stochastic properties of mobility models in mobile ad hoc networks

Stochastic Properties of MobilityModels in Mobile Ad Hoc Networks

Seema Bandyopadhyay, Member, IEEE, Edward J. Coyle, Fellow, IEEE, and

Tillmann Falck, Student Member, IEEE

Abstract—The stochastic model assumed to govern the mobility of nodes in a mobile ad hoc network has been shown to significantly

affect the network’s coverage, maximum throughput, and achievable throughput-delay trade-offs. In this paper, we compare several

mobility models, including the random walk, random waypoint, and Manhattan models on the basis of the number of states visited in a

fixed time, the time to visit every state in a region, and the effect of the number of wandering nodes on the time to first enter a set of

states. These metrics for a mobility model are useful for assessing the achievable event detection rates in surveillance applications

where wireless-sensor-equipped vehicles are used to detect events of interest in a city. We also consider mobility models based on

Correlated Random Walks, which can account for time dependency, geographical restrictions, and nonzero drift. We demonstrate that

these models are analytically tractable by using a matrix-analytic approach to derive new, closed-form results in both the time and

transform-domains for the probability that a node is at any location at any time for both semi-infinite and finite 1D lattices. We also

derive first entrance time distributions for these walks. We find that a correlated random walk 1) covers more ground in a given amount

of time and takes a smaller amount of time to cover an area completely than a random walk with the same average transition rate,

2) has a smaller first entrance time to small sets of states than the random waypoint and random walk models, and 3) leads to a

uniform distribution of nodes (except at the boundaries) in steady state.

Index Terms—MANET, mobility models, correlated random walk, random walk, random waypoint model.

Ç

1 INTRODUCTION

A mobile ad hoc network (MANET) consists of mobilenodes that rely only on wireless connections to

communicate with each other. These networks are stillbeing developed, so few of them have actually beendeployed. Researchers have thus had to rely on analysisand simulations to predict how these networks willperform. From this preliminary work, it is clear that themobility patterns of the nodes will have a very significantimpact on the network coverage, maximum throughput,and achievable throughput-delay trade-offs. The accuracyof the predictions made about network performance in real-life scenarios will depend on how well the mobility modelsused in the simulations captured the real-life behavior ofnodes. It is therefore important to have realistic mobilitymodels. Hence, many mobility models have been proposedin the literature. A survey of mobility models that havebeen proposed for MANETs can be found in [1], [2]. Thesemodels vary widely in terms of their complexity, analyticaltractability, correlation structure, and restrictions on move-ments. No single one, however, is detailed enough to beappropriate in all situations.

Several past studies [3], [4], [5] have explored the impactof mobility models on network performance and havefound that they significantly affect the protocol perfor-mance and delay-capacity trade-offs. In this paper, wecompare widely used mobility models like the randomwalk, random waypoint, and Manhattan mobility models interms of the number of states visited in a fixed time, thetime to visit every state in a region, and the effect of thenumber of wandering nodes on the time to first enter a setof states. The full coverage time and first entry times of amobility model are useful for assessing the achievable eventdetection rates in surveillance applications where wireless-sensor-equipped vehicles are used to detect events ofinterest in a city. These results provide guidelines for thedensity with which the nodes should be distributed in anarea to achieve event detection within a specified amount oftime, given the information about their mobility behavior.

We also propose and investigate a Correlated RandomWalk (CRW)-based mobility model that is designed tocharacterize the movements of nodes, such as wireless-equipped vehicles, on the streets of a city. The streets aremodeled as a finite, 2D square grid, and each point on thegrid corresponds to an intersection of streets, as shown inFig. 1. A node’s motion is controlled by the following rules:

Two-Dimensional CRW Mobility Model:

. A node takes a step in the same direction as itsprevious step with probability p.

. It takes a step in the opposite direction withprobability q.

. It takes a step in the two orthogonal directions withprobability r where pþ q þ 2r ¼ 1.

. On reaching the boundary of the square region, it isreflected back with probability one.

1218 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 11, NOVEMBER 2007

. S. Bandyopadhyay is with the Department of Computer and InformationScience and Engineering, CSE Building, University of Florida, Gainesville,FL 32611-6120. E-mail: [email protected].

