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Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks

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Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks Hongbin Luo, Student Member, IEEE, Lemin Li, and Hongfang Yu

Abstract—Reliability has been well recognized as an important design objective in the design of modern high-speed networks. While traditional approaches offer either 100% protection in the presence of single link failure or no protection at all, connections in real networks may have multiple reliability requirements. The concept of differentiated reliability (DiR) has been introduced in the literature to provide multiple reliability requirements in protection schemes that provision spare resources. In this paper, we consider the problem of routing connections with differentiated reliability in wavelength-division multiplexing (WDM) mesh networks when backup sharing is not allowed. Our objective is to route connections with minimum network cost (e.g., network resources) while meeting their required reliability. We assume connections arrive dynamically one-at-a-time and a decision as to accept or reject a connection has to be made without a priori knowledge of future arrivals. Since sharing cannot be used for achieving efficiency, the goal is to achieve efficiency by improved path selection. In this paper, we first present an integer linear programming (ILP) formulation for the problem. By solving the ILP formulation, we can obtain an optimal solution with respect to the current network state for each dynamical arrival. To solve the ILP formulation, however, is time consuming for large networks. We thus propose two approximation algorithms for the problem. The first one, called shortest-path-pair-based auxiliary graph (SPPA), can obtain an ε -approximation solution whose cost is at most 1 + ε times the optimum in O((n 2 (n + 1) + 2mn)(log log(2n) + 1/ ε )) time, where n and m are the number of nodes and links in a network, respectively. To reduce the computational complexity of the first algorithm, the second algorithm, called auxiliary graph-based two-step approach (ATSA), is proposed and can obtain a near optimal solution with cost at most 2 + ε times that of the optimal solution in O(mn(log log n + 1/ε )) time. Results from extensive simulations conducted on two typical carrier mesh networks show the efficiency of the two algorithms. Index Terms—wavelength-division multiplexing (WDM), mesh networks, reliability, differentiated reliability, routing algorithm. Manuscript received April 25, 2006, revised May 23, 2007. This work was supported in part by the National Science Foundation of China (NSFC) under Grant No. 60302010 and 60473001, and in part by the National Basic Research Program of China (“973 Program”) under Grant No. 2007CB307100 and 2007CB307104. Hongbin Luo is with the School of Electronic and Information Engineering, Beijing Jiaotong University, China, 100044. He was with the School of Communication and Information Engineering, University of Electronic Science and Technology of China (UESTC), Chengdu, China, 610054 (email: luohb@ uestc.edu.cn). Lemin Li and Hongfang Yu are with the Key Lab of Broadband Optical Transmission and Communication Networks, UESTC, Chengdu, China, 610054 (email: [email protected], [email protected]).

NOMENCLATURE

AP AS b BP BS DiR E G(V,E) ILP MADR r ra s, d V

O

Active path. Active segment. Bandwidth requirement of the current connection. Backup path. Backup segment. Differentiated reliability. The set of links. A graph representing a mesh network. Integer linear programming. Maximum acceptable downtime ratio. The required MADR of the current connection. the actual MADR offered to the current connection ` by the network. The source and destination of the current connection. The set of nodes. I. INTRODUCTION

PTICAL MESH networks employing wavelength-division multiplexing (WDM) technology have been recognized as the core networks of the next generation Internet. In such networks, any single link failure may lead to huge data loss since each fiber can provide a bandwidth of several terabits. Thus reliability is of paramount importance in such networks. Traditionally, optical networks offer two degrees of service reliability: full protection in the presence of a single fault in the network, and no protection at all. The current development trend, however, is gradually driving the design of networks towards a unified solution that will jointly support voice and data services, as well as a variety of novel multimedia applications. Evidence of this trend over the last decade is the introduction of concepts such as quality of service (QoS) [1, 2] and differentiated services [3, 4] that provide multiple levels of service performance in the same network [5, 6]. To capture this trend, the concept of differentiated reliability (DiR) [5] was introduced for the design of static protectionbased networks. According to the DiR concept, each connection at the network layer under consideration is guaranteed a minimum absolute reliability degree allowed for that connection. The reliability of each connection is individually chosen to match the application requirements by the client [5]. If a connection multiplexes multiple applications, the reliability is chosen to satisfy the reliability of the application whose reliability requirement is the most stringent.

Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks For simplicity, however, we in this paper only consider the case that a connection only multiplexes a single application. Connections are then created and assigned spare resources by each connection in such a way that the required reliability by each connection is met. For this purpose, an active path (AP) and, if necessary, a protection topology (in this paper, we call the backup path or backup segments (BSs) derived for protecting an AP as a protection topology) have to be derived. The ultimate objective of DiR is to satisfy the reliability degree required by a connection while minimizing the network cost (e.g., network resources) consumed by it. In what follows, we will outline previous work and motivate our study. A. Previous Work The problem of routing connections with DiR requirements has been studied by several researches in the literature. In [5], the concept of DiR was first introduced and applied to provide multiple reliability degrees in WDM rings using dedicated path protection switching. While the study in [5] assumes that each network node is equipped with wavelength converters, the study in [6] further extends the concept of DiR to the design of WDM rings without wavelength converters. In both studies, the authors assume that a set of connections is given and the objective is to minimize the total network cost such that all connections can be routed while guaranteeing their required reliability. Furthermore, preemption is allowed in both studies. An efficient algorithm, called difficult-reuse-first (DRF) has been proposed for the problem with/without wavelength converters. The basic idea lies in DRF is that, prior to being routed, connections are sorted taking into account how difficult it is for their corresponding APs to reuse any of the already provisioned BPs connections that are more difficult to handle in that regard are routed first. While the studies in [5] and [6] place their focuses on WDM ring networks, the study in [7] consider the problem of routing connections with DiR requirements in WDM mesh networks. Similar to [5] and [6], the authors in [7] assume that a set of connections is given and the objective is to minimize the total network cost. A two-step solution based on simulated annealing, called shared path protection with DiR (SPP-DiR), was proposed. With SPP-DiR, as the first step, a conventional SPP design problem is solved to determine the AP and BP for each connection. Since the outcome of the first step can provide 100% survivability degree against any single link failure, the second step of SPP-DiR is to reduce the reliability degree of some connections with the objective of further reducing the network cost. In both steps of SPP-DiR, simulated annealing is used. While above studies all focus on protection schemes, the authors in [8] further extend the concept of DiR to restoration schemes and propose three restoration schemes. While above studies all assume that there is only a single failure in the network, the following studies consider the case that the network links fail independently to each other. In [9], the concept of quality-of-protection (QoP) is proposed. With QoP, several classes of QoP are predefined and each connection request is assigned a QoP class. The connection

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request is then routed according to the reliability of its QoP class. In [10], a partial backup scheme is proposed. With the partial backup scheme, a single backup segment (BS) will be derived to protect the AP if a single AP cannot satisfy the reliability requirement. In [11], connections are classified into two categories. In the first category, connection requests can be routed using a single AP. While in the second category, connection requests cannot be accommodated using one AP. For each of the two categories, an integer linear programming (ILP) formulations is proposed. The drawback of this approach is its computation complexity. A partial backup scheme similar to that in [10] is proposed in [12]. Different from that in [10], where only one BS is allowed to protect an active segment (AS), the study in [12] allows to protect an AS using two or more overlapping BSs (here, overlapping means that two BSs simultaneously protect at least one same link in the AP). Furthermore, spare capacity sharing is allowed in [12]. As pointed out in [12], however, to allow the sharing of spare capacity will dramatically increase the computation complexity of an algorithm since the sharing of a BS reduces the reliability offered to those existing connections that use the BS as backup. In [13], the authors study the problem of routing connections with both reliability and restoration time requirement. B. Motivation In this paper, we turn back to consider the case that there is only a single link failure in the network. As stated above, prior work on DiR all focus on the optimal network design aspect under static traffic environment. That is, they all assume that connections are given and the design objective is to minimize the total network cost. In this paper, we study the problem of routing connections with DiR in WDM mesh networks when backup sharing is not allowed. To our knowledge, this is the first time to study the problem. As is widely used in the literature [7, 14 - 16], we assume that only one link fault may occur in the network at once, i.e., the probability that two or more links are down at the same time is assumed to be negligible. Our objective is to minimize network cost (e.g., network resources) for one-at-a-time arrivals while meeting their required reliability. We also assume that connections arrive dynamically and there is no a priori knowledge of future arrivals and a decision as to accept or reject a connection has to be made according to the current network state. Notice that, although sharing of spare capacity [17 - 21] can often lead to higher resource efficiency, a complex restoration process is required and the restoration time cannot be guaranteed. Thus in this paper we focus on the non-sharing case, i.e., sharing of spare capacity is not allowed. Since we cannot achieve efficiency by sharing, our objective here is to improve performance by improved path selection. Also notice that in the path selection algorithms presented here, we will only be concerned with link-disjointedness. However, if protection against single node failures is required, the selected paths have to be node-disjoint. The node-disjoint path selection problem can however be transformed to the link-disjoint path selection problem by node splitting, and therefore our

Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks algorithms are applicable in that case too. C. Contributions In this paper, we first explore some important properties of an optimal solution. We next present an integer linear programming (ILP) formulation for the problem. By solving the ILP formulation, we can obtain an optimal result with respect to the current network state for each dynamical arrival. To solve the ILP formulation, however, is time consuming for large networks. To address the high computational complexity, we further propose two approximation algorithms for the problem. The first algorithm, called shortest-path-pair-based auxiliary graph (SPPA), can obtain an ε -approximation solution whose cost is at most 1+ ε times the optimal one in O((n 2 (n + 1) + 2mn)(log log(2n) + 1/ ε )) time, where n and m are the number of nodes and links in a network, respectively. To reduce the computational complexity of the first algorithm, we propose the second algorithm, called auxiliary graph-based two-step approach (ATSA). The ATSA algorithm can obtain a near optimal solution with cost at most 2 + ε times that of the optimal solution in O(mn(log log n + 1/ε )) time. Results from extensive simulations conducted on two typical carrier mesh networks show the efficiency of the two algorithms. The rest of this paper is organized as follows. Section II formally presents the network model under consideration. Section III presents some important properties of an optimal solution. Section IV presents an ILP formulation for the problem. Section V describes in detail the proposed two novel and efficient algorithms. Section VI presents numerical results and Section VII concludes this paper. II.

NETWORK MODEL AND PROBLEM STATEMENT

In this section, we describe the network model and the problem addressed in this paper. A. Network Model We represent the mesh WDM network under consideration by an undirected graph G(V, E), where V is the set of network nodes representing optical cross-connects (OXCs) and E is the set of network links representing optical fibers. Let n and m denote the number of nodes and links in the network, respectively. We assume that each link can only accommodate one optical fiber, which carries W wavelengths. Undirected means that each link in the physical topology is bi-directional. We further assume that a link fault disrupts connections in both directions of propagation. Each link (i, j) is characterized by two parameters: the cost c(i, j) of providing one fiber on that link and the link failure probability Pf(i, j). The link failure probability is defined as the conditional probability that the considered link is faulted given the occurrence of a single fault. Due to the single fault assumption made earlier, this probability numerically equals the link downtime ratio, i.e., Pf(i, j) = τ ⋅ length(i, j ) / SFP , where τ is the link downtime ratio normalized to the link length and can be estimated according network statistics, length(i, j) is the length of link (i, j), and under the single failure assumption, SFP is the overall network

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1

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6

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0.1 0.1 3

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5

Fig. 1. An example network.

single fault probability, i.e., SFP =

∑ (i, j )∈E length(i, j )

⋅τ .

Under the single failure assumption, for a path P in G, the failure probability of a path P is the sum of the link failure probability of links along the path P. Without loss of generality, the cost of a path P, say C(P), is the sum of the link cost of links along it. That is (1) C ( P ) = ∑ c(i, j ) ( i , j )∈P

We consider the connection to be defined by a quadruple (s, d, b, r), where s and d are the ingress and egress of the connection, respectively, b is the bandwidth (in number of wavelengths) required by the connection, and r is the required MADR of the demand (in this paper, we interchangeably use the terms connection and demand without any difference). The connection downtime ratio is the fraction of time during which the connection is unavailable due to a network failure [7]. In simple terms, the connection downtime ratio is the probability that the demand fails to work due to a network fault, and in this paper, a link failure. In this paper, we assume that connections arrive online, one-at-a-time and there is no a priori knowledge of future arrivals. We also assume that splitting of a connection along multiple routes is not allowed. Our goal is to minimize the network cost for the arrived connection while meeting the required MADR of the connection. That is, the actual MADR, say ra, offered by the network to the connection should be less than the required MADR, r. For this purpose, an AP, and if necessary, a protection topology have to be derived such that the required MADR can be met or a decision to reject the demand has to be made according to the current network state. The following example explains why a protection topology is required for a connection in some cases. Consider the network shown in Fig. 1, where the number next to each link is the conditional link failure probability of that link and each link has a cost of 1. Now assume a connection request – (1, 6) with a bandwidth requirement of 1 wavelength channel – arrives at the network. If this demand requires an MADR of 0.3, we can choose the path 1 – 2 – 4 – 6 with a least cost of 3 as the AP of the demand since the actual MADR offered to the demand is exactly the required MADR of the demand. If the demand requires an MADR of 0.2, however, there is not a single path that meets the required MADR of the demand. In this case, we have to derive an AP for the demand and then protect the whole AP or protect part of the AP such that the required MADR can be met. In general, to protect the whole AP requires higher network cost than to protect part of

Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks the AP. In this example, the least cost link-disjoint path pair (i.e., 1 – 2 – 4 – 6 and 1 – 3 – 5 – 6) from node 1 to node 6 has a network cost of 6 and offers an actual reliability of 0 to the demand. If we choose the path 1 – 2 – 4 – 6 as the AP for the demand and the BS 3 – 5 – 6 to protect the active segment (AS) 4 – 6, as shown in Fig. 1, however, the network cost is reduced to 5 and the required MADR can also be met with an actual offered MADR of 0.2. Of course, if the first demand requires an MADR of 0, we have to derive two link-disjoint paths (e.g., 1 – 2 – 4 – 6 and 1 – 3 – 5 – 6) for this demand with a cost of at least 6 since any single path cannot meet the required MADR of the demand. Notice that, under the single link failure assumption, the actual MADR offered to a demand is determined by the failure probability of those links (in the AP) that are not protected, since the failure of those links that have been protected can be restored quickly by using the protection topology derived for the AP. That is, the actual MADR offered to a demand can be expressed as follows: (2) ra = ∑ Pf (i, j ) ( i , j )∈ AP

where AP is the set of unprotected links in the AP. Therefore, the failure probability of a link-disjoint path-pair is zero, irrespective the failure probabilities of the paths, since all links in the link-disjoint path-pair are protected. With above definitions, we formulate the problem addressed in this paper as follows. Problem PWD (Protection with DiR): Given the current network state and a connection request (s, d, b, r), to derive an AP, and if necessary, a protection topology for the demand such that: 1) The cost of the AP and the protection topology is minimized; 2) ra ≤ r. In general, the problem PWD is NP-hard since its special case that a protection topology is not allowed is just the wellknown restricted shortest path (RSP) problem, which is NPhard [22]. In what follows, we first explore some important properties of an optimal solution, which is followed by an ILP formulation and two approximate algorithms. III.

PROPERTIES OF AN OPTIMAL SOLUTION

In this section, we first explore some important properties of an optimal protection topology with respect to a given AP. With these properties, we next investigate some properties of an optimal solution for problem PWD. Property 1: Any two different BSs in an optimal protection topology cannot be nestled, one within the other. Proof: Let BSi and BSj are two different BSs in an optimal protection topology and their corresponding ASs are ASi and ASj, respectively. Without loss of generality, we assume that BSi is nestled in BSj, as shown in Fig. 2. From Fig. 2, one can see that any link l in ASi is protected by BSj. Thus there is no need to include BS BSj in the optimal protection topology.  Property 2: Two BSs in an optimal protection topology cannot share any common node except the nodes in the AP. Proof: Let BSm = {sm, ···, tm} and BSn = {sn , ··· , tn} are two different BSs in an optimal protection topology. Their ASs are

4 BS i BS j

sj

si

tj

ti

Fig.2. Illustration for two BSs to be nestled.

vi

BS m

sm

BS n

tn

sn = t m = v j (a) vi

BS m

sm

sn

BS n

(b)

tm

tn

Fig. 3. Illustration for two backup segments to share a node.

ASm and ASn, respectively. We further assume that C(BSm) and C(BSn) are their costs, respectively. Obviously, there are two cases for two BSs to share a node, as shown in Fig. 3. In both cases, we can obtain a new BS, by concatenating the path from sm to vi and the path from vi to tn, with smaller network cost than the sum of C(BSm) and C(BSn), which contradicts the assumption that the protection topology is optimal.  From property 1 and property 2, we immediately have the following property. Property 3: A link in an AP cannot be protected by three or more BSs in an optimal protection topology. Property 3 means that links in an AP can be classified into the following three classes: 1) Protected by one BS; 2) Protected by two BSs, or 3) Unprotected. Definition 1: a link is said to be 2-protected if it is protected by two BSs, and is 1-protected if it is protected by one BS. Consider the simple network shown in Fig. 4, where the bold lines constitute the AP and the dotted lines form an optimal protection topology. In this example network, the link (v4, v5) is unprotected while the link (v1, v2) is 2-protected. The rest links in the AP are 1-protected. While above properties describe the characteristics of an optimal protection topology with respect to a given AP, the following theorem states that each link in an optimal solution for problem PWD can be either 1-protected or unprotected. Theorem 1: Each link in an optimal solution for problem PWD can be either 1-protected or unprotected. Proof: We first notice that links in an optimal solution for problem PWD fall into the categories: unprotected, 1-protected, or 2-protected. Thus we only need to prove that links cannot be

Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks v10

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Fig. 4. Example of a network used for illustrating definition 1.

2-protected. In the following, we prove this by contradiction. Indeed, if a link in an optimal solution is 2-protected, it must be true that the link (vi, vj) is protected by two BSs, say BS1 and BS2, as shown in Fig. 5. Since the solution is optimal, the sum of the cost of the two BSs and that of the path P = v0–…– vi– vj– …–vn must be minimized among all possible APs and protection topologies from node v0 and vn. Let us consider the two paths P1 = v0 – vi – v3 – vn and P2 = v0 – v4 – vj – vn. Obviously, C(P1) + C(P2) < C(BS1) + C(BS2) + C(P) since the cost of link (vi, vj) is not counted by the left term. Furthermore, the two paths must be link-disjoint since the two BSs are link-disjoint; otherwise, the two BSs will share a common node, which contradicts property 2. Therefore, if we select path P1 as the AP and P2 as the protection topology from node v0 to node vn, the cost of the new AP and protection topology formed by P1 and P2 must be less than the old one, which contradicts the assumption that the cost of the old one is minimized. We thus conclude that any link in an optimal solution for problem PWD can be either 1-protected or unprotected.  Definition 2: An AS together with its BS is called a protection domain. Furthermore, we say that the consecutive unprotected links in an optimal solution form an unprotected segment. We further say that the node at which a protection domain and an unprotected segment intersect is an end node of an unprotected segment or a protection domain. It is notable that there may be no protection domain(s) or unprotected segment(s) in an optimal solution. If there do exist both protection domains and unprotected segments, from theorem 1, they must interchangeably appear in the optimal solution. Here, interchangeably means that a protection domain is immediately followed by an unprotected segment, or that an unprotected segment is immediately followed by a protection domain. Furthermore, from theorem 1, we have that a protection domain can be formed by a shortest link-disjoint path pair between the two end nodes of the protection domain. Formally, we have the following corollary. Corollary 1: The protection domains and unprotected segments must interchangeably appear in the optimal solution if there are both protection domains and unprotected segments in an optimal solution. Furthermore, a protection domain in an optimal solution is formed by a shortest link-disjoint path pair between the two end nodes of the protection domain. From theorem 1, we further observe that we can form a path from s to t by concatenating the ASs and the unprotected segments in the optimal solution. Similarly, we can form another path from s to t by concatenating the BSs and the unprotected segments in the optimal solution. This observation

vn

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Fig. 5. Example of a network used for proof of theorem 1.

v9 v0

v1

v4 v2

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Fig. 6. Example of a network used for illustration of the two corollaries.

is formally presented in the following corollary. Corollary 2: We can form a path from s to t by concatenating the ASs and the unprotected segments in the optimal solution. Similarly, we can form another path from s to t by concatenating the BSs and the unprotected segments in the optimal solution. The following example illustrates the two corollaries. Consider the network shown in Fig. 6, where the bold lines form the optimal solution for problem PWD. In this example, there are two protection domains and two unprotected segments. The first protection domain is formed by links (v0, v1), (v1, v2), (v0, v9) and (v9, v2) while the second one is formed by links (v4, v5), (v5, v6), (v6, v7), (v4, v11) and (v11, v7). The first unprotected segment is formed by links (v2, v3) and (v3, v4) while the second one is formed by the link (v7, v8). As shown, the protection domains and the unprotected segments are interchangeably appeared. Furthermore, two paths can be formed. The first one is the path P1 = v0 – v1 – v2 – v3 – v4 – v5 – v6 – v7 – v8 and the second one is the path P2 = v0 – v9 – v2 – v3 – v4 – v11 – v7 – v8. IV.

MATHEMATICAL (ILP) FORMULATION

In this section, we will present an elegant ILP formulation for the problem PWD. The basic idea, which stems from Corollary 2, of the ILP formulation is to find two paths from s to t such that 1) the total cost of the two paths is minimized (if a link is in both paths, its link cost is counted only once); and 2) the sum of the link probability of those links that are in paths is less than the required MADR of the current connection request. To proceed, we first introduce the following notations used in our ILP formulation. i, j endpoints of a physical fiber link that might occur in a lightpath. s, d source and destination of the current connection request. b bandwidth requirement (in number of wavelengths) of the current connection request. r the required MADR of the current connection request. k index of path number. k = 1, 2 in this paper. Given:

Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks the number of free wavelengths in link (i, j). Fij c(i, j) the cost of link (i, j) to carry unit bandwidth from node i to node j. Pf(i ,j) the failure probability of link (i, j). N number of nodes in the network. Variables: X ijk a Boolean variable indicating whether or not link (i, j) is in path k (= 1, 2). If link (i, j) is in path k, X ijk = 1; otherwise, X ijk = 0. Yij

a Boolean variable indicating whether or not link (i, j) is in both paths. If link (i, j) is in both paths, Yij = 1; otherwise Yij = 0. Minimize: (3) ∑∑ X ijk ⋅ c(i, j ) − ∑ Yij ⋅ c(i, j ) ij

k

ij

Subject to: ⎯ Flow conservation constraints: ∀k ,

∑

X sjk

∀k ,

=1

(4)

∑ X isk = 0

(5)

∑ X idk = 1

(6)

∑ X djk = 0

(7)

i

∀k ,

i

∀k ,

j

∀k , ∀i ≠ s, d ,

∑ X ijk = ∑ X kji j

⎯ ⎯

(8)

j

Bandwidth constraints: ∀k , ∀i, j , X ijk ⋅ b ≤ Fij

(9)

MADR constraints: ∀k , ∀i, j , Yij ≤ X ijk ∀i, j ,

∑ X ijk ≤ Yij + 1

(10) (11)

k

∑ Yij ⋅ Pf (i, j ) ≤ r

MADR can be met. That is, the actual reliability offered to the connection request should be less than the required. Number of variables and constraints: Since the complexity of the ILP model is proportional to the number of variables in the system, we count them to obtain an insight into the complexity of the formulation. It is a straightforward matter to verify that the number of variables in the formulation is O(n2), which is a count that grows quadratically with the number of nodes in the network. The number of constraints in the above ILP formulation is bounded by O(n2), which is also a count that grows quadratically with the number of nodes in the network. By solving the above ILP formulation, we can obtain an optimal result with respect to the current network state for each arrival of requests. To solve the ILP formulation, however, is time consuming for large networks. We thus in the next section propose two approximation algorithms for the problem. V.

j

(12)

ij

Explanation of Equations: The above equations are based on principles of conservation of flows and resources. Equation (3) shows the optimization objective function. Equations (4) – (8) are flow conservation constraints. Equation (4) states that, for each path, the source node has exactly one outgoing edge. Equation (5) says that, for each path, the source node has no incoming edge. Equation (6) ensures that, for each path, the destination node has exactly one incoming edge. Equation (7) says that, for each path, the destination node has no outgoing edge. Equation (8) ensures that, for each path, its intermediate node has equal number of incoming edge and outgoing edge. Equation (9) is the bandwidth constraint. It guarantees that, if a link is employed by a path, its free bandwidth should be larger than the bandwidth requirement. Equations (10) - (12) are MADR constraints. Equation (10) says that, if a link is in both paths, it must be in each path. Equation (11) shows that if a link is only in one path, it cannot on both paths. Equation (12) guarantees that the required

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APPROXIMATION ALGORITHMS

In this section, we proceed to present the two approximation algorithms proposed for the problem PWD. We begin with the first algorithm, which is called shortest-path-pair-based auxiliary graph for the problem PWD (SPPA) and obtains an approximation solution whose cost is at most 1 + ε times that of the optimal one in O((n 2 (n + 1) + 2mn)(log log(2n) + 1/ ε )) time. In order to reduce the computational complexity required to obtain such an approximate solution, we then propose the second approximate algorithm, which is called auxiliary graph-based two-step approach (ATSA) and can obtain a near optimal solution whose cost is at most 2 + ε times that of the optimal solution in O(mn(log log n + 1/ε )) time. A. The SPPA Algorithm From Corollary 1, we have that each protection domain in an optimal solution must be formed by a shortest link-disjoint path pair between the two end nodes of that protection domain. Inspired by this corollary, we in this subsection propose to derive a near optimal solution for problem PWD by generating a shortest path pair based auxiliary graph. In what follows, we describe in detail how to generate such an auxiliary graph. For a physical network, we generate an auxiliary graph AG, which has two layers, say shortest path pair (SPP) layer and physical layer. The physical layer represents the residual network, which can be obtained by pruning from the initial network those links whose free capacity is less than the bandwidth requirement. On the other hand, the SPP layer represents whether or not there is a shortest path pair between each node pair in the residual network. In order to generate such an auxiliary graph, we split each node in the physical network into two vertices, say SPP vertex and physical vertex. Each SPP vertex represents a vertex in the SPP layer while each physical vertex represents a vertex in the physical layer. Furthermore, we introduce the following three types of edges. 1) SPP edge, which is an edge between two SPP vertices; 2) Physical edge, which is an edge between two physical

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(1, 0.1) (1, 0.1)

C' E A'

(1, 0.2)

(1, 0.2) (1, 0.1)

F'

(1, 0.1) B'

(a)

physical layer

D'

E' (b)

Fig. 7. An illustrative example for constructing an auxiliary graph in the SPPA algorithm. (a) A six-node physical network, where the numbers next to each link represent the link cost and the failure probability of that link. It is assumed that each link (except the one with bold line, which has no free capacity) has enough free capacity for the current connection request. (b) The generated auxiliary graph, where the numbers next to each edge represent the cost and the failure probability of that edge.

vertices; Connecting edge, which is an edge between an SPP vertex and a physical vertex. Upon a connection request arrival, we generate an auxiliary graph as follows. 1) Generate the SPP layer: First, for each node on the physical network, add an SPP vertex to the SPP layer. Next, for each vertex pair in the SPP layer, add an SPP edge to the SPP layer if there is a shortest path pair between the node pair (corresponding to the vertex pair) in the residual network. At last, for each SPP edge, assign it a failure probability of 0 and an edge cost, which equals to the total cost of the shortest path pair corresponding to the SPP edge. 2) Generate the physical layer: First, for each node in the physical network, add a physical vertex to the physical layer. Next, for each physical vertex pair, add a physical edge to the physical layer if the free capacity of the link between the node pair (corresponding to the vertex pair) is larger than the bandwidth requirement. At last, for each physical edge, assign it a failure probability, which equals to the failure probability of the link corresponding to the edge, and an edge cost, which equals to the cost of the link corresponding to the edge. 3) Connecting the two layers: Within each node in the physical network, add a connecting edge between its SPP vertex and its physical vertex. For each connecting edge, assign it a cost of 0 and a failure probability of 0. As an example, consider a six-node network shown in Fig. 7 (a), where the numbers next to each link represent the link cost and the failure probability of that link. We assume that each link (except the one with bold line, which has no free capacity) has enough free capacity for the current connection request, say (A, F) with a reliability of 0.1. Fig. 7 (b) shows the generated 3)

auxiliary graph upon arrival of the connection request. As shown, the physical layer represents the residual network by pruning the link (B, E), which has no sufficient free capacity for the current connection request. In the SPP layer, each edge represents a shortest path pair. For example, the edge (A”, B”) represents a shortest path pair, say path A – B and path A – C – B between node A and node B, in the physical network. Since the cost of the shortest path pair is 3, the edge (A”, B”) is assigned a cost of 3. Furthermore, a link failure probability of 0 is assigned to it. After the auxiliary graph has been generated, we now derive an approximate solution for problem PWD. Notice that, in the auxiliary graph, each edge has two parameters: the edge cost and its failure probability. Obviously, those parameters are additive. Our objective is now to compute an approximate solution for problem PWD such that the cost of the approximate solution is as small as possible while the required MADR is met. From the perspective of multi-constraint routing, this is just the restricted shortest path problem, which has been extensively studied in the literature [23 - 25] and several approximate algorithms have been proposed for it. In this paper, we employ the simple efficient approximation (SEA) algorithm proposed in [23] to obtain a solution for problem PWD. Let the path obtained by the SEA algorithm be P. The SPPA algorithm then maps each link in path P back to the physical network. This step is required because some links in path P may represent a shortest path pair in the physical network. By this way, the approximation solution can be obtained. We show the main steps of the SPPA algorithm in Fig. 8. In the example shown in Fig. 7, we can obtain a solution (the path constituted by the dashed lines) using the SEA algorithm in the auxiliary graph. By mapping the links in the solution back to the physical links, we can obtain the approximation solution. For example, the link (A”, C”) in the path P should be mapped back to the links (A, B), (A, C) and (B, C) since the

Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks SPAA algorithm Input: A network G(V,E), the free capacity Fij on each link and the failure probability of each link. An source node s and a destination node d between which a connection of b units of bandwidth and ra MADR have to be routed. Output: An AP and its protection topology for the connection such that the required MADR can be met while the total cost is minimized. Algorithm: 1. Generate an auxiliary graph according to current network state; 2. Run the SEA algorithm on the auxiliary graph to get an approximation solution. Let the path got at this step be P 3. Map each link in the path P back to the physical networks.

Fig. 8. SPPA algorithm.

link (A”, C”) in the auxiliary graph represents the shortest path pair from node A to node C in the physical network. Theorem 2: The SPPA algorithm returns an approximate solution whose cost is at most 1 + ε times the optimum in O(( n 2 (n + 1) + 2mn)(log log(2n) + 1/ ε )) time. Proof: We have to prove two aspects, i.e., 1) the cost of the approximate solution is at most 1 + ε times the optimal, and 2) the computational time is O((n 2 (n + 1) + 2mn)(log log(2n) + 1/ ε )) . The proof for the first aspect can be easily obtained from Corollary 1 and the correctness of the SEA algorithm. For the second aspect, we notice that there are two phases in algorithm SPPA. The first phase is to generate the auxiliary graph and the second phase is to run the SEA algorithm in the auxiliary graph. In the first phase, we need to compute, for each node pair in the physical network, a link-disjoint shortest path pair. By using the Suurballe’s shortest path pair algorithm [18], to compute a link-disjoint shortest path pair entails a time of O(n2). Since there are n(n-1)/2 node pair, the first phase then entails a time of O(n4). However, if those shortest path pairs are periodically computed offline, the first phase entails only a time of O(n2). Since the auxiliary graph generated above has exactly 2n vertices and at most n(n + 1) / 2 + m edges, the computational time of second phase is O((n 2 (n + 1) + 2mn)(log log(2n) + 1/ ε )) . Therefore, the computational complexity of the SPPA algorithm is O((n 2 (n + 1) + 2mn)(log log(2n) + 1/ ε )) in the worst case.  B. The ATSA Algorithm Although the algorithm SPPA can obtain an approximation solution whose cost is at most 1 + ε times the optimum, the computational time required by it, however, is high. In this subsection, we propose the ATSA algorithm, which can improve the computational time by a factor of n but with a looser bound. The ATSA algorithm employs a two-step approach. In the first step, a least cost path, called AP, is computed without considering the MADR requirement. In the second step, an auxiliary graph with respect to the AP is generated. The SEA algorithm then runs on the auxiliary graph to obtain a near optimal protection topology for the AP such that the required MADR can be met. Assume that the AP has been derived using a shortest path

8 v10

v9

d = v8 s = v0 v1

v2 v11

v3

v4

v5

v12

v6

v7

v13

(a) Initial graph G with bold lines constituting the AP

v10

v9

d = v8 s = v0 v1

v2 v11

v3

v4

v5

v12

v6

v7

v13

(b) The auxiliary graph G

v10

v9

d = v8 s = v0 v1

v2 v11

v3 v12

v4

v5

v6

v7

v13

(c) The resulting BP (composed by the dotted lines) Fig. 9. An example used for illustrating the ATSA algorithm.

