Designing Connection Oriented Networks for Multi-Domain Path ...

J Netw Syst Manage (2010) 18:374–394 DOI 10.1007/s10922-009-9155-z

Designing Connection Oriented Networks for Multi-Domain Path Resilience Angelos Lenis • Vasilis Merekoulias • Vasilis Maglaris

Published online: 6 January 2010 Ó Springer Science+Business Media, LLC 2010

Abstract A challenge in network management and control is the ability to account for multi-domain requirements in the network planning process. Especially in Connection Oriented Networks where multi-domain path finding is a critical issue, networks need to be designed in such a manner as to restrict the possibility of erroneous inter-domain path selection. In this paper we propose per-domain topology design considerations that would leverage accurate disjoint path computations in a multi-domain environment, covering requirements of end-to-end path resiliency. In multi-domain environments state information between domains is heavily aggregated, hiding internal topology details dictated by scalability concerns, but also by restrictive domain administration policies for privacy, and security. Disjoint path finding is strongly affected by the aggregation techniques, since they do not provide information on path overlap. To handle this issue we introduce a metric, the Overlap Factor (OF), that quantifies path overlap in domains. The OF can be passed as an additional parameter of the inter-domain information exchange model to evaluate disjoint end-to-end paths. Alternatively, if domains were appropriately designed, this additional parameter might not be needed in evaluating resilient pairs of inter-domain paths. We based our recommended topology design algorithm on exploiting locally known OF values within the context of Genetic Algorithms. Extensive simulations confirm that domains designed using our proposed algorithm, result into accurate multi-domain disjoint path identification, with A. Lenis (&)  V. Merekoulias  V. Maglaris Network Management & Optimal Design Laboratory (NETMODE), School of Electrical & Computer Engineering, National Technical University of Athens (NTUA), Zografou, 157 73 Athens, Greece e-mail: [email protected] V. Merekoulias e-mail: [email protected] V. Maglaris e-mail: [email protected]

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a high success ratio compared to networks that are designed without inter-domain considerations. Keywords Network design  Multi-domain  Topology aggregation  Disjoint path finding  Genetic algorithms

1 Introduction Network planning and design is a process aimed at ensuring that a network meets requirements expressed by its subscribers and administrators. The process takes into account many parameters like traffic requirements, scalability requirements, availability and resilience requirements, geographical considerations and cost considerations. Each parameter influences the quality of the services that the network supports and must be carefully selected. Decisions entail selecting appropriate technologies and equipment as well as topological considerations. Although end-to-end requirements may often traverse multiple administrative domains, network design and management is traditionally focused on intra-domain requirements. Thus, the effect of internal decisions on inter-domain routing behavior is neglected, a fact that may be crucial especially in the context of emerging connection oriented networks involving large amounts of data rates. In Connection Oriented Networks [1] technologies such as MPLS [2] and more recently GMPLS [3], offer a new set of services that can guarantee a level of quality per end-to-end connection within a single domain. Power-users are gradually enabled with the ability to substitute best-effort IP with high speed—high bandwidth circuit switched services ranging from MPLS VPNs [4] & Gigabit Ethernet Provider Bridge Backbones (PBB) [5] to DWDM lightpaths forming 10– 100 Gbps Optical Private Networks—OPNs [6]. End-to-end circuits based on these technologies often span multiple administrative areas. Thus, enabling multi-domain cooperation with appropriate inter-domain protocols and network design considering multi-domain requirements becomes of increasing importance. A backbone network designed to provide advanced services requires an enhanced control plane implementing three main components: A routing protocol, a signaling protocol and a set of path finding algorithms, all three incorporating Traffic Engineering (TE) parameters [7]. Examples of routing and signaling protocols are OSPF-TE [8] and RSVP-TE [9]. Path finding can be based on distance vector or link state implementations [10] for shortest, k-shortest, constraint-based and disjoint path finding [11]. The success of the best-effort global Internet was partially due to the scalability exhibited by BGP, based on a distance vector algorithm. However, for cases of constraint based path finding or disjoint path finding, link state algorithms are required since network-wide information is necessary to guarantee conformance to preset constraints. Demanding power users (e.g., large distributed computing communities) request resilient end-to-end connectivity with performance guarantees across multiple networking domains. Over the last decades a wealth of protection and restoration mechanisms have been suggested and implemented for single domain profiles.

