Ant colony optimization for dynamic routing and wavelength assignment in WDM networks with sparse wavelength conversion

(1)

Ant colony optimization for dynamic routing and wavelength assignment

in WDM networks with sparse wavelength conversion

Ming-Tsung Chen

a,c

, Bertrand M.T. Lin

b,n

, Shian-Shyong Tseng

c,d a

Telecommunications Strategy and Marketing Research Department, Telecommunication Laboratory, Chunghwa Telcom Co., Ltd., Taiwan

b_{Institute of Information Management/Department of Information and Finance Management, National Chiao Tung University, Taiwan} c

Department of Computer and Information Science, National Chiao Tung University, Taiwan

d

Department of Information Science and Applications, Asia University, Taiwan

a r t i c l e

i n f o

Article history:

Received 17 November 2009 Received in revised form 28 May 2010

Accepted 31 May 2010 Available online 22 June 2010 Keywords:

Routing and wavelength assignment Communication cost

Delay bound Approximate solution Ant colony optimization

a b s t r a c t

Since optical WDM networks are becoming one of the alternatives for building up backbones, dynamic routing, and wavelength assignment with delay constraints (DRWA-DC) in WDM networks with sparse wavelength conversions is important for a communication model to route requests subject to delay bounds. Since the NP-hard minimum Steiner tree problem can be reduced to the DRWA-DC problem, it is very unlikely to derive optimal solutions in a reasonable time for the DRWA-DC problem. In this paper, we circumvent to apply a meta-heuristic based upon the ant colony optimization (ACO) approach to produce approximate solutions in a timely manner. In the literature, the ACO approach has been successfully applied to several well-known combinatorial optimization problems whose solutions might be in the form of paths on the associated graphs. The ACO algorithm proposed in this paper incorporates several new features so as to select wavelength links for which the communication cost and the transmission delay of routing the request can be minimized as much as possible subject to the speciﬁed delay bound. Computational experiments are designed and conducted to study the performance of the proposed algorithm. Comparing with the optimal solutions found by an ILP formulation, numerical results evince that the ACO algorithm is effective and robust in providing quality approximate solutions to the DRWA-DC problem.

1. Introduction

Optical networks are a type of high-capacity telecommunica-tion networks that can provide routing, grooming, and restoratelecommunica-tion at wavelength level (Green, 1992). The technology of Wavelength Division Multiplexing (WDM) networks is mainly based on optical wavelength-division multiplexing on optical ﬁbers for forming a number of multi-communication channels at different wavelengths with an electronic processing speed (Lowe, 1998). WDM networks provide connectivity among optical components to let optical communication meet the increasing demands for high channel bandwidths and low communication delays. The utilization of wavelengths to route data is referred as wavelength routing, and an optical switch employing the technique is referred as a routing switch. Therefore, in a wavelength-routing WDM network that is constructed using optical ﬁber links to connect input ports and output ports in wavelength-routing switches, data can be routed to other optical switches based on

wavelengths of optical ﬁbers. If the transmission between input port and output port involves two different wavelengths, the switch should have the capacity of wavelength conversion (Ramamurthy and Mukherjee, 1998) and gives rise to transmis-sion delay and deployment cost. Deploying a part of switches with wavelength conversion in networks can be a viable alternative to balance the cost of constructing networks and network efﬁciency. Networks of this type are referred as WDM networks with sparse wavelength conversion.

In (wavelength-routing) WDM networks, a light-path (Chlamtac et al., 1992) can be set up in a similar way as a circuit-switched network to carry data among switches at wavelength level without optical-to-electrical and electrical-to-optical conversions, and then the data can be transmitted according to the trail of the light-path. Since different set-up light-paths will occupy different resources (e.g., switches, wave-lengths) and end-to-end transmission time in network commu-nication, the communication cost and the transmission delay of a light-path are usually used as the criterion for evaluating the efﬁciency of a light-path. The communication cost may be the numbers or the costs of utilized ﬁbers and switches included in a light-path. The transmission delay is the sum of transmission delays of all switches and links in the light-path. The routing and Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/engappai

Engineering Applications of Artiﬁcial Intelligence

n

Corresponding author. Tel.: +886 3 5131472.

E-mail addresses: [email protected], [email protected] (B.M.T. Lin).

(2)

wavelength assignment (RWA) problem known to be NP-hard (Karasan and Ayanoglu, 1998) is defined as follows: Given a set of connection requests, each of which is specified to transmit data from a source to a destination, the problem is to find a light-path from the source to the destination for each request and to assign a wavelength to each link included in the light-path.

The RWA problem can be categorized into two types, static RWA (SRWA) and dynamic RWA (DRWA). The SRWA problem is to determine the logical topology which can be used to configure all switches according to the found light-paths and assigned wavelengths in the given network. The objective is usually to maximize the routing throughput (Krishnaswamy and Sivarajan, 2001) or to minimize the number of required wavelengths. The DRWA counterpart is an on-line version in which connection requests arrive one at a time, and the requests must be routed in real time under the current environment of the network. For minimizing blocking probability (Shen et al., 2001; Qin et al., 2003; Ngo et al., 2006) and wavelengths, the objective is to find a survival light-path when some links fail, and to maximize the carried traffic (Kavian et al., 2007). For setting up and tearing down light-paths to minimize the number of blocked connections, a distributed control scheme for establishing reliability-con-strained least-cost light-paths and four heuristics were proposed inSaradhi et al. (2007).

Many new network applications, such as videoconferencing, video on demand system, on-line gaming, etc., have inspired the demands for new communication models. Moreover, to guarantee that video and audio signals can be efﬁciently transmitted in interactive multimedia applications, transmission delays from a source to a destination will be limited under a given delay bound, where the delay bound may be decided according to the degree of emergence, data priority, or application type of data. Therefore, transmitting data with delay bound constraints is realistic to reﬂect the demand about data transmission in the future. A request with delay bound dictates that it needs to be successfully transmitted before its given delay constraint is violated. The issue of routing this type of requests is referred as the RWA with delay constraints problem (RWA-DC) in WDM networks.

In most cases, switches with wavelength conversion are reserved to provide the imperative of converting wavelength in the light-path such that the data can be successfully transmitted. In order to avoid using this type of switches in the light-path not requiring wavelength conversion, it is necessary to incur extra costs for using this type of switches. Although most of previous research neglected the communication cost of wavelength conversion in switches so as to simplify the problem and thus reduce the complexity, the communication cost of wavelength conversion is taken into consideration in this paper to ensure that the light-path can make better use of the switches without conversion capability. Besides, because transmission delay occurs in the case that wavelength is converted in a switch with wavelength conversion, it is therefore crucial to take transmission delay into account. In this paper, we assume communication cost and transmission delay in switches are incurred when the wavelength is converted between the input and output ports. In summary, the dynamic RWA-DC (DRWA-DC) problem involves the following three features: (1) some of the switches can provide wavelength conversion, (2) data transmission through switches using different wavelengths at input and output ports incurs communication cost and transmission delay, and (3) requests are associated with delay bounds. The goal is to minimize the total communication cost.

