Analyzing the robustness of an array of wireless access points to mobile jammers

Accepted Manuscript Analyzing the Robustness of an Array of Wireless Access Points to Mobile Jammers David Schweitzer, Ruholla Jafari-Marandi, Hugh Medal PII: DOI: Reference:

S0360-8352(17)30083-9 http://dx.doi.org/10.1016/j.cie.2017.02.022 CAIE 4652

To appear in:

Computers & Industrial Engineering

Received Date: Accepted Date:

9 November 2016 27 February 2017

Please cite this article as: Schweitzer, D., Jafari-Marandi, R., Medal, H., Analyzing the Robustness of an Array of Wireless Access Points to Mobile Jammers, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/ 10.1016/j.cie.2017.02.022

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Title Page

Analyzing the Robustness of an Array of Wireless Access Points to Mobile Jammers

David Schweitzer*a

Ruholla Jafari-Marandib

Hugh Medalc

Department of Industrial & Systems Engineering, Mississippi State University, PO Box 9542, Mississippi State, MS 39762, United States of America a: [email protected] b: [email protected] c: [email protected] *: corresponding author

Analyzing the Robustness of an Array of Wireless Access Points to Mobile Jammers February 24, 2017

Abstract We present an approach for measuring the vulnerability of a wireless network. Our metric, nRobustness, measures the change in a network's total signal strength resulting from the optimal placement of n jammers by an attacker. Toward this end, we develop a multi-period mixedinteger programming interdiction model that determines the movement of n jammers over a time horizon so as to minimize the total signal strength of users during a sustained jamming attack. We compared several solution approaches for solving our model including a Lagrangian relaxation heuristic, a genetic algorithm, and a stage decomposition heuristic. We tested our approach on a wireless trace dataset developed as part of the Wireless Topology Discovery project at the University of California San Diego. We found found that the Lagrangian approach, which performed best overall, nds a close-to-optimal solution while requiring much less time than solving the MIP directly. We then illustrate the behavior of our model on a small example taken from the dataset as well as a set of experiments. Through our experiments we conclude that the total signal power follows a sigmoid curve as we increase the number of jammers and access points. We also found that increasing access points only improves network robustness initially; after that the benet levels o. In addition, we found that the problem instances we considered have an n-Robustness of between 39 and 69 percent, indicating that the value of the model parameters (e.g., number of jammers, number of time periods) has an eect on robustness.

Keywords : networks, mixed-integer programming, security, interdiction, location analysis 1

Introduction

The goal of this paper is to develop a new computational tool for measuring the vulnerability of a wireless network. By considering a set of wireless access points (signal senders), demand points (signal receivers or users), the connectivity between the two, and how the loss of a connection can lead to the reassignment via a second connection, we examine the robustness of a wireless network's performance in the event of an attack on the network by a set of jamming devices in which the goal of the attacker is to minimize the total signal strength. We develop a multi-period mixed-integer programming (MIP) model that minimizes the total signal strength over a time horizon which takes

1

into account the mobility of demand points. We then dene a metric, how robust a network is when under attack by

n

n-Robustness, for determining

jammers. This metric, computed using our MIP

model, measures the ratio of the total remaining signal power on a network because of jamming against its total signal power when jammers are not present.

1.1

Motivation

As wireless networks require limited necessary infrastructure, needing only access points to establish the connections, it is important to defend them from attacks.

The rst step to increasing the

robustness of a system is understanding its weakness and proneness to attacks. In many contexts, wireless service is provided by a set of spatially-distributed access points, or transmitters.

In these contexts, the loss of one of these access points can result in the loss

of service for a large number of demand points.

This paper examines this idea by studying the

optimal placement of jammers within the service area of a set of access points in order to minimize its connectivity.

Understanding strategies for eective attacking will provide insights as to the

robustness of a given network. These insights can lead to the increased defense of the network by examining or re-examining the locations of signal access points.

1.2

Relevant Literature

Technological aspects of wireless communication security have been well studied, and they comprise descriptions of jamming attacks (Noubir et al., 2003, Law et al., 2009) and the strategies against jamming attacks such as instance channel-hopping (Navda et al., 2007) and spread spectrum techniques (Liu et al., 2010). Also, studies related to anti-jamming protocols for dierent network layers (Awerbuch et al., 2008, Richa et al., 2011) and key management (Du et al., 2004) are other focuses in the literature. Operational considerations have also been studied, such as methods for responding to a jamming attack (Jiang et al., 2009) and creating ecient approaches toward determining the placement of one or a set of jammers (Pelechrinis et al., 2009, Feng et al., 2014, Vadlamani et al., 2014, Commander et al., 2009). wireless networks from the

Each of these technological studies investigates the security of

micro-level. In this paper we take a macro-level approach.

The jamming placement problem is related to the network interdiction problem, which has been been extensively studied.

There are studies to interdict networks in order to achieve objectives,

such as maximizing the shortest path (Ramirez-Marquez, 2010 ,Cappanera et al., 2011), minimizing the maximum ow (Aksen et al., 2014), and, similar to this study, minimizing network connectivity (Nguyen et al., 2013).

Interdiction models have applications in a variety of areas, such as drug

smuggling prevention (Wood, 1993), drug-related law enforcement (Malaviya et al., 2012), waterway commodity ows (Baroud et al., 2014), vulnerability analysis of power grids (Salmerón et al., 2009), and interdicting nuclear plans projects (Brown et al., 2009, Morton et al., 2007). Recently, researchers have been expanding on dierent variations of the problem, including: 1) multiple ow commodities (Lim et al., 2007), 2) multiple time periods (Malaviya et al., 2012), 3) dynamic interdiction (Rad and Kakhki, 2013), 4) dynamic attacker/defender interactions (Lunday and Sherali,

2

2010), 5) stochastic network interdiction (Morton, 2011), including random network topology (Held et al., 2005) and random adversary characteristics (Feng et al., 2008), 6) deception tactics (Salmerón, 2012), and 7) disruption intensity levels (Losada et al., 2012). The literature shows that interdiction models have been studied extensively in physical systems with several (aforementioned) applications. Despite this fact, there is little literature in focusing on the interdiction of networks within the context of communication networks such as wireless networks. Nevertheless, a few studies have focused on the placement of jammers so as to minimize connectivity (Commander et al., 2009, Vadlamani et al., 2014, Feng et al., 2014), and there is some literature showing the optimal defense of networks knowing an attack is imminent in a gametheoretic approach (Basdere et al., 2013); however, these studies consider multi-hop wireless mesh networks. No studies to date have considered jamming in the context of a single-hop network, such as a network of access points.

1.3

Contributions

This paper is the rst to study the jammer placement problem in an array of wireless access points, i.e., a wireless local area network (WLAN). Specically, this paper makes several contributions. 1) We present a mixed-integer programming model for the problem of placing jammers in order to jam access points. 2) We investigate several approaches, exact and heuristic, for solving the model and compare their run times and solution quality. 3) We analyze how the model responds to changes in parameters; this analysis helps generate insights for wireless network security.

2

Problem Description

Our problem consists of a set of users (demand points), signal towers (access points), a number of jammers, and a list of possible jammer locations. Users link with a signal tower through some device (e.g., laptop, smart device), and the device receives an associated signal strength (received signal strength indicator, or RSSI). Users may link with multiple signal towers if they are in range of more than one, but a connection will only be established with the signal tower coinciding with the strongest signal, i.e., the largest RSSI. We seek to determine the optimal location of placing jammers such that the total signal strength is minimized. As users change their location over time, we also consider how the jamming devices themselves may relocate over multiple time periods. We also consider that users may be reassigned to a dierent signal tower (with a reduced RSSI or weaker signal) should the current signal be jammed.

Although many types of jammers exist,

such as jammers that only jam a fraction of the signal strength, jammers that deceive networks by transmitting legitimate information while distorting the eect of actual demand points, and others (each proposed in Xu et al. 2005), we assume that the jammer blocks the signal completely whenever an access or demand point is within its jamming radius. We consider this type of jammer because we are seeking to examine the robustness of a worst-case scenario, where signals are completely blocked and no connection to the network can be found between some demand points and some

3

access points. Figure 1 shows a wireless network that is being jammed. Squares represent the access points (a wireless router or cell phone tower, for example), circles represent demand points (people trying to connect to an access point), and the triangles represent the jammers. The circular elds around the triangles show the jamming radius of a jammer.

The lines between users and access points

represent connections, either actual or potential; the numbers on the lines represent the signal strength received by the user from the access point.

Thus, in this paper we distinguish between

potential signal strength and actual signal strength. Solid lines represent the strongest received signal

strength by a user and thus an established connection; the dashed lines represent backup connections which have less signal strength. Under jamming, a user may need to reassign to a backup connection if the primary established connection is jammed. The way a jammer works is that it emits some omnidirectional signal power over a radius. If an access point falls within this radius, it is jammed, severing all connections any demand points have with it. If a demand point falls within the radius, any connection it has made and any possible connections it can make with other access points are also jammed. On a graph consisting of nodes representing access and demand points with edges representing a connection between a demand point and access point, potential or actual, jamming is eectively removing edges from the graph. However, in the case of a jammed link, the demand point is reassigned to another access point if one is available and the demand point is capable of establishing a connection (is not within the radius of a jammer). For example, consider demand point

3

in Figure 1a, which is linked to access points

The demand point's strongest signal links it with access point can reassign itself to access point demand point

3

2.

1,

but if access point

So to eectively block demand point

needs to be jammed outright or both access points

1

and

3's 2

1

4

2.

signal entirely, either

need to be jammed, as

(b) Jammed network.

Figure 1: Toy illustration.

and

is jammed, it

is the case in Figure 1b.

(a) Network without jamming.

1

3

Mathematical model

3.1

Mixed-integer programming model

To model the problem described in Section 2 we present the following mixed-integer programming (MIP) model. The core of the MIP model has been adopted from the

r-interdiction

median model,

which has applications in identifying critical infrastructures in supply chain management (Church et al., 2004). The

r-interdiction

median problem is essentially the reverse of the famous

problem. In the latter, the problem entails locating the facilities to

minimize

p-median

the total weighted

distance to a set of demand points, while the former problem is to choose a subset of facilities to remove with the purpose of

r-interdiction

maximizing

weighted distance.

In our problem, the facilities in the

median problem correspond to access points.

Specically, our model seeks to jam

(remove) a set of access points in order to minimize the total signal strength received by demand points. This objective function is similar to those used in the facility location literature (Church et al. 2004 and Wood, 1993). In order to formulate an optimization model, the following notation is considered.

Indices

i

index for demand points (locations change over time)

j

index for access points

k

index for locations of jammers

l

index for jammers

t

index for time periods

Parameters

Sij

the signal power from access point

Dijklt

1 if jammer

l

located in position

and demand point

i

at period

t;

k

j

for the demand point

i

can disturb the connectivity between access point

j

0 otherwise

dkl

the Euclidean distance between the two locations

R

the greatest distance between any two possible jammer locations a jammer may relocate to between time periods

n

the number of jammers available

5

k

and

l

Sets

I

the set of demand points

J

the set of access points

K

the set of potential locations for jammers

L

the set of existing jammers (not necessarily identical)

T

the set of time periods

Ait = {jJ | 0 < Sijt } the set of access points that has some signals for demand point i at period t n o Tijt = qAit | n 6= j , Siqt ≤ Sijt the set of existing access points (not including j ) that have at most the same signal from access point

Pk = {mK | m 6= k, dkm > R} from

k

j

for demand point

i

at period

the set of locations of jammers (not including

is greater than

k)

t

whose distance

R

Decision variables

xijt

1 if the demand point

yklt

1 if the jammer

l

i

is connected to access point

is located at position

k

in period

j

t;

at time

t;

otherwise 0

otherwise 0

Using the above notation, our MIP is the following:

M in Z =

XXX i

j

Sij xijt

(1a)

t

subject to:

X

xijt = 1 ∀iI, tT

(1b)

yklt = 1 ∀lL, tT

(1c)

jAit

X k

xijt ≤ (1 − Dijklt yklt ) ∀iI, jJ, kK, , lL, tT X

xint ≤

nTijt

X

X X (Dijklt yklt ) ∀iI, jJ, tT k

(1d)

(1e)

l

ym,l,t+1 ≤ 1 − yklt

mPk

6

∀kK, lL, tT

(1f )

yklt {0, 1} ∀kK, lL, tT

(1g)

The objective function of this model (1a) strives to minimize the network connectivity (in terms of total received signal strength) and thus maximize the impact of jammer placement. The set of constraints (1b) maintains that every demand point in every time period has connection to only one access point. In reality, there are cases that an access point is not in the range of any demand points or simply has been jammed.

