Levy Models Of Survivor Movement

Published Date: 02 Nov 2017

Disasters are events that occur in our everyday living. They are either man made, such as terrorist

attacks, or natural, such as earthquakes and tsunamis. It is important that when disasters occur, the

victims in the disaster area are rescued quickly to avoid massive casualties. This is because a

disaster area leaves victims stranded for food, water, shelter and medical help. The goal of the

work outlined in this thesis is to study the movement of survivors towards rescue devices when

barriers are involved. Barriers are obstacles that prevent both survivors and crewmembers from

moving freely in a disaster area; thereby; slowing down rescue operations. It would therefore be

necessary to study movements of survivors with barriers included in the simulations so as to

produce more realistic results to be used when rescuing survivors in disaster areas. Sensitivity

analysis simulations are also be performed to determine how changing the step length parameter

of the Levy walk model affect how long it would take survivors to reach dropped beeping phones

to call for help and how many survivors get to the beeping phones.

Acknowledgements

I would like to take this opportunity of thank the people who have made this thesis possible. First

and foremost, I would like to thank Professor Oliver Ibe for giving me the opportunity and

guidance needed to complete this thesis. I am proud to have worked with such a dedicated and

motivated faculty member and the experience is something I will remember for years to come.

Secondly, I would like to thank Dr. Ram G. Lakshmi Narayanan for the support and information

he provided me to develop this project.

Lastly, I would like to thank my family and friends who have been provided me with assistance to

continue with my education.

Table of Contents

Abstract 2

Acknowledgements 3

Chapter 1 - Introduction 5

1.1 – Introduction 5

Chapter 2 – Overview of the PDRN 7

Chapter 3 – Introduction to Levy Walk 10

3.1 – The Special Cases 11

3.2 – Levy Walk versus Levy Flight 13

3.3 – Levy Walk Models of DRN Survivors 14

Chapter 4 – PDRN with Barriers 18

4.1 – Barrier Model 18

4.2 – Simulation Results 21

Chapter 5 – Sensitivity Analysis 31

5.1 – Simulation Results 31

Chapter 6 – Conclusion and Recommendations 41

References 43

Chapter 1 – Introduction

1.1 - Introduction

In recent times, disasters have occurred and there were survivors that needed to be rescued in time

so as to avoid major casualties. The disasters range from terrorist attacks, which are man-made

disasters, to natural disasters such as earthquakes and tsunamis. Examples of such recent disasters

include the Indian Ocean tsunami of 2004, Hurricane Katrina of 2005 in the Gulf Coast of the US,

and of recent, the Hurricane Sandy of 2012 in the Mid-Atlantic Coast of the US. These disasters

have reminded us of the need to have reliable disaster recovery and relief operations. These

operations will have to be planned ahead of time to increase efficiency of rescuing survivors with

the availability of reliable communication networks. A disaster recovery network should be able

to provide assistance to both the disaster victims and the rescue crewmembers. Presently, this

process is carried out manually. In a short period of time and with the limited crewmembers that

are deployed to search for survivors, it is very time consuming and inefficient. To ensure efficiency

is at its best, a mechanism is used that involves the survivors reporting their locations to a command

center thereby making it easier for crewmembers to be able to locate survivors quickly. The

disaster recovery network has to be wireless in order for the deployment of crewmembers to a

disaster area to occur within a short period of time. A number the existing wireless networks

require complicated infrastructure for deployment and also the emphasis is less on disaster

services. It is therefore necessary to understand how to build a disaster recovery network that can

be used anywhere in the world and is easy to operate, thereby making search and rescue operations

more efficient. An example of such a network is the Portable Disaster Recovery Network (PDRN)

[1, 2].

The models used in [1, 2] discussed the movements of survivors in the PDRN using the Levy walk

models and the random walk models towards dropped beeping phones but did not consider barriers

when the simulations were run. The need to study survivor movement with barriers considered is

important as real life disaster situations have obstacles in the way of survivors and crewmembers

preventing both parties from communicating with each other efficiently. If barriers are not

considered, the results tend to not give a full description of what actually goes on in real life

disaster locations. Without barriers, it is assumed that the walker has a free path to get to a rescue

phone to call for help, which is not so as barriers are present in virtually all disaster areas. The

exclusion of barriers gives rescue teams a vague idea of the time needed to rescue survivors and

limits them to plan for high efficient operations.

The purpose of this thesis is to extend the work in [1] by studying the movement of survivors in

the PDRN using the Levy walk models and the random walk models when barriers are present.

Barriers are obstacles that prevent survivors or crewmembers from moving freely around a disaster

area. If both survivors and crewmembers are unable to move freely in a disaster, rescue operations

are affected in the sense that the survivors cannot get to the beeping phones on time or are unable

to reach beeping phones and also crewmembers are unable to rescue survivors quickly. With this

in mind, it is necessary to study how long it takes survivors to reach a beeping phone and how

many survivors reach a beeping phone within a time period. Furthermore, if barriers are not put

into consideration, the results tend to be somewhat inaccurate as real life disaster situations tend

to have obstacles in the way of survivors and crewmembers trying to communicate with each other

to facilitate rescue operations.

The thesis also studies the sensitivity analysis of various Levy walk jumps made by a survivor to

see how the number of survivors arriving at a beeping phone is affected when the mean length

parameter taken by survivors is changed.

The remainder of the thesis is organized as follows. In Chapter 2 we give an overview of PDRN.

An introduction to the Levy walk is discussed in Chapter 3. The PDRN with barriers along with

the simulation results is discussed in Chapter 4. Sensitivity analysis is discussed in Chapter 5, and

concluding remarks are made in Chapter 6.

Chapter 2 - Overview of the PDRN

In this chapter we give a brief summary of the operation of the PDRN. A more detailed discussion

is given in [1]. The PDRN can work as a standalone system and can also be included in a cellular

network and data network. It enables smooth communication between survivor and crewmembers

during a rescue operation. The PDRN architecture consists of one or more Access Points (APs),

Gateway (GW) node, Disaster Recovery Network Phones (DRNP) and on-site Command Center

(CC). An AP can be a Fixed Access Point (FAP) or a Mobile Access Point (MAP). The AP has

two wireless network interfaces, one is a cellular network interface and the other is a backhaul

network interface. Figure 1 shows one of the possible deployment situations that use Access Points

to provide wireless network coverage. In this scenario, the Access Points (AP-1 to AP-6) are fixed

for the duration of the PDRN network and are called FAPs. They can be deployed by helicopter

or by some other means. The APs can be deployed in a random manner in and around disaster

area. After an AP is deployed it establishes communication with the onsite Command Center either

directly or via other APs. If for some reason one of the APs (say, AP-5) is not able to establish

communication with the Command Center and with other APs, then additional APs (say, AP-6)

can be deployed along the path between the AP-5 and Command Center. In this way the AP-5 can

establish communication with Command Center via AP-6.

Figure 1. PDRN architecture [1]

After a disaster has occurred in an area, the PDRN is deployed to cover the affected area with the

aim of rescuing survivors in an efficient manner. The APs are dropped in the disaster area by either

using helicopters or dropping them manually to cover a perimeter of the disaster area. As soon as

the APs hit the ground, they start communicating with the Command Center and notify the

Command Center once a survivor picks up a PDRN phone. The PDRN is a cheap device that has

the following features:

Distinctive ring/beep: To get the attention of survivors who are in the disaster area, the PDRN phones can be configured to produce continuous distinctive ring that is different from a normal phone ring. There is a maximum radius within which the ringing of each phone can be heard. This radius is referred as beeping radius of the phone.

Flashing LED: To enable the survivors to locate the phone at night time, the DRN phones are equipped with LED. The PDRN phone LED flashes continuously.

Cabinet sensor: The PDRN phones have special cabinet sensors. As soon a survivor picks up a phone, the flashing LED and distinctive ring will stop, indicating that the phone is ready to be used.

