Cost optimal allocation of amplifiers and DCMs in WDM ring networks

Designing metropolitan wavelength division multiplexing (WDM) ring networks with minimum deployment cost is a demanding issue in Telecommunication Network planning . We have already presented amplifier placement methods to minimize the number of amplifiers in WDM rings for the case all amplifiers follow a unique gain model. In this paper, we take into account different types of amplifiers with predefined fixed characteristics and costs. We also formulate fiber dispersion limitations on the ring design, and present two efficient methods for placing amplifiers and Dispersion Compensation Modules (DCMs) in WDM rings to minimize the total deployment cost of the system. The first method deals with both linear and nonlinear equations and uses a mixed integer nonlinear programming (MINLP) solver where the second method applies the linear approximation of nonlinear constraints, and uses a mixed integer linear programming (MILP) solver to minimize the total cost of the system. We carry out Simulation experiments to confirm the applicability of the methods and compare the results for various network configurations. © 2006 Optical Society of America OCIS codes: (060.2330) Fiber optics communications; (060.4250) Networks References and links 1. A. Saleh and J. Simmons, “Architectural principles of optical regional and metropolitan access networks,” J. Lightwave Technol. 17, 2431 – 2448 (1999). 2. M. Borella, J. Jue, D. Banerjee, B. Ramamurthy, and B. Mukherjee, “Optical components for WDM lightwave networks,” Proceedings of the IEEE 85, 1274 – 1307 (1997). 3. R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows (Prentice Hall, New Jersey, 1993). 4. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical perspective (Morgan Kaufmann, San Francisco, CA, 1998). 5. B. Sanso and P. Soriano, Telecommunications Network Planning (Kluwer Academic, Norwell, MA, 1999). 6. T. E. Stern and K. Bala, Multiwavelength optical Networks: A Layered approach (Addison Wesley, Reading, MA, 1999). 7. C.-S. Li, F.-K. Tong, C. Georgiou, and M. Chen, “Gain equalization in metropolitan and wide area optical networks using optical amplifiers,” Proceedings IEEE INFOCOM ’94 1, 130 – 137 (1994). 8. B. Ramamurthy, J. Iness, and B. Mukherjee, “Optimizing amplifier placements in a multiwavelength optical LAN/MAN: the equally powered-wavelengths case,” J. Lightwave Technol. 16, 1560 – 1569 (Sept. 1998). 9. B. Ramamurthy, J. Iness, and B. Mukherjee, “Optimizing amplifier placements in a multiwavelength optical LAN/MAN: the unequally powered wavelengths case,” IEEE/ACM Transactions on Networking 6, 755 – 767 (1998). #74401 $15.00 USD Received 24 August 2006; revised 10 October 2006; accepted 10 October 2006 (C) 2006 OSA 30 October 2006 / Vol. 14, No. 22 / OPTICS EXPRESS 10278 10. J. Iness and B. Mukherjee, “New optical amplifier placement schemes for broadcast networks,” European Transactions on Telecommunications 11, 117 – 124 (2000). 11. A. Fumagalli, G. Balestra, and L. Valcarenghi, “Optimal amplifier placement in multi-wavelength optical networks based on simulated annealing,” Proceedings of the SPIE The International Society for Optical Engineering 3531, 268 – 279 (1998). 12. L. Zhong and B. Ramamurthy, “Optimization of amplifier placements in switch-based optical networks,” ICC 2001. IEEE International Conference on Communications 1, 224 – 228 (2001). 13. A. Tran, R. Tucker, and N. Boland, “Amplifier placement methods for metropolitan WDM ring networks,” J. Lightwave Technol. 22, 2509 – 2522 (2004). 14. A. Minagar and M. Premaratne, “Cost Optimal Configuration of Optical Networks,” J. Lightwave Technol. 24 3295 – 3302 (2006). 15. P. Saengudomlert, E. Modiano, and R. Gallager, “On-line routing and wavelength assignment for dynamic traffic in WDM ring and torus networks,” IEEE/ACM Transactions on Networking 14, 330 – 340 (2006). 16. K. Mosharaf, “Optimal Resource Allocation and Fairness Control in All-Optical WDM Networks,” IEEE Journal on Selected Areas in Communications 23, 1496 – 1507 (2005). 17. B. Mukherjee, “WDM Optical Communication Networks: Progress and Challenges,” IEEE Journal on Selected Areas in Communications 18, 1810 – 1824 (2000). 18. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley-Interscience, New York, 2002). 19. W. Cornwell and I. Andonovic, “Interferometric noise for a single interferer: comparison between theory and experiment,” Electron. Lett. 32, 1501 – 1502 (1996). 20. R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, 2nd ed. (Duxbury Press, Toronto, 2003). 21. “MINLP solver on NEOS server,” URL http://neos.mcs.anl.gov/neos/solvers/minco:MINLP/AMPL.html. 22. “User Manual for MINLP,” URL http://www.maths.dundee.ac.uk/ ̃ sleyffer/MINLP manual.ps.Z. 23. “MINTO solver on NEOS server,” URL http://neos.mcs.anl.gov/neos/solvers/milp:MINTO/AMPL.html.