For the RWA problem, Karasan and Ayanoglu (1998)

proved the NP-hardness. Integer linear programming (ILP) models (Krishnaswamy and Sivarajan, 2001), statistics (Ngo et al., 2006), and meta-heuristics including heuristics (Shen

et al., 2001; Qin et al., 2003) and genetic algorithm (GA) (Kavian et al., 2007) were proposed to cope with different problem settings. Although the ILP model can be deployed to ﬁnd optimal solutions, the execution time is not affordable for large-scale networks. Moreover, the DRWA-DC problem exhibits much more complicated structures; it is unlikely to follow the ILP approach to produce optimal solutions in an acceptable time.

In this paper, we address a design of ant colony optimization (ACO), which is a meta-heuristic developed in the early 1990s (Dorigo et al., 1991). The ACO uses natural metaphor inspired by the behavior of ant colonies to solve complex combinatorial optimization problems for ﬁnding near-optimal solutions. It has demonstrated signiﬁcant strengths in many application areas, such as the traveling salesman problem (Dorigo and Gambardella, 1997), generalized minimum spanning tree problem (Shyu et al., 2003), scheduling problems (Shyu et al., 2004a; Lin et al., 2008; Udomsakdigool and Kachitvichyanukul, 2008; Sabuncuoglu et al., 2009), optimization of chaotic systems (Wang and Ip, 2005), minimum weight vertex cover problem (Shyu et al., 2004b), layout design of satellite modules (Sun and Teng, 2003), assembly line balancing (McMullen and Tarasewich, 2006), and distributed optimization of a logistic system (Silva et al., 2006), just to name a few. The details of the ACO design for solving the RWA problem will be described later.

In Varela and Sinclair (1999), Garlick and Barr (2002), and

Kwang and Weng (2003), the ACO has been used to solve the RWA problem, but communication cost, wavelength conversion cost and delay bound were not incorporated in their studies. To the best of our knowledge,Varela and Sinclair (1999)is the ﬁrst paper applying the ACO to cope with the SRWA problem. In their design, each ant keeps a tabu list of previously visited nodes to avoid dead-ends and cycles and to allow backtracking, where back-tracking means that an ant will reversely pop out its previous location to alter the visited nodes when the already found partial tour is blocked. Garlick and Barr (2002) extended the ACO application to the DRWA problem by using length and congestion information in making routing decisions to reduce the possibility of network blocking in the tour-constructing phase. InKwang and Weng (2003), a survey and comparison on ACO applications to routing and load-balancing issues were presented. Varela and Sinclair (1999)andGarlick and Barr (2002)both used the shortest path algorithm and the minimum number of edges of paths to ﬁnd a light-path as the heuristic ingredient of the ACO. However, delay bounds make these heuristics inappropriate for solving the DRWA-DC problem. Moreover, the realistic concerns about wavelength conversion in switches give rise to a more compli-cated problem. According to the three characteristics: delay bound, wavelength conversion, and objective function,Table 1shows the comparisons of previous research papers: Krishnaswamy and Sivarajan (2001),Shen et al. (2001),Garlick and Barr (2002),Qin et al. (2003),Ngo et al. (2006), andKavian et al. (2007).

The DRWA-DC problem is relatively difﬁcult because many issues need to be simultaneously taken into account: the request is associated with a delay bound, the parts of switches have a wavelength conversion capability, and a light-path is evaluated by communication cost and wavelength conversion cost. Since the DRWA-DC problem is computationally challenging, it is very unlikely to optimally solve it in polynomial time. While the ACO has been applied to solve some speciﬁc RWA, no results have been reported to the complex, but realistic, problem involving delay bound and wavelength conversion simultaneously. In this paper, we shall design new ACO features to produce solutions to the studied problem.

The rest of this paper is organized as follows. Section 2 is dedicated to a formal formulation of the DRWA-DC problem. In Section 3, we shall introduce the basic structure of the ACO and

(3)

then present several features that can nicely shape the DRWA-DC problem into a graph-based framework that is suitable for the development and application of ACO algorithms. Section 4 is dedicated to the computational experiments designed to evaluate the performance of the proposed ACO algorithm. Numerical results and analysis are also included. Section 5 summarizes the results and gives some concluding remarks.

2. Problem formulation

Before proceeding to the problem statements and formulation, we introduce the notation that will be used throughout this paper.

Notation

W set of wavelengths available for data transmission in the given WDM network

n number of nodes in the WDM network

m number of different wavelengths in W Ii wavelength label at the input port of node i

Oi wavelength label at the output port of node i

r(s, d,

D

) transmission request r from source s to destination d subject to delay bound

D

eij directed edge from node i to node j

eijl directed wavelength link of wavelength l on eij

c(eij) communication cost on eij

d(eij) transmission delay on eij

^cðiÞ wavelength conversion cost at node i ^dðiÞ wavelength conversion delay at node i

Tc(eijl) communication cost for routing from node i to node j

using wavelength l

Td(eijl) transmission delay for routing from node i to node j

using wavelength l

wi binary variable dictating whether node i provides

wavelength conversion or not; i.e., wi¼1, if yes; 0,

otherwise

l

ijl binary variable dictating whether wavelength link eijlis

feasible or not to represent the wavelength l in eijcan be

to be used to transmit data or not, i.e.,

l

ijl¼1, if yes; 0,

otherwise

A network is represented by a weighted graph G(V, E), where V is the set of switches and set E contains directed edges corresponding to the directed optical links among the switches. 9V9¼n denotes the number of nodes in the network. Binary variable wi indicates whether the node iAV is associated with

wavelength conversion, annotated by wi¼1 or 0. The directed

edge from node i to node j is denoted by eij. c(eij) and d(eij)

represent the communication cost and the transmission delay of edge eij, respectively. At node i, wavelength communication cost

and wavelength transmission delay are denoted by ^cðiÞ and ^dðiÞ,

respectively. The set of wavelengths available on the optical links is denoted by W with cardinality 9W9 ¼m as the number of different wavelengths. The m wavelengths on each eij can be

viewed as m wavelength links eijl, 1rlrm, to represent the

wavelength-based connections, where c(eijl)¼c(eij) and

d(eijl)¼d(eij). Therefore, when a light-path includes two

wave-length links eijl and ejkl0 (lal0), the switch j must provide the

wavelength conversion capacity such that the signal passing from eij to enter the input port of j using wavelength l can be

transmitted to switch k from the output port of j using wavelength l0_.

A request under a delay bound

D

is represented by r(s, d,

D

) indicating that there is data originating from source s to be routed to destination d and the transmission delay of the complete routing session from s to d must be smaller than or equal to the delay bound

D

. Each request may be different from any of the others in respects of different sources, different destinations, and different delay bounds, which usually are determined by its priority, degree of emergence or other criteria.