Automatic registration: As soon as a PDRN phone touches the ground, it performs location registration with any nearby PDRN AP. The PDRN phone has inbuilt GPS systems, and forwards the location information to the AP and the latter forwards it to the Command Center. At the completion of this registration operation the Command Center will include the phone among those that have fallen in wireless coverage region and thus can be used to locate at least one survivor.

After a PDRN phone has communicated with an AP and the Command Center, it will start ringing

continuously. This allows survivors within the area to hear the beep and move towards the phone

to call for help. As soon as a survivor picks up a phone, the beeping sound stops and this allows

the Command Center to know the exact location of a survivor, thereby allowing them to send

rescue crewmembers to the exact location.

Chapter 3 - Introduction to Levy Walk

A random walk is derived from a sequence of Bernoulli trials. It is used in many fields including

thermodynamics, biology, physics, chemistry, and economics where it is used to model

fluctuations in the stock market. Different random walk models are discussed in [1]. In this chapter

we describe one of the models, namely the symmetric two-dimensional random walk.

In a classical random walk, a walker takes a small step that adds to the total walk and there is a

possibility that he will return to the starting point. Let denote the length of the kth step, which

is also called the kth flight length, and assume that has zero mean and a finite variance σx2.

The position of the walker after N steps is given by According

to the central limit theorem, the PDF of the scaled random variable in the limit as

is the normal PDF, which follows from the central limit theorem. Thus,

The central limit theorem assumes that the variance of the is finite.

Mathematically, a random variable X obeys a power law if the complement of its CDF is of the

type [3]

where α is a constant parameter of the distribution known as the exponent or scaling parameter, C

is a normalization constant and is the cumulative distribution function (CDF) of X.

A random variable X is said to be stable (or have a stable or Levy distribution) if it satisfies the

following property. Let and be independent copies of X. Then X is said to be stable if for

any constants a and b the random variable has the same distribution as where c

and d are some constants. The distribution is said to be strictly stable if this condition holds with

. The most general representation of these distributions is through their characteristic

functions which are defined by the following equation:

where C is a positive constant and the four characterizing parameters and are defined by

, is the Levy index or index of stability that determines the weight in the tails. The smaller the value of, the greater the frequency and size of the extreme events. Thus, the parameter describes the asymptotic decay of the distribution. Levy distribution is not normalized or gets negative values when, which violates the requirement that the PDF must be nonnegative. Also, when both the expectation and variance of the Levy distribution do not exist, as discussed earlier.

, is a skewness parameter. A value of implies that the distribution is symmetric. Negative means that the PDF is skewed to the left while a positive means that it is skewed to the right. Thus, describes the symmetry of the distribution.

is positive and measures dispersion. It is similar to the variance of a normal distribution. It is a scale parameter that is a measure of the width of the distribution.

is a real number that may be thought of as a location measure. It is similar to the mean of a normal distribution. It is called the shift parameter.

The parameters and describe the shape of the distribution since defines the asymptotic

decay of the PDF while defines the asymmetry of the distribution. The scale and shift

parameters play lesser role in the sense that they can be eliminated by proper scale and shift

transformations [3, 4]. The PDF corresponding to equation (3) is given by

3.1 - The Special Cases

Generally, rigorous analysis of stable distributions is lacking due to their mathematical

intractability. Consequently, there is no general explicit solution for the form of

. However, the closed form expressions for these distributions are known only

in the following three cases:

When and is irrelevant, we obtain the normal or Gaussian distribution with variance and mean. For the case of we have

When and we obtain the Cauchy distribution, which is also called the Lorentz distribution:

When and we get the Levy-Smirnov distribution:

Figure 2 shows the plots of the PDFs of the three α-stable distributions for various α, β, μ and σ

values.

Figure 2: α-stable density functions for various α, β, σ and μ values

3.2 - Levy Walk versus Levy Flight

The difference between the two walks lies in the velocity. In a Levy flight, the walker visits only

the endpoints of a jump and the notion of velocity does not come up and the jumps take very little

time. This means that in a Levy flight, the walker is only either at the end of the jump or at the

beginning, there is no stop in between the jump. However, in a Levy walk, the walker follows a

continuous trajectory from the beginning of the walk to the end and this leads to a finite time being

needed to complete the walk. The Levy walk takes a flight time that is proportional to the flight

length. Both can also be looked at by the mobility of speed. A Levy flight is a fast mobility model

while a Levy walk is a slow mobility model. Levy flights are unrealistic in many physical systems

because physically realizable systems cannot have diverging moments. Moreover, Levy walks are

realistic and do not violate any physical laws.

Levy walks have been applied to a diverse range of fields such as those that describe animal

foraging patterns [5, 5, 7, 8], the distribution of human travel [9], the stock market [10], some

aspects of earthquake behavior [11], anomalous diffusion in complex systems [12, 13, 14],

epidemic spreading [15, 16] and human mobility [17].

One of the advantages of the Levy walk over the random walk is that the probability of a Levy

walker returning to a previously visited site is smaller than in the random walk. Also, the number

of sites visited by n random walkers is much larger in the Levy walk than in the random walk. The

n Levy walkers diffuse so rapidly that the competition for target sites among themselves is greatly

reduced compared to the competition encountered by n random walkers. The latter typically

remain close to the origin and hence close to each other [18, 19, 20, 21]. This feature is

advantageous in PDRN network where factors such as the concentration of survivors and the

distribution of phones are considered. Irrespective of survivors’ locations each survivor’s

trajectory is likely to be different, and hence when competing to reach the dispersed phones, a

Levy walk model ensures that the probability that two or more of them are heading for the same

phone will be greatly reduced. Thus, with respect to the PDRN, the walker will occasionally take

long steps and thus is more likely to reach the vicinity of a beeping phone than a random walker.

3.3 - Levy Walk Models of PDRN Survivor

The reason for using the Levy walk or any other type of random walk to study the movement of a

survivor in a disaster area is because survivors initially move aimlessly looking for help. In the

PDRN, the survivor is able to reduce walking aimlessly as a result of hearing the beeping of the

phones. The Levy walk types considered are:

Classical Levy walk, which involves four parameters, namely the step size, time taken during each step, the waiting time between steps and the direction for the next step. The step size is based on the Levy distribution and a survivor chooses a random location that is uniformly distributed between 0 and 360 degrees from the current position. The step time is based on Levy distribution, and waiting time between steps is random time and there is no correlation between the four parameters.

Symmetric Levy walk, which is essentially a Levy lattice walk. In this walk the survivor moves in one of the four directions: east, west, north or south. After each step, the survivor chooses the next location with equal probability, and moves in one of the four directions based on probability outcomes. The step size and step duration are based on the Levy distribution, and after each step a survivor waits a random time before the next one. The step size, step duration, direction and waiting time are independent parameters.

Figure 3 shows the simulated trajectories of the different Levy walk types with α-stable densities

and the following parameters: α = 0.5, β = 0, σ =1 and μ = 0.

Figure 3: Simulated trajectories of the different Levy walk types with α = 0.5, β = 0, σ =1and μ = 0

Figures 4 and 5 illustrate the trajectories for the different Levy walk types for different values of

α generated from 4096 points.

Figure 4: Classical Levy walk when β = μ = 0 for various values of α

Figure 5: Symmetric Levy walk when β = μ = 0 for various values of α

In a Levy walk simulation, a survivor is likely to leap over barriers and also leading to the survivor

to leap over the beeping area. To solve this problem, a hybrid model is used that utilizes the Levy

walk until a survivor comes in the district of a beeping zone where he switches over to a form of

the random walk [1]. The phones that are dropped in disaster area are at discrete locations.

Chapter 4 – PDRN with Barriers

In [1] it is assumed that there are no barriers in the disaster area and as a result the survivors are

free to move about anywhere in the area. In a more realistic environment the survivor movement

and that of the crewmembers is hampered by the presence of obstacles, such as fallen trees and

debris. In this chapter we discuss the details of the extension of the work reported in [1].

Specifically, we assume that there are barriers in the disaster area. In the remainder of the chapter

we discuss how the barriers are modeled and discuss the simulation results.

4.1 – Barrier Model

In order to see how efficiently survivors can be rescued in a disaster area through communication

with the dropped phones, simulations were run to see how various combinations of walk types,

phone distributions and survivor distributions affect the result in trying to rescue a survivor with a

barrier in place. These simulations were run on MATLAB software. Various random walks types

were studied with the distributions altered in the walk types that were studied. The area studied

was also varied between a 1km by 1km perimeter and a 3km by 3km perimeter.

During simulations, a walker is assumed to take a step of 0.5m at a time while performing the

Symmetric Random walk model. A walker is assumed to take a jump of 0.75m while performing

the Levy walk model. Upon taking the first step which leads to the survivor arriving at a location

in the disaster area, the survivor does not wait at that location if there is no beeping phone available.

This is because the survivor is assumed to be in a hurry to flee from the disaster area regardless of

if he or she is performing the reward-based walk or the non-reward based walk. As described

earlier, the reward-based walk is when the survivor is walking within a beeping zone with the aim

of looking for a beeping phone to call for rescue while the non-reward based walk is when the

survivor is walking aimlessly.

The barrier as a simulation parameter is assumed to be normally distributed in airspace. It is

assumed that there are ten barriers within an area of distribution with each barrier having a length

of 6m. This means that the barriers are centered on the disaster area. A practical example can be

seen when a big mall falls down, collapsing of a bridge and just about any obstacle that is centered

on the disaster area to prevent easy movement to the rescue phones. The barriers are assumed to

be reflecting as the survivor is not expected to stop walking in a disaster area without

communicating with the Command Center. During the reward-based walk, it is assumed that when

a survivor reaches a barrier, he or she turns either left or right in search of a different route and

continues walking in search of a beeping phone. There is equal probability of ½ in the direction

the survivor turns, left or right, after arriving at the barrier [22]. The probability distribution of

how the survivor turns is based on the reflection principle [24-26] or the case of δ = 1 and p = 1/2

[27, 28]. As it is reward-based, the survivor probably can hear a beeping phone and as a result of

a barrier present, he or she has to find a way to get to the beeping phone by looking for an alternate

route or try and locate another beeping phone within the vicinity. The non-reward based walk

would mean that the survivor simply turns around and looks for another route to keep walking.

The time it takes for a survivor to move around an obstacle in search of a different route depends

on how many steps or jump the survivor takes. As mentioned earlier, the survivor is not expected

to stop until he or she has arrived at a beeping phone and made communication with the Command

Center. Obstacles present slow down survivors from arriving at a beeping or reduce the amount of

survivors arriving at a beeping phone.

The random walk type combinations considered are symmetric random walk to symmetric random

walk (SRW_SRW), Levy walk to symmetric random walk (LEVY_SRW) and Levy walk to Levy

walk (LEVY_LEVY). In the case of SRW_SRW, the survivor is assumed to have an initial walk

type of symmetric random walk and after the survivor has arrived in a beeping zone, the survivor

continues with the same symmetric random walk until he or she finds an alternate route in search

of a beeping phone. The same can be said of the LEVY_LEVY combination walk. However, the

LEVY_SRW does not want the survivor to continue with the same walk upon arriving in a beeping

zone. The survivor changes his or her walk type, in search of alternate route, to the symmetric

random walk type model to search for a beeping phone. The beeping zone is assumed to be a 6m

perimeter.

There are different configurations of survivor and phone distributions considered and these

include:

survivors and phones uniformly distributed

survivors and phones normally distributed

The configurations also cover when the number of phones is fixed and the number of survivors

varied as well as the number of survivors fixed and the number of phones being varied. The reason

for this configuration is to understand how dropping a fixed number of phones in a disaster area

regardless of not knowing how many survivors are in the area affects the efficiency of rescuing

survivors and also how dropping as many phones as possible in a disaster area when there is a

knowledge of how many survivors are in the area affects the efficiency of rescuing survivors.

