunLab)
End1
l1
R
l2
End2
l3
End3
Deterministic
Link attributes (e.g. delay) are considered
unknown, constant
 Goal: estimate constants
 Link attributes are typically time varying
 method is suitable for periods of local
‘stationarity’

Fundamentals of Networking Lab(FunLab)
Stochastic

Link attribute specified by a suitable probability
distribution


e.g. link delay follows a Gaussian distribution
Estimation problem: unknown model parameters
based on path observation in the presence of
additive noise
Fundamentals of Networking Lab(FunLab)
Deterministic vs. Stochastic
Methods

Stochastic

Bayesian - requires a prior distribution



incorrect choice leads to biases in the estimates
More computationally intensive
Deterministic

Lower complexity but suffers from generic nonidentifiability
Fundamentals of Networking Lab(FunLab)
Link Failure Model
l1
End1
l2
l3
R1
Define an indicator
function for status of
each link
R2
0
x li  
1
y end 1 end 2
0

1
Fundamentals of Networking Lab(FunLab)
End2
l i is ok
l i is congested
all of l1 , l 2 , l 3 is ok
o .w .
Binary Deterministic Model
l1
End1
l2
R1
l3
R2
y end 1 end 2  x l1 or x l 2 or x l 3
y = Ax
A: N-by-M binary routing matrix
x: M-by-1 binary vector, the status of each link
y: N-by-1 binary vector, the status of each path (measurements)
Fundamentals of Networking Lab(FunLab)
End2
Failure Monitoring



Network G(V,E) with set of paths P
|E |
|P |
x  {0 ,1} , y  {0 ,1}
x, y are binary vectors
A path is congested if at least one of its links
is congested l l l
End1
1
 y1  End 1  End 2  1
 y 2   End 1  End 3  1

 
 y 3  End 2  End 3  0
 y 1   x l1 ( OR ) x l 2 
 y 2    x l ( OR ) x l 
1
3

  
 y 3   x l 2 ( OR ) x l 3 
2
1
0
1
3
0   x l1 
 
1  x l2  ,

1   x l 3 
Fundamentals of Networking Lab(FunLab)
l1
x l1  { 0 ,1}
Router
l2
End2
l3
End3
Identifiability y = Ax

Problem: Estimate x from y with



A (N-by-M) : binary routing matrix
x (M-by-1) : binary link failure status
y (N-by-1) : end-to-end measurements
6 links, 3 End-to-End routes  N=6, M=3

Identifiability: a network is identifiable if y = Ax has
a unique solution

Usually, M ( # of links in network) >> N (# of
measurements), so network is generically NOT identifiable.
Fundamentals of Networking Lab(FunLab)
Identifiability: Binary Model

Solution: limit (maximum) number of failed links
inside the network


Suppose at most k links can fail simultaneously
Defn: k-Identifiability

Network is k-identifiable if
x | E | 1
 x 1 , x 2 s.t. x 1

0
0
k
 k, x2
x 6 1
0
 k , x 1  x 2  Ax 1  Ax
2
0
1
Only one
link can be
congested
from end-to-end observation it is possible to uniquely
identify up to k congested links
Fundamentals of Networking Lab(FunLab)
Example of 1-identifiability

x 6 1
0
1
l1
y1 2
y1 3
y 2 3
-0
0
0
l1 l2 l3 l4 l5 l6
1 1 0 0 0 0
1 0 1 1 0 1
0 0 0 0 1 1
l2
Fundamentals of Networking Lab(FunLab)
l5
l4
1

A 1

 0
l3
l6
1
0
1
0
0
1
0
0
0
0
1
1
0

1

1 
Example: k=2 identifiability

x 6 1
0
2
l1
Ambiguity
 y 1 2   1 

  
y  y 1 3  1

  
 y 2  3   0 
 y 1 2   1 

  
y  y 1 3  1

  
 y 2  3   0 
Fundamentals of Networking Lab(FunLab)
l2
l5
l4
1

A 1

 0
l3
l6
1
0
1
0
0
1
0
0
0
0
1
1
0

1

1 
1-Identifiability

A network with an intermediate
degree two node is not 1-identifiable


End1
`
l1
If path End1End2 is congested, it is
impossible to determine which link
among l1 and l2 is congested .
l2
Necessary but not sufficient!
End2
`
x l1  1  y End 1 End 2  1
x l 2  1  y End 1 End 2  1
Fundamentals of Networking Lab(FunLab)
k=1 Identifiability