Introduction
Optical ring networks deploying Wavelength Division Multiplexing (WDM) technology has gained prominence in present-day metro-area technology.Amplifier placement in these rings is a demanding design issue.As the distance between two adjacent nodes in WDM ring networks is less than 30 km, usually a single amplifier is sufficient to compensate the fiber loss [1].However, using one amplifier at each node increases the system cost.On the other hand, fiber dispersion is another factor limiting the ability to utilize system bandwidth.So, dispersion Compensation Modules (DCMs) need to be placed in WDM rings to compensate the dispersion incurred by the fibers.
There are variety of published studies in the literature on optical components and WDM networks design [2][3][4][5][6].There have been some amplifier placement methods to minimize the number of amplifiers in star and switched networks [7][8][9][10][11][12].We have previously introduced amplifier placement methods [13] that minimize the number of amplifiers in WDM rings considering a unique gain model for all amplifiers.But in the design process, the designer has access to a large number of amplifier types with different gain models and costs.Moreover, the fiber dispersion and the placement of DCMs was not considered.We have also proposed some methods [14] for the placement of amplifiers and DCMs in optical links based on dynamic programming methods.But WDM Ring networks are different to optical links due to the problems of ring lasing, recirculating amplified spontaneous emission (ASE) noise, and effect of optical add-drop multiplexer (OADM) crosstalk on the bit-error rate (BER) of the received signals.This paper presents two methods to find the cheapest configuration of amplifiers and DCMs in WDM rings from sets of different available amplifier and DCM types.The first method contains both linear and nonlinear constraints and is solved by a mixed integer nonlinear programming (MINLP) solver.The second method linearly approximates the nonlinear constraints of the first method and is solved by a mixed integer linear programming (MILP) solver.
The paper is organized as follows: In Section 2, the problem of cost optimal allocation of amplifiers and DCMs in WDM rings is explored and formulated.In Section 3, the solution methods are presented whereas its application to WDM ring design is demonstrated using series of examples in Section 4. After discussing the results, this paper is concluded in Section 5. A typical unidirectional WDM ring network is shown in Fig. 1 with N nodes and L links (L = N).We assume that each pair of nodes in WDM metro-area rings have a unique wavelength to communicate with.Therefore, the total number of wavelengths in the ring is W = N(N − 1)/2, and there is an OADM in each node that can add and drop (N − 1) wavelengths to communicate with the other nodes.However, in order to decrease the number of required wavelengths in networks with larger number of nodes (i.e.wide-area networks), different wavelength strategies should be used [15,16], or multiple rings need to be interconnected together to provide large geographical coverage [17].