The configurations furthermore include the reward-based system for which survivors are able to

reach the rescue phones and also a non-reward based system, which is the aimless walk performed

by a survivor. The reward-based system involves a survivor walking towards a direction he or she

hears a louder beeping sound of a rescue phone or walking in the opposite direction if the beeping

sound of a rescue phones is reducing. The reason for this configuration is that without having a

reward-based system, the survivors will just walk aimlessly in the disaster area without searching

for beeping phones. It is important to consider this configuration as one can see the difference in

efficiency in rescuing survivors when they are walking with the aim of finding a beeping phone as

to when they are walking around aimlessly.

The simulation results are the mean first passage time (MFPT) and the percentage of survivors

who reach a beeping phone and are rescued. MFPT is the mean time it takes for a survivor to reach

a beeping phone from the beginning of the search process. The battery life of the phones affects

the simulation results and is varied from 2 hours to 8 hours.

Lastly, there is a sensitivity analysis which shows how long on the average it takes for a survivor

to get to a phone and how many survivors get to the rescue phone when the step length assumed

to be taken by a survivor is altered. This analysis is carried out to see how changing a parameter

in the Levy walk model of a survivor can affect the outcome of rescuing survivors. The outcome

is the mean time it takes for survivors to reach a beeping phone and the percentage of survivors

arriving at beeping phones.

4.2 - Simulation Results

Figures 6 through 13 show the results that were obtained from the various simulations that were

run. From the graphs it can be seen that when a survivor switches from a Levy walk type to a

rewarded Levy walk, there is a performance improvement unlike when the survivor is switching

to a non-rewarded Levy walk type. This means that when a survivor switches to a non-rewarded

walk type, he or she does not walk with the intent of reaching the beeping zone of the rescue phone,

rather the survivor walks aimlessly with the hope of stumbling across a rescue phone. The aimless

walk leads to the performance being poor.

The same can be seen in the other walk type combinations, which include a survivor switching

form a symmetric random walk type to a Levy walk type with reward and without reward and a

survivor switching from a symmetric random walk type to a rewarded or non-rewarded symmetric

random walk type. The walk types studied just show different walk types that can be performed

by survivors within a disaster area. The results obtained tend to be different depending on the walk

type studied as different walk types require survivors to take different distances when attempting

the walk.

The type of distribution used affects both the MFPT and the percentage of survivors rescued. The

better results are obtained when the survivors and phones and normally distributed when compared

to the both of them being uniformly distributed.

The battery life has an impact on the number of survivors that will be rescued in a given area. This

is because as the battery life increases, the number of rescued survivors increases. However, as the

battery life increases, the MFPT decreases and later remains constant with little decrease.

Finally, when the survivors are fixed during a simulation and the phones are also fixed in another

simulation run, it can be seen that performance improves if the number of survivors is fixed in a

given area. Performance at the beginning of the simulation run is slow to show improvement, but

as the number of phones dropped increases, there performance improvement can be seen. If the

number of phones is fixed in a given area, the percentage of survivors rescued will decrease as the

number of survivors increases. This is because if the number of phones is constant and the

survivors in the area begins to increase, there will not be enough phones for the survivors to use.

The poor performance of the LEVY_LEVY model combination is because the model does not take

advantage of its entry into a beeping zone. Since a Levy walk tends to move away from its starting

point and since the probability of landing exactly at a beeping phone is negligibly small, this model

tends to wander about aimlessly without reaching a beeping phone. The other models take

advantage of the beeping zone by switching to a random walk model that has the property of

returning to its starting point with probability 1 in the one-dimensional and two-dimensional walk

[4]. In general, the non-reward models just like the LEVY_LEVY give poor performance in

comparison with the reward-based models of the various walk combinations.

The results from a 1km by 1km disaster area and that of a 3km by 3km disaster area are similar to

each other.