1-identifiability Theorem:
End-to-End probe based
measurements can detect a unique
congested link in a network if and
only if there are no two identical
columns in the network routing
matrix
P1
P1  1
P3
Fundamentals of Networking Lab(FunLab)

0

0

0
0

 0
P3
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0

0

0

0
0

1 
k- identifiability

k-identifiability Theorem:
End-to-End probe based
measurements can detect a unique
congested link in a network only if
there are no k+1 dependent
columns in the network routing
matrix
Fundamentals of Networking Lab(FunLab)
1

0

0

0
0

 0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0

0

0

0
0

1 
Example: k=2 identifiability

x 6 1
0
2
l1
Ambiguity
 y 1 2   1 

  
y  y 1 3  1

  
 y 2  3   0 
 y 1 2   1 

  
y  y 1 3  1

  
 y 2  3   0 
Fundamentals of Networking Lab(FunLab)
l2
l5
l4
1

A 1

 0
l3
l6
1
0
1
0
0
1
0
0
0
0
1
1
0

1

1 
Shortest Path Routing Revisited

Packets are sent on shortest path between two
end nodes
- sub-graphs = tree starting from a boundary
(source) node


Node 4 has degree two in all graphs
But node 4 has degree four in the original
network
Fundamentals of Networking Lab(FunLab)
Revisiting Shortest Path Routing

What if we could change routing
matrix ?
Example: in place of shortest path
routing, route packets through longer
paths, e.g. n1l2l4n2


Now network is 1-identifiable !
Intrinsic limitation for end-to-end
measurement methods based on
shortest path routes

probes transmitted along such paths
contain only minimum information
Fundamentals of Networking Lab(FunLab)
Solution



Look to exchange probes between boundary
nodes via other (non-shortest) paths?
Changing the routing tables violates tomography
assumption
Use Network Coding; exploit broadcast nature
of network coding, a transmitted probe will
traverse almost every path between two
boundary nodes
Fundamentals of Networking Lab(FunLab)
Network Coding: Short Review


Present: routers just forward incoming packets, i.e. copy
the packets on an input link onto the output links
Proposed: What if each node in a network performs
some computation on received data prior to forwarding?
y1
y2
y1
y1
y2
y2
Fundamentals of Networking Lab(FunLab)
y3
f1(y1,y2,y3)
f2(y1,y2,y3)
How does NC work? (1)
sender s
receiver t2
A
C
B



D
receiver t1
“Butterfly” network: All links have the same capacity 1 b/s
s wants to send data bits a, b to both t1 and t2
Bottleneck is CD
Fundamentals of Networking Lab(FunLab)
How does NC work?(2)
sender s
A
a
b
B
XOR
D
a+b
b

receiver t2
a
receiver t1
Node C XORs received messages on each of its links
Fundamentals of Networking Lab(FunLab)
How does NC work?(3)
sender s
A
a
b
B
XOR
a+b
b


receiver t2
a
D
a+b
a+b
receiver t1
t1 and t2 know both a and b
Now s can send data at rate 2 b/s/receiver
Fundamentals of Networking Lab(FunLab)
Linear Network Coding




Network Coding is a coding at layer three
The coding is conducted over the finite field Fu, u=2q
each coded symbol can be represented by q-bits
within an IP layer frame
Signal Y(j) on an outgoing link j of node v, is a linear
combination of signals Y(i) on incoming link i of v:

We assume there is no process generated at node v
Y ( j) 
  Y (l )
l
{ l :d ( l )  v }
Fundamentals of Networking Lab(FunLab)
Received Symbols


Pi : i-th route from source to destination
Source sends α over Pi
y     l   i ( G ),   F 2 q
l P
 i (G ) 
i

l P

l
Path NC Coef.
i
βi depends on topology G hence βi(G)
y   1 2   1 ( G )
α
γ1
S
γ3
γ4
Fundamentals of Networking Lab(FunLab)
γ2
D
γ5
Received Symbols: Linear Model



ek one of source outgoing links
Pek : collection of all paths between source
and destination starts at ek
Source sends αk over ek. By superposition
destination receives
Pe1
α1
S
γ1
y  k
γ2
e1
i
P  Pe
γ3
γ4
 
