Title: TCOM 540
1(No Transcript)
2TCOM 540
3Agenda
- Review Session 2 assignment and Quiz
- Economies of Scale
- Traffic and Cost Generation Techniques
- Case Study of Traffic Generation
4Economies of Scale
- Highly important in telecommunications
- Big pipes often (but not always!) cheaper than
small ones per unit capacity - Big pipes carry traffic more efficiently lower
blocking/more effective capacity
5Big Pipes are (Usually) Cheaper per Unit capacity
- FTS2001 price for dedicated circuit from
- Falls Church, VA to Englewood, CO
Access Transport Access Total Cost/DS0
DS0 40 276 59 375 375
4xDS0 155 798 175 1128 282
T1 155 1787 175 2117 88
3xDS1 1813 6025 1245 9083 126
6Efficiency of Big Pipes
- Example Max minutes per circuit for p 0.03
7Traffic Models
- Topics
- Uniform
- Random
- Population power
- Modified population power
- Normalized model
- Asymmetric model
8Uniform
- Traffic from site a to site b is
- T(a,b) C
- Not realistic for most situations
9Random
- T(a,b) R
- Where R is a random number generated on a defined
interval Tmin, Tmax - This simple model is useful in some applications
- WWW-type traffic
- As one component of a more complex model
10Population Power
- If the sites a and b have populations Pa and Pb,
and are distance Da,b apart, then - T(a,b) a(papb)b/Da,bg
- where a, b, g are suitably-chosen constants
11Modified Population Power
- Large, close sites can dominate the simple
population power model - Fails if D 0
- Use offsets Doff and poff
- T(a,b) a(papb poff)b/(Da,b Doff )g
12Normalization
- Choose a so as to give the desired level of
traffic on the network by matching - Total traffic on network
- Traffic from each site (row normalization)
- Traffic to and from each site (row and column
normalization) - Must have S traffic in S traffic out for this
to be possible! - Algorithmic iterative approach can be used
13Asymmetric Traffic
- Models considered so far are symmetric
- T(a,b) T(b,a)
- Real traffic is often not symmetric
- E.g., WWW access
- Introduce concept of Levels
- Each site is assigned to a level Li, i1, , n
- Matrix of multipliers M(Li, Lj)
14Asymmetric Traffic (2)
- If M ( ) then traffic from a level 1 node
to a level 2 node will be one-third of the
traffic from a level 2 node to a level 1 node - Revised model is then
- T(a,b) aM(La, Lb)(papb poff)b/(Da,bDoff
)g
0 1 3 0
15More Complex Models
- Introduce a random element into the previous
model - Superimpose multiple components representing
different types of traffic - Redefine the distance function
- Organizational distance
16Tariff Structures
- Fundamental distinction
- Fixed cost per month
- Private lines, PVCs, some internet access
- Cost may also depend on bandwidth, distance,
quality, - Usage based
- Switched pipes e.g., switched voice
- Price may also depend on distance, bandwidth,
- Data per packet
17Tariff Structures (2)
- Additional fees may include
- Initiation charge
- Cancellation charge
- Features charges
- Access charges
18Tariff Structures (3)
- Tariff structures are not simple
- Depend on level of competition, administrative
and other boundaries, other factors - Best deals are not tariffed
- Usually competed/negotiated by large customers
19Tariff Structures (4)
- In some cases (especially international), tariffs
may not exist, or may be for half-circuit only - I.e., to a notional mid-ocean meeting point
- Commercial tariff services (e.g., Valucom, CCMI)
are not cheap
20Linear Distance-Based Charges
- Underlying tariff structure for many dedicated
circuits and PVCs is distance-based, e.g., - Cost a bdistance
- Rates for individual location-pairs may vary
- Carrier may have excess capacity on certain
routes gt may be cheaper - Carrier may have to buy capacity from others gt
may be more expensive
21Piecewise-Linear Charges
Cost/month
Distance
22Step Function
Cost/month
Distance
23Cost Generators
- Cahn provides 5 cost generators
- 1. Linear (TARIFF-UNIVERSAL)
