Optimal Placement of FSO Relays for NetworkDisaster Recovery

Farshad Ahdi and Suresh SubramaniamDepartment of Electrical and Computer Engineering

George Washington University

e-mail: {ahdi,suresh}@gwu.edu

Abstract—Free Space Optics (FSO) relays can be used torecover a network which is partially disconnected due to naturaldisasters or terrorist attacks. Rapid and efficient recovery can beachieved thanks to FSO technology being wireless and providinghigh bandwidth. However, placement of such relays is a challeng-ing problem as FSO links greatly depend on weather conditions.In this paper, we find the minimum number of transceiversand their optimal placement which guarantees the recovery ofa certain fraction of network capacity in the worst weatherconditions and maximizes the throughput in the best weatherconditions through transceiver reconfiguration. The problem isformulated as an integer linear program (ILP) which takes thelink availability prediction as an input and guarantees fairnessto all existing traffic flows. To avoid the complexity of the ILP,an efficient probabilistic heuristic that computes the placementof FSO transceivers is proposed. We show through extensivesimulations that the heuristic performs within 12% of the optimalperformance.

I. INTRODUCTION

Access to highly reliable networks, and especially the In-

ternet, is essential for the success and continuity of businesses

and corporations these days. Natural disasters, terrorist attacks,

and emergency situations, on the other hand, are generally

unpredictable. A flexible solution which can be deployed

quickly if such phenomena damage the network infrastructure

and that can provide at least a fraction of the network capacity

is expected from network operators. Otherwise, access to the

outside world is compromised which may cost them thousands

of dollars even if it happens for a short period of time.

Recovery of the network can be accomplished quickly

using Free Space Optics (FSO) technology. For this purpose,

multiple FSO relays may be placed in the network to re-

cover the connections existing before a disaster. Despite being

wireless, which allows very fast deployment even on a tripod

for temporary purposes, FSO relays provide high bandwidth

which is comparable to their wire-line counterparts. Moreover,

the interference-free characteristic of FSO links provides high

scalability of the networks so that required bandwidth demands

can be met. In addition, unlike RF technologies which suffer

from security issues due to their broadcast nature, FSO links

are highly secure. Therefore, an appropriate placement of the

transceivers can guarantee recovery of the network at a fraction

of its capacity within a few hours.

The major drawback of using FSO technology is its relia-

bility, especially its dependence on weather. In other words,

different weather conditions (e.g., fog, cloud, air turbulence)

Fig. 1. Disaster recovery of network after earthquake using FSO underdifferent weather conditions (clear, cloudy, light rain, and dense fog).

may lead to failure of FSO links which makes the recovered

network vulnerable. For example, as can be seen in Fig. 1,

as the weather condition becomes worse the links become

shorter. One effective approach is to reconfigure the FSO

transceivers if any obscuration is predicted to happen. Re-

configuration of FSO transceivers to establish new links is

achievable in a relatively short period of time. We investigated

FSO transceiver reconfiguration to improve hybrid FSO/RF

network reliability in [1]. In [2], a topology control scheme

is proposed to make the FSO/RF network survivable under

hostile weather conditions. Bloom et al. discussed the suit-

ability and limitations of an FSO system depending on the

application and atmospheric conditions in [3]. Another barrier

against the widespread deployment of FSO network is the

costly transceivers. An integrated topology control and routing

scheme was suggested by Kashyap where the cost of FSO

transceivers is considered as a major constraint in their NP-

hard formulation [4].

In this paper, considering the fact that FSO transceivers

can be rapidly reconfigured [5], we suggest a new method

to find the optimal placement in a partially disconnected

network. Especially, we are interested in finding the strategic

locations for deployment with two main features. First, in the

worst weather conditions there exists an orientation for the

transceivers which guarantees at least an acceptable fraction

of network capacity (say 20%) for each connection. Second, in

the best weather conditions the overall network throughput is

maximized over all possible deployments for the same number

of transceivers.

