1

TCOM 541

Session 1

2

Agenda

• Overview

• Recap TCOM540 material needed for this course

3

Introduction

• Text: Wide Area Network Design by Robert S. Cahn, Publ. Morgan Kaufmann, ISBN 1-55860-458-8

• Other supplementary readings• Web site

http://teal.gmu.edu/ececourses/tcom540/TCOM540541.htm

4

Introduction (2)

• Approximate schedule for TCOM 541– Week 1 – Introduction and recap TCOM 540 material– Week 2 – Automated design tools, mesh network

design– Week 3 – Augmenting and merging networks– Week 4 – Buying services vs. buying equipment and

circuits– Weeks 5 and 6 – Practical vs. theoretical considerations– Week 7 – Review– Week 8 – TCOM 541 final

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Introduction (3)

• Evaluation weightings– Homework 25%– Term paper/project 25%– Finals 25%– Class Participation 25%

• Quizzes approximately every two weeks

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Network Optimization …

• Is generally not possible …– Conflicting objectives– Combinatorial explosion defeats exact solutions– Inadequate /inaccurate information– Rate of change, especially for data networks

• Generally have to settle for a “pretty good” design

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Conflicting Objectives

• Cost

• Performance

• Reliability

• Trade-offs are inevitable!

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540 Recap Topics

• Queuing and blocking

• Graph theory

• Minimum Spanning Trees

• Shortest Path Trees

• Tours

• Traffic Models

• Access and Backbone

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Voice Traffic Distribution and Measurement

• Traffic peaks around 11 am and 2 pm• Assume 20% of traffic in “busy hour”• (Voice) traffic is measured in Erlangs

– Erlangs = arrival rate/departure rate

– E.g., if calls arrive at 2 per minute and hold for 3 minutes, then:

• Arrival rate = 2 per minute

• Departure rate = 0.3333 per minute

• Traffic = 2/(0.3333) = 6 Erlangs

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Erlang Recursion

• Let B(E, n) = blocking when E Erlangs of traffic offered to n lines

• Then

B(E, n) = E*B(E, n-1)/(E*B(E, n-1) + n)

This is generally the easiest way to calculate blocking. In real life there is software to do this.

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Example Erlang Blocking Calculation

Assume 5 lines, 3 Erlangs of traffic. Then:

B(3,0) = 1

B(3,1) = 3*B(3,0)/(3*B(3,0)+1) = 3/4

B(3,2) = 3*B(3,1)/(3*B(3,1)+2) = 2.25/(2.25+2) = 0.5294

B(3,3) = 3*B(3,2)/(3*B(3,2)+3) = 1.5882/(1.5882+3) = 0.3461

B(3,4) = 3*B(3,3)/(3*B(3,3)+4) = 1.0383/(1.0383+4) = 0.2061

B(3,5) = 3*B(3,4)/(3*B(3,4)+5) = 0.6183/(0.6183+5) = 0.1101

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Queuing Theory (1)

• Needed to understand/calculate link delays

• Store-and-forward

• Service time = packet size/link speed

• Delay determined by packet size distribution and packet arrival distribution

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Queuing Theory (2)

• Service time is N/S, where:– N = number of bits/packet– S = link speed in bits/sec

• Assume interarrival and packet length PDFs are of form c*e (-c*x)

• Called an M/M/1 queue

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• Define = ratio of arrival rate () to service rate ()

• Average waiting time Tw = ( • Average service time Ts = 1/

• Average total time = Ts + Tw

= (1/

Queuing Theory (3)

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Set Theory

• A set is a collection of (mathematical) objects– E.g., {A, B, C}– E.g., {1,2,3, …, 99}

• sS means “s is a member of S”• s ~ S means “s is not a member of S”• T is a subset of S if every member of T is also a

member of S– It’s a proper subset if there is at least one member of S

that is not a member of T

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Set Theory (2)

• The set of all subsets of S is denoted as 2s

• If S has n members, then 2s has 2n members

• Union of two sets is the set of all their members

• Intersection of two sets is the set of common members

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Set Theory (3)

Intersection

Union

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Set Theory (4)

• Cartesian Product of two sets, S and T, is the set SxT – Elements are (s,t) where s S and t T

• The graph of a function f:S T is the subset of SxT that consists of

{(s,t) f(s) = t)}

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Graphs

• A graph consists of a set of vertices (or nodes) V and a set of edges E

A

D

I

ZC

B

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Some Definitions for Graphs

• Each edge connects two vertices (may be same – then it’s called a loop)

• Two edges are called parallel if they connect the same vertices

• A graph is simple if it has no loops or parallel edges

• The degree of a vertex is the number of edges it has

• Two nodes are adjacent if there is an edge that has them as endpoints

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Some Definitions for Graphs (2)

• A path between vertices v1 and vn is a set of edges (e1, e2, …, en) such that ei and ei+1 have a common endpoint, and v1 is an endpoint of e1 and vn is an endpoint of en

• A cycle is a path from a vertex to itself• A graph is connected if for any two nodes there is

a path between them

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A Small (But Not Simple) Graph

A

D

I

ZC

B

Parallel edges

LoopNode degree 3

Adjacent nodes

(DZ), (ZB), (BI), (ID) is a cycle.