Problem formulation
Each link can have up to one amplifier and one DCM.For practical and economical considerations, it is beneficial to place the amplifiers and DCMs at the end of the links.This strategy enhances network maintainability and reduces the deployment cost.In this paper, whenever a DCM is required to be placed, we assume that it is placed after the amplifier in the link.This helps the signal power to be boosted to the DCMs input power range, and to contract the losses in the DCMs.
One of the popular but simple gain models for optical amplifiers is [13] where G is the saturated gain in linear scale, P sat is the internal saturation power in watts, P in is the total input power in watts, and G 0 is the small signal gain in linear scale.Fig. 2 shows the amplifier gain G (dB) versus input power P in (dBm) for P sat = 20 dBm and G 0 = 30 dB.The dotted line is the exact gain function given by equation 1 and the solid line is a linear approximation of that.It can be assumed that the amplifier has a flat gain over the left piece of the solid line.Different types of amplifiers have different values of G 0 and P sat which lead to different gain functions, flat gain regions and costs.In this paper we reference each amplifier of type j with its maximum gain G amp, j (dB), maximum input power P max amp, j and minimum input power P min amp, j to operate in the unsaturated mode, and its cost C amp, j .As amplifiers with larger gains have higher costs, it's beneficial to use amplifiers in their flat gain regions and bring the total cost down.We assume that all amplifiers have a constant gain for all wavelengths being amplified and there are N amp types of amplifiers and N DCM types of DCMs available to choose from.We also assume that fibers have negative dispersion and DCMs have positive dispersion compensation values.The parameters and specifications of all devices including OADMs, amplifiers, and DCMs are shown in Table 1, whereas the network parameters and variables are summarized in Table 2 and Table 3 respectively.
Constants A and B are defined to calculate the ASE noise power (in dBm) using 0.1 nm and 20 nm bandwidths respectively [13], i.e., A = 10 log 10 (2N sp hνB 0 10 3 ) (2) where N sp = 2 is the spontaneous emission factor, h = 6.63 × 10 −34 Js is the Planck's constant, ν = 193.1 THz is the signal frequency, B 0 = 12.5 GHz is the noise bandwidth equivalent to 0.1 nm, B 1 = 2.5 THz is the noise bandwidth equivalent to 20 nm.We use constant A to calculate the total ASE noise as part of OSNR and constant B to calculate the total ASE noise contributing to the total system power.The objective function of this placement method is to minimize the total cost of the system which is the cost of amplifiers and DCMs, i.e., minimize subject to and Minimum allowed input power of DCM type j dBm We assume that amplifiers are in ascending order in accordance with their maximum gain, and amplifiers with higher gains have a higher cost.So, amplifier j is selected for link i if the required gain value in link i The exact value of G i is then achieved by tuning the amplifier or by use of attenuators.This is formulated as The total power in watts is the sum of all channel powers and the total ASE noise power on that link.The total ASE noise power in dBm at the beginning of link i is equal to P ASE SCR i − β 1 NODE SCR i − A + B where P ASE SCR i is the total ASE noise power at the end of the previous link using 0.1 nm bandwidth, β 1 NODE SCR i is the insertion loss for through channels at the source The total input power to amplifiers (P tot i ) should be within amplifiers input range.So we impose the inequality We define Λ i as the attenuation of DCM on link i, i.e., and Ω i as the dispersion compensation value of DCM on link i, i.e., The total input power to DCMs (P tot i +G i ) should be within DCMs input range.So we impose the inequality The total power at any point in the fibers should be lower than a maximum value P nonlinear to prevent nonlinear effects such as four-wave mixing and self-phase modulation.So, the total power at the beginning of the link (P tot i +αL i ) and the total power after the amplifiers (P tot i +G i ) should not be greater than P nonlinear , i.e., The total dispersion value of each channel on link i need to be within the minimum and maximum dispersion allowed in fibers.So, the total dispersion at the beginning of the the link (D i,k ) should be less than D max , and the total dispersion before the DCM should be greater than D min , i.e., The spectral density function of ASE noise S(ν) for an amplifier with gain value G in watts is calculated as [18] S(ν) = (G − 1)n sp hν (17) To calculate the total ASE noise power at the end of link i we should add the ASE noise power from the amplifier on link i and the total ASE noise power from the previous link in watts, i.e., If the channel at wavelength λ k in link i is not dropped at the destination node, the power of wavelength λ k at the beginning of the following link (P DEST i ,k ) is calculated as and the dispersion of wavelength λ k at the beginning of the following link (D DEST i ,k ) is calculated as If the channel at wavelength λ k in link i is dropped at the destination node, the OSNR of the received signal should be greater than or equal to the Desired OSNR, i.e., Furthermore, the power of dropped wavelengths should be no less than the receivers minimum detectable power (P minr ) and within the receivers dynamic range, i.e., (22) and the dispersion of dropped wavelengths should be within the minimum and maximum dispersion limits of the receiver, i.e., If the channel at wavelength λ k is added at the node NODE DEST i , its power at the beginning of the following link (P DEST i ,k ) is related to the transmitted power (P xmit NODE DEST i ,k ) as The transmitted power is limited to a minimum and a maximum value, i.e., However, the transmitted dispersion of added channels is set to 0, i.e., As it is shown in Fig. 3, for all wavelengths (λ k ) that are both added and dropped at each node j, the leakage at the add/drop ports from the input wavelength to the output wavelength (IX1), and from the add wavelength to the drop wavelength (IX2) cause input-output crosstalk (27), and add-drop crosstalk respectively (28).These crosstalks both need to be no greater than -25 dB to achieve a power penalty below 1 dB [19], i.e., In DROP ADD