D
γ5
k
l P
i
| Pe |
k
l
  k   i ,ek (G )
i 1
y   1 (  1 2   1 3  5 )   1 (  1, e1   2 , e1 )
Fundamentals of Networking Lab(FunLab)
Received Symbols: Linear Model

Source sends out symbols αk over ek using
superposition once more
| Pe |
K
y
k
  
k
k 1
i 1
i ,ek
(G )
Pe1
α1
S


y=αtβ(G)
γ1
e1
α2
In vector format:
γ
β(G) is total network coding vector
Fundamentals of Networking Lab(FunLab)
γ2
4
γ3
D
γ5
Received Symbols: Linear Model

Source sends symbols in M succ. time slots:
y M 1  AM  N  ( G ) N 1
 ( G ) N 1

   1, e1  2 , e1   N 1 , e1
  
   

Pe

1
Fundamentals of Networking Lab(FunLab)
 1, e
  N 2 ,e2
    
1
Pe
2


 N K ,e K 
 

Pe

K
t
Link Failure Model


If a link is severely congested, packets are
significantly delayed and assumed lost at the
destination
We model the network with link l in
congestion state by its edge deleted
subgraph denoted by Gl(V,El)
γ1
S
γ3
γ4
Fundamentals of Networking Lab(FunLab)
D
γ5
Link Failure Model

Total network coding vecor of Gl(V;El), β(Gl) is
different from β(G)
 i ,e

k
i

if l  Pe k ( d )
  i ,ek (G )
(G l )  

o .w .
0
if the congested link doesn’t belong to i-th path from
source to destination, Pi, it will not affect packets
going through those paths

It is zero otherwise
 1 ( G )   1 2
 2 (G )   4 5
γ1
 1 (G l )  0
S
 2 (G l )   2 (G )
1
1
Fundamentals of Networking Lab(FunLab)
γ2
e1
e2
γ4
γ3
l1
γ5
D
Link Failure Model



Training sequence is A
yl : vector of symbols observed at the
destination in M time slots with link l
congested
l
y M 1  A M  N  ( G l ) N 1
Potential for identifying: received symbols
change uniquely in response to link
l
congestion
y
 y
Fundamentals of Networking Lab(FunLab)
M 1
M 1
l1
y M 1
l2
y M 1

Example

1 
 
 (G )  3
 
 1 
1
A
3
1
3
 1, e  1  1  1
2

3
1
 2 ,e  1  2  2  3
1
 2 ,e  3  2  1
2
-- e1 e2 l1 l2 l3
1st time slot 0 2 2 3 1 1
2nd time slot 2 3
1 0 1 3
Fundamentals of Networking Lab(FunLab)
Pe1
S
Pe 2
1
1
e1
e2
3
D
2
2
Theorem 1: Sufficient Conditions

If Rank(A)= deg(S), and

for all Pek set of paths between source and
destination starting at ek
| Pe |
k

j 1
j
 j ,e  0   j  0  j
i
then
A  (G )  A  (G l )
A  ( G l1 )  A  ( G l 2 )
l  E
 l1 , l 2  E
Fundamentals of Networking Lab(FunLab)
(more next slide)
Theorem 1
| Pe |
k

Condition  
j 1

j

j ,ei
 0  
 0 j
j
means
For a set of paths having ek in common, Pek , NC
coefficient of the paths are independent !
 Independent

 2 , e1   N 1 , e1
1 , e1
  
   

Pe

1
Independent
 1, e
  N 2 ,e2
    
 1 , e   1 2
1
 2 ,e
1
 1, e
2


 independen
  1 3  5 

  4 5
Fundamentals of Networking Lab(FunLab)
1
Pe

2
Pe1
t
S
Pe 2

 N K ,e K 
 

Pe

K
γ1
γ2
e1
e2
γ4
t
γ3
D
γ5
Example

1 
 
 (G )  3
 
 1 
1
A
3
1
3
Independent
 1, e  1  1  1
2
1
 R ank ( A )  2  deg( S )  2 , e  1  2  2  3
1
3
 2 ,e  3  2  1
2
-- e1 e2 l1 l2 l3
1st time slot 0 2 2 3 1 1
2nd time slot 2 3
1 0 1 3
Fundamentals of Networking Lab(FunLab)
Pe1
S
Pe 2
1
1
e1
e2
3
D
2
2
Complexity/Speed

