**1.2 Problem definition**

Formally, the proposed model that combines the network survivability and network reliability approaches is the following:

**115**

*A Survivable and Reliable Network Topological Design Model*

• *G* = (*V,E*) as a nondirected simple graph.

municating bearing *i* and *j* in the solution.

The remainder of this chapter is structured as follows:

• *T* ⊆ *V* as a subset of distinguished nodes (denominated terminals).

• *C* = {*cij*}(*i,j*)∈*E* as a nonnegative cost matrix associated with the edges of *G.*

• *R* = {*rij*}*i,j*∈*T* as a node-connectivity requirements matrix among pairs of terminal nodes in which at least *rij* node-disjoint paths are required to be com-

In addition to the above, suppose that the edges of *E* and the nodes of *V*\*T* (usually called Steiner nodes) have associated operation probabilities given by the vectors: *PE* = {*pe*}*e*∈*E* and *PV*\*T* = {*pv*}*v*∈*V* \*T*, respectively, where the failures are assumed to be statistically independent. Given a certain probability *pmin* set as reliability lower threshold, the objective is to find a subgraph *GS* ⊆ *G* of minimum cost that satisfies the node-connectivity requirement matrix *R* and furthermore its *T*-terminal reliability. The latter is defined as the probability that the partial graph *GR* ⊆*GS*, obtained by randomly dropping edges and nodes from *GS* with probabilities given by 1−*pe* and 1−*pv*, respectively, connects all nodes in *T* [2]*.* The *T-*terminal reliability R*T* (*GS*) has to satisfy R*T* (*GS*) ≥ *pmin* (i.e., the probability that all nodes in *T* are connected by working edges exceed *pmin*). This model will be referred to as "generalized Steiner problem with survivable and reliable constraints," GSP-SRC.

• Sections 2 and 3 present a background of the meta-heuristic algorithm applied

• Section 5 deals with the experimental results obtained over a set of heterogeneous test instances as well as the most important contributions and conclu-

Greedy randomized adaptive search procedure (GRASP) is a well-known metaheuristic, i.e., a particular method to find sufficiently good solutions to optimization problems, that has been successfully used to solve many difficult combinatorial optimization problems. It is an iterative multi-start process which operates in two

In the construction phase, a feasible solution is built whose neighborhood is then explored in the local search phase [3]. A neighborhood of a certain solution S is a set of solutions that differ from S in well-defined forms (e.g., replacing any link by a different one, replacing "stars" by "triangles," and so on). Regarding optimization, different neighborhoods of S will not, in general, share the same local minimum. Thus, local optima trap problems may be overcome by deterministically changing

and the means to compute the network reliability, respectively.

• Section 4 introduces the algorithm required to solve the GSP-SRC.

**2. Greedy randomized adaptive search procedure (GRASP)**

phases, namely, the construction and the local search phases.

*DOI: http://dx.doi.org/10.5772/intechopen.84842*

Consider:

**1.3 Chapter organization**

sions of this work.

the neighborhoods [4, 5].

*A Survivable and Reliable Network Topological Design Model DOI: http://dx.doi.org/10.5772/intechopen.84842*

Consider:

*Reliability and Maintenance - An Overview of Cases*

ture even in probabilistic terms.

**1.1 Aim and objectives**

nal nodes).

**1.2 Problem definition**

network reliability approaches is the following:

In view of the above, the networks must continue to be operative even when a component (link or central office) fails. In this context, survivability means that a certain number of pre-established disjoint paths among any pair of central offices must exist. In this case, node-disjoint paths will be required, which show a stronger constraint than the edge-disjunction ones. Assuming that both the links and the nodes have associated certain operation probabilities (elementary reliability), the main objective is to build a minimum-cost sub-network that satisfies the nodeconnectivity requirements. Moreover, its reliability, i.e., the probability that all sites are able to exchange data at a given point in time, ought to surpass a certain lower bound pre-defined by the network engineer. In this way the model takes into account the robustness of the topology to be designed by acknowledging its struc-

The aim of this chapter is to introduce an algorithm that combines the two approaches so as to achieve the design of robust networks and test it on a number of instances that are representative of communication network problems. On the one hand, the network must be highly reliable from a probabilistic point of view (its reliability) assuming that the probabilities of failure of all links and sites are known. On the other hand, the network structure must be topologically robust; for this, node or edge connectivity levels between pairs of distinguished nodes are required. This means that between all pairs of distinguished nodes there exists a given number of edge paths or disjoint nodes. Then, once a minimum threshold for

i.Constructs feasible low-cost solutions that satisfy the connectivity levels (disjoint paths) between pairs of distinguished nodes of the network (termi-

ii.The reliability of the network built meets the pre-established threshold, thus

Performed research indicates that literature pertaining to algorithms on design topologies which consider both approaches (survivable networks and network reliability) is scarce. The works on the design of robust networks in general fix a level of node/edge global connectivity of the network and try to design a network at the lowest possible cost that satisfies that level (e.g., 2-node-connectivity) [1]. Nevertheless, there are contexts where this combination of approaches is imperative and demanded. For example, in the context of military telecommunication networks, it is required that the networks are topologically very robust (e.g., 3-nodeconnectivity) and at the same time that they are extremely reliable from the network reliability approach's point of view, surpassing very high reliability levels. Another example of the application of the combined model is the logical distribution of highly dangerous merchandise on a country's roads. In such a context, two things are desirable: high reliability in the connection of points of distribution (i.e., "reliable" roads) and high levels of connectivity between the points that must exchange cargo (availability of alternative roads to possible road cuts, traffic saturation, etc.).

Formally, the proposed model that combines the network survivability and

achieving fault-resistant networks according to both approaches.

reliability (e.g., 0.98) is set, the algorithm here introduced:

**114**


In addition to the above, suppose that the edges of *E* and the nodes of *V*\*T* (usually called Steiner nodes) have associated operation probabilities given by the vectors: *PE* = {*pe*}*e*∈*E* and *PV*\*T* = {*pv*}*v*∈*V* \*T*, respectively, where the failures are assumed to be statistically independent. Given a certain probability *pmin* set as reliability lower threshold, the objective is to find a subgraph *GS* ⊆ *G* of minimum cost that satisfies the node-connectivity requirement matrix *R* and furthermore its *T*-terminal reliability. The latter is defined as the probability that the partial graph *GR* ⊆*GS*, obtained by randomly dropping edges and nodes from *GS* with probabilities given by 1−*pe* and 1−*pv*, respectively, connects all nodes in *T* [2]*.* The *T-*terminal reliability R*T* (*GS*) has to satisfy R*T* (*GS*) ≥ *pmin* (i.e., the probability that all nodes in *T* are connected by working edges exceed *pmin*). This model will be referred to as "generalized Steiner problem with survivable and reliable constraints," GSP-SRC.

## **1.3 Chapter organization**

The remainder of this chapter is structured as follows:

