A metaheuristic approach for optical network optimization problems

Applied Soft Computing 13 (2013) 981–997

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

A metaheuristic approach for optical network optimization problems Urmila Bhanja a,∗ , Sudipta Mahapatra b,1 a b

Department of Electronics and Communication Engineering, Indira Gandhi Institute of Technology, Sarang, Odisha 759146, India Department of E&ECE, Indian Institute of Technology, Kharagpur 721302, West Midnapur, West Bengal, India

a r t i c l e

i n f o

Article history: Received 6 September 2011 Received in revised form 6 May 2012 Accepted 12 September 2012 Available online 5 October 2012 Keywords: QoS routing DRWA problem Linear impairments Non-linear impairments Evolutionary programming algorithm Convergence rate Fitness function

a b s t r a c t Two of the most complex optimization problems encountered in the design of third generation optical networks are the dynamic routing and wavelength assignment (DRWA) problem under the assumptions of ideal and non-ideal physical layers. Both these problems are NP-complete in nature. These are challenging due to the presence of multiple local optima in the search space. Even heuristics-based algorithms fail to solve these problems efficiently as the search space is non-convex. This paper reports the performance of a metaheuristic, that is, an evolutionary programming algorithm in solving different optical network optimization problems. The primary motivation behind adopting this approach is to reduce the algorithm execution time. It is demonstrated that the same basic approach can be used to solve different optimization problems by designing problem-specific fitness functions. Also, it is shown how the algorithm performance can be improved by integrating suitable soft constraints with the original constraints. Exhaustive simulation studies are carried out assuming the presence of different levels of linear impairments such as switch and demultiplexer crosstalk and non-linear impairments like four wave mixing to illustrate the superiority of the proposed algorithms. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The major issues in dense optical wavelength division multiplexing (DWDM) networks include quality of service (QoS) routing in multihop lightwave networks, traffic grooming, optimal routing and wavelength assignment in wavelength division multiplexing (WDM) network, survivability in WDM mesh networks, physical layer impairment aware (PLI aware) routing and wavelength assignment, and many others. However, optical telecommunication network problems are characterized by multiple local optima, thereby leading to a number of challenging design and optimization problems. Various approaches have been employed to solve these problems, including classical approaches like mathematical programming techniques, and heuristics or metaheuristics-based approaches. Traditional techniques are able to give accurate results for simple or small sized problems; but, to solve complex and large sized problems, these techniques incur too much computational time. One of the alternative approaches used for complex optimization problems with multiple objectives is the use of evolutionary programming algorithm. Evolutionary programming

∗ Corresponding author. Tel.: +91 9437142056/6764 221744. E-mail addresses: [email protected] (U. Bhanja), [email protected] (S. Mahapatra). 1 Tel.: +91 9434037863/3222 283561. 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.09.011

based techniques converge to an optimal or near optimal solution even if there is non-linear interaction among the variables [9]. The basic evolutionary programming algorithm (BEP) discussed in the literature is used only in continuous optimization domain [11]. Also, this algorithm is found to give the best solution for unconstrained problems. For a constrained optimization problem it is not guaranteed to give good solutions. For few of the constrained benchmark optimization problems, it takes a long time to find a feasible solution. It is observed that most of the optical network optimization problems do need to be solved in the discrete domain. The constraints and the associated variables in the optical domain are mostly integers and the fitness function is either in integer domain or real domain depending on the specific problem. Hence, the major objective of the work reported in this paper is to introduce certain novelty into the evolutionary programming approach so as to efficiently solve difficult optical network optimization problems found to be NP-complete. The major contribution of the work reported in the paper is in terms of demonstrating how the evolutionary programming approach can be used to solve difficult optimization problems encountered in the discrete domain of optical networks. In the proposed evolutionary programming algorithm, a chromosome or individual is represented in integer domain. Moreover, unlike the basic evolutionary programming approach (BEP), the proposed approach uses an initial population consisting of a single individual only, which is then mutated a fixed number of times to introduce diversity into the search space. Also, only mutation is used for both exploration

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and exploitation of the search space, unlike genetic algorithm (GA) that uses both crossover and mutation. This reduces the program complexity. It is shown that by properly tuning the fitness function the proposed algorithm can be used to solve difficult optimization problems, even those found to be NP-complete in nature. In the recent years, QoS applications have attracted enormous attention from the research community. In such applications, each of the users specifies a QoS requirement in terms of delay, bandwidth, jitter, and packet loss ratio or throughput. To test the efficacy of the proposed scheme for a constrained optimization problem, it is first used to discover parallel disjoint paths for routing of multiple concurrent requests in a circuit switched telecommunication network [4]. In this problem, bandwidth or link capacity is taken as a hard constraint and hop count is taken as a soft constraint. The solution has been illustrated on a 20 node network taken from [2] and on a 50 node Bernoulli random graph, in which a bidirectional edge exists between any two nodes with a probability of 0.5 [4]. The developed algorithm is shown to outperform both the genetic algorithm (GA) and the simulated annealing (SA) in terms of obtaining a better solution and a lower algorithm execution time. This algorithm has a low memory requirement and is computationally less expensive than the other existing approaches like the GA. Moreover, to reduce the execution time of the algorithm, soft constraints are introduced in the form of empirically determined threshold times. The second problem considered in this paper is one of the most complex optimization problems encountered in the field of DWDM optical networks, that is, the dynamic routing and wavelength assignment (DRWA) problem, which involves the establishment of wavelength continuous paths in a network of wavelength routing nodes (WRNs) for a set of dynamic requests. The fact that the problem has to be solved under multiple constraints makes it more challenging. It is shown that the developed evolutionary programming based routing algorithm can be integrated with suitable wavelength assignment approaches to address the DRWA problem for an ideal physical layer. In this work, the wavelength continuous route is discovered online unlike other bio-inspired algorithms available in the literature for solving the DRWA problem [8,15]. The proposed scheme uses a novel fitness function that is designed with a view to overcome the drawbacks of the existing techniques, to minimize the blocking of requests, and to achieve proper load balancing while solving the DRWA problem [5]. The third and fourth problems addressed in this paper are the DRWA problem for a non-ideal physical layer, respectively, with linear and non-linear impairments, commonly referred as PLI aware DRWA problems [3,17]. It is shown that the proposed evolutionary programming algorithm can be successfully applied to solve the problems by including the noise variances, corresponding to the different linear and non-linear impairments, in the fitness functions. Only the dominant linear impairments, such as amplified spontaneous emission noise (ASE noise), switch and demultiplexer crosstalk are considered for the third problem. Similarly, four wave mixing related noise (FWM noise), which is a non-linear impairment affecting a lightpath even at low to medium data rates and low transmission power, is considered for the fourth problem. However, linear impairments like polarized mode dispersion (PMD) and nonlinear impairments like simulated Brillouin scattering (SBS) and simulated Raman scattering (SRS), which occur mainly at high data rates such as 10 Gbps to 40 Gbps, or at a high transmission power, are ignored [17,19]. For each of the problems considered, the superiority of the proposed evolutionary programming algorithm is established by estimating suitable performance measures through extensive simulation studies. The proposed routing approach is shown to outperform two other metaheuristic approaches such as GA and SA. It is also shown to outperform GA in solving the DRWA problem

for an ideal physical layer. Similarly, for impairment aware DRWA problem also the proposed schemes perform better than the existing heuristic and metaheuristic approaches. The rest of the paper is organized as follows: Section 2 briefly introduces the problem model, network model and describes the four problems considered in this paper, Section 3 presents the bit error rate evaluation model used in the third and fourth problems and Section 4 describes the proposed evolutionary programming algorithm. In Section 5, the algorithm implementation is discussed and Section 6 presents the simulation results along with the algorithm complexity analysis. Section 7 finally concludes the paper. 2. Problem model 2.1. Problem definition Each of the four problems is described briefly in this section. The first problem is a QoS routing problem that assumes only the bandwidth and initial hop bound as constraints to present a feasible solution to the shortest path problem [4]. It is assumed that a set of concurrent requests may arrive at a centralized controller at any time to establish dedicated paths through a telecommunication network under bandwidth and hop count constraints. The purpose of the problem is to establish constrained disjoint paths for the set of concurrent requests. The bandwidth constraint is taken as a hard constraint whereas the initial hop count bound is used as a soft constraint. The proposed algorithm also generates parallel suboptimal paths during the process of generating the best path. One of these suboptimal paths can be used as a backup path if it is link disjoint with all the primary paths for the concurrent requests. The second problem reported in this paper is a call admission problem in an optical network where lightpath requests are initiated dynamically, that is, the dynamic routing and wavelength assignment (DRWA) problem assuming an ideal physical layer. The call arrival process is assumed to follow a Poisson’s distribution. A lightpath is a path in the network that satisfies the wavelength continuity constraint, or uses the same wavelength in all the links constituting the path. Each lightpath request is assumed to be specified by three attributes: S, D, and Th , which, respectively, represent the source node, the destination node, and the holding time for the request. The holding time defines the time period during which a lightpath and the associated resources in that path remain engaged. It is possible to formulate this problem as a constrained optimization problem by combining wavelength assignment constraints with least-cost routing [5]. The objective function is designed so as to minimize the blocking of connection requests, reduce the set up time of the lightpath and to achieve network load balancing. The third and fourth problems reported in this paper are DRWA problems while assuming a non-ideal physical layer, respectively, with linear and non-linear impairments. These problems are addressed by integrating linear and non-linear physical layer constraints with the routing and wavelength constraints used in the DRWA algorithm described for the second problem. As the assumed data rate is 2.5 Gbps, the important sources of quality of transmission (QoT) degradation considered in this work are linear impairments like ASE noise, node crosstalk and non-linear impairments like FWM. 2.2. The network model The N node network considered in this work for all the four problems can be modeled as a graph G(V, E), in which V is the set of nodes representing routers or switches, and E is the set of edges representing connectivity between the nodes. The link existing between a pair of nodes is assumed to be bidirectional in nature, that is, the

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Fig. 1. Architecture of a wavelength routing node (WRN) [18].

