A routing and wavelength assignment scheme considering full optical carrier replication in multi-carrier-distributed optical mesh networks with wavelength reuse

Optical Switching and Networking 28 (2018) 23–35

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A routing and wavelength assignment scheme considering full optical carrier replication in multi-carrier-distributed optical mesh networks with wavelength reuse

MARK

⁎

Praphan Pavarangkoona, , Eiji Okib,1 a b

Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan

A R T I C L E I N F O

A BS T RAC T

Keywords: Wavelength division multiplexing Multi-carrier light source Optical carrier regeneration Routing Wavelength assignment

This paper proposes a routing and wavelength assignment (RWA) scheme that considers full optical carrier replication to minimize the number of required wavelengths for wavelength-reusable multi-carrier-distributed (WRMD) mesh networks. Unlike the conventional wavelength division multiplexing networks where each node contains multiple laser diodes, the WRMD networks use a light source, called the multi-carrier light source, to ease the difficulty of controlling many light source devices. The optical carrier replication at any node of optical carrier or any end node of lightpath, called full optical carrier replication, improves the performance of the conventional schemes, which do not consider full optical carrier replication. This paper first formulates the RWA problem considering full optical carrier replication as an integer linear programming problem that minimizes the number of required wavelengths to satisfy the given lightpath setup requests. A heuristic RWA scheme is then proposed to solve the RWA problem in practical times. Simulation results show that our proposed heuristic RWA scheme for the WRMD network achieves a near-optimum number of wavelengths. In addition, it is able to reduce the number of required wavelengths for lightpath establishment, compared to the conventional scheme.

1. Introduction With the rapid growth of both the Internet and available bandwidth, wavelength division multiplexing (WDM) technology has been as a promising candidate for the next generation network (NGN). WDM technology has the potential to meet rising demands for high bandwidth and low latency communication [1]. Conventional WDM networks attempt to meet the explosive demand for network bandwidth by using more laser diodes (LDs) to provide sufficient wavelengths. This will increase network energy consumption and implementation cost. Moreover, the complexity of optical carrier management increases with the number of wavelengths [2]. In other words, it will be difficult to adequately control the wavelengths of huge numbers of LDs, since each wavelength of each LD has to be adjusted individually to satisfy the extremely narrow channel spacings demanded. A multi-carrier-distributed optical network with wavelength reuse capability [3,4] was introduced as a solution. This network is called the wavelength-reusable multi-carrier-distributed (WRMD) network. As

⁎

1

shown in Fig. 1, the WRMD network places a multi-carrier light source (MCLS) in an MCLS node, as the communication light source device. A number of MCLSs are reported [5,6], and high-capacity, 2000 km longdistance WDM transmission experiments using MCLS are demonstrated [7]. In general, the MCLS node consists of an MCLS, wavelength selective switch (WSS), and multiple wavelength converters (WCs) to avoid wavelength collisions in the network. The MCLS is able to generate stable and multiple optical carriers at the same time over long periods [5–9]. The generated multiple optical carriers are utilized not only as distributed carriers of the network, but also continuouswave (CW) probes of the WCs. The individual wavelengths are used as optical carriers. The MCLS generates the optical carriers and passes them to all requesting source nodes for lightpath establishment. By replacing many widely dispersed LDs with the single MCLS, the difficulties posed by monitoring and controlling a large number of LDs are greatly simplified. Each node in the WRMD network, which is called a regeneration point [3], consists of an optical add-drop multiplexer (OADM), an

Corresponding author. E-mail addresses: [email protected] (P. Pavarangkoon), [email protected] (E. Oki). This work was supported in part by JSPS KAKENHI, Japan. This work was performed in part when Eiji Oki was with The University of Electro-Communication, Tokyo, Japan.

https://doi.org/10.1016/j.osn.2017.12.001 Received 11 May 2016; Received in revised form 30 November 2017; Accepted 15 December 2017 Available online 21 December 2017 1573-4277/ © 2017 Elsevier B.V. All rights reserved.

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Fig. 1. WRMD network architecture.

optical carrier regenerator (OCR), multiplexers (MUXs)/demultiplexers (DMUXs), external modulators (MODs), and receivers (RXs). The OCR allows the nodes to reuse a wavelength to satisfy multiple disjoint lightpath requests. However, the number of transmission spans and the transmission length limit the available number of OCR nodes [7]. The scalability of the WRMD ring network is experimentally investigated in [7], while the WRMD mesh network is assessed from a transmission point of view in [10]. Considering the WRMD network architecture, the latency of the packet sent by the regeneration point is neither different from that of the conventional WDM nor reconfigurable optical add/drop multiplexer (ROADM) node. The analysis of cost and power consumption was presented in [2]. It observes that the WRMD network has lower cost and power consumption than the conventional WDM network when the number of wavelengths becomes large. For large-scale networks, which must support an increasing number of lightpaths, there may be a need to have more than one MCLS node to use wavelength resources efficiently. Wavelength management in the WRMD network is more complex than that in the conventional WDM network, since the routing and wavelength assignment (RWA) schemes in the WRMD network must take into account both optical carrier connections and requested lightpaths while maximizing the reuse of the optical carrier connections. An optical carrier connection connects the MCLS node and a requested lightpath, or between two requested lightpaths. For this reason, an RWA scheme satisfying these constraints is needed for the WRMD network. This paper assumes that lightpath setup requests are statically given in advance, and focuses on RWA schemes under the static scenario. The RWA problems in the WRMD network are often decoupled into two separate subproblems in order to make the solution more tractable. The first subproblem is routing, which can be classified into fixed routing and alternate routing. The second subproblem is wavelength assignment, which obeys the following three rules:

• • •

Fig. 2. WRMD mesh network rules and conditions.

carrier replication for lightpath establishment. There are two works considering the optical carrier replication [10,11]. Both works take into account the optical carrier replication only at any MCLS node or any regeneration point. In other words, the optical carrier can be split into several other optical carriers at any MCLS node or any regeneration point, and each optical carrier is used to establish another requested lightpath. Since each node in the WRMD network is equipped with the OCR, it is possible to replicate optical carriers at any node of optical carrier or any regeneration point, which is called full optical carrier replication, to reduce the number of required wavelengths. Note that there is no additional hardware cost on full optical carrier replication. The optical power of the carrier needs to be considered for full optical carrier replication. As each node includes both pre- and post-amplifiers, the optical power loss of the split carrier signal can be compensated at each node. An optical amplifier in the WRMD network is able to control its amplifier gain. However, no study has addressed the use of full optical carrier replication in the WRMD mesh networks. This paper proposes an RWA scheme that considers full optical carrier replication for WRMD mesh networks to minimize the number of required wavelengths for lightpath establishment. A heuristic RWA scheme is introduced to solve the RWA problem in this paper. This scheme consists of a routing algorithm and a wavelength assignment algorithm, which are run separately. We introduce lightpath selection

Each requested lightpath uses only one wavelength, as shown in Fig. 2(a), Each requested lightpath uses an optical carrier generated by an MCLS node or a reused optical carrier from another established lightpath, as shown in Fig. 2(b), To avoid wavelength collision, optical carriers and requested lightpaths on the same fiber link must be assigned different wavelengths, as shown in Fig. 2(c).