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Examples of protection and restoration techniques have been proposed. Protection rings, protection cycles, p-cycles, redundant trees, resilient routing layer and disjoint paths/segment/links are among them. Extensive overviews can be found in [12–14], covering resiliency methods from legacy SDH to lightpath backbones and from static to adaptive implementations. These methods are mostly based on Graph Theory with each network modeled as a connected graph. Path finding algorithm, applied on the network’s graph representation, results to selecting a couple of disjoint paths, one being active (primary) and the other reserved for back-up (secondary). These single domain approaches may fail in multi-domain environments, since domains do not exchange full topology data or internal protection details, hindering multi-domain disjoint path finding. As stated in Yannuzzi et al. [15] in networks where the number of available disjoint path options is low accurate disjoint path calculation becomes a requirement. The limitations posed by today’s mechanisms for TE in multi-domain environments led IETF to create the Path Computation Element (PCE) WG (http://www.ietf.org/dyn/wg/charter/pce-charter. html). A PCE is capable and responsible for computing disjoint paths within a domain based on state information gathered and stored into a Traffic Engineering Database. The major drawback of the PCE approach is that it does not deal with a holistic view of the multi-domain environment. It acts more like a domain broker, thus it may point to disjoint end-to-end paths that are either invalid or far below optimal. Disjoint path finding, as a link state algorithm, is strongly influenced by the accuracy of the topological description of the multi-domain network. Multi-domain link state algorithms were proposed for ATM networking i.e., PNNI, using domain abstractions referred to as Topology Aggregation (TA) [16]. TA aimed at reducing the size of the state information used by routing algorithms allowing link state algorithms to be executed in large scale networks. TA targets at (a) hiding the internal structure of each domain, (b) advertizing reduced domain information for scalability reasons, while (c) allowing for accurate path-finding. Topology aggregation techniques like Full Mesh, Star and Spanning Trees typically aim at providing accurate solutions for single path selection with up to two objectives [17], e.g., minimizing delay or maximizing available bandwidth or both. Identifying a pair of disjoint paths adds a third constraint in path selection, in that the two paths discovered must not share any resources (e.g., links). This information is not considered in existing TA techniques that hide too many topology details. As a result, inconsistencies between the aggregated and real state of the network may often lead to erroneous disjoint path selection. Our work focuses on properly designing domain topologies that limit the possibility of mishaps, when computing disjoint paths based on the aggregated view of the domains. The topological design of such networks is considered a complicated problem with no existing polynomial algorithms that can produce optimal results. In this paper we adopted the heuristic approach of Genetic Algorithms [18, 19] that are inspired by evolutionary concepts such as natural selection, mutation, and survival of the fittest. Genetic heuristics are found to be well suited for network optimization problems [20–22].

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Genetic Algorithms define a fitness function to grade generated topologies and select the fittest ones that lead towards optimal network layouts. We selected a fitness function based on our proposed metric, namely the Overlap Factor that quantifies path overlap in a network topology. We then applied the optimization algorithm to determine domain topologies that always have primary and backup— disjoint—path options with no diverging weights in the real versus the aggregated views of the network. Hence, a valuable planning tool is provided to domain designers, that will lead to near-optimal disjoint end-to-end path finding in a distributed multi-domain network operation. The rest of the paper is outlined as follows. In Sect. 2 we define the problem, in Sects. 3 and 4 we describe our design algorithm and in Sect. 5 we provide simulation results. Section 6 summarizes conclusions.

2 Problem Definition The techniques presented in this paper are based on a graph transformation of a backbone network, with switching devices and their connections represented by nodes and links.1 A directed link from node I to node J is expressed as (I, J). The full network is the set of all domains and inter-domain links. We distinguish two types of backbone nodes within a domain: Border nodes that interconnect different domains and internal switching nodes. Border nodes also include the end-nodes of paths (ingress/egress nodes), that interconnect subscriber access networks to the multi-domain backbone. We will refer to these as source and target nodes. A path from source node S to target node T is defined as the connected ordered list of backbone links, starting from border node S and ending at border node T. Each link (I, J) is associated with a set of parameters, that define its performance metrics. Paths may inherit link parameters in three modes: Additive, restrictive and multiplicative. Typical examples are delay, bandwidth and packet loss ratio, respectively. 2.1 Topology Aggregation Constraint based path finding in connection oriented networks require state information exchange. This includes both static (e.g., the capacity of a link) and dynamic information (e.g., the currently available capacity and delay). Scalability and privacy issues in multi-domain environments led to distance vector, instead of link state algorithms, i.e., use of BGP in the global Internet. Alternatively when link state information is required, as in our case, Topology Aggregation (TA) techniques have been suggested to summarize or abstract domain topological details. TA techniques involve removing internal nodes and links, while preserving border nodes and inter-domain links, which are anyhow visible to other 1

Recall that backbone nodes are assumed to implement enhanced control plane functionality in terms of routing, signaling and path-finding, e.g. (G)MPLS enabled Label Switched Routers (LSRs) [2]. Our model is agnostic to data plane (transmission) technologies e.g. WDM, SDH, PBB; such considerations can be embedded within link weights, along with other network design constraints.