Fig. 2. The model of a transmission lightpath [18].

existence of a link e = (i,j) from node i to j implies the existence of another link e’ = (j,i) for any pair of nodes (i,j)∈E. For the DRWA problem with ideal as well as non-ideal physical layers, V is the set of nodes representing routers or WRNs, and E is the set of fiber links representing physical connectivity between the nodes. Each link is assumed to be bidirectional with fixed number of wavelengths per fiber. For the physical impairment aware DRWA problem, each wavelength routing node (WRN) consists of a cross connect switch (XCS), transmitter and receiver arrays, optical taps and erbium doped fiber amplifiers (EDFA) as shown in Fig. 1 [18]. The wavelength routing switches (WRSs) in the XCS are assumed to employ non-blocking active splitter/combiner architecture. The XCSs transfer each wavelength in an input fiber into the same wavelength in one of the output fibers. A transmission lightpath follows the model illustrated in Fig. 2 [18]. No inline amplifier is assumed to be present throughout the lightpath. The span length is assumed to vary from 60 km to 300 km. A tap is present at the input and output of each XCS to monitor the signal condition. The EDFA at the input side compensates for the fiber loss and the tap loss and the EDFA at the output side compensates for the switch loss. In Fig. 2, WRN(1) represents the source node, WRN(m) represents the destination node, and WRN(k) represents the kth intermediate node. Arrays of transmitters and receivers are present in each of the nodes for locally adding or dropping the traffic. In the adopted network model, each XCS consists of an array of demultiplexers followed by a set of WRSs and a set of multiplexers as described in Fig. 2. All the signals that are demultiplexed and have identical wavelengths are directed to the

corresponding WRS tuned to the same wavelength and the switch redirects the signal to the desired output port; the multiplexers then combine signals with different wavelengths and redirect them to the output fibers. The number of WRSs in an XCS depends on the number of input wavelengths, and the number of input and output ports of a WRS depends on the number of input and output fibers. The demultiplexers present in each XCS may also introduce in-band crosstalk [22] because of the leakage by non-ideal filters present in the demultiplexers as depicted in Fig. 3. In this figure LP1

Fig. 3. The model of a non-ideal demultiplexer in XCSs [22].

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on i goes from input 1 to output 1. LP2 on j and LP3 on i enter input 2 and there is signal leakage between LP2 and LP3 because of the non-ideal filter isolation. In the proposed work, signal leakage is assumed to be present only between adjacent wavelengths. Hence, LP2 on j carries some interference power from LP3 on i provided i and j are adjacent to each other. When LP2 exits at output 1, the interference power on i generates crosstalk with LP1 that has an identical wavelength. For the FWM aware DRWA problem, the network model is same as that of the previously described model only with a difference that the WRN’s are connected through non-zero dispersion shifted optical fibers (NZDSF). In this work, the effect of signal leaks in the optical cross connect switches and the effect of non-ideal filtering at the demultiplexers are neglected. 2.3. The problem formulation and the routing model The routing models used for all the four problems are nearly identical. For the QoS constrained routing problem F is assumed to be the set of flows existing at any time and f is any unicast flow or request belonging to F. The variable If i,j is set to 1 if link (i,j) is used by flow f; otherwise, If i,j is set to 0. A Path from the source S to destination D for a flow f is represented as Path(f) and is the collection of all the links belonging to the flow from S to D. Any link e  E has a bandwidth, bandwidth(e): E → R+ , associated with it, R+ being the set of positive real numbers. The bandwidth from any source to any destination, for a flow f, is denoted by bandwidth(f), and is defined as: bandwidth(f ) = min{bandwidth(e)|e ∈ Path(f ),

f ∈ F.

Each link (i,j) has an associated cost Ci,j , which may represent any measure such as the link length, capacity of the link, etc. This is a static value and is assumed to be specified. The link costs in the network are specified by the cost matrix C = [Ci,j ] and Ci,j denotes the cost of transmitting a packet from source S to destination D, where the sum is taken over all the links that belong to Path(f). It is possible to formulate the least-cost QoS constrained routing problem as a combinatorial optimization problem minimizing the objective function as follows: Minimize





f ∈F

(i, j) ∈ E

f

Ci,j Ii,j

(1.1)

i= / j Subject to the following constraints:





(i,j) ∈ E

f

Ij,i = −1, if i = D, f ∈ F



(j,i) ∈ E f

Ii,j −

(i,j) ∈ E  f (i,j) ∈ E

f

Ij,i = 1, if i = S, f ∈ F

(1.2)

(j,i) ∈ E  f

Ii,j −





f

Ii,j −

Ij,i = 0, if i = / S, i = / D, f ∈ F

(j,i) ∈ E f Ii,j

≤ 1, if i = / D, f ∈ F

i= / j

f1 f2 = / Ii,j ∀(i, j) ∈ E, ∀f1 ∈ F, ∀f2 ∈ F, f1 = / f2 Ii,j

 

(i, j) ∈ Path(f ) f Ii,j

Ii,j ≤ h0 ,

f

for t ≤ T

≤ (N − 1),

for t > T

.

(1.5)

(1.6)

(i, j) ∈ Path(f )

The objective function given by (1.1) minimizes the cost for each of the network flows. Equations (1.2) and (1.3), respectively, represent the flow conservation constraint and ensure that the path is without any loops [2]. The constraint given by (1.4) ensures that bandwidth of each link in the path or flow should satisfy the minimum bandwidth requirement B0 of a request [4]. Constraint (1.5) implies that the paths corresponding to any two flows are link disjoint. Finally, (1.6) represents the hop count constraint that is however a soft constraint. For the algorithm execution time t less than a threshold time T, a hop count bound, referred to as h0 , is specified for a path. After this time, however, the bound is relaxed up to a maximum value of (N − 1). Also, the threshold time is different for the initialization and mutation phases of the algorithm. In the initialization phase the threshold time is set to T1 , whereas while executing the mutation process the threshold time is set to T2 . The assumptions made in the constrained routing model in the DRWA problem with an ideal and non-ideal physical layer are identical as those described in the QoS constrained routing model. The only difference is that flow set F is replaced by the set of lightpaths LP, flow f is replaced by the lightpath lp and the variable If i , is replaced by Ilp ij . Ilp ij is equal to one when the link (i, j) is used by the lightpath lp, and zero otherwise. This is considered to be a positive variable if the lightpath leaves the node, and negative if the lightpath is incident on the node. For the DRWA problems, the routing models do not incorporate the link-disjoint constraint and the bandwidth constraint. Rather, for these problems another constraint, in the form of a wavelength continuity constraint, is incorporated in the routing model to ensure the use of a single wavelength along a path as explained in the following section. 2.4. The wavelength assignment model 2.4.1. The wavelength assignment for the DRWA problem For solving the DRWA problem with an ideal physical layer, different wavelength assignment approaches are integrated with the evolutionary programming routing algorithm to discover a wavelength continuous path through the network. In the proposed fitness function, a free wavelength factor, Wx , is updated after the wavelength assignment phase. In the proposed wavelength assignment model, the variable Iij lp is equal to one when the link (i, j) is used by the lightpath lp, and zero otherwise. The additional variables used are, Iijw lp , the lightpath wavelength indicator that shows whether the lightpath lp uses wavelength ‘W’ on link (i, j), Iijw lp(x, y) , the lightpath wavelength link indicator that is one when the lightpath uses wavelength ‘W’ on link (i, j) between the nodes x and y, and, l(x, y) which equals one if a physical link exists between the nodes x and y. The wavelength continuity constraints are as follows [21]:



W −1 lp

(i, j) ∈ E



f Ii,j

= 0, if i = D, f ∈ F

(1.3)

Iij =

lp

Iijw ∀(i, j)

(2.1)

lp

(2.2)

w=0

i= / j

Iijw

(i, j) ∈ E

 lp(x,y)

bandwidth(f ) > B0 , f ∈ F

lp(x,y)

≤ Iijw , ∀(i, j), ∀(x, y), ∀w

Iijw

(1.4)

i,j

≤ 1∀(x, y), w

(2.3)

U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997

  lp(x,y)

W −1

Iijw

x

w=0 W −1

  lp(x,y) Iijw

  lp(x,y) Iijw

w=0

x

Iijw

l(x,y) −

  lp(y,x) Iijw

  lp(y,x) Iijw

w=0

lp

l(y,x) = −Iij , y = i

(2.4)

x

w=0 W −1

l(x,y) −

lp

l(y,x) = Iij , y = j

x

w=0 W −1

x

w=0 W −1

  lp(y,x)

