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Optical Switching and Networking 7 (2010) 95–107

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Optical Switching and Networking

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

Topology and routing optimization for congestion minimization inoptical wireless networksIradj Ouveysi a,1, Feng Shu b,1, Wei Chen c, Gangxiang Shen d,1, Moshe Zukerman e,∗,1a EEE Department, The University of Melbourne, Victoria 3010, Australiab IMEC Nederland, The Netherlandsc Eindhoven University of Technology, The Netherlandsd 418 Hillview Dr. Apt 301, Linthicum, MD, 21090, United Statese Electronic Engineering Department, City University of Hong Kong, Kowloon, Hong Kong

a r t i c l e i n f o

Article history:Received 18 November 2009Accepted 18 February 2010Available online 23 May 2010

Keywords:Optical wirelessMixed integer linear programmingTopology controlRoutingSurvivable networks

a b s t r a c t

Optical wireless networks have appealing features such as very high broadband datarates and cost effectiveness. They represent a potential alternative to the last mile (firstmile) wireless access problem. However, they are also highly vulnerable to externaldisturbances such as adverse weather and building sway. In this paper, we develop robustand efficientmethods for outdoor opticalwireless networks by jointly considering topologyoptimization and survivability strategies. We propose linearized congestion minimizationschemes with working and protection paths (LCM–WP), in which a mixed integer linearprogram is formulated to choose the optimal working and protection paths for everyOD pair such that the network congestion is minimized. In particular, the objective isto minimize the maximum amount of traffic on the links. To solve realistically sizedproblems, we consider a restricted version of the LCM–WP, in which only limited sets ofcandidate working and protection paths are considered. A simple algorithm is developedto find candidate working and protection paths for each origin-destination (OD) pair.Implementation of our LCM–WP schemes demonstrates the efficiency of our approach interms of the number of constraints and solution time. It also shows that our approach isapplicable to realistically sized networks.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The technology of optical wireless, also known as free-space optics (FSO) communication, has been continuouslygrowing as an attractive solution for the provision ofhigh data rates over short distances in recent years [1–5].

∗ Corresponding author.E-mail addresses: [email protected] (I. Ouveysi),

[email protected] (F. Shu), [email protected] (W. Chen), [email protected](M. Zukerman).1 The work reported in this paper was partly conducted whenI. Ouveysi, F. Shu, G. Shen and M. Zukerman were with the Electrical andElectronic Engineering (EEE) Department, The University of Melbourne,Australia.

An optical wireless link consists of a transmitter, the prop-agation channel and a receiver. FSO links typically havea maximum transmission range of a few kilometers anddirect line-of-sight (LOS) is imperative in optical wirelesscommunications. Invisible, eye-safe light beams are usedbetween optical tranceivers to provide optical bandwidthconnections that can send and receive voice, video, anddata information [6].In an optical wireless network, connectivity between

end users is achieved using optical transceivers installedon windows, building rooftops or exterior walls. The limi-tations of imperative LOS and transmission range force FSOtransceivers to have direct links only with relatively close-by neighbors. This leads to typically low nodal degrees,and in turn, imposes constraints on the number of disjoint

1573-4277/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.osn.2010.02.002


96 I. Ouveysi et al. / Optical Switching and Networking 7 (2010) 95–107

Fig. 1. An optical wireless transceiver.Source: HCS Technologies [9].

Fig. 2. A fictitious optical wireless network illustration in an urban areaof Hong Kong.

paths between origin-destination (OD) pairs which are im-portant for network survivability.Commercially available optical wireless transceivers

(see Fig. 1 for a typical example) can provide high-speeddata rates in the range of 100 Mbps to 2.5 Gbps, and adata rate as high as 160 Gbps has been reported in demon-stration systems [7]. Potential applications of the opticalwireless technology include: last mile network access [8],enterprise connectivity, fiber backup, and disaster recov-ery. A fictitious urban optical wireless network in HongKong is illustrated in Fig. 2.However, the performance of optical wireless links is

highly vulnerable to external disturbances [10], such as:• atmospheric disturbances (e.g., fog, absorption, scatter-ing, scintillation), which could result in high bit errorrate (BER) and transmission delays [11,12];• sway of buildings, which leads to a difficulty in main-taining the strict line-of-sight requirement [3,12].

Consequently, the progress of widely accepting opticalwireless technology has been slow. Innovative technolo-gies and architectures are still required to provide solu-tions for high-speed, resilient and robust optical wirelessnetworks.Many methods have been proposed to increase the

accuracy and robustness of optical wireless links. Kwok

et al. [13] proposed a temporal domain diversity recep-tion scheme that reduces BER and improves the reliabil-ity of optical wireless transmission links. Such a schemeis especially important under atmospheric turbulence ef-fects. Ho et al. [14] introduced an alignment approach toimprove pointing performance using the geometrical de-pendence between cameras and transceivers in mid- andlong-range scenarios. In [15,16], Muhammad et al. pro-posed an FSO system with increased resilience to varyingweather conditions, by adopting suitable modulation andcoding schemes. Uysal et al. [17] analyzed the error per-formance for coded FSO communication systems operat-ing in atmospheric turbulence channels. O’Brien et al. [18]introduced a solid-state architecture and seven-channeltracking system to design integrated and accurate opticalwireless transceivers.It has been shown that optical wireless links can be

expected to be available for more than 99% of the time onlinks up to 1 km in length, and high performance radiofrequency (RF) provides backup connectivity in adverseconditions [19]. Hybrid system architectures that exploitthe complementary nature of optical wireless and RFtechnologies have attracted significant research efforts inrecent years [20–25]. In [20], experimental results of anFSO/RF network deployed in the five campuses of AnkaraUniversity were reported. Kukshya [21] characterized thelink performance of a hybrid FSO and millimeter wave(MMW) system under different weather conditions.In this paper, we address the issues of robustness

and efficiency of optical wireless networks by consideringthe problems of topology optimization and survivabilitystrategies in a joint manner. It is important to incorporatesurvivability strategies as well as optimal topology controlinto the network design to guarantee the high-speed datadelivery performance of optical wireless networks. Thisgives rise to new optimization problems related to FSOnetworks that are different from existing optimizations ofwireline optical fiber networks.The design of a dynamic topology controlwhich is adap-

tive to real-time network changes has been extensivelystudied for wavelength-division multiplexed (WDM) all-optical networks (see e.g., [26–29]). For example, a mesh-based traffic grooming ILP model for sub-wavelengthtraffic grooming overWDMnetworkswas developed and alightpath-based virtual topology was designed in [30]. Theimpact of number of transceivers at each switch node wasalso evaluated. However, for FSO networks, there are farfewer studies dedicated to topology design. In [31], Desaiand Milner extended the optimization problem of topol-ogy control to the domain of opticalwireless networks. Theauthors of [31] formulated the congestion minimizationproblem as a mixed integer program (MIP) to dynamicallyoptimize the network topology in response to link statechanges, and proposed various heuristics to find near op-timal topologies to solve the problem in an efficient man-ner. However, they did not consider network survivabilityin their topology optimization formulation.Network survivability is of paramount importance for

an optical wireless network owing to its many challengesin free space physical transmission [3,12,13]. In topol-ogy design for an optical wireless network, it is neces-sary to jointly consider it with survivability. Extensive


I. Ouveysi et al. / Optical Switching and Networking 7 (2010) 95–107 97

research has been performed for the survivability ofwired communication networks, which can be catego-rized into classes of predesigned protection and onlinerestoration [32–38]. Predesigned protection schemes areable to immediately change the route to pre-allocatedbackup paths when network failures on the workingpaths are detected. By contrast, alternative paths aredynamically established in dynamic restoration schemeswhen the working path is affected. Therefore, predesignedprotection schemes can react quickly to network failures atthe cost of reservingmore spare capacity. Dynamic restora-tion schemes are more efficient than protection schemesin terms of the utilization of network resources, but takelonger to recover network connections.Meanwhile, due to the popularity of GMPLS/MPLS net-

works, much research has been focussed on GMPLS/MPLSnetwork survivability. In [39], a simple path protectionmechanism based on a reverse notification tree structurewas proposed forMPLS networks. In [40], a survey onMPLSprotection methods and their utilization in combinationwith online routingmethods was presented. In [41], an ap-proach called design-based routing (DBR)was proposed forGMPLS networks to employ optimized paths computed of-fline to guide online path setups. Also, for virtual topologydesign in the context of aWDMoptical network, a compre-hensive ILPmodel was developed for a dynamic survivablenetwork jointly consideringworking andprotection capac-ity. For fast online reconfiguration, a heuristic based on LP-relaxation to release the constraint of integer values for thevariables of the original ILP model was also proposed [42].Despite the importance of network survivability and

its extensive investigation for a wired communicationnetwork, little work has been reported on designing ahighly reliable FSO network. In this paper, we focus onthis type of design. We employ the 1 + 1 protectiontechnique to support FSO network survivability. Although1 + 1 protection is well-studied for the traditional wirednetworks, the present study has several novel aspects.Firstly, in the traditional wired networks, a physicaltopology (e.g., a fiber network) that provides networkcapacity is often predefined. The 1 + 1 survivable designfollows the existing link trail of the physical topology tochoose link-disjoint working and protection path pairs foreach pair of nodes. In contrast, in the present study, inaddition to finding a pair of working and protection paths,we also aim to optimize the topology for the FSO network.Thus, the present problem is more challenging than thewired 1 + 1 survivable network design. Secondly, for thetopology design, the traditional approaches often do notconsider the physical length limitation of a topology link,which means that each node is not bounded by a nodaldegree [30,42]. In contrast, in the current design problem,the nodal degree of each node is constrained by thetransmission distance of an FSO link. (Only the neighboringnodes that are within the FSO link transmission distanceare allowed to establish direct communication links with anode.)In this work, we first simplify the optimization prob-

lem for topology control by linearizing the formulation of[31]. We then propose an optimal topology control strat-egy for survivable optical wireless networks by taking net-work survivability into account based on a path-oriented

predesigned protection approach. We consider only fail-ures associated with a single FSO link. Simultaneous mul-tiple link failures including node failures are beyond thescope of this paper. As we discuss below, an RF networkwill back-up the FSO network when multiple-link failureoccurs, which may happen in the case of heavy fog.The remainder of the paper is organized as follows. In

Section 2, the network model and the assumptions usedin this paper are described. The linearized formulationof the traditional problem of topology control based oncongestion minimization is discussed in Section 3. InSection 4, we first formulate the linearized congestionminimization problem with working and protection pathsfor optimal wireless networks, and analyze its complexity.We then introduce a restricted version of the problemin Section 5, in which only limited sets of candidateworking and protection paths are considered. We includein an Appendix a multi-hour extension of this restrictedproblem. We also propose an algorithm to find thecandidate working and protection paths for each OD pair.Numerical results and related discussions are provided inSection 6. The paper is concluded in Section 7.

2. Network model and assumptions

Consider a network of N nodes as a digraph G =〈V , E〉, where V and E designate the sets of nodes andarcs, respectively. In the context of an FSO network, anarc represents an FSO link which is a directed bit pipefrom a transmitting node to a receiving node. Such an FSOlink uses a laser diode to transmit and a photodetectorto receive multi-channel signals [43]. Therefore an FSOlink carries many wavelengths and its total capacity canbe used by many OD pairs simultaneously, each of whichcarries the traffic of many users.It is assumed that the traffic matrix R of size N × N ,

where its [s, d]th entry Rsd denotes the effective bandwidthfor OD pair [s, d], is known a priori. In Appendix A, we de-scribe a method to derive effective bandwidth. We fur-ther assume that traffic of a given OD pair that uses anarc along the path can also use a subsequent arc if ca-pacity is available there. Such an assumption requires thatfull wavelength conversion is available at any intermediatenode along the path. Availability of fullwavelength conver-sion is justified, as conventional FSO systems use optical-electronic-optical (OEO) conversion in the nodes [44], sowavelength conversion can be used for independent data-format transmission.

3. Linearized formulation for congestion minimization

The problem of congestion minimization has beenstudied in the context of logical topology design forwavelength-routed optical networks. Desai and Milner[31] extended the research to optical wireless networks,and formulated this problem as a non-linear MIP. To solvethe non-linearMIP problem, the authors of [31] consideredvarious heuristics, and sub-optimal solutions were found.However, the non-linearMIP can be in fact reformulated asa linear MIP that finds optimal solutions of the non-linear



MIP in significantly less computation time. We first definethe indicator function Iij as:

Iij =1, if nodes i and j can see each other,0, otherwise.

We introduce the following notation.• λsdij : the amount of traffic on arc eij from origin destina-tion (OD) pair [s, d].• bij: binary decision variable represents the selection ofarc eij.• ∆i: the degree constraint on node i.

