Design of Capacitated Survivable Networks With a Single Facility

Author : Hervé Kerivin, Dritan Nace, and Thi-Tuyet-Loan Pham

R97725025 張耀元， R97725036 李怡緯

IEEE transactions on networkingPublication Date: April 2005

Hervé Kerivin received the Ph.D. degree in combinatorial optimization from the University Blaise Pascal, Clermont-Ferrand, France, in November 2000.

Dritan Nace received the degree in mathematics from the University of Tirana, Albania, in 1991, and the M.Sc. (DEA) degree in computer science and the Ph.D. degree in computer science, both from University of Technology of Compiègne, Compiègne Cedex, France, in 1993 and 1997, respectively.

Thi-Tuyet-Loan Pham received the B.Sc. Degree from the Technology University of Ho Chi Minh City, Vietnam, the M.Sc. degree from the Francophone Institut for Computer Science, and the Ph.D. degree from University of Technology of Compiègne, Compiègne Cedex, France.

Outline AbstractIntroductionMathematical formulationSolution methodComputational resultConclusion

ABSTRACT

Abstract Single-facility capacitated survivable network

design problem SFCSND.We optimize the network topology and the

link dimensioning in order to rout all traffic commodities.

We also consider rerouting strategies to deal with link failure

We present a mix-integer linear programming model solved by combining several approaches.

INTRODUCTION

Introduction Given ◦a set of nodes

Link Dimension

◦a set of single type facilities with constant capacity◦a set of commodities (OD-pair and required

bandwidth)We consider only the problem of designing

survivable networks when a single link failure arises

IntroductionThe problem involves designing the topology

and dimensioning the linksThe installed capacities will be sufficient to

route all traffic demands for◦ Nominal state: all network hardware is operational (without fail).◦ Reroute interrupted traffic for failure state

The problem determines the lowest cost resource◦Link installation cost◦A unit facility loading cost

IntroductionIn order to reduce cost, the spare capacity

devoted to protection is usually shared among several rerouting paths: Shared reroute mode

local rerouting: tries to reroute traffic locally between the extremities of the failed link

end-to-end rerouting: propagates failure information to the destination nodes of traffic demands, in order to set up rerouting paths between them

MATHEMATICAL FORMULATION

End-to-end reroutingLocal rerouting

Mathematical FormulationWe formulate both end-to-end rerouting

and local rerouting

Local rerouting schemes have in theory a higher bandwidth overhead than end-to-end rerouting schemes

G(V,E) Graph with vertex and edgea Installation cost associated with the edge of GXe={0,1} Topology variableK={1,2,…|K|} Commodities for OD-pair & Bk Traffic demand (required bandwidth)M Maximum number of modularity

λ Capacity of given facility ye Dimension variable Ce A cost corresponds to the loading of a single

facility onto edge eL={1,2,…|L|} Link failure indexes where |L| |E|p P(k) P(k) is the finite set of elementary paths of

commodity kNominal flow variable

q Q(l,k) Q(l,k) is the set of elementary paths of failed commodity k in fail index lEnd-to-end reroute flow variable

ko kd

kpf

k,lqh

Mathematical Formulation

Mathematical FormulationLocal reroute strategy: interrupted traffic must be

rerouted between the extremity nodes of the failed link

GelGraph composed of failed link el

qQ(l) Q(l) is the finite set of elementary paths of Gel

local rerouting flow variablelqg

Mathematical Formulation

We rewrite some constrains:

Mathematical FormulationThe size of both mixed-integer linear programs

may be very large because of the huge number of paths in P(k), Q(l,k), Q(l) .

The working paths P(k) and rerouting path Q(l,k), Q(l) are independent so decomposition method (such as Benders’ decomposition) can be used to obtain near optimal solution.

