Design of Capacitated Survivable Networks With a Single Facility Author : Hervé Kerivin, Dritan Nace, and Thi-Tuyet- Loan Pham R 張耀元， R 李怡緯

IEEE transactions on networking Publication Date: April 2005

Hervé Kerivin received the Ph.D. degree in combinatorial optimization from the University Blaise Pascal, Clermont-Ferrand, France, in November 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 Abstract Introduction Mathematical formulation Solution method Computational result Conclusion

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

Introduction The problem involves designing the topology and dimensioning the links The 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

Introduction In 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 rerouting Local rerouting

Mathematical Formulation We 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 edge aInstallation cost associated with the edge of G X e ={0,1}Topology variable K={1,2,…|K|}Commodities for OD-pair & BkBk Traffic demand (required bandwidth) MMaximum number of modularity λCapacity of given facility yeye Dimension variable CeCe A cost corresponds to the loading of a single facility onto edge e L={1,2,…|L|}Link failure indexes where |L| |E| p  P(k) P(k) is the finite set of elementary paths of commodity k Nominal flow variable q  Q(l,k) Q(l,k) is the set of elementary paths of failed commodity k in fail index l End-to-end reroute flow variable

Mathematical Formulation

Local reroute strategy: interrupted traffic must be rerouted between the extremity nodes of the failed link GelGel Graph composed of failed link e l q  Q(l) Q(l) is the finite set of elementary paths of G el local rerouting flow variable

Mathematical Formulation We rewrite some constrains:

Mathematical Formulation The 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 problem B. Capacity feasibility problem C. Topology and dimensioning problem D. Implementation detail

Solution method We 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 (B K ), the more critical this distance becomes

Solution method

Implementation detail

COMPUTATIONAL RESULT

Computational Results A 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 C CPLEX 7.1 Machine configuration: ◦ Sun Enterprise 450 ◦ Solaris 2.6 ◦ Quadri-UltraSparcII 400 MHz ◦ 1 GB RAM

Problem Instances Three (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) ) / 2 Run all of the tests with four different facility capacities ◦ 2400 ◦ 1800 ◦ 1200 ◦ 600 All links are subject to failure ◦ L = E

Facility Capacity Results obtained with four different facility capacities for the single-facility capacitated network design problem: ◦ λ: constant facility capacity ◦ F: number of installed facilities ◦ C i : 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 plays a significant role in the nature of the SFCSND problem The major difference between nonsurvivable and survivable networks is the number of used links

Obtained Network Topology Average 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 requirements Main 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 Cost Consider link installation costs and the cost of capacity loading Gaps 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 Time The 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.”

Computational Time (Cont.) The computational time spent for solving the super master program takes often more than 50% of global time ◦ The result justify our strategy to reduce the number of calls through introducing as much as possible valid inequalities in order to approach faster to the optimal solution before checking for the feasibility of assigned capacities

CONCLUSION

Conclusion Survivable 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 rerouting Presented mixed-integer linear programming models Proposed 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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