Fig. 3. Insertion losses and leakage paths at a node in the ring
To prevent lasing and accumulation of ASE noise in the ring, the total gain in the ring needs to be less than the total loss by a margin, i.e.,

Solution method
This problem contains some linear and integer constraints as well as two nonlinear constraints (8) and ( 18) which result the problem to fall into the category of MINLP problems which are highly computational complex and hard to solve.
In the first method, the problem was programmed in the AMPL modeling language [20] and solved by the MINLP solver on NEOS server [21].The MINLP solver implements a branchand-bound algorithm searching a tree whose nodes correspond to continuous nonlinearly constrained optimization problems.The continuous problems are solved using filter SQP, a Sequential Quadratic Programming solver which is suitable for solving large nonlinearly constrained problems.The software guarantees to find global solutions, if the problem is convex [22].It is also effective to solve non-convex MINLP problems.However, no guarantee is given that a global solution is found in this case.
Due to presence of high number of variables and constraints as well as nonlinear equations in this optimization problem, The running time of the first method is too long and cannot solve the problem for a large number of nodes.So, in the second method, we first estimate the two nonlinear constraints ( 8) and (18) with two linear constraints with acceptable accuracy and then solve the remaining MILP problem.
Using the same scheme as [13], constraint (8) can be written in the form of y = 10 log 10 where We use Least-Squares Fit (LSF) to find a best fit approximation of (30) by the following linear function and our mission is to find h T which is a (W + 2) × 1 vector.In order to find h, we need M data samples taken from each x k in (30) and we define the (W + 2) × 1 vector x j to note each set of data samples, i.e., We also represent the value of y corresponding to each set of data samples in Equation ( 30) by y j y j = 10 log 10 To linearly approximate (30) using LSF method the following function C needs to be minimized with respect to h, i.e., where h T is the transpose of vector h.We can expand (34) as where When h is calculated, the nonlinear constraint (8) can be approximated by the following linear constraint  The solution from the MINTO solver (Method 2) for the same network topology and the same available amplifiers and DCMs, is exactly the same as the solution of Method 1.However, the running time of Method 2 is only 24 seconds whereas the running time of Method 1 is 13.1 minutes.It shows that using the linear approximations of the nonlinear constraints can significantly decrease the running time of the algorithm with acceptable accuracy of the solutions.
Table 6 shows the running time, total cost, and solution configuration of Method 1 and Method 2 under various network topologies.The results in Table 6 confirm that by increasing the link lengths the total cost of the system increases as amplifiers and DCMs with higher values of gain and dispersion are required which are more expensive.However, in some instances there is a small difference between the costs calculated by Method 1 and those calculated by Method 2. This is due to the error between the actual and approximate values of P tot i in ( 8) and (37), and P ASE i in (18) and (42).However, using tighter margins in Method 2 can insure the validity and applicability of the solutions.The results also confirm that using these placement methods, the number of required amplifiers and cost has reduced compared to placing one amplifier at each node.
The running time of Method 1 and Method 2 increase sharply if the number of nodes in the ring exceeds two dozen.In this case, an approximation of Method 2 could be employed in which the depth of the branch-and-bound algorithm is constrained to a maximum number.This strategy leads to a shorter running time, however, as some part of the search space is cut, the final solution may only be quasi-optimal.However, such methods are widely used in practice due to the acceptable quality of results.For instance, by limiting the branch-and-bound search to 5000 nodes in the previous example, the results are consistent as Method 2 for the 6-node ring topologies, but different for the 10-node ring topologies (see Table 7).
The running time of the Approximation of Method 2 is much shorter than that of Method 2 for 10-node ring topologies as it is shown in Table 7.However, for 10-km spacing, the approxi-  -30-20-10-30 km) mation of Method 2 could not find the optimal solution of 277.8 K$, but it found the solution of 5 × A1 + 2 × A2 + 6 × D1 + 2 × D2 with the total cost of 286.2 K$.By increasing the branchand-bound search nodes, more accurate results can be found.However, the running time will then be closer to that of Method 2.