. E.J. Coyle is with the School of Electrical and Computer Engineering, 465Northwestern Ave., Purdue University, West Lafayette, IN 47907-2035.E-mail: [email protected].

. T. Falck is with the Department of Electrical Engineering, Ruhr-Universitat, 44780 Bochum, Germany. E-mail: [email protected].

Manuscript received 14 Mar. 2006; revised 11 Nov. 2006; accepted 15 Dec.2006; published online 7 Feb. 2007.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-0075-0306.Digital Object Identifier no. 10.1109/TMC.2007.1014.

1536-1233/07/$25.00 � 2007 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS

. The time instant at which the node takes its steps isgoverned by a Poisson process of intensity �. Thismeans that a node (vehicle) moves from oneintersection to a neighboring intersection with arandom velocity.

Existing mobility models like the random walk mobilitymodel and the Manhattan mobility model [3] are specialcases of the above mobility model. The random walk mobilitymodel is a 2D CRW with p ¼ q ¼ r ¼ :25 and the Manhattanmobility model is the above model with p ¼ :5, q ¼ 0, andr ¼ :25. In the Manhattan model, there are additionalconstraints: The speed of a node in a step is correlated withits speed in the previous step and is restricted by the speed ofany node preceding it on the same street.

We derive analytical results on both the transient andsteady state behavior of this CRW mobility model. Suchan analysis is important to assessments of the validity ofsimulation results on the performance of protocols thatare almost always obtained by taking an average over atime period.

The main contributions of this paper are:

. Proposal of a mobility model called the CorrelatedRandom Walk model that accounts for temporaldependence, captures such geographic restrictionsas the motion of nodes on streets in a city, and allowsnonzero drift.

. Use of a new technique for transient and steady stateanalysis of CRW mobility models on finite and semi-infinite lattices in both one and two dimensions.

. Derivation in the transform domain of the first-entrytime distributions for 1D CRW mobility models.

. Simulation of the proposed CRW mobility modeland comparisons with the random walk, the randomwaypoint and the Manhattan mobility models interms of first-entry times and time to full coverage.

An initial version of these results can be found in [6].

2 RELATED WORK

2.1 Background in Mobility Models

Many mobility models have been proposed in the literature.A survey of these models can be found in [1], [2], where theauthors categorized the existing mobility models into fourclasses: random models, models with temporal dependency,models with spatial dependency, and models with geo-graphic restrictions. Temporal dependency means that a

node’s movements are correlated in time. Spatial depen-dence refers to correlations between the movements of nodesthat are close to each other. By mobility models withgeographic restrictions, they mean that the movements ofnodes are bounded by streets and obstacles in the region.Models such as random walks and the random waypointmodel and its variations fall under the category of randommodels; they do not consider geographical restrictions,temporal correlation, or spatial correlation. Gauss-Markovmobility models and Smooth Random models considertemporal dependence but do not consider spatial depen-dence or geographical restrictions. Group mobility models[1], [7] capture spatial dependence but do not account fortemporal dependence or geographical restrictions. Pathwaymobility models and Obstacle mobility models restrict thenodes’ movements to pathways in the simulation field butdo not consider temporal or spatial dependence. In [3], twonew mobility models, the Freeway and the Manhattanmobility models, were proposed as models of the mobilitypatterns on highways and city streets. In [8], Hsu et al.proposed a modified version of the random waypoint modelin which there is a weight (that depends on both the currentlocation and time) and a pause-time distribution associatedwith destinations in order to model the popularity of certaindestinations. Boudec and Vojnovic [9] proposed the randomtrip model which contains the random waypoint (RWP) andrandom walk models as special cases. In [10], Jardosh et al.proposed a mobility model in which the nodes move in anarea with obstacles and the destination and pathways of thenodes are determined by the location of these obstacles.Recent research has also focused on studying how realisticthese models are [3]. Researchers have also explored thepossibility of using WLAN traces to develop more realisticmobility models [11], [12].