W −1

l(x,y) −

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2.5.2. DRWA problem with an ideal physical layer (i) The N routers or nodes do not have any wavelength conversion capability. (ii) There is a centralized controller to update the database of available wavelengths for each of the links. Each node in the network has the capability to access the controller whenever there is a need to ascertain the availability of wavelengths in any link.

l(y,x) = 0, y = / i, y = / j

x

Equation (2.1) implies that the wavelength used by a lightpath is unique; equation (2.2) ensures that the same wavelength is used over all the links constituting any lightpath. Equation (2.3) ensures that two lightpaths using the same link cannot be assigned identical wavelengths. Equation (2.4) expresses the conservation of wavelengths at the end nodes of the physical links traversed by a lightpath. Different wavelength assignment techniques such as Random, First fit, and Round robin are utilized to satisfy the above mentioned wavelength constraints [7,25]. 2.4.2. Wavelength assignment for the linear impairment aware DRWA problem A new wavelength ordering assignment technique is proposed in this work for reducing the crosstalk at the nodes. The proposed new wavelength ordering scheme is efficiently designed to allocate wavelengths for the incoming requests that help in reducing the in-band crosstalk in the switches and in the demultiplexers. The objective of wavelength spectrum separation between any two consecutive requests is to minimize the in-band crosstalk both at the switch and at the demultiplexer. If rj is the wavelength assigned to the jth request, the wavelength to be assigned to the (j + 1)th request rj + 1 is obtained as follows:

2.5.3. Linear and the non-linear impairment aware DRWA problem The additional assumptions made here are as follows: (i) The WRSs and demultiplexers present in the network model are non-ideal and assumed to have a high crosstalk ratio [18,22]. (ii) The data rate is 2.5 Gbps and hence, the effect of pulse broadening because of chromatic dispersion (CD) and PMD are negligible. (iii) There are no inline amplifiers in the span and no wavelength converters present in the switches. (iv) For the FWM aware DRWA problem, the partial FWM crosstalk powers, FWM crosstalk products are calculated offline for unit link length. (v) FWM crosstalk powers over the links are assumed to be independent of each other. (vi) The total FWM crosstalk power over a path is assumed to be the sum of the FWM crosstalk powers of all the links involved. (vii) There is equal power on each channel. (viii) The dispersion slope is assumed to be the same for all the wavelengths with equal channel spacing.

if rj = v rj+1 = b where b = ( + W ≥ 2m ,

m is an integer, and m ≥ 2,

W 2k

− 1)mod W, k = log2 (

W ). 4

W is the total number of wavelengths,

(3)

v being

an integer.1 ≤ v ≤ W. 2.4.3. Wavelength assignment for the FWM aware DRWA problem In this work also, after finding feasible paths a wavelength is assigned on each of the links by suitable assignment techniques that satisfy the wavelength constraints. In this work, an FWMaware priority based wavelength assignment technique that helps to reduce the FWM related noise is proposed to solve the FWM aware DRWA problem. As the FWM crosstalk power will be more at the center of the transmission window compared to that of at the edges [1], in the FWM aware priority based wavelength assignment technique a higher priority is given to wavelengths near the two edges of the transmission window compared to those at the center of the window.

(ix) For the linear impairment aware DRWA problem, links are assumed to be bidirectional and assumed to be a single span consisting of standard single mode fiber. (x) For the FWM aware DRWA problem, links are assumed to be bidirectional and assumed to be a single span consisting of NZDSF. (xi) All the EDFAs used for both the work have a flat gain spectrum as the wavelength spectrum used ranges from 1544 nm to 1557 nm [12] and therefore, the EDFA gain saturation and the wavelength dependence of EDFA gain are absent.

2.5. Assumptions and system description for the problems 2.5.1. QoS constrained routing Each call request is specified by five attributes: source node, destination node, start time, duration of request, and the bandwidth requirement. All the QoS requests have the same priority. All the requests have the same start time and the duration of all the requests are the same in the network. If the bandwidth constraint is not satisfied within certain threshold times in the initialization and mutation phases of the algorithm the call request is dropped.

3. Online bit error rate evaluation model for the PLI aware DRWA problems 3.1. Online signal power and noise power evaluation for the PLI aware DRWA problems This is an extra model used for the PLI aware DRWA problem in order to evaluate the bit error rate. In this work, in the context of evolutionary computation (EC) an individual or chromosome represents a possible solution as explained in Section 4. Any arbitrary solution is referred to as the xth individual. In the presence of both

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linear and non-linear impairments, the calculation of received signal power, switch crosstalk power, ASE noise power, demultiplexer crosstalk power and the FWM crosstalk power along a lightpath, during the call admission step, at the output of the kth intermediate node for the xth individual are computed recursively and are expressed as below [14,18,22].

In the above equation, c is the speed of light and 0 is the zero dispersion wavelength. The FWM efficiency depends on the channel frequency separation, fiber chromatic dispersion Dc , dispersion slope (dDc /d), and the fiber length L. In a multichannel system, a

2 psigx (k, i ) = psigx (k − 1, i )Lfx (k − 1, k)Gin (k)Ldm (k)Lsw (k)Lmx (k)Gout (k)Ltap

(4.1)

2 pxtx (k, i ) = pxtx (k − 1, i )Lfx (k − 1, k)Gin (k)Ldm (k)Lsw (k)Lmx (k)Gout (k)Ltap jk 

+

(4.2) Xsw pinx (j, k, i )Lsw (k)Lmx (k).Gout (k)Ltap

j=1 2 pasex (k, i ) = pasex (k − 1, i )Lfx (k − 1, k)Gin (k)Ldm (k)Lsw (k)Lmx (k)Gout (k)Ltap

+2sp [Gin (k) − 1]hi Bo Ldm (k)Lsw (k)Lmx (k).Gout (k)Ltap

(4.3)

+2sp [Gout (k) − 1]hi Bo Ltap. 2 pmtx (k, i ) = pmtx (k − 1, i )Lfx (k − 1, k)Gin (k)Ldm (k)Lsw (k)Lmx (k)Gout (k)Ltap Zk 

+

(4.4) Xadj px (z, k, i )Ldm (k)Lsw (k)Lmx (k)Gout (k)Ltap .

z=1

The FWM crosstalk power depends on the number of different signals present in a link and the length of the associated links that a lightpath traverses during the call admission phase. In a multichannel system, each lightpath traverses H hops or links until it reaches its destination node. The accumulated FWM crosstalk power at the destination node, PDN , is the sum of all the crosstalk components generated in the links traversed by the lightpath. FWM crosstalk power per link at the kth node for the xth chromosome due to these three co-propagating signals is expressed below [6]. 2 Pr,l,m,x (k) = ((Pfwm (fr , fl , fm )/e−˛L )e−˛L(k−1,k) Gin (k)Ltap

Lmx (k)Ldm (k)Lsw (k)Gout (k))/9.0

(4.5)

In the above equations, L(k − 1,k) denotes the length of the link between the (k − 1)th and the kth node. The FWM power generated in a link due to the presence of frequencies fr , fl and fm is expressed as: Pfwm (fr , fl , fm ) =

 2 2 . D  2 Pr Pl Pm e−˛L Leff 9 rlm

Leff

2␲n2 , = Aeff

(4.8)

where, n2 is the fiber non-linear refractive index, Aeff is the effective fiber core area and  is the wavelength. The FWM efficiency is given by:

ˇ=



1+

H   q=1

fr

fl

(4.11)

Pr,l,m (q).

fm

In the above equation, Pr,l,m (q) is the FWM power due to the three coexisting signal frequencies fr ,fl ,fm in the hop number q. Pr,l,m (q) is same as Pfwm (fr ,fl ,fm ) obtained from equation (4.6).The summation over q varying from 1 to H indicates that the crosstalk components in each of the hops in the lightpath are to be added up. After computing the signal power and noise power in the discovered paths the bit error rate is estimated for each of the paths as described in Section 3.2. 3.2. Online noise and crosstalk variance evaluation for the PLI aware DRWA problems For the linear impairment DRWA problem the combined noise in the presence of signal can be modeled as a Gaussian random process with mean zero and variance considering only the dominant beat noise components is as expressed in equation (4.12). 2 2 2 2 2 12 (x) = sig−ase (x) + sig−xtalk (x) + thermal + shot (x) + sig−demux (x)

(4.7)

The non-linear coefficient of the fiber is given by:

˛2 = 2 ˛ + ˇ2

PDN =

(4.6)

In the above equation, Pr , Pl , and Pm are, respectively, the input light powers at frequencies fr , fl , and fm , and  represents the FWM efficiency that depends on the phase mismatch factor ˇ, expressed later; Drlm is the degeneracy factor, which takes the value three for r = l and is equal to six otherwise;  and ˛ are the non-linear and the attenuation coefficients of the fiber, L is the fiber length, and Leff denotes the effective length given by: 1 − e−˛L = . ˛

lightpath goes through H hops or links until it reaches its destination node, and the accumulated FWM noise power at the destination node is the sum of all the crosstalk components along the lightpath [16]. The FWM noise power at the destination is expressed as:

4e−˛L sin2 (ˇL/2) (1 − e−˛L )

2



(4.9)

2 2 dDc 2␲0 (fr − fm )(fl − fm )(Dc + ( 0 )( )[(fr − f0 ) + (fl − f0 )]). c 2c d (4.10)

(4.12) The noise variances for the xth individual are expressed as below [18,22], 2 sig−xtalk (x) = 2εpol R2 bsig (x)sigprec(x) xtalkprec(x)

(4.13)

2 sig−ase (x) = 4R2 bsig (x)sigprec(x) aseprec(x) be /bo

(4.14)

2 (x) = 2qR (bsig (x)sigprec(x) + xtalkprec(x) + aseprec(x) shot

+ demux xtalkprec(x) )be

(4.15)

2 = th be thermal 2 sig−demux (x)

=

(4.16) εpol R2 bsig (x)sigprec(x) demux

xtalkprec(x) .