algorithm (such as Dijkstra’s shortest path algorithm [26]). We now place our focus on how to construct an auxiliary graph G = (V, E) used for computing the near optimal protection topology. Let V = V and the link set E is constructed as follows. 1) For each link in the AP, add two directed links into G: one is along the direction of the AP and the other is reverse to the direction of the AP. Assign each of those links along the direction of the AP with a link cost of zero and a link failure probability as that of the corresponding links in the initial graph G. Further assign those links reverse to the direction of the AP with a link cost of zero and a link failure probability of zero. 2) For the rest links, add them into the auxiliary graph G and assign each link a link failure probability of zero. The link cost of those links is equal to that of the corresponding links in G. The reason we assign each of those links a link failure probability of zero is that, if a link is selected by an algorithm as part of the protection topology, it is used to protect the AP. That is, the failure of those links cannot lead to the failure of a connection. For example, we can construct an auxiliary graph G (as shown in Fig. 9 (b)) with respect to the initial graph G and the AP (bold lines) shown in Fig. 9 (a). Notice that each link in the auxiliary graph G has two additive parameters: the link cost c(i, j) and the link failure probability Pf(i, j). Our objective is now to derive a least cost path BP in G such that the cost of BP is minimized while the failure probability of the path is not greater than the required MADR. This is again the well-known RSP problem. We now prove that the BP derived above can meet the required MADR.

Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks Theorem 3: The BP derived by the ATSA algorithm meets the required MADR. Proof: For any BP derived as above, there are three possible cases: 1) There are no common links between the BP and the AP. In this case, all links in the AP can be protected by the BP. Thus the actual MADR offered to the current connection is zero. This implies that the required MADR can be met. 2) There are common links between the AP and the BP, but these links are in a reverse direction. In this case, these common links are 2-protected. For example, the AP and BP in Fig. 9 share the common link (v1, v2) in a reverse direction, as shown in Fig. 9 (c). Notice that the link failure probability in each of these links is assigned to be zero. Thus these links do not contribution to the failure probability of the BP since they can be protected from any single link failure. 3) There are common links between the AP and the BP and these links are in the same direction. For example, the AP and BP in Fig. 9 share the common link (v4, v5) in the same direction, as shown in Fig. 9 (c). In this case, each of these common links is unprotected. Thus these links will contribute to the actual MADR offered to a connection. From the above analysis and the way of generating the auxiliary graph, we can see that if the failure probability of the derived BP meets the required MADR, the actual MADR offered to a connection must be no greater than the required MADR.  Assuming that the BP has been derived, we now can obtain the protection topology by pruning the 2-protected and unprotected links from the BP. We show the pseudo-code for the ATSA algorithm in Fig. 10. Theorem 4: The ATSA algorithm obtains for problem PWD an approximation solution whose cost is at most 2 + ε times the optimum in O(mn(log log n + 1/ε )) time. Proof: Let OPT be the optimal solution and C(OPT) be the cost of OPT. In the first step of the ATSA algorithm, a least cost path, say AP, is computed in O(n2) time. Obviously, the cost of the AP is not greater than that of the OPT, i.e., C(AP) ≤ C(OPT). In the second step, the SEA algorithm runs on the auxiliary graph and obtains a BP whose cost is at most 1 + ε times the optimum in O(| E || V | (log log | V | + 1/ε )) time. Notice that |V| = |V| = n and |E| = |E| = m. Thus the time required by the second step is O(mn(log log n + 1/ε )) . Further notice that the cost and failure probability of each link in G equal to or less than those of each link in G. The cost of an optimal BP, say BPopt, in G must be less than that of the OPT. That is, C(BPopt) ≤ C(OPT). Since C ( BP) ≤ (1 + ε ) ⋅ C ( BP opt ) , we have that the cost of the BP derived in the second step must satisfy C ( BP) ≤ (1 + ε ) ⋅ C (OPT ) . Therefore, the cost of the solution obtained by running the ATSA algorithm is at most 2 + ε times the optimum since C(AP) + C(BP) ≤ C (OPT ) + (1 + ε ) ⋅ C (OPT ) = (2 + ε ) ⋅ C (OPT ) .

9

Fig. 10. ATSA algorithm.

The computational time of the ATSA algorithm depends on three parts: the time required to derive to least cost path AP, the time required to construct the auxiliary graph, and the time required to run the SEA algorithm. The times required by the first and third parts have been shown above. For the second part, we can easily verify that the time required to construct such an auxiliary graph is in the order of the number of edges in the initial graph G. That is, the time required by the second part is O(m). Therefore, the computational complexity of the ATSA algorithm is O(mn(log log n + 1/ε )) .  C. Resource Allocation and Release Once an algorithm has computed a feasible route for each dynamic arrival, we need to allocate resources for this arrival. For this purpose, we only need to decrease the number of free wavelengths along the route by b. After a certain duration, the connection request may depart the network. Thus we need to release to resources allocated to the connection. To this end, we also only need to increase the number of free wavelengths along the route of the connection by b. VI.

NUMERICAL RESULTS

In this section, we present numerical results for our solution approaches through experiments, which are coded in C++ and run on a lightly-loaded PC with 1.8 GHz CPU and 512 MB memory. We use the CPLEX [29] software to solve the ILP formulations. Our experiments are divided into two parts. The first part investigates the effect of reliability requirements on the average cost for routing a connection. The second part compares the two approximation algorithms, SPPA and ATSA proposed in Section V, with the ILP formulation proposed in Section III. The performance metrics that we use are the average cost for accommodating a connection by an algorithm and the blocking probability, which is the ratio of the number of blocked connections to the total number of connections injected into a network.


1

120 100

5

65

2 65

4

14 85

6.4

80

85 19 65 65 55 20 65 15 55 65 65 55 65 21 55 65 55 75 60 12 75 16 55 22 7 9 60 55 6560 80 60 60 17 60 23 13 80 80

65 65 6 55 3 60

70

10

18 6.2

90

60 65 11

8

5.8

50

40 0

40

30 4

40 50

50

8

30 60

18

50

4.8 4.6

0.04

0.08

0.12

Distribution 1 Distribution 2 Distribution 3

6.2 6.0 5.8

15

65

70

80 40 19

0.20

(a) US-NET

20

10 12

0.16

p3

6.4

9

70 90

14

6

30

40

110

5.0

4.0 0.00

60 40

5.2

3

30

30

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11

110

40

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2

5.4

4.2

45 13

75 17

80

80 16

40 80 20

(b) Italian network (21 nodes and 72 links) Fig. 11. Networks used for simulation.