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Border Node Internal Transit Note Subscriber Access Network Subscriber

Fig. 1 Full mesh aggregation

networks. The internal structure of each domain is replaced with a virtual one that preserves its general characteristics, while hiding per-domain details, thus reducing the complexity of advertized domain topology and preserving privacy of domain administrative internals. Examples of TA techniques include full mesh, star, spanning trees, De Brujin graphs, Shufflenets and combinations like full mesh star [23]. Nevertheless, they all aim at single path selection with up to two objectives [24], e.g., identifying the shortest path passing through the aggregated network, that meets delay and available bandwidth requirements. Our work builds upon the Full Mesh (FM) aggregation model, the most common TA technique and the first step for many others. FM transforms a domain network to a set of fully connected border nodes via virtual links, as in Fig. 1. Virtual links are obtained through a shortest path algorithm based on a single selected metric (e.g., minimum border-to-border additive path delay or maximum restrictive path bandwidth). This type of transformation captures closely the details for single path selection in terms of the selected metric. 2.2 Disjoint Path Finding The problem of finding two paths, that do not share common resources, is referred to as disjoint path finding [25]. Disjoint paths can be link disjoint, if node sharing is allowed, or node disjoint for complete redundancy. When establishing a resilient end-to-end connection in a communications network two paths are identified, one referred to as the primary while the second as the backup alternate route. We are considering path optimization related to an additive metric. This was motivated by our experience in over-provisioned optical backbone networks [26] where one-way delays (an additive parameter) are the major design concern to meet user requirements e.g., latency.

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We will refer to the additive parameter as the weight w(I, J) for link (I, J); W(S, T) or W(P) will denote the total weight from source to target (S, T) or path P. We formulate the problem as the discovery of two link disjoint paths P1, P2 from a source node S to a target node T minimizing the total disjoint weight DW(S, T) = W(P1) ? W(P2), the sum of the path weights. A standard algorithm for finding disjoint paths is due to Suurballe and Tarjan (S&T) [25]. We illustrate the algorithm using the example in Fig. 2. Assume that a request is made for two disjoint paths from node S to node T. The algorithm proceeds in five steps: 1. 2.

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Identify the shortest path from S to T, using a link-state algorithm like Dijkstra’s, P1 = {S,C,E,T}. Modify the graph of the network reversing the links on P1 and setting the weight of the reversed links to the negative value of the original links. In the example of Fig. 2 reverse links (S,C), (C,E), (E,T) and set their weights to w(C,S) = -w(S,C), w(E,C) = -w(C,E) and w(T,E) = -w(E,T). In the modified graph, identify the shortest path P2 = {S,B,F,E,C,D,G,T} from S to T via an algorithm that handles negative weights e.g., Bellman Ford’s [10]. Perform post-processing on the two paths as follows: 4.1.

4.2. 5.

Remove from P1, P2 common links used in opposite directions. In Fig. 2 remove link (C,E) from P1 and (E,C) from P2 resulting to P1 = {S,C}U{E,T} and P2 = {S,B,F,E}U{C,D,G,T}. Swap the remaining path segments after the removed link resulting to P1 = {S,C,D,G,T} and P2 = {S,B,F,E,T}.

Restore the graph and return the results.

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An alternate formulation might aim at minimizing the maximum value of the weight of the two paths. However, this is an NP-Hard problem [27] while a Polynomial algorithm minimizing the weight sums provides adequate balance in most practical case. Our simulations, reported in Sect. 4 below, confirmed this statement for domains of reasonable size. 2.3 Effect of Full Mesh Aggregation on Multi-Domain Disjoint Path Finding Multiple domain path-finding services are based on an aggregated view of the real network. This view contains border nodes and inter-domain links, while internal domain topologies are hidden and exported as FM (Full Mesh) abstractions. An aggregated domain exposes virtual links representing shortest paths between pairs of border nodes. Algorithms executed on the aggregated view of each domain assume that domain virtual links do not overlap with each other. Figure 3, however, shows that this assumption is not always true. Figure 3 shows a case where three different networks produce the same FM abstraction. In each FM network, the virtual link (A,B) represents the corresponding shortest path from A to B in the real network. In Fig. 3a this is the real link (A,B), in Fig. 3b and c the path (A,E,F,B). Similarly, virtual link (C,D) represents the real paths (C,D), (C,E,F,D) and (C,E,F,D) in Fig. 3a, b and c, respectively. A disjoint path finding algorithm on the aggregated network will assumes virtual links (A,B) and (C,D) are disjoint with a total disjoint weight of DW = 3 ? 3 = 6. This is correct for the network in Fig. 3a, where the corresponding shortest paths are also disjoint, but it is not for the domains in Fig. 3b and c. In these cases corresponding shortest paths overlap as they pass through the real link (E,F).