The DRWA-DC problem seeks to ﬁnd an assigned light-path P that consists of a sequence of connected wavelength links. Let variables Iiand Oirepresent the used wavelength labels at the

input and output ports passing through switch i, respectively. The transmission delay passing through switch i is jIiOij

m l m ^dðiÞ, where IiOi j j m l m

is 0 or 1 depending on whether Ii¼Oior not; that is, the

transmission delay ( ^dðiÞ) exists only in the case that the wavelengths are different between input and output ports (IiaOi). Considering wavelength conversion in general, the

communication cost Tc(eijl) and the transmission delay Td(eijl) of

using wavelength link eijlto node j could be calculated as follows:

TcðeijlÞ ¼ 1 if wi¼0 and laIi, cðeijlÞ þwi 9Iil9 m ^cðiÞ otherwise, 8 > < > : ð1Þ TdðeijlÞ ¼ 1 if wi¼0 and laIi, dðeijlÞ þwi 9Iil9 m ^dðiÞ otherwise : 8 > < > : ð2Þ

The formula is computed based on the used wavelength links to reduce the complexity. Therefore, the overall communication cost and the transmission delay incurred in the assigned light-path P are cðPÞ ¼PeijlAPTcðeijlÞ and dðPÞ ¼

P

eijlAPTdðeijlÞ,

respec-tively. The assigned light-path P will be a feasible solution for routing the request r(s, d,

D

) when the following three conditions are all satisﬁed:

(1) the origin of P is s;

(2) the destination node of P is d; and

(3) the transmission delay of P is no greater than the delay bound (i.e., d(P)r

D

).

Table 1

Comparisons of related research.

Delay bound

Wavelength conversion

Objective

Krishnaswamy and Sivarajan (2001)

No No Maximizing the number of connections and minimizing the number of required wavelengths

Shen et al. (2001) No Yes Minimizing the blocking probability

Garlick and Barr (2002) No No Minimizing the blocking connections

Qin et al. (2003) No Yes Maximizing the number of connections and reducing the number of required conversions

Ngo et al. (2006) No No Minimizing the blocking probability

Kavian et al. (2007) Yes No Maximizing the number of connections

(4)

In the rest of paper, for notational convenience, assigned light-path and feasible solution will be replaced with light-light-path and solution if no confusion would arise.

3. ACO design for DRWA-DC

In this section, we develop several features for the deployment of the ACO. The notation used in our ACO design is given in the following:

b number of ants

x

percentage of the b ants to distribute at s and d Ak set of nodes accessible to ant k

t

ijl initial pheromone on wavelength link eijl

t

ijl dynamic desirability measure (pheromone intensity) on

wavelength link eijl

Z

k

ijl static desirability measure about link eijl based on a

heuristic value for ant k pk

ijl probability that ant k moves from node i to node j using

wavelength l (i.e., using eijl)

Pk _{light-path traversed by ant k}

c(Pk₎ _{communication cost of P}k_{, cðP}k_{Þ ¼}P

eijlAPkTcðeijlÞ

d(Pk₎ _{transmission delay of P}k_{, dðP}k_{Þ ¼}P

eijlAPkTdðeijlÞ

The ACO is a family of meta-heuristics that are inspired by the natural optimization mechanism conducted by real ants. The general framework of the ACO algorithm is shown inFig. 1. In the ACO framework, the underlying environment for the ant colony to explore through is a directed graph, possibly with weights assigned to the edges. Therefore, a studied problem is usually represented by a weighted graph. The ACO system starts by distributing a set of artificial ants onto the graph. Each ant will construct a tour that corresponds to a solution to the original problem. When all ants attain solutions, they share their information via pheromone and then next iteration commences. The process is repeated until some pre-specified criterion is satisfied. The optimization mechanism of the above-mentioned process is carried out by two important features: state transition rules and pheromone updating rules. A state transition rule is used for an ant to determine which node it will visit next (Step 2.2.1). The pheromone updating rules dynamically updates the pheromone intensities (or, in simple words, the degree of preference) on the edges (Steps 2.2.2 and 2.3). For general discussion on the philosophy and design detail, the reader is referred toDorigo et al. (1991).

Although the ACO has been applied to deal with SWRA (Varela and Sinclair, 1999) and DRWA (Garlick and Barr, 2002), these proposed approaches do not work for the DRWA-DC problem. For example, the backtracking method for avoiding dead-ants in

Varela and Sinclair (1999)andGarlick and Barr (2002)cannot be

used in DRWA-DC because transmission delays need to be taken into account. The existence of delay bounds of requests stipulates the global pheromone updating rule to test whether some ants arrive at the destinations successfully qualifying by the delay bound. In this section, we propose and design an ACO algorithm that can produce approximate solutions with all the addressed realistic constraints incorporated.

3.1. Initialization of ACO

In our ACO design for the DRWA-DC problem, the initialization phase includes two parts, dispatching ants to nodes and initializing pheromone on edges. For the ﬁrst part, the trail of an ant can be viewed as a light-path. Therefore, it is reasonable to expect that an ant will start from the source and stop at the destination of the request. The optimal trail with the minimum communication cost indicates that it is an optimal routing light-path for the request. The strategy that ﬁnds trails from source towards destination is called forward searching. On the other hand, backward searching refers to the strategy starting from the destination. Combining both strategies, we can let the ants begin their searching sessions randomly at either the source s or the destination d. In this paper, parameter

x

is given to adjust the percentage of b ants to be initially dispatched to s; that is, the numbers of ants initially positioned at the source and the destination are

x

b and (1

x

)b, respectively.

For the second initialization task, applying some heuristics will provide informative guidance to determine the initial pheromone, and possibly shorten the time required by ﬁnding a near optimal or even optimal solution. Considering the objective of minimizing the total communication cost of routing a given request, the initial pheromone

t

ijlon each wavelength

link eijlis deﬁned as

t

ijl¼

0 if

l

ijl¼0,

1þP 1=cðeijlÞ

x A V

l

ixl=cðeixlÞ

if

l

ijl¼1: 8 > < > : ð3Þ

Recall the deﬁnition of

l

ijl¼0, which indicates that wavelength

link eijlis infeasible. In order to prevent an ant from traversing an

infeasible wavelength link, the initial pheromone of that wave-length link is set to 0 (Eq. (3)). If the wavewave-length link is viable, the initial pheromone will be set as in Eq. (3) to let a wavelength link with less communication cost have a higher intensity of initial pheromone than the others.

3.2. State transition rule

The state transition rule presented in this paper features the following two aspects. First, unlike the ACO research (Shyu et al., 2004b) proposed solutions through the exploration of the power

(5)

set of the vertex set, the solution in this paper will obtain a path of wavelength links from the source to the destination. Depending on whether the switches provide wavelength conversion or not, each ant can choose wavelength links using the same or different wavelengths to route data to the next switch. That is, the solution to DRWA-DC can be viewed as a sequence of wavelength links, and the preference information (including pheromone intensity and local heuristic value) is deposited on the wavelength links. Secondly, the local heuristic used in most of previous research is static (that is, the value will not change during the optimization process). Shyu et al. (2004b) deployed a dynamic heuristic to reﬂect the situation that the access preference for a wavelength link changes over time depending on which wavelength links have been already selected. Due to the above concerns, we modify the state transition rule deﬁning the probability that ant k at node i uses wavelength link eijlto route data to node j as follows:

pk ijl¼

1 if qoq0 and /j,lS ¼arg max r A Ak,t A W

flirttirtðZkirtÞ

b_g,

0 if qoq0 and /j,lSaarg max r A Ak,t A W

flirttirtðZkirtÞ

b_g,

lijltijlðZkijlÞ

b P r A AklirltirlðZ k irlÞ b if q Z q0, 8 > > > > > > > < > > > > > > > : ð4Þ where Akdenotes the set of accessible nodes for ant k to visit such

that no node can be traversed for more than once,

t

ijl is the

dynamic desirability measure about the access to the wavelength link eijl,

Z

kijl is the static desirability measure about the same

wavelength link based on a problem-speciﬁc local heuristic, and

b

is the parameter controlling the relative signiﬁcance between the two measures. Following the same line of reasoning in Eq. (3),

l

ijl

is added to each equation to guarantee that any infeasible wavelength link, i.e.,

l

ijl¼0, cannot be chosen.