978-1-4673-3122-7/13/$31.00 ©2013 IEEE

IEEE ICC 2013 - Optical Networks and Systems

3921

The contributions of this paper are two-fold. First, we

minimize the number of FSO transceivers through an integer

linear program (ILP) to avoid unnecessary cost. Then we find

the optimal placement of the transceivers such that the network

throughput is guaranteed in the worst weather conditions and

is maximized in the best weather conditions, if reconfigured.

Second, to overcome the complexity of the ILP, we propose a

simple and scalable heuristic to address the problem. Simula-

tions show the effectiveness of our heuristic when compared

with the optimal results.

The rest of this paper is organized as follows. In Sec. II,

we present the network model and in Sec. III, the problem

is presented and formulated as an ILP. We also present our

heuristic in this section. In Sec. IV, some experimental results

are presented to compare the optimal algorithm with our

heuristic. Sec. V concludes the paper.

II. NETWORK MODEL

We denote the network by a directed graph G = (V,E)where V and E represent the sets of nodes (transceivers)

and the links between them, respectively. After a disaster,

depending on the severity of the event some of the links from

E are failed and the remaining links are denoted by Eorg. In

this paper, we consider only link outages and assume they may

happen with a given probability. Node failure can be modeled

by assuming all links around the node are failed. The list of

notations used in this paper is shown in Table I.

We assume that FSO placement is done only on the existing

nodes of the graph, V . Each FSO node consists of at least one

transmitter or one receiver and depending on the number of

interfaces that it has, it can establish FSO links. Each pair of

FSO transmitter and receiver is capable of creating one link

which exists until the orientation of its transceivers changes

as decided by our algorithm. We denote the number of FSO

transmitters and receivers assigned to node v by txv and rxv ,

respectively. The vectors of Tx = (txv : v = 1, 2, . . . , |V |)and Rx = (rxv : v = 1, 2, . . . , |V |) will be determined later

through optimization.

In this paper, the disaster recovery problem is defined as:

find the minimum number of FSO transceivers, Nmin, and

their placement that allows fair routing of all connections

which existed before the disaster. In particular, we define

a solution is α−recovery if it guarantees fair routing of at

least a fraction α of all traffic flows (before disaster). The

origin-destination pairs can be any pair of nodes in V and the

traffic routing happens in a multihop manner. In this paper,

we use the terms origin-destination pair, connection, and flow

interchangeably. It is assumed that the traffic matrix, Ω, is

given where each connection ω has a traffic demand of dω .

For the sake of implementation, some RF transceivers might

be used for signaling. The link availability information for

each FSO transceiver is collected and routed through this RF

control plane to databases in central office for optimization

purposes. Therefore, all decision-making and optimizations are

done in a centralized manner and the computed transceiver

orientations are sent back to each node through the RF control

TABLE ISYMBOL DESCRIPTION

Symbol DescriptionG = (V,E) network graph = (transceivers, communication links)Eorg, Efso remaining links after disaster, potential FSO linkstxv, rxv number of FSO transmitters and receivers at node vA, B, R the set of active paths, blocked paths, remaining pathsβ, α ratio of net. capacity to that in clear, foggy weatherN ,Nmin number of TX (RX) used, minimum N for α−recoveryΩ = {S,D} set of origin-destination pairs = {origin set, dest set}dω , fω , F traffic demand, flow of connection ω, minimum flowλωe traffic flow of connection ω on link e

Ce nominal capacity of link eXe binary var for existence of FSO link e in the solutionS(t) t-th path schedule for the heuristic algorithmEB

fso, EWfso potential FSO links in the Best, Worst weather conditions

ϕω ,ϕiω all paths and i-th path of connection ω

plane. The physical layer and link reconfiguration framework

are described in the following section.

A. Physical layer

We assume coherent FSO systems which can be reconfig-

ured at a very high speed. Such transceivers can be tuned in

less than 1 ns, which is fast enough to build any physical

topology in a short period of time [5]. Therefore, the tuning

delay caused by switching the network topology is negligible.

We do not go into the details of the tuning procedure and

assume it is done before any reconfiguration.

The unpredictable attenuation of the laser beam in the

atmosphere changes the availability of the FSO link for a given

distance from the transmitter. Visibility information can be

used to predict the attenuation of the laser beam. For example,

the amount of attenuation can vary from 0.2 dB/km in very

clear weather to 350 dB/km in very dense fog [6]. Depending

on the requirement of the network operators on link availability

(e.g., 99.999%), the maximum range of an FSO link can be

found for a given transceiver. Using these maximum ranges,

the potential network topology graph (i.e., the graph consisting

of links that are shorter than the computed maximum range)

can be constructed. Let this set of links be denoted by Efso.

The nominal capacity of FSO transceivers using narrow laser

beams can be up to 2.5 Gbps over a distance of up to 4 km

[4]. In this paper, we consider a lower rate of 1.2 Gbps [7].

B. Routing and reconfiguration

In our ILP solution, we consider reconfiguring the network

if there is a change in weather conditions. Therefore, the

network configuration is fixed as long as weather conditions

do not change. Due to weather dependence of the FSO links,

their ranges change from nominal long range to very short

range for the same availability. Clearly, in the worst weather

conditions (e.g., dense fog), the links are short and as a result

the connectivity graph becomes sparser. In that case, the traffic

is routed through paths with more hops as opposed to in

clear weather conditions. The solution to our problem finds

the best configuration of FSO transceivers as a complement to

the surviving network links such that the network throughput

is fairly maximized. Moreover, it provides the routing of the

3922

traffic flows which needs to be followed in order to obtain the

expected throughput.

In our heuristic, we change the network topology dynam-

ically by reconfiguring the transceivers. For this, we assume

time to be divided into T slots during which we determine

the configuration of the networks. For each time slot we

provide a set of schedules which are repeated periodically to

realize our TDMA-based approach. These schedules determine

the orientation of the transceivers in addition to routing of

the flows and they are fixed until a change happens in the

availability of FSO links. In the following, we formulate the

problem based on these assumptions.

III. PROBLEM FORMULATION

In this section, the problem is stated and formulated as an

ILP. For the sake of brevity, a subset of constraints is combined

to form the convex polytope of the problem, and then the

formulation is presented.

Problem statement: Find the optimal placement and orien-tations of FSO relays for a partially disconnected networkgiven topology and link availability so that the networkis α−recovered in the worst weather conditions and per-connection throughput is fairly maximized in the best weatherconditions.

A. Convex polytope determination

The convex polytope of the problem is defined as a com-

bination of several linear constraints and is denoted by P.

This polytope is completely determined by its input variables

namely, G, Ω, and N . In the following, the constraints of this

problem are provided.

Flow constraint: Let us denote the portion of traffic from

connection ω carried by link e by λωe . For each node in the

network we have,

∑

e∈Eo(v)

λωe −

∑

e∈Ei(v)

λωe =

⎧⎨

⎩

fω v ∈ S−fω v ∈ D

0 o.w.(1)

where S and D are the sets of origins and destinations and the

sets of outgoing and incoming links from node v are denoted

by Eo(v) and Ei(v), respectively.

Link capacity constraint: For any existing link e in the

network, the total amount of traffic passing through it cannot

exceed its capacity denoted by Ce. Therefore, we must have∑

ω∈Ω

λe(ω) ≤ Ce. (2)

Since the actual locations of FSO transceivers are not

determined before solving the problem, let us assume the

binary variable Xe indicates the existence of an FSO link

in the resulting solution. In other words, it is 1 if an FSO

transmitter and a receiver are placed at the end-points of eand they are steered along it, and 0, otherwise. Therefore, the

capacity of link e in (2) is calculated from CeXe. It should be

noted that the capacity of FSO links are assumed to be given

as input to our problem. In the final formulation, we use the

compact inequality of∑

λe(ω) ≤ CeXe.

FSO placement: The number of outgoing (incoming) links

from a node is limited to the number of transmitters (receivers)

at that node. For the outgoing flows from node v, we have:∑

e∈Eo(v)

Xe ≤ txv (3)

where txv is the number of transmitters at node v. A similar

inequality can be obtained for the incoming links.

Tx/Rx number: The total number of transmitters and re-

ceivers is limited to N . Therefore, we have:∑

v

txv =∑

v

rxv ≤ N. (4)

The combination of the above constraints leads to the

convex polytope shown in the following box.

Def: Convex polytope [FΩ, TxV , RxV ] = P(G,Ω, N)

∑

e∈Eo(v)

λωe −

∑

e∈Ei(v)

λωe =

⎧⎨

⎩

fω v ∈ S−fω v ∈ D

0 o.w.(ω, v) ∈ Ω× V

∑

ω

λωe ≤ CeXe (ω, e) ∈ Ω× E

∑

e∈Eo(v)

Xe ≤ txv ,∑

e∈Ei(v)

Xe ≤ rxv v ∈ V

∑

v

txv =∑

v

rxv ≤ N v ∈ V

B. Optimal relay placement

Using the convex polytope notation in addition to the

fairness constraint which is described below, we can find the

minimum number of FSO transceivers for α−recovery.

Fairness constraint: In order to guarantee fairness to each

connection, the concept of weighted fairness is taken into

account. We assign the weights based on per-connection

demand. For example, given fω as the allocated rate (flow)

of connection ω, the normalized flow is fω/dω . We maximize

the minimum normalized flow over all connections denoted

by F . Therefore, we have

F ≤ fω/dω. (5)

To find the minimum number of FSOs, Nmin, the ILP

shown in Box # 1 can be solved on G1(E1, V ) where

E1 = Eorg ∪ EWfso. In the latter EW

fso denotes the set of

potential FSO links in the worst weather conditions. Solving

the ILP shown in Box # 2 using Nmin provides the optimal

placement of the relays which give α− and β−recovery in

worst and best weather conditions, respectively. In this box,

we have G2(E2, V ) and E2 = Eorg ∪EBfso. In the latter, EB

fso

denotes the set of potential FSO links in the best weather

conditions.

Box #1 Input : G1, α, Ω, dΩ Output : Nmin, Tx1V , Rx1

V

Nmin = arg min N

[F 1Ω, Tx1

V , Rx1V ] = P(G1,Ω, N) , dΩ × α ≤ F 1

Ω

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Box #2 Input : G1, G2, Nmin, α, Ω, dΩ Output : Tx2V , Rx2

V

max β

[F 1Ω, Tx1

V , Rx1V ] = P(G1,Ω, Nmin) , dΩ × α ≤ F 1

Ω

[F 2Ω, Tx2

V , Rx2V ] = P(G2,Ω, Nmin) , dΩ × β ≤ F 2

Ω

Tx2V = Tx1

V , Rx2V = Rx1

V

It should be noted that the solution to this problem

also provides us with the orientations of the transceivers and

rerouting of traffic flows, given the predicted link availability

and expected traffic demands at each node.

C. Probabilistic greedy algorithm

Solving the ILPs is quite computationally intensive, and

hence the ILPs are not scalable to realistically sized networks.

We present heuristic algorithms based on their relaxed versions

(converting integer variables to real) which are easy to solve.

1) Heuristic to estimate Nmin: In order to find Nmin,

we relax the ILP in Box # 1 by converting all integer

variables to real. This conversion makes the program solvable

in polynomial time at the price of providing solution variables

which are not necessarily integer. For example, let us assume

that Eo(v) = (3, 5, 20, 22) and the solution of the ILP

determines txv = 2 and Xe = (1, 1, 0, 0) for e ∈ Eo(v)which means two transmitters are to be placed at node v and

steered along links 3 and 5. Assume the solution of the relaxed

ILP gives us txv = 1.6 and Xe = (0.7, 0.3, 0.4, 0.2). As

a result, this solution does not determine the orientations of

the transmitters precisely; however, it gives some meaningful

information about the utility of the links.

One easy approach is to round up all elements of Xe which

requires four transmitters in the above node. This approach is

not efficient as most of the transceivers might be underutilized.

Another approach which can be used is to add one link,

or equivalently transmitter-receiver pair, at a time and then

solve the relaxed ILP again for the resulting new graph. Our

criteria to add the link is to use the information regarding

the utility of the links from the Xe vector. In particular, we

use a probabilistic approach where the utility of the links are

used as their weights for selection. For instance, in the above

example the chance of link 20 being selected is twice that for

link 22. After adding this link to the existing link set, Eorg,

and solving the relaxed ILP, we get a smaller value for Nmin.

We keep adding the links iteratively until the resulting Nmin

is close to zero or reasonably small. This approach is efficient

and its performance is shown by simulation.

2) Heuristic to find the placement: After estimating Nmin,

we need to find the placement of transceivers. We solve the

relaxed version of ILP shown in Box # 2 such that all integer

variables except Tx and Rx vectors are converted to real.

Clearly, the relaxed version of this ILP is much less complex

computationally as discussed later. The solution of this relaxed

ILP provides the exact locations of the transceivers as these

variables are integer. This solution, however, does not give us

the exact orientations of the transceivers due to Xe being real.

Similarly, we may consider the utilities of the links as the

probabilities of their transmitters being pointed in different

directions denoted by pe. For instance, we can steer the

transmitters in the above example along links 3, 5, 20 and 22for fractions of times (i.e., time proportional to 70%, 30%, 40%and 20%, respectively). Obviously, this approach uses the fact

that FSO transceivers can be steered as needed in a short

period of time to make a dynamic topology to realize these

fractions. We need to design an algorithm which utilizes the

links in a way such that the their utility fractions become the

solution of the relaxed ILP.

We adopt a TDMA-based approach and divide the unit of

time (e.g., one or a few seconds) into T slots during which we

determine the configuration of the network. In other words, we

find T schedules one for each time slot which determine the

status of each link in terms of being active or inactive. These Tschedules are repeated every unit of time to realize our TDMA-

based heuristic until a change happens in the availability of

FSO links. By this way, if a link is active for n time slots, the

utility fraction of that link becomes nT . The algorithm finds

a subset of links to be activated in each time slot and the

transceivers to be steered accordingly. The main constraint in

our algorithm is that the number of outgoing and incoming

links cannot exceed the number of transceivers at each node.

In order to ensure that each connection is routed properly,

we find the path for each origin-destination pair using a

subgraph of G for which we have λωe > 0. We find all possible

paths for connection ω denoted by ϕω = ∪ϕiω where ϕi

ω is

the ith path of connection ω. The utility of each computed

path can be easily found using the utility fraction of each link,

pe(ω), which is equal to λωe /Ce. Obviously, the utility fraction

of each link, for a given connection indicates the degree of

involvement of that link in our final solution. For example,

if we have p1(ω) = 0.3 and p2(ω) = 0.6, connection ωshould use one of the links twice as much as the other link. A

similar observation applies to the paths between each origin-

destination pair and this is the main idea of our heuristic. In

other words, we pick the possible path for each connection

based on their utility fraction weight at random. In order

to guarantee fairness to such connections we normalized the

utility fraction, p̄(ϕiω) = p(ϕi

ω)fωF . This enhances the chance

of lower demand nodes to compete fairly in the following

process. We use an iterative algorithm to find the selected

paths for a time slot at each stage.

In the following, we explain our algorithm to find the

schedule of the links for the first time slot and then we proceed

to the remaining time slots. At any point in the procedure the

set of paths not yet processed, which is called remaining set,

is denoted by R. This set is initialized by R = ∪ϕω and the

selection is performed at random with selection probability

proportional to the path utility fraction. The function which

performs this task is called RandomPick() in the algorithm

shown below. For example, if the utility fraction of the paths

in ϕω is (0.5, 1.5), this function may pick one of them with

probabilities of 0.25 and 0.75, respectively. The selected path,

say ϕiω , is added to the set of active paths denoted by A. The

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remaining set of paths is updated, R = R\ϕω , so that no

more paths from the same connection is activated in the same

time slot. Moreover, the set of paths from R which exceed the

number of transceivers at each node if they are selected, B,

are removed from R. This procedure is repeated among the

remaining links until R is empty.

So far, we are able to find the schedule of the links for

the first time slot. In order to ensure the highest throughput in

each time slot we maximize the minimum flow for the selected

connections using a simple max-flow algorithm. In order to

get the remaining schedules, we repeat the algorithm T − 1more times. Then the utility fraction of that path is updated

as follows,

ϕiω ∈ A : p̄(ϕi

ω) = p̄(ϕiω)−

f̄(ω)

fω(6)

where f̄(ω) is the allocated flow to connection ω at time slot

t. The algorithm is summarized in Algorithm 1. Using the

above algorithm, we might not be able to realize every link

utility fraction to get the optimal results. However, we show

later that the performance of the algorithm is quite close to the

optimal. It should be noted that the above algorithm must be

run for each weather conditions separately to find the proper

schedule for each one.

Algorithm 1: Probabilistic algorithm

begint←− 1for t ≤ T do

R←− ∪ϕω A←− ∅ B ←− ∅while R �= ∅ do

ϕiω = RandomPick(p̄, R)

A←− A ∪ {ϕiω}

B = {b|b ∈ R, b exceeds # RX or TX}R←− R\{ϕω , B}fω = MaxMinF low(A)

p̄(ϕiω) = p̄(ϕi

ω)− f̄(ω)fω

S(t)←− At←− t+ 1

3) Performance and complexity: In terms of complexity,

the first heuristic which is used to estimate Nmin includes a

few iterations each consisting of a linear program solved in

polynomial time. The number of iterations depends on the net-

work size and the recovery level, α. Based on our observations

the algorithm converges in close to Nmin iterations.

The placement heuristic is more complex as it needs solving

an ILP with a few integer constraints. However, the number

of integer constraints is significantly lower and it is of O(|V |)while that for the original ILP is O(|V |+ |E|). The algorithm

part is also simple with the complexity of O(|E|T ) and it is

also polynomial in time. T directly impacts the complexity

of our heuristic and it also plays a key role in terms of

performance. As T increases, a higher granularity of link

activity periods is achievable. Therefore, we can get closer

to the utility fraction of the links computed from the LP,

and achieve throughput values close to the optimum. Such an

improvement is expected to diminish as T increases beyond

a certain value. There is also a practical consideration in the

choice of T . A very large value of T may require too frequent

switching of the transceivers which might cause a reduction

in bandwidth efficiency.

IV. SIMULATION RESULTS

We assume that a disaster (e.g., earthquake) damages a large

part of the network as shown in a randomly generated graph

in Fig. 2. In this figure, we only show the potential FSO links

which can be established by steering the transceivers in those

directions. Depending on the weather conditions the range of

the FSO links varies and it can be predicted from visibility

information. To model disruption due to weather, International

Visibility Code is used. For example, the link range for a given

FSO transmitter, under thick fog, moderate fog, thin fog, light

haze, and very clear weather conditions for 99.999% avail-

ability are assumed to be (330, 610, 880, 1020, 1300) meters

for a commercial system [6]. As can be seen in Fig. 2, in

clear weather each node has the chance to communicate with

more nodes and the potential graph is denser. As the weather

conditions change to light haze, light fog, and moderate fog

most of the longer range links become unavailable. As a

result, the graphs become sparser and in the worst case, thick

fog, the network is mostly disconnected (not shown). For the

simulation, we consider the moderate fog as the worst case

and very clear as the best case. Clearly, this assumption does

not change the problem formulation and any two conditions

may be considered for optimization.