- 2. Piecewise linear (2 pieces)
(TARIFF-UNIVERSAL) - 3. Piecewise linear (limited international)
(TARIFF-NATIONAL) - 4. International half-circuit (TARIFF-HCKT)
- 5. Exceptions (TARIFF-OVERRIDE)
24Case Study of Traffic Generation
25Outline
- Background and Problem Statement
- Current Academic Researchers in this Problem
- Proposed Algorithm
- Sample NetHealth Data Description
- Numerical Example with Proposed Algorithm
- O-D Matrix Tool Interface
- O-D Matrix Tool Outputs
- Next Steps
- References
26Background and Problem Statement
- Background
- Client uses Concords NetHealth to monitor
performance - NetHealth only generates link statistics such as
link utilization - When what if analyses on the network are
required, the origin destination (O - D)
traffic matrix that generated the measured link
utilization reported by NetHealth is required - The O-D traffic is the matrix of offered loads
that originates at one node and is destined for
another node1 - Problem
- To estimate the O-D traffic matrix given
aggregate link utilization - 1 Note nodes are groups of users that enter a
router on a common interface, not single users
27Problem Background
- Vardi (1996) first used the term network
tomography to refer to this problem due to the
similarity between network inference and medical
tomography1. - There are two forms of network tomography (our
problem is the 2nd) (Coates, 2001) - Link level parameter estimation based on
end-to-end, path level traffic measurements, or - Sender-receiver path-level traffic intensity
estimation based on link-level traffic
measurements (antithesis of first form) - 1 Tomography a method of producing a
three-dimensional image of the internal
structures of a solid object (as the human body
or the earth) by the observation and recording of
the differences in the effects on the passage of
waves of energy impinging on those structures
(Merriam-Webster Dictionary).
28Current Academic Researchers in this Problem
- Bell Labs Cao, Cleveland, Lin, Sun, Vander Wiel,
Davis, Yu, Zhu - UC Berkeley Coates, Hero, Nowak, Yu
- Vardi (Rutgers)
29Current Academic Approaches Used
- Frequently a linear model is assumed for the O-D
traffic matrix estimation problem (Vardi (1996),
Coates et al. (2001), Cao et al. (2001), Cao et
al. (2000)) - y A x
- where
- y (y1, , ynL) is the observed column vector of
incoming and outgoing byte counts for each of nL
links - x (x1, , xnP) is the unobserved vector of byte
counts for all OD nP pairs in the network, where
nP n(n-1) in a network of n nodes - A nL x nP routing matrix (0/1)
- Elements aij of A are 1 if link i belongs to
the directed path of the O-D pair j
30Current Academic Approaches Used
- Often the matrix A has a very large dimension
(thousands of rows and columns for a moderate
number of sites), and thus iterative algorithms
are used - Although the model is linear, it is not a typical
linear regression because of the the
nonnegativity constraints on the parameters x - Also, because nP is typically larger than nL,
identifiability of the parameters is a problem
31Current Academic Approaches
A matrix
32Current Academic Approaches (2)
- Vardi (1996) assumes the O-D byte counts are
Poisson - Maximum likelihood via the Expectation-maximizatio
n1 (EM) algorithm is used to solve for O-D matrix - O-D byte counts can be assumed Normal as an
approximation to Poisson may allow simpler
solution techniques - A moment method for estimation is also proposed
- Cao et al. (2000) embellish the above Poisson
model by assuming all quantities are time-varying - Maximum likelihood estimation is done via a
combination of the EM algorithm and a
second-order global optimization method - 1 EM algorithm is an algorithm for finding
likelihood estimators from incomplete data. It is
an iterative algoirthm in which a starting
estimate is updated using a transformation called
the EM operator. (Vanderbei and Iannone, 1994).