Figure 6: MFPT of survivors reaching a target when the Area is 1km by 1km, with both Phones and Survivors normally distributed and a constant number of Survivors

Figure 7: MFPT of survivors reaching a target when the Area is 1km by 1km, with both Phones and Survivors normally distributed and a constant number of Phones

Figure 8: Percentage of Survivors reaching a target when the Area is 1km by 1km with both Phones and Survivors normally distributed and a constant number of Survivors

Figure 9: Percentage of Survivors reaching a target when the Area is 1km by 1km with both Phones and Survivors normally distributed and a constant number of Phones

Figure 10: MFPT of survivors reaching a target when the Area is 1km by 1km, with both Phones and Survivors uniformly distributed and a constant number of Survivors

Figure 11: MFPT of survivors reaching a target when the Area is 1km by 1km, with both Phones and Survivors uniformly distributed and a constant number of Phones

Figure 12: Percentage of Survivors reaching a target when the Area is 1km by 1km with both Phones and Survivors uniformly distributed and a constant number of Survivors

Figure 13: Percentage of Survivors reaching a target when the Area is 1km by 1km with both Phones and Survivors uniformly distributed and a constant number of Phones

Chapter 5 – Sensitivity Analysis

Sensitivity analysis deals with how the different results of an output are affected by the

corresponding change in the input. The approach taken to perform the sensitivity analysis in this

thesis was to change the mean length parameter of the Levy walk model assumed to be taken by a

walker and observe how such a change affects the time taken for the walker to get to a beeping

phone and how many walkers arrive at the beeping within the given battery life of the phone.

The simulation of the sensitivity analysis was run using similar parameters as discussed in chapter

4 except that only the Levy walk model was observed and the mean length of the walk was altered.

The reward-based model of the Levy walk was studied alone as the non-reward based walk model

would mean that survivors take very long jumps and tend to walk aimlessly with a low probability

of arriving at beeping phones. This would also lead to very poor performance as discussed in

chapter 4. The simulation was done by varying the mean length parameter of the Levy walk type

with the aim of seeing how such an input parameter would affect the overall outcome of the

simulation.

The varied mean length parameters used were, the original 0.75m, which was used for the previous

simulations as well as the sensitivity analysis, 1.5m, 3m and 5m.

5.1 – Simulation Results

From the sensitivity analysis charts, Figures 14 through 21, the increase of the mean length of the

Levy walk type of a survivor shows that the time taken for survivors to reach a beeping reduces

and the percentage of survivors arriving at the beeping phones increases. This means that as the

mean length is increased, there is a performance improvement and it is as a result of the long jumps

taken by survivors which are related to the Levy walk models as stated earlier in this paper. This

is mainly due to the survivor using a reward based walking model. If there was no reward and the

survivor was walking aimlessly, the performance would be very poor as longer jumps would be a

very low probability of arriving at rescue devices. Also, as stated earlier on in this paper, the normal

distribution combination produces better results.

The battery life also has an impact on the survivors that will be rescued in a given area as was seen

in the simulation results discussed in chapter 4. This is because as the battery life increases, the

number of rescued survivors increases. However, as the battery life increases, the MFPT decreases

and later remains constant with little decrease.

Finally, when the survivors are fixed during a simulation and the phones are also fixed in another

simulation run, it can be seen that performance improves if the number of survivors is fixed in a

given area. Performance at the beginning of the simulation run is slow to show improvement but

as the number of phones dropped increases, there performance improvement can be seen. If the

number of phones is fixed in a given area, the percentage of survivors rescued will decrease. This

is because as the phones remain constant, the survivors in the area start to increase and there

becomes not enough phones for the survivors to make use of. This was the same as the simulation

results discussed in chapter 4.

The results from a 1km by 1km disaster area and that of a 3km by 3km disaster area are similar to

each other.