Fig. 3 Topology aggregation effect on disjoint path selection

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Such mishaps motivated our work, namely to develop a design method pointing towards domain topologies, which allow accurate multi-domain disjoint path calculation. This is achieved by reducing the possibility of overlapping virtual links. Towards this we defined the Overlap Factor, a metric that quantifies virtual link overlap. Based on this metric we implemented a Genetic algorithm, which produces topologies with reduced OF values and thus, reduced chance of erroneous disjoint path selection.

3 Overlap Factor In Sect. 2.3 we pointed out that errors appear in disjoint path calculation when the sum of the weights of two shortest paths, with entry border nodes {Bi, Bk} and exit border nodes {Bj, Bl}, is lower than the sum of the weights of two shortest disjoint paths with the same entry and exit border nodes. DW½ðBi ; Bj Þ; ðBk ; Bl Þ  WðBi ; Bj Þ þ WðBk ; Bl Þ

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In disjoint path calculation inconsistencies occur after identifying the primary path. Primary path, as outlined in S&T’s algorithm, is the shortest path from source node to target node and thus is unaffected by Full Mesh (FM) aggregation. Backup path calculation is executed in a modified graph, where path weights update is affected by primary path selection. With FM aggregation, the default modification in S&T’s algorithm will produce graphs of wrong weights since path overlap is not considered. This may occur while identifying secondary paths via domains, which already host primary path links. B ;B We define as Overlap Factor ofBki;Blj ; for two pairs of border nodes, a factor that modifies virtual link weights to a correct value in order to select a disjoint backup route. This is achieved by multiplying the weight of the candidate for backup B ;B shortest path. W(Bk, Bl), by ofBki;Blj : This results to a new value of the backup weight, which takes into account the extra constrain for a candidate backup path, as of being disjoint with the primary. B ;B

DW½ðBi ; Bj Þ; ðBk ; Bl Þ ¼ WðBi ; Bj Þ þ ofBki;Blj  WðBk ; Bl Þ

ð2Þ

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DW½ðBi ; Bj Þ; ðBk ; Bl Þ  WðBi ; Bj Þ WðBk ; Bl Þ

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Equation (3) suggests that we can calculate ofBki;Blj by calculating the shortest path weights and the total disjoint paths weight of corresponding border node pairs. The shortest path weights W(Bi,Bj) calculation is straight forward by running a shortest path finding algorithm, like Dijkstra’s, from Bi to Bj. To evaluate DW[(Bi,Bj),(Bk,Bl)] we propose the following algorithm:

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The algorithm introduces two additional virtual nodes S, T and connects them with zero weight directed links to the two pairs of border nodes. Calculating now disjoint paths from S to T, is equivalent to calculating disjoint paths from ingress border nodes, Bi, Bk, to egress nodes, Bj, Bl, independently of inter-domain links and other domains. This is illustrated in Fig. 4. Figure 4 depicts a domain with border nodes A,B,C,D. We can easily identify the shortest paths from A to B and C to D to be {A,E,F,B} and {C,E,F,D}, respectively. Their weights are W(A,B) = 3 and W(C,D) = 3. In order to calculate the disjoint weight for paths using A and C as ingress and B, D as egress nodes we add the virtual nodes S and T. A disjoint path finding algorithm, like S&T’s, from S to T can identify the disjoint paths P1 = {S,A,B,T} and P2 = {S,C,E,F,D,T} with disjoint weight DW(P1,P2) = W(P1) ? W(P2) = 100 ? 3 = 103. The Overlap Factor can now be calculated for these border pairs as: Fig. 4 Overlap factor calculation

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DWðP1 ; P2 Þ  WðA; BÞ 103  3 ¼ ¼ 33:33 WðC; DÞ 3

A value close to the lower bound 1 indicates that there is a small variance between the weights of the shortest paths and the total weight of corresponding disjoint paths. This also suggests that the shortest paths do not overlap with each other. Increased of values indicate large overlaps while a value of infinity indicates lack of disjoint paths. It becomes apparent that designing network topologies with reduced Overlap Factors will limit the possibilities of errors when calculating disjoint paths based on the FM representation of domains. For our Topology Design approach below, we selected a fitness function increasing with respect to the Overlap Factor, so that reducing the fitness function will lead towards candidate domain topologies with minimal path overlap.

4 Topology Design Method 4.1 Genetic Algorithm Our topology design method is based on a Genetic Algorithm (GA) [28, 29]. Genetic Algorithms are global optimization techniques, inspired by the principles of natural selection and evolutionary theory [18, 19]. They work by transforming the problem (network topology) to a suitable representation usually referred to as ‘‘chromosome’’. A GA transforms a population of chromosomes to a new generation where the Darwinian principle of survival of the fittest applies. Fitter chromosomes are identified by applying a fitness function. Fitness functions are calculated on a carefully selected subset of the properties of each chromosome. In our case this translates to selecting topologies based on required optimization objectives such as cost and resiliency. Next generations are produced by applying genetic operators like crossover and mutation. Each generation successively produces chromosomes closer to the optimal solution. In pseudo code this is described in three steps: 1. 2.

Randomly create an initial population of chromosomes (topologies). Iteratively perform the following sub-steps on the current generation of the population until the termination criterion has been satisfied. 2.1. 2.2. 2.3.

3.

Assign fitness value to each solution using the fitness function. Select parents to mate. Create children from selected parents by crossover and mutation.

Identify the best-so-far solution for the last iteration of the GA.

4.1.1 Topology Representation Each network topology in our approach is comprised of N nodes with Cartesian coordinates (x,y). Edges connecting two nodes (i,j) are assigned with a weight equal

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to the Cartesian distance of the two nodes i.e., wði; jÞ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi xi  xj Þ2 þ ðyi  yj

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Chromosome representation of each network topology is an N x N connection matrix. An entry in position (i,j) of the matrix represents that nodes i and j are connected. A matrix representation of chromosomes is common in Gas, since connection matrices are easy to apply crossover and mutation techniques. 4.1.2 Random Topology Generation The first step of the algorithm is to produce an initial population comprised K random solutions, each with N nodes. To this end we implemented a topology generator that creates random directed connected graphs. Our approach works in two steps. The first step is to produce a random tree connecting all nodes to ensure connectivity and then as a second step to add a random number of edges from 0 to the maximum value of a full meshed network 2 N(N - 1). Edges are added by selecting two random nodes as endpoints. The algorithm for producing random trees is the following:

A set of K random graphs generated with the above procedure institutes the initial population. 4.1.3 Crossover and Mutation The algorithm runs for M generations. For each generation we selected the random keys crossover scheme of Bean [30] for mating (i.e., pairing edges from two topologies to generate an offspring topology) and mutation (i.e., introducing random deviations to offspring topologies), which has already been applied in network design [29] with good results. Other crossover and mutation schemes may be selected offering different characteristics in terms of how fast they converge, the level of processing power etc. All schemes will eventually converge to network topologies with the same characteristics, as the quantification of the final solution is dependent on the way the fitness function is calculated and on how it is reduced in each intermediate generation.

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Solutions in a generation are sorted based on a fitness value, related to our optimization objective, and then classified into three classes A, B and C. We will refer to NA as the number of solutions with the smallest fitness value assigned to the ‘‘elite’’ class A and NC to solutions with the largest fitness value assigned to the ‘‘poor’’ class C. The remaining NB : N - NA - NC solutions are assigned to class B. Succeeding generations are produced through the following procedure: Chromosomes (topologies) in class A are automatically promoted, as is, to the next generation. Chromosomes in class C are discarded and replaced by random ones in the next generation. The remaining NB chromosomes of the next generation are replaced with offsprings. To produce each offspring, one parent p1 is selected at random from class A and the other p2 from class B. Mating parents from class A only entails the problem of converging to a local optimum solution. Mating parents from class B only entails the problem of never converging to a solution. Thus, mixing these two classes produces offsprings with the ‘‘elite’’ genes from class A and demonstrating enough variance from class B. Each edge in the offspring’s connection matrix is selected from p1 with probability p1, from p2 with probability p2 \ 1 - p1, or is randomly selected with probability 1 - p1 - p2. The above generation producing algorithm leads towards eventual improvement of chromosomes, or in network planning terms towards topologies that exhibit better (lower) fitness function [30]. The algorithm will lead to optimal (or close to optimal) solutions depending on proper parameter selection. Parameter setting is guided by the following rules: (1) The larger the population size K, the longer will each generation take to be computed but will provide solutions more close to the global optimum. (2) The number of new generations M affects the quality of the solution, providing better solutions as it increases. (3) The mating/mutation probabilities p1 and p2 should be such that p1 [ ‘, p2 \ 1 - p1 and p1 ? p2 & 1. NA, NB, and NC should be such that NB [ NA [ NC. 4.2 Fitness Function: Overlap Factor Proper selection of the fitness function results to domain topologies with desired characteristics. In our case the goal is to minimize the probability of virtual link overlap in domains that leads to erroneous selection of disjoint paths. This is achieved by minimizing OF values of all virtual links in each domain. We define the domain OF as the maximum value of all virtual links of’s as in (4) n o OF ¼ max of Bi;Bj ð4Þ Bk;Bl ðBi;BjÞðBk;BlÞ

Based on the domain OF we used the following fitness function (5): X wðI; JÞ eaOF þ

ð5Þ

ðI;JÞ

Minimizing the maximum value of results to upper bounded of values of all possible combinations of border pairs. The first exponential term, eaOF ; aims at reducing the OF for a particular domain. The value of a depends on the network. As a general

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P rule, a is set to a value which for full mesh network results to ea1  PðI;JÞ wðI; JÞ. In our experiments a assumed a value of 5 (five). The second term, ðI;JÞ wðI; JÞ i.e., the sum of all edge weights, is added to reduce the number of edges in each network, thus avoiding solutions like fully connected domains. Based on our experimental results in Sect. 5 below, minimization of the fitness function in (5) results to network topologies with overlap factors close to the minimum, usually close to the lower bound, 1 (one). Their connectivity is dense enough to allow for pairs of non-overlapping shortest paths, while avoiding excessive over designed solutions. The algorithm exhibited no sensitive behavior in selecting parameter a, provided that the first term in (5) dominates the value of the fitness function.

5 Experimental Results 5.1 Simulation Setup For evaluating our Topology Design Method, we used the BRITE topology generator [31] to produce randomly generated multi-domain network topologies. We selected the Barabasi–Albert’s method [32] to generate multi-domain graphs and the Waxman’s method [33] for comparisons with randomly generated intradomain topologies.2 The Baraba´si–Albert algorithm generates scale-free random graphs. Scale free networks refer to a network structure that preserves its topology characteristics independently of the number of its nodes. In such networks a few nodes are highly connected hubs forming the core of the network, while the rest are low connected forming its edges. The algorithm proceeds by adding new nodes to the network, one at a time. Each new node is connected to M existing ones with a probability that is proportional to the number of links of the existing node. The probability PI that the new node is connected to node I is kI PI ¼ P kJ J

where kI is the degree of node I. Heavily linked nodes (hubs) tend to quickly accumulate more links, while nodes with only a few links are unlikely to be chosen as the destination for a new link. The resulting degree distribution follows the power law Pk ffi k3 [34].

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The Barabasi-Albert’s method is a standard procedure to model intra-domain graphs, which follows the power-law [34]. We also performed a series of simulations on evaluating overlap factors towards identification of disjoint shortest paths. Apart from domain topologies generated via the power—law, we tried our algorithms on domains drawn from real ISPs topologies as reported in [35, 36]. Our experiments demonstrated that real ISP topologies lead to similar results (successful identification of inter-domain disjoint paths) as with topologies generated via the Barabasi–Albert algorithm.

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The Waxman method refers to generation of random topologies using the Waxman probability model for interconnecting the nodes. It proceeds by placing new nodes in 1 9 1 squares of a grid and connecting them to m existing ones according to the formula PU;V ¼ a  edðU;VÞ=bL where d(U,V) is the Euclidean distance between nodes U and V, a and b are parameters 0 \ a, b \ 1 and L is the maximum distance between two nodes. In our simulations we used the default values in BRITE, a = 0.15 and b = 0.1. This method produces domain topologies that imitate real network connections. Using the above methods, a number of random topologies can be generated with parameters (1) the number of domains N, (2) the number of nodes per domain n, (3) the intra-domain number of links per new node m and (4) the inter-domain number of links per new node M. The last two parameters directly affect the link density of the corresponding graphs. For purposes of our evaluation we compared multi-domain disjoint path calculations between random (based on the Waxman’s model) domain topologies and domain topologies designed as indicated by our Genetic Algorithm. Comparisons were performed on randomly generated multi-domain topologies. Interdomain connectivity and node placement were kept unmodified, but intra-domain edges were selected with the two models above. For all generated topologies, we randomly selected a set of nodes in different domains to serve as end-points for disjoint path calculations. With these nodes as end-points, we identified disjoint paths. Calculations were performed for pairs of end-points, assuming that domains announce Full Mesh (FM) abstractions (except the domain hosting the source node where full topology information is always available). These calculations are performed for both random and genetic variations over identical inter-domain topologies. For comparison with the actual optimal solution, they were repeated assuming full topology information across domains. We will refer to the disjoint paths calculated with full topology knowledge as optimal disjoint paths whereas the ones calculated based on the aggregated view as virtual disjoint paths. Virtual disjoint paths are composed of actual inter-domain links and virtual intra-domain links, the ones advertised by FM abstracted domain. Virtual paths are post-processed with virtual links replaced by the actual installed real paths they represent. Overall, for each pair of end-points we calculate three pairs of disjoint paths (virtual paths, their corresponding installed paths and ideal optimal ones) for (1) intra-domain topologies with randomly generated connectivity and (2) optimized connectivity via our genetic algorithm. We end up with six pairs of disjoint paths, with the same end-points, but calculated under different conditions. Based on the above setup we counted the following cases: 1.

False Positives: Selecting a virtual disjoint path but failing to create a corresponding installed one or selecting a virtual disjoint path with total weight different than the total weight of the corresponding installed disjoint path.