Average cost

1

5.6

4.4

(a) US-NET (24 nodes and 86 links)

60

Distribution 1 Distribution 2 Distribution 3

6.0

Average cost

55 0 65

10

5.6 5.4 5.2 5.0 4.8 4.6 4.4 4.2 4.0 0.00

0.04

0.08

0.12

0.16

0.20

p3 (b) Italian network

We conduct our experiments on two sample networks: US-NET that has 24 nodes and 86 unidirectional links (shown in Fig. 11(a)) and the Italian network that has 21 nodes and 72 unidirectional links (shown in Fig. 11(b)). We simulate the two networks since they represent two typical carrier networks and are widely used in the literature such as [28]. Each link in these networks represents two unidirectional fiber links in opposite directions. A link length is labeled on each link in kilometers. The failure probability of a link is assumed to be proportional to its link length. We consider two types of link cost. In the first type, the link cost of each link is equal to one. This kind of cost may correspond to the number of wavelengths consumed on each link. In the second type, the link cost of each link is equal to its link length. This kind of cost may correspond to the cost of burying a fiber between the end nodes of the link. In the experiments, connection requests are uniformly distributed among all node pairs and divided to form three classes. 1) Class 1: This class of demands requires a stringent MADR(1) = p1. 2) Class 2: This class of demands requires a moderate MADR(2) = p2. 3) Class 3: This class of demands requires a loose MADR(3) = p3. The bandwidth required by each connection is assumed to be one wavelength channel. It is important to notice, however, that

Fig. 12. Average cost for routing a connection as a function of p3 when the link cost is equal to one.

experiments carried out with various other bandwidth distributions also yielded results similar to those shown below and these results are not shown here due to space limitation. A. Investigation of the Effect of Reliability Requirement on the Average Cost for Routing a Connection In this subsection, we investigate the effect of reliability requirement on the average cost for routing a connection. We have simulated three kinds of traffic distributions: 1) Distribution 1: Class 1 : Class 2 : Class 3 = 1 : 1 : 1; 2) Distribution 2: Class 1 : Class 2 : Class 3 = 1 : 2 : 3; 3) Distribution 3: Class 1 : Class 2 : Class 3 = 3 : 2 : 1. In all traffic distributions, we assume that the parameters p1, p2, and p3 are the same to those in [7]. That is p1 = 0.0, p2 = 0.06 and p3 varies from 0.0 to 0.20. With these settings, we generate 10,000 random connections into the two sample networks and record the average cost consumed by each connection request by solving the ILP formulation proposed in Section III. Fig. 12 (a) and Fig. 12 (b) show the average cost for routing a connection as a function of the MADR for Class 3 traffic demands for the US-NET and Italian network, respectively, when the link cost is equal to one. It is notable that, the results shown in Fig. 12 (a) and Fig. 12 (b) are average values of 10

420

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380

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Average cost

Average cost


340 320 300 280

Distribution1 Distribution2 Distribution3

11

ILP SPPA ATSA

5.1 5.0 4.9 4.8 4.7

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1

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Average cost

320

Average cost

5

300 280 260

5.2

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220 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

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Trial number

(b) Italian network

(b) Italian network

Fig. 13. Average cost for routing a connection as a function of p3 when the link cost is equal to its link length.

Fig. 14. Comparison of solution approaches for traffic Distribution 1 when the link cost is equal to one.

test runs for each experiment, where each test corresponds to a different demand pattern (generated by a different random seed). One can easily see from these figures that, as expected, the average cost decreases as the reliability requirement of Class 3 traffic demands becomes less stringent, i.e., p3 increases. This is because a connection request with a stringent reliability requirement often needs to compute not only an AP but also a protection topology for it. On the other hand, a connection request with a loose reliability requirement may only need to compute an AP. One can also notice from Fig.12 (a) and Fig. 12 (b) that, when the reliability requirement of Class 3 traffic demands reaches a threshold (e.g., 0.16 for US- NET and 0.18 for the Italian network), the average cost for routing a connection request does not decrease any more. This is because there is no need to compute a protection topology for a connection request if its reliability requirement is loose enough such that a single AP can meet its reliability requirement. Another observation from these two figures is that the average cost increases with the increase of percentage of Class 1 traffic, which has stringent reliability requirements. For example, the average cost for routing a connection request in traffic distribution 3 is evidently higher than that in traffic distribution 1. This again can be explained as above.

Fig. 13 (a) and Fig. 13 (b) further plot the average cost for routing a connection as a function of the MADR for Class 3 traffic for the US-NET and Italian network, respectively, when the link cost of each link is equal its link length. From these figures, we can make similar observations to those from Fig. 12. B. Comparison of the Solution Approaches In this subsection, we compare the solution approaches, the SPPA algorithm, the ATSA algorithm and the ILP formulation, through experiments. We show the results for traffic Distribution 1 with p1 = 0.0, p2 = 0.06 and p3 = 0.20. With these network settings, Class 1 may correspond to the case that a connection requires a link-disjoint path pair such that it can be protected from any single link failure. Class 3 may correspond to the case that a connection does not require protection, while Class 2 lies between. It is notable that, however, experiments carried out with various other traffic distributions and reliability requirements also yielded results similar to those shown below. Recall that the SPPA algorithm and the ATSA algorithm use an approximation parameter ε . In our experiments we chose ε to be a fairly small constant.


12

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210

0.50 0.45

Blocking probability

Average cost

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ILP SPPA ATSA

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ILP SPPA ATSA

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(b) Italian network

(b) Italian network

Fig. 15. Comparison of solution approaches for traffic Distribution 1 when the link cost is equal to its link length.

Fig. 16. Comparison of solution approaches for traffic Distribution 1 when the link cost is equal to one.

1) Average Cost Comparison In order to compare the average cost of the three solution approaches, we generate 10,000 random connections into the two sample networks and record the average cost consumed by three solution approaches. Fig. 14 compares the average cost of the three solution approaches proposed in this paper for ten different trials when the link cost is equal to one. As expected, we observe from these two figures that the cost obtained by running the SPPA algorithm is very close to that obtained by solving the ILP formulation, which can obtain the optimal solution for the problem PWD. Specifically, by averaging the cost obtained from the ten trials, the cost obtained by using the SPPA algorithm is about 2.5% (for the Italian network) and about 3.38% (for the US-NET) higher than that obtained by solving the ILP formulation. On the other hand, the cost obtained by running the ATSA algorithm is significantly higher than those obtained by the rest two solution approaches. Indeed, by averaging the ten trials, the cost obtained by running the ATSA algorithm is about 11% higher (for the Italian network) and about 16.9% (for the US-NET) than that obtained by solving the ILP formulation. Fig. 15 further compares the average cost of the three

solution approaches proposed in this paper for ten different trials when the link cost is equal to its link length. From this figure, we can make similar observations to those from Fig. 14. 2) Blocking Probability Comparison We also compare the three solution approaches in terms of blocking probability in a dynamic network environment, where connections arrive dynamically one after another. We assume that connections (each with a bandwidth requirement of one wavelength channel) arrive at a network according to Poisson process with an average rate λ , and the holding times are negative exponentially distributed with unit mean. We further assume that each link carries 16 wavelengths. Without loss of generality, we assume that each connection has a bandwidth requirement of one wavelength. Notice that, however, results from other network settings are similar to those shown below and are not shown here due to space limitation. Fig. 16 shows the blocking probability of the three solution approaches as a function of p3 when the link cost is equal to one and the traffic load offered to the networks is 100 Erlang. From this figure, we immediately see that, in most cases, the blocking probability of the ILP formulation is the lowest. This is because it can obtain the optimal routing and consumes the fewest network resources (i.e., wavelengths) among the three


13 TABLE I

AVERAGE TIME REQUIRED FOR RUNNING EACH SOLUTION APPROACH IN THE TWO SAMPLE NETWORKS (IN ms)

0.55 0.50


0.45

ILP SPPA ATSA

0.40 0.35

ILP

SPAA

ATSA

Italian

196

15.6

0.954

US

207

17.1

1.067

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

the average time to run the ATSA algorithm is significantly shorter than the SPPA algorithm. It is important to note that, although we do not show the actual offered reliability, the MADR requirement can be met using any solution approaches proposed. VII.

p3 (a) US-NET 0.45 0.40

ILP SPPA ATSA

0.35


Network

0.30 0.25 0.20 0.15 0.10 0.05

0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

p3 (b) Italian network Fig. 17. Comparison of solution approaches for traffic Distribution 1 when the link cost is equal to its link length.

solution approaches. On the other hand, the blocking probability of the ATSA algorithm is the highest in most cases. This is because it consumes more network resources to route connections. Another observation from Fig.16 is that the three approaches have the same blocking probability when p3 reaches a threshold. For example, in US-NET, the threshold in US-NET is 0.16. This is because we only need a single path to route each connection when p3 reaches the threshold, as have been seen from Fig. 12 and Fig. 13. Fig. 17 further plots the blocking probability of the three solution approaches as a function of p3 when the link cost is equal to the link length and the traffic load offered to the networks is 100 Erlang. From this figure, we can also make similar observations to those from Fig. 16. It is notable that, however, although the ILP formulation can obtain the optimal solution, the required time to run it is significantly longer than the required time to run the other two algorithms. As shown in Table I, the average time to run the ILP formulation is 196 ms for the Italian network and 207 ms for the US-NET. By solving the SPPA algorithm, however, the average time is 15.6 ms for the Italian network and 17.1 ms for the US-NET. Furthermore, we can observe from Table I that

CONCLUSIONS

The increasing importance of concepts such as qualityof-service and differentiated services has led to the introduction of differentiated reliability, which can provide multiple levels of reliability in the same network. With reliability being a necessity in optical networks and an important aspect of MPLS networks, dynamic routing of connection requests with DiR requirements is an important area of study. In order to meet the DiR requirement of a connection request, an AP and in some cases a protection topology are required, with the protection topology being possibly shared for efficiency. Since a complex restoration process is required in shared protection, the restoration time cannot be guaranteed upon failure. Thus in some cases, sharing may not be possible. In these cases, performance efficiencies can be obtained only be good path selection. In this paper, we proposed an ILP formulation and developed two approximation algorithms with proven performance guarantees. The simulations conducted on two sample networks verify the efficiencies of these solution approaches. In this paper, we have assumed that each connection only multiplexes one application. In practice, however, the bandwidth requirement of a single application may be less than the bandwidth of a wavelength (e.g., 10 Gb/s or 40 Gb/s). How to route application requests with DiR requirement in that case is an ongoing research and the research results will be appeared in a future paper. ACKNOWLEDGEMENTS The authors would like to thank the editor, Prof. Malathi Veeraraghavan, and the anonymous reviewers for their valuable comments that help improving this paper. REFERENCES [1] [2] [3] [4]

M. A. Marsan, A. Bianco, E. Leonardi, A. Morabito, and F. Neri, “All-optical WDM multi-rings with differentiated QoS,” IEEE Commun. Mag., vol. 37, no. 2, pp. 58–66, Feb. 1999. A. Jukan, A. Monitzer, and H. R. Van As, “QoS-restorability in optical networks,” in Proc. 24th Eur. Conf. Optical Commun. (ECOC), vol. 1, 1998, pp. 711–712. IEEE Networks (Special Issue on Integrated and Differentiated Services for the Internet), vol. 13, Sept./Oct. 1999. N. Golmie, T. Ndousse, and D. Sue, “A differentiated optical services model for WDM networks,” IEEE Commun. Mag., vol. 38, pp. 68–73, Feb. 2000.

Routing Connections with Differentiated Reliability Requirements in WDM Mesh Networks [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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[29]

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Hongbin Luo received the B.S. degree from Beihang University (formerly known as Beijing University of Aeronautics and Astronautics), Beijing, China, in 1999. He received the M.S. (with honors) and Ph.D. degrees in Communications and Information Science from University of Electronic Science and Technology of China (UESTC), Chengdu, China, in June 2004 and March 2007, respectively. From July 2002 to March 2007, he was a research associate at the Key Lab of Broadband Optical Fiber Transmission and Communication Networks, UESTC. In March 2007, he joined the School of Electronic and Information Engineering, Beijing Jiaotong University, where he is a lecturer. He is the first author of more than 20 refereed technical papers published in prestigious international journals including IEEE Journal on Selected Areas in Communications, IEEE Communications Letters, Journal of Optical Networking, Photonic Network Communications, Computer Communications, etc., and international conferences such as GLOBECOM, ICC, OFC, ICCCN, APOC, etc. His research interests are in the wide areas of optical networking and wireless networking, with a focus on routing and survivability. Dr. Luo is a student member of IEEE and OSA.

Lemin Li received the B.S. degree in electrical engineering from Shanghai Jiaotong University, Shanghai, China, in 1952. From 1952 to 1956, he was with the Department of Electrical Communications at Shanghai Jiaotong University. Since 1956, he has been with the Chengdu Institute of Radio Engineering (now the University of Electronic Science and Technology), where he is a now full professor at the Key Lab of Broadband Optical Fiber Transmission and Communication Networks, UESTC. From 1980 to 1982, he was a visiting scholar in the Department of Electrical Engineering and Computer Science, University of California at San Diego, USA. He has authored/coauthored over 200 research papers in the areas of communication networks. His previous research interests were in the area of digital information transmission and communication networks and are now in the areas of communication networks including broadband networks and wireless networks. Professor Li is a member of the Chinese Academy of Engineering (CAE).

Hongfang Yu received the B.S. degree in Electrical Engineering from Xidian University, Xi’an, China, in 1996. She received the M.S. and Ph.D. degrees in Communication and Information Engineering from University of Electrical Science and Technology of China (UESTC), Chengdu, China, in 1999 and 2006, respectively. She is currently with the Key Lab of Broadband Optical Fiber Transmission and Communication Networks, UESTC, where she is an associate professor. In recent years, she has authored/coauthored over 40 research papers on international journals and conferences. Her research interest includes optical network survivability and traffic engineering etc.