This graph is connected

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Definitions (3)• A subgraph G* of a graph G with vertices V and edges E

is a pair (V*, E*) where – V* is a subset of V – E* is a subset of E– If an edge belongs to E* then both its endpoints must belong to

V*

• A component of a graph is a maximal connected subgraph• Two graphs G1, G2 are isomorphic if there is a 1-to-1

mapping f:V1 V2 such that (v1, v2) is a member of E1 iff (f(v1), f(v2)) is a member of E2

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Another Graph

A

D

I

ZC

N

K

M

J

e1

e2

e3

e4

e5

e6

e7

e8

( (D, Z, B, M, J), (e2, e8, e4)) is a subgraph

It is not a component.

The two right-hand components are isomorphic

X

Q

R H

e11

e99

Pe15

e16

e12B

e13

e9

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Definitions (4)

• A tree is a connected simple graph without cycles

• A star is a tree in which exactly one node has degree >1

• A chain is a tree in which no node has degree greater than 2

• Define N(G) = number of nodes in G

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Tree, Star, Chain

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V&H Coordinates• V&H coordinate grid covers U.S.

• Approx 10,000 x 10,000

• Distance between points (v1, h1) and (v2, h2) for tariff calculations is sometimes defined as:

• A simpler formulation is:

• Note V&H assumes earth is flat …

Dist = 1+int{[(dv2+9)/10+(dh2 +9 )/10]0.5}

Where dv = v1-v2 and dh = h1-h2

Dist = 1 + int[(dv2+dh2)/10]

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Weighted Graph

• A weighted graph is a graph G where each edge e has a weight w(e)– Denoted by (G, w)– Generally w(e) > 0– Weight of a subgraph G* is sum of weights of

edges in G*

• Real networks are weighted graphs– Weight may be cost, delay, or other parameter

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Minimum Spanning Tree

• A Minimum Spanning Tree (MST) is a connected subgraph with minimum weight

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Kruskal’s Algorithm for MST

Is G connected?yes no stop

Sort edgesin ascending

order ofweight

Mark each node as separate

component

Loop on edgesLet e be candidate edgeIf ends of e are in different components, accept e

Stop when number of edges = N(G) - 1

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Prim’s Algorithm for MSTStart with all nodesunconnected and

Label = infinity

Select rootnode

Scan neighbors, update Labels =min edge to tree

Add closest neighbor

(smallest Label)

Stop whenN(G) –1added

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Limitations of MSTs

• No redundancy– One link failure separates the network into two

disconnected components– Big problem for large networks

• May involve very long paths in large networks

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MSTs Do Not Scale

• Number of hops between nodes n1 and n2 is the number of edges in the path chosen by the routing algorithm

• Average number of hops is traffic-weighted

= (n1,n2traffic(n1,n2)*hops(n1,n2))/n1,n2traffic(n1,n2)

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Definitions

• For a weighted graph (G,w), and nodes n1 and n2, the shortest path P from n1 to n2 minimizes ePw(e)

• The shortest-path tree (SPT) rooted at node n1 is a tree T such that for any other node n2 the path from n1 to n2 is a shortest path

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Dijkstra’s Algorithm for Shortest-Path Trees

1. Mark each node unscanned, assign label infinity2. Set label of root to 0, and predecessor to self3. Loop through nodes

• Find node n with smallest label• Mark as scanned• Examine all adjacent nodes m, see if distance through n < label

• If so, update label, update predecessor(m) = n

Note that a link may drop out of the tree if a shorter route is found

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Characteristics of SPTs

• In a complete graph, SPT is a star*– High performance and reliability– But likely implies low link utilization, high

expense

* Unless triangle inequality does not hold

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Prim-Dijkstra Trees

• We play with the definition of the Label

• Prim’s Label

= minneighborsdist(node, neighbor)

• Dijkstra’s Label

= minneighbors[dist(root, neighbor) + dist(neighbor, node)]

• Prim-Dijkstra Label =

= minneighbors[*dist(root, neighbor) + dist(neighbor, node)]