In the following, we reformulate the problem of [31] as alinear MIP.Objective function: minimize zSubject to:1. Flow conservation at node i:∑

j

Iijλsdij −∑j

Ijiλsdji =

+Rsd, if i = s−Rsd, if i = d0, otherwise

∀i ∈ V ,∀[s, d].2. Bound on variable z:

z ≥∑[s,d]

λsdij , ∀(i, j) ∈ E.

3. Bound on traffic flow variables:

0 ≤ λsdij ≤ Rsdbij, ∀(i, j) ∈ E,∀[s, d].4. Constraint on degree of each node:∑

j

Iijbij ≤ ∆i, ∀i ∈ V .

5. Bound on decision variable bij:

bij ∈ 0, 1, ∀(i, j) ∈ E.

The objective function of the new formulation is linear,and the number of integer and linear variables remains thesame as in [31]. In such congestionminimization problems,the optimal topology and optimal routing are searchedsimultaneously. Throughout this paper, this scheme isreferred to as ‘‘linearized congestionminimization’’ (LCM).In LCM the objective is to minimize the maximum amountof traffic on the links.In practice, one might be more interested in finding the

optimal topologies given the set of possible routes for eachODpair. In thiswork,wepropose a simple algorithm to findsets of candidateworking and protection paths for each ODpair, and then the congestion minimization is performedto find the optimal working and protection paths such thatthe congestion of the entire network is minimized. We re-fer to this scheme as ‘‘linearized congestion minimizationwith working and protection paths’’ (LCM–WP). The LCMscheme will be used to illustrate the efficiency of our pro-posed LCM–WP scheme, and the results of the evaluationwill be presented later in Section 6.

4. Linearized congestion minimization with workingand protection paths

Herewe provide a formulation of the linearized conges-tion minimization problem with working and protection

paths. In the proposed scheme, our optimizationwill selectfor eachODpair, oneworking path and one protection path(which is arc-disjoint with the working path) such that theoverall selection of paths will globally optimize the objec-tive function which is to minimize network congestion.In the predesigned protection schemes, the working

and protection paths for a given OD pair can operate ineither ‘‘1 + 1’’ protection or ‘‘1:1’’ protection [36]. In the‘‘1 + 1’’ approach, data are sent on the working pathsand duplicate data are sent on the protection paths. Thereceiver can choose the stronger of the two signals. In‘‘1:1’’ protection, no traffic is transmitted on the backuproutes unless network failures are detected. In this work,we assume that our algorithms operate under the ‘‘1+ 1’’protection scheme. We introduce the following additionalnotation.

• eij: an arc between two nodes i and j.• λ

sd,Wij : the amount of traffic on arc eij of the workingpath.• λ

sd,Pij : the amount of traffic on arc eij of the protectionpath.• λij: total traffic on arc eij ∈ E.

We now formulate the congestion minimization problemwith working and protection paths. For a node-set D ⊆ V ,we define the cut induced by D in G as

δG(D) = eij ∈ E : i ∈ D, j ∈ V \ D

where δG(D) is the set of all arcs with one end-node in Dand other end-node in V \ D. Define:

φsdij =

1, if arc eij is an arc on the working pathfor OD pair [s, d],

0, otherwise

and

γ sdij =

1, if arc eij is an arc on the protection pathfor OD pair [s, d],

0, otherwise.

The following is our MIP problem:

minimize z

Subject to:For each OD pair [s, d] and for any [s − d] cut induced

by D ⊆ V in Gwe need to have:

χ sdλ (δG(D)) = Rsd (1)∀D ⊆ V : s ∈ D, d 6∈ D, φ 6= D 6= V

where for anyM ⊆ E,

χ sdλ (M) =∑eij∈M

λsd,Wij .

To ensure that at each [s−d] cut, only one arc is a memberof any working path for OD pair [s, d] traffic, the followingset of constraints is introduced:

χ sdφ (δG(D)) = 1 (2)∀D ⊆ V : s ∈ D, d 6∈ D, φ 6= D 6= V ,


χ sdφ (M) =∑eij∈M

φsdij .



Weneed to have a protection path aswell for OD pair [s, d].Similar to (1), for the protection path, we set the followingconstraints:

χ sdλ (δG(D)) = Rsd (3)∀D ⊆ V : s ∈ D, d 6∈ D, φ 6= D 6= V ,


χ sdλ (M) =∑eij∈M

λsd,Pij .

As in (2),

χ sdγ (δG(D)) = 1 (4)∀D ⊆ V : s ∈ D, d 6∈ D, φ 6= D 6= V ,


χ sdγ (M) =∑eij∈M

γ sdij .

For each arc eij, we have the following constraints forworking paths

λsd,Wij ≤ φsdij Rsd, ∀[s, d] ∈ G (5)

and the following set of constraints for protection paths

λsd,Pij ≤ γ

sdij Rsd, ∀[s, d] ∈ G. (6)

These constraints ensure that unless an arc has been se-lected to be a member of a path, no traffic flow could beassigned for that arc.Since the working and protection paths for an OD pair

[s, d] need to be arc-disjoint, we introduce the followingset of constraints:

φsdij + γsdij ≤ 1, ∀ [s, d], and ∀ eij ∈ E. (7)

The total capacity required on each arc eij is

λij =∑∀[s,d]

λsd,Wij +

∑∀[s,d]

λsd,Pij . (8)

The remaining constraints are:

z ≥ λij, ∀eij ∈ E. (9)∑j

bij ≤ ∆i, ∀i ∈ V . (10)

φsdij ≤ bij, ∀[s, d] ∈ G and ∀eij ∈ E. (11)

γ sdij ≤ bij, ∀[s, d] ∈ G and ∀eij ∈ E. (12)

λij ≥ 0, λsd,Wij ≥ 0, λsd,Pij ≥ 0, ∀eij ∈ E and ∀[s, d]. (13)

bij, φsdij , γsdij ∈ 0, 1, ∀eij ∈ E,∀[s, d]. (14)

Now we have formulated an MIP problem in which wejointly consider optimal topology design and routing foroptical wireless networks. The number of binary variables(φ and γ ) in this NP-hard optimization problem is of theorder ofN4. This is becausewe haveO(N2) binary variablesfor each arc and the total number of arcs is in the orderof O(N2) as well. Furthermore, the number of constraintspresented by (2) and (3) is O(2N−2N2). This is explainedby considering the fact that the total number of OD node

pairs is O(N2) and the number of constraints in (2) and(3) for each OD pair is O(2N−2)— this is because excludingthe origin and destination nodes we have N − 2 remainingnodes and total number of combinations of choosing allpossible subsets out of a set of N − 2 remaining nodesis 2N−2. This would lead to having a total number ofconstraints to be O(2N−2N2). We can conclude that suchan optimization problem is very hard to solve for realisticproblems where the value of N normally exceeds 20.Therefore, we consider a restricted version of it, which

we call the restricted MIP problem. In our restricted MIPproblem, we use a limited set of potential working andprotection paths, and subsequently decrease the numberof binary variables to the order of O(N2). This is due tothe fact that the binary variables in this path optimizationmethod are defined based onworking and protection pathsfor each OD node pair. This implies that the number ofsuch binary variables in our restricted case is in the orderof O(kN2) and since k is bounded, then the total numberof binary variables in the restricted case is in the orderof O(N2). Note that in both the arc-based and path-basedoptimization methods, we ignored the number of binaryvariables bij as the total number of such variables in bothcases is dominated by the other binary variables (φ and γ ).Also note that our paper addresses a pure FSO network

and provides optimal design solutions that can copewith afailure of any single arc. For places and situations whereit is expected that many arcs fail at once, e.g. heavyfog, a backup is necessary. One such possible backupcan be achieved by directional RF connections. These RFconnections are sensitive to different weather conditionsand are thus complementary to the FSO network. Forexample, the FSO network is sensitive to heavy fog butcan operate well in rain, while the RF is sensitive to heavyrain and operates well in fog. In case of heavy rain, wherethe RF connections cannot provide backup, still a singlearc obstruction may occur in the FSO network and ourproposed optimal design will guarantee survivability.

5. Restricted linearized congestion minimization prob-lem

In this section, we first describe our algorithm that findsthe candidate working and protection paths, and presentits complexity analysis. We then provide a formulationfor the restricted LCM–WP problem which searches theoptimal working and protection paths for each OD pair.Define• W sd: the set of working paths for OD pair [s, d].• wsd: an arbitrary path in setW sd.• P sd: the set of protection paths for OD pair [s, d].• psd: an arbitrary path in set P sd.

5.1. The k-shortest paths problem

The k-shortest paths (KSP) problem is generalized fromthe shortest path problem, and has been well-studied inrecent decades. Numerous algorithms have been proposedto find not one but several shortest paths [45–48]. In thiswork, we use the approach proposed in [47] to find thecandidates of working and protection paths for each ODpair. To find the shortest path, the Dijkstra’s algorithm isused as a main subroutine in the KSP approach proposedin [47].



5.2. Algorithm for finding candidate working and protectionpaths

We propose a new method to find a set of candidateworking and protection paths for each OD pair in a givennetwork. The purpose of the procedure is to identify anumber of potential primary and backup paths for each ODpair, from which our optimization scheme is able to selectthe optimal working and protection paths such that thetotal network congestion is minimized.Assuming it is required to find k candidate working

paths and k candidate protection paths for OD pair [s, d],we consider the following algorithm.

Step 1: Use Dijkstra’s algorithm to find the shortest pathfor OD pair [s, d];

Step 2: Disable all the arcs on the shortest path found inStep 1;

Step 3: Use the KSP algorithm to find k protection paths forOD pair [s, d], and store them to P sd;

Step 4: Disable all the arcs on all the k paths found in Step3;

Step 5: Restore the shortest path found in Step 1;Step 6: Use the KSP algorithm to find k working paths for

OD pair [s, d], and store them toW sd.

In our algorithm, the shortest path is always included asa working path candidate. In order to get high qualityprotection paths, the next k shortest paths (which are arc-disjoint with the shortest path) are allocated to the set ofprotection paths. Finally, the other k−1 shortest paths areobtained for the candidate working path by disabling allthe arcs of the protection paths. In thisway, it is guaranteedthat any path inW sdmust be arc-disjointwith a path in P sd.Without loss of generality, our analysis applies to the casewhere the numbers of candidate working and protectionpaths are different. For simplicity of notation, we considerhere the casewhere the numbers of candidateworking andprotection paths are the same and it is denoted k.Existing work on node disjoint paths (e.g. [49]), may

also be, generally speaking, used to obtain arc-disjointpaths. Since in FSO, the electronic/optical nodes are de-signed for a very high degree of availability, we are inter-ested here in arc-disjoint paths and not in node disjointpaths. Furthermore, given the limitations of imperativeLOS and transmission range for FSO links, the nodal degreesin FSO networks are typically not very high, and it may be-come infeasible to find, for example, two candidate work-ing paths and two candidate protection paths for each ODpair that are node-disjoint.The procedures of the algorithm are repeated for each

OD pair of a given network (digraph) to find the requirednumber of candidate working and protection paths. Thenetworks are randomly generated and we assume thatall arcs in a network are assigned the same weights. Anexample of the algorithm is illustrated in Fig. 3.

5.3. Analysis of algorithm complexity

In the following, we derive the complexities of ouralgorithm and show that it is polynomial. Letm denote thenumber of potential arcs in the network, i.e.,m = |E|.

Fig. 3. Algorithm for finding candidate working and protection paths.k = 2.

Step 1 of the algorithm has the complexity of Dijkstra’salgorithm implemented using the binary heap method,which has complexity of O(m + N logN) to obtain theshortest path between any pair of nodes in the network.Steps 2, 4 and 5 have a complexity of O(m). Steps 3and 6 are the more greedy parts of the algorithm, whichimplement repeatedly the shortest path algorithm manytimes. They have the complexity ofO(m+N logN+k) [47].Therefore, the overall complexity of the algorithm isO(m+N logN + k).

5.4. Formulating the restricted MIP problem

As stated in Section 4, our original optimizationmay notbe solvable for practically sized problems.Wepropose herea restricted constraint space, and formulate the restrictedMIP problem.It is beneficial to have a fixed number of practical

paths to be chosen as the set of candidate working andprotection paths. Consider a network that is currently inoperation. Every time when a new customer is added tothe network, or an existing customer changes its usagesignificantly, then we need to solve the problem again.And if we do not dictate the feasible sets of working andprotection paths, then according to the new circumstances,the algorithm may choose totally different working andprotection paths from the ones before the change. Suchdynamic routing implies the pre-emption of all (ormost) ofthe existing paths and establishment of new working andprotection paths at each iteration. This will cause serviceinterruption and incur a significant administrative burdenwhich is normally unacceptable to telecom companies.Furthermore, such an approach that gives freedom to thealgorithm to select the paths would not provide verysignificant cost reduction. Notice that in our approach onemay increase the sizes of the path sets to include goodpaths during optimization.The restricted congestion minimization problem with

working and protection paths in our approach can then beformulated as:Objective function:

minimize z (15)

The constraints of this optimization problem are describedas follows:

5.4.1. Flow conservation constraintsTo address the flow conservation constraints, we first

define Ω sdW and ΩsdP as the set of nodes that appear as the



second node on a path inW sd and P sd, respectively:

Ω sdW = v : (s, v) ∈ Wsdw ∈ W

sd.

Similarly, for the protection paths, we have

Ω sdP = v : (s, v) ∈ Psdp ∈ P

sd.

The flow conservation constraints for working paths are asfollows:∑v∈ΩsdW

λsd,Wsv = Rsd. (16)

Similarly, for protection paths,∑v∈ΩsdP

λsd,Psv = Rsd. (17)

If a path is selected as a working or protection path fora specific OD pair, the traffic flow on an arbitrary arc(FSO link) of this path is the same as on any other arcof the same path. Therefore, on any working path wsd =s, esv, v, evg , g, . . . , u, eud, d, where euv denotes the arcbetween node u and v, we have the following flow con-servation constraints for working paths:

λsd,Wsv = λsd,Wvg = · · · = λsd,Wut . (18)

Similarly on any protection path, psd = s, esv, v, evg , g,. . . , u, eud, d, we have,

λsd,Psv = λsd,Pvg = · · · = λ

sd,Pud . (19)

5.4.2. Bounds on traffic flow variablesThe following binary decision variable is defined:

φsdw =

1, if pathwsd is selected as the working pathfor OD pair [s, d]

0, otherwise.

and

γ sdp =

1, if path psd is selected as the protection pathfor OD pair [s, d]

0, otherwise.

Sincewe only choose oneworking path and one protectionpath for each OD pair, we use the following constraints:∑w

φsdw = 1 (20)

and∑p

γ sdp = 1. (21)

If aworking path is not selected, all the traffic flowdecisionvariables associated with that are set to zero:

λsd,Wuv ≤ φsdw Rsd, for all OD pair [s, d] (22)

where euv ∈ wsd ∈ W sd. Similarly, for protection paths,

λsd,Puv ≤ γsdp Rsd, for all OD pair [s, d] (23)

where euv ∈ psd ∈ P sd.

5.4.3. Bound on variable zThe total traffic amount on arc euv is given by:

λuv =∑[s,d]

λsd,Wuv +∑[s,d]

λsd,Puv , ∀ euv ∈ E.

The bound on variable z is given by:z ≥ λuv. (24)

5.4.4. Constraint on degree of each node:∑j

bij ≤ ∆i, ∀i ∈ V . (25)

5.4.5. Bounds on decision variablesThe decision variables are binary, thus we have

bij ∈ 0, 1, ∀ eij ∈ E. (26)

If a path wsd is chosen as the working path, φsdw is set toone and the decision variables of all the arcs on this pathwill be set to one. On the other hand, if a path wsd is notchosen as the working path, φsdw is set to zero, but it is stillpossible that working paths for other OD pairs include arceij. Therefore, we have the following constraint:

φsdw ≤ bij, ∀ eij ∈ wsd, (27)

and similarly for protection paths:

γ sdp ≤ bij, ∀ eij ∈ psd. (28)

As a matter of fact, the restricted problem provides asolution that bounds from above the optimal solutionof the MIP (since the restricted problem has a smallerfeasible region than itsMIP equivalent). Aswedemonstratein Section 6, the restricted MIP problem is solvable forproblems of realistic size.The level and the distribution of traffic tend to vary

in a telecommunications network. The traffic variabilitymay be partly attributable to the stochastic nature of cus-tomer demand and some of the variability may be due tochange in usage pattern over the course of a day. Duringworking hours, business usage dominates and in theevening residential usage dominates. A telecommunica-tion network should be designed to take into account themulti-hour traffic profile and to be able to carry the trafficover all hours in a cost-effective manner [50–55]. Obtain-ing a minimum-cost network for multi-hour traffic pro-file is a large-scale optimization problem. We provide themulti-hour networkdesignmodeling for our restrictedMIPin Appendix B.

6. Numerical evaluation and discussion

We now evaluate the proposed methods using numeri-cal examples. The results were obtained with CPLEX 6.0 ona 3 GHz Pentium 4 with 1 GB RAM.Networks up to 40 nodes with potential arcs (FSO

links) were randomly generated. The degree constraint ofnode ∆i is set to be six for all nodes in the networks.For a randomly generated network of N nodes, we alsorandomly generated N×N traffic matrices with uniformly[0, 1] chosen entries, each of which represents a possibleeffective bandwidth for OD pair [s, d]. The shortest pathsare determined based on the number of hops; in otherwords, all arcs in the networks are assigned the sameweights.



Table 1OD traffic matrix of a 15-node network.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 0 0.55 0.88 0.46 0.74 0.94 0.04 0.27 0.57 0.70 0.80 0.96 0.25 0.99 0.402 0.15 0 0.69 0.67 0.35 0.36 0.87 0.46 0.33 0.19 0.17 0.13 0.07 0.12 0.813 0.11 0.41 0 0.04 0.71 0.14 0.84 0.07 0.65 0.61 0.06 0.43 0.90 0.84 0.134 0.80 0.54 0.51 0 0.82 0.61 0.20 0.41 0.71 0.40 0.67 0.82 0.01 0.36 0.595 0.40 0.36 0.25 0.88 0 0.01 0.25 0.43 0.99 0.30 0.64 0.61 0.43 0.51 0.556 0.74 0.89 0.31 0.50 0.70 0 0.83 0.04 0.90 0.91 0.82 0.04 0.07 0.66 0.397 0.93 0.38 0.48 0.63 0.15 0.09 0 0.31 0.68 0.39 0.81 0.92 0.11 0.30 0.028 0.37 0.25 0.67 0.36 0.50 0.42 0.79 0 0.31 0.15 0.14 0.72 0.51 0.99 0.079 0.04 0.57 0.64 0.51 0.26 0.55 0.66 0.64 0 0.75 0.43 0.89 0.35 0.92 0.9710 0.39 0.10 0.30 0.74 0.73 0.55 0.65 0.75 0.05 0 0.12 0.54 0.46 0.04 0.6711 0.05 0.89 0.72 0.75 0.24 0.32 0.20 0.79 0.23 0.51 0 0.96 0.94 0.96 0.5412 0.23 0.31 0.42 0.40 0.49 0.26 0.66 0.31 0.27 0.82 0.93 0 0.10 0.42 0.0513 0.65 0.39 0.74 0.46 0.99 0.57 0.71 0.72 0.38 0.77 0.98 0.82 0 0.73 0.6214 0.90 0.17 0.34 0.15 0.39 0.67 0.38 0.38 0.22 0.56 0.04 0.14 0.84 0 0.0115 0.96 0.13 0.81 0.01 0.60 0.03 0.08 0.25 0.53 0.95 0.73 0.41 0.37 0.22 0

Fig. 4. The topology of a 15-node network example.

6.1. Comparison between LCM and the restricted LCM–WP

Fig. 4 shows the topology of an example network con-sisting of 15 nodes. The randomly generated traffic matrixof the network is shown in Table 1.Figs. 5 and 6 demonstrate comparisons between LCM

and our restricted LCM–WP schemes. Fig. 5 illustratesa comparison in terms of number of constraints whenN varies from five to 25. It shows that the number ofconstraints in the LCM grows exponentially as the networksize N increases. By contrast, the number of constraintsin our LCM–WP schemes increases almost linearly. Forinstance, when N is 20, the number of constraints inLCM–WP is less than one fifth of that in LCM.The solution time of the LCM–WP schemes is signifi-

cantly less than that of LCM. This is because the optimalroutes are searched only within the path candidates in therestricted LCM–WP schemes. Fig. 6 illustrates a compar-ison of solution time between LCM–WP and LCM. It canbe observed that the solution times are similar when thenetwork size is small (e.g., N = 5). When the networksize grows, the LCM scheme can take more than ten timeslonger than the LCM–WP schemes in most cases.

6.2. Validating optimality for problems of realistic size

Evaluation of the optimality of our results is a challengefor realistic size problems because the general MIP is not

Fig. 5. Comparisons of the number of constraints in LCM versus ourLCM–WP schemes (k = 2).

Fig. 6. Comparisons of the solution time of LCM versus that of ourLCM–WP scheme (k = 2).



Table 2The impact of k on the values of the objective function for N = 5. Here EB represents the maximum error bound on the objective function value.

N = 5 Results RMA OriginalInstance k = 2 k = 3 k = 4 k = 5

1 Z 0.89 0.89 0.89 0.89 0.753EB 0.00% 0.00% 0.00% 0.00% 0.00%

2 Z 1.07 0.8 0.8 0.8 0.756EB 0.00% 0.00% 0.00% 0.00% 0.00%

3 Z 1.79 0.72 0.72 0.72 0.69EB 0.00% 0.00% 0.00% 0.00% 0.00%

4 Z 0.98 0.86 0.86 0.86 0.691EB 0.00% 0.00% 0.00% 0.00% 0.00%

5 Z 1.53 0.89 0.89 0.89 0.776EB 0.00% 0.00% 0.00% 0.00% 0.00%

6 Z 1.25 0.85 0.85 0.85 0.722EB 0.00% 0.00% 0.00% 0.00% 0.00%

7 Z 0.77 0.77 0.77 0.77 0.657EB 0.00% 0.00% 0.00% 0.00% 0.00%

8 Z 1.47 0.94 0.94 0.94 0.817EB 0.00% 0.00% 0.00% 0.00% 0.00%

9 Z 0.86 0.86 0.86 0.86 0.71EB 0.00% 0.00% 0.00% 0.00% 0.00%

10 Z 1.53 0.99 0.99 0.99 0.64EB 0.00% 0.00% 0.00% 0.00% 0.00%

solvable for such problems. Therefore, in order to evaluatethe optimality of our results, we consider a relaxed versionof our linearized congestion minimization problem withworking and protection paths described in Section 4whichwe call the relaxed MIP. In our relaxed MIP we considerthe MIP of Section 4, but removing a set of constraints(7) that guarantees that the working and protection pathsfor an OD pair [s, d] are arc-disjoint. We know that asolution of the relaxed MIP provides a lower bound to ouroptimal solution. To estimate the solution of the relaxedMIP we use a modified version of our algorithm describedin Section 5 where we do not aim to achieve that theworking and the protection paths for an OD pair [s, d] arearc-disjoint. We call this modified algorithm the relaxedMIP approximation (RMA) algorithm.This procedure of the RMA algorithm is described as

follows.

Step 1: Use the KSP algorithm to find k working paths forOD pair [s, d], and store them toW sd.

Step 2: Disable all the arcs on the shortest path found inStep 1;

Step 3: Use the KSP algorithm to find k protection paths forOD pair [s, d], and store them to P sd.

In the RMA, the shortest path is always reserved as acandidate working path, and aworking pathmay share thesamearcwith a protectionpath. TheRMAalgorithmshouldnot be used to find optimal working and protection pathsfor an OD pair, as we do not have protection in a situationwhere failures happen to a common arc.The RMA algorithm for a large k value will find an op-

timal solution which approximates the optimal solution ofthe RelaxedMIP problembecause for large k values, almostall paths are allowed and there is no restriction of havingarc-disjoint paths. Running the RMA algorithm for a large kis not computationally difficult. Therefore, we increase thevalue of k until the marginal benefit diminishes, or untilthe running times are excessive. This provides an approxi-mation to the relaxed MIP problem. Then we compare the

Table 3The impact of k on the values of the objective function for N = 10.

