Cost optimal allocation of amplifiers and DCMs in WDM ring networks

Amir Minagar; Malin Premaratne; An V. Tran

doi:10.1364/OE.14.010278

1. 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 [2–6]. There have been some amplifier placement methods to minimize the number of amplifiers in star and switched networks [7–12]. 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.

2. Problem formulation

Fig. 1. A typical unidirectional WDM ring network

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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].

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]

G = 1 + \frac{P_{sat}}{P_{in}} \ln (\frac{G_{0}}{G})

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 $P_{amp, j}^{max}$ and minimum input power $P_{amp, j}^{min}$ 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.

Fig. 2. Amplifier gain G (dB) versus input power P _in (dBm) for P _sat = 20 dBm, G ₀ = 30 dB

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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} (2 N_{sp} h v B_{0} 10^{3})

B = 10 \log_{10} (2 N_{sp} h v B_{1} 10^{3})

where N _sp = 2 is the spontaneous emission factor, h = 6.63×10^-34 Js is the Planck’s constant, v= 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.,

min imize \sum_{i = 1}^{L} (\sum_{j = 1}^{N_{amp}} n_{i, j} C_{amp, j} + \sum_{j = 1}^{N_{DCM}} m_{i, j} C_{DCM, j})

subject to

\sum_{j = 1}^{N_{amp}} n_{i, j} \leq 1, n_{i, j} \geq 0, i = 1, \dots, L

and

\sum_{j = 1}^{N_{DCM}} m_{i, j} \leq 1, m_{i, j} \geq 0, i = 1, \dots, L

Table 1. Device Parameters

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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 (G_i ) falls between G _amp,j-1 and G _amp,j (G _amp,j-1 < G_i ≤ G _amp,j). The exact value of G_i is then achieved by tuning the amplifier or by use of attenuators. This is formulated as

\sum_{j = 1}^{N_{amp}} n_{i, j} G_{amp, j - 1} < G_{i} \leq \sum_{j = 1}^{N_{amp}} n_{i, j} G_{amp, j} i = 1, \dots, L

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 $P_{SCR}^{ASE}$ _i -β1_{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, β1_{NODE_SCR_i} is the insertion loss for through channels at the source node of the link, and -A+B is to convert from 0.1 nm to 20 nm bandwidth. So the total power at the input to the amplifier in dBm is

P_{i}^{tot} = 10 \log_{10} (\sum_{k = 1}^{W} 10^{\frac{P_{i, k}}{10}} + 10^{\frac{p_{{SCR}_{i}}^{ASE} - β 1_{NODE_{SCR}_{i}} - A + B}{10}}) - α L_{i}

Table 2. Network Parameters

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The total input power to amplifiers ( $P_{i}^{tot}$ ) should be within amplifiers input range. So we impose the inequality

P_{amp, j}^{min} \leq P_{i}^{tot} \leq P_{amp, j}^{max} \forall n_{i, j} = 1 j = 1, \dots, N_{amp}

Table 3. Variables

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We define Λ _i as the attenuation of DCM on link i, i.e.,

Λ_{i} = \sum_{j = 1}^{N_{DCM}} m_{i, j} γ_{DCM, j}

and Ω_i as the dispersion compensation value of DCM on link i, i.e.,

Ω_{i} = \sum_{j = 1}^{N_{DCM}} m_{i, j} D_{DCM, j}

The total input power to DCMs ( $P_{i}^{tot}$ + G_i ) should be within DCMs input range. So we impose the inequality

P_{DCM, j}^{min} \leq P_{i}^{tot} + G_{i} \leq P_{DCM, j}^{max} \forall m_{i, j} = 1 j = 1, \dots, N_{DCM}

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_i^tot + αL_i) and the total power after the amplifiers ( $P_{i}^{tot}$ + G_i) should not be greater than P_nonlinear, i.e.,

P_{i}^{tot} + α L_{i} \leq P_{nonlinear}

P_{i}^{tot} + G_{i} \leq P_{nonlinear}

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.,

G_{i, k} \leq D_{max}

D_{i, k} + δ L_{i} \geq D_{min}

The spectral density function of ASE noise S(v) for an amplifier with gain value G in watts is calculated as [18]

S (v) = (G - 1) n_{sp} h v

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.,

P_{i}^{ASE} = 10 \log_{10} [\frac{10^{\frac{A}{10}} (10^{\frac{G_{i}}{10}} - 1)}{10^{\frac{Λ_{i}}{10}}} + 10^{\frac{p_{{SCR}_{i}}^{ASE} + G_{i} - Λ_{i} - β 1_{NODE_{SCR}_{i}} - α L_{i}}{10}}]

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

P_{{DEST}_{i}, k} = P_{i, k} + G_{i} - Λ_{i} - α L_{i} - β 1_{NODE_{DEST}_{i} \forall λ_{k} \in {NONDROP}_{NODE_{DEST}_{i}}}

and the dispersion of wavelength λ_k at the beginning of the following link (D _{DEST
_i
,k}) is calculated as

P_{{DEST}_{i}, k} = D_{i, k} + δ L_{i} + Ω_{i} \forall λ_{k} \in {NONDROP}_{NODE_{DEST}_{i}}

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.,

P_{i, k} + G_{i} - Λ_{i} - α L_{i} - P_{i}^{ASE} \geq Desired_OSNR \forall λ_{k} \in {DROP}_{NODE_{DEST}_{i}}

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.,

{P_{minr} \leq P}_{i, k} + G_{i} - Λ_{i} - α L_{i} - β 2_{NODE_{DEST}_{i}} \leq P_{minr} + DR \forall λ_{k} \in {DROP}_{NODE_{DEST}_{i}}

and the dispersion of dropped wavelengths should be within the minimum and maximum dispersion limits of the receiver, i.e.,

{D_{minr} \leq D}_{i, k} + δ L_{i} + Ω_{i} \leq D_{max r} \forall λ_{k} \in {DROP}_{NODE_{DEST}_{i}}

If the channel at wavelengthλ_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

P_{{DEST}_{i}, k} = P_{NODE_{DEST}_{i,} k}^{xmit} - β 3_{NODE_{DEST}_{i}} \forall λ_{k} \in {ADD}_{NODE_{DEST}_{i}}

The transmitted power is limited to a minimum and a maximum value, i.e.,

P_{min t} = P_{NODE_{DEST}_{i,} k}^{xmit} \leq P_{max t} \forall λ_{k} \in {ADD}_{NODE_{DEST}_{i}}

However, the transmitted dispersion of added channels is set to 0, i.e.,

P_{{DEST}_{i}, k} = 0 \forall λ_{k} \in {ADD}_{NODE_{DEST}_{i}}

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.,

(P_{i, k} + G_{i} - Λ_{i} - α L_{i} + IX 1_{NODE_{DEST}_{i}}) - (P_{NODE_{DEST}_{i}, k}^{xmit} - β 3_{NODE_{DEST}_{i}}) \leq - 25

(P_{NODE_{DEST}_{i}, k}^{xmit} - IX 2_{NODE_{DEST}_{i}}) - (P_{i, k} + G_{i} - Λ_{i} - α L_{i} - β 2_{NODE_{DEST}_{i}}) \leq - 25

Fig. 3. Insertion losses and leakage paths at a node in the ring

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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.,

(\sum_{i = 1}^{L} (α L_{i} + Λ_{i}) + \sum_{j = 1}^{N} β 1_{j}) - \sum_{i = 1}^{L} G_{i} \geq MARGIN

3. 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 branch-and-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} (\sum_{k = 1}^{W + 1} 10^{\frac{x_{k}}{10}})

where y = $P_{i}^{tot}$ + αL_i , x_k = P_i,k (k = 1,…,W), and x _W+1 = $P_{SCR i}^{ASE}$ - β1_{NODE_SCR_i} -A+B. We use Least-Squares Fit (LSF) to find a best fit approximation of (30) by the following linear function

y = \sum_{k = 1}^{W + 1} h_{k} x_{k} + h_{W + 2}

and our mission is to find h = [h ₁ h ₂ … h _W+1 h _W+2]^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.,

x^{j} = {[x_{1}^{j} x_{2}^{j} \dots x_{W + 1}^{j} + 1]}^{T} \forall j = 1, \dots, M

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} (\sum_{k = 1}^{W + 1} 10^{\frac{x_{k}^{j}}{10}})