33Current Academic Approaches (2)
- Cao et al. (2001) describe a divide-and-conquer
approach that can be used for large networks - O-D pairs are partitioned into a number of
disjoint sets - Clustering methods used to group the O-D pairs
- For each O-D set, a corresponding set of links
are selected for estimation (not disjoint) - Heuristics used to select links, balancing
estimation accuracy and computational cost - Parameters are estimated for each O-D set, which
aggregates the remaining rates for other O-D
pairs not in the current set - Computational complexity can be reduced from
O(Ne5) to O(Ne2), where Ne is the number of edge
nodes
34Current Academic Approaches (2)
- ? Exceeds requirements, highly desirableÂ
- è Meets requirements
- ? Does not meet requirements, less desirable
- Â
- 1 This method can also be used with parallel
computing
35Need for New Algorithm
- Academic methods are not feasible due to number
of nodes in Clients network (approximately 1000
nodes) - Proposed algorithm is a heuristic faster
36Proposed Algorithm
- Compute the probability of originating at node i
and terminating at node j - p(i,j) is the fraction of all (unidirectional)
network traffic coming in to node j, if i to j
within a certain number of hops - Estimate the total load originating at each node
as the outgoing load from each node - Compute TM(i,j), the estimate of packets per
second originating at node i and terminating at
node j, using estimate of the total load
originating at each node times the probability
p(i,j) - Route traffic via Enhanced Interior Gateway
Routing Protocol (EIGRP)
37Proposed Algorithm (2)
- Sum up the estimated load on each link from
TM(i,j), compare to given link loads - Compute an adjustment factor based on the ratio
of the given link loads to the estimated link
loads - Adjust the estimate of total load originating at
each node using the ratio of the given link loads
to the estimated link loads, then re-estimate
TM(i,j) - Iterate above until convergence in the link loads
is achieved - Final adjustment if load originating at each node
is greater than total load at node
38Sample NetHealth Data Description
- Net data from January 2001
- Data
- 1169 unique links
- 991 nodes
- Data fields for each link record
- Originating and terminating nodes for each link
- Link utilizations in each direction
- Link speed
39O-D Matrix Tool Interface
- Input Parameters
- Max Hops max number of hops between nodes for
nonzero traffic probability - Link Factor maximum deviation of estimated from
measured link loads
40Numerical Example with Proposed Algorithm
- Input Parameters
- Link Factor 1.1
- Number of Hops varied from 2 to 10
41Numerical Example with Proposed Algorithm (2)
- Results (run times on 866 Mhz PC)
- Problem with Backbone Links only runs quickly (1
min) - Full Network problem about 4 hours
- Error in link utilizations under 1 percent for
either problem - Small increase in run times with number of hops
(e.g., 20 increase as number of hops doubles
from 4 to 8)
42O-D Matrix Tool Outputs
- Network Performance
- Average link percentage error
- Expected packet delay
- Average number of hops
- Link Performance
- Estimated and measured packets per second in each
direction - Expected packet delay
- Node Performance
- Originating packets per second
- Total packets per second in and out of each node
- Number connected to each node
43Sample Map Output
Client CONUS Network
44References
- A Scalable Method for Estimating Network Traffic
Matrices from Link Counts, Bell Labs Tech
Report, 2001, Jin Cao, D. Davis, Scott Vander
Wiel Bin Yu, and Zhengyuan Zhu. - Time-Varying Network Tomography Router Link
Data, Journal of the American Statistical
Association, 95, 1063-1075, 2000, Jin Cao, D.
Davis, Scott Vander Wiel and Bin Yu. - Mark Coates, Alfred Hero, Robert Nowak, and Bin
Yu (2001). Large scale inference and tomography
for network monitoring and diagnosis. Technical
Report 604, Stat Dept, UC Berkeley. August, 2001.
- Y. Vardi, "Network Tomography Estimating
Source-Destination Traffic Intensities From Link
Data", Journal of the American Statistical
Association March 1996,Vol.91 No 433, Theory and
Methods - R.J. Vanderbei and J. Iannone, "An EM approach to
OD matrix estimation," Technical Report,
Princeton University, 1994
45Assignment Session 4
For the following set of sites and traffic,
design a minimum-cost network with reasonable
performance
Sites
Traffic (kbps)
46Assignment Session 4 (2)
- You have two types of links available for this
design - Capacity 1.5 Mbps, cost 80 8/mile
- Capacity 64kbps, cost 10 1/mile
- You may use multiple links to satisfy demand.
- Note What is the maximum utilization you will
allow per link?