First condition of Theorem 1:
Rank( AM  N )  deg( S ) implies




M  deg( S )
In previous example M=2=deg(S)
Number of time slots: at least the number of
outgoing links of source
Is it possible to decrease number of time
slots?  faster monitoring
Possible by increasing number of bits in LNC
coeff.  more complexity
Fundamentals of Networking Lab(FunLab)
Example


q=3
A=[1 1 4]
1
-- e1 e2 l1 l2 l3
1st time slot 6 4 2 5 7 1
Fundamentals of Networking Lab(FunLab)
S
1
e1
e2
3
D
2
2
Theorem 2: Complexity/Speed
tradeoff






Ni=|Pi|
q bits per symbol are used in network coding
M number of (desired) time slots
Let Z={1,2,…,K}
S
K degree of source
ZM: collection of all partitions
of Z with size M
M
K links
Z M  {{ H 1 , H 2 ,..., H M } |  H i  Z , H i  H j   }


i 1
K=3, 2  Z={1,2,3}
ZM={ {{1,2},{3}} , {{1,3},{2}} , {{2,3},{3}} }
Fundamentals of Networking Lab(FunLab)
Theorem 2: Complexity/speed
tradeoff

Network is 1-identifiable if
q

min
{ H i , i  1 ,..., M } Z M
Rank(A)=M
Fundamentals of Networking Lab(FunLab)
max
i
N
j H i
j
Theorem 3: Random LNC




Random linear network coding is a distributed
approach achieving capacity asymptotically
Intermediate node choose their NC coefficients
uniformly from the elements of Fu (u=2q)
1 M
P ( G is 1-identifiable)  1  | E | (| E |  1)(
)
q
2
Exponential increase with q (number of bits) and M
(number of time slots)
Quadratic decrease with size of network
Fundamentals of Networking Lab(FunLab)
Multi-source Multi-destination


So far, considered only Single source Single
destination
Easily extendable to Multi-source Multidestination
Fundamentals of Networking Lab(FunLab)
Simulation

Simulation environment



OPNET 14.5
MATLAB 7.1 (finite field operations)
Evaluation

University of Washington’s Electrical Engineering
network



Thirteen subnets
3 backbone routers
Full Duplex Ethernet links
Fundamentals of Networking Lab(FunLab)
Simulation Set-Up


Implementation of Network Coding (NC) within
OPNET
We employ network coding at transport layer (instead
of IP layer)


Routers model is modified to distinguish between nonNC/NC packets through the use of a flag bit within the
UDP header



Easier to implement
NC packets are sent for separate processing
non-NC packets are processed normally
We assign a q-bit field called LNC field within the
TCP/UDP header, for linear network coding.
1
LNC field
Fundamentals of Networking Lab(FunLab)
UDP packet
RECEIVE/SEND interface


Inherently network coding operates on
unidirectional links
Each interface within a router mode is
designated as a SEND or RECEIVE interface
only for the network coded packets


operating regularly with non-network coded
packets
Finite field operation is done in MATLAB

Using MATLAB API within OPNET
Fundamentals of Networking Lab(FunLab)
RECEIVE/SEND
Fundamentals of Networking Lab(FunLab)
Evaluation
Fundamentals of Networking Lab(FunLab)
UW EE Network
Fundamentals of Networking Lab(FunLab)
UW EE Network-lookup table
Fundamentals of Networking Lab(FunLab)
Fundamentals of Networking Lab(FunLab)
Network Tomography: A
Stochastic Model [1]



Passage of probes can be
modeled as two stochastic
process: {Xl(i)} and {Zl(i)} for
each node k
Zl(i) time delay process of link k
Xl(i) called bookkeeping
process: cumulative probe from
root to k
[1] V. Arya, N. Duffield, D. Veitch “ Temporal Delay Tomography”,
IEEE Infocom 2008
Fundamentals of Networking Lab(FunLab)
Network Tomography: Stochastic
Method



l’
Discretize delay D={0,b,2b,…,mb,∞}
mb is delay threshold
Xl(i)=Xl’(i)+Zl(i)
k-1
l
Xl(i)
0


Pr[ X l ( i )  d | X l ( i )  v ]  
1
 Pr[ Z ( i )  d  v ]
k

Fundamentals of Networking Lab(FunLab)
Xl’(i)
Zl(i)
k
if d  v
if d  v  
o .w .