For the simulation, we generate 15 graphs at random, each

consisting of 35 nodes which are distributed randomly in a

square of side 3 km. We consider 10 randomly selected pairs

among the non-neighboring nodes as origins and destinations

of the traffic flows. The traffic demand is selected at random

in the range of [60, 90] Mbps. In our simulation, we consider

T = 50 so the frequency of topology change is 50 Hz.

We assume that the capacity of each link before disaster is

comparable to that of an FSO link which is assumed to be 1.2Gbps [7]. All links are assumed to have the same capacity [4].

Fig. 2. Solid lines are the remaining links from the original graph and dashlines are the potential FSO links to recover the network affected by earthquakeunder different weather conditions.

3925

α α α

(a) (b) (c)

Fig. 3. (a) Computed average number of FSO transceivers for α−recovery, Nmin, in the worst case weather conditions. (b) The fairness index for ourheuristic. (c) Average throughput (Mbps) per connection.

In order to simulate the effect of a widespread disaster such as

earthquake we assume the majority of the links, say 80% (used

in our simulation), are failed at random and only the remaining

links can be used in addition to FSO links. CPLEX 12.2 is

used to solve the integer linear programs. Figure 3(a) shows

the average Nmin computed for α−recovery of the networks

where α ranges in [0.1, 0.4] in the steps of 0.1. In this figure

OPT and HRS stand for the optimal and heuristic, respectively.

As it can be seen, the required number of FSO transceivers

increases with α similarly for both OPT and HRS. Based on

this figure, in all cases the number computed by HRS is almost

two units larger than for OPT which is only around 12%; this

shows the efficiency of our heuristic.

In order to evaluate the fairness of our heuristic we em-

ploy Jain’s fairness index defined as follows. For a vec-

tor of n elements, z, the fairness index is calculated by

η(z1, z2, ..., zn) =(∑

zi)2

n∑

z2i

where zi is the resource share for

the ith element [8]. For complete fairness this index is equal

to 1, and to 1/n, for complete unfairness. We generalized

this index for non-equal demands for each node so that

we have zω = fω/dω . Figure 3(b) shows the calculated

fairness index for our heuristic for both cases of clear and

moderate fog weather conditions. The error bars are used to

show the variability of this index for different graphs and

they are calculated from the standard deviation. Variability

of this index, however, is very small in our simulation. For

our heuristic, the fairness index is always above 0.67 as can

be seen in this figure. This index shows different trends for

different weather conditions. Loosely speaking, it is higher for

moderate fog weather conditions when α is small and vice

versa when weather is very clear.

Figure 3(c) shows the average per-connection throughput

obtained by OPT and HRS. As can be seen, the effective

throughput increases with α almost linearly for the HRS

similar to OPT in moderate fog, which is expected. The reason

is that if the network is α-recovered it guarantees α fraction

of throughput before disaster for every connection. In case

that the network is 2α-recovered we expect the average per-

connection throughput doubles, which is verified in the figure.

The gap between them, however, increases with α and in

the worst case (α = 0.4) it is around 80% of guaranteed

value. Another observation from this figure is that the effective

throughput achieved in the clear weather conditions by HRS

is generally higher than that obtained by OPT. The reason is

mainly due to the gain we achieved through reconfiguration of

the FSO transceivers which was not the case for OPT. Also,

the ratio of the throughput obtained by HRS in clear weather

conditions to that obtained for moderate fog ranges from 1.7at α = 0.4 to around 4 at α = 0.1, which is significant.

V. CONCLUSION

In this paper, we proposed the problem of optimal place-

ment of FSO relays for network disaster recovery using the

minimum number of transceivers, and presented a solution.

We took the weather dependence of FSO links into account to

find the strategic locations which guarantee a given network

throughput in the worst weather conditions. Moreover, the

network throughput is maximized under the best weather

conditions if the FSO transceivers are reconfigured in the

appropriate directions. We formulated the problem in the form

of an integer linear program and proposed a heuristic to avoid

its complexity of computation. We showed that the proposed

heuristic is efficient and its performance is close to optimal.

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