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False Negatives: Not finding a disjoint pair of paths in the aggregated networks when one exists in the un-aggregated network. True Negatives: Identifying correctly that a pair of disjoint paths does not exist, based on the aggregated topology. True Positives: The virtual disjoint paths in the aggregated topology and the corresponding installed ones have equal weights. These are the cases where the algorithm worked flawlessly.

In what follows we present numerical results for multi-domain topologies with different generation parameters. 5.2 Numerical Results We run two sets of experiments, the first with randomly sized domains in terms of number of nodes and link density and the second with all domains of identical node size and link density. 5.2.1 First Set of Experiments Our first set of experiments entailed domains of randomly varying generation parameters in terms of their size n, and their density as affected by parameter m with (n,m) 2{(12,1), (25,1), (50,1), (12,2), (25,2), (50,2), (25,4)}. The inter-domain network was composed of N = 30 such domains with inter-domain link density set to M = 2. In such environments there are domains that are dense with a lot of options for disjoint paths and domains that may not be able to offer backup options. Parameter m is only used in random domains and does not affect the genetically designed ones. The variety in size and density of domains results to networks similar to real ones, allowing us to simulate how our topology design approach will behave in real life conditions. For this set we generated 100 different multi-domain topologies (inter-domain links and node placement). On each topology, we executed on the average 20 disjoint path calculations (total of approximately 2,000 experiments, each identifying 6 variations of multi-domain disjoint path pairs, 3 for the random and 3 for the genetic intra-domain topologies as explained in Sect. 5.1 above). Table 1 depicts the results from the first set of experiments. We used the ideal optimal results as a basis for evaluating success (or failure) metrics for random and genetically designed domains. With FM aggregation the ability to discover the non-

Table 1 Overall results for the first set of experiments Italic values emphasize the cases where disjoint paths were successfully identified for better comparison between the two cases, random and genetically optimized

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Random (%)

Genetical (%)

False positive

37.42

2.71

False negative

0.00

0.00

True negative

0.00

0.00

True positive

62.58

97.29

100.00

100.00

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Fig. 5 Random and genetic deviation against optimal path selection

existence of disjoint path pairs can be severely reduced. So, as expected, there are no False or True Negative cases. False Positive cases occur when selecting virtual disjoint path pairs that cannot be installed or when the disjoint weight is not calculated correctly from the aggregated view of the network. In random domains both cases can occur, while in genetically designed domains identified virtual disjoint paths always correspond to a feasible installed disjoint path pair, mostly with equal weights; the number of cases with different weights is significantly reduced compared to random cases. If Fig. 5 we illustrate the average disjoint weight deviation from the ideal global optimal for installed paths if FM aggregation is used. Both random and genetically designed domains deviations are plotted. The deviation in both cases is quite small: On the average, the random case deviates 0.18% while the optimized case 0.08% Note, however, that we did not include in our average metric false positives that are due to erroneous assumption that disjoint pairs exist (this amounts to 14% for the random case; in the genetically designed case this is zero). 5.2.2 Second Set of Experiments As a second set of experiments, we created multi-domain topologies composed of domains with identical generation parameters n, m. The scope of this experiment is to study how the size of the domain affects disjoint path calculations. An increased domain size offers more flexibility in edge selection, reducing the effect of node placement. Recall that edge (link) weights were taken equal to the Cartesian distance of nodes to emulate real over-provisioned networks in which delay (our minimization metric) is roughly proportional to their distance (see Sect. 4.1.1), thus node placement is a key factor. We expect that as domain size increases the genetic algorithm will perform better given the larger set of decisions to minimize the fitness function that depends on the Overlap Factor for all border node pairs and on the weights of internal edges. Parameters m, M are assigned a value greater than 2 to avoid sparse topologies with no feasible disjoint path options. We executed approximately 8,000 experiments, on 100 multi-domain topologies (inter-domain

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links and node placement) for varying domain size n [ {12, 25, 50, 100} and for 20 disjoint path calculations per topology. As in Sect. 5.2.1 above, each run resulted into 3 pairs of paths (virtual, installed, optimal) for both random and genetically designed domains (total 6 pairs). In Fig. 6 the False Positive ratio per domain size is depicted. We omitted False and True Negative that are always zero with FM abstraction. True Positives are just the complement of False Positive percentages. As expected, the advantage of using genetically designed domains drastically increases with respect to domain sizes. Abstraction in randomly generated domains tends to blur internal path weights as the number of nodes increases. Note that the number of border nodes is determined by the inter-domain topology power law model. In all our experiments, the power law resulted into a maximum of 12 border nodes and this restricts the size of the FM aggregation; e.g., with five border nodes domains advertize 20 values, regardless of whether they have 12 or 100 internal nodes. This results to more possibilities for inconsistencies between real and aggregated topology. Figure 7 shows the domain Overlap Factor values in relation to domain size for random and genetically designed domains. The latter exhibit OF values very close to the lower bound of one for all cases, approaching it as size increases. Random