(4.17)

U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997

 2 sig-xtalk ,  2 sig-ase (x),  2 shot (x),  2 thermal ,  2 sig-demux (x) represent the signal-crosstalk beat noise variance, signal-ASE beat noise variance, shot noise variance, thermal noise variance, signal-demultiplexer crosstalk beat noise variance, respectively, for the xth individual. εpol is the polarization mismatch factor and is taken to be 1/2 [18], R␭ is the responsivity of the photodetector, bsig (x) is 2 for a bit ‘1’ and 0 for a bit ‘0’ for the xth individual. sigprec(x) , xtalkprec(x) , aseprec(x) , demux xtalkprec(x) is the received signal power, switch crosstalk power, ASE noise power and demultiplexer crosstalk power, respectively, at the destination for the xth individual and are computed from the equations (4.1), (4.2), (4.3), and (4.4), respectively. be and bo are the electrical and optical bandwidths respectively, and th is the spectral density of the thermal noise current in the optical receiver. Equation (5) below represents the variance corresponding to bit ‘0’ for the xth chromosome while considering linear impairments only: 2 02 (x) = thermal .

2 2 2 2 12 (x) = sig−ase (x) + ase−ase (x) + sig−fwm (x) + fwm

1 −ase

1

2 +thermal

(x)

2 (x) . + shot

(6)

In equation (6),  2 ase–ase (x)„  2 sig–fwm1 (x),  2 fwm1–ase (x) denote ASE–ASE beat noise, signal–FWM beat noise, FWM–ASE beat noise and, are expressed below in equation (6.1) [14,23]. Other terms are already explained before and are expressed in equation (6.1). 2 (2b b − b2 ) 22 (x) = 4R02 b(i)Psr (x)Ssp (be /bo ) + R02 Ssp e o e 2 2 +sig−fwm (x) + fwm

1 −ase

1

(x) + (4KB T/RL )be

+(2eR0 (b(i)Psr (x) + Ssp

(6.1)

+ PDN be ).

The first term of the above equation represents signal–ASE beat noise variance, the second term represents ASE-ASE beat noise variance, the fifth term represents thermal noise variance, and the sixth term represents shot noise variance. In the above equation: Ssp = SP (Ga − 1)hf

2 (x) sig−fwm1

=

(6.2)

2Psr (x)R02 [

+

1  Pr,l,m,x (k) 8

Ps r (x)

r= / l= / m

1  Pr,l,m,x (k) 4

r=l = / m

Psr (x)

2 (2b b − b2 ) +  2 (x) 02 (x) = R02 Ssp e o e ase−fwm 0

(4KB T/RL )be + (2eR0 Ssp be bo )

1 4

 r= / l= / m=s

1 + 4

 r= / l= / m=s

2 ase−fwm (x) = 4R02 Ssp be 2Psr (x)[ 0

+

1 4

1 8

 Pr,l,m,x (k) r= / l= / m

 Pr,l,m,x (k) r=l = / m

Psr (x)

Psr (x) (6.7)

].

In the above equations, Ssp is the power spectral density of ASE noise. Ps r (x) is the signal power received at the receiver. e is the electronic charge, be , bo , sp , h are defined earlier, KB is the Boltzmann constant, T is the temperature, RL is the load resistance, f is the frequency of operation, Pr,l,m,x (k) is the average power of the FWM components generated by the rth, lth, and mth channel, respectively, satisfying r + l – m = s. The fractional numbers associated with equations (6.3), (6.4) and (6.7) represent the probability of the coexistence of multiple signals among the four wavelengths, including the signal channel itself and is illustrated here only as an example. The fractional numbers vary with the number of different coexisting signals in a channel. In the proposed model, Pr,l,m,x (k) is calculated using equation (4.5).While computing the FWM crosstalk power, the FWM crosstalk power is calculated partially for all the possible combinations of active lightpaths and stored in an offline database. During the online evaluation of a lightpath, FWM efficiency (), the phase mismatching factor (ˇ), the FWM crosstalk power per link (Pr,l,m,x (k)), the FWM crosstalk power per path (PDN ) are computed online for any request. The system parameters and the values used in the model are shown in Tables 1 and 2 for solving DRWA problem with linear and nonlinear impairments, respectively [14,18]. The receiver BER due to the xth individual can be expressed as in [18]: Is1 (x) − Ith (x) I (x) ) + erfc( √ th )]. √ 21 (x) 20 (x)

(7)

Is1 (x) = R sigprecx bsig (x), represents the photocurrent of the receiver for the xth chromosome, and Ith (x) =

(6.3)

]εpol

(6.6)

The first term of the above equation represents ASE–ASE beat noise variance, the second term represents ASE–FWM beat noise variance, the third term represents thermal noise variance, and the fourth term represents shot noise variance corresponding to bit ‘0’ for the xth chromosome, where  2 ase-fwm0 (x) is expressed as:

x = 0.25[erfc( Pr,l,m,x (k) Psr (x)

.

0 (x)Is1 (x) 0 (x)+1 (x)

represents the

threshold current of the receiver for the xth chromosome. 4. The proposed algorithm

2 (x) = 4R02 Ssp be 2Psr (x)[ fwm1−ase

+

In equation (6.5),  2 ase–fwm0 (x) represents ASE–FWM beat noise variance for bit ‘0’ for xth chromosome, which is expressed in equation (6.6) [14].

(5)

For the FWM aware DRWA problem in the presence of ASE noise and FWM crosstalk power the combined noise can be modeled as a zero mean Gaussian random process with a variance given by:

987

1 8

 Pr,l,m,x (k) r= / l= / m

Pr,l,m,x (k) 1 + 4 Psr (x)

Psr (x)

 Pr,l,m,x (k) r=l = / m

Psr (x)

]. (6.4)

Equations (6.3) and (6.4) represent signal-FWM beat noise variance and FWM–ASE beat noise variance, respectively. Ga is the gain of the amplifier. f represents the optical frequency. Equation (6.5) below represents the variance corresponding to bit ‘0’ for the xth chromosome: 2 2 2 2 02 (x) = ase−ase (x) + ase−fwm0 (x) + thermal + shot (x).

(6.5)

This section briefly explains the mechanism of the proposed evolutionary programming algorithm. The step by step methodology of the proposed evolutionary programming algorithm (EP) is shown in Fig. 4 below. This iteratively works on a set of initial solutions, referred to as a population, and finally converges to the best solution. The distinguishing characteristics of the proposed algorithm are: (i) it depends on the mutation operator alone for creating and exploring the search space (ii) it uses an initial population consisting of only a single individual, and (iii) it assumes the a soft constraint with a set of problem-specific hard constraints to arrive at an optimal or near optimal solution quickly. The flow chart of the proposed algorithm used for solving the multi-constrained routing problem is shown in Figs. 5 and 6. Steps of the proposed algorithm:

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U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997

Table 1 System parameters and their values [18]. Parameter

Values

Number of wavelengths (W) Wavelength spacing Wavelengths (in nm)

16 100 GHz (1544.53,1545.32, 1546.12,1546.92, 1547.72,1548.51,1549.32,1550.12, 1550.92,1551.72,1552.52,1553.33, 1554.13,1554.94,1555.75,1556.55) 2.5 Gbps Bit rate per channel (r) 1.875 GHz Electronic bandwidth (Be ) Multiplexer loss (Lmx ) 4 dB 4 dB Demultiplexer loss (Ldm ) 1 dB Switch element insertion loss (Ls ) 1 dB Waveguide/Fiber coupling loss (Lw ) 2log2 NLs + 4Lw dB (for a N, X, N Switch Loss (Lsw ) switch) 1 dB Tap loss (Ltap ) 0.2 dB/km Fiber loss (Lf ) 22 dB Input EDFA Gain (Gin ) 18 dB Output EDFA Gain (Gout ) 1.5 ASE factor (sp ) √ √ √ RMS thermal current/ bandwidth ( th ) 5.3 × 10−12 A/ Hz 0 dBm Max laser power (Pl ) −30 dB Switch crosstalk ratio (Xsw ) 1 A/W Responsivity (R ) Optical bandwidth (Bo ) 3.375 GHz Adjacent wavelength rejection ratio (Xadj ) −30 dB Fig. 4. General flow of EP.