To support the network scalability, the RWA scheme for the WRMD network needs to use of the available resources efficiently, e.g., optical

24

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node to use wavelength resources efficiently. The work in [11] presented the RWA scheme supporting multiple MCLS nodes to minimize the number of wavelengths in the WRMD mesh network. The mathematical model and heuristic algorithm for the RWA scheme are introduced. Moreover, the two lightpath selection policies, namely nearest optical carrier first (NCF) and less number of required wavelengths first (LWF), are introduced for the route and wavelength assignment. However, the schemes of both [10,11] considers only the optical carrier replication at any MCLS node or any regeneration point. They take into account the common source node of requested lightpaths in each MCLS node and regeneration point. The RWA scheme is able to utilize the optical carrier connections by replicating them along the carrier lightpaths for establishing the requested lightpaths as well.

policies for the wavelength assignment algorithm. There are two policies: nearest optical carrier first with full optical carrier replication (NCF-FCR), and less number of required wavelengths first with full optical carrier replication (LWF-FCR). The contribution of this paper is to show how to decide both routes and wavelengths of lightpaths so as to reduce the number of required wavelengths in the WRMD network. In addition, we also investigate the number of MCLS nodes and the locations of the MCLS nodes to reduce the number of required wavelengths in the WRMD mesh network. Simulation results show that the proposed heuristic scheme for the WRMD network determines the near-optimum number of wavelengths in practical time. Furthermore, optimizing MCLS node location also reduces the number of required wavelengths. The remainder of this paper is organized as follows. Section 2 describes related works briefly. Section 3 presents the network model and RWA problem in the WRMD network. Section 4 describes our proposed heuristic RWA scheme to solve the RWA problem and reduce the number of wavelengths required for lightpath establishment. Section 5 presents numerical results under a variety of parameter settings. Finally, Section 6 summarizes the key points.

3. Routing and wavelength assignment scheme considering full optical carrier replication 3.1. System model and assumptions The objective of the RWA problem in the WRMD network is to minimize the number of wavelengths required for establishing the requested lightpaths. In other words, to solve the RWA problem, all routes and wavelengths of requested lightpaths are determined with the minimal number of wavelengths. The following notations are introduced to describe the RWA problem mathematically. A network is represented as undirected graph G = (V , E ), where V is the set of network nodes and E is the set of bidirectional links. Let (i , j ) ∈ E be a link between two network nodes. Let E′ be the set of bidirectional links including self links, E′ = E ∪ {(v, v )}, v ∈ V . A self link (v, v ) means that a link for an optical carrier connection between the node v itself. Let W be the set of wavelengths generated by the MCLS. Let w be wavelength index, where w ∈ W (w = 1, 2, …, wmax ). r ∈ R indicates the number of times an optical carrier is reused, where R = {0, 1, …, Rmax }. Rmax is the maximum number of times an optical carrier can be reused. r = 0 means that the optical carrier is directly generated from the MCLS node. p ∈ P indicates a lightpath request, where P is the set of lightpath requests. c ∈ C indicates an optical carrier connection, where C is the set of optical carrier connections. Let sp ∈ V and dp ∈ V be the source and destination nodes of lightpath p ∈ P , respectively. Let sc ∈ V be the source node of optical carrier connection c ∈ C . Assumptions made for addressing the RWA problem are as follows.

2. Related works Several studies have examined the RWA problem for WRMD networks. Both ring and mesh topologies are investigated. Wavelength assignments for the WRMD ring networks were introduced in [4,12]. There is one MCLS node in the WRMD ring network and none of the source nodes includes an LD. Each requested lightpath directly receives a generated optical carrier from the MCLS node or a reused optical carrier from a regeneration point, which is the destination node of other requested lightpath. In other words, an optical carrier connection connects the MCLS node and a requested lightpath, or between two requested lightpaths. The work in [13] presented a wavelength assignment method for all-to-all broadcast in WRMD WDM linear array and ring. All-to-all broadcast is to disseminate a unique message from each node to every other node in a network. The minimum number of wavelengths required to support all-to-all broadcast is derived. Since there are only two possible paths: the clockwise and anticlockwise directions, it is simpler to select the optical carrier connection in the ring topology than in the mesh topology [14]. On the other hand, in the mesh topology, there are several possible paths for each optical carrier connection. A path for an optical carrier connection is called a carrier lightpath. Therefore, the mesh topology makes distributing optical carriers and assigning wavelengths much more complex than the ring topology [15]. The work in [16] presented a mathematical model for wavelength assignment to minimize the number of required wavelengths in the WRMD mesh network. This model provides reference values, including upper and lower bounds, which are useful for benchmarking purposes. However, the schemes of [4,16,13] considered only predefined routes of optical carrier connections and requested lightpaths. In practical cases, all routes of optical carrier connections and requested lightpaths are required to be designed to minimize the number of wavelengths. Note that the works in [4,16,13] provided only a wavelength assignment scheme, the routing of optical carrier connections and requested lightpaths was not considered. Therefore, the number of wavelengths required for lightpath establishment obtained by these works is not minimized. In order to minimize the number of wavelengths, the RWA scheme that decides both routes and wavelengths of lightpaths to minimize the number of wavelengths is needed. The work in [10] presented the RWA scheme that decides both routes and wavelengths of lightpaths to minimize the number of wavelengths in the WRMD mesh network. The mathematical model and heuristic algorithm for the RWA scheme supporting one MCLS node are introduced. To support increasing numbers of lightpaths for large-scale networks, there may be a need to have more than one MCLS

• • • •

The number of nodes is given. Bi-directional connection is realized by two connections having opposite directions. The set of lightpath requests, P, is given. The maximum number of times an optical carrier can be reused, Rmax , is given for each wavelength.

3.2. Integer linear programming for RWA Problem We formulate the RWA problem considering full optical carrier replication for the WRMD network with multiple MCLS nodes as an ILP problem. For full optical carrier replication, an optical carrier at any node of optical carrier or any regeneration point is able to replicated, or split, into several other optical carriers and each optical carrier is used to establish another requested lightpath. Moreover, the loop problem may occur in the mesh topology when a source node receives a reused optical carrier from another established lightpath or a regeneration point, if we consider only flow conservation constraints, which are usually adopted in the conventional network design. We solve this loop problem by adding constraints that reflect the characteristics of the WRMD network. The following notations are used to describe the ILP problem. Let 25

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∑ ∑

N = |V | be the number of nodes, H = |E| be the number of links, and M be the set of MCLS nodes (MNs). Let qp(p , w, r ) be a binary decision variable that is set to one if lightpath request p ∈ P uses wavelength w ∈ W with r ∈ R , and zero otherwise. Let qc(c , w, r ) be a binary decision variable that is set to one if optical carrier c ∈ C uses wavelength w ∈ W with r ∈ R , and zero otherwise. Let f (p , i ) be an integer decision variable that denotes the ranking of node i ∈ V on lightpath p ∈ P . Let h(c, i ) be an integer decision variable that denotes the ranking of node i ∈ V on optical carrier c ∈ C . Let x (p , i , j ) be a binary decision variable that is set to one if lightpath request p ∈ P is routed on (i , j ) ∈ E , and zero otherwise. Let z(c, i , j ) be a binary decision variable that is set to one if optical carrier c ∈ C is routed on (i , j ) ∈ E′, and zero otherwise. Let y(w ) be a binary decision variable that indicates the usage of wavelength w, where w ∈ W . This variable is set to one if wavelength w is used at least once, and zero otherwise. Let a(p , i , j , w, r ) be a binary decision variable that is set to one if lightpath request p ∈ P is routed on (i , j ) ∈ E using wavelength w ∈ W with r ∈ R , and zero otherwise. Let b(c, i , j , w, r ) be a binary decision variable that is set to one if optical carrier c ∈ C is routed on (i , j ) ∈ E′ using wavelength w ∈ W with r ∈ R , and zero otherwise. Let d (i , j ) be a transmission length of (i , j ) ∈ E′. Let u(p , r ), which is a given parameter, be the maximum allowable transmission length of lightpath p ∈ P for r ∈ R . Let u(c, r ), which is a given parameter, be the maximum allowable transmission length of optical carrier c ∈ C for r ∈ R . Let (c, p , r ) ∈ U be a triplet that indicates the prohibition of transmission between optical carrier c ∈ C and lightpath p ∈ P for r ∈ R , where U is a set of triplets (c, p , r ). The prohibition is determined by considering the maximum allowable transmission length of optical carrier and lightpath for each optical carrier regeneration. The notations of parameters and variables are summarized in Tables 1 and 1, 2, respectively. The objective function is represented by,

∑

(2a)

∑ ∑

qc(c , w, r ) ≤ 1, ∀ c ∈ C

(2b)

r∈R w∈W

∑j :(i, j )∈ E x (p , i , j ) − ∑j :(j, i )∈ E x (p , j , i ) = ∑r ∈ R ∑w ∈ W qp(p , w, r ), ∀ p ∈ P, i = sp

∑

∑

x (p , i , j ) −

j :(i, j )∈ E

∑

(2c)

x (p , j , i ) = 0, ∀ p ∈ P, i ≠ sp , dp (2d)

j :(j, i )∈ E

z (c , i , j ) ≥

j :(i, j )∈ E

∑

∑ ∑

qc(c, w, r ), ∀ c ∈ C , i = sc

r∈R w∈W

(2f)

f (p , j ) − f (p , i )≥ ∑r ∈ R ∑w ∈ W qp(p , w, r ) − V (∑r ∈ R ∑w ∈ W qp(p , w, r ) − x (p , i , j )) , ∀ p ∈ P , (i , j ) ∈ E

(2g)

f (p , sp ) = 0, ∀ p ∈ P

(2h)

h(c, j ) − h(c, i )≥ ∑r ∈ R ∑w ∈ W qc(c , w , r ) − V (∑r ∈ R ∑w ∈ W qc(c , w, r ) − z(c, i , j )), ∀ c ∈ C , (i , j ) ∈ E

(2i)

h(c, sc ) = 0, ∀ c ∈ C

(2j)

∑r ∈ R {a(p , i , j , w, r ) + a(p′, i , j , w, r )+ b(c, i , j , w, r ) + b(c′, i , j , w, r )} ≤ y(w ), ∀ p , p′(p ≠ p′) ∈ P, ∀ c , c′(c ≠ c′) ∈ C , (i , j ) ∈ E , w ∈ W

y(w ).

(2l)

(1)

∑

This ILP minimizes the number of required wavelengths for lightpath establishment. The constraints are as follows.

b(c, i , j , w, r )d (i , j ) ≤ u(c, r ), ∀ c ∈ C , w ∈ W , r ∈ R (2m)

(i, j )∈ E ′

qp(p , w, r ) ≤ ∑c ∈ C ⧹{c :(c, p, r )∈ U} ∑j :(j, i )∈ E ′: i = s b(c, j , i , w, r ), p

∀ p ∈ P, w ∈ W , r ∈ R Table 1 Summary of parameters. Parameters

Description

V E E′

sp ∈ V

Set of network nodes Set of bidirectional links Set of bidirectional links including self links, E′ = E ∪ {(v, v )}, v ∈ V Set of MCLS nodes Set of lightpath requests Set of optical carrier connections Set of wavelengths generated by MCLS Set of number of times an optical carrier is reused Set of prohibition of transmission between an optical carrier and a lightpath for a regeneration Undirected graph G with |V | network nodes and |E | bidirectional links Source node of lightpath p ∈ P

dp ∈ V

Destination node of lightpath p ∈ P

sc ∈ V N = |V | H = |E | d (i , j ) u (p , r )

Source node of optical carrier connection c ∈ C Number of nodes Number of links Transmission length of (i, j ) ∈ E′ Maximum allowable transmission length of lightpath p ∈ P for r∈R Maximum allowable transmission length of optical carrier c ∈ C for r ∈ R Prohibition of transmission between optical carrier c ∈ C and lightpath p ∈ P for r ∈ R

G = (V , E )

u (c , r )

(c , p , r ) ∈ U

(2k)

a(p , i , j , w, r )d (i , j ) ≤ u(p , r ), ∀ p ∈ P , w ∈ W , r ∈ R

(i, j )∈ E

w∈W

M P C W R U

(2e)

z(c, j′, i ) − z(c, i , j ) ≥ 0, ∀ c ∈ C , (i , j ) ∈ E′, i ≠ sc

j ′:(j ′, i )∈ E

∑ min

qp(p , w, r ) = 1, ∀ p ∈ P

r∈R w∈W

qc(c, w, r ) ≤ ∑p ∈ P : d

(2n)

q (p , w, r − 1),

p = sc p

∀ c ∈ C , w ∈ W , r ∈ R⧹{0}

(2o)

qc(c , w, 0) = 0, ∀ w ∈ W , c ∈ C: sc ∉ M

(2p)

z(c, j , i ) = 0, ∀ c ∈ C , (j , i ) ∈ E: i ∈ M

(2q)

y(w ) ≥ y(w + 1), ∀ w ∈ W ⧹{wmax}

(2r)

a(p , i , j, w, r ) ≤ x (p , i , j ), ∀ p ∈ P, (i , j ) ∈ E , w ∈ W , r ∈ R

(2s)

a(p , i , j, w, r ) ≤ qp(p , w, r ), ∀ p ∈ P, (i, j ) ∈ E , w ∈ W , r ∈ R

(2t)

a(p , i , j, w, r ) ≥ x (p , i, j ) + qp(p , w, r ) − 1, ∀ p ∈ P , (i , j ) ∈ E , w ∈ W , r ∈ R

(2u)

b(c, i , j , w, r ) ≤ z(c, i , j ), ∀ c ∈ C , (i , j ) ∈ E′, w ∈ W , r ∈ R

(2v)

b(c, i, j , w, r ) ≤ qc(c, w, r ), ∀ c ∈ C , (i , j ) ∈ E′, w ∈ W , r ∈ R

(2w)

b(c, i , j , w, r ) ≥ z(c, i , j ) + qc(c , w, r ) − 1, ∀ c ∈ C , (i , j ) ∈ E′, w ∈ W , r ∈ R

(2x)

Eq. (2a) ensures the assignment of lightpaths to all connection requests. Eq. (2b) ensures that each optical carrier connection is established at most once with at most one wavelength. Eqs. (2c) and (2d) are the flow conservation constraints on the incoming and outgoing flows at each node for lightpaths. Eqs. (2e) and (2f) are the 26

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Table 2 Summary of variables. Variables

Description

qp(p , w, r )

Binary decision variable used to designate wavelength w ∈ W with r ∈ R for lightpath request p ∈ P

qc(c, w, r )

Binary decision variable used to designate wavelength w ∈ W with r ∈ R for optical carrier c ∈ C

f (p , i ) h (c , i ) x (p , i , j ) z (c , i , j ) y (w ) a (p , i , j , w , r ) b (c , i , j , w , r )

Integer decision variable used to denote the ranking of node i ∈ V on lightpath p ∈ P Integer decision variable used to denote the ranking of node i ∈ V on optical carrier c ∈ C Binary decision variable used to designate the route of lightpath request p ∈ P on (i, j ) ∈ E Binary decision variable used to designate the route of optical carrier c ∈ C on (i, j ) ∈ E′ Binary decision variable used to indicate the usage of wavelength w ∈ W Binary decision variable used to designate wavelength w ∈ W with r ∈ R for the route of lightpath request p ∈ P on (i, j ) ∈ E Binary decision variable used to designate wavelength w ∈ W with r ∈ R for the route of optical carrier c ∈ C on (i, j ) ∈ E′

flow conservation constraints on the incoming and outgoing flows at each node for optical carrier connections. Eqs. (2g) and (2h) are the loop-prevention constraints [17,18] for lightpaths, and Eqs. (2i) and (2j) are those for optical carrier connections. Eq. (2g) expresses the rank of each node, where the ranking of the upstream node on lightpath p ∈ P must be lower than that of the downstream node. Eq. (2h) expresses that the rank of the source node of lightpath p ∈ P is set to zero. Eq. (2i) expresses the rank of each node, where the ranking of the upstream node on optical carrier connection c ∈ C must be lower than that of the downstream node. Eq. (2j) expresses that the rank of the source node of optical carrier connection c ∈ C is set to zero. Eq. (2k) ensures that different lightpaths and optical carrier connections must use different wavelengths for each link. Eq. (2l) is the maximum allowable transmission length constraint on lightpath p ∈ P for r ∈ R . Eq. (2m) is the maximum allowable transmission length constraint on optical carrier connection c ∈ C for r ∈ R . Eq. (2n) ensures that a lightpath is established if a source node receives an optical carrier. Moreover, Eq. (2n) also ensures that the transmission length between optical carrier connection and lightpath for a connection request must not exceed the maximum allowable transmission length for each optical carrier regeneration. This limitation reflects the constraint that the number of transmission spans and the transmission length limit the available number of OCR nodes. Eq. (2o) ensures that an optical carrier is reused if a connection request (lightpath) is established. On the other hand, an optical carrier with r should be received by another optical carrier with r − 1. Eq. (2p) ensures that optical carrier connection c ∈ C that is not generated from any MCLS nodes must not produce any optical carrier with r = 0 . In other words, an optical carrier with r = 0 must be generated only from any MCLS nodes. Eq. (2q) ensures that optical carrier connection c ∈ C that is duplicated or regenerated from any nodes must not produce any optical carrier to any MCLS nodes. Eqs. (2n) to (2q) guarantee the prevention of loop generation between lightpath and optical carrier connection. Eq. (2r) states that wavelengths are used in ascending order of wavelength index w. Eqs. (2s) to (2u) are represented as a Boolean expression of a(p , i , j, w, r ) = x (p , i, j )*qp(p , w, r ) with linear forms with binary variables, where a(p , i , j , w, r ) is set to one only when both x (p , i , j ) = 1 and qp(p , w, r ) = 1. Eqs. (2v) to (2x) are represented as a Boolean expression of b(c, i , j , w, r ) = z(c, i , j )*qc(c , w, r ) with linear forms with binary variables, where b(c, i , j , w, r ) is set to one only when both z(c, i , j ) = 1 and qc(c , w, r ) = 1. The RWA decision problem in the WRMD network, called RWAWRMD, is NP-complete [11]. When the size of the ILP problem as formulated above becomes large, the ILP problem is not solvable within practical time. Moreover, due to memory constraint, a feasible solution of the ILP problem for large-scale networks is not obtained. In order to overcome the time computational difficulty and the memory constraint, a heuristic RWA approach is needed.

replication for overcoming the difficulty of the ILP problem. The heuristic RWA scheme is based on two lightpath selection policies in [11], which are nearest optical carrier first (NCF) and less number of required wavelengths first (LWF). Both policies are improved by including full optical carrier replication, which replicates optical carriers at any node of optical carrier or any regeneration point, for establishing the requested lightpaths. 3.3.1. Overview The heuristic RWA scheme consists of a routing algorithm and a wavelength assignment algorithm, which are performed separately. The routing algorithm provides the routes of optical carrier connections and requested lightpaths. We employ the alternate routing using the k-shortest path (KSP) algorithm [20] to find the first k shortest paths from the light-source node or regeneration point to source for the optical carrier connection and source to destination for the requested lightpath. The time complexity of the KSP algorithm is O(kN (H + N logN )) [21]. The wavelength assignment algorithm requires the routes before it runs. There are two steps in the wavelength assignment algorithm. In the first step, chains of lightpaths are created. In this step, a requested lightpath is selected to establish a connection, based on a lightpath selection policy. An optical carrier is generated from the MCLS node, and travels along a carrier lightpath to the selected requested lightpath. The optical carrier is regenerated at the end node of the selected requested lightpath. Another requested lightpath is selected. The optical carrier travels along the other carrier lightpath to the requested lightpath. A path from the MCLS node to the end node of the last requested lightpath, including carrier lightpaths and requested lightpaths, is called a chain of lightpaths. Moreover, due to the property of optical carrier replication, the optical carrier can be split into several other optical carriers and each optical carrier is used to establish another requested lightpath. In the second step, each lightpath chain is assigned a wavelength. Wavelength assignment is then solved as a graph coloring problem. We use a heuristic algorithm, called the largest degree first (LDF) [19], to solve the graph coloring problem, since it is widely used in graph coloring research and its effectiveness has been confirmed. The time complexity of the LDF algorithm is O( Vgcp + Egcp ), where Vgcp and Egcp are a set of vertices and edges in a graph coloring problem, respectively. As shown in Fig. 3, the heuristic RWA scheme first decides the routes by the routing algorithm. Wavelengths are then assigned by the wavelength assignment algorithm. 3.3.2. Lightpath selection policies to create lightpath chains In the heuristic RWA scheme, the NCF and LWF policies are modified by integrating a full optical carrier replication to create the lightpath chains. To describe the heuristic RWA scheme, additional terms are defined. Let T = {t1, t2, …, tN } be a set of light sources, which are defined as MCLS nodes or regeneration points. Let gt be the nodal degree of light source t. Let L = {l1, l2, …, lP} be a set of lightpaths. Let flt be the distance from light source t to lightpath l, and fl be the length

3.3. Heuristic RWA scheme We develop a heuristic RWA scheme to consider full optical carrier 27

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Fig. 3. Flowchart of heuristic RWA scheme.

of lightpath l. We define an optical connection as a lightpath associated with an optical carrier. Let O = {ol1t1, olt21, ol1t2 , …, olPtN } be a set of optical connections, where oltji is an optical connection from light source ti ∈ T to lightpath l j ∈ L . The two lightpath selection policies are described as follows. At the beginning, each MCLS node and requested lightpath are indexed.

• • • •

3.3.2.1. Nearest optical carrier first with full optical carrier replication (NCF-FCR) policy. The aim of the NCF-FCR policy is to create each chain of lightpaths with the shortest length, since a longer transmission length has larger transmission loss and results in stricter limits being placed on optical carrier regeneration [7]. This policy first selects the requested lightpaths that are the nearest to any MCLS node in case regeneration is not used. In case of carrier regeneration, it selects the remaining requested lightpaths that are possibly nearest to the regeneration point to fulfill both carrier regeneration and wavelength resources. The NCF-FCR policy considers the optical carrier replication to use the resources effectively when it selects the requested lightpaths. The overall time complexity of this policy is O(kP 2NR ) . The lightpath selection process is as follows.

• • • •

• • • • •

Step 1: Set r ∈ R to 0. Step 2: Add MCLS node m ∈ M that is the nearest to lightpath p ∈ P to T. Step 3: Select a light source t ∈ T . Step 4: If r = 0 , then add lightpath p ∈ P that is the nearest to light source t and is less than or equal to u(p , r ) to L. Otherwise, add

lightpath p ∈ P that is less than or equal to u(p , r ) to L considering full optical carrier replication. Step 5: Select lightpath l ∈ L . Step 6: Add an optical connection from light source t ∈ T to lightpath l ∈ L using optical carrier c ∈ C that is not in prohibition (c, l , r ) ∈ U to O and then remove lightpath l from L. Step 7: If L is not empty, then repeat from step 5. Otherwise, remove light source t from T. Step 8: If T is not empty, then repeat from step 3. Otherwise, sort O on gt in descending order as the first key, flt in ascending order as the second key, and fl in ascending order as the third key. Step 9: Select the optical connection o ∈ O that has the highest rank. Step 10: If the selected optical connection from light source t to lightpath l does not collide with each other, then set up this optical connection, add destination node of lightpath l to T as well as remove optical connection o with lightpath l from O, and lightpath l from P. Otherwise, remove optical connection o from O. Step 11: If O is not empty, then repeat from step 9. Otherwise, increase r by one. Step 12: If r ≤ Rmax , then go to step 3. Otherwise, a chain of lightpaths is created; reset T and L to empty. Step 13: If P is not empty, then go to step 1. Otherwise, the lightpath selection process is finished.

3.3.2.2. Less number of required wavelengths first with full optical carrier replication (LWF-FCR) policy. The aim of the LWF-FCR policy is to create each lightpath chain to avoid unnecessary carrier regeneration while minimizing the number or required wavelengths. 28

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lightpath i = 5 travels on route 4 → 6. We assume that node 1 is the MN, which is named as MN . Moreover, the allowable carrier regeneration number is one and the alternate routing algorithm considers the maximum number of paths with k = 1. In the NCF policy, the policy first sets r to 0 for initialization. Next, the policy adds the MCLS node that is the nearest to each requested lightpath to T. As a result, T includes MN (step 2). Next, the policy selects MN and then adds the requested lightpath i = 1 that is the nearest to MN to L (step 4). The requested lightpath i = 1 is processed in steps 5-10. T is then set with the destination node of requested lightpath i = 1, which is node 4 (step 10). Next, r is increased by one in step 11. The policy selects the regeneration point (node 4) and then adds all requested lightpaths i=2,3,4, and 5. The requested lightpaths are processed to add optical connections to O in steps 5–7. The policy sorts O on nodal degree of regeneration point in descending order as the first key, distance from regeneration point to requested lightpath in ascending order as the second key, and length of requested lightpath in ascending order as the third key (step 8). The sorted O is related to the requested lightpaths i=4,5,2, and 3. All requested lightpaths i = 4,5,2, and 3 can be selected. The first chain of lightpaths is then created on route 1 → 4, 4 → 1, 4 → 5 → 6, 4 → 2 → 3 → 5, and 3 → 6 (step 12), as shown in Fig. 4(b). In the LWF policy, the policy first sets r to 0 for initialization. Next, all MNs are first added to T (step 2). Next, the policy selects MN and then adds all requested lightpaths to L. Next, it selects MN and adds all the requested lightpaths to L. Notice that the policy adds all combinations between the MNs and the requested lightpaths to O in steps 3–7. Next, the policy sorts O on nodal degree of regeneration point in descending order as the first key, distance from regeneration point to requested lightpath in ascending order as the second key, and length of requested lightpath in ascending order as the third key (step 8). Next, it selects an optical connection from MN to the requested lightpath i = 1 and selects optical connections from MN to the requested lightpaths i = 2 and 3. The requested lightpaths i = 1, 5, 6, and 2 are processed in steps 5–10. The optical carrier replication is used to select an optical connection from node 2 to the requested lightpath i = 4 and 5. The first chain of lightpaths is then created on route 1 → 4, 1 → 2 → 3 → 5, 3 → 6, 2 → 4 → 1, and 4 → 5 → 6 → 1 (step 12), as shown in Fig. 4(b).

This policy selects the requested lightpaths that do not collide with each other regardless of the number of transmission spans and transmission length if regeneration is not used. If carrier regeneration is used, it selects the requested lightpaths that are the nearest to any regeneration point to minimize the transmission length. The LWF-FCR policy considers the optical carrier replication to use the resources effectively when it selects the requested lightpaths. The overall time complexity of this policy is O(kP 2NR ). The lightpath selection process is as follows.

• • • • • • • • • • • • •

Step 1: Set r ∈ R to 0. Step 2: Add MCLS node m ∈ M to T. Step 3: Select light source t ∈ T . Step 4: If r = 0 , then add lightpath p ∈ P that is less than or equal to u(p , r ) to L considering full optical carrier replication. Otherwise, add lightpath p ∈ P that is the nearest to light source t and is less than or equal to u(p , r ) to L. Step 5: Select lightpath l ∈ L . Step 6: Add an optical connection from light source t ∈ T to lightpath l ∈ L using optical carrier c ∈ C that is not in prohibition (c, l , r ) ∈ U to O and then remove lightpath l from L. Step 7: If L is not empty, then repeat from step 5. Otherwise, remove light source t from T. Step 8: If T is not empty, then repeat from step 3. Otherwise, sort O on gt in descending order as the first key, flt in ascending order as the second key, and fl in ascending order as the third key. Step 9: Select the optical connection o ∈ O that has the highest rank. Step 10: If the selected optical connection from light source t to lightpath l does not collide with each other, then set up this optical connection, add destination node of lightpath l to T as well as remove optical connection o with lightpath l from O, and lightpath l from P. Otherwise, remove optical connection o from O. Step 11: If O is not empty, then repeat from step 9. Otherwise, increase r by one. Step 12: If r ≤ Rmax , then go to step 3. Otherwise, a chain of lightpaths is created; reset T and L to empty. Step 13: If P is not empty, then go to step 1. Otherwise, the lightpath selection process is finished.

4. Performance evaluation NaN. Example of lightpath selection policies Fig. 4 shows examples of the two lightpath selection policies. There are five requested lightpaths: first, requested lightpath i = 1 travels on route 1 → 4; second, requested lightpath i = 2 travels on route 3 → 5; third, requested lightpath i = 3 travels on route 3 → 6; fourth, requested lightpath i = 4 travels on route 4 → 1; and finally, requested

We evaluate the performance of the heuristic RWA scheme using two metrics. First, the number of required wavelengths demanded by the proposed ILP approach and heuristic RWA scheme is compared with that demanded by the conventional ILP approach and heuristic RWA scheme in [11] and that demanded by the theorem of bidirectional ring in [13].

Fig. 4. Example of NCF-FCR and LWF-FCR policies.

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replication at any start node of optical carrier or any end node of lightpath. The proposed ILP approach with full optical carrier replication, named ILP-FCR approach, gives the minimum number of required wavelengths. Figs. 7 and 8 show the number of required wavelengths (indicated by W) and running times (indicated by RT) of the conventional ILP approach and heuristic RWA scheme without full optical carrier replication and those of the proposed ILP approach and heuristic RWA scheme with full optical carrier replication for Network 1 and Network 2, respectively, and the three different regeneration numbers. The ILP-FCR approach gives the minimum number of required wavelengths. Therefore, the ILP-FCR approach is able to provide reference values for further analysis. We observe that the heuristic RWA scheme approaches the optimal number of required wavelengths. For Network 1, the heuristic RWA scheme with LWF-FCR policy outperforms that with NCF-FCR policy in the no-regeneration case, while the heuristic RWA scheme with NCF-FCR policy outperforms that with LWF-FCR policy in the cases of one and two regenerations. We observe that the number of required wavelengths decreases as the number of MCLS nodes increases. Moreover, the ILP-FCR approach reduces the number of required wavelengths, compared to the conventional ILP approach. Similarly, the heuristic RWA scheme with NCFFCR and LWF-FCR policies reduces the number of required wavelengths, compared to the conventional heuristic RWA scheme with NCF and LWF policies, respectively. For Network 2, the heuristic RWA scheme with NCF-FCR and LWF-FCR policies approaches the optimal number of wavelengths as the maximum number of paths, k, increases. Furthermore, we note that the running times of the ILP approach are significantly higher than those of the heuristic RWA scheme, especially when the numbers of network nodes and lightpath requests increase. The running times of the heuristic RWA scheme with both policies are slightly different in each network topology. However, the ILP approach is hard to solve in any practical time, especially when the network topology is large. Therefore, we investigate the performance of the heuristic RWA scheme for large-scale networks in the next subsection.

Second, the number of required wavelengths by the proposed heuristic RWA scheme under different parameters is compared with that demanded by the conventional schemes in [10,11]. In our evaluation, we assume that there is no limitation of transmission length of optical carrier connections and requested lightpaths, and no prohibition of transmission between optical carrier and lightpath in order to focus on the minimum number of required wavelengths for all networks. u(c, r ) and u(p , r ) are set to the maximum of possible transmission length (or the summation of transmission length of all links) of optical carrier connections and requested lightpaths, respectively. (c, p , r ) ∈ U is set to an empty set, ∅. 4.1. Comparison of RWA schemes without and with full optical carrier replication The RWA problem is to determine both route and wavelength for each lightpath request. Therefore, all routes of optical carrier connections and requested lightpaths are required to be designed to minimize the number of wavelengths. In order to evaluate the effectiveness of the RWA scheme, the number of required wavelengths is investigated. First, the number of wavelengths required under the proposed ILP approach and heuristic RWA scheme with full optical carrier replication is compared with that under the conventional ILP approach and the heuristic RWA scheme without full optical carrier replication in [11] and that under the theorem of bidirectional ring in [13] in the WRMD ring network, as shown in Fig. 5(a). Second, the number of wavelengths required under the proposed ILP approach and heuristic RWA scheme with full optical carrier replication is compared with that under the conventional ILP approach and the heuristic RWA scheme without full optical carrier replication in [11] in the WRMD mesh networks, as shown in Fig. 6. The proposed heuristic RWA scheme with full optical carrier replicaton takes into account the first k shortest paths from the light-source node or regeneration point to the source node of requested lightpath for the optical carrier connection and the source to the destination for the requested lightpath in the WRMD mesh network. Fig. 5(b) shows the number of required wavelengths of the theorem of bidirectional ring in [13], the conventional ILP approach and heuristic RWA scheme without full optical carrier replication, and the proposed ILP approach and heuristic RWA scheme with full optical carrier replication for 4-node bidirectional ring network and the three different regeneration numbers. As the theorem of bidirectional ring in [13], the number of required wavelengths needed to support all-to-all broadcast in an N-node bidirectional ring is (1/2)N (N − 1) without regeneration and ((N 2 − 1)/4) if N is odd and (N 2 /4) if N is even, under one or more regeneration. This theorem demands more required wavelengths than the others, since it does not consider optical carrier

4.2. Effectiveness of heuristic RWA scheme We evaluate the effectiveness of the heuristic RWA scheme using the two lightpath selection policies for large-scale networks, namely European COST239 network and U.S. long distance network as shown in Fig. 9. The number of requested lightpaths is set to 200 and 300. We average the values over the numbers of required wavelengths for 100 randomly generated, different sets of requested lightpaths. The allowable number of carrier regenerations is two since, based on a literature

Fig. 5. Comparison of theorem of bidirectional ring, ILP approach, and heuristic RWA scheme.

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Fig. 6. Network topologies for comparison of ILP approach and heuristic RWA scheme.

Fig. 7. Comparison between ILP approach and heuristic RWA scheme for Network 1.

scheme without regeneration demands a large number of required wavelengths; the number decreases as the allowable carrier regeneration number increases. The heuristic RWA scheme with one regeneration reduces the number of required wavelengths by at least 32% and 35% from that without regeneration for the European COST 239 and U.S. long distance networks, respectively. Furthermore, the heuristic RWA scheme with two regenerations reduces the number of required wavelengths by at least 37% and 38% from that without regeneration for the European COST 239 and U.S. long distance networks, respectively. In addition, the heuristic RWA scheme is able to reduce the number of required wavelengths when the number of MCLS nodes increases. The results show that the heuristic RWA scheme with two MCLS nodes reduces the number of required wavelengths by at least 13% and 22% from that with one MCLS node for the European COST 239 and U.S. long distance networks, respectively. On the other hand, the heuristic RWA scheme with three MCLS nodes reduces the number of required wavelengths by at least 30% and 30% from that with one MCLS node for the European COST 239 and U.S. long distance networks, respectively. Additionally, the number of required wavelengths decreases under the heuristic RWA scheme as the maximum number of paths, k, increases. The number of required wavelengths of the conventional WDM networks also decreases as the maximum

survey [7], at most two regenerations are reported. Regenerating carriers more than two times results in high degradation in optical signal-to-noise ratio (OSNR) of the transmission signal. The alternate routing algorithm considers the maximum number of paths with k = 3. The number of required wavelengths of the heuristic RWA scheme is compared with that of the conventional WDM network, the scheme with near MCLS node (NM) policy [10], and the conventional scheme with NCF and LWF policies [11]. The conventional WDM network, which has its own multiple LDs at each node, employs the k shortest path algorithm as the routing decision and the largest degree first as the wavelength assignment algorithm to calculate the number of required wavelengths. The scheme with the NM policy in [10], which supports only one MCLS, is based on the shortest path algorithm and the largest degree first. Figs. 10 and 11 show the number of required wavelengths and running times of the heuristic RWA scheme with the two different policies and those of the scheme with the NM, NCF, and LWF policies for the European COST 239 and U.S. long distance networks, respectively, with 200 requested lightpaths. Moreover, the number of required wavelengths of the conventional WDM network is also shown in each table. We observe similar behavior with regard to the number of required wavelengths and the running times. The heuristic RWA

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Fig. 8. Comparison between ILP approach and heuristic RWA scheme for Network 2.

carrier replication reduces the number of required wavelengths from the conventional scheme. As shown in Figs. 10 and 11, the running times of the NM policy increase significantly as the allowable carrier regeneration number increases, since it takes more time to find the shortest distance between the regeneration point and the source of requested lightpaths. For the networks with multiple MCLS nodes and without carrier regeneration, the running times of the LWF and LWF-FCR policies increase significantly as the number of MCLS nodes and the maximum number of paths, k, increase, since it takes more time to avoid the overlapping of optical carrier connections and requested lightpaths. For the networks with carrier regeneration, the running times of the NCF and NCF-FCR policies increase as the allowable carrier regeneration number and the maximum number of paths, k, increase, since it takes more time to reuse optical carriers. Note that the LWF-FCR policy increase the running times, compared to the LWF policy, in case regeneration is not used. In case of carrier regeneration, the NCF-FCR policy increase the running times, compared to the NCF policy. The reason is that the NCF-FCR and LWFFCR policies take more time to replicate optical carriers, compared to the

number of paths, k, increases. We note that the number of required wavelengths in the WRMD network approaches that in the conventional WDM network if the number of MCLS nodes, the allowable number of carrier regenerations, and the maximum number of paths, k, are increased. That is, the WRMD network with the suitable number of MCLS nodes, the suitable number of carrier regenerations, and the suitable number of paths, k, has almost the same performance as the conventional WDM networks in terms of the network resource occupation. Besides, we observe that the heuristic RWA scheme with the NCF-FCR policy and the maximum number of paths with k = 1 outperforms the scheme with the NM policy in terms of the number of required wavelengths. For the European COST 239 network, the heuristic RWA scheme with the NCF-FCR and LWF-FCR policies reduces the number of required wavelengths by at least 2% and 10% from that with the NCF and LWF policies, respectively. For U.S. long distance network, the heuristic RWA scheme with the NCF-FCR and LWF-FCR policies reduces the number of required wavelengths by at least 2% and 30% from that with the NCF and LWF policies, respectively. As a result, the heuristic RWA scheme considering optical

Fig. 9. Network topologies examined.

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Fig. 10. Comparison of NM, NCF, and LWF policies for European COST 239 network with 200 requested lightpaths.

and 9% from that with the NCF and LWF policies, respectively. For U.S. long distance network, the heuristic RWA scheme with the NCF-FCR and LWF-FCR policies reduces the number of required wavelengths by at least 2% and 28% from that with the NCF and LWF policies, respectively.

NCF and LWF policies, respectively. Similarly, Figs. 12 and 13 show the number of required wavelengths and running times of the heuristic RWA scheme with the two different policies and those of the scheme with the NM, NCF, and LWF policies for European COST 239 and U.S. long distance networks, respectively, with 300 requested lightpaths. The results show that for the European COST 239 network, the heuristic RWA scheme with the NCF-FCR and LWFFCR policies reduces the number of required wavelengths by at least 3%

4.3. Dependency of location of MCLS nodes We examine the impact of MCLS node location since their placement

Fig. 11. Comparison of NM, NCF, and LWF policies for U.S. long distance network with 200 requested lightpaths.

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Fig. 12. Comparison of NM, NCF, and LWF policies for European COST 239 network with 300 requested lightpaths.

is employed with each regeneration condition. In the case of noregeneration, the best MCLS node locations in the European COST 239 network are node 3 for one MCLS node, nodes 4 and 7 for two MCLS nodes, and nodes 2, 4, and 10 for three MCLS nodes. For the U.S. long distance network, the best MCLS node locations are node 20 for one MCLS node, nodes 6 and 28 for two MCLS nodes, and nodes 6, 27, and 28 for three MCLS nodes. If regeneration is used, the best MCLS node locations in the

in the WRMD mesh network affects the number of required wavelengths. We observe the most and fewest number of required wavelengths, which are averaged over those with 100 randomly generated and different sets of requested lightpaths, for all combinations of MCLS node locations in each network. We use the heuristic RWA scheme with both policies with k, the maximum number of paths, set to 3 for the European COST 239 and U.S. long distance networks with 300 requested lightpaths, as shown in Fig. 9. From Table 3 we present the best locations, where a suitable policy

Fig. 13. Comparison of NM, NCF, and LWF policies for U.S. long distance network with 300 requested lightpaths.

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Table 3 Comparison of locations of MCLS nodes with 300 requested lightpaths. Network topology

Number of MCLS nodes

Policy

Worst location

Best location

Average of combinations

European COST 239 network

1

LWF-FCR without regeneration (k = 3) NCF-FCR with two regenerations (k = 3) LWF-FCR without regeneration (k=3) NCF-FCR with two regenerations (k = 3) LWF-FCR without regeneration (k = 3) NCF-FCR with two regenerations (k = 3)

43 26 40 27 38 27

32 21 26 21 23 20

40 23 31 23 28 23

LWF-FCR without regeneration (k=3) NCF-FCR with two regenerations (k=3) LWF-FCR without regeneration (k = 3) NCF-FCR with two regenerations (k = 3) LWF-FCR without regeneration (k = 3) NCF-FCR with two regenerations (k = 3)

85 60 83 48 83 47

58 44 45 43 43 43

74 46 60 45 53 45

2 3

U.S. long distance network

1 2 3

European COST 239 network are node 2 for one MCLS node, nodes 2 and 7 for two MCLS nodes, and nodes 1, 2, and 10 for three MCLS nodes. For the U.S. long distance network, the best MCLS node locations are node 27 for one MCLS node, nodes 25 and 28 for two MCLS nodes, and nodes 16, 26, and 28 or nodes 24, 27, and 28 for three MCLS nodes.

[3]

[4]

[5]

5. Conclusion This paper proposed a routing and wavelength assignment scheme for WRMD mesh networks that consider optical carrier replication to minimize the number of wavelengths required for lightpath establishment. We focused on the static scenario, which assumes that lightpath setup requests are statically given in advance. The heuristic RWA scheme consists of a routing algorithm and a wavelength assignment algorithm; they are run separately to solve the RWA problem. We use the KSP algorithm to realize alternate routing. The wavelength assignment algorithm has two steps. The first step is to create chains of lightpaths and the second step is to assign a wavelength to each lightpath chain. Moreover, two original lightpath selection policies, NCF-FCR and LWF-FCR, are introduced to create the lightpath chains considering optical carrier replication. The results showed that the heuristic RWA scheme with optical carrier replication and either of the lightpath selection policies reduces the number of required wavelengths by at least 2% and at most 37% from the conventional scheme. The scheme with the LWF-FCR policy achieves better performance than that with the NCF-FCR policy if regeneration is not used, while the scheme with the NCF-FCR policy outperforms that with the LWFFCR policy in the cases of one and two regenerations. In addition, we observed that optimizing MCLS node location also reduces the number of required wavelengths.

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13] [14]

[15] [16]

Acknowledgments

[17]

We would like to thank Motoharu Matsuura for his helpful comments on system and experimental aspects of our work.

[18]

[19]

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