A significant effort is being made to study how mobilitymodels affect the performance of different MANET proto-cols [3], [4], [10], [13], [14], [15]. In [3], Bai et al. proposed aframework to evaluate the performance of different routingprotocols when the nodes move according to differentmobility models. They observed that the protocol perfor-mance is highly influenced by the mobility models. Bai et al.[4], Trivino-Cabrera et al. [13], Han et al. [14], and Tsenget al. [15] studied how the distribution of path duration in anetwork depends on the mobility model and which in turnaffects the network protocol performance. Jardosh et al. [10]proposed a mobility model in which the nodes move in anarea with obstacles and studied the effect of this model onnetwork performance taking the signal fading in presenceof obstacles into consideration.

In [5], [16], Lin et al. focused on how mobility modelsaffect achievable capacity-delay trade-offs. Gupta andKumar [17] found that per-node capacity decreases asOð1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffin lognp

Þ as the number of nodes n increases in staticwireless multihop networks. Grossglauser and Tse [18]showed that node mobility can improve the per-nodecapacity in multihop wireless networks, but at the cost ofincreased delay. In [5], [16], Lin et al. found that the delay-capacity trade-offs are radically different for different(Brownian motion, i.i.d., and the random way-point)mobility models.

Recently, researchers have also studied the effect ofmobility on various other performance measures in

BANDYOPADHYAY ET AL.: STOCHASTIC PROPERTIES OF MOBILITY MODELS IN MOBILE AD HOC NETWORKS 1219

Fig. 1. Model for streets in a city. The horizontal and vertical motions ofthe node combine to create diagonal movements along the solid linesthat are shown. These diagonal solid lines form the grid of city streets.

MANETs. Lassila et al. [19] and Liu et al. [20] studied theimpact of mobility on connectivity and coverage inMANETs. In [21], Chin et al. studied the effect of plannedmobility on coverage in sensor networks in the presence ofnoise and obstacles. In [22], Chin et al. studied the latency intarget detection by sensors moving according to anunccordinated radom walk (sensors choose a randomlocation within radius r around its current location in eachmeasurement period). In [23], Kesidis et al. have analyzedthe distribution of time to detect an object by sensorsmoving according to a Brownian Motion mobility model.

In the last few years, researchers have also paid attentionto the study of stochastic properties of mobility models thatare widely used in simulation studies. Getting a betterunderstanding of how the models behave as time evolves insimulation studies is very important in order to avoidmisinterpretations of simulation results. Examples of suchresearch are [24], [25], [26], [27], [28], [29], [30], [31], inwhich the authors have looked at the steady-state char-acteristics of the random waypoint and random directionmodel. In [32], Yoon et al. showed that mobility models likethe RWP have a transient period which, if not taken intoaccount, can impair the accuracy of simulations results.They then developed stationary mobility models that caneliminate these transient periods and, hence, lead to reliablesimulation results. Boudec and Vojnovic [9] proposed therandom trip model, which contains the random waypoint(RWP) and random walk models as special cases and foundconditions for a stationary regime to exist. Garetto andLeonardi [33] analyzed the relation between the mobilitymodel parameters and the duration of the transient periodand proposed a generalized random direction model toachieve the desired steady state node distribution.

To the best of our knowledge, none of the existing studieshave compared mobility models in terms of their time tofirst entry to a given region and their time to full coverageof an area, which is the focus of the study in this paper.Also, no earlier paper has considered and analyzed thestochastic properties of a correlated random walk-basedmobility model as general as the one proposed in this paper.

2.2 Background in Correlated Random Walks

The earliest available work on the subject of correlatedrandom walks appears to be that of Goldstein [34], whogave limiting distributions under various conditions for thediscrete-time, 1D CRW on ð�1;þ1Þ. In a 1D correlatedrandom walk, the walker takes a step in the same (opposite)direction as its previous step with probability p1 or p2

(q1 ¼ 1� p1 or q2 ¼ 1� p2) depending on whether itsprevious step was in the positive or the negative direction.He also derived formulae for the moments and obtained anasymptotic estimate of the distribution function in terms ofhyper-geometric functions. Researchers [35], [36], [37], [38]have obtained generating functions or exact combinatorialexpressions for the probability of being at any lattice pointat the nth step for a discrete-time CRW on a 1D lattice thatis unrestricted1 or has one or two absorbing boundaries.Zhang [39] obtained explicit expressions for the absorbingprobability and expected duration of the discrete-timeasymmetric2 CRW (ACRW) in the presence of elastic or