• Now is a parameter that we choose, between 0 and 1

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Varying Alpha

0.000

0.200

0.400

0.600

0.800

1.000

1.200

0 0.5 1

Alpha

No

rmali

zed

Valu

e

Normalized Avg Hops

Normalized delay

Normalized Cost

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Tours

• A tree design may be unreliable

• A tour adds one link to significantly increase reliability

• A tour of a set of vertices (v1, v2, …, vn) is a set of n edges such that each vertex has degree 2 and the graph is connected

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Tours (2)

• Leads to the (in)famous Traveling Salesman Problem (TSP)– Given a set of vertices (v1, v2, …, vn) and a

distance function d(vi,vj) between vertices, find the tour T(vti) such that d(vti,ti+1) is minimized

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Reliability of a Tree

• Reliability = probability that functioning nodes are connected by working links

• For a tree, reliability = (1-p)n-1, where– p = probability of a link failing– n = number of nodes

• P(failure) = 1- reliability = 1 - (1-p)n-1

(n-1)*p

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Reliability of a Ring

• A ring can tolerate one failure

• For a ring,

P(failure) = 1- (1-p)n – n*p*(1-p)n-1

0.5*n*(n-1)*p2 if p is small

X

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A Simple Algorithm for Building a Tour

• Denote a root node, set current node = root

• Loop through nodes– Find closest node (not in tour) to current node– Add an edge to it– Reset current node to be this node just added

• Create an edge between last node and root

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Creditable Solutions and Creditability Tests

• A solution is creditable if it is a local optimum– I.e., it is not creditable if, by some method, we

can manipulate the solution to a better one

• Cahn uses a crossing test to determine creditability of the simple tour-building algorithm– It does not do well ….

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A Better Tour-Building Algorithm

• Look for closest neighbor to any node in the partial tour (not just the last one added)

• Insert between two adjacent nodes in tour in “best” place– Minimum increase in partial tour length

• Also “farthest neighbor” heuristic – Avoids stranding distant nodes

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A Difficulty

• TSP tours do not scale– Similar to trees in this respect– Average number of hops increases O(n)

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2-Connectivity

• A vertex v of a connected graph G = (V, E) is an articulation point if removing the vertex and all attached edges disconnects the graph

• If a connected graph has no articulation points, it is said to be 2-connected

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Connecting 2-Connected Graphs

• Suppose G1 = (V1, E1) and G2 = (V2, E2) are two disjoint 2-connected graphs. Take v1 and v2 from G1 and v3 and v4 from G2 and add the edges (v1,v3) and (v2,v4). The resulting graph is 2-connected

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Connecting 2-Connected Graphs

• Suppose G1 = (V1, E1) and G2 = (V2, E2) are two disjoint 2-connected graphs. Take v1 and v2 from G1 and v3 and v4 from G2 and add the edges (v1,v3) and (v2,v4). The resulting graph is 2-connected

• Roughly, if you connect 2 pairs of vertices from two 2-connected graphs, the resulting graph is 2-connected

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Heuristic Based on Partitioning

• Divide set of nodes into multiple “clusters”

• Use nearest-neighbor algorithm to build TSP tour on each cluster

• Connect clusters, ensuring no connectors have a common vertex

• Resulting graph is 2-connected

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Traffic Tables

• Matrix vs. List

From\To A B … Z

A TAA TAB TAZ

B TBA TBB TBZ

…

Z TZA TZB TZZ

From To Traf.

A A TAA

A B TAB

… … …

Z Z TZZ

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Traffic Models

• Topics:– Uniform– Random– Population power– Modified population power– Normalized model– Asymmetric model

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Uniform

• Traffic from site a to site b is

T(a,b) = C

• Not realistic for most situations

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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

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Population Power

• If the sites a and b have populations Pa and Pb, and are distance Da,b apart, then

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

where , , are suitably-chosen constants

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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) = *(pa*pb+ poff)/(Da,b+ Doff )

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Normalization

• Choose 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 traffic in = traffic out for this to be

possible!• Algorithmic iterative approach can be used

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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)

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Asymmetric 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) = *M(La, Lb)*(pa*pb+ poff)/(Da,b+Doff )

0 13 0

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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”

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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

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Tariff Structures (2)

• Additional fees may include– Initiation charge– Cancellation charge– Features charges– Access charges

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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

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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

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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

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Piecewise-Linear Charges

• Each segment is linear

Cost/month

Distance

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Step Function

Distance

Cost/month

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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)

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Access and Backbone

• To over-simplify, access lines provide local connectivity, backbone provides long-distance transport

• Balance between access and backbone costs can vary widely

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Access and Backbone (2)

Access

Backbone

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Access and Backbone (3)

IXC1 POP

IXC2 POP

LECCentralOffice

LocalLoops

IXC1 Backbone

IXC2 BackboneAccess lines to IXCs

To otherLEC COs

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Esau-Williams Algorithm

• Esau-Williams creates a Capacitated Minimum Spanning Tree (CMST)