N = 10 Results RMAInstance k = 2 k = 3 k = 4 k = 5

1 Z 2.87 1.76 0.98 0.98EB 0.00% 0.00% 0.00% 0.00%

2 Z 2.43 2.17 0.98 0.98EB 0.00% 0.00% 0.00% 0.00%

3 Z 3.02 1.05 1.05 1.12EB 0.00% 0.00% 0.00% 10.71%

4 Z 2.89 2.89 1.57 1.22EB 0.00% 0.00% 0.00% 9.02%

5 Z 3.07 1.19 0.99 9.9EB 0.00% 0.00% 0.00% 0.00%

6 Z 1.98 1.15 0.95 9.5EB 0.00% 0.00% 0.00% 0.00%

7 Z 2.1 1.08 0.99 0.99EB 0.00% 0.00% 0.00% 0.00%

8 Z 3.21 2.12 0.98 0.98EB 0.00% 0.00% 0.00% 0.00%

9 Z 2.98 1.01 0.95 0.95EB 0.00% 0.00% 0.00% 0.00%

10 Z 4.4 1.93 1.07 1.07EB 0.00% 0.00% 0.00% 0.00%

value of the objective function obtained by our original al-gorithm with the values obtained by the RMA algorithm.The results are presented in Tables 2–7. In these tables thevalues of the best objective function for each problem hasbeen provided with the corresponding error bound (EB)provided by the CPLEX optimizer package. EB is the maxi-mum error of the corresponding objective function value.Please note that the length of running time of the optimizerpackage for each problemwas less that 20 min in all cases.Table 2 demonstrates that in the cases considered, our

original algorithm does provide objective function valuesthat are close to those of the RMA algorithm. The resultsin Tables 2–7 demonstrate that our heuristic approachprovides better solutions (lower objective function values)as k increases. Please note that due to non-scalability of the





1 Z 8 5.26 5.26 2.95EB <1% <1% <1% 4.15%

2 Z 3.83 2.34 2.11 2.1EB <1% 1.28% <1% <1%

3 Z 1.78 1.62 1.56 1.26EB <1% 1.23% 21.15% 19.05%

4 Z 5.75 3.11 1.65 1.52EB <1% <1% 9.7% 1.97%

5 Z 3.86 1.93 1.79 1.85EB <1% <1% 21.79% 24.32%

6 Z 4.92 2.13 2.2 2.05EB <1% <1% 15.91% 17.56%

7 Z 4.79 3.56 3.59 2.54EB <1% <1% 1.21% 2.23%

8 Z 5.72 4.36 4.36 1.8EB <1% <1% <1% 1.85%

9 Z 5.21 2.2 1.76 1.63EB <1% 2.05% 15.48% 19.94%

10 Z 4.08 3.49 2.34 2.01EB <1% 5.44% 12.15% 12.08%



1 Z 6.16 5.3 3.82 3.55EB <1% <1% 6.37% 7.79%

2 Z 3.1 2.17 2.45 2.43EB <1% 1.23% 12.52% 11.8%

3 Z 8.19 6.07 6.07 6.07EB <1% <1% <1% <1%

4 Z 4.17 3.86 3.73 3.61EB <1% 5.18% 7.42% 9.76%

5 Z 8.93 4.18 4.18 4.19EB <1% <1% <1% 0.24%

6 Z 3.5 2.52 2.33 2.07EB <1% <1% 16% 16.84%

7 Z 9.07 5.4 4.6 3.71EB <1% 2.41% <1% 8.18%

8 Z 7.48 6.63 6.34 6.27EB <1% <1% 2.6% 1.52%

9 Z 6 3.79 3.58 3.43EB <1% <1% 7.91% 7.73%

10 Z 6.74 4.77 4.77 4.76EB <1% <1% <1% <1%

original problem we did not perform its implementationsfor larger N values as the number of constraints in theoriginal modeling is computationally prohibitive.It is also important to mention that in the case of the

40-node restricted MIP problem, the objective functionvalues corresponding to the feasible integral solutions pro-vided by the CPLEX optimizer package within the startingminutes of running time had an error bound value of nomore than 10%. This behavior was not observed for smallernetworks. This indicates a potential to develop heuristicsfor very large networks for which optimal solution is notobtainable. The results presented in Table 7 for the case ofa 40 node network (k = 40) particularly demonstrate thatour method is scalable to provide optimal solutions for re-alistic networks.



1 Z 5.49 3.84 3.37 3.2EB <1% 10.49% 12.93% 10.68%

2 Z 4.37 3.15 2.78 2.87EB <1% 11.24% 13.13% 15.85%

3 Z 6.29 5.8 5.42 5.31EB <1% 6.03% 7.51% 6.59%

4 Z 7.26 6.11 4.14 3.46EB <1% 1.64% 8.21% 14.16%

5 Z 4.88 4.82 4.39 4.25EB 2.25% 6.95% 8.7% 7.06%

6 Z 3.73 3.25 2.42 2.33EB 2.77% 13.7% 16.63% 16.68%

7 Z 4.44 3.89 3.02 2.93EB <1% 9.77% 12.14% 13.82%

8 Z 8.37 7.04 6.5 6.57EB <1% 4.71% 6.35% 7.35%

9 Z 4.23 4.1 3.47 3.18EB 1.13% 9.82% 11.77% 11.66%

10 Z 2.72 2.92 2.82 2.78EB 2.57% 12.7% 12.28% 12.1%



1 Z 16.35 13.39 10.43 9.47EB <1% <1% 1.2% 3.12%

2 Z 15.96 12.44 10.64 9.87EB <1% <1% 2.54% 3.04%

3 Z 7.75 6.77 6.23 5.92EB 1.31% 7.37% 8.8% 8.95%

4 Z 17.67 16.72 16.75 12.39EB <1% <1% 0.18% 1.13%

5 Z 12.88 12.76 12.61 6.49EB <1% <1% <1% 8.99%

6 Z 14.33 13.75 13.74 9.26EB <1% <1% <1% 3.13%

7 Z 7.22 6.45 5.97 5.22EB 2.94% 8.65% 10.94% 9.56%

8 Z 11.22 9.54 8.15 7.92EB <1% 4.01% 5.52% 7.23%

9 Z 9.61 8.61 8.67 6.29EB <1% 4.26% 4.92% 9.1%

10 Z 9.35 8.85 8.25 7.96EB <1% 3.56% 3.64% 5.8%

7. Conclusion

Wehave proposed an optimal topology control scheme,called LCM–WP, forminimizing congestion in optical wire-less networks, in which topology and routing optimizationstrategies are jointly considered.We then formulated a lin-earized congestion minimization problem in the form ofMIP that aims to select optimal working and protectionroutes for each OD pair such that the total network con-gestion is minimized.As the MIP is not scalable, we proposed to use a re-

stricted version of theMIP problemwhere the sets ofwork-ing and protection paths are limited, and introduced analgorithm that efficiently chooses candidate working andprotection paths for each OD pair in the network.We have evaluated the efficiency of our proposed

schemes in comparison to the LCM scheme. As compared



to LCM, in our LCM–WP scheme the number of constraintsis significantly lower and hence more tractable for realsize problems. The implementation results show that thevalue of the objective function decreases (improves) as kincreases (granting more routing flexibility).