4.1. Chromosome or individual representation A chromosome represents a route or a path encoded from source to destination as described in [4,5]. In this work, the possible paths between a source and destination pair are represented as a set of chromosomes. Each chromosome is a sequence of nodes that is randomly generated while satisfying the topology of the particular network [2,4,5]. The chromosomes are of variable length, each of which is the encoding of a path from the source node, S, to the destination node, D, as shown in Fig. 7. The first gene always represents S, the second gene, k1 , represents one of the physically connected nodes to S, the third gene, k2 , is one of the physically connected nodes to the second node in the path, and so on, till D is reached. While

Table 2 System parameters and their values [14]. Parameter

Values

Number of wavelengths Wavelength spacing Wavelengths (in nm)

16 100 GHz (1544.53,1545.32, 1546.12,1546.92, 1547.72,1548.51,1549.32,1550.12, 1550.92,1551.72,1552.52,1553.33, 1554.13,1554.94,1555.75,1556.55) 1.875 GHz 4 dB 4 dB 10 dB 1 dB 0.2 dB/km 18 dB 16 dB 1.5 √ 5.3 × 10−12 A/ Hz 0 dBm

Electronic bandwidth (Be ) Multiplexer loss (Lmx ) Demultiplexer loss (Ldm ) Switch loss (Lsw ) Tap loss (Ltap ) Fiber loss (Lf ) Input EDFA gain (Gin ) Output EDFA gain (Gout ) ASE factor (sp ) √ √ RMS thermal current/ bandwidth ( th ) Max laser power (Pl ) equal power on all channels Responsivity (R ) Optical bandwidth (Bo ) Fiber chromatic dispersion at 1.55 ␮m (Dc ) Dispersion slope (dDc /d) Fiber non-linear coefficient ()

0.9 A/W 3.375 GHz 3.0 ps/nm/km 0.09 ps/nm2 /km 2.3 W−1 km−1

generating the chromosome, it is ensured that the nodes are not repeated from any source to any destination, thereby avoiding loops.

4.2. Population initialization The population that consists of a single individual is generated using a Lazy random walk approach [10] so as to satisfy the given constraints.

4.3. Mutation When a chromosome undergoes mutation, the mutation site is chosen randomly and an alternate path is chosen from that site to the destination following a lazy random walk approach. The mutation of chromosomes takes place by computing a new partial path from the node corresponding to the mutation site to the destination node, D. In the proposed algorithm mutation probability is kept equal to one. The chromosome that is generated initially undergoes the mutation loop fifteen times, each time generating one offspring [4]. Mutation site of the parent chromosome is chosen randomly. A different path is generated from that site to the destination node based on the topology database. Before mutation the encoding of chromosome is as shown in Fig. 8. In the mutation process first the chromosome length is calculated. Let the mutation site comes out to be the locus four where node k4 is located. Mutation site is chosen randomly. From the source node S to the node k4 located at the mutation site the nodes are kept intact as the parent chromosome as explained below. As shown in Fig. 9, the encoding of the chromosome remains same till node k4 and from k4 onwards the nodes are randomly generated based on the topology database information till the destination node D is reached. In this example, the two chromosomes are different, having different fitness functions. This way a single parent is made to produce fifteen offsprings. The chromosome with the best fitness value among the parent and offsprings survives for the next generation.

U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997

989

Start

Call request

Start timer

Initialization

No Time< T1

No

Hop constraints satisfied?

Yes

Relax hop constraints

Yes Yes

Hop constraints satisfied? No

No Time< T2

Yes BW constraints satisfied

Yes

Mutation

No

No

No

BW constraints satisfied

Time < T3

Yes

Drop request

Yes Links are cut topology change

Mutation

Fig. 5. Initialization steps of the proposed scheme for multi-constrained routing problem.

4.4. Fitness calculation for the QoS constrained routing model

4.5. Fitness calculation for the DRWA model assuming an ideal physical layer

The objective function or the fitness function is so designed that it evaluates the quality of a chromosome in the entire population. The fitness function for the given QoS constrained shortest path routing problem is formulated as follows: Fitnessi =

1 ki −1



.

(8)

Cgi (j),gi (j+1)

The fitness function for the proposed DRWA algorithm is formulated as follows: dx =

Wx



kx −1

Cgx(j),gx(j+1)

+

W

x

Hij

+

Wx . Tx

(9)

(i,j) ∈ E

j=1

j=1

Here Fitnessi represents the fitness value of the any arbitrary chromosome or individual, i, ki is the length of the ith chromosome, and gi (j) and gi (j + 1), respectively, represent the jth and (j + 1)th gene of the ith chromosome. Cgx(j),gx(j+1) represents the cost of the link between jth and (j + 1)th gene of the ith chromosome and the denominator of the fitness function represents summation of link costs over a path and therefore, the total cost of the path.

In the above equation, dx represents the fitness value of the xth chromosome. The denominator of the first term represents the total cost of the path, the denominator of the second term represents the total number of hops in the path, and the denominator of the third term represents the set up time of the lightpath. As explained earlier, the numerator of all the three terms, the binary variable Wx , represents the free wavelength factor for the lightpath. The variable kx is the length of the xth chromosome,

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U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997

Offspring generation

No Time< T4

Hop constraints satisfied?

Yes No

No

Relax hop constraints

Yes Yes

Hop constraints satisfied?

Links are cut Yes BW constraints satisfied

Time< T5 Yes No

No

No

No

Time< T6

Yes Yes

BW constraints satisfied Yes

No of offspring < 15 No Fitness evaluation Select the best offspring Links are cut topology change

No Stop

Yes No of request < total req.

Call request

Fig. 6. Mutation steps of the proposed scheme for multi-constrained routing problem.

S

k1

Locus

Chromosome

K2

1

S

2

K1

Kr-1

Kr

3

K2

D

m-2

m-1

Kr-1

Kr

Fig. 7. A chromosome representation.

m

D

Drop request

U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997

S

S

K2

K1

K4

K2

K1

K4

K9

K8

K8

K10

K9

991

D

K10

D

Fig. 8. Chromosome representation before mutation.

4.6. Fitness calculation for the PLI aware DRWA problems (linear impairment)

14

2 2 2 Hijx + sig−xtalk (x) + sig−ase (x) + sig−shot (x)

(i,j) ∈ E

j=1

(10)

4

1 80

60

2

2 2 +thermal + sig−demux (x)).

75

4.7. Fitness calculation for the PLI aware DRWA problems (non-linear impairment)

3 15 0

The fitness functions for the PLI aware DRWA problem assuming only linear impairment is formulated as follows: bx = Wx [

Cgx(j),gx(j+1) +



2 2 Hijx + shot (x) + sig−ase (x)

(11)

(i,j) ∈ E 2 2 2 2 (x) + thermal + sig−fwm1 (x) + fwm1−ase (x)]. +ase−ase j=1

In equations (10) and (11), fx and bx represent the fitness values of the xth chromosome for linear and FWM aware EP algorithm, respectively. In both the equations, the first term represents the total cost of a path, the second term represents the total number of hops in the path, and the rest of the terms represent different beat noise variances for the lightpath. Wx ,kx and gx(j) and gx(j+1) are already defined in equations (8) and (9). 5. The algorithm implementation The lightpath requests are assumed to arrive at the network dynamically according to a Poisson process with an average arrival rate of . The source and the destination for each request are randomly chosen according to a uniform distribution. The holding times for lightpath requests are assumed to be exponentially distributed with fixed mean, T hold = E[Th ]. The network load, the mean blocking probability and the mean execution time are defined as Network load =  × T hold; Mean blocking probability = (Number of requests blocked/Total number of requests processed); Mean execution time = (Total simulation time/Total number of requests). The proposed algorithm is simulated using Microsoft Visual C++ on an Intel Core 2 processor (2.8 GHz clock and 2 GB RAM). The reduced version of 14 node NSF network topology is used as depicted in Fig. 10 and the cost values are randomly assigned [8].

S

S

K2

K1

K1

K2

8

9

60

11

12

120

10

300

6

7

150

Fig. 10. Physical topology of NSFNET [8].

K4

K4

K6

K6

K7

K7

D

D

The threshold times T1 and T2 are fixed at 0.5 s and 1.5 s, for initialization and mutation, respectively. The hop count bound, h0 , is kept at four [4,5]. 6. Results and discussions In this simulation work, Figs. 11 and 12 depict the rate of convergence of the proposed evolutionary programming algorithm used for routing purpose on different pairs of source and destination nodes on an arbitrary twenty node network [2] and the comparison of the proposed EP algorithm with the other metaheuristic approaches, such as GA [2] and SA [9] in terms of fitness deviation and mean execution time. Fig. 13 depicts the performance metric of the DRWA problem, such as mean blocking probability exhibited by the proposed algorithm for an ideal physical layer. 0

10

Average fitness Score for 3 different sd pair



kx −1

60

5

80

60

75



13

105

Cgx(j),gx(j+1) +

0

75

x

30

120

k −1 

fx = Wx (

300

75

1 The fitness functions for the PLI aware DRWA problem assuming only linear impairment is formulated as follows:

80 12 0

-1

10

-2

10

First source destination pair Second source destination pair Third source destination pair

-3

10

0

5

10

15

20

Generations Fig. 9. Chromosome representation after mutation.

Fig. 11. Average fitness score of EP for a 14 node network.

25

992

U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997

0.01 0.009

0

0.008 0.007

Mean blocking probability

Fitness deviation for EP,GA,SA

10

GA EP SA

0.006 0.005 0.004 0.003 0.002

10

-1

Random First fit Round robin Wavelength ordering

0.001 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Average execution time (ms)

10

-2

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

Switch crosstalk and adjacent channel rejection ratio(dB)

Fig. 12. Fitness deviation of EP, GA, and SA for a 14 node network.

Fig. 14. Mean blocking probability for a fixed network load.

10

2

Average fitness score

Fig. 14 shows the variation in the blocking probability assuming different values of switch crosstalks and adjacent wavelength rejection ratios, for a fixed traffic load of 60 Erlang. In this simulation work, the proposed wavelength ordering assignment technique is based on efficient spectrum allocation so as to reduce the crosstalk at each of the nodes. Figs. 15 and 16 depict the rate of convergence of the proposed QoT aware evolutionary programming algorithm and the mean execution time exhibited by the algorithm for all the wavelength assignment techniques, respectively. Fig. 17 shows the performance exhibited by the FWM aware evolutionary programming algorithm in terms of mean blocking probability. Figs. 18 and 19 depict the mean execution time obtained by the proposed FWM aware evolutionary programming algorithm and convergence rate of the proposed fitness function of the algorithm, respectively. For the NSF network topology, the convergence rate of EP was determined on a constrained routing problem [4] for each of three different source destination pairs and is plotted in Fig. 11. The program was run twenty times and the estimated margin of error was found to be ±.00959127 at the point of convergence for the first pair of source and destination. To test the efficacy of EP on a constrained routing problem [4], it is compared with other metaheuristic approaches. Fig. 12 is a plot of the fitness deviation versus

1.9

Round robin Wavelength ordering

1.8

Random First fit

1.7 1.6 1.5 1.4 1.3 1.2 1.1 1

1

10

10

4

5

6

7

8

Fig. 15. Average fitness for a fixed network load of 60 Erlang, Xsw = −30 dB, Xadj = −30 dB.

0.8

0

Mean execution time (second)

Mean blocking probability

10

3

Generations

0.7

10

2

-1

-2

-3

First fit Random Round robin

0.5

0.4

0.3

0.2

0.1

0 50

-4

1

0.6

2

3

Wavelength ordering First fit Random Round robin

4

5

6

7

8

Number of generations Fig. 13. Mean blocking probability of the proposed scheme for the DRWA problem.

60

70

80

90

100

Network load (Erlang) Fig. 16. Mean execution time for Xsw = −30 dB, Xadj = −30 dB.

110

U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997 0

Mean blocking probability

10

-1

10

-2

10

Random Round robin First fit FWM aware priority based wavelength assignment -3

10

50

60

70

80

90

100

110

Network load (Erlang) Fig. 17. Mean blocking probability for different network loads. 1.4 Round robin First fit

Mean execution time (second)

1.2

Random FWM aware priority based wavelength

1

0.8

0.6

0.4

0.2

0 50

60

70

80

90

100

110

Network load (Erlang) Fig. 18. Mean execution time for different network loads. 10

5

Average fitness score

First fit FWM aware priority based wavelength assignment Round robin Random

10

10

10

4

3

2

1

2

3

4

5

6

Generations Fig. 19. The fitness convergence curve.

7

8

993

the execution time incurred by EP, GA, and SA [2,9]. Fitness deviation is defined as the deviation of the estimated fitness value from the optimum fitness value. In this experiment, the optimum fitness value was first estimated for a particular source–destination pair using GA [2]. For this estimation, the program was executed twenty times and then, the average fitness value was computed. The proposed approach EP was also applied to the same source–destination pair to find out the actual average fitness value incurred by it and then to estimate the fitness deviation. Fig. 12 shows that EP gives the opportunity for a trade-off between an optimal solution and the worst case algorithm execution time and it is shown to exhibit better performance compared to GA and SA. EP is first integrated with different wavelength assignment techniques (the combined approach referred to as evolutionary programming algorithm for routing and wavelength assignment or ERWA) to solve the DRWA problem with an ideal physical layer and subsequently the algorithm was applied on a DRWA problem assuming a non-ideal physical layer. The mean blocking probabilities obtained by ERWA for the three wavelength assignment techniques are plotted in Fig. 13, assuming exponential holding times distribution. For a fair comparison of the proposed approach with the approach suggested by Bisbal et al. [8], an identical set of parameters were used in the simulation. The result is shown assuming a traffic load of 60 Erlang and eight wavelengths per link and was simulated for one million call requests. It is observed from the plot that at the seventh generation or at the point of convergence, the mean blocking probability of the order of 10−4 with the estimated margin of error ±(1.12730 × 10−4 ) obtained by ERWA for this experiment found to yield a value lower than that exhibited by other approaches [8,15]. The margin of error was computed for the Round robin assignment technique when integrated with the proposed ERWA. EP was also, utilised to solve the DRWA problem assuming the existence of few linear and non-linear physical layer impairments. In this paper, this algorithm is referred as QOT aware evolutionary programming algorithm while assuming only linear physical layer impairments. In this the work also, the statistical results were obtained for one lakh call requests while assuming linear physical layer impairments. As switch crosstalk and demultiplexer crosstalk are assumed to be the dominant physical impairments in this work, it is important to determine how the algorithm performs under different switch and demultiplexer crosstalk power levels. Fig. 14 shows the variation in the blocking probability assuming different values for each of switch crosstalk and adjacent wavelength rejection ratio, for a fixed traffic load of 60 Erlang. The mean blocking probability is estimated in each case by executing the program ten times and then by computing the average. As per expectation, the mean blocking probability decreases for a reduction in each of the switch crosstalk ratio and the adjacent wavelength rejection ratio. It is also observed that for a low value of the switch crosstalk ratio as well as the adjacent wavelength rejection ratio, the algorithm maintains a blocking probability at an acceptable level. As depicted in figure, the proposed wavelength ordering assignment technique incurs the lowest blocking probability for all types of switch crosstalk ratios. The simulation experiments show that the proposed algorithm assuming the new wavelength ordering based wavelength assignment technique exhibits a very good blocking performance for the NSFNET network topology. This is true at a value of −30 dB for each of switch and adjacent wavelength rejection ratios. The margin of error was estimated assuming only the wavelength ordering assignment technique and is found to be ±0.0059854 for 95% of time. A mean blocking probability of the order of 10−2 exhibited by the proposed algorithm integrated with the new wavelength ordering based technique with margin of error, as shown in the plot, is found to be far better compared to that of exhibited by the heuristic approach [22]. This demonstrates that the

994

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performance of the algorithm will be still better for MEMS or fiber Bragg grating based WRS devices, which exhibit very low switch crosstalk ratios. Fig. 15 depicts the fitness convergence curve. The curves are obtained by choosing an individual randomly and plotting the best fitness value for that particular individual versus the number of generations. Initially, the curves show instability in the value of the fitness function for that particular individual; the fitness value however stabilizes at later generations. The margin of error was estimated for only wavelength ordering assignment technique as it offers the lowest mean blocking probability in comparison to the existing assignment techniques and was found to be ±0.00156820 for 95% of time around fourth generation or at the point of convergence. When the switch and the demultiplexer both are non-ideal, the first fit assignment technique exhibits the lowest execution time among all the assignment techniques and the proposed wavelength ordering technique incurs a higher execution time compared to the First fit technique as shown below in Fig. 16. But, the order of the execution time allows the proposed technique to be used in real time applications [20] when the network load is less than 90 Erlang and the margin of error was estimated for network load of 80 Erlang and was found exhibit a value of ±.0098998108 for 95% of time. The last part of the simulation work involves the solution of the DRWA problem while assuming the non-linear physical layer impairment of four wave mixing (FWM) that is prevalent even at a moderate data rate and small input power. The mean blocking probability is obtained for all the existing wavelength assignment techniques and also for a newly developed assignment technique, referred in this paper as FWM aware priority based wavelength assignment technique. Fig. 17 depicts the blocking probability for all the wavelength assignment techniques for a total of 50,000 requests. Among all the assignment techniques, as expected the FWM aware priority based wavelength assignment technique provides the lowest blocking probability. Among the other wavelength assignment techniques, the Random and the First fit wavelength assignment techniques offer the same order of blocking probability, even though the Random assignment technique shows slightly better network performance in terms of the blocking probability. The round robin wavelength assignment technique exhibits the worst performance in terms of the blocking probability. As anticipated, the first fit technique tries to use the first available wavelength always and hence, there is less chance of using the central part of the transmission window that gives rise to more FWM crosstalk. The round robin technique explores the wavelengths uniformly and hence this approach uses all the wavelengths in the transmission window equally providing the worst blocking probability. The mean blocking probability of the order of 10−2 even with the estimated margin of error of value ±9.59527519 × 10−4 for 95% of time obtained by the proposed algorithm integrated with the FWM aware priority based wavelength assignment technique is found to be better compared to that of heuristic [16] and metaheuristic approaches [24] at network load of 90 Erlang. Fig. 18 describes the mean execution time obtained by different wavelength assignment techniques. The Random wavelength assignment technique provides the least mean execution time for different network loads and can be used for real time applications as for these applications an execution time of about 3 ms is acceptable [20]. Wavelength priority based technique can also be used for real time applications when the network load is less than 90 Erlang. The margin of error was estimated for the wavelength priority based technique as this technique is found to exhibit the best network performance compared to others in terms of mean blocking probability and found to be ±0.0098998108 for a network load of 80 Erlang for 95% of time. It is observed from the plot that even with this margin of error the execution time exhibited by the wavelength

priority based technique less than 3 ms and is suitable for real time application. It is concluded that the first fit and the round robin techniques exhibit the worst performance in terms of the mean execution time and can never be used for real time applications. Fig. 19 above shows the fitness convergence curve for the FWM aware evolutionary programming algorithm for the different wavelength assignment techniques. While executing the program a particular request is chosen and the average fitness score is recorded for that request. It is observed from the experiment that the fitness of a chromosome converges almost for all the wavelength assignment techniques around the third generation except in the case of the Random assignment technique. The random assignment technique shows instability up to the sixth generation and then converges. For this experiment, the program is run ten times and the average of the best chromosomes is chosen at each generation to get the results. The estimated margin of error was found to be ±0.001568204 for the FWM aware wavelength assignment technique at fourth generation or at the point of convergence for 95% of time. 6.1. Algorithm time complexity analysis for the DRWA problems In this section, the complexity of the developed evolutionary programming algorithms for the DRWA problem assuming an ideal physical layer, and in the presence of linear impairments and for FWM aware DRWA problem are estimated for a random Bernoulli graph (Erdos–Renyi graph). A random graph is represented by Gp (N), where N is the number of nodes in the graph and p is the probability of an edge being present between any two nodes. It is assumed that p is independent of the presence of any other link. The expected number of links is E(l) for a geometric random graph as given in [13] under the assumption that N nodes are uniformly distributed over a two-dimensional area with size ˝. Using a dissection technique, which assumes that the two-dimensional area where the nodes are distributed is covered with m small squares, each of which is a placeholder of size ˝ containing at most one node, the expression obtained for the expected number of links is as follows: E(l) =

m m   N(N − 1)  

m(m − 1)

p(rij ),

i=1 j=i+1

where rij is the distance between two placeholders i and j. For a random Bernoulli graph p(rij ) = p as the edge probability is independent of the distance between any two nodes. Therefore, the expected number of links for a random Bernoulli graph is expressed below [6]: E(l) =

N(N − 1)p . 2

The expected number of links for a fully connected mesh network is expressed below [6]: E(l) =

N(N − 1) . 2

The average node degree for a random Geometric graph is: E(d) =

2E[l] . N

Consequently, for a fully connected random Bernoulli graph, the mean node degree, E(d), equals (N − 1). For a fully connected random Bernoulli graph the expected number of hops is expressed as: ¯ = H

ln(N) ≈ 1. ln(N − 1)

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In order to evaluate the time complexity of the proposed evolutionary programming algorithm for different cases, it is assumed that G denotes the number of iterations, C denotes the number of offsprings, and W is the number of wavelengths per link. In the worst case the proposed evolutionary programming algorithm performs the following steps. 6.2. Initialization The initial population consists of a single chromosome. The initialization process of the proposed algorithm follows a lazy random walk approach. As cited in reference [10], the cover time of a lazy random walk is upper bounded by O(dmax N3 log2 N) = O(N4 log2 N), where dmax is the maximum node degree of the graph. In the worst case the initialization step of the proposed algorithm is assumed to be upper bounded by O(N3 log2 N) since the initialization step finds a path between any two randomly generated nodes. 6.3. Mutation During the mutation phase, a single parent is mutated to produce C offsprings, the complexity involved being O(CN3 log2 N) time units as the lazy random walk approach is adopted while generating the offsprings. This step is repeated G times and hence the complexity is O(GCN3 log2 N). 6.4. Fitness evaluation It is assumed that in the worst case all the W wavelengths are examined in every link of the route. The average or the expected numbers of links is (N(N − 1)/2) for an N node network. Hence, the cost of fitness evaluation is O(GW(C + 1)(N(N − 1)/2)) ≈ O(GWCN2 ) time units. 6.5. Selection The best solution is chosen out of total (C + 1) solutions involving C comparisons, each of which takes O(C) time units. As the operation is performed G times, the complexity of the selection process is O(GC). In the proposed approach G and C have fixed values. Therefore, the overall complexity is dominated by the mutation phase and is equal to O(GCN3 log2 N) ≈ O(N3 log2 N) for the DRWA problem assuming an ideal physical layer. The additional step that is required while estimating the algorithm time complexity for QoT aware evolutionary programming algorithm is the complexity involved in evaluating the online signal power, ASE noise power, and switch and demultiplexer crosstalk power and the steps are described as below.

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turn depends on the average hop count. In this step also the ASE noise power is evaluated for all the (C + 1) chromosomes and this is repeated G times. Hence, the complexity is: ¯ = O(GC). O(G(C + 1)H)

6.8. Online switch crosstalk power evaluation For a chromosome the switch crosstalk power estimated at the destination node depends on the number of XCSs traversed by a lightpath from any arbitrary source node to any destination node, which in turn depends on the average hop count. It also depends on the number of in-band signals present at the switch input. At any one particular WRS, the maximum number of in-band signals in the worst case is limited by the average input node degree and this equals to average node degree E(d). In this step the switch crosstalk power is evaluated for all the (C + 1) chromosomes and this is repeated G times for each of the W wavelengths. Hence, the complexity is: O(G(C + 1)HWE(d)) = O(GC(N − 1)W ) ≈ O(GCNW ).

6.9. Online demultiplexer crosstalk power evaluation For a chromosome the demultiplexer crosstalk power estimated at the destination node depends on the number of adjacent wavelengths present at the demultiplexer and number of in-band signals present at the output port. At the demultiplexer input, W comparisons are needed to know the presence of adjacent wavelengths and at the output port, W comparisons are needed to know the presence of in-band signals. The maximum number of adjacent wavelength signals in a demultiplexer and the maximum number of in-band signals at the output port in the worst case is limited by the average node degree E(d). In this step the demultiplexer crosstalk power is evaluated for all the (C + 1) chromosomes and this operation is repeated G times for each of the wavelengths. Hence, the complexity is O(G(C + 1)W2 E(d)) ≈ O(GC(N − 1)W2 ) ≈ O(GCNW2 ). As the DRWA problem assuming an ideal physical layer involves with initialization, mutation, fitness evaluation and selection steps and therefore, the overall complexity is dominated by the mutation phase is equal to O(GCN3 log2 N) ≈ O(N3 log2 N). For the QoT aware evolutionary programming algorithm incorporating both the switch crosstalk and the demultiplexer crosstalk, the overall complexity is still dominated by the mutation phase and is equal to O(GCN3 log2 N) time units ≈ O(N3 log2 N). The additional step that is required while estimating the algorithm time complexity for FWM aware evolutionary programming algorithm is the complexity involved in evaluating the online FWM crosstalk power and the step is described as below [6].

6.6. Online signal power evaluation 6.10. Online FWM crosstalk power evaluation The time a signal takes to reach a destination node from any arbitrary source node depends on the average number of hops it traverses. Since during the mutation process a total of C offsprings are generated and the signal power evaluation takes place G times for each of the chromosomes, the complexity involved is: ¯ = O(GC). O(G(C + 1)H) 6.7. Online ASE noise power evaluation For a chromosome the ASE noise experienced at the destination node depends on the number of EDFAs present at both the input and output of a wavelength routing node in a lightpath, which in

During the online FWM crosstalk evaluation, initially the partial FWM crosstalk power stored in the data base is retrieved by simple search techniques by matching the wavelengths present in the link. This involves a complexity of O(W4 ). The average or the expected number of links is (N(N − 1)/2) for a fully connected mesh network of N nodes. Hence, the complexity of evaluating the FWM crosstalk power for all the (C + 1) solutions for G generations O(GCW4 N(N − 1)/2) ≈ O(W4 N2 ). Therefore, the overall algorithm complexity for FWM aware evolutionary programming algorithm is found to be O(N3 log2 N) if the number of wavelengths W is much smaller than the number of nodes. Otherwise, the algorithm complexity is found to be O(W4 N2 ).

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7. Conclusion This paper reports the results of an extensive simulation study conducted to validate a set of evolutionary programming algorithms for the solution of important optimization problems encountered in the field of optical networks. The advantages of the proposed algorithm are that the initial search space is very small since the initial population consists of only a single chromosome. Encodings of the chromosomes are random and simple and these are of variable length. No penalty parameter is used to penalize infeasible chromosomes. The proposed algorithm discards illegal and infeasible chromosomes during chromosome generation; generating only feasible chromosomes. No crossover operation is used and it is shown that only mutation is sufficient for generating good solutions. It is shown that the proposed algorithm for multi-constrained routing problem assuming multiple concurrent requests exhibits a superior performance compared to both GA and SA in terms of the fitness function and mean execution time. The proposed evolutionary algorithm works simultaneously on multiple points in the search space during the mutation process. So it generates many alternative solutions along with the best solution that is treated as the primary path. One of the alternative solutions may be used as a backup path in case of a failure in the original path provided this path is link disjoint with the primary paths of all the concurrent requests. The evolutionary programming algorithms for solving the DRWA problem in WDM all-optical networks is shown to exhibit a lower computational complexity compared to the other existing metaheuristic approaches such as GA and ACO. The overall complexity is equal to O(GCN3 log2 N) ≈ O(N3 log2 N).The proposed fitness function provides better load balancing, thereby leading to a low blocking probability. Use of an initial hop count bound, taken to be a soft constraint, and use of the set up time during the mutation phase help in lowering the execution time. Unlike other existing metaheuristic approaches, the proposed algorithm does not involve any offline computation. The proposed routing scheme when integrated with the existing wavelength assignment techniques exhibits a superior performance in terms of the blocking probability compared to the other existing algorithms for the DRWA problem for an ideal physical layer. Unlike the existing heuristic or metaheuristic approaches for the QoT aware DRWA problem, the proposed scheme integrates the routing and wavelength assignment parts to discover a wavelength continuous path online. Also, linear impairments like node crosstalk and ASE noise are incorporated into the designed fitness function so that the best path is sure to provide an acceptable level of received signal quality. The proposed scheme also provides certain degree of flexibility in the network design. The simulation experiments show that the proposed algorithm along with the new wavelength ordering based wavelength assignment technique referred in this work as wavelength ordering technique exhibits a very good blocking performance for a reduced version of NSFNET network topology. Given that the experiments are carried out assuming a high value of −30 dB for each of switch and adjacent wavelength rejection ratios, the performance of the algorithm will be still better for MEMS or fiber Bragg grating based devices, which exhibit very low switch crosstalk ratios. The complexity of the proposed algorithm, in the worst case, is found to be O(N3 log2 N). In the FWM aware DRWA problem, a novel fitness function is proposed that minimizes the physical length of a path, the total number of hops a lightpath takes to reach the destination, including the FWM noise, and amplifier ASE noise variance in a path. The simulation experiments show that the proposed algorithm, along with the FWM aware wavelength assignment technique, exhibits a very

good blocking performance for the NSFNET network topology compared to other heuristic and metaheuristic approaches. The overall algorithm complexity for FWM aware evolutionary programming algorithm is found to be O(N3 log2 N) if the number of wavelengths W is much smaller than the number of nodes. Otherwise, the complexity is found to be O(W4 N2 ). Finally, it can be stated that this paper addresses only few of the important optimization problems encountered in the field of optical networks. Specifically, it is shown how it is possible to use the same basic evolutionary approach to solve different optimization problems by incorporating problem-specific information into the designed fitness function and the use of suitable hard and soft constraints. References [1] A. Adhya, D. Datta, Design methodology for WDM backbone networks using FWM-aware heuristic algorithm, Optical Switching and Networking 6 (1) (2009) 10–19. [2] C.W. Ahn, R.S. Ramakrishna, A genetic algorithm for shortest path routing problem and the sizing of populations, IEEE Transactions on Evolutionary Computation 6 (6) (2002) 566–579. [3] S. Azodolmolky, M. Klinkowski, E. Marin, D. Careglio, J. Sole Pareta, I. Tomkos, A survey on physical layer impairments aware routing and wavelength assignment algorithms in optical networks, Computer Networks 53 (7) (2009) 926–944. [4] U. Bhanja, R. Roy, S. Mahapatra, An evolutionary programming algorithm for finding constrained optimal disjoint paths for multihop communication networks, International Journal of Metaheuristics 1 (2) (2010) 132–155. [5] U. Bhanja, S. Mahapatra, R. Roy, A novel solution to the dynamic routing and wavelength assignment problem in transparent optical network, International Journal of Computer Networks and Communications 2 (2) (2010) 119–130. [6] U. Bhanja, S. Mahapatra, R. Roy, FWM aware evolutionary programming algorithm for transparent optical networks, Photonic Network Communications 23 (3) (2012) 285–299. [7] Bijja, P., Wavelength assignment for all optical networks for mesh topologies, A thesis submitted for the degree of M.S. in electrical engineering, Louisiana State University, 2003. [8] D. Bisbal, I.D. Miguel, J.B. Gonzalez, J.C. Aguado, P. Fernandez, J. Duran, R. Duran, R.M. Lorenzo, E.J. Abril, M. Lopez, Dynamic routing and wavelength assignment in optical networks by means of genetic algorithms, Photonic Network Communications 7 (1) (2004) 43–58. [9] K. Deb, Optimization for Engineering Design Algorithms and Examples, PHI, INDIA, 1998. [10] Denysyuk, O., Rodrigues, L., Random walk on directed dynamic graphs, http://www.gsd.inesc-id.pt/∼ler/reports/dynas10.pdf [11] D.B. Fogel, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence, 2nd ed., IEEE press, Piscataway, NJ, USA, 2000. [12] A. Ghatak, K. Thyagarajan, An Introduction to Fiber Optics, Cambridge University Press, Cambridge, UK, 2008. [13] R. Hekmat, P.V. Mieghem, Degree distribution and hopcount in wireless ad-hoc networks, ICON (2003) 603–609. [14] G. Kaur, M.L. Singh, M.S. Patterh, Theoretical investigations of the combined effect of fiber nonlinearities, amplifier and receiver noise in a DWDM transmission system, Journal of Optical and Fiber Communications Research 6 (1–10) (2009) 1–10. [15] V.T. Le, X. Jiang, S.H. Ngo, S. Horiguchi, Dynamic RWA Based on the Combination of Mobile Agents Technique and Genetic Algorithms in WDM Networks with Sparse Wavelength Conversion, IEICE Transactions on Information and Systems E88-D (9) (2005) 2067–2078. [16] A. Marsden, A. Maruta, K. Kitayama, Reducing the lightpath establishing time of FWM-aware dynamic RWA for wavelength-routed optical networks, Photonic Network Communications 18 (2009) 183–190. [17] G.S. Pavani, H. Waldman, Adaptive routing and wavelength assignment with power constraints using ant colony optimization, in: Proceedings of International Telecommunications Symposium, Fortaleza, Brazil, 2006, pp. 637–642. [18] B. Ramamurthy, D. Datta, H. Feng, J.P. Heritage, B. Mukherjee, Impact of transmission impairments on the teletraffic performance of wavelength-routed optical Networks, Journal of Lightwave Technology 17 (10) (1999) 1713–1723. [19] R. Ramaswami, K.N. Sivarajan, Optical Networks: A Practical Perspective, Morgan Kaufmann Publishers, San Francisco, 2000. [20] R. Ramjee, K. Murakami, R.W. Buskens, Y.-J. Lin, T.F. La porta, Design, implementation, and performance of a cluster mobile switching center, wireless multimedia network technologies, in: R. Ganesh, K. Pahlavan, Z. Zvonar (Eds.), The International Series in Engineering and Computer Science, Springer, USA, 2006. [21] P.P. Sahu, A new shared protection scheme in optical network, Current Science 91 (9) (2006) 1176–1183. [22] V. Saminadan, M. Meenakshi, In-band crosstalk performance of WDM optical networks under different routing and wavelength assignment algorithms, IWDC, Lecture Notes in Computer Science 3741 (2005) 159–170.

U. Bhanja, S. Mahapatra / Applied Soft Computing 13 (2013) 981–997 [23] S.P. Singh, S. Kar, V.K. Jain, Effect of four wave mixing on optimal placement of optical amplifier in WDM star networks, Fiber Integrated Optics (2006) 111–140. [24] S.C. Tan, F.M. Abbou, H.T. Ewe, Four wave mixing aware wavelength assignment using ant-based algorithm, Journal of Applied Sciences 7 (23) (2007) 3796–3800. [25] H. Zang, J.P. Jue, B. Mukherjee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks, Optical Network Magazine 1 (1) (2000) 47–60. Urmila Bhanja received her M. Tech and Ph.D. degrees from the Dept. of E&ECE, Indian Institute of Technology (IIT), Kharagpur, West Bengal, India. Currently she is working as Assistant Professor in the Department of Electronics and Communication Engineering, Indira Gandhi Institute of Technology, Sarang, Orissa, 759146, India. Her research interest includes computer network design, resource optimization using bio-inspired algorithms, development of different soft computing approaches for function optimization, and wireless sensor networks.

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Sudipta Mahapatra received his M. Tech. and Ph.D. degrees in computer engineering from the Dept. Of E&ECE, Indian Institute of Technology (IIT), Kharagpur, India in 1992 and 1997, respectively. From 1993 to 2002, he worked in various capacities in the Computer Science and Engineering Department, of NIT, Rourkela, India. From March 1999 to March 2000, he was with the Electronic Systems Design Group of Loughborough University, UK, as a BOYSCAST Fellow of the Department of Science and Technology, Government of India. Currently, he is working as an Associate Professor in the Department of Electronics and Electrical Communication Engineering, IIT, Kharagpur, India. His areas of research interest include parallel and distributed computing, computer networking and data compression hardware.