absorbing barriers. Seth [40] and Nain and Sen [41]obtained generating functions of the probabilities for thetime to the first and jth passage to the origin or any otherlattice point in a discrete-time, unrestricted ACRW. In [42],Nain and Sen proposed the use of transition probabilitymatrices associated with discrete-time CRWs to obtainvarious probability distributions for CRWs, even when thesteps are of unequal lengths. Lal and Bhat [38] used thematrix-geometric results [43] to obtain the steady stateprobabilities for a discrete-time, 1D asymmetric CRW withone or two reflecting boundaries. They further obtained theprobability of first passage to the boundaries at the nth stepin a discrete-time ACRW on a 1D lattice with two absorbingboundaries. Gillis [44] studied the discrete-time CRW on ad-dimensional lattice in which the walker takes a step in thesame direction as its previous step with probability p, in thedirection opposite to its previous step with probability q,and in one of the directions orthogonal to his previous stepwith probability r and pþ q þ 2ðd� 1Þr ¼ 1. He obtained agenerating function for the probability that the walker is ata particular lattice point at the nth step and the probabilityof return to the origin at the nth step for the case p ¼ r.

To the best of our knowledge, none of the existingstudies provide transient probabilities and first passagetimes to any lattice point starting from 0 for a continuous-time CRW on a finite lattice in one or two dimensions; wederive these quantities in this paper.

3 STOCHASTIC PROPERTIES OF CRW MOBILITY

MODELS

Before comparing the stochastic properties of variousmobility models, we derive the transient and steady stateprobabilities for the 2D CRW mobility model described inSection 1.

3.1 Transient Probability Distributions for CRW onOne-Dimensional Lattice

In this section, we first derive the transient probabilitydistributions of a continuous-time 1D CRW on ½0;1Þ with areflecting boundary at 0.

In a 1D continuous-time CRW, the walker movesaccording to the following rules:

. The walker takes a step in the same direction as itsprevious step with probability p1 or p2 or dependingon whether its previous step was in the positive orthe negative direction.

. It takes a step in a direction opposite to its previousstep with probability q1 ¼ 1� p1 or q2 ¼ 1� p2

depending on whether its previous step was in thepositive or the negative direction.

. On reaching a reflecting boundary, it takes a step inthe opposite direction with probability one.

. The time at which it takes its next step is governedby a Poisson process of intensity �.

Then, we propose a mapping of a continuous-time CRWon ½0;1Þ to a continuous-time CRW on ½0; N � withreflecting boundaries at 0 and N . This mapping allowsus to obtain the Laplace transform of the transientprobability distributions in closed form for the CRW on½0; N � with two reflecting boundaries.


1. The term “unrestricted” is used to mean random walks on ð�1;þ1Þ.2. A CRW is called asymmetric if p1 6¼ p2 and symmetric otherwise.

3.1.1 Transient Analysis of a CRW on ½0;1ÞAn asymmetric CRW on ½0;1Þ with a reflecting boundary

at 0 can be modeled as a quasi-birth-death (QBD) process

with state transition diagram shown in Fig. 2. The state n�

(or nþ) at any level n is reached when the walker reaches

the location n from the location nþ 1 (or n� 1). Note that

the boundary location 0 has only one state 0�. If Q denotes

the generator for this QBD process, the Laplace transform

ð�ðsÞÞ of the transient probabilities �ðtÞ satisfies the

equations [43], [45], [46]:

�ðsÞðQ� sIÞ ¼ ��ð0Þ; �ð0Þ ¼ �ðtÞjt¼0: ð1Þ

For this QBD process,

Q ¼

�� 0 0 0 0 0 0 . . .q1� �� 0 p1� 0 0 0 0 . . .p2� 0 �� q2� 0 0 0 0 . . .0 0 q1� �� 0 p1� 0 0 . . .0 0 p2� 0 �� q2� 0 0 . . .0 0 0 0 q1� �� 0 p1� . . .0 0 0 0 p2� 0 �� q2� . . .

..

. ... ..

. ... ..

. ... . .

.

2666666666664

3777777777775:

We assume that the initial position of the walker is 0; i.e.,

�ð0Þ ¼ ½1; 0; 0; . . .�. Let �iðsÞ denote the vector of Laplace

transforms of the probability of being in the states at level i;

i.e., �0ðsÞ ¼ ½�0�ðsÞ� and �iðsÞ ¼ ½�iþðsÞ;�i�ðsÞ� for i > 0.