• Given a central node N0 and a set of other nodes (N1, N2, … Nn), and a set of weights (w1, …, wn) for each node, the capacity of a link W, and a cost matrix Cost(i,j) find a set of trees T1, …, Tk such that each Ni belongs to exactly one Tj and each Tj contains N0 and

iTj wi < W

trees l Links Cost(end1l, end2l) is a minimum

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Esau-Williams Algorithm (2)

• Central concept is tradeoff function• Build “good” trees• Each tree starts off as one node

– Component (graph theory meaning) Comp(Ni)

• Tradeoff function is Tr() where

Tr(Ni) = minj[Cost(Ni,Nj)] –Cost (Comp(Ni),N0)

• Computes cost of linking to neighbor vs. cost of going to center

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Esau-Williams Algorithm (3)

• Negative value of Tr means it is preferable to link to neighbor tree rather than running a link to the center

• Must check that the design is feasible – i.e., does not exceed link capacity:

W(Comp(Ni)) +W(Comp(Nj)) < W– Algorithm limitation – often desirable to

increase link capacities in real life

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Multiple Link Speeds

• In real problems, almost always have a variety of link speeds to choose from– DS0 @ 64kbps– N x DS0– T1 @1.5 Mbps– T3 @ 45 Mbps– Etc.

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Multiple Link Speeds (2)

• Intuitively, we’d like the access tree to use higher speeds closer to the root, and lower speeds out towards the edges

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Predecessor Function

• A tree T rooted at node Root can be represented uniquely by a predecessor function pred:V V on the set of vertices:– pred(Root) = Root– No other node is its own predecessor– For any node N there is an n>0 such that

predn(N) = Root

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Ancestors

• Given a tree T and the associated predecessor function, the ancestors of N are all the nodes N* where

predn(N*) = N for some n > 0

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Multispeed CMST Definition

• Given:– A set of nodes N0, N1, …, Nn

– A set of weights (w1, …, wn) for each node

– A set of link types L1, L2, …, Lm

– Capacities W1, W2, …, Wm

– A cost matrix C(i,j,k) for the cost of link type Lk between Ni and Nj

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Multispeed CMST Definition (2)

• Then the multispeed CMST problem is to find the tree rooted at N0 with link assignments such that

ancestors(N) w(i) < WLink(N, pred(N))

And Linksc(end1L, end2L, typeL) is minimized

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MSLA Algorithm for Multispeed CMSTs

1. Assign each node n the smallest link l to connect it to root. Compute spare_capacity(n) = Wl – wn

2. Create tradeoff heap for n (similar to E-W) – tradeoffs represent savings by connecting site n to site i rather than to the root

Tradeoffn(i) = c(n,i,L) + Upgrade (i, wn) – c(n,0,L)

The function Upgrade() computes the cost of adding wn units to the links that connect i and 0 by following back the predecessors

3. Add edges as long as tradeoffs are less than or equal to 0

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Another Definition

• A Forest, F = (V,E) is a simple graph without cycles– Note it doesn’t say connected

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Multicenter Local Access (MCLA) Problem

• Given– A set of backbone sites (B0, …, Bm) = B

– A set of access nodes (N1, …, Nn) = N

– A set of weights (w1, …, wn) for each access node

– A cost matrix Cost(i,j) giving the costs between each backbone/access pair of sites

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Multicenter Local Access (MCLA) Problem (2)

• MCLA is to find a set of trees T1, …, Tk such that– Exactly one backbone site belongs to each tree

– Ni Tj wi < W

– Trees L LinksCost(end L1, endL2) is minimum

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Multicenter Esau-Williams (MCEW)

• Developed by Kerschenbaum and Chou (1974)

• Changes the tradeoff function

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MCEW (2)

• EW Tradeoff function is Tr() where

Tr(Ni) = minj[Cost(Ni,Nj)] –Cost (Comp(Ni),N0)

• Computes cost of linking to neighbor vs. cost of going to center

• MCEW Tradeoff function isTr(Ni) = minjCost(Ni,Nj) – dist(Comp(Ni), Center(Nj))

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MCEW (3)

• Initially, set Center(Ni) to be closest center

• If merge Ni with Nj, update Center(Ni) = Center(Nj)

• Note: Tradeoff function merges cost and distance functions

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MCEW (4)

• MCEW produces more creditable results than NNEW– Produces a better solution much more often– But cost advantage is surprisingly small

• < 1% for large numbers of sites

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Practical Issues

• Real problems often involve additional, sometimes quirky, constraints, such as– Limit on number of nodes in an access tree– Limit on number of hops– Limit on number of connections at a site– Unreliable links or sites

90

Assignment

• Read Cahn Chapter 8 (and preceding chapters if you didn’t attend TCOM540)