Topology Design for Service Overlay Networks with Bandwidth

Guarantees

Sibelius Vieira* Jorg Liebeherr**

*Department of Computer ScienceCatholic University of Goias, Brazil

**Department Computer ScienceUniversity of Virginia

Service Overlay Networks

Provisioning of end-to-end QoS across multiple autonomous systems (ASs) requires a level a cooperation that is difficult to achieve in the current architecture.

Service Overlay Networks can avoid these difficulties

We define a Provider Network as a value-added overlay network that supports end-to-end bandwidth guarantees to a collection of subscribers

Problem studied in this paper: Building a topology for a provider network

Endsystems and Provider Nodes

Internet

Endsystems ProviderNode

Sprint MCI Level 3

Transport LinkAccess Link

ProviderNode

UUNet

Level 3

MCI

Internet

Provider network = Provider nodes + Endsystems

Provider nodes and endsystems gain access to the Internet through ISPs

Provider network buys bandwidth from ISPs and sells bandwidth guarantees to endsystems

Provider nodes, endsystems and ISPs

ProviderNode 1

ProviderNode 2

ProviderNode 3

ProviderNode 4

ISP A ISP B ISP C

Endsystem 1 Endsystem 4Endsystem 3 Endsystem 6Endsystem 2 Endsystem 5

Two provider nodes and/or endsystems can establish a link between themselves if they have a common ISP

Access link

Transport link

Topology Design Problem

ProviderNode 1

ProviderNode 2

ProviderNode 3

ProviderNode 4

ISP A ISP B ISP C

Endsystem 1 Endsystem 4Endsystem 3 Endsystem 6Endsystem 2 Endsystem 5

Given the connectivity of endsystems, provider nodes, and ISPs Given the bandwidth requests between endsystems

How to construct a “good” topology ?

Solution to the topology of the provider network

ProviderNode 1

ProviderNode 2

ProviderNode 3

ProviderNode 4

ISP A ISP B ISP C

Endsystem 1 Endsystem 4Endsystem 3 Endsystem 6Endsystem 2 Endsystem 5

For each endsystem, select an ISP to connect endsystem to a provider node

Connect provider nodes, so that there are end-to-end paths for traffic between endsystems

Resulting topology:

ProviderNode 2

ProviderNode 3

ProviderNode 1

ProviderNode 4

Endsystem 3 Endsystem 2

Endsystem 1

Endsystem 4

ISP B

Endsystem 5

Endsystem 6

ISP B

ISP A

ISP A

ISP C

ISP CISP A

ISP CISP B

ISP B

Formal problem statement M Number of endsystems N Number of provider nodes

ESi Endsystem i PNj Provider node j αij Cost of reserving one Mbps from ESi to PNj, through

the ISP which provides the minimal cost (access cost) lij Cost of reserving one Mbps between PNi to PNj

through the ISP that provides the minimal cost of connecting the two provider nodes (transport cost)

ωij Required bandwidth from ESi to ESj Ωj Total bandwidth for traffic generated at ESj (Ωj = j

ωij).

Formal problem statement Each endsystem must be assigned to one provider node via

an access link Provider nodes must be connected by transport links Cost of a link is weighted by the traffic sent over the link

Total cost of network = Costs of the access links + transport links

Goal: Minimize total cost of network

Irrespective of the amount of traffic, traffic between two provider nodes is sent at lowest cost if it is sent on the least-cost path between the two provider nodes

Let rnm denote the least-cost path between PNn to PNm

Cost of the least-cost path per unit of reserved bandwidth from PNn to PNm is bnm = (ij) rnm lij .

Optimization problem

Let yij be a 0-1 decision variable that indicates if ESi is assigned to PNj

Solving the topology design problem requires:

Minimize

i k Ωi αik yik + i j kl yij ykl ωik bjl + jl Ωj αjl yjl

subject to j yij = 1 for i = 1,..,M

Ingress access costs

(from endsystems to

provider nodes)

Transport cost Egress access costs

(from provider nodes to

endsystems)

Each endsystem is

connected to one provider

node

Complexity

Minimize

i k Ωi αik yik + i j kl yij ykl ωik bjl + jl Ωj αjl yjl

subject to j yij = 1 for i = 1,..,M

Bad news: The optimization is a variant of the NP-hard quadratic assignment problem (QAP)

Good news: In some special cases, the problem can be much simplified Heuristics optimizations (e.g., simulated annealing) seem to

work well for this problem

The optimization problem can be expressed as an equivalent matrix-combination problem

Define: u(i) = j, iff yij = 1.

Then: u = (u(1),u(2), ..,u(M)) is assignment of endsystems to provider nodes.

We can write optimization as:

Minimize Z(u) = i j ωij (αiu(i) + bu(i)u(j) + αju(i))

Side conditions of the original problem are implicitly given via the definition of the u(i)´s.

Finding simpler solutions Special case

Choose v(i) such that αiv(i) = minj{αij}.

Consider the following conditions:(C1) bij ≤ bik + bkj for all i,j,k ≤ N.(C2) αij ≥ αiv(i) + bv(i)j for all i ≤ M and j, v(i) ≤ N.