The value of pk

ijlcan be decided according to a random number

q drawn from the open interval (0, 1). If q is less than a speciﬁed threshold q0, the wavelength link eijlwith the maximal product

l

irt

t

irtð

Z

kirtÞ

b _{is always selected (see Eq. (4)); otherwise, the}

wavelength link is selected according to the probability given in Eq. (4). That is, the state transition rule is a controlled trade-off scheme between the exploitation search and the exploration search of the problem space. Note that the probability value pk

ijl

depends on which wavelength link the ant uses to construct the light-path (trail) and that the transmission delay of the light-path is constrained by the delay bound. When the trail exceeds the delay bound, the value of

Z

k

ijlwill be set to be 0 in Eq. (5), which

will be deﬁned in the next paragraph. Therefore, such a wavelength link will not be selected in Eq. (4). It thus highly suggests that the quality and feasibility of a solution depend on the wavelength links selected; that is, the communication cost and the transmission delay of light-path reﬂect the quality of the solution found by some ant and whether the solution is feasible or not, respectively. The value of variable

t

ijl, which gradually

reﬂects the global preference for link eijl, is updated according to

the quality of the final solution constructed at the end of each cycle and will be described in the next subsection. Local preference is incorporated to reflect the objective of communica-tion cost minimizacommunica-tion subject to transmission delay bounds. When there is no feasible solution found by the ant colony, determining a feasible solution, if exists, becomes more crucial. Therefore, the local preference needs to reflect the status of whether a feasible solution has been found thus far. During the solution-seeking session, if no feasible solution has been encoun-tered, the local preference will center on how to find a feasible solution according to the transmission delays of links; otherwise, the aspect of communication cost is considered. Therefore, the value of variable

Z

k

ijl, which evaluates the local preference of ant k

for wavelength link eijl, changes dynamically and is given by

Z

k

ijl¼

0 if dðPkÞ þTdðeijlÞ4

D

,

l

ijl

TdðeijlÞ

if no feasible solution has been found,

l

ijl TcðeijlÞ otherwise, 8 > > > > > > > < > > > > > > > : ð5Þ where

Z

k

ijlcan be seen as the inverse value of transmission delay

(Eq. (5)) or the inverse value of communication cost (Eq. (5)), depending on whether the ACO system has explored some feasible solution or not. In the sequel, the proposed dynamic local heuristic favors the feasible wavelength link that has either minimum transmission delays or minimum communica-tion costs.

3.3. Pheromone updating rule

In the proposed system, we apply global and local pheromone updating rules as follows. First, at the end of each cycle we keep track of the best feasible solution Pbest_{and the worst infeasible}

solution Pworst _{encountered by the colony. Our idea is to}

encourage the ants to follow links in Pbest_{and avoid links in P}worst

in the following cycles. This idea is realized by reinforcing (respectively, lessening) the intensities of the pheromone cur-rently left on the wavelength links in Pbest_{(respectively, P}worst_).

The pheromone

t

ijlon wavelength link eijlis updated according to

the following global updating rule:

t

ijl¼ ð1

r

Þ

t

ijlþ

r

X k

t

ijlu

r

X k

t

00 ijl, ð6Þ where

t

_ijlu ¼ 1=TcðeijlÞ P e A Pbest 1 TcðeÞ if eijlAPbest, 0 otherwise, 8 > > < > > : ð7Þ and

t

00 ijl¼ TdðeijlÞ P e A PworstTdðeÞ if eijlAPworst, 0 otherwise : 8 > < > : ð8Þ

Parameter

r

A(0, 1) simulates the evaporation rate of the pheromone intensity and enables the algorithm to reduce the signiﬁcance of inferior links or forget the bad decisions previously made.

Secondly, we activate the local pheromone updating rule to shufﬂe the solutions and prevent early convergence, i.e., all the ants make the same decisions. The local updating rule is performed at the end of each step when each ant selects a new wavelength link eijl. The pheromone intensity on link eijl is

updated by

t

ijl¼ ð1

j

Þ

t

ijlþ

jt

ijl, ð9Þ

where

j

A(0, 1) is a parameter adjusting the current pheromone previously laid on eijland

t

ijlis the initial value of pheromone laid

on eijl. Note that the local updating rule decreases the pheromone

intensity on the link just visited by an ant and makes the selected links less attractive to other ants. The effect of the process will direct the exploration session of an ant toward the links that have not yet been visited by other ants.

(6)

3.4. Stopping criterion

Like other meta-heuristics, different types of stopping criteria can be used for the ACO. We cite the following four for the reader’s interest:

C1: The number of iterations is greater than a speciﬁed iteration limit.

C2: The execution time is longer than a speciﬁed CPU time limit.

C3: The averages of communication costs of ants in some consecutive iterations remain unchanged.

C4: The number of consecutive iterations in which no improvement attained on the incumbent solution is greater than a speciﬁed limit.

In this paper, we use the last one (C4) as the stopping criteria of our ACO design.

4. Computational experiments

This paper focuses on determining an assigned light-path of low communication cost such that switches in the network can be set up to route a request. To study the performance of the proposed approach, we designed and conducted a series of computational simulations. The scheme used in our simulation can be referred to Waxman (1988). In the scheme, there are n nodes randomly distributed over a rectangular grid with integer coordinates. In a network topology generated for experiments, each directed link from node u to node v is associated with the probability function p(u,v) ¼

l

exp( p(u,v)/

g

d

), where p(u,v) is the coordinate distance between u and v,

d

is the maximum distance between each two nodes, and

l

and

g

are control variables selected from interval (0, 1].

The communication cost from node u to node v is deﬁned by taking the integer value of the distance between them on the grid. The transmission delay is an integer randomly generated from interval [1, 5]. For each request r(s, d,

D

), s and d are generated in a random manner. The delay bound

D

must be reasonable, for otherwise it is unlikely for a feasible light-path to be found. To generate a request with a reasonable delay bound, we use the value according to the minimum transmission delay between s and d found by applying Dijstra’s shortest path algorithm (Dijstra, 1959), and set

D

to be equal to

w

times of the derived minimum transmission delay, where

w

is a control parameter dictating the tightness between delay bound and minimum transmission delay. In our experiments, we set

l

¼0.7,

g

¼0.7, and size of rectangular grid¼100 to simulate the networks with different numbers of nodes. Moreover, 15% of the nodes are equipped with wavelength conversion. As for the parameters of the ACO algorithm, preliminary experiments suggest

x

¼0.5,

b

¼1,

r

¼0.7,

j

¼0.9, and qo¼

a

¼0.5.