In these cases, the demand points will be associated to a

dummy access point which has zero signal strength for all of the demand points and can never be jammed; this point is illustrated in Figure 1 with access point

4

in the bottom right corner. The

set of constraints (1c) makes certain that all jammers are active and are positioned in one of the possible locations. The set of constraints (1d) forces all of the interdicted connections to be zero. The set of constraints (1e) prevents assignments from demand points signals than what

i

has from

j,

i to access points with smaller

unless the connection between the access point and the demand

point has been interdicted (jammed). Thus, demand point

i

will be forced to be assigned to the

access point with the strongest signal. Finally, the set of constraints (1f), the jammer movement constraints,establishes the restrictions on the jammers not being able to move farther than a distance limit from one period of time to the next.

3.2

n-Robustness

We dene

metric

n-Robustness

as the robustness of a network when

as to maximize the total actual signal jammed. We dene all demand points without jamming for a time period

t

under the optimal jamming attack consisting of

n

t.

Tt

n

jammers are placed optimally so

as the total actual signal strength of

The total actual signal strength in period

jammers is dened as

Rtn .

Eectively,

Rtn

is

the remaining actual signal after the network has been jammed, and we note that our MIP model calculates

Rtn

directly. Finally, our metric,

n-Robustness,

is dened as

P n R n − Robustness = Pt t × 100% t Tt So if a network has a

3-Robustness

the total signal power remains after

3

equal to

75%

for a

10-period

horizon, this means

75%

of

jammers have been introduced to the network and placed

optimally for the ten time periods.

4

Dataset Construction

In this section we describe how we modied a publicly-available dataset in order to construct a dataset for our problem.

7

4.1

WTD Dataset

We used a modied version of a wireless trace dataset developed as part of the Wireless Topology Discovery project at UC San Diego (UCSD) (McNett and Voelker, 2005) that provided traces of user-to-access point connections over a period of time. This data contained Cartesian coordinates for all access points and a list of pings for each demand point. A ping is a documented record of a connection a demand point has with an access point (or an attempt to connect with an access point, in the case of multiple connections). Our pings contained a time of the ping and an associated signal strength (which we took as received signal strength indicator, or RSSI) to each access point the demand point attempted to establish a connection with, one row per connection. Several assumptions were made in the modication of this dataset because of a lack of suitable information. The rst was that we assumed the space to be free, i.e., no obstacles or topographical interference from hills or buildings. This assumption came from the fact that, over all the data,

z

the

Cartesian coordinate did not vary signicantly, and there was no information regarding any

possible obstructions or details on landmarks. Another assumption made was that each individual time period in our model represented an interval of time in our datasets. For our purposes, we used one hour to represent each time period. For some of the demand points there were some time periods without any pings. In this case we assumed a linear signal power change between the two time periods with ping data. For our experiments

150

demand points (clients) were used for

10

time periods.

200,

dataset contained much more data: the total number of access points was of demand points was

275,

and the total number of pings was

13, 215, 413.

The WTD

the total number

To select a subset of

data for our experiments, the demand points and pings associated with those demand points were randomly chosen.

4.2

Estimated Received Signal Strength

The UCSD contained signal strengths for each instance that a demand point pinged an access point.

However, our model requires distances.

Thus, we used the following approach to convert

signal strength to distance. The data used was interpreted as a set of demand points with accompanying received signal strength indications (RSSI) with values that ranged from

0

(no connection) to

tion), and we assumed them to be wi signals. We rst converted the

0-33

33

(strong connec-

RSSI values to a signal

quality percentage, which is eectively a measure of how clear the signal is with regards to outside noise. With signal quality ranging from

0 to 99 percent, the signal quality percentage was converted

into a signal-to-noise ratio (SNR), which is the power ratio between the signal strength and the noise level, measured in dBm, using the equation

SN R = [quality/2] − 100 We note that signal-to-noise ratio is an absolute power quantity, meaning from here, we could

8

estimate a distance between the demand point and access point by solving for

d

in the following

equation

SN R = −10blog10 (d) + M Here,

b is the path loss exponent.

2 (free space). M

Taken from (Farahani and Shahin, 2008), we assumed it to be

is the RSSI value at a reference distance. This is typically taken from a point one

meter away from a source with no obstacles between them. For the purposes of our model, because of a lack of explicit information, we took this to be the largest reported RSSI value in the dataset,

33,

which when converted from the rst steps, was equal to

−50

dBm. Thus our equation became

d = 10(−50−SN R)/20 . 4.3

Placing Users Via Trilateration

With the signal strength converted into a distance, each demand point was separated into one of three categories: those linked (i.e., have a signal strength value with) with one, two, or more access points.

For a given demand point, we could represent each signal strength distance as a radius

around each access point. Given the radius and location of each access point, we use trilateration, the process of using known locations and distances to determine another location, to estimate the locations of the demand points.

We assumed a linear rate of change among the

z

coordinates

(height) of each access point because given only access point locations and RSSI, true height could not be determined exactly; since height values did not vary signicantly, we did not include the

z -dimension

in our analysis.

In the case of three or more access points, we only used the three access points with which the demand point has the strongest signal to, coinciding with a best estimate that also guarantees the demand point, in the case of an attack, will connect to the next strongest available signal as required in the set of constraints (1e). Alongside the case of two access points, these two problems could be solved exactly by calculating either the intersection point of three circles or the external tangent point of two circles. Both of these ideas are illustrated in Figures 2a and 2b. For the case of only one access point, the demand point was assigned a random location on the circumference of the given circle. However, care was made that this location's choice was such that it was not accidentally within the range of some other access point, or else the dataset would have provided the necessary ping information. If such a random point had been chosen, a new one was selected, and the cycle was repeated until a suitable location was found. In Figure 2c, we represent the unsuitable locations within the radius of another access point with dashed lines..

4.4

Set of Potential Jammer Locations

To form the set

K

of potential jammer locations, we considered the largest and smallest

coordinates of each access point to give the boundaries of a region and divided this into a

9

x

and

y

12 × 12

(a) Three access points.

(b) Two access points.

(c) One access point.

Figure 2: Three, two, and one point trilateration. Squares represent access points, circles represent signal radius, and solid points represent demand points.

evenly spaced grid, where the corner points of the cells are the possible locations. We ltered out potential jamming locations that were able to jam only a small number of connections (less than

5%).

We assumed a jamming radius of

1, 000

locations of access points (averaging around points. Our total region measured

5

feet for each jammer, a choice coming from the given

800 feet in distance) and estimated locations of demand

11, 295 × 8, 301

in feet.

Solution Methodologies

The presented problem is an NP-hard problem because it is a more general case of the

r-interdiction

median problem [Church et al., 2004] with added constraints to account for the jammers' movements and connectivity which is itself an extension of the

p-median

problem, which was proven to

be NP-hard [Megiddo and Supowit, 1984].

As such, its solve time exponentially increases when

the dimensions of entering instances grow.

Although commercial general solvers can solve small

instances of the problem, we also employed Lagrangian relaxation, a stage decomposition heuristic approach, and a genetic algorithm for faster but sub-optimal solutions.

5.1

Lagrangian Relaxation Heuristic

As an alternative to solving our MIP formulation exactly, we employed a Lagrangian relaxation heuristic.

The general Lagrangian Relaxation procedure dualizes one or more of the constraints

in the model and then uses a subgradient technique to nd the optimal dual multiplier vector. Although this procedure is optimal for convex problems, it is not guaranteed to be optimal for integer programs, which are non-convex. In our Lagrangian procedure we dualized the set of constraints (1d), obtaining the following Lagrangian subproblem:

10

L(λ) = M in Z =

XXX i

xijt × Sijt +

XXXXX [λijklt (1 − Dijklt × yklt − xijt )]

t

j

i

j

k

(2a)

t

l

subject to:

X

xijt = 1 ∀iI, tT

(2b)

yklt = 1 ∀lL, tT

(2c)

jAit

X k

X

xiqt ≤

qTijt

X

(Dijklt × (

X

k

X

yklt )) ∀iI, jJ, tT

(2d)

l

yml(t+1) ≤ 1 − yklt

∀kK, lL, tT

(2e)

mPk The Lagrangian relaxation procedure involved repeatedly solving the Lagrangian subproblem (2), continually updating the Lagrangian multiplier vectors

λ, which must be non-positive, to maxi-

mize the resulting bound. In other words, a Lagrangian-based lower bound was obtained by solving the following optimization problem:

max L(λ).

(3)

λ≤0

To obtain a feasible we set the values of

xijt

x

vector for a

y

vector returned by solving a Lagrangian subproblem (3),

so that each demand point was assigned to a non-jammed access point with

the largest signal strength in each period. This gave a feasible solution to the original problem (1).

5.1.1 Subgradient Optimization To search for the optimal multipliers, we employed the sub-gradient method. Through trial-anderror we chose

−0.0125

as the initial value of each element of the

λ

multiplier vector. We then used

the following iterative scheme to update the multipliers:

r λr+1 ijklt = λijklt + φr (Dijklt × yklt + xijt − 1) The parameter

φr ,

(4)

a step size, was updated using the following equation (see Fisher, 1985):

φr = tr (L(λr ) − Z ∗ )/

XXXXX i

j

k

l

r (1 − Dijklt × yklt − xrijt )

(5)

t

L(λr ) is the objective value of the Lagrangian model (2) at the current multiplier value λr , ∗ and Z is the objective value of the current best known feasible solution to the original model (1). where

11

The parameter

tr ,

a scalar between

0

and

2,

started at

2

and was halved whenever

L(λr )

failed to

improve after 30 consecutive iterations.

5.2

Stage Decomposition Heuristic

The formulated problem has two distinct parts: one is the placement of jammers, and the other is the association of demand points and access points. The connecting point of the two parts of the problem in the original program is the set of constraints (1e).

Removing these constraints

allowed the problem to decompose into two separate problems: the location of the jammers and the assignment of demand points to access points.

In the rst part, maximization was done by

just taking the jamming placement into consideration. This task was accomplished by applying the jammers' placement in the set of constraints (1d) and updating the set of constraints (6c). On the other hand, the objective function of the second part of the model objective function attempted to maximize the possible impact of jamming placement when the assignments were already decided. The two decomposed models are proposed as follows:: 1. Maximize the connectivity while the jammer location variables are xed (the are the decision variables and the

yklt

xijt

variables

variables are xed):

M ax

XXX

Sijt xijt

(6a)

t

j subject to:

X

xijt = 1 ∀iI, tT

(6b)

jAit

xijt ≤ (1 − Dijklt yˆklt ) ∀iI, jJ, kK, , lL, tT

(6c)

2. Minimize the connectivity by maximizing the amount of jamming while the assignments are xed (the

xijt

variables are xed and the

M ax

yklt

variables are the decision variables):

XXXXX l

t

Sijt Dijklt )ˆ xijt yklt

(7a)

j

subject to:

X

yklt = 1 ∀lL, tT

(7b)

k

X

yml(t+1) ≤ 1 − yklt

mPk

12

∀kK, lL, tT

(7c)

Figure 3: A sample chromosome for the genetic algorithm.

Thus, the procedure was as follows:

yklt

1. Randomly assign values for solution as

that are feasible to constraints (7b); denote this random

yˆ.

2. Optimize the rst mixed-integer program with the

y

3. Optimize the second mixed-integer program with the 4. If the values of the vector

x ˆ

variables xed and denoted as

x

yˆ..

variables xed and denoted as

x ˆ.

have been found as a solution to a previous solution to model 6,

report the smallest objective function for model 6 found thus far. If not, go to step 2. With regards to step 4, it was possible that the algorithm may have converged to a sub-optimal solution. Specically, if the objective value increased between the rejected the (i

5.3

+ 1)st

ith

and (i

solution, terminated the algorithm, and reported the

+ 1)st

ith

iterations, we

solution.

Genetic Algorithm approach

In addition, we also tested a genetic algorithm for this problem. Genetic algorithms have long been used with success on a variety of combinatorial optimization problems. The fact that the GA is a random and population based algorithm helps it escape from local optima in the search for global optima.

5.3.1 Chromosome Encoding GA's chromosome based feature whose structure allows for the immediate realization of some of the constraints has made the algorithm popular. The algorithm works with a population of solutions, represented as chromosomes. We used the following chromosome structure to represent solutions to our problem. For each time period, each jamming location had an element in the chromosome. The element for a jamming location at a particular time period contained an integer that represented the jammer's location at that time period. Figure 3 shows an example of our chromosome representation for a problem with

3 jammers, 8 possible locations for the jammers with 2 periods of time.

to the solution represented by this chromosome, the rst jammer was at locations for periods one and two, the second jammer at locations

2

and

1.

