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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
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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
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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