Fig. 6 Second set of experiments—false positive ratio per domain size (n)

Fig. 7 Second set of experiments—average overlap factor per domain size (n)

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Fig. 8 Second set of experiments—random and genetic deviation against optimal path selection per domain size (n)

domains are not designed with inter-domain disjoint path considerations so average OF values are much higher explaining mishaps in disjoint path selection. In Fig. 8 we plot the deviation of the disjoint weight values compared to the optimal ideal cases. Again we conclude that our design optimization guidelines result into small deviations 0.1% with the random case exhibiting 0.2%.

6 Conclusions End-to-end path resiliency is a common requirement in high speed connection oriented networks where a single cut or a failure strongly affects critical services or a large number of users. Multi-domain disjoint path computation is strongly influenced by the accuracy of network state information exchanged between domains. Thus, distance-vector based algorithms (like the ones used in BGP) are inadequate in this environment and link-state algorithms need to be adopted. The selected path finding algorithm needs to account for scalability concerns, but also for domain administration policies regarding privacy and security. Topology aggregation techniques are proposed as adequate solutions but they need extensions to address path resilience. To handle this issue we introduced a metric, the Overlap Factor (OF) that quantifies path overlap in domains. The OF can be passed as an additional parameter of the inter-domain information exchange model to evaluate disjoint endto-end paths. In this paper we used this metric as the basis of a topology design algorithm that if used by domain planners will create domain networks where this OF parameter is already minimized and thus not required to be announced. A network planner does not need any information on other domain internals or interdomain specifics to apply the proposed design algorithm. Once adopted, domains announce an abstract model of their network topology, based on the Full Mesh (FM) model. Announcements include a single parameter for each node pair, the virtual path weight, i.e., the shortest path weight amongst all border pairs of nodes (border nodes interconnect domains and interface with subscriber access networks).

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In suggesting design guidelines to domain planners, we propose a procedure based on Genetic optimization concepts. We selected a fitness function that strictly increases with OF. Based on the fitness function we specified a Genetic Algorithm that selects which domain nodes to connect as to minimize the fitness function’s value. The result is domain topologies that (1) always have disjoint primary and backup paths for each target node and (2) leverage FM aggregation with minimized overlap, thus limiting inconsistencies in disjoint weight advertisements between the actual and aggregated network. Results from our extensive simulations indicated that our design process exhibits a high success rate in multi-domain disjoint path selection compared to networks generated without inter-domain considerations. If domains adopt inter-domain design criteria they will significantly improve services offered to an increasing number of their subscriber base requesting multi-domain resilient connections. Acknowledgments This work was partially supported by the Greek Research & Technology Network (GRNET) within the GN2 Project (GE´ANT2) of the 6th Framework Program on Research & Technological Development, European Commission. The authors wish to express their gratitude to Afrodite Sevasti of GRNET and Dimitrios Kalogeras of NTUA for their insightful comments. Cees de Laat of the University of Amsterdam provided significant insight on network abstraction modeling.

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Author Biographies Angelos Lenis is a PhD Student at the National Technical University of Athens (NTUA). He is a researcher in the NETwork Management and Optimal DEsign laboratory (NETMODE) of the National Technical University of Athens. His research interests include network management, routing architectures and distributed systems. He has a Diploma degree in Electrical and Computer Engineering from National Technical University of Athens. Vasilis Merekoulias holds an Engineering Degree from the National Technical University of Athens— NTUA (1996). He is currently a full time researcher at the Network Management & Optimal Design Laboratory (NETMODE), School of Electrical & Computer Engineering, NTUA. Since his graduation, he was involved in several National & European projects on regulation and policy in the telecommunications

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sector, e.g. broadband market analysis, universal servise provisioning and interconnection models. Apart from regulatory issues, his research interests cover broadband technologies, peer-to-peer and ad-hoc networking. Vasilis Maglaris is Professor of Electrical & Computer Engineering at the National Technical University of Athens (NTUA) teaching and performing research on Computer Networks. He completed his studies in Athens and New York and held industrial and academic positions in the USA for ten years, before joining NTUA in 1989. In 1994, he was responsible for establishing GRNET, the Greek National Research & Education Network, serving as its Chairman from its inception (1995) to 2004. Since October 2004, he serves as the Chairman of the National Research & Education Networks Policy Committee (NREN PC). The NREN PC harmonizes policies amongst 34 NRENs in the extended European Research Area; it is also responsible for governance of their Pan-European interconnect GE´ANT/ GE´ANT2.

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