Also, �ðsÞ ¼ ½�0ðsÞ;�1ðsÞ; . . .�. Then, from (1), we get

�ð�þ sÞ�0ðsÞ þ�1ðsÞq1�

p2�

� �¼ �1; ð2Þ

��0ðsÞ þ�1ðsÞ�ð�þ sÞ

0

� �¼ 0; ð3Þ

�nðsÞBðsÞ þ�nþ1ðsÞCðsÞ ¼ 0; n � 1; ð4Þ

where

BðsÞ ¼ 0 p1��ð�þ sÞ q2�

� �and

CðsÞ ¼ q1� �ð�þ sÞp2� 0

� �:

From (4), we get

�nþ1ðsÞ ¼ �nðsÞWðsÞ; ð5Þ

where

WðsÞ ¼ �BðsÞ½CðsÞ��1 ¼p1��þs � p1q1�

p2ð�þsÞq2��þs

�þsp2�� q1q2�

p2ð�þsÞ

" #:

Observe that, to obtain the complete transient probabilitydistribution, we need to obtain the three boundary variables,�0�ðsÞ, �1�ðsÞ, and �1þðsÞ; the rest of the entries in vector�ðsÞ can then be easily obtained by using the recursiverelation in (5). But, also observe that we have only twoequations, (2) and (3), in these three variables. Hence, weneed another equation to be able to solve for the boundaryvariables. We resort to the results in [46], which prove thatthe boundary variables also satisfy the following equation:

�1ðsÞp1ðsÞ ¼ 0; ð6Þ

where p1ðsÞ denotes the right eigenvector corresponding tothe eigenvalue of WðsÞ whose magnitude is greater than orequal to one for all possible values of s. We find that WðsÞ isdiagonizable and its eigenvalues are given by

�kðsÞ ¼fðsÞ � ð�1Þk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðfðsÞÞ2 � 4p1p2�2ð�þ sÞ2

q2p2�ð�þ sÞ

;

where fðsÞ ¼ ðp1 þ p2Þ�2 þ 2s�þ s2, for k ¼ 1; 2. The righteigenvectors pkðsÞ, k ¼ 1; 2 corresponding to the eigenva-lues �1ðsÞ and �2ðsÞ are given by

pkðsÞ ¼def pk1ðsÞ

pk2ðsÞ

� �

¼fðsÞ � 2p1p2�

2 þ ð�1ÞkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðfðsÞÞ2 � 4p1p2�2ð�þ sÞ2

q2p2ðp2 � 1Þ�2

1

2664

3775:

We find numerically that, for all possible values of p1, p2,�, and s, the magnitude of �1ðsÞ is greater than or equal to 1.Hence, (6) implies that �1�ðsÞ ¼ ��1þðsÞp11ðsÞ. Using thisequation with (2) and (3), we find

�1þðsÞ ¼2�ð1� p2Þ

ðp1 þ p2 � 1Þ�2 þ ð1� 2p2Þðsþ �Þ2 þ gðsÞ;

where

gðsÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsþ�Þ4� ½ð1� 2p1Þð1� 2p2Þþ 1��2ð�þ sÞ2þðp1þ p2� 1Þ2�4

q:

Without loss of generality, we assume � ¼ 1 from thispoint. It can be proven that the inverse transform of �1þðsÞis given by

�1þðtÞ ¼ 2q2e�t�p1 þ p2 � 1

4p2ðp2 � 1Þ t � sinhðtÞ þ 1� 2p2

4p2ðp2 � 1Þ sinhðtÞ

� 1

4p2ðp2 � 1Þ ðe�t ffiffiffiffiffiffiffip1p2p

J0ðtffiffiffiffiffiffi�cp

ÞÞ

� ðetffiffiffiffiffiffiffip1p2p


ÞÞ þ ð1�2p1Þð1�2p2Þ4p2ðp2�1Þ sinhðtÞ

� ðe�tffiffiffiffiffiffiffip1p2p


ÞÞ � ðetffiffiffiffiffiffiffip1p2p


ÞÞ

� ðp1 þ p2 � 1Þ2

4p2ðp2 � 1Þ t � sinhðtÞ

� ðe�tffiffiffiffiffiffiffip1p2p


ÞÞ � ðetffiffiffiffiffiffiffip1p2p


ÞÞ�;

where q2 ¼ 1� p2, c ¼ ð1� p1Þð1� p2Þ, and J0ðtÞ is thezeroth order Bessel function of first kind. Note that the


