Topology Design for Service Overlay Networks with Bandwidth Guarantees

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Topology Design for Service Overlay Networks with
Bandwidth Guarantees

• Sibelius Vieira
• Jorg Liebeherr
• Department of Computer ScienceCatholic
  University of Goias, Brazil
• Department Computer Science
• University of Virginia

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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

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Endsystems and Provider Nodes

• 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

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Provider nodes, endsystems and ISPs

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

Transport link
Access link
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Topology Design Problem

• Given the connectivity of endsystems, provider
  nodes, and ISPs
• Given the bandwidth requests between endsystems
• How to construct a good topology ?

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Solution to the topology of the provider network

• 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

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Resulting topology
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Formal problem statement

• M Number of endsystems
• N Number of provider nodes
• ESi Endsystem i
• PNj Provider node j
• aij 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)
• ?ij Required bandwidth from ESi to ESj
• Oj Total bandwidth for traffic generated at ESj
  (Oj ?j ?ij).

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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

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• 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
  .

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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 Oi aik yik ?i ?j ?k?l yij ykl
  ?ik bjl ?j?l Oj ajl yjl
• subject to ?j yij 1 for i
  1,..,M

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Complexity

• Minimize
• ?i ?k Oi aik yik ?i ?j ?k?l yij ykl
  ?ik bjl ?j?l Oj ajl 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

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Finding simpler solutions Special case

• 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 ?ij (aiu(i) bu(i)u(j)
  aju(i))
• Side conditions of the original problem are
  implicitly given via the definition of the u(i)s.

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Finding simpler solutions Special case

• Choose v(i) such that aiv(i) minjaij.
• Consider the following conditions
• (C1) bij bik bkj for all i,j,k N.
• (C2) aij aiv(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)

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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

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Finding simpler solutions Greedy Algorithm

• Greedy assignment assign endsystems to provider
  nodes with lowest access cost, i.e.,
• yiv(i)1 iff. aiv(i) minjaij
• When (C2) holds, greedy assignment yields the
  optimal solution
• The algorithm performs well when access costs
  dominate transport costs

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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 Oi of each endsystem ESi and number
  of desired regions, R.
• Output R cluster centers (centroids) and
  assignment of each endsystem to each centroid.

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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 ? Rk ri. Oi / ?i ESi ? Rk Oi
• sk ?i ESi ? Sk si. Oi / ?i ESi ? Sk Oi
• 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.

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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
  pa100 of provider nodes
• Access costs Uniform5,50
• Transport costs Uniform5,50
• Traffic matrix Uniform10,20 Mbps

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Evaluation of Simulated Annealing
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

• 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

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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 N

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

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Evaluation of Heuristic Algorithms

• Plots show cost of network relative to Greedy
  algorithm

P 0.1
P 0.5
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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 N10

pa 0.5
pa 0.9
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Impact of Clustering

• Network of M100 endsystems and N 10 provider
  nodes.
• Number of regions is 10 100
• Solution method Simulated annealing

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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