To linearly approximate (30) using LSF method the following function C needs to be minimized with respect to h, i.e.,

C = \sum_{j = 1}^{M} {∥ h^{T} x^{j} - y^{j} ∥}^{2}

where h^T is the transpose of vector h. We can expand (34) as

C = h^{T} Dh - 2 Eh + \sum_{j = 1}^{M} {(y^{j})}^{2}

where D = $\sum_{j = 1}^{M}$ [x ^j(x ^j )^T], and E = $\sum_{j = 1}^{M}$ (y^j (x ^j) ^T ). It can be easily seen that C is minimized when h = D ^-1 E ^T. Therefore,

h = {[\sum_{j = 1}^{M} (x^{j} {(x^{j})}^{T})]}^{- 1} [\sum_{j = 1}^{M} (y^{j} x^{j})]

When h is calculated, the nonlinear constraint (8) can be approximated by the following linear constraint

P_{i}^{tot} = \sum_{k = 1}^{W} (h_{k} P_{i, k}) + h_{W + 1} (P_{{SRC}_{i}}^{ASE} - β 1_{NODE_{SRC}_{i}} - A + B) + h_{W + 2} - α L_{i}

For a 6-node ring where (W = 15), h was calculated in MATLAB using M = 10000 data samples. Due to device limitations as shown in Table 1, P _i,k and P ^ASE _{SCR_i} samples are taken randomly from -30 to -5 and from -50 to -20 respectively assuming a uniform distribution. For this instance, (37) is written as

P_{i}^{tot} = \sum_{k = 1}^{15} 0.048 P_{i, k} + 0.023 (P_{{SCR}_{i}}^{ASE} - β 1_{NODE_{SCR}_{i}} - A + B) + 12.025 - α L_{i}

The average error between the approximate and the actual values is 0.931 dB.

In order to linearly approximate (18), we assume that the amplifier gains are much greater than 1 and ( $10^{\frac{G_{i}}{10}} - 1$ ) ≈ $10^{\frac{G_{i}}{10}}$ . So, (18) can be approximated as

P_{i}^{ASE} = 10 \log_{10} [\frac{10^{\frac{A}{10}} 10^{\frac{G_{i}}{10}}}{10^{\frac{Λ_{i}}{10}}} + 10^{\frac{P_{{SCR}_{i}}^{ASE} + G_{i} - Λ_{i} - β 1_{NODE_{SRC}_{i}} - α L_{i}}{10}}]

Further simplifying (39) gives

P_{i}^{ASE} = G_{i} - Λ_{i} + {10 \log}_{10} [10^{\frac{A}{10}} + 10^{\frac{P_{{SRC}_{i}}^{ASE} - β 1_{NODE_{SRC}_{i}} - α L_{i}}{10}}]

(40) can now be rewritten as

y = 10 \log_{10} (10^{\frac{x_{1}}{10}} + F)

where y = $P_{i}^{ASE}$ - G_i + Λ _i , x ₁ = $P_{SRC i}^{ASE}$ - β1_{NODE_SRC_i}, - αL_i , and $F = 10^{\frac{A}{10}}$ . Using a similar method as for (30), we can approximate (41) as y = h ₁ x ₁ + h ₂ where h ₁ and h ₂ are derived from (36). Finally, the linear approximation of (18) is

P_{i}^{ASE} = G_{i} - Λ_{i} + h_{1} (P_{{SRC}_{i}}^{ASE} - β 1_{NODE_{SRC}_{i}} - α L_{i}) + h_{2}

Implementing this method in MATLAB with M = 10000 data samples, $P_{SRC i}^{ASE}$ samples taken randomly from -50 to -20, and 30-km node spacing, the Equation (42) is written as

P_{i}^{ASE} = G_{i} - Λ_{i} + 0.541 (P_{{SRC}_{i}}^{ASE} - β 1_{NODE_{SRC}_{i}} - α L_{i}) + - 19.156

The average error between the approximate and the actual values is 1.456 dB.

Having replaced nonlinear constraints (8) and (18) by linear approximations (37) and (42), the second method is programmed in AMPL modeling language and solved by the MINTO solver on NEOS server [23]. MINTO solves mixed-integer linear programs by a branch-and-bound algorithm with linear programming relaxations.