The experiments consist of three parts: (1) introduction of transmission delays to the ILP formulation (Chen and Tseng, 2003), (2) comparisons between the ACO algorithm and the ILP formulation, and (3) investigation on the number of iterations exerted in the ACO algorithm. The codes were written in C + + . The platform is a personal computer with an Intel P4 2.4 GMHz CPU and 1 GB RAM.

4.1. Introduction of transmission delays to the ILP formulation The ILP formulation used in the simulation to solve DRWA-DC is adapted from that proposed inChen and Tseng (2003). It was

implemented using the linear programming tool ILOG’s CPLEX 7.1. Three types of networks were tested: 40 switches (n ¼40), 50 switches (n ¼50), and 60 switches (n¼60), for each of which 200 different requests were randomly generated. Five wavelengths were provided for the networks. The delay bound was set to be

w

(

w

¼3.0, 2.0, 1.5, 1.4, 1.3, 1.2, 1.1) times of the minimum transmission delay between the source and the destination in each request. For each combination of values of

w

and network types, the elapsed run times, each of which are averaged over 200 requests, are summarized inTable 2. The experimental results suggest that the elapsed execution times increase sharply as the number of switches grows or the delay bound becomes tighter (i.e., smaller values of

w

). For example, when

w

¼1.1, the average execution time is more than 1380 s. Therefore, the ILP formulation cannot solve the DRWA-DC problem well when the number of switches is more 70 or the speciﬁed delay bound of a request is close to the minimum transmission delay.

4.2. Comparisons between the ACO and the ILP formulation In this part, we deﬁne the stopping criterion for the ACO algorithm to be that 2000 iterations are reached or the incumbent value is equal to the optimal one. The same experiment settings were also applied to observe the solutions found by the ACO approach. Experimental results are shown inTables 3–5for the networks with 40, 50, and 60 switches and different

w

values, respectively. The solutions found by the ILP formulation are used as the baseline for comparisons. In these tables, the ﬁrst two columns show the value of

w

and the number of ants. We kept track of the scenarios of the ACO algorithm at iteration 1000 and iteration 2000. Recall that the algorithm will stop before entering later iterations if it encounters an optimal solution. Consider the major column entitled ‘‘1000 Iterations’’. Four sub-columns summarize the computational statistics at the end of the 1000th iteration:

#Fea: number of requests for which feasible solutions are found;

#Opt: number of requests that are optimally solved; Dev: average communication cost deviation of the found solutions from the optimal ones; and

ET: average execution time.

Dev is deﬁned as follows: Dev ¼ P rcðP feas r ÞcðProptÞ=cðPoptr Þ 100% #Fea , ð10Þ

where cðPfeasr Þ and cðProptÞ are the communication cost of the

feasible solution Prfeas found at the end of some iteration in the

ACO algorithm and the communication cost of the optimal solution Poptr found by the ILP formulation, respectively. The

second part reports the results at the end of 2000 iterations. When the ACO algorithm ﬁnished processing 200 requests, we also keep

Table 2

Average execution time (s) for differentwvalues and different networks.

w n¼ 40 n¼ 50 n ¼60 3.0 0.216 0.369 1.987 2.0 0.583 0.682 3.496 1.5 1.463 1.691 9.889 1.4 2.245 4.192 19.382 1.3 3.664 9.758 55.751 1.2 4.679 10.746 79.421 1.1 32.789 395.394 1380.914

(7)

track the number of requests that have been optimally solved (column #Opt). The sub-columns Iter and ET contain the average number of iterations and the average execution time required to produce these optimal solutions. The last major column Non-Opt records information on those test cases for which no optimal solutions were found. Sub-column Iter records the iteration at which the best feasible solution was encountered.

From the numerical results, we have the following observations:

(1) For the case that

w

has a tight value, it is guaranteed to ﬁnd a feasible solution with less iterations or fewer ants. According to the following three sets of experimental results in

w

¼1.1 of Table 4, (i) #Fea¼190 in b¼ 20 and at the end of 1000 iterations, (ii) #Fea¼200 in b¼100 and at the end of 1000 iterations, and (iii) #Fea¼198 in b¼20 and at the end of 2000 iterations, the first and the second sets of results indicate that more ants can benefit to find feasible solutions. This is due to wider and diversified explorations within the solution space. Moreover, the second and the third sets of results demon-strate that execution with more cycles will have a higher probability of finding feasible solutions.

(2) When a request with a tight delay bound which is close to the minimum transmission delay, it seems to take less execution

time because the ants were soon trapped and because a tight delay bound diminishes the number of viable wavelength links. It is thus less possible to compose feasible solutions. This reasoning is evinced in the numerical results. For example, for

w

¼1.1 and ¼1.5 inTable 5, we have #Fea ¼196 and #Fea¼200, and ET ¼0.657 s and 1.483 s at the end of 1000 iterations for b¼60.

(3) According to the comparisons from Tables 2 to 5, the execution time of the ACO algorithm is not sensitive to the change of the number of switches and the tightness of delay bound; but the time required by the ILP formulation highly depends on the change of the two features. For example, the ET values of ACO for

w

¼1.1, n ¼40, 50, and 60 are less than 1 s, but the corresponding ET values of ILP are more than 32, 395, 1380 s. This demonstrates the robustness and superiority of the ACO algorithm for the DRWA-DC problem.

(4) Although a larger number of iterations and ants deployed in ACO can reduce the communication cost of feasible solutions, the long execution time may be inefﬁcient. The maximum average numbers of iterations to optimally solve optimally and non-optimally requests are 124 and 318 for n¼40, 123, and 440 for n¼50, and 169 and 420 for n¼60. Therefore, the stopping criterion adopts the combination of that a given number of consecutive iterations within which no improvement on solutions is attained and a given limited number of iterations,

Table 3

Results of 200 requests routing in 40 nodes (n ¼40).