SOLUTION METHOD

A. Break down the problemB. Capacity feasibility problemC. Topology and dimensioning problemD. Implementation detail

Solution methodWe break down into 2 consecutive stages of

optimization◦Topology and link dimension ◦Capacity feasible

This breaking down of the problem has a drawback: there are some distance for optimal to our solution

The higher is the basic capacity of the facility (λ) in relation to a single traffic demand (BK), the more critical this distance becomes

Solution method

Implementation detail

COMPUTATIONAL RESULT

Computational ResultsA series of computational experiments were

performed to compare and analyze the survivability cost based on end-to-end and local rerouting strategies

Compare effectiveness of both restoration strategies (end-to-end and local rerouting):◦Total capacity installed in the network◦Topology◦Global cost with respect to the obtained network

Computational Results (Cont.)Algorithm implemented in CCPLEX 7.1Machine configuration:◦Sun Enterprise 450◦Solaris 2.6◦Quadri-UltraSparcII 400 MHz◦1 GB RAM

Problem InstancesThree (undirected) network instances

considered to perform the numerical experiments:◦net1 is generated randomly◦net2, net3 are furnished by France Telecom

R&D Correspond to real telecommunication networks

Problem Instances (Cont.)Main parameters of the network:

◦ |V|: number of nodes◦ |E|: number of potential links◦ |K|: number of traffic demands◦d(G): average nodal degree◦T(k): average demand traffic

Problem Instances (Cont.)Considered all possible traffic demands◦The number of traffic demand:

|K| = ( |V| * (|V|-1) ) / 2Run all of the tests with four different facility

capacities ◦2400◦1800◦1200◦600

All links are subject to failure◦L = E

Facility CapacityResults obtained with four different facility

capacities for the single-facility capacitated network design problem:◦λ: constant facility capacity◦F: number of installed facilities◦Ci: percentage of the whole capacity

that is idle (unused)◦d: average nodal degree in the optimal network ◦f: average link facility in the optimal network

Facility Capacity (Cont.)

Facility Capacity (Cont.)Facility capacity plays a significant role

in the nature of the SFCSND problemThe major difference between

nonsurvivable and survivable networks is the number of used links

Obtained Network TopologyAverage nodal degree for the obtained

network depends on the value of facility capacity λ, regardless of the survivability requirement

Obtained Network Topology (Cont.)Small values of λ are of the same

magnitude order to some traffic demands◦Often more cost-effective to create a link than

to use long paths to carry this traffic◦Obtained network is highly meshed

Obtained Network Topology (Cont.)Sufficiently large values of λ may

therefore enable us to obtain the minimal topology for both the nonsurvivable case and for the survivable case problems

Obtained Network Topology (Cont.)If we stipulate survivability, the obtained

network always has an average nodal degree strictly superior to that obtained in the nonsurvivable case (about 20% on average)

Obtained Network Topology (Cont.)Survivable networks need spare links in order

to meet the survivability requirementsMain difference between partial end-to-end

rerouting without recovery and local rerouting:◦Local rerouting tends to be slightly more meshed◦Local rerouting generally uses longer rerouting

paths than other rerouting strategies◦Meshed network permits a better use of resources

when addressing failure situations

Obtained Network Topology (Cont.)The obtained network topology is

sometimes the same for both restoration strategies

Network CostConsider link installation costs and the

cost of capacity loadingGaps between the global costs for the

networks:◦ : end-to-end rerouting and nonsurvivable

case◦ : local rerouting and nonsurvivable case◦ : gap related to the global costs between

two rerouting strategies

Network Cost (Cont.)The cost for a survivable network can be

almost 70% more than the cost for a nonsurvivable network◦We need to optimize simultaneously the

dimensioning problem for the nominal state and the spare capacity computation, in order to reduce this gap

Network Cost (Cont.)The cost of survivable networks based on

a local rerouting strategy is slightly greater than the cost for an end-to-end rerouting strategy◦Local rerouting may be used without bringing

about a significant impact in terms of cost

Computational TimeThe computational time becomes

generally greater as the facility capacity becomes smaller◦Large combinatory of the problem with

respect to the choice for installing links and assigning capacities

◦The case with large capacity facility where the number of links to be installed is obviously lower and the choice “easier.”

CONCLUSION

ConclusionSurvivable network design problem with single

facility:◦Survivability requirements are expressed in terms of

the spare capacity required to address link failures◦Various rerouting strategies:

Local and end-to-end reroutingPresented mixed-integer linear programming

modelsProposed an appropriate decomposition approach◦Allows to accelerate the resolution time

Conclusion (Cont.)Reported some numerical results in terms

of overall network cost for:◦Restoration schemes◦Nonsurvivable case

Main result is that survivable networks designed on basis of local restoration may be used without bringing about a significant impact in terms of cost

Conclusion (Cont.)Result could be very useful for

telecommunication operators ◦Restoration strategy is already known to be

quite simple and efficient in operational terms.

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