Fig. 2. State transition diagram of a correlated walk on ½0;1Þ.

arguments of the Bessel functions in the above equation arecomplex-valued. Also, it can be proved that

�0�ðtÞ ¼ �0ðtÞ � �1þðtÞ;�ðnþ1ÞþðtÞ ¼ p1�nþðtÞ þ q2�n�ðtÞ;

�ðnþ1Þ�ðtÞ ¼ �ðp1q1=p2Þ�nþðtÞ þ1

p2�0ðtÞ � �n�ðtÞ �

q1q2

p2�n�ðtÞ:

Special Case: When p1 ¼ p2 ¼ p,

�1þðtÞ ¼ e�t

2p� 1

2pþ 1

2pI0ðtÞ � I0ðð2p� 1ÞtÞ � ð2p� 1Þ2

2pt

� I0ðtÞ � I0ðð2p� 1ÞtÞ!;

�0�ðtÞ ¼ e�t2p�1

2p þ 12p I1ðtÞ � I0ðð2p� 1ÞtÞþ 1

2p I0ðð2p�1ÞtÞ

� ð2p�1Þ22p I0ðtÞ � I0ðð2p� 1ÞtÞ

0@

1A;

�1�ðtÞ ¼ e�t

ð2p�1Þðp�1Þ2p2 t� 8p2�10pþ3

2p2

h iI0ðtÞ � I0ðð2p� 1ÞtÞ

þ ð2p�1Þ2ðp�1Þ2p2 t � I0ðtÞ � I0ðð2p� 1ÞtÞ

þ 14p2 I2ðtÞ � I0ðð2p� 1ÞtÞ þ 2p�1

2p I1ðð2p� 1ÞtÞ

0BBB@

1CCCA;

�ðnþ1ÞþðtÞ ¼ p�nþðtÞ þ q�n�ðtÞ; �ðnþ1Þ�ðtÞ ¼ �q�nþðtÞ

þ 1

p�0ðtÞ � �n�ðtÞ �

q2

p�n�ðtÞ;

where IvðtÞ is the vth-order modified Bessel function of

the first kind.

3.1.2 Transient Analysis of a CRW on ½0; N�The CRW on ½0; N� with reflecting boundaries can be

modeled as a finite QBD process with the state transition

diagram shown in Fig. 3.The transient probability distribution for such a walk

can be obtained by solving the equation �ðtÞ ¼ �ð0ÞeQfint,

where Qfin is the generator of the finite QBD process. But,

as the number of states in the process increases, it becomes

increasingly difficult to solve this equation. Hence, in this

paper, we take a different approach that allows transient

analysis even when the number of states is large. We map

a CRW on ½0;1Þ to a CRW on a finite lattice and use the

transient probabilities obtained for CRW on ½0;1Þ to

obtain the transient probability distributions for contin-

uous-time CRW on a finite lattice with reflecting bound-

aries at the two ends.The correlated random walk on ½0; N � with reflecting

boundaries at 0 and N can then be obtained by mapping thelocations ð2lN � iÞ, l ¼ 1; 2; . . . in the correlated randomwalk on ½0;1Þ with a reflecting boundary at 0 to the