Conclusion
In this paper, two cost optimal equipment placement methods for WDM ring networks were presented.Unlike the previous methods that tried to minimize the total number of amplifiers in the ring (considering only a single type of amplifier) and neglected the fiber dispersion, these methods deal with different types of amplifiers and DCMs with predefined fixed characteristics and costs, and their objective function is to minimize the total cost of the system.Method 1 contains both linear and nonlinear constraints and is solved by a MINLP solver where Method 2 linearly approximates the nonlinear constraints, so they can be solved by a MILP solver.The methods were tested for various ring topologies and it was revealed that Method 2 has a shorter run time compared to Method 1.However, both method solutions are close to each other.The results confirm that Method 2 or an approximation of Method 2 can be used to find solutions for large ring topologies with practically acceptable results.

Fig. 4 .
Fig.4.Amplifier and DCM placement solution for a six-node ring with 10 km node spacing using both Method 1 and Method 2

Table 1 .
Device Parameters

Table 2 .
Network Parameters Node at the beginning of link i NODE SCR i ∈ NODES NODE DEST i Node at the end of link i, NODE DEST i ∈ NODES i Set of wavelengths on link i λ k ∈ W i , k = 1, ...,WADD j Set of wavelengths added at node j DROP jSet of wavelengths dropped at node j DROP j = ADD j in this ring topology NONDROP j Set of wavelengths passing through node j

Table 3 .
VariablesTransmitted power at the add channel at wavelength λ k at the dBm add port of node j, λ k ∈ ADD j m i, jBinary variable indicating number of DCMs of type j on link i D i,kTotal dispersion value of the channel at wavelength λ k at

Table 4 .
Set of available amplifiersAmplifier Type j G amp, j (dB) C amp, j (1000$) P min amp, j (dB) P

Table 6 .
Running time, total cost, and solution configuration of Method 1 and Method 2 for different ring topologies

Table 6
also shows that Method 2 is always faster than Method 1.Moreover, for the 6-node ring with 10 km node spacing, Method 1 was unable to find a solution due to the MINLP limitations on stack size and maximum task time.

Table 7 .
Running time and total cost of Method 2 and the approximation of Method 2 for different ring topologies