Note: (C1) always holds by construction of the least-cost paths, and (C2) is satisfied if the cost structure is such that access costs outweigh transport costs.

Lemma 1. Under (C1) and (C2), Z(u) is minimized for the mapping u(i)=v(i)

Finding simpler solutions: Special case

Finding simpler solutions: Heuristic solutions

Without (C2), exact solutions can be obtained only for problems up to 30 endsystems and provider nodes

Here, heuristic optimizations are necessary

Simulated annealing has been shown to provide good results for QAP type problems.

See paper for details of the simulated annealing algorithm

Finding simpler solutions: Greedy Algorithm

Greedy assignment: assign endsystems to provider nodes with lowest access cost, i.e.,

yiv(i)=1 iff. αiv(i) = minj{αij}

When (C2) holds, greedy assignment yields the optimal solution The algorithm performs well when access costs dominate

transport costs

Finding simpler solutions: Clustering Cluster endsystems into groups (regions) and assign complete

regions to a provider node

Rules for clustering: Endsystems that are geographically close are likely to be assigned to the

same region Endsystems with higher traffic load are given more consideration when

regions are being formed

Use the k-means clustering algorithm to assign endsystems into regions: Input – M endsystems with position (ri,si) and traffic load

Ωi of each endsystem ESi and number of desired regions, R. Output – R cluster centers (centroids) and assignment of

each endsystem to each centroid.

Clustering Algorithm for Endsystems If Rk is the set of endsystems assigned to the kth centroid, the centroid

position is given by: rk = i: ESi є Rk ri. Ωi / i: ESi є Rk Ωi

sk = i: ESi є Sk si. Ωi / i: ESi є Sk Ωi

After establishing the new centroid position, re-associate each endsystem with a region by reassigning each endsystem to the closest centroid, until the algorithm converges.

Endsystem Centroid Region

Region 1

Region 2 Region 3

(c)(a) (b)

Numerical Evaluation Questions

How well do the heuristic algorithms perform? How does cost change with the number of provider nodes? What is the impact of the clustering algorithm?

Evaluation with random graphs Connectivity of provider nodes is determined by random

graph (using the GT-ITM, ‘Pure Random’ model) Each endsystem can access a randomly subset of pα·100%

of provider nodes Access costs = Uniform[5,50] Transport costs = Uniform[5,50] Traffic matrix = Uniform[10,20] Mbps

Evaluation of Simulated Annealing

Comparson with optimum solution for a small network (M = 9, N = 9)

Repetition factor (Repmax) controls the number of solutions evaluated by simulated annealing

Conclusion: Simulated annealing seems to work well

Repetition factor

(Repmax)

Average deviation

from minimum

(%)

Number of optimal

solutions found

(from 100)

10 6.59 % 1

20 4.44% 3

30 1.41% 4

40 0.02% 7

50 0.02% 9

Evaluation of Simulated Annealing

Enforce condition (C2) optimum solution can be computed for large networks

Here: Simulated annealing always gets close to optimum solution

Set: M = N10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600

700

800

900

1000

Number of Provider Nodes

Rep

etiti

on F

acto

r

Value of “Repetition Factor” (Repmax) needed to get simualted annealing within 1% of optimal solution

Evaluation of Heuristic Algorithms General network (i.e.,do not assume (C2)) Number of endsystems and provider nodes: 10 to 100 Prob. of transport link between provider nodes: P = 0.1, 0.5,

0.9.

Comparison of: simulated annealing greedy algorithm random assignment

Evaluation of Heuristic Algorithms

10 20 30 40 50 60 70 80 90 1000.5

1

1.5

2

2.5

3

Number of Provider Nodes

Rel

ativ

e T

opol

ogy

Cos

t

Simulated AnnealingRandomGreedy

Plots show cost of network relative to “Greedy algorithm”

10 20 30 40 50 60 70 80 90 1000.5

1

1.5

2

2.5

3

Number of Provider Nodes

Rel

ativ

e T

opol

ogy

Cos

t

Simulated AnnealingRandomGreedy

P = 0.1 P = 0.5

Impact of the Number of Provider Nodes

Network with M = 100 endsystems and N= 10-100 provider nodes Solution method: Simulated annealing Costs normalized to network with N=10

10 20 30 40 50 60 70 80 90 1000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Number of Provider Nodes

Rel

ativ

e T

opol

ogy

Cos

t

P = 0.1P = 0.5P = 0.9

10 20 30 40 50 60 70 80 90 1000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Number of Provider Nodes

Rel

ativ

e T

opol

ogy

Cos

t

P = 0.1P = 0.5P = 0.9

p = 0.9 p = 0.5

Impact of Clustering

Network of M=100 endsystems and N= 10 provider nodes. Number of regions is 10 – 100 Solution method: Simulated annealing

10 20 30 40 50 60 70 80 90 1000.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

Number of Regions

Rel

ativ

e T

opol

ogy

Cos

t

Conclusions

Formaluated network topology design problem for a service overlay network with QoS guarantees

We showed that the general problem is NP-hard But when the underlying network satisfies certain

conditions, the problem has only linear complexit Developed and evaluated several heuristic methods

Caveat: Different cost structure may give different results and may require a different solution approach