4. Results and discussion

The solution from the MINLP solver (Method 1) for a 6-node ring with 10 km node spacing, 2 types of available amplifiers shown in Table 4, and 5 types of available DCMs shown in Table 5, is demonstrated in Fig. 4. It shows that the minimum deployment cost of the system is 148.5 K$ corresponding to 1 amplifier of Type 1 (A1), 2 amplifiers of Type 2 (A2), and 5 DCMs of Type 1 (D1). The MINLP solver provides the transmitted power of all add channels ( $P_{j, k}^{xmit}$ ) too.

Table 4. Set of available amplifiers

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Table 5. Set of available DCMs

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

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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 $P_{i}^{tot}$ in (8) and (37), and $P_{i}^{ASE}$ in (18) and (42). However, using tighter margins in Method 2 can insure the validity and applicability of the solutions.

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

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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.

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 approximation 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 branch-and-bound search nodes, more accurate results can be found. However, the running time will then be closer to that of Method 2.

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

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5. 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.

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Symbol	Description	Value/Unit
β1_j	Insertion loss for through channels at node j	10 dB
β2 _j	Insertion loss for drop channels at node j	5dB
β3_j	Insertion loss for add channels at node j	5dB
IX1_j	Leakage at the add/drop ports from the input wavelength to the output wavelength at node j	-40 dB
IX2_j	Leakage at the add/drop ports from the add wavelength to the drop wavelength at node j	-100 dB
DR	Dynamic range of the receiver for reliable operation	25 dB
P _maxt	maximum output power of a transmitter at each wavelength	0 dBm
P _mint	minimum output power of a transmitter at each wavelength	-25 dBm
P _minr	Minimum detectable signal power at a receiver	-30 dBm
D _maxr	Maximum dispersion acceptable at the receiver	200 ps/nm
D _minr	Minimum dispersion acceptable at the receiver	-200 ps/nm
N _amp	Number of available amplifier types
C _amp,j	Cost of amplifier type j	$
G _amp,j	Gain of amplifier type j	dB
$P_{amp, j}^{max}$	Maximum input power to operate in the unsaturated mode of amplifier type j	dBm
$P_{amp, j}^{min}$	Minimum input power to operate in the unsaturated mode of amplifier type j	dBm
N _DCM	Number of available DCM types
γ _DCM,j	Attenuation of DCM type j	dBm
C _DCM,j	Cost of of DCM type j	$
D_DCM,j	Dispersion compensation value of DCM type j	ps/nm
$P_{DCM, j}^{max}$	Maximum allowed input power of DCM type j	dBm
$P_{DCM, j}^{min}$	Minimum allowed input power of DCM type j	dBm

Symbol	Description	Value/Unit
L_i	Length of link i	km
L	Number of links in the ring, L = N
N	Number of nodes in the ring
LINKS	Set of links in the ring
NODES	Set of nodes in the ring
SCR_i	Source of link i, also a link, SCR_i ∈ LINKS
DEST_i	Destination of link i, also a link, SCR_i ∈ LINKS
NODE_SCR_i	Node at the beginning of link i NODE_SCR_i ∈ NODES
NODE DESTi	Node at the end of link i, NODE DESTi ? NODES
W	Total number of wavelengths in the ring W = N(N-1)/2
W_i	Set of wavelengths on link i λ_k ∈ W_i , k = 1,…,W
ADD_j	Set of wavelengths added at node j
DROP_j	Set of wavelengths dropped at node j DROP _j = ADD _j in this ring topology
NONDROP_j	Set of wavelengths passing through node j
α	fiber attenuation	0.2 dB/km
A	Constant to calculate ASE noise power using 0.1 nm bandwidth	-51.9 dBm
B	Constant to calculate ASE noise power using 20 nm bandwidth	-28.9 dBm
Desired_OSNR	Minimum OSNR allowed for the received signal	20 dB
MARGIN	Margin to prevent lasing in the ring	10 dB
P _nonlinear	Maximum total power allowed in fibers to prevent nonlinear effects	25 dBm
δ	fiber dispersion	-18ps/(nm.km)
D _max	Maximum total dispersion allowed in fibers	1500ps/nm
D _min	Minimum total dispersion allowed in fibers	-1500ps/nm