location i, i ¼ 1; 2; . . . ; N in the correlated random walk on

½0; N �. By this, we mean that the walker of CRW on finite

lattice is at location i, i ¼ 1; 2; . . . ; N when the walker of the

CRW on ½0;1Þ is at any one of the locations in the set

fj : j ¼ 2lN � i; l ¼ 1; 2; . . .g. This mapping is shown in

Fig. 4 for N ¼ 6. In Fig. 4, the black dots represent the

lattice points and the number adjacent to these dots denote

the location of the walker of the CRW on ½0;1Þ. The walker

of the CRW on [0, 6] is at location 2 when the walker of the

CRW on ½0;1Þ is at any one of the locations in the set

f2; 10; 14; . . .g.Hence, the probability that the walker of the CRW on

½0; N � is at location i, i ¼ 1; 2; . . . ; N at any time t is the

sum of the probabilities that the walker of the CRW

on ½0;1Þ is at any one of the locations in the set

fj : j ¼ 2lN � i; l ¼ 1; 2; . . .g. Let ��iðsÞ denote the Laplace

transform of transient probability distributions of being

at the ith location for the correlated random walk on

½0; N � with reflecting boundaries at both ends. Then,

using (5) recursively, we get

��iðsÞ ¼X1l¼0

�2lNþiðsÞ þX1l¼1

�2lN�iðsÞ

¼X1l¼0

�1ðsÞ½WðsÞ�2lNþi�1þX1l¼1

�1ðsÞ½WðsÞ�2lN�i�1ðsÞ; ð7Þ

0 < i < N;

��0ðsÞ ¼X1l¼0

�2lNðsÞ ¼ �0ðsÞ þX1l¼1

�1ðsÞ½W ðsÞ�2Nl�1; ð8Þ

��NðsÞ ¼X1l¼0

�2lNþNðsÞ ¼X1l¼0

�1ðsÞ½WðsÞ�2NlþN�1: ð9Þ

Since the matrix WðsÞ is diagonalizable,

½WðsÞ�n ¼ XDnX�1; where D ¼�1ðsÞ 0

0 �2ðsÞ

� �

and X ¼p11ðsÞ p21ðsÞp12ðsÞ p22ðsÞ

� �:

In the above equation, X�1 denotes the inverse of

the matrix X, �iðsÞ are the eigenvalues of WðsÞ, and

½pk1ðsÞ; pk2ðsÞ�T ; k ¼ 1; 2 are the corresponding right eigen-

vectors. It can be proved that, for any n, �1ðsÞ½W ðsÞ�n ¼ð�2ðsÞÞn�1ðsÞ: Hence, (7), (8), and (9) reduce to


Fig. 3. State transition diagram of a correlated walk on ½0; NÞ.

Fig. 4. Mapping of a correlated random walk on ½0;1Þ to a correlated

random walk on [0, 6].

��iðsÞ ¼�1ðsÞð�2ðsÞÞi�1

1� ð�2ðsÞÞ2Nþ 1

�2ðsÞð Þiþ1

1

1� ð�2ðsÞÞ2N� 1

! !;

ð10Þ

��0ðsÞ ¼�0�ðsÞ þ�1ðsÞ1

�2ðsÞ1

1� ð�2ðsÞÞ2N� 1

!eT; ð11Þ

��NðsÞ ¼X1l¼0

�1ðsÞ½W ðsÞ�2NlþN�1¼�1ðsÞð�2ðsÞÞN�1

1� ð�2ðsÞÞ2NeT; ð12Þ

where e ¼ ½1; 1�T . The Laplace transforms of the transient

probabilities obtained above could not be inverted analy-

tically, but they can be inverted numerically by any of the

many algorithms available in the literature [47], [48], [49].

Fig. 5 shows the transient probabilities obtained whenthe Gaver-Wynn-Rho (GWR) algorithm [49] was used tonumerically invert (10) for N ¼ 12 and for different valuesof p and i. The initial position of the walker is 0 and the rateof the Poisson process is set to � ¼ 10. In Fig. 5, for p ¼ 0:99,we observe that the occupation probabilities for the selectedstates first oscillate for a while and then settle to theirsteady state values. For the first few steps, the behavior isalmost deterministic because there are very few changes ofdirection. Hence, the first peaks are visible at t ¼ i=�, wherei denotes the state index. As the probability p becomessmaller, we still have overshoots in all states, but there arealmost no oscillations. Also, the steady state is reachedmuch earlier; at about t ¼ 4 instead of at t ¼ 9. The last twofigures ðp ¼ :8; p ¼ :6Þ illustrate that the overshoots are onlyvisible in the very first states with decreasing correlation.


Fig. 5. Transient probability distribution of a CRW on [0, 12]. Different curves correspond to different locations on the lattice. (a) pR ¼ pL ¼ 0:99.(b) pR ¼ pL ¼ 0:9. (c) pR ¼ pL ¼ 0:8. (d) pR ¼ pL ¼ 0:6. For a color version of this figure, please see http://www.cise.ufl.edu/~seema/papers/StochasticPropertiesTMC.pdf.