7 and 8,

5 and 4 respectively

and the third jammer at locations

Thus, this chromosome translated into a solution for each variable

13

According

yklt .

5.3.2 Fitness function The tness function in our GA calculated the tness/goodness of any chromosome, serving the same purpose as the objective value in our MIP, and this value was the basis for sorting the population and giving each member the proportional level of chance to undergo crossover or mutation. However,

xijt

could not be directly computed from the chromosome as the encoding only extracted the values

for

yklt

as the calculation of the tness function determined the association between a demand

point and some access point. Every time a chromosome was passed to the tness function, decoded from the chromosome, and then

xijt

yklt

was

was assigned appropriately based on the following

procedure. The access points with which the connection was not jammed based on the calculated

yklt

and had the highest signal was associated to the considered demand points, unless there was

no unjammed signal between the demand point and any access points, in which case the demand point was associated with the dummy access point. With the calculated

xijt

values, the total signal

strength (1a) was the tness value for the entering chromosome, and every chromosome in the population had its tness value calculated independent of its genes' structure.

5.3.3 Operators: Crossover and Mutation In our GA implementation we used two procedures for ensuring population diversity: crossover and mutation. Diversity is necessary because without it the algorithm does not necessarily yield new solutions. While crossover allows for the population to move towards optimality, mutation prevents premature convergence and nding local optima.

Also, unlike mutation, crossover operates on

two chromosomes, attempting to combine the two chromosomes' characteristics to produce better ospring; to this end, we used the uniform crossover procedure (Mills et al., 2014). During mutation, a drastic change is inicted upon the chromosomes' structure with the goal of producing better solutions; here, a traditional rate-controllable mutation procedure was adopted, meaning the level of mutation was controlled and was based on each generation's solution.

5.3.4 Feasibility Check An important issue for both the crossover and mutation operations is the need for a feasibility check of exiting chromosomes (ospring in case of crossover). Often, when the population is randomly initialized, the way the procedure is designed makes sure that infeasible solutions will not be created, such as placing multiple jammers in the same location as an example of our chromosomes' structure. In this paper, the initialization of chromosomes was done in a way that preserved feasibility, but the chromosomes that had undergone crossover or mutation were prone to violation of the jammer movement constraints (1f). Therefore, the existence of a feasibility check after every mutation and crossover was crucial. At initialization of the population (except for the rst time period), the jammer placement was randomly assigned from the set of possible locations based on the parameter

R.

Furthermore, for the feasibility check that ran for the outputs of crossovers and

mutations, the jammers' placements that infringed the jammers' ranges of movement were again

14

randomly assigned from the set of possible locations. The set of constraints (1c) were enforced by the chromosome structure and all other constraints (1b,1d, and 1f) were enforced by the way tness function calculation was designed.

5.3.5 Evolutionary Iterations Figure 4 shows the main procedure for the genetic algorithm. In every iteration, the population was sorted based on the chromosomes' tness values.

Chromosome selection was based on the

roulette-wheel function, where chromosomes higher on the list had a better chance to be selected to go through crossover and mutation operations. Roulette-wheel selection was used for chromosome selection for both operators. The algorithm continued until there had been no improvement for a prespecied number of iterations.

Figure 4: Genetic algorithm owchart.

15

6

Example illustration

In this section, we present a visualization of how the model works. We considered a set of points,

15

demand points, and

points, and triangles jammers.

F.

4

time periods.

6

access

Squares represent access points, circles demand

For the sake of legibility, demand points

10-15

are labeled

A-

We include solid lines representing an established connection and a dashed line is a potential

reassignment; the numbers on line lines denote the associated RSSI. As time moves from period to

4,

1

the demand points change their position according to the data from the WTD dataset. In

pursuit of the demand points, the jammers, with a jamming radius of 250 feet, also move in order to disrupt the maximal signal strength.

The circles around each of the jammers represent their

jamming radii.

Figure 5:

An example illustration.

Time period

16

0,

the network without jamming.

Figure 6:

(a) Time period

1.

(b) Time period

2.

(c) Time period

3.

(d) Time period

4.

The example illustration, continued.

Demand points move over the four periods,

while the access points remain stationary. The jammers also move in pursuit of the demand points, seeking to minimize the total signal strength over the four periods.

Initially, in time period points.

0,

we had in our unjammed network

113.

access points and

15

demand

Demand points were each connected to their closest access points, which maximized the

total signal strength. Access point of

6

Access point

and access point

1

4

2 was connected to four demand points for a total signal strength

supplied a total signal strength of

supplied a total of

43.

112,

access point

The total signal power was

5

supplied a total of

63,

375.

During the rst time period, jammers were initially placed, causing demand points to be reas-

17

signed to other access points. In the optimal jammer placement, jammers were placed near access points

2

and

4,

likely because these were supplying a large signal strength as seen in Figure 5.

5

Notice that although demand point

has a jammed connection to access point

2,

this was not

its strongest connection, so its potential connection being jammed does not count in a total signal jammed. Also notice that demand point

1 could not obtain a signal from any demand point because

its only available access point was jammed. In addition, demand point

3

had no signal even though

its second-closest access point (6) was not jammed; this was because it was within the jamming radius of jammer period was

2;

The total signal power for time period

1

was

108.

The

3-Robustness

for

1

time

(108/375) × 100% = 28.8% 1, several demand

During the second time period, all three jammers relocated. Near access point points moved closer together which gave jammer

1

a better means of jamming several signals at

once by jamming them all directly. The total signal power for time period total unjammed signal power was

348.

2

increased to

154.

The

3-Robustness for 2 time periods was (108 + 154)/(375 +

The

348) × 100% = 36.2% During the third time period, jammer follow the movement of demand points

F,

2

5, C ,

did not move, while jammers and

E.

1

and

3

did in order to

Although other demand points, such as

B

and

moved, they did not move in such a way so as to force the jammer to move. The total signal

power for time period

3

decreased to

129.

The total unjammed signal power was

the same calculation pattern as for the previous two periods, the

3-Robustness

389. 3

for

Following

time periods

35.2%.

was

During the nal time period, all three jammers relocated again, and in this case, no access point was directly jammed.

Instead, all jammers were actively following the demand points and

jamming only them. Jammer

104.

jammed demand points

The total unjammed signal power was

3-Robustness

the

1

for

4

time periods was

380.

33.2%.

1, 2,

and

7.

The total signal power was

Similarly calculated from the previous periods,

Thus, the presence of jammers caused a

2/3

drop

in network performance.

7

Numerical experiments

7.1

Problem tractability

In this section we investigated how changing the number of access points, demand points, jammers, jamming locations, and time periods aected the run time of our solution methodologies. Table 1 shows the run time for each of the algorithms used in our project for dierent values of the number of demand points periods

|T |.

|I|,

access points

|J|,

potential jamming locations

For all experiments, we took

R = 60,

|K|,

jammers

|L|,

and time

the maximum distance a jammer may move

between two time periods. We also assumed that all jammers had the same power (and radius), but we note that the model can be easily generalized to encompass multiple types of jammers. The number of access points depended on the number of demand points because of both actual and potential connections. If demand point

1

was connected to access points

18

1, 2,

and

3,

we were

forced to include all three access points. The reverse was not true. If access point

1, 2,

with demand points

and

3,

1 had connections

we were not required to experiment on demand points

2

and

3

necessarily. Thus, our choice of the number of access points was dependent on our choice of demand points, both how many and which ones were chosen. By running all of the remaining possible combinations of considering a small or large parameter value, which added up to

16

experiments, many

insights could be gained about the computational capability of the algorithms used in this paper. For each combination, we listed the total run time for each of the four algorithms implemented: the exact approach (Exact), the stage decomposition heuristic (SDH), the genetic algorithm (GA), and the Lagrangian relaxation (LR). Bold are the best run times for each experiment. Finally, we included the percentage over the optimal solution that the SDH, GA, and LR algorithms provided, and we bold the best method. For GA, the population, crossover percentage, mutation percentage, and mutation rate values were taken to be iterations was set to

30, 70, 30, 0.4,

respectively. The maximum number of

40.

To solve the problem instances we used a Lenovo Ideapad Y510 running 64-bit Windows 8.1 on 4 dedicated Intel Core i7-4700MQ processors running at 2.4 GHz with 8 GB RAM. The Genetic Algorithm was encoded using MATLAB. The MIP model was solved using Gurobi 6.0.5 with its Python API. All other algorithms were encoded in Python 3.4, with subproblems solved with Gurobi 6.0.5. Table 1: Run times and solution percentage deviations of the algorithms. Run time (seconds) No.

|I|

|J|

|K|

|L|

|T |

1

50

27

10

2

1

2

50

27

10

2

10

3

50

27

10

5

1

4

50

27

10

5

10

5

50

27

40

2

1

6

50

27

40

2

10

7

50

27

40

5

1

8

50

27

40

5

10

9

150

63

10

2

1

10

150

63

10

2

10

11

150

63

10

5

1

12

150

63

10

5

10

13

150

63

40

2

1

14

150

63

40

2

15

150

63

40

16

150

63

40

Exact 2.6 12.5 5.4 33.6

SDH

0.2 0.5 0.3 1.3

4.4

1.0

30.4

4.6

4.2

1.3

50.5

9.7

502.6

11.2

1,702.4

115.0

607.1

14.3

2,015.4

132.6

989.3

40.0

10

2,770.5

302.5

5

1

1,021.1

86.7

5

10

2,047.5

819.2

Percentage over optimal

GA

LR

1.1

1.3

1.2

1.8

1.1

1.8

1.1 1.0 2.1 1.7

2.7 6.5 18.1 12.3 31.0 28.6 67.1 32.3 207.1

SDH

GA

LR

16.4

1.7

18.2

1.9

13.3

1.9

4.0

0.0 0.0 0.0 1.3 0.5 0.4

4.7

2.1

6.7

5.5

2.7

8.1

19.2

3.3

10.2

44.1

3.0

11.3

20.2

14.2

7.3

54.4

13.3

5.4

51.6

13.9

5.5

123.2

14.4

4.9

75.2

15.7

3.4

242.6

14.8

2.7

2.2 3.6

17.3

1.4

9.1

1.7

8.4

1.6

1.2 1.3 1.5 2.0 1.5 1.3 0.9 1.1 0.7 0.9

For small problems, the SDH algorithm ran the fastest and produced the best quality solutions. For larger problems, however, the GA ran the fastest while the LR yielded the best quality solutions. Overall, the LR heuristic yielded the best combination of solution quality and run time, especially for larger problems.

19

7.2

n-Robustness

metric computation

Here, we computed the

n-Robustness

metric from the exact solution to display how we dened

the robustness of our dataset. Table 2 shows the combinations.

That is, rows with

|L| = 2

show

n-Robustness

value for the dierent parameter

2-Robustness,

and rows with

|L| = 5

show

5-

Robustness. As the results show, there was good robustness for a single time period. Further, the robustness value decreased as more jammers and possible jammer locations were added to the network. However, as time periods increased in number, the robustness increased. Similarly, as demand points and access points increased, robustness increased as more signal connections (actual and potential) were possible and jammers had to work harder under a movement distance restriction to optimally jam the total signal strength.

Table 2:

n-Robustness metric, as %. P P n |L| |T | n−Robustness t Tt t Rt

No.

|I|

|J|

|K|

1

50

27

10

2

1

1,194

762

63.8

2

50

27

10

2

10

12,387

8,457

68.3

3

50

27

10

5

1

1,194

468

39.2

4

50

27

10

5

10

12,387

5,376

43.4

5

50

27

40

2

1

1,194

730

61.1

6

50

27

40

2

10

12,387

7,892

63.7

7

50

27

40

5

1

1,194

433

36.3

8

50

27

40

5

10

12,387

4,732

38.2

9

150

63

10

2

1

2,733

1,878

68.7

10

150

63

10

2

10

28,963

20,593

71.1

11

150

63

10

5

1

2,733

1,145

41.9

12

150

63

10

5

10

28,963

13,236

45.7

13

150

63

40

2

1

2,733

1,473

65.3

14

150

63

40

2

10

28,963

19,492

67.3

15

150

63

40

5

1

2,733

1,060

38.8

16

150

63

40

5

10

28,963

11,643

40.2

In Figure 7, we've also shown how the robustness changed as the distance restriction by having plotted

2-Robustness

over

10

time periods.

possible distance a jammer could move, in intervals of

R

changed

The horizontal axis shows the maximum

50

feet, and the vertical axis shows the

overall robustness for one time period. We produced results using the values for the variables from experiments

2, 6, 10,

and

14.

As the gure shows, as this distance increased, robustness decreased

along a sigmoid curve. Initially, jammers appeared to nd it dicult to pursue moving signals if their movement is too limited; however, as their limitations decreased, they could better move to minimize signal strength, until eventually they yielded diminishing returns in their ecacy.

20

Figure 7: Robustness versus maximum distance of jammers' possible movements.

7.2.1 Eect of access point topology on robustness We wished to explore the eects that the geographical distribution of access points has on network robustness. To this end, we compared the dataset's access point distribution to three other access point topologies. For all three topologies, we provided output for only experiments

4 and 12 because

the overall trend, as in Figure 7, remained consistent across experiments for each of the three experiments.

The rst topology distributed the access points uniformly around the perimeter.

Since this is a rectangular region, this means each side had per side and

6

or

15 or 16 (for experiment 12) access point per side.

7

(for experiment

4)

access points

We named this pattern Perimeter.

The second topology was a cross pattern, where half the access points were aligned vertically and half were aligned horizontally (13 or

14

each for experiment

4, 31

or

32

each for experiment

12),

each column or row in the center of the region; we named this pattern Cross. The third topology had a completely random distribution, and to avoid clustering, we required that every access point remained at least

150

feet away from every other access point; we named this pattern Random.

Figure 8 illustrate these distributions with

27

access points (experiment

4).

Table 3 shows the

robustness results. For all three experiments, we kept the demand point location and movement patterns the same as from the dataset. Finally, for ease of comparison, we re-provide the results of our original experiments one from experiment

4

4

and

12,

which we named Original. We made bold the two best results,

and one from experiment

12.

21

(a) Perimeter pattern

(b) Cross pattern

(c) Random pattern

Figure 8: Topology distributions

Table 3:

5-Robustness

comparison (as %) across four dierent network topologies

5 t Rt

Pattern

Experiment Number

|I|

|J|

|K|

|L|

|T |

P

Original

4

50

27

10

5

10

12,387

5,376

Perimeter

4

50

27

10

5

10

4,006

2,185

t Tt

P

5−Robustness 43.4

54.5

Cross

4

50

27

10

5

10

14,111

4,516

32.0

Random

4

50

27

10

5

10

7,227

3,830

53.0

Original

12

150

63

10

5

10

28,963

13,236

Perimeter

12

150

63

10

5

10

9,239

4,509

48.8

Cross

12

150

63

10

5

10

35,892

12,067

33.6

Random

12

150

63

10

5

10

19,454

7,332

37.7

The columns follow the same naming conventions as in Table 2.

45.7

The rst column gives the

topology name and the experiment number. The results indicate that the placing the access points along a perimeter yields the best robustness. The high robustness of the Perimeter pattern was the result of the demand distribution and movement patterns from the original dataset; because access points are on the perimeter, demand points were able to more easily access a variety of additional access points. In the original dataset the low number of connections for most demand points meant that only a few signals needed to be jammed to completely remove a demand point's connection (or minimize it if demand points could reassign to other access points); as a result, several demand points could be removed from the network. In the Perimeter pattern, however, very few demand points were able to be completely removed from the network. However, as this is a preliminary analysis, we oer the following caveats. First, the number of jammers is signicantly overshadowed by the number of access points, and having such a distribution of access points with limited movements between the jammers while demand points may move freely suggests that the jammers might not move much at all between time periods as there was not a

22

signicantly dierent total signal strength to jam, and we note that with our dataset this point was true. Also, and as an extension to the previous point, there was very low overall signal strength and thus overall low connectivity as most of the demand points were clearly not near the perimeter, suggesting further that jammers not only did not move much but did not need to move much to continue jamming maximum signal strengths. More analysis is needed not only in the topology of access point locations, but also that demand distributions and movement patterns be reected more realistically (where demand points may seek stronger signals if they do not immediately have access to any, such as if they are in the center of a region where all access points are located along the perimeter) in determining the robustness of a network in terms of its physical conguration.

7.3

Problem sensitivity analysis

7.3.1 Experimental design We also ran experiments to analyze how certain parameters inuence the optimal solution. Specically, we wished to examine how the number of jammers and possible jamming locations inuenced the total signal jammed based on where they were located. Again, we used a simple grid to dene where jammers could have been located as in Section 4.4. In all experiments, we only considered time period. For the rst two experiments, we considered a base problem instance of and

25

demand points.

1

10 access points

For the second problem instance, we considered the eects of increasing

the number of access points from

10

to

20.

For our nal problem instance, we reconsidered only

access points but increased our demand points from

25

to

50

10

and reported the third results. Thus,

relative to our base instance, we saw the impact of the numbers of jammers and their locations as we attempted to increase access points, simulating the expansion of a network, and as we attempted to increase demand points, simulating increased network activity. Respectively, we've referred to all problem instances as base, rst, and second instances.

7.3.2 Eect of the number of jammers In our rst experiment, we examined how increasing the number of jammers aects the total signal jammed, considering increasing to

30

possible locations. Figure 9 shows the results. Starting with

1

jammer and

29 by adding 2 for each run, our base trial gave us an apparent exponential curve.

Our

rst trial showed a signicantly smoother exponential curve, and our second showed a signicantly sharper exponential curve.

These results should have been expected; the rst few jammers will

immediately be placed in locations where there are the most connections, or the most activity. Once these sites were jammed, the remaining possible locations had considerably fewer signals to jam, so we saw diminishing results.

23

Figure 9: Total signal jammed versus number of jammers.

7.3.3 Eect of the number of possible jammer locations In our second experiment, we examined how increasing the number of possible jammer locations aected the total signal jammed, considering with

10

locations and increasing to

three experiments.

80

by

5

10

jammers. Figure 10 shows the results. Starting

for each run, we again saw exponential curves for all

However, these curves were signicantly atter, possibly indicating that the

number of locations were not as stressful on a network as the number of jammers themselves. In this case, the atness of the curve resulted from how increasing the number of potential locations for a jammer made the model more accurate for the jammer.

Figure 10: Total signal jammed versus number of jamming locations.

24

7.3.4 Eect of the numbers of access points and jammers In our third experiment, we examined the inuence of total signal jammed by the eects of changing both the number of access points and the number of jammers, simulating the attempts at countering a possible jamming threat by adding more access points which allows for continued network connectivity. We considered

50

demand points and

30

possible jammer locations, and we gradually

increased the number of jammers relevant to the number of access points. Specically, we began with

3

and

5

jammers as a base, and we proceeded to add

5

jammers until the number of jammers

equaled the number of access points. For example, in Figure 11a, there are increased the jammers to from

5

to

10, 15, 20,

10.

In Figure 11d, there are

and nally to

25,

10

access points, so we

25 access points, so we increased the jammers

and so forth for the others. Figures 11a through 11e show

the results.

25

(a)

10

access points.

(c)

(d)

25

20

(b)

15

access points.

(e)

30

access points.

access points.

access points.

Figure 11: Total signal jammed versus total number of jammers and access points, with points,

30

jammer locations, and

1

50 demand

time period.

Initially, in Figures 11a through 11c the plots continue to follow an exponential curve. However,

26

from Figures 11c through 11e, the curve began to take a more sigmoid (concave then convex) shape. Indeed, the gradual change can be seen through all

5 gures.

In terms of signal strength, this meant

that a smaller number of jammers initially were unable to counter the eects of the growing number of access points because the access points could be placed as they need to be, but once the jammers were placed, there were only so many locations they could be moved to. Then, once all access points were in place, jammers were quickly able to locate optimal areas of placement. The result, then, was that the number of access points mitigated the total signal lost.

However, this was only an

initial conclusion; as the number of jammers increased, the total signal lost followed the results of the previous experiments.

8

Conclusions

8.1

Discussion

Wireless networks play such an important role in areas of business, recreation, and defense that ensuring their safety is an absolute priority.

This study examines the robustness of a wireless

network via a mathematical model that considers the optimal placement of jammers to jam a network consisting of a set of xed access points and mobile demand points. In doing so, a mixed-integer programming model is developed. To solve the model, we employ a stage decomposition heuristic, Lagrangian relaxation (LR) heuristic, and a genetic algorithm (GA) as means of approaching suboptimal solutions in a shorter run time. The LR approach gives a much better sub-optimal solution, but the GA is signicantly faster in computational time. Overall the LR heuristic yields the best combination of run time and solution quality. We illustrate the model through an example, showing an initial network comprised of demand and access points and connections between them and, over four time periods, showing the optimal placement of jammers to minimize the total signal. The example demonstrates that, because jammers have a constraint on how far they can move in a given time period, initial placement of jammers is important,. Finally, we oer a metric,

n-Robustness,

should it come under attack by

n

jammers.

that gives insight to how robust a network can be We dene it in such a way that users of the metric

can have an idea on how functional a network will continue to be after or during a jamming attack as the metric shows the percentage of what total signal power remains. Our results show that our network overall robustness is in the range of

39-69%.

Also, robustness increases as time periods,

demand points, and access points increase. With regards to time periods, the jammers' distance restriction is what allows for the robustness to increase. Initially, jammers may be placed anywhere in the set of possible locations. However, although demand points may move as they are able to, jammers' distance restriction forces them to lose their ecacy if they are not allowed to keep up with the moving demand points. We present a sigmoid trend showing how robustness changes with a jammer's distance's restriction, showing that the jammers' movement only slightly decreases until their movement restriction is relaxed signicantly, and then their ability to decrease robustness

27

yields diminishing returns. The results in Section 7 give a strong indication that the number of jammers is more critical to a network's security than their locations, as evidenced by a sharper drop in total signal with fewer jammers than with fewer location possibilities. However, although the possible locations of a jammer may not directly inuence total signal loss, the atness seen in Figure 10 compared to Figure 9 suggests something else. Namely, by improving the accuracy of optimal placement for a jammer, a jammer that may be just out of range of a possible link may be able to jam said link with more accurate location choices.

Also, a set of jammers can better follow roaming demand

points, being able to continue to jam large signals as these signal carriers move. On the other hand, attempts to mitigate such an attack by increasing the number of access points only works to a point: eventually, the number of jammers will become too strong that increased access points cease to mitigate the total jammed signal with any real signicance.

8.2

Future Work

With these points in mind, we address possible future works. First, the potential location of jammers needs to be addressed. In our case, we used a simple grid approach. Placing potential locations in intelligent areas, perhaps based on signal activity density or total signal power, should be investigated. In addition, our model only considers minimizing total signal power, which is important when critical devices must maintain connectivity; another possibility is to consider disrupting the total number of access-demand point connections, regardless of total signal strength. From the defender's point-of-view, intelligent placement of jammers leads to the problem of intelligent network design such that jammers do not have easy access to key locations. In our dataset, we assumed free space, or that jammers could move freely in any direction with no obstructions and were limited only by how far they could move in any time period. Taking note of this fact, the defender could focus on the intelligent placement of obstructions, particularly if there is an additional constraint that required access points have a load capacity, meaning an access point can only establish connections with a certain number of demand points. Our model assumed no such constraint. Similarly, the placement of access points may greatly mitigate an attack. dataset, we considered a region with boundaries based on maximum and minimum

With our specic

x and y Cartesian

coordinates. Generalizing this idea, building more access points around and outside the perimeter would signicantly reduce the lost signal by an attack if many demand points are concentrated near the boundaries. Thus, it would be useful to optimize the location of access points within a defender-attacker-defender model. This idea coincides with the previously mentioned proposition of adding in a load capacity constraint, where if access points can only connect with a specic number of demand points, their locations with regards to demand point density becomes of greater importance. In reality, network connectivity consists of two primary players: attackers and defenders. Although our model considers only stationary access points with moving jammers and demand points,

28

there is also the consideration of mobile access points. In this case, a game theory approach may demystify what can happen if both attacker and defender have the means of intelligently moving their players: jammers for the attacker and access points for the defender.

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Analyzing the Robustness of an Array of Wireless Access Points

Highlights 1. 2. 3. 4. 5.

A risk metric for a wireless network under attack by jammers is proposed. A multi-period mixed integer programming interdiction model is developed. Signal strength trend is negative-exponential relative to jammers and locations. UCSD dataset indicates a fairly robust network in spite of jamming. Lagrangian relaxation often finds near-optimal solutions for jammer placement.