Tunable Survivable Spanning Trees

Jose Yallouz, Ori Rottenstreich and Ariel Orda

Department of Electrical EngineeringTechnion, Israel Institute of Technology

Proceedings of ACM Sigmetrics 2014

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Quality of Service (QoS)

• The Internet was developed as a Best Effort network.

• What is Quality of Service (QoS)?• “The collective effect of service performance which determines

the degree of a user satisfaction of the service.” (ITU)

• QoS common criteria:• Delay• Jitter• Bandwidth

• QoS metric classification:• Bottleneck• Additive

• Packet loss• Out of order• Survivability

Introduction

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Survivability

• Survivability – The capability of the network to maintain service continuity in the presence of failures.

• Recovery Schemes• Restoration is a post-failure operational process, i.e. a backup

solution is calculated only after the failure occurrence. • Typical recovery times range from seconds to minutes.

• Protection is a pre-failure planning process, i.e. a backup solution is calculated in advance before the failure occurrence. • Typical recovery times are in the range of milliseconds.

• According to many standards, a single failure recovery operation must be performed within 50 ms.

• These two techniques are often implemented together.• “First Failure Protection, Next Failures Restoration”

Introduction

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Single Failure Model

• Single Failure Model: assumes that at most one failure can be handled in the network

• Under the single link failure model, only the links that are common to all paths can fail the connection.

common link

Introduction

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• Broadcasting - a method of transferring a message to all recipients simultaneously.

Broadcasting Methods

Spanning-Tree BroadcastFlooding Broadcast

Motivation

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Tunable Survivability

• Full survivability - (100%) protection against network single failures. • Establishment of link-disjoint spanning trees. • This scheme is often too restrictive.

=0.01=0.99

• Tunable survivability allows any desired degree of survivability in the range 0% to 100%.

Motivation

common link

-7-

𝑇 2

𝑇 1

Model Formulation• Network represented by an undirected graph • : bandwidth of link e • : independent failure probability of link e• Given a network , a k-survivable spanning connection is a tuple of k

spanning trees (not necessarily disjoint).

2-survivable spanning connection

Formulation

𝑝𝑒=0 .01

𝑏𝑒=5

𝑝𝑒=0 .01

𝑏𝑒=5𝑏𝑒 =10

𝑏𝑒 =2 0

𝑏𝑒 =10

𝑏 𝑒=10

𝑝 𝑒=0 .01

𝑝 𝑒=0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0.01

a

b c d

e

-8-

Model Formulation

• The survivability level of is defined as:• The probability that all common links are operational• )• 1 ()

𝑆 (𝑇1 ,𝑇2 )=1−0 .01=(0 .99)

Formulation

𝑝𝑒=0 .01

𝑏𝑒=5

𝑝𝑒=0 .01

𝑏𝑒=5𝑏𝑒 =10

𝑏𝑒 =2 0

𝑏𝑒 =10

𝑏 𝑒=10

𝑝 𝑒=0 .01

𝑝 𝑒=0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0.01

a

b c d

e

𝑇 2

𝑇 1

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Model Formulation

• The bandwidth of is defined:• The bandwidth of the bottleneck link across all spanning trees.

𝑆 (𝑇1 ,𝑇2 )=0 .99𝐵 (𝑇 1,𝑇 2 )=2

Formulation

𝑝𝑒=0 .01

𝑏𝑒=5

𝑝𝑒=0 .01

𝑏𝑒=5𝑏𝑒 =10

𝑏𝑒 =2 0

𝑏𝑒 =10

𝑏 𝑒=10

𝑝 𝑒=0 .01

𝑝 𝑒=0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0.01

a

b c d

e

𝑇 2

𝑇 1

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Optimization Problems

• Constrained Bandwidth Max-Survivability (CBMS) Problem:Find a k-survivable spanning connection such that:

• Constrained Survivability Max-Bandwidth (CSMB) Problem:Find a k-survivable spanning connection such that:

Formulation

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𝑇 1

𝑇 4𝑇 2

𝑇 3

Survivability

Bandwidth

𝑝𝑒=0 .01 𝑝𝑒=0 .01

Example

𝑏𝑒=50 𝑏𝑒=50

𝑏𝑒 =100

0

00

𝑏𝑒 =100

𝑏 𝑒=100

𝑝 𝑒=0 .01

𝑝 𝑒=0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0.01

a

b c d

e

𝑏𝑒=1

Characterization

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How Many Spanning Trees?• What is the maximum level of survivability which can be achieved

for a given a network ?• A bridge is a link whose deletion increases the number of connected

components.• is the set of all bridges in the network.• Theorem: The maximum level of survivability of satisfies .