Figure 14: MFPT of survivors reaching a target when the Area is 1km by 1km, with both Phones and Survivors normally distributed and a constant number of Survivors with altered step lengths

Figure 15: MFPT of survivors reaching a target when the Area is 1km by 1km, with both Phones and Survivors normally distributed and a constant number of Phones with altered step lengths

Figure 16: Percentage of Survivors reaching a target when the Area is 1km by 1km with both Phones and Survivors normally distributed and a constant number of Survivors with altered step lengths

Figure 17: Percentage of Survivors reaching a target when the Area is 1km by 1km with both Phones and Survivors normally distributed and a constant number of Phones with altered step lengths

Figure 18: MFPT of survivors reaching a target when the Area is 1km by 1km, with both Phones and Survivors uniformly distributed and a constant number of Survivors with altered step lengths

Figure 19: MFPT of survivors reaching a target when the Area is 1km by 1km, with both Phones and Survivors uniformly distributed and a constant number of Phones with altered step lengths

Figure 20: Percentage of Survivors reaching a target when the Area is 1km by 1km with both Phones and Survivors uniformly distributed and a constant number of Survivors with altered step lengths

Figure 21: Percentage of Survivors reaching a target when the Area is 1km by 1km with both Phones and Survivors uniformly distributed and a constant number of Phones with altered step lengths

Chapter 6 – Conclusion and Recommendations

In this thesis, we have proposed different Levy walk and Random walk models that show the

movement of survivors in the Portable Disaster Recovery Network, which is a communication

infrastructure that enables survivors to communicate with rescue crewmembers to enable them to

be rescued quickly. The rescue devices are dropped in a disaster area from helicopters and start

beeping immediately they reach the ground. It can be seen that the movement of survivors in a

disaster area when modeled by Levy walk is faster than that of Random walk because of the

continuous jumps assumed to be made by the survivors. It becomes faster when the step length

parameter of the Levy walk model is increased.

However, with the Levy walks, survivors have a tendency to leap over the phones without seeing

them hence, a reward-based model was introduced. In this reward-based model, the survivor

switches from the Levy walk to a random walk within a beeping zone and the movement is biased

in favor of directions with louder beeps within the zone. This enables the survivor to move towards

a beeping sound as he approaches a beeping zone with the hope of getting to a phone to make a

call to be rescued.

The performance measures of the model considered include the mean first passage time, which is

the average time it takes a survivor to reach a phone, and the percentage of survivors that reach a

phone and are, therefore, rescued. It can be observed that the distribution of survivors and the

phones in the disaster area has an impact on the two performance measures mentioned. The best

results are obtained when both the survivors and phones are normally distributed followed by when

both are uniformly distributed. The results indicate that when the number of survivors is fixed, the

performance improves as the number of phones being dropped increases. If the number of phones

is fixed in a given area, the percentage of survivors rescued will decrease as the number of

survivors in the disaster area increases. This is because as the phones remain constant, the survivors

in the area start to increase and not enough phones are available for the survivors to use. Finally,

the percentage of survivors rescued increases as the battery life increases.

In the sensitivity analysis graphs, it can be seen that with the increase in mean length, the time

taken for survivors to reach a beeping phone decreases and the percentage of survivors arriving at

the beeping phones increases. This is true because an increase in step length results in longer jumps

taken by the walker resulting in the walker moving faster to get to a destination (beeping phones).

The results shown are of a 1km by 1km area and this does not change when a 3km by 3km area is

studied.

There is still a future with this thesis. This thesis can be further developed using the cluster analysis

in a disaster area. When survivors arrive at a beeping phone, they report their location to the control

center at different times and from different locations. If the disaster area is very larger and a limited

number of crewmembers are available, there would need to be a strategy in place to rescue

survivors efficiently. To do this, the disaster area would need to be partitioned. The need and

number of partition needed involves cluster analysis. The number of clusters in a disaster area

depends upon concentration of survivors, distribution of phone, battery life of phones and other

scenarios one can think of. However, due to time constraint, we were unable to extend the

simulations for this case.

Another area to look into would be if the barriers are uniformly distributed. It would be important

to see how performance is affected when the barriers are uniformly distributed. Real life examples

of uniform distributed can be seen from parts of planes scattered around the disaster area, large

earthquakes and just about any obstacle that is scattered around the disaster area. When the barriers

are uniformly distributed they can form partitions within the disaster area thereby preventing

survivors from moving within the disaster area.