We also observe that the time to reach the steady state doesnot reduce any further. That is because only the first graphshows an oscillation that has to be damped. The rise time tothe limiting value is almost identical for all probabilities.Note that for all but the first and the last state, theprobabilities converge to the same value 1/12, and the firstand last converge to 1/24; this is in accordance with theanalytical steady state probability results presented inSection 3.2. This is expected as boundary states can onlybe reached from one direction.

3.2 Steady-State Probability Distribution for CRWon ½0; N�

For the sake of completeness, we provide the steady state

probability of being at any location n found in [38] below:

��0 ¼1

2

ðp1=p2Þ � 1

ðp1=p2ÞN � 1; ��N ¼ ��0ðp1=p2ÞN�1;

��n ¼ ��0 1þ p1

p2

� �p1

p2

� �n�1

; n ¼ 1; 2; . . . ; ðN � 1Þ:

For the special case of p1 ¼ p2 ¼ p, ��0 ¼ ��N ¼ 1=ð2NÞ and

��n ¼ 1=N; 1; 2; . . . ; ðN � 1Þ.

3.3 First Passage Times

In this section, we obtain the distribution of the first passage

time to any lattice point in such walks assuming that the

walker was at 0 at time t ¼ 0. To obtain the first passage

probabilities, let us first define the following:

. Tij: time of first passage from state i to state j.

. fijðtÞ: pdf of the variable Tij.

. qij: the ½i; j�th element of the generator Q of the QBDprocess modeling the walk.

. LijðsÞ: Laplace transform of the first passage time tolattice point j from lattice point i.

The Laplace transform of Tij is given by

LijðsÞ ¼defZ 1

0

e�stfijðtÞdt ¼ E½expð�sTijÞ�

¼ qijs� qii

þXk 6¼j

qiks� qii

Lkj;ð13Þ

which implies

ðqii � sÞLijðsÞ þXk6¼j

qikLkjðsÞ ¼ �qij: ð14Þ

Now, since the walker can reach the state n� only after it

has already reached the state nþ, we have L0nðsÞ ¼Lð0�ÞðnþÞðsÞ: Also, define LjþjþðsÞ ¼ 1. Let QðjÞ denote

the submatrix which is obtained by deleting the columns

numbered ðj�Þ and beyond and deleting the rows ððj� 1Þ�Þand beyond from the generator Q. For j ¼ 4,

Qð4Þ ¼

�� 0 0 0 0 0 0q1� �� 0 p1� 0 0 0 0p2� �� 0 q2� 0 0 0 00 0 q1� �� 0 p1� 0 00 0 p2� �� 0 q2� 0 00 0 0 0 q1� �� 0 p1�

26666664

37777775:

Then, (14) reduces to ðQðjÞ � sIÞLjðsÞ ¼ 0,

LjðsÞ ¼ ½Lð0�ÞðjþÞðsÞ; Lð1þÞ;ðjþÞðsÞ; Lð1�Þ;ðjþÞðsÞ; . . . ;

Lððj�1Þ�Þ;ðjþÞðsÞ; LðjþÞ;ðjþÞðsÞ�T ;

which can be rewritten as the following set of equations:

�þ s�

L0�;jþðsÞ ¼ L1þ;jþðsÞ; ð15Þ

Ln�;jðsÞLðnþ1Þþ;jðsÞ

" #¼ �C�1B

Lðnþ1Þ� ;jðsÞLðnþ2Þþ ;jðsÞ

" #

¼ �WðsÞLðnþ1Þ�;jðsÞLðnþ2Þþ;jðsÞ

" #; 0 � n � j� 3; ð16Þ

Lðj�1Þþ;jþðsÞ ¼q1�

�þ sLðj�2Þ�;jþðsÞ þp1�

�þ s ; ð17Þ

where

BðsÞ ¼ 0 p1��ð�þ sÞ q2�

� �;

CðsÞ ¼ q1� �ð�þ sÞp2� 0

� �;

and, hence,

�WðsÞ ¼�þsp2�

� 1�p2

p2

1�p1

p2

ðp1þp2�1Þ�p2ð�þsÞ

" #:

From (15), (16), and (17), we get

L0� ;jