session4 by shuifanglj


									TCOM 540

 Session 4
•   Review Session 2 assignment and Quiz
•   Economies of Scale
•   Traffic and Cost Generation Techniques
•   Case Study of Traffic Generation
          Economies 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
Big 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
       Efficiency of Big Pipes
• Example: Max minutes per circuit for p = 0.03

    Circuits       Minutes   Min/Ckt
               1         250       250
               2        2300      1150
               3        5850      1950
               4      10300       2575
               5      15300       3060
               6      20800       3467
              Traffic Models
• Topics:
  –   Uniform
  –   Random
  –   Population power
  –   Modified population power
  –   Normalized model
  –   Asymmetric model
• Traffic from site a to site b is
  T(a,b) = C
• Not realistic for most situations
• T(a,b) = R
  – Where R is a random number generated on a
    defined interval [Tmin, Tmax]
• This simple model is useful in some
  – WWW-type traffic
  – As one component of a more complex model
           Population Power
• If the sites a and b have populations Pa and
  Pb, and are distance Da,b apart, then

  T(a,b) = a*(pa*pb)b/Da,bg

  where a, b, g are suitably-chosen constants
    Modified 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*(pa*pb+ poff)b/(Da,b+ Doff )g
• 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
     • Must have S traffic in = S traffic out for this to be
     • Algorithmic iterative approach can be used
          Asymmetric 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, i=1, … , n
• Matrix of multipliers M(Li, Lj)
         Asymmetric Traffic (2)

• If M =  ( )
            0 1
            3 0  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) = a*M(La, Lb)*(pa*pb+ poff)b/(Da,b+Doff
       More Complex Models
• Introduce a random element into the
  previous model
• Superimpose multiple components
  representing different types of traffic
• Redefine the distance function
  – “Organizational distance”
             Tariff 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
           Tariff Structures (2)
• Additional fees may include
  –   Initiation charge
  –   Cancellation charge
  –   Features charges
  –   Access charges
         Tariff 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
         Tariff 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
 Linear Distance-Based Charges
• Underlying tariff structure for many dedicated
  circuits and PVCs is distance-based, e.g.,
  Cost = a + b*distance
• Rates for individual location-pairs may vary
   – Carrier may have excess capacity on certain routes =>
     may be cheaper
   – Carrier may have to buy capacity from others => may
     be more expensive
        Piecewise-Linear Charges



   • Each segment is linear
             Step Function


            Cost Generators
• Cahn provides 5 cost generators
  – 1. Linear (TARIFF-UNIVERSAL)
  – 2. Piecewise linear (2 pieces) (TARIFF-
  – 3. Piecewise linear (limited international)
  – 4. International half-circuit (TARIFF-HCKT)
  – 5. Exceptions (TARIFF-OVERRIDE)
  Case Study of
Traffic Generation
• 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
              Background and Problem
• Background
     – Client uses Concord’s 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 node 1
• 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
                Problem 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).
Current Academic Researchers in
         this Problem
• Bell Labs: Cao, Cleveland, Lin, Sun,
  Vander Wiel, Davis, Yu, Zhu
• UC Berkeley: Coates, Hero, Nowak, Yu
• Vardi (Rutgers)
    Current Academic Approaches
• 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 =(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
  Current Academic Approaches
• 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
• 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 n L,
  identifiability of the parameters is a problem
Current Academic Approaches



                           b                    d

A matrix:                                    O-D pairs
                               1 2 3 4 5 6 7 8 9 10 11 12
                               ab ac ad ba bc bd ca cb cd da db dc
                  1 (a to b)    1 0 0 0 0 0 0 0 0 0 0 0
                  2 (a to c)    0 1 1 0 0 1 0 0 0 0 0 0
                  3 (b to a)    0 0 0 1 0 1 1 0 0 1 0 0
            Links 4 (b to c)    0 0 0 0 1 0 0 0 0 0 0 0
                  5 (c to b)    0 0 0 0 0 0 1 1 0 1 1 0
                  6 (c to d)    0 0 1 0 0 1 0 0 1 0 0 0
                  7 (d to c)    0 0 0 0 0 0 0 0 0 1 1 1
Current Academic Approaches (2)
• Vardi (1996) assumes the O-D byte counts are Poisson
     – Maximum likelihood via the Expectation-maximization1 (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).
Current 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
Current Academic Approaches (2)
                                                        Cao et al. Cao et al.
                                                          (2000)     (2001)
                                                         ("Time-   ("Scalable
                   Feature                    Vardi    Varying…") Method...")
Scalability                                                          
                                                                     
Computational Complexity
Mathematical/Programming Complexity                                  
Robustness to model misspecification                                 
Usable w/ missing data                                               

 Exceeds requirements, highly desirable
 Meets requirements
 Does not meet requirements, less desirable
1 This   method can also be used with parallel computing
     Need for New Algorithm
• Academic methods are not feasible due to
  number of nodes in Client’s network
  (approximately 1000 nodes)
• Proposed algorithm is a heuristic; faster
              Proposed 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)
         Proposed 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
• Final adjustment if load originating at each node is greater
  than total load at node
       Sample NetHealth Data
• 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
     O-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
       Numerical Example with
        Proposed Algorithm
• Input Parameters:
   – Link Factor = 1.1
   – Number of Hops varied from 2 to 10
         Numerical 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)

                                         Setup: Analysis:
                                Link    CPU time CPU time
                 Problem      Error (%)   (min)    (min)
             Backbone Only        0.1%        0.1      1.1
             Full Network         0.6%       26.5    229.4
       O-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
Sample Map Output
          NIPRNet: Network
• “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,
• 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
• R.J. Vanderbei and J. Iannone, "An EM approach to OD matrix
  estimation," Technical Report, Princeton University, 1994
               Assignment Session 4
    For the following set of sites and traffic, design a minimum-cost
    network with reasonable performance:
         V           H              A          B           C         D         E
A              156        998   A                    250       250       400
B             1929       1537   B        250                   250       350       200
C             2112        542   C        250         250                 450       150
D             1526       1090   D        400         350       450
E             2727       1210   E                    200       150

             Sites                                 Traffic (kbps)
       Assignment Session 4 (2)
You have two types of links available for this design:

1.   Capacity 1.5 Mbps, cost $80 + $8/mile
2.   Capacity 64kbps, cost $10 + $1/mile

You may use multiple links to satisfy demand.

Note: What is the maximum utilization you will allow per

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