TCOM 540 Session 4 Agenda • 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 Uniform • Traffic from site a to site b is T(a,b) = C • Not realistic for most situations Random • 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 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 Normalization • 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 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 )g 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 Cost/month Distance • Each segment is linear Step Function Cost/month Distance Cost 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) Case Study of Traffic Generation Outline • 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 Statement • 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 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=Ax 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 Current 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 n L, identifiability of the parameters is a problem Current Academic Approaches a c 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 1 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 achieved • Final adjustment if load originating at each node is greater than total load at node Sample 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 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) Data 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 Client CONUSCONUS References • “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 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 link?
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