Symbol	Description	Unit
n_i,j	Binary variable indicating number of amplifiers of type j on link i
G_i	Gain of amplifier on link i	dB
$P_{i}^{tot}$	Total power at the input to the amplifier on link i	dBm
$P_{i}^{ASE}$	Total ASE noise power at the end of link i calculated using 0.1 nm bandwidth	dBm
P _i,k	Power of the channel at wavelength λ_k at the beginning of link i	dBm
$P_{j, k}^{xmit}$	Transmitted power at the add channel at wavelength λ_k at the add port of node j, λ_k ∈ ADD3	dBm
m _{i, j}	Binary variable indicating number of DCMs of type j on link i
D _i,k	Total dispersion value of the channel at wavelength λ_k at the beginning of link i	ps/nm
Λ_i	Attenuation of DCM on link i	dB
Ω_i	Dispersion compensation value of DCM on link i	ps/nm

DCM type j	D _DCM,j (ps/nm)	γ _DCM,j (dB)	C _{DCM, j} (1000$)	$P_{DCM, j}^{min}$ (dB)	P _DCM,j (dB)
D1	200	3.1	11.7	-25	25
D2	310	4.0	13.8	-25	25
D3	440	4.9	15.2	-25	25
D4	600	7.0	19.2	-25	25
D5	760	7.8	20.2	-25	25

Ring Topology	Method 1	Method 2
6-node 10 km node spacing	Running time=13.1 minutes Cost=148.5 K$ Amplifiers: 1×A1+2×A2 DCMs: 5×D1	Running time=0.4 minutes Cost=148.5 K$ Amplifiers: 1×A1+2×A2 DCMs: 5×D1
6-node 30 km node spacing	Running time=2.1 minutes Cost=250.8 K$ Amplifiers: 6×A1 DCMs: 3×D3+3×D4	Running time=0.1 minutes Cost=242.4 K$ Amplifiers: 3×A1+2×A2 DCMs: 3×D3+3×D4
6-node (link length: 1-2-3-4-5-6 =10-20-30-20-10-30 km)	Running time=2.3 minutes Cost=183 K$ Amplifiers: 3×A1+1×A2 DCMs: 1×D1+3×D3+1×D4	Running time=0.1 minutes Cost=174.6 K$ Amplifiers: 3×A2 DCMs: 1×D1+3×D3+1×D4
10-node 10 km node spacing	unable to solve	Running time=43.5 minutes Cost=277.8 K$ Amplifiers: 2×A1+4×A2 DCMs: 6×D1+2×D2
10-node 30 km node spacing	Running time=91.1 minutes Cost=462.5 K$ Amplifiers: 5×A1+5×A2 DCMs: 4×D3+6×D4	Running time=34.3 minutes Cost=438.2 K$ Amplifiers: 8×A1+2×A2 DCMs: 4×D3+6×D4
10-node (link length: 1-2-3-4-5-6-7-8-9-10 =10-20-30-10-20-20 -30-20-10-30 km)	Running time=75.2 minutes Cost=371 K$ Amplifiers: 8×A1+1×A2 DCMs: 3×D1+7×D3	Running time=30.6 minutes Cost=342.4 K$ Amplifiers: 6×A1+2×A2 DCMs: 3×D2+2×D3+3×D4

Cost optimal allocation of amplifiers and DCMs in WDM ring networks

Abstract

1. Introduction

2. Problem formulation

3. Solution method

4. Results and discussion

5. Conclusion

References and links

Cited By

Figures (4)

Tables (7)

Equations (45)

Optics Express

Ring Topology	Method 2	Approximation of Method 2
10-node 10 km node spacing	Running time=43.5 minutes Cost=227.8 K$	Running time=5.2 minutes Cost=286.2 K$
10-node 30 km node spacing	Running time=34.3 minutes Cost=438.2 K$	Running time=4.9 minutes Cost=438.2 K$
10-node (link length: 1-2-3-4-5-6-7-8-9-10 = 10-20-30-10-20-20 -30-20-10-30 km)	Running time=30.6 minutes Cost=342.4 K$	Running time=7.3 minutes Cost=342.4 K$