w b 1000 iterations 2000 iterations Opt Non-opt

#Fea #Opt Dev (%) ET #Fea #Opt Dev (%) ET Iter ET Iter ET

3.0 20 200 156 1.49 0.654 200 169 0.70 1.045 205 0.463 619 4.218 40 200 169 0.71 0.992 200 179 0.35 1.503 177 0.763 516 7.807 60 200 178 0.37 1.132 200 184 0.26 1.604 128 0.767 390 11.236 80 200 178 0.33 1.467 200 182 0.26 2.146 121 0.980 434 13.944 100 200 185 0.26 1.190 200 190 0.15 1.655 92 0.854 344 16.870 2.0 20 200 153 4.39 0.558 200 163 3.74 0.906 229 0.403 429 3.124 30 200 149 3.17 0.760 200 161 2.02 1.281 202 0.451 471 4.708 60 200 164 1.45 1.046 200 174 1.00 1.668 166 0.714 264 8.052 80 200 168 1.65 1.341 200 173 1.29 2.082 138 0.820 390 10.171 100 200 172 0.98 1.346 200 180 0.62 2.149 133 0.941 405 13.021 1.5 20 199 152 3.73 0.438 200 161 2.31 0.748 174 0.235 513 2.868 40 199 161 2.34 0.699 200 170 1.69 1.090 160 0.406 357 4.964 60 200 160 2.69 0.943 200 169 1.94 1.534 149 0.521 242 7.055 80 200 172 1.81 0.985 200 178 1.11 1.558 123 0.589 389 9.393 100 200 172 2.21 1.104 200 177 1.97 1.817 96 0.561 253 11.483 1.4 20 200 152 5.43 0.420 200 155 4.28 0.732 130 0.170 342 2.667 40 200 168 2.54 0.569 200 171 2.40 0.943 112 0.269 392 4.917 60 200 159 3.20 0.952 200 167 2.23 1.581 156 0.559 364 6.753 80 200 171 1.95 0.882 200 179 1.25 1.420 117 0.556 335 8.779 100 200 172 2.31 1.088 200 177 1.53 1.804 110 0.605 337 11.031 1.3 20 200 159 4.79 0.352 200 168 3.01 0.593 163 0.210 270 2.604 40 200 173 2.71 0.432 200 180 1.58 0.682 119 0.286 201 4.247 60 200 171 2.93 0.669 200 177 2.62 1.062 115 0.399 142 6.167 80 200 170 2.57 0.818 200 175 2.10 1.349 79 0.353 577 8.316 100 200 184 0.76 0.757 200 186 0.60 1.138 86 0.472 184 9.987 1.2 20 200 164 4.19 0.349 200 169 3.59 0.549 132 0.182 189 2.549 40 200 167 3.39 0.465 200 177 2.14 0.764 135 0.301 366 4.331 60 200 170 2.93 0.645 200 175 2.34 1.077 102 0.334 301 6.276 80 200 173 2.04 0.783 200 176 1.87 1.303 81 0.346 173 8.325 100 200 178 2.10 0.845 200 181 1.03 1.346 83 0.435 267 10.026 1.1 20 199 180 1.96 0.204 199 182 1.90 0.318 69 0.102 69 2.502 40 199 178 2.07 0.315 200 182 1.75 0.517 72 0.157 328 4.161 60 200 184 1.85 0.382 200 188 1.21 0.596 72 0.243 52 6.127 80 200 185 1.56 0.430 200 188 1.34 0.694 62 0.259 43 7.515 100 200 186 1.57 0.464 200 191 0.77 0.737 68 0.328 181 9.421

(8)

which may be more reasonable. The comparisons about the number of iterations will be discussed in following section.

4.3. Comparisons of iterations

This part is dedicated to investigating the number of consecutive iterations within which no improvement is attained on solution values. Average experimental results of Dev, ET, #Fea, and #Opt of different

w

values (

w

¼3.0, 2.0, 1.5, 1.4, 1.3, 1.2, and 1.1) are shown inFigs. 2–7for different numbers of consecutive iterations (Iter¼200, 400, 600, 800, and 1000) and different networks (n ¼40, 50, and 60). According to the experimental results, the type of stopping criterion can provide the performance with less execution time and approximated deviation in average. The number of consecutive iterations could be determined by the response time and the number of ants. Nevertheless, deviation and execution time seem to be the reasonable factors. From different criteria, we make several observations:

(1) From the experimental results inFigs. 2–4, the value of Dev decreases steadily for the increase of the number of ants and the increase of the number of iterations. For example, inFig. 2, the average values of Dev are 7.38, 5.14, 4.52, 3.93, and 3.57%

for 200, 400, 600, 800, 1000 iterations in 20 ants (b ¼20), and are 4.82, 4.19, 3.84, 2.77, and 2.93% for 40, 60, 80, 100, and 110 ants in 200 iterations. More ants collaborate through a longer execution course would accumulate and share more knowledge (in the differentiation of pheromone densities over edges) through extensive explorations. Nevertheless, it is not clear which factor’s increase has impacts on the decrease the Dev values.

(2) According to the experimental results inFig. 5, the elapsed execution time is proportional to the numbers of ants and iterations. This is due to the fact that the algorithmic steps required in the ACO algorithm are proportional to the ant population and the number of cycles. Besides, the ACO algorithm needed a larger number of consecutive iterations and fewer ants seem to provide lower deviation and to take longer run time than the ACO algorithms that used a smaller number of consecutive iterations and more ants. For example, inFigs. 4and5(n¼60), the values of ET and Dev are 3.622 s and 3.84% in b¼40 and 1000 iterations, and 2.165 s and 4.85% in b¼130 and 200 iterations.

(3) For the approximate elapsed execution time, it is more likely for the ACO algorithm needed a larger number of consecutive iterations and fewer ants to construct feasible solutions than the ACO algorithms using less consecutive iterations and more ants. Nevertheless, for the opportunity of attaining

Table 4

3.0 20 200 169 0.98 0.614 200 174 0.61 1.010 139 0.367 832 5.313 40 200 172 0.73 0.922 200 181 0.46 1.455 122 0.585 572 9.743 60 200 178 0.49 1.204 200 185 0.31 1.868 123 0.883 739 14.014 80 200 181 0.46 1.408 200 185 0.30 2.223 99 0.979 694 17.562 100 200 181 0.44 1.618 200 185 0.26 2.543 80 0.907 835 22.724 2.0 20 200 152 3.78 0.621 200 156 2.21 1.089 129 0.256 706 4.045 30 200 156 2.21 1.017 200 165 1.28 1.697 166 0.609 747 6.825 60 200 164 1.50 1.342 200 174 0.79 2.009 168 0.880 515 9.561 80 200 167 1.89 1.516 200 176 0.87 2.397 145 0.958 625 12.951 100 200 170 0.94 1.762 200 175 0.80 2.825 109 0.907 378 16.248 1.5 20 200 143 8.61 0.618 200 149 6.71 1.084 142 0.247 506 3.527 40 200 151 5.50 0.948 200 158 4.67 1.655 143 0.418 367 6.308 60 200 154 3.88 1.160 200 161 2.95 2.066 120 0.492 535 8.566 80 200 152 3.82 1.665 200 159 2.69 2.875 123 0.740 461 11.154 100 200 155 2.98 1.865 200 161 2.48 3.271 108 0.785 489 13.532 1.4 20 200 142 8.71 0.587 200 153 7.02 1.013 171 0.286 274 3.379 40 200 156 6.11 0.843 200 161 4.11 1.457 123 0.369 521 5.950 60 200 162 4.98 1.024 200 170 3.56 1.732 144 0.541 427 8.479 80 200 165 4.27 1.362 200 172 3.03 2.183 145 0.788 422 10.752 100 200 165 3.51 1.379 200 170 2.44 2.452 85 0.574 411 13.091 1.3 20 198 155 6.75 0.478 199 164 5.80 0.826 161 0.285 231 3.287 40 200 158 7.11 0.751 200 166 5.24 1.291 140 0.393 519 5.675 60 199 162 5.16 0.960 199 172 3.87 1.649 157 0.648 322 7.800 80 200 170 3.74 1.004 200 175 2.67 1.688 112 0.545 299 9.692 100 200 164 4.00 1.274 200 169 2.75 2.265 90 0.510 435 11.830 1.2 20 197 154 6.80 0.432 199 165 5.63 0.739 163 0.259 269 3.003 40 200 161 5.82 0.622 200 171 4.72 1.018 137 0.328 304 5.084 60 199 166 5.74 0.772 200 173 4.34 1.333 106 0.369 380 7.514 80 200 169 5.11 1.042 200 173 4.66 1.743 111 0.512 294 9.625 100 200 171 4.23 1.109 200 174 3.76 1.927 77 0.450 188 11.813 1.1 20 190 169 4.37 0.347 198 178 4.61 0.547 150 0.247 370 2.972 40 198 170 4.56 0.488 199 176 3.68 0.830 106 0.279 272 4.878 60 198 182 2.43 0.545 198 187 1.52 0.847 102 0.426 199 6.908 80 199 181 2.03 0.660 199 183 1.86 1.074 70 0.378 61 8.571 100 200 185 2.73 0.620 200 186 2.39 0.978 53 0.299 188 10.003