Characterization

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How Many Spanning Trees?• How Many Spanning Trees are necessary in order to achieve this

maximum level of survivability?• Theorem: Let , the number of sufficient spanning trees which satisfies

maximum level of survivability is bounded by

⌈ ¿ �̌�∨ ¿|𝐸|−|𝑉|+1

⌉=⌈10

10−5+1⌉=2¿

(b) A clique demonstrating a tight lower bound

example

¿𝑉∨¿5

(a) A cycle demonstrating an tight upper bound

example

Characterization

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Algorithmic Scheme

• Constrained Bandwidth Max-Survivability (CBMS) Problem:Find a k-survivable spanning connection such that:

• Minimum Cost Edge Disjoint Spanning Tree Problem:Given an undirected weighted network G(V,E) . Find a k Edge Disjoint Spanning Trees of minimal total cost.

•Polynomial solution by Roskind and Tarjan – “A note on finding minimum-cost edge-disjoint spanning trees”, 1985.

Optimization

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Algorithmic Solution

𝑝𝑒 =0 .01

𝑝𝑒=0 .01𝑏𝑒=5

𝑝𝑒=0 .01𝑏𝑒=5

𝑏𝑒 =10

𝑏𝑒 =2 0

𝑏𝑒 =10

𝑏 𝑒=10

𝑝 𝑒=0 .01

𝑝 𝑒=0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0.01

a

b c d

e

𝑏𝑒 =1

• Find a 2-survivable spanning connection such that:

Optimization

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Algorithmic Solution

𝑝𝑒 =0 .01

𝑝𝑒=0 .01𝑏𝑒=5

𝑝𝑒=0 .01𝑏𝑒=5

𝑏𝑒 =10

𝑏𝑒 =2 0

𝑏𝑒 =10

𝑏 𝑒=10

𝑝 𝑒=0 .01

𝑝 𝑒=0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0.01

a

b c d

e

𝑏𝑒 =1

• Each link with a bandwidth

• Each link with a bandwidth :

𝒃 e ,𝒑 eDiscard the link

𝒘 𝒆𝟏=− 𝒍𝒏(𝟏−𝒑e)

𝒘 𝒆𝒌=𝟎

𝒘 𝒆𝟐=𝟎

Original Network Auxiliary Network

𝑤 𝑒=−𝑙𝑛0 .99

𝑤 𝑒=0

𝑤𝑒 =−𝑙𝑛0 .99

𝑤𝑒 =0

𝑤𝑒=−𝑙𝑛0 .99

𝑤𝑒=0

a

b c d

e

𝑤𝑒=−𝑙𝑛0 .99

𝑤𝑒=0

𝑤𝑒 =−𝑙𝑛0.99

𝑤𝑒 =0

𝑤 𝑒=−𝑙𝑛0 .99

𝑤 𝑒=0

𝑤𝑒 =−𝑙𝑛0 .99

𝑤𝑒 =0

Optimization

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𝑝𝑒 =0 .01

𝑝𝑒=0 .01𝑏𝑒=5

𝑝𝑒=0 .01𝑏𝑒=5

𝑏𝑒 =10

𝑏𝑒 =2 0

𝑏𝑒 =10

𝑏 𝑒=10

𝑝 𝑒=0 .01

𝑝 𝑒=0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0 .01

𝑝𝑒 =0.01

a

b c d

e

𝑏𝑒 =1

Algorithmic Solution

• In the Auxiliary Network, find 2 Edge Disjoint Spanning Trees utilizing the minimum cost edge disjoint spanning tree algorithm.

Original Network Auxiliary Network

𝑤 𝑒=−𝑙𝑛0 .99

𝑤 𝑒=0

𝑤𝑒 =−𝑙𝑛0 .99

𝑤𝑒 =0

𝑤𝑒=−𝑙𝑛0 .99

𝑤𝑒=0

a

b c d

e

𝑤𝑒=−𝑙𝑛0 .99

𝑤𝑒=0

𝑤𝑒 =−𝑙𝑛0.99

𝑤𝑒 =0

𝑤 𝑒=−𝑙𝑛0 .99

𝑤 𝑒=0

𝑤𝑒 =−𝑙𝑛0 .99

𝑤𝑒 =0

Optimization

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Maximum survivability level ratio versus the number of spanning trees k for different bandwidth requirements

SimulationSimulation

• - maximum survivability level that can be obtained by a -survivable spanning connection with a bandwidth requirement of

• - maximum survivability level of the network with a bandwidth requirement of

𝑘

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Bandwidth ratio versus the survivability level requirement

Simulation

X12 times improvement

𝑆0

Simulation

• - maximum bandwidth of a -survivable spanning connection with a survivability level of at least

• - maximum bandwidth of a fully disjoint spanning connection

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Conclusion

• The establishment of a comprehensive methodology for efficiently providing tunable survivability.• Ron Banner and Ariel Orda. “The power of tuning: A novel approach

for the efficient design of survivable networks”. In IEEE/ACM Trans. Networking, 2007.

• Jose Yallouz and Ariel Orda. “Tunable QoS-aware network survivability”. In IEEE Infocom, 2013.

• Jose Yallouz, Ori Rottenstreich and Ariel Orda. “Tunable Survivable Spanning Trees”. In ACM Sigmetrics, 2014.

Conclusion

Introduction

Characterization

Formulation

Simulation

Optimization

Question?

Motivation

Thank You!