Traffic Matrix
Estimation for Traffic
Engineering
Mehmet Umut Demircin
Traffic Engineering (TE)

Tasks
 Load
balancing
 Routing protocols configuration
 Dimensioning
 Provisioning
 Failover strategies
Particular TE Problem

Optimizing routes in a backbone network
in order to avoid congestions and failures.
 Minimize
the max-utilization.
 MPLS (Multi-Protocol Label Switching)

Linear programming solution to a multi-commodity
flow problem.
 Traditional

shortest path routing (OSPF, IS-IS)
Compute set of link weights that minimize
congestion.
Traffic Matrix (TM)



A traffic matrix provides, for every ingress point i
into the network and every egress point j out of
the network, the volume of traffic Ti,j from i to j
over a given time interval.
TE utilizes traffic matrices in diagnosis and
management of network congestion.
Traffic matrices are critical inputs to network
design, capacity planning and business
planning.
Traffic Matrix (cont’d)

Ingress and egress points can be routers or
PoPs.
Determining the Traffic Matrix

Direct Measurement:
TM is computed directly by collecting flowlevel measurements at ingress points.
Additional infrastructure needed at routers.
(Expensive!)
 May reduce forwarding performance at routers.
 Terabytes of data per day.

Solution = Estimation
TM Estimation

Available information:
 Link
counts from SNMP data.
 Routing information. (Weights of links)
 Additional topological information. ( Peerings,
access links)
 Assumption on the distribution of demands.
Traffic Matrix Estimation:
Existing Techniques and New
Directions
A. Madina, N. Taft, K. Salamatian, S.
Bhattacharyya, C. Diot
Sigcomm 2003
Three Existing Techniques

Linear Programming (LP) approach.


Bayesian estimation.


O. Goldschmidt - ISMA Workshop 2000
C. Tebaldi, M. West - J. of American Statistical Association, June
1998.
Expectation Maximization (EM) approach.

J. Cao, D. Davis, S. Vander Weil, B. Yu - J. of American
Statistical Association, 2000.
Terminology




c=n*(n-1) origin-destination (OD) pairs.
X: Traffic matrix. (Xj data transmitted by OD pair
j)
Y=(y1,y2,…,yr ) : vector of link counts.
A: r-by-c routing matrix (aij=1, if link i belongs to
the path associated to OD pair j)
Y=AX
r<<c => Infinitely many solutions!
Linear Programming

Objective:

Constraints:
Statistical Approaches
Bayesian Approach
Assumes P(Xj) follows a Poisson distribution
with mean λj. (independently dist.)

needs to be estimated. (a prior
is needed)
 Conditioning on link counts: P(X,Λ|Y)
Uses Markov Chain Monte Carlo (MCMC)
simulation method to get posterior distributions.
 Ultimate goal: compute P(X|Y)

Expectation Maximization (EM)

Assumes Xj are ind. dist. Gaussian.

Y=AX implies:

Requires a prior for initialization.
Incorporates multiple sets of link measurements.
Uses EM algorithm to compute MLE.


Comparison of Methodologies




Considers PoP-PoP traffic demands.
Two different topologies (4-node, 14-node).
Synthetic TMs. (constant, Poisson, Gaussian,
Uniform, Bimodal)
Comparison criteria:
 Estimation
errors yielded.
 Sensitivity to prior.
 Sensitivity to distribution assumptions.
4-node topology
4-node topology results
14-node topology
14-node topology results
Marginal Gains of Known Rows
New Directions

Lessons learned:
 Model
assumptions do not reflect the true nature of
traffic. (multimodal behavior)
 Dependence on priors
 Link count is not sufficient (Generally more data is
available to network operators.)

Proposed Solutions:
 Use
choice models to incorporate additional
information.
 Generate a good prior solution.
New statement of the problem

Xij= Oi.αij
 Oi :
outflow from node (PoP) i.
 αij : fraction Oi going to PoP j.
Equivalent problem: estimating αij .

Solution via Discrete Choice Models
(DCM).
 User
choices.
 ISP choices.
Choice Models




Decision makers: PoPs
Set of alternatives: egress PoPs.
Attributes of decision makers and alternatives:
attractiveness (capacity, number of attached
customers, peering links).
Utility maximization with random utility models.
Random Utility Model
Uij= Vij + εij : Utility of PoP i choosing to
send packet to PoP j.
 Choice problem:
 Deterministic component:


Random component: mlogit model used.
Results
Two different models (Model 1:attractiveness,
Model 2: attractiveness + repulsion )

Fast Accurate Computation of
Large-Scale IP Traffic Matrices from
Link Loads
Y. Zhang, M. Roughan, N. Duffield, A. Greenberg
Sigmetrics 2003
Highlights
Router to router traffic matrix is computed
instead of PoP to PoP.
 Performance evaluation with real traffic
matrices.
 Tomogravity method (Gravity +
Tomography)

Tomogravity

Two step modeling.
 Gravity
Model: Initial solution obtained using
edge link load data and ISP routing policy.
 Tomographic
Estimation: Initial solution is
refined by applying quadratic programming to
minimize distance to initial solution subject to
tomographic constraints (link counts).
Gravity Modeling

General formula:

Simple gravity model: Try to estimate the
amount of traffic between edge links.
Generalized Gravity Model

Four traffic categories
 Transit
 Outbound
 Inbound
 Internal



Peers: P1, P2, …
Access links: a1, a2, ...
Peering links: p1,p2,…
Generalized Gravity Model
Generalized Gravity Model
Tomography

Solution should be consistent with the link
counts.
Reducing the computational
complexity
Hundreds of backbone routers, ten
thousands of unknowns.
 Observations:

Some elements of the BR to BR matrix are empty.
(Multiple BRs in each PoP, shortest paths)
 Topological equivalence. (Reduce the number of
IGP simulations)

Quadratic Programming

Problem Definition:
Use SVD to solve the inverse problem.
 Use Iterative Proportional Fitting (IPF) to
ensure non-negativity.

Evaluation of Gravity Models
Performance of proposed algorithm
Comparison
Robustness
Measurement errors
x=At+ε
ε=x*N(0,σ)

Questions?