Acknowledgements

The authors wish to thank HCS Technologies for theirpermission to allow us to publish the image presented inFig. 1. The authors also extend their thanks to Wendy Zuk-erman for providing the photograph of Hong Kong used inFig. 2. Thework of I. Ouveysi, G. Shen andM. Zukermanwaspartly supported by the Australian Research Council andthe work of F. Shu was partly supported by the NationalICT Australia, Victoria Laboratory. The work described inthis paperwas also partially supported by a grant fromCityUniversity of Hong Kong (Project No. 9380044).

Appendix A. Effective bandwidth evaluation

Analysis of measured traffic streams taken from a widerange of sources has shown that traffic streams exhibitLong Range Dependent (LRD) and self-similar character-istics in VBR video [56,57], Ethernet traffic [58,59], andmetropolitan area traffic [60]. It has been shown that theGaussianmodel accurately represents aggregation ofmanytraffic streams due to the central limit theorem [61]. Oneway to evaluate effective bandwidth is to use the resultsin [61] to derive an optimal service rate value indepen-dently for any given OD pair. We introduce the followingnotation.

• Ssd: a decision variable that represents a service rate forOD pair [s, d]which we aim to optimize.• Rsd: the effective bandwidth, which represents theminimal value of the service rate subject to a quality ofservice (QoS) constraint.• Dsd: the mean variable traffic that arrives for OD pair[s, d].• σsd: the standard deviation of the variable traffic thatarrives for OD pair [s, d].• msd: the mean of the net traffic input, i.e., msd = Dsd −Ssd.• t: a buffer content threshold (expressed in data units,e.g., packets), which is used to define buffer overflow.• Pof : overflow probability, which is defined as theprobability that the number of data units in the bufferexceeds a specified threshold t .• H: Hurst parameter.

The overflow probability in the LRD case based on theGaussian model is approximated by [61]:

Pof (t) =

√2πσsd

ψ(−msd, σsd)es∗(t)t (29)

where

s∗(t) = −12σ 2sd|1− H|−2

(H

|msd(1− H)|

)2Ht1−2H (30)

and

ψ(x, σsd) =σsd√2πe−x2

2σ2sd −x2erfc

(x

σsd√2

). (31)

It has been validated that (29) is accurate for a stationaryLRDGaussian queuewhere 0.5 < H < 1 [62].We consideronly the busiest hour in a day and thus assume the arrivalprocess is stationary.We define the following optimizationproblem to obtain the effective bandwidth Rsd.

1. Objective function: Rsd = min Ssd2. Subject to: Pof (t) ≤ εqos

where εqos is the preassigned QoS measure, e.g., εqos canbe set to 10−5. Since s∗(t) < 0, Pof (t) is a monotonicallydecreasing continuous function, the optimal value Rsd canbe obtained by bisection.

Appendix B. Multi-hour optimization model

While LRD traffic modeling is appropriate for di-mensioning links and evaluating effective bandwidth intimescales of micro seconds to a few hours, for longertimescales, metropolitan area networks traffic exhibitmulti-hour behavior [50]. This non-stationary periodictraffic behavior is a result of people movements. Whileintra CBD traffic and suburbs-CBD traffic peak during thebusiness hours, it almost diminishes in the evening whentraffic shifts to suburbs. A commonmethodology to dimen-sion such networks is based on multi-hour traffic profiles.Accordingly, wemay consider only two time intervals dur-ing a day. Namely, the busiest hour during business hoursand the busiest residential traffic activity.Wemay consideradditional time intervals if traffic intensity, for example, intheweekend, justifies their considerations. In the formula-tion below we assume a general number of time intervals.We define the following decision variables:

• λsd,Wij,t : the amount of traffic on arc eij of working pathswsd at time interval t .• λ

sd,Pij,t : the amount of traffic on arc eij of protection pathspsd at time interval t .• λtij: total traffic on arc eij ∈ E at time interval t .

Assume that end-to-end effective bandwidth require-ments are provided for Nt time intervals. This implies thatwe have Nt matrices denoted as

Rt =(Rtsd).

The multi-hour optimization is formulated as follows.

minimize z (32)

Subject to:



Flow conservation constraints

The flow conservation constraints for working paths fortime interval t are as follows:∑v∈ΩsdW

λsd,Wsv,t = R

tsd. (33)

Similarly, for protection paths,∑v∈ΩsdP

λsd,Psv,t = R

tsd. (34)

We have the following flow conservation constraints forany working pathwsd = s, esv, v, evg , g, . . . , u, eud, d:

λsd,Wsv,t = λ

sd,Wvg,t = · · · = λ

sd,Wud,t . (35)

Similarly we have the following flow conservation con-straints for any protection path, psd = s, esv, v, evg , g,. . . , u, eud, d:

λsd,Psv,t = λ

sd,Pvg,t = · · · = λ

sd,Pud,t . (36)

Bound on traffic flow variables

The following binary decision variables are defined:

φsdw,t =

1, if pathwsd is selected as the working pathfor OD pair [s, d] at time interval t .

0, otherwise.

γ sdp,t =

1, if path psd is selected as the protection pathfor OD pair [s, d] at time interval t .

0, otherwise.

Sincewe only choose oneworking path and one protectionpath for each OD pair, we use the following constraints:∑w

φsdw,t = 1 (37)∑p

γ sdp,t = 1. (38)

If aworking path is not selected, all the traffic flowdecisionvariables associated with that are set to zero:

λsd,Wuv,t ≤ φsdw Rsd, ∀ OD pair [s, d], (39)

where euv ∈ wsd ∈ W sd. Similarly, for protection paths,

λsd,Puv,t ≤ γ

sdp Rsd, ∀ OD pair [s, d]. (40)

where euv ∈ psd ∈ P sd.

Bound on z

The total traffic amount on arc euv is given by:

λtuv =∑[s,d]

λsd,Wuv,t +

∑[s,d]

λsd,Puv,t , ∀ euv ∈ E.

The bound on variable z is given by:

z ≥ λtuv. (41)

Constraint on degree of each node:∑j

bij ≤ ∆i, ∀i ∈ V . (42)

Bounds on decision variables

bij ∈ 0, 1, ∀eij ∈ E. (43)

φsdw,t ≤ bij, ∀ eij ∈ wsd. (44)

γ sdp,t ≤ bij, ∀ eij ∈ psd. (45)

z, λtuv, λsd,Wuv,t , λ

sd,Puv,t ≥ 0. (46)

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