(9)

optimal solutions, the situation is reversed. For example, for the results of the values of #Fea shown in Fig. 6 and Opt shown in Fig. 7 (n ¼60), ET¼2.165, #Fea¼198.7, and #Opt ¼150.6 for 200 iterations in b¼130, and ET¼2.171, #Fea ¼199.0, and #Opt¼ 148.9 for 400 iterations in b¼ 60. Nevertheless, the phenomenon is inconspicuous.

(4) Although it is hard to decide the most appropriate ant population and the number of iterations, the colony with a

greater number of ants seem to let the ACO algorithm ﬁnd feasible solutions in a more efﬁcient way. This could be attributed to the fact that the increase of ant populations can better facilitate the mechanism of information or knowledge sharing. To route the requests with less deviation and higher success probability, we suggest that the ant population may be set as the number of nodes in the network plus 20, and that the value of consecutive iterations is set as large as possible.

Table 5

3.0 20 200 152 1.40 1.147 200 159 1.04 1.847 220 0.681 731 6.371 40 200 156 0.93 1.793 200 168 0.62 2.870 236 1.262 597 11.313 60 200 158 0.92 2.288 200 168 0.64 3.751 168 1.373 592 16.237 80 200 168 1.11 2.550 200 177 0.37 4.020 161 1.704 621 21.847 100 200 174 0.47 2.643 200 181 0.32 4.098 133 1.768 603 26.297 2.0 20 200 133 4.83 1.015 200 144 3.12 1.768 237 0.597 668 4.778 30 200 141 3.65 1.473 200 150 1.94 2.579 187 0.802 500 7.910 60 200 146 2.51 2.018 200 157 1.92 3.370 194 1.190 411 11.331 80 200 149 2.61 2.445 200 159 1.90 4.172 185 1.479 475 14.618 100 200 150 1.90 2.961 200 164 1.52 4.781 213 2.058 382 17.188 1.5 20 200 140 7.15 0.869 200 151 5.66 1.416 223 0.491 465 4.268 40 200 149 6.18 1.246 200 159 4.77 2.020 210 0.730 468 7.025 60 200 158 4.21 1.483 200 164 3.35 2.464 165 0.831 498 9.905 80 200 158 3.52 1.803 200 169 2.16 2.872 177 1.140 441 12.311 100 200 163 2.78 1.958 200 172 1.72 3.085 179 1.273 380 14.214 1.4 20 200 139 7.61 0.784 200 152 5.95 1.298 252 0.500 315 3.828 40 200 150 6.33 1.081 200 160 3.37 1.801 199 0.639 522 6.445 60 200 153 4.66 1.506 200 161 4.21 2.460 188 0.841 302 9.144 80 200 165 3.51 1.473 200 174 2.27 2.323 173 0.981 377 11.307 100 200 156 3.84 2.065 200 171 1.98 3.367 210 1.498 789 14.388 1.3 20 199 143 6.40 0.550 200 155 4.31 0.927 187 0.297 330 3.098 40 200 148 6.12 0.718 200 158 4.18 1.221 154 0.352 252 4.488 60 200 159 4.59 0.848 200 164 3.86 1.453 130 0.418 376 6.167 80 200 165 2.74 1.073 200 172 2.05 1.657 156 0.640 237 7.907 100 200 172 3.03 1.083 200 178 2.21 1.682 143 0.704 277 9.594 1.2 20 199 146 6.61 0.458 200 156 5.01 0.773 163 0.227 336 2.711 40 199 158 5.40 0.636 200 167 2.81 1.020 151 0.352 570 4.402 60 200 160 4.18 0.822 200 167 2.74 1.381 142 0.447 367 6.111 80 200 162 3.69 0.975 200 165 3.11 1.703 98 0.381 340 7.936 100 200 165 3.79 1.135 200 169 2.59 1.951 112 0.568 425 9.493 1.1 20 193 167 2.77 0.366 196 170 2.74 0.595 127 0.251 225 2.548 40 196 170 2.77 0.506 198 174 2.39 0.827 113 0.317 258 4.239 60 196 170 2.74 0.657 197 177 1.96 1.076 114 0.469 196 5.748 80 198 178 2.31 0.650 199 181 2.08 1.030 77 0.360 262 7.415 100 198 179 1.65 0.763 199 189 1.02 1.146 132 0.693 106 8.936 0 1 2 3 4 5 6 7 8 20 Dev (%) Number of Ants (b) Iter=200 Iter=400 Iter=600 Iter=800 Iter=1000 30 40 50 60 70 80 90 100 110

Fig. 2. The values of Dev for different numbers of ants in 40 nodes (n¼ 40).

0 1 2 3 4 5 6 7 8 9 Iter=200 Iter=400 Iter=600 Iter=800 Iter=1000 120 Dev (%) Number of Ants (b) 30 40 50 60 70 80 90 100 110

(10)

5. Conclusions

In this paper, a meta-heuristic scheme based upon ant colony optimization has been proposed to compose approximate solu-tions to the DRWA-DC problem, which is already known to be computationally intractable. Some heuristics for the SRWA problem and the DRWA problem have been developed in the literature, but few of them have applied the ACO approach or have addressed the issue concerning delay bounds. In this study, we have designed and implemented an ACO approach for solving the DRWA-DC problem. To adjust the ACO approach to meet the speciﬁc characteristics of the studied problem, a wavelength-link-based graph is constructed for the ants to traverse on. The effectiveness and robustness of the ACO approach have been examined by extensive experiments. We have also implemented the ILP formulation as a baseline to study the performance of the proposed ACO approach.

The experimental results have clearly evinced that our proposed ACO algorithm can ﬁnd approximate solutions with average deviations of less than 4% from the optimal ones with an average elapsed execution time only about 0.1% of that required by an ILP formulation. Moreover, the ACO algorithm still works well in solving the DRWA-DC problem for large-scale networks, for which the ILP formulation fails to provide optimal solutions in a reasonable time.

The purpose of this paper is not to address the superiority the ACO over other meta-heuristic in solving the RWA problem. Our focus is set on addressing the applicability as well as the capability of the ACO algorithm in dealing with the DRWA-DC problem. Our study has not only extended the application areas of the ACO approach but also suggested a new viable method for coping with the complex optimization problems arising from the WDM domain. For further research, it is of potential interest to apply the ACO approach to solve the static RWA-DC problem or the logical network topology design problems. Besides, the multicast routing and wavelength assignment may be another interesting research direction.

References

Chen, M.T., Tseng, S.S., 2003. Multicast routing and wavelength assignment with delay constraint in WDM networks with heterogeneous capabilities—new ILP. In: Proceedings of the International Symposium on Communications (ISCOM), Tao-Yuan, Taiwan.

Chlamtac, I., Ganz, A., Karmi, G., 1992. Light-path communications: approach to high bandwidth optical WAN. IEEE Transactions on Communication 40, 1171–1182.

Dijstra, E.W., 1959. Anode on two problems in connexion with graph. Numerische Mathematik 1, 269–271.

Dorigo, M., Maniezzo, V., Colorni, A., 1991. Positive feedback as a search strategy. Technical Report, 91-016, Politecnico di Milano, IT.

Dorigo, M., Gambardella, L.M., 1997. Ant colonies for the traveling salesman problem. BioSystems 43, 73–81.

Garlick, R.M., Barr, R.S., 2002. Dynamic wavelength routing in WDM networks via ant colony optimization. In: Proceedings of the Third International Workshop on Ant Algorithms, Brussels, Belgium, pp. 250–255.

Green, P.E., 1992. Fiber-optic Networks, Cambridge, MA. Prentice-Hall.

Karasan, E., Ayanoglu, E., 1998. Performance of WDM transport networks. IEEE Journal of Selected Areas in Communications 16, 1081–1096.

Kavian, Y.S., Rashvand, H.F., Ren, W., Leeson, M.S., Hines, E.L., Naderi, M., 2007. RWA problem for designing DWDM networks—delay against capacity optimization. Electronics Letters 2 (43), 892–893.

Krishnaswamy, R.M., Sivarajan, K.N., 2001. Algorithms for routing and wavelength assignment based on solutions of LP-relaxations. IEEE Communications Letters 5/10, 435–437.

Kwang, M.S., Weng, H.S., 2003. Ant colony optimization for routing and load-balancing: survey and new directions. IEEE Transactions on Systems, Man, and Cybernetics—Part A: Systems and Human 33/5, 560–572.

Lin, B.M.T., Lu, C.Y., Shyu, S.J., Tsai, C.Y., 2008. Development of new features of ant colony optimization for ﬂowshop scheduling. International Journal of Produc-tion Economics 112/2, 742–755.

Lowe, E., 1998. Current European WDM development trends. IEEE Communica-tions Magazine 36/2, 46–50. 190 191 192 193 194 195 196 197 198 199 200 40 #Fea Number of Ants (b) Iter=200 Iter=400 Iter=600 Iter=800 Iter=1000 50 60 70 80 90 100 110 120 130 Fig. 6. #Fea for different numbers of ants in 60 nodes (n¼ 60).

100 110 120 130 140 150 160 170 180 40 #Opt Number of Ants (b) Iter=200 Iter=400 Iter=600 Iter=800 Iter=1000 50 60 70 80 90 100 110 120 130 Fig. 7. #Opt for different numbers of ants in 60 nodes (n¼ 60). 1 2 3 4 5 6 7 8 9 Iter=200 Iter=400 Iter=600 Iter=800 Iter=1000 Dev (%) Number of Ants (b) 40 50 60 70 80 90 100 110 120 130

Fig. 4. The values of Dev for different numbers of ants in 60 nodes (n¼60).

0 1 2 3 4 5 6 7 8 9 10 11 40 ET (sec.) Number of Ants (b) Iter=200 Iter=400 Iter=600 Iter=800 Iter=1000 50 60 70 80 90 100 110 120 130 Fig. 5. ET for different numbers of ants in 60 nodes (n ¼60).

(11)

McMullen, P.R., Tarasewich, P., 2006. Multi-objective assembly line balancing via a modiﬁed ant colony optimization technique. International Journal of Produc-tion Research 44, 27–42.

Ngo, S.H., Jiang, X., Horiguchi, S., 2006. An ant-based approach for dynamic RWA in optical WDM networks. Photonic Network Communications 11, 39–48. Qin, H., Zhang, S., Liu, Z., 2003. Dynamic routing and wavelength assignment for

limited-rang wavelength conversion. IEEE Communications Letters 7, 136–138. Ramamurthy, B., Mukherjee, B., 1998. Wavelength conversion in WDM

network-ing. IEEE Journal of Selected Areas in Communications 16, 1061–1073. Sabuncuoglu, I., Erel, E., Alp, A., 2009. Ant colony optimization for the single model

U-type assembly line balancing problem. International Journal of Production Economics 120/2, 287–300.

Saradhi, C.V., Gurusamy, M., Zhou, L., 2007. Distributed network control for establishing reliability-constrained least-cost lightpaths in WDM mesh net-works. Computer Communications 30, 1546–1561.

Shen, G., Bose, S.K., Cheng, T.H., Lu, C., Chai, T.Y., 2001. Efﬁcient Heuristic algorithms for light-path routing and wavelength assignment in WDM networks under dynamically varying loads. Computer Communications 24, 364–373. Shyu, S.J., Yin, P.Y., Lin, B.M.T., Haouari, M., 2003. Ant-tree: an ant colony system

for the generalized minimum spanning tree problem. Journal of Experimental and Theoretical Artiﬁcial Intelligence 15, 103–112.

Shyu, S.J., Lin, B.M.T., Yin, P.Y., 2004a. Application of ant colony optimization for no-wait ﬂowshop scheduling to minimize the total completion time. Computers and Industrial Engineering 47/2–3, 181–193.

Shyu, S.J., Yin, P.Y., Lin, B.M.T., 2004b. An ant colony optimization algorithm for the minimum weight vertex cover problem. Annals of Operations Research 13, 283–304.

Silva, C.A., Sousa, J.M.C., Runkler, T.A., da Costa, J.M.G.S., 2006. Distributed optimisation of a logistic system and its suppliers using ant colonies. International Journal of Systems Science 37/8, 503–512.

Sun, Z.G., Teng, H.F., 2003. Optimal layout design of a satellite module. Engineering Optimization 35, 513–529.

Udomsakdigool, A., Kachitvichyanukul, V., 2008. Multiple colony ant algorithm for job-shop scheduling problem. International Journal of Production Research 46, 4155–4175.

Varela, G.N., Sinclair, M.C., 1999. Ant-colony optimization for virtual wavelength-path routing and wavelength allocation. In: Proceedings of the Congress on Evolutionary Computation (CEC’99), Washington, DC, USA, pp. 1809–1816. Wang, D., Ip, W.H., 2005. Ant search based control optimisation strategy for a class

of chaotic system. International Journal of Systems Science 36, 951–959. Waxman, B.M., 1988. Routing of multipoint connections. IEEE Journal of Selected