A mathematical model for wavelength assignment in wavelength division multiplexing mesh networks with wavelength reuse Bonar Sitorus a), Nattapong Kitsuwan, and Eiji Oki Department of Communication Engineering and Informatics, The University of Electro-Communications, 1 5 1 Chofugaoka, Chofu-shi, Tokyo, 182 8585 Japan a) bonar.sms@uec.ac.jp Abstract: This letter presents a mathematical model that minimizes the number of wavelengths required for wavelength assignment in a wavelength-reusable multi-carrier-distributed (WRMD) wavelength division multiplexing mesh network. None of the source nodes in the WRMD network is equipped with laser diodes. It receives carrier wavelengths from a multi-carrier light source or reuses carrier wavelengths from other established lightpaths. The loop problem may occur in mesh topologies when a source node selects a reused carrier wavelength from a destination lightpath node, if we consider only flow conservation constraints, which are usually adopted in conventional network design. We solve the loop problem by formulating constraints that reflect the special characteristics of the WRMD network. To support the development of heuristic algorithms towards large-scale networks, our mathematical model, which can provide reference values including upper and lower bounds, is useful for benchmarks. Keywords: optical network, wavelength assignment, multi carrier light source, optical carrier regenerator Classification: Fiber-Optic Transmission for Communications References [1] M. Matsuura and E. Oki, Optical Carrier Regeneration for Carrier Wavelength Reuse in a Multicarrier Distributed WDM Network, IEEE Photonics Technology Letters, vol. 22, no. 11, pp. 808 810, May 2010. [2] M. Keri, E. Oki, and M. Matsuura, Wavelength Assignment in Multi- Carrier Distributed Optical Ring Networks with Wavelength Reuse, Opt. Commun. Netw., vol. 3, no. 4, April 2011. [3] Y. Miyagawa, T. Yamamoto, H. Masuda, M. Abe, H. Takahashi, and H. Takara, Over-10000-channel 2.5 GHz-spaced ultra-dense WDM light source, Electron. Lett., vol. 42, no. 11, pp. 655 657, 2006. 125
[4] M. Sharma, H. Ibe, and T. Ozeki, WDM ring network using a centralized multiwavelength light source and add drop multiplexing filters, J. Lightw. Technol., vol. 43, no. 19, pp. 1040 1042, Sept. 2007. [5] D. Cavendish, A. Kolarov, and B. Sengupta, Routing and Wavelength Assignment in WDM Mesh Networks, IEEE Communications Society, Globecom, 2004. [6] I. Chlamtac, A. Ganz, and G. Karmi, Lightpath Communications: An Approach to High Bandwidth Optical WAN s, IEEE Trans. Commun., vol. 40, no. 7, July 1992. 1 Introduction Wavelength division multiplexing (WDM) technology has been identified as a suitable candidate for future wide area network (WAN) environments due to its potential ability to meet rising demands for high bandwidth and low latency communication. In conventional WDM networks without wavelength reusable capability, see Fig. 1 (a), more laser diodes (LDs) are needed to provide sufficient wavelengths to meet the explosive demand for network bandwidth. This, unfortunately, will raise energy consumption and development cost. Moreover, the complexity of wavelength management increases with the number of wavelengths. One solution is shown in Fig. 1 (b), the multi-carrier distributed optical network with wavelength reuse capability [1, 2]. This network is called the wavelength reusable multi-carrier distributed (WRMD) network. The WRMD network places a multi-carrier light source (MCLS) in an MCLS node, as the light source. The MCLS emits several optical line spectra with uniform frequency intervals [3]. The individual wavelengths are used as carrier wavelengths. MCLS generates the carrier wavelengths and passes them to all requesting source nodes for lightpath establishment. Furthermore, each node in the WRMD network is equipped with an optical carrier regenerator Fig. 1. Network architectures. 126
(OCR) [1]. The OCR allows the nodes to reuse a wavelength to satisfy multiple disjoint lightpath requests. Wavelength assignment for the WRMD network was proposed in [2]. None of the source nodes include a LD, and instead each directly receives a generated carrier wavelength from the MCLS node or a reused carrier wavelength from other destination lightpath request node. Therefore, the source node has several sources from which it can receive a carrier wavelength. Carrier distribution has to be managed to minimize the number of wavelengths, but only a ring topology was considered in the original paper. This letter presents a mathematical model that minimize the number of wavelengths required for wavelength assignment in a WRMD network with mesh topology. In the ring topology, a carrier wavelength connection, which connects the MCLS node and a lightpath or two lightpaths, is uniquely determined because the connecting direction is limited. On the other hand, in the mesh topology, several of carrier wavelength connection candidates must be considered. Therefore, the mesh topology makes distributing carrier wavelengths and assigning wavelengths much more complex than is true with the ring topology. The loop problem may occur in the mesh topology when a source node selects a reused carrier wavelength from a destination lightpath node, as shown in Fig. 1 (a), if we consider only flow conservation constraints, which are usually adopted in conventional network design. The challenge of this work is to provide a mathematical model that by solves the loop problem by considering the special characteristics of the WRMD network. The solution is based on ensuring that all carrier wavelengths used in the network are generated by the MCLS device, as shown in Fig. 1 (b). Fig. 2. Loop problem and our solution. 2 Integer linear programming model 2.1 Rules of wavelength assignment Wavelength assignment in a WRMD mesh network must obey these three rules: 1. Each wavelength can be used to satisfy several lightpath requests. 127
2. Each lightpath request is satisfied by using a generated carrier wavelength from the MCLS node or reused carrier wavelength from another established lightpath. 3. To avoid collision, carrier wavelengths and lightpath wavelengths on the same link must be assigned different wavelengths. 2.2 ILP formulation The objective of ILP is to minimize the number of wavelengths used in the WRMD network to establish the requested lightpaths. We assume that lightpath requests are given and the routes of lightpaths and carrier wavelength connections are designed in advance. A carrier wavelength connection connects MCLS with the starting point of a lightpath, or the end point of a light path to the starting point of another lightpath by using a carrier wavelength. The starting and end points of a carrier wavelength connection can be the same node, where no link is used to carry a carrier wavelength. To describe our ILP model, the following terminologies are used. Let W be a set of wavelengths. y(r) is a binary variable that is set to one if wavelength r is used, and otherwise zero. s S indicates the number of times a carrier wavelength is reused, where S = {0, 1,,S max }. S max is the maximum number of times a carrier wavelength can be reused. s =0 means that the carrier wavelength is directly generated from the MCLS node. Let P be a set of lightpath requests, and C be a set of carrier wavelength connections. q p (i, r, s) is a binary variable that is set to one if lightpath request i P uses wavelength r W with s S, otherwise zero. q c (k, r, s) is a binary variable that is set to one if carrier wavelength connection k C uses wavelength r W with s S, otherwise zero. Let E be a set of links. p(i, e) is a binary parameter that is set to one if lightpath request i P is routed on link e E, otherwise zero. c(k, e) is a binary parameter that is set to one if carrier wavelength connection k C is routed on link e E, otherwise zero. a(i, k) is a binary parameter that is set to one if the end point of lightpath request i P and the starting point of carrier wavelength connection k C belong to the same node, otherwise zero. a(i, k) =1means that lightpath request i P is connected to carrier wavelength connection k C. b(k, i) is a binary parameter that is set to one if the end point of carrier wavelength connection k C and the starting point of lightpath request i P belong to the same node, otherwise zero. b(k, i) =1means that carrier wavelength connection k C is connected to lightpath request i P. C RE is the set of carrier wavelength connections that reuse carrier wavelengths. Carrier wavelength connection k C RE is not generated from the MCLS node. The optimization problem is formulated as the following ILP problem: Objective min y(r) r q p (i, r, s) =1, i P (1a) Subject to s S r W (1b) 128
q c (k, r, s) 1, k C s S r W {p(i, e)q p (i, r, s)+p(j, e)q p (j, r, s)+ s S c(k, e)q c (k, r, s)+c(l, e)q c (l, r, s)} y r, i, j(i j) P, k, l(k l) C, r W, e E q p (i, r, s) r W {a(i, k) qc (k, r, s)}, i P i P, r W, s S q c (k, r, s) r W {b(k, i) qp (i, r, s 1)}, k C r W, s S\{0} q c (k, r, 0) = 0, r W, k C RE (1c) (1d) (1e) (1f) (1g) The objective in Eq. (1a) is to minimize the number of wavelengths needed to establish all lightpaths in the network. Eq. (1b) ensures the establishment of lightpaths of all connection requests. Eq. (1c) ensures that each carrier wavelength connection is established at most once with at most one wavelength. Eq. (1d) ensures that different lightpaths and carrier wavelength connections must use different wavelengths for each link. Eq. (1e) ensures that a lightpath is established if a source node receives a carrier wavelength. Eq. (1f) ensures that a carrier wavelength is reused if a lightpath is established. On the other hand, a carrier wavelength with s should be replaced by another carrier wavelength with s 1. Eq. (1g) ensures that carrier wavelength connection k C RE that is not generated from the MCLS node must not produce any carrier wavelength with s = 0. Eqs. (1e) to (1g) guarantee the prevention of loop generation. 3 Numerical results We demonstrate our mathematical model with three network topologies, as shown in Fig. 3. All network models consist of four nodes, and five lightpath requests. Our results confirms that no loop is permitted. With one regeneration, the mathematical model achieves the minimum number of wavelengths in each sample network. 4 Conclusion This paper has presented and demonstrated a mathematical model for wavelength assignment that minimizes the number of wavelengths needed in WRMD mesh networks. Mesh topologies can trigger the loop problem since a source node can select a reused carrier wavelength from a destination lightpath node. We solved the loop problem by formulating constraints that reflect the special characteristics of the WRMD network. For a large networks, some heuristic approaches/algorithms need to be developed and our presented mathematical model, which can provide reference values including upper and lower bounds, is useful for benchmarks. 129
Fig. 3. Numerical results from various network models. Acknowledgments This work was supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B) 23360168, and the Support Center for Advanced Telecommunications Technology Research (SCAT). 130