All-Optical Monitoring Path Computation Based on ...

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All-Optical Monitoring Path Computation Based on Lower Bounds of Required Number of Paths Nagao Ogino and Hajime Nakamura KDDI R&D Laboratories Inc. 2-1-15 Ohara, Fujimino-shi, Saitama 356-8502 JAPAN {ogino, nakamura}@kddilabs.jp Abstract —To reduce the cost of fault management in the alloptical networks, it is investigated to detect the degradation of optical signal quality solely at the terminal points of all-optical monitoring paths. The monitoring paths must be routed so that all single-link failures can be localized using routes information of monitoring paths where signal quality degradation is detected. However, routes computation for the monitoring paths that satisfy the above condition is time consuming. This paper proposes a procedure for deriving the lower bounds of the required number of monitoring paths to localize all the singlelink failures, and an efficient monitoring path computation method based on the derived lower bounds. The proposed method repeats the route computation for the monitoring paths until feasible routes can be found, while the assumed number of monitoring paths increases starting from the lower bounds. With the proposed method, the minimum number of monitoring paths with the overall shortest routes can quickly be obtained when the possible terminal nodes of monitoring paths are arbitrarily given. Thus, the proposed method can minimize the operational cost of monitoring the degradation of signal quality and can reduce the overhead traffic transferred through the monitoring paths. Keywords —all-optical network; monitoring path; single-link failure; path computation; lower bounds of number of paths

I. INTRODUCTION All-optical networks can relieve the electronic bottleneck and increase data transmission rates. They can also reduce the network cost of optoelectronic regenerators and the operational cost of power consumption [1-2]. However, fault detection and localization is more complex in all-optical networks [3]. To reduce the cost of fault management in all-optical networks, it is investigated to detect the degradation of signal quality solely at the terminal points of all-optical monitoring paths [4-5]. The number of monitors required to localize all the single-link failures may be reduced by monitoring only key paths, compared with having one monitor attached to each link. Alloptical monitoring paths must be routed so that all single-link failures can be localized from the route information on the monitoring paths where degradation of optical signal quality is detected. However, the optimum-routes computation for the monitoring paths that satisfy the above condition is time consuming [6-11]. This paper proposes a procedure for deriving the lower bounds of the required number of monitoring paths to localize all the single-link failures, and a monitoring path computation method based on the derived lower bounds. By the proposed

method, the minimum number of monitoring paths with the overall shortest routes can be obtained quickly when the possible terminal nodes of monitoring paths are arbitrarily given. This paper verifies the effectiveness of the proposed method by comparison with the existing method. Moreover, desired placement of the terminal nodes is also clarified. Section II presents the background of this paper. Section III proposes a procedure for deriving lower bounds for the required number of monitoring paths and a monitoring path computation method based on the lower bounds. In section IV, the proposed method is evaluated and its effectiveness is verified. Finally, section V concludes this paper. II. FAULT MANAGEMENT SCHEME AND RELATED WORKS A. Considered Fault Management Architecture In all-optical networks, fault management is complex because detecting fault-caused degradation of signal quality is difficult in the optical domain. Therefore, it is considered that the degradation of signal quality is only detected at the terminal points of all-optical monitoring paths. Figure 1 shows an example of such fault management architectures. Supervisory devices are connected to several nodes composing the managed all-optical network. The required number of all-optical monitoring paths is established between the supervisory devices. The optical signal quality in each monitoring path is supervised by the supervisory device terminating the monitoring path.

Figure 1. An example of fault management architecture

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

The fault management system computes the optimum routes of monitoring paths, and requests that supervisory devices setup monitoring paths. The supervisory devices notify the management system which monitoring paths show degradation of signal quality. The fault management system localizes single-link failures from the routes information of the monitoring paths where signal quality degradation is detected. Thus, the monitoring paths must be routed so that all the single-link failures can be localized from their routes information. However, the optimum routes computation for such monitoring paths is generally a burden on computational resources. B. Existing Works Related to Monitoring Path Computation The monitoring structure required to localize all single-link failures has been a key concern. The concept of a monitoring cycle (m-cycle), a class of monitoring paths whose source and destination node is identical, was proposed [6-9]. To design mcycle solutions, heuristic algorithms and ILP (Integer Linear Programming) models have been proposed [7-9]. The necessary and sufficient conditions for localizing all single-link failures from one terminal point have been derived [7]. The optimum placement for the minimum number of terminal points was also clarified for any arbitrary network topology [8]. Recently, the concept of more general monitoring paths called m-trails was proposed [10-11]. An m-trail may have different source and destination nodes and may pass through an identical node multiple times. The optimum m-trails computation has been formulated using an ILP model with assumed upper bounds for the required number of m-trails [10]. However, the upper bounds are only given in ring networks and mesh networks assuming that all the nodes can be terminal points. Moreover, this approach is time consuming since a large-scale ILP problem needs to be solved. Thus, a heuristic approach to computing sub-optimum m-trails has been proposed [11]. However, this approach assumes that all the nodes can be the terminal points of m-trails. This paper proposes an optimum monitoring path computation method based on the lower bounds of the required number of paths. The proposed method also needs to solve the ILP problem. However, the ILP problem in the proposed method is significantly smaller than that in the literature [10] because the assumed number of monitoring paths is small when starting from the lower bounds. The proposed method can compute the optimum monitoring paths when any possible terminal points are arbitrarily selected. III. MONITORING PATH COMPUTATION METHOD A. Principle of Localizing Single-Link Failure Figure 2 illustrates the principle of localizing a single-link failure from the routes information of monitoring paths. In Fig. 2, the same two monitoring paths P1 and P2 traverse the link N1-N0 and the link N0-N4. Therefore, the signal quality in both the paths P1 and P2 degrades simultaneously when either the link N1-N0 or the link N0-N4 fails. This means that single-link failures in the link N1-N0 and the link N0-N4 cannot be localized. However, the monitoring path P3 traverses the link

N2-N0 and the link N0-N5, and the monitoring path P4 traverses the link N3-N0 and the link N0-N5. In this case, the signal quality in path P3 only degrades when link N2-N0 fails. The signal quality in path P4 only degrades when link N3-N0 fails. The signal quality in both paths P3 and P4 degrades when link N0-N5 fails. Therefore, single-link failures in links N2-N0, N3-N0, and N0-N5 can be localized based on the combination of monitoring paths in which the signal quality degrades.

Figure 2. Principle of localizing single-link failures

The combination of monitoring paths traversing each link must be different in order to localize all the single-link failures. In Fig. 2, two monitoring paths passing through a node can localize single-link failures in two incoming and one outgoing link or one incoming and two outgoing links. This is the most efficient case in which more links connecting to a node can be discriminated by fewer monitoring paths passing the node. B. Lower Bounds of Required Number of Monitoring Paths In this subsection, the required number of monitoring paths for discriminating links connecting to multiple nodes is derived from the sizes of the multiple nodes and the network topology. Each monitoring path passes through a node only once except for the case where the monitoring path has an identical source and destination node. For simplicity, each node is assumed to be square. Two necessary conditions that the required number of monitoring paths must satisfy are introduced for each combination of multiple nodes. Strict lower bounds of the required number of monitoring paths can be derived from the two necessary conditions for all combinations of multiple nodes. However, this is time consuming because the total number of combinations of multiple nodes is so large. In this subsection, lower bounds of the required number of monitoring paths are only derived from a combination of nodes when the size of the combination is given. For the purpose of obtaining stricter lower bounds, the nodes are selected preferentially from those with more incoming and outgoing links discriminated by the monitoring paths. The above node selection procedure is explained as follows. Figure 3 shows the stage where the first four nodes are selected. Basically, all the nodes are selected in descending order of size. When multiple nodes are identical in size, the node with the fewest links connecting to the already selected nodes is preferentially selected. In Fig. 3, the node N3 has two links connecting to the nodes N1 and N2 while the node N4 has four links connecting to the nodes N1 and N2. Therefore, the node N3 is selected as a higher priority than node N4 even if both nodes

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are the same size. More links that the monitoring paths must discriminate are added by selecting node N3. When multiple nodes have identical sizes and the same number of links connecting to the already selected nodes, a node is randomly selected.

combinations of two monitoring paths. Therefore, the following inequality holds in each stage of node selection: K

∑ ( yik + yok ) − l 2K ≤ M C2

k =1

K =1 ~ N

(4)

Here, the symbol l2K means the number of links connecting between two of the K selected nodes and discriminated by a combination of two monitoring paths. Generally, the following inequality holds in each stage: K ⎧ K ⎫ K ⎨2 ∑ (xik + xok ) + ∑ ( yik + yok )⎬ − ∑ (xik + xok ) k =1 ⎩ k =1 ⎭ k =1 K

K

K

k =1

k =1

k =1

= ∑ (xik + xok ) + ∑ ( yik + yok ) ≤ 2 ∑ nk

K = 1 ~ N (5)

The first necessary condition for the value of M is given by substituting inequalities (2) and (3) into inequality (5): K

4 ∑ nk − l1K ≤ (2K + 1)M k =1

Figure 3. Number of monitoring paths passing through multiple nodes

The two necessary conditions that the required number of monitoring paths must satisfy for each number of selected nodes K are introduced as follows. It is assumed that each link is ideally discriminated by a combination of fewer than four monitoring paths since the number of combinations of fewer than four monitoring paths is sufficiently larger than the total number of links. The total number of nodes is denoted by N, and the size of the k-th selected node is denoted by nk. The numbers of incoming and outgoing links connecting to the k-th selected node and discriminated by a monitoring path are indicated by xik and xok respectively. The numbers of incoming and outgoing links connecting to the k-th selected node and discriminated by a combination of two monitoring paths are indicated by yik and yok respectively. The required number of monitoring paths M must be more than or equal to the number of monitoring paths passing through each selected node. Therefore, the following inequality holds: ⎧xik + 2 yik + 3(nk − xik − yik ) ≤ M ⎨ ⎩xok + 2 yok + 3(nk − xok − yok ) ≤ M

k =1~ N

(1)

From inequality (1), the following inequality holds in each stage of node selection: K

K

K

k =1

k =1

k =1

6 ∑ nk + 2KM ≤ 2 ∑(xik + xok ) + ∑ ( yik + yok ) K = 1 ~ N

(2)

Since the number of links discriminated by a monitoring path is not more than the total number of monitoring paths, the following inequality holds in each stage of node selection: K

∑ (xik + xok ) − l1K ≤ M

k =1

K =1 ~ N

(3)

The symbol l1K means the number of links connecting between two of the K selected nodes and discriminated by a monitoring path. The number of links discriminated by the combination of two monitoring paths is not more than the total number of

k =1~ N

(6)

The second necessary condition for the value of M is given by substituting inequalities (3) and (4) into the inequality (2): K

6 ∑ nk − (2l1K + l 2K )≤ M C2 + 2(K + 1)M k =1

k =1~ N

(7)

The value of M that satisfies inequalities (6) and (7) may increase as more nodes are selected sequentially. After the value of l1K increases toward the value of M, the value of l2K increases toward the value of MC2. The final value of M corresponds to the derived lower bounds of the required number of monitoring paths. If the size of the first selected node is denoted by n, the lower bounds derived from the first necessary condition (6) become ⎡4n / 3⎤ . Here, ⎡x⎤ means the smallest integer greater than or equal to x .This means that at least ⎡4n / 3⎤ monitoring paths need to pass through the first selected node. This corresponds to the most efficient case mentioned in the previous subsection, where more links connecting to a node can be discriminated by fewer monitoring paths passing through the node. C. Proposed Monitoring Path Computation Method In the proposed method, the optimum routes computation for the monitoring paths is repeated until feasible routes can be found, while the assumed number of monitoring paths increases starting from the derived lower bounds of the required number of monitoring paths LB. Figure 4 shows the procedure for the proposed method. First, the assumed number of monitoring paths M is increased by step-size d from LB − 1 (Step 1). If feasible routes are found when the value of M is increased k times, the least number of monitoring paths with feasible routes is searched for using the bisection method between M 0 = ( LB − 1) + (k − 1)d and M 1 = ( LB − 1) + kd (Step 2). The minimum number of monitoring paths to localize all the single-link failures can be obtained with certainty using the above procedure.

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source node and the terminal destination node. A virtual link also connects the virtual source node and each terminal source node, and each terminal destination node and the virtual destination node. The length of each virtual link is disregarded. The symbols used in the ILP model are defined as follows: l: Link; n: Node; s: Virtual source node; d: Virtual destination node; l(s): Starting node of link l; l(t): Terminal node of link l; (l1, l2): A pair of links excluding virtual links; k: Identification of monitoring path; The constants and sets are defined as follows: M: Assumed number of monitoring paths Dist l: Distance of link l; L: Set of links including virtual links; VL: Set of virtual links; N: Set of nodes; n in: Set of incoming links in node n; n out: Set of outgoing links in node n; LP: Set of link pairs excluding virtual links; The variables are defined as follows: Xl(k): Integer variable indicating whether the monitoring path k (1 ~ M) traverses the link l (=1) or not (=0); Y(l1,l2)(k): Integer variable indicating whether the monitoring path k (1 ~ M) traverses the link pair (l1, l2) (=1) or not (=0); Zn(k): Integer variable indicating the voltage in the node n for the monitoring path k (1 ~ M); Here, Zs(k)=0. The constraints in the ILP model are given as follows. First, the following constraints hold as the route preservation rule:

∑ X l(k ) = 1; 　　 ∑ X l (k ) = 1; 　　　

l ∈ s out Figure 4. Procedure for monitoring path computation

∑ X l (k ) = ∑ X l (k ) ≤ 1;

l ∈ n in

∀k = 1　~ M

(8)

l ∈ d in

∀n ≠ s, d ∈ N　 , 　 ∀k = 1　~ M

l ∈ n out

The following constraint is necessary to avoid loop in the monitoring paths:

Zl ( s )(k ) + 1 ≤ Zl ( t )(k ) + A × (1 − X l (k )); ∀l ∈ L, 　 ∀k = 1　~ M

(9)

Here, the constant A has a sufficiently large value. The condition for detecting all single-link failures is as follows: M

∑ X l (k ) > 0; ∀l ∈ L - VL

k =1

(10)

The condition that monitoring path k traverses link pair (l1, l2) is indicated as follows: Figure 5. Extended network for the shortest monitoring path computation

Each shortest monitoring path computation involved in the above procedure can be formulated using an ILP model. When the ILP model is applied, the network under consideration is extended as shown in Fig. 5. A pair of virtual source and destination nodes is added to the managed network. Each terminal node is divided into a pair of terminal source and destination nodes, and a virtual link connects the terminal

M

M

Y ( l1, l 2)(k ) ≥ ∑ X l1(k ) + ∑ X l2(k ) − 1; k =1

k =1

∀(l1,l2)∈ LP, 　 ∀k = 1　~ M

(11)

The combinations of monitoring paths traversing all the link pairs must be different in order to localize all the single-link failures. Thus, the following constraint holds:

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M

M

M

∑ X l1(k ) + ∑ X l2(k ) > 2 × ∑Y (l1, l 2)(k ); 　 ∀(l1,l2)∈ LP

k =1

k =1

k =1

(12)

If necessary, the constraint on the distance of each monitoring path can be added easily in the ILP model. The objective function to be minimized is the total length of monitoring paths: M

TotalLengt h = ∑ ∑ (Dist l × X l ( k ) ); k =1 l∈L

(13)

The final values of Xl(k) indicate the optimum route for the k-th monitoring path. The terminal nodes of each monitoring path can be also known from the virtual links that the monitoring path traverses. By this ILP model, the overall shortest routes for the monitoring paths can be solved for when the possible terminal nodes of the monitoring paths are arbitrarily given. IV.

EVALUATION OF MONITORING PATH COMPUTATION

A. Evaluated Networks Figure 6 shows two kinds of all-optical managed networks evaluated in this section. Each edge shown in Fig. 6 is composed of a pair of directional links. Fig. 6(a) shows a random network with 16 nodes and 64 links. Fig. 6(b) shows a scale-free network with 16 nodes and 64 links. This scale-free network is based on the BA model [12]. In both the networks, nodes with fewer than three incoming links and outgoing links are removed when the networks are generated. This is because the monitoring paths need to pass through a node multiple times to discriminate all the links if such nodes are included in the network. All the nodes in both the networks are identified in the order of node selection for deriving the lower bounds of the required number of monitoring paths. In both the networks, the length of every link is regarded as 1.0.

(a) Random network

(b) Scale-free network Figure 6. Evaluated networks

B. Derivation Results for Lower Bounds The lower bounds of the required number of monitoring paths are derived using the procedure shown in subsection IIIB. Table I shows the required number of monitoring paths M for the network shown in Fig. 6(a). The value of M satisfies the two necessary conditions (6) and (7) in each stage of node selection. At first, the incoming and outgoing links in the selected nodes are discriminated by the combination of one or two monitoring paths from the assumed number of monitoring paths. However, as the number of selected nodes increases, the combination of three monitoring paths becomes necessary to discriminate the additional links. This may cause the number of monitoring paths required to pass through the new selected node to be larger than the assumed number of monitoring paths. Thus, the required number of monitoring paths may have to be increased depending on the size of the new selected node. In Table I, the required number of monitoring paths derived from necessary condition (7) is actually increased when the number of selected nodes is 9. The lower bounds for the random network in Fig. 6(a) are 10 as shown in Table I. TABLE I.

DERIVATION OF LOWER BOUNDS FOR RANDOM NETWORK

Scale-free networks usually include a node with a quite large size. Therefore, in most cases, the lower bounds of the required number of monitoring paths are given by necessary condition (6) when only the first node is selected. The lower bounds number 14 for the scale-free network shown in Fig. 6(b) and are given by necessary condition (6) when only the largest node with 10 incoming links and 10 outgoing links is selected. C. Evaluation of Monitoring Path Computation Time In this subsection, routes computation time for the monitoring paths is compared between the proposed method and the existing method [10]. In the existing method, the minimum number of monitoring paths required for localizing all the single-link failures is obtained by solving an ILP problem assuming the upper bounds of the required number of monitoring paths. Next, the overall shortest routes for the minimum number of monitoring paths are computed as another ILP problem. Here, the upper bounds of the required number of monitoring paths are assumed to number half of the total number of links, i.e. 32. Several ILP problems need to be solved for obtaining the minimum number of monitoring paths with the overall shortest routes in both the proposed and existing methods. Since solving ILP problems exactly is time consuming, a tabu search approach is adopted to obtain a sub-optimum solution [13]. The

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sub-optimum solution for each ILP problem is obtained using 50 randomly generated initial solutions. Starting from each initial solution, 106 neighbor searches are repeated to obtain the best solution. Table II shows eight evaluation scenarios in which the evaluated network and the placement of possible terminal nodes are different. Table III shows the routes computation time needed to obtain the minimum number of monitoring paths with the overall shortest routes. Table III shows the total computation time per initial solution required to solve all the ILP problems in each method. In the proposed method, the assumed number of monitoring paths in each ILP problem is small, starting from the lower bounds. Therefore, the total computation time can be reduced by the proposed method with step-sizes d of 2 and 4 in all the evaluation scenarios. TABLE II.

EVALUATION SCENARIOS

is reduced when the distance between two possible terminal nodes is larger and the number of incoming and outgoing links to and from the possible terminal node is larger. However, the required number of monitoring paths is almost equal in each scenario when the evaluated network is identical. From Table IV, nodes with larger sizes and far from each other should be selected as the terminal nodes of monitoring paths. V. CONCLUSIONS This paper proposed a procedure for deriving the lower bounds of the required number of all-optical monitoring paths to localize all single-link failures and an efficient monitoring path computation method based on the derived lower bounds. Using the proposed method, the minimum number of monitoring paths with the overall shortest routes can be computed quickly using small ILP models when the possible terminal nodes of monitoring paths are arbitrarily given. Thus, both the operational cost of monitoring the degradation of optical signal quality and the overhead traffic transferred through the monitoring paths can be minimized. REFERENCES [1] [2]

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TOTAL ROUTES COMPUTATION TIME (UNIT: SEC) [4]

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TABLE IV.

RESULTS OF MONITORING PATH COMPUTATION

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D. Evaluation of Computed Monitoring Paths Table IV shows the results of monitoring path computation. In the evaluated scenarios, the total length of monitoring paths

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