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Tunable Survivable Spanning Trees. Jose Yallouz , Ori Rottenstreich and Ariel Orda Department of Electrical Engineering Technion , Israel Institute of Technology Proceedings of ACM Sigmetrics 2014. Quality of Service ( QoS ). Introduction. - PowerPoint PPT Presentation
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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
-2-
Quality of Service (QoS)
โข The Internet was developed as a Best Effort network.
โข What is Quality of Service (QoS)?โข โThe collective effect of service performance which determines
the degree of a user satisfaction of the service.โ (ITU)
โข QoS common criteria:โข Delayโข Jitterโข Bandwidth
โข QoS metric classification:โข Bottleneckโข Additive
โข Packet lossโข Out of orderโข Survivability
Introduction
-3-
Survivability
โข Survivability โ The capability of the network to maintain service continuity in the presence of failures.
โข Recovery Schemesโข Restoration is a post-failure operational process, i.e. a backup
solution is calculated only after the failure occurrence. โข Typical recovery times range from seconds to minutes.
โข Protection is a pre-failure planning process, i.e. a backup solution is calculated in advance before the failure occurrence. โข Typical recovery times are in the range of milliseconds.
โข According to many standards, a single failure recovery operation must be performed within 50 ms.
โข These two techniques are often implemented together.โข โFirst Failure Protection, Next Failures Restorationโ
Introduction
-4-
Single Failure Model
โข Single Failure Model: assumes that at most one failure can be handled in the network
โข Under the single link failure model, only the links that are common to all paths can fail the connection.
common link
Introduction
-5-
โข Broadcasting - a method of transferring a message to all recipients simultaneously.
Broadcasting Methods
Spanning-Tree BroadcastFlooding Broadcast
Motivation
-6-
Tunable Survivability
โข Full survivability - (100%) protection against network single failures. โข Establishment of link-disjoint spanning trees. โข This scheme is often too restrictive.
=0.01=0.99
โข Tunable survivability allows any desired degree of survivability in the range 0% to 100%.
Motivation
common link
-7-
๐ 2
๐ 1
Model Formulationโข Network represented by an undirected graph โข : bandwidth of link e โข : independent failure probability of link eโข Given a network , a k-survivable spanning connection is a tuple of k
spanning trees (not necessarily disjoint).
2-survivable spanning connection
Formulation
๐๐=0 .01
๐๐=5
๐๐=0 .01
๐๐=5๐๐ =10
๐๐ =2 0
๐๐ =10
๐ ๐=10
๐ ๐=0 .01
๐ ๐=0 .01
๐๐ =0 .01
๐๐ =0 .01
๐๐ =0.01
a
b c d
e
-8-
Model Formulation
โข The survivability level of is defined as:โข The probability that all common links are operationalโข )โข 1 ()
๐ (๐1 ,๐2 )=1โ0 .01=(0 .99)
Formulation
๐๐=0 .01
๐๐=5
๐๐=0 .01
๐๐=5๐๐ =10
๐๐ =2 0
๐๐ =10
๐ ๐=10
๐ ๐=0 .01
๐ ๐=0 .01
๐๐ =0 .01
๐๐ =0 .01
๐๐ =0.01
a
b c d
e
๐ 2
๐ 1
-9-
Model Formulation
โข The bandwidth of is defined:โข The bandwidth of the bottleneck link across all spanning trees.
๐ (๐1 ,๐2 )=0 .99๐ต (๐ 1,๐ 2 )=2
Formulation
๐๐=0 .01
๐๐=5
๐๐=0 .01
๐๐=5๐๐ =10
๐๐ =2 0
๐๐ =10
๐ ๐=10
๐ ๐=0 .01
๐ ๐=0 .01
๐๐ =0 .01
๐๐ =0 .01
๐๐ =0.01
a
b c d
e
๐ 2
๐ 1
-10-
Optimization Problems
โข Constrained Bandwidth Max-Survivability (CBMS) Problem:Find a k-survivable spanning connection such that:
โข Constrained Survivability Max-Bandwidth (CSMB) Problem:Find a k-survivable spanning connection such that:
Formulation
-11-
๐ 1
๐ 4๐ 2
๐ 3
Survivability
Bandwidth
๐๐=0 .01 ๐๐=0 .01
Example
๐๐=50 ๐๐=50
๐๐ =100
0
00
๐๐ =100
๐ ๐=100
๐ ๐=0 .01
๐ ๐=0 .01
๐๐ =0 .01
๐๐ =0 .01
๐๐ =0.01
a
b c d
e
๐๐=1
Characterization
-12-
How Many Spanning Trees?โข What is the maximum level of survivability which can be achieved
for a given a network ?โข A bridge is a link whose deletion increases the number of connected
components.โข is the set of all bridges in the network.โข Theorem: The maximum level of survivability of satisfies .
Characterization
-13-
How Many Spanning Trees?โข How Many Spanning Trees are necessary in order to achieve this
maximum level of survivability?โข Theorem: Let , the number of sufficient spanning trees which satisfies
maximum level of survivability is bounded by
โ ยฟ ๏ฟฝฬ๏ฟฝโจ ยฟ|๐ธ|โ|๐|+1
โ=โ10
10โ5+1โ=2ยฟ
(b) A clique demonstrating a tight lower bound
example
ยฟ๐โจยฟ5
(a) A cycle demonstrating an tight upper bound
example
Characterization
-14-
Algorithmic Scheme
โข Constrained Bandwidth Max-Survivability (CBMS) Problem:Find a k-survivable spanning connection such that:
โข 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.
โขPolynomial solution by Roskind and Tarjan โ โA note on finding minimum-cost edge-disjoint spanning treesโ, 1985.
Optimization
-15-
Algorithmic Solution
๐๐ =0 .01
๐๐=0 .01๐๐=5
๐๐=0 .01๐๐=5
๐๐ =10
๐๐ =2 0
๐๐ =10
๐ ๐=10
๐ ๐=0 .01
๐ ๐=0 .01
๐๐ =0 .01
๐๐ =0 .01
๐๐ =0.01
a
b c d
e
๐๐ =1
โข Find a 2-survivable spanning connection such that:
Optimization
-16-
Algorithmic Solution
๐๐ =0 .01
๐๐=0 .01๐๐=5
๐๐=0 .01๐๐=5
๐๐ =10
๐๐ =2 0
๐๐ =10
๐ ๐=10
๐ ๐=0 .01
๐ ๐=0 .01
๐๐ =0 .01
๐๐ =0 .01
๐๐ =0.01
a
b c d
e
๐๐ =1
โข Each link with a bandwidth
โข Each link with a bandwidth :
๐ e ,๐ eDiscard the link
๐ ๐๐=โ ๐๐(๐โ๐e)
๐ ๐๐=๐
๐ ๐๐=๐
Original Network Auxiliary Network
๐ค ๐=โ๐๐0 .99
๐ค ๐=0
๐ค๐ =โ๐๐0 .99
๐ค๐ =0
๐ค๐=โ๐๐0 .99
๐ค๐=0
a
b c d
e
๐ค๐=โ๐๐0 .99
๐ค๐=0
๐ค๐ =โ๐๐0.99
๐ค๐ =0
๐ค ๐=โ๐๐0 .99
๐ค ๐=0
๐ค๐ =โ๐๐0 .99
๐ค๐ =0
Optimization
-17-
๐๐ =0 .01
๐๐=0 .01๐๐=5
๐๐=0 .01๐๐=5
๐๐ =10
๐๐ =2 0
๐๐ =10
๐ ๐=10
๐ ๐=0 .01
๐ ๐=0 .01
๐๐ =0 .01
๐๐ =0 .01
๐๐ =0.01
a
b c d
e
๐๐ =1
Algorithmic Solution
โข In the Auxiliary Network, find 2 Edge Disjoint Spanning Trees utilizing the minimum cost edge disjoint spanning tree algorithm.
Original Network Auxiliary Network
๐ค ๐=โ๐๐0 .99
๐ค ๐=0
๐ค๐ =โ๐๐0 .99
๐ค๐ =0
๐ค๐=โ๐๐0 .99
๐ค๐=0
a
b c d
e
๐ค๐=โ๐๐0 .99
๐ค๐=0
๐ค๐ =โ๐๐0.99
๐ค๐ =0
๐ค ๐=โ๐๐0 .99
๐ค ๐=0
๐ค๐ =โ๐๐0 .99
๐ค๐ =0
Optimization
-18-
Maximum survivability level ratio versus the number of spanning trees k for different bandwidth requirements
SimulationSimulation
โข - maximum survivability level that can be obtained by a -survivable spanning connection with a bandwidth requirement of
โข - maximum survivability level of the network with a bandwidth requirement of
๐
-19-
Bandwidth ratio versus the survivability level requirement
Simulation
X12 times improvement
๐0
Simulation
โข - maximum bandwidth of a -survivable spanning connection with a survivability level of at least
โข - maximum bandwidth of a fully disjoint spanning connection
-20-
Conclusion
โข The establishment of a comprehensive methodology for efficiently providing tunable survivability.โข Ron Banner and Ariel Orda. โThe power of tuning: A novel approach
for the efficient design of survivable networksโ. In IEEE/ACM Trans. Networking, 2007.
โข Jose Yallouz and Ariel Orda. โTunable QoS-aware network survivabilityโ. In IEEE Infocom, 2013.
โข Jose Yallouz, Ori Rottenstreich and Ariel Orda. โTunable Survivable Spanning Treesโ. In ACM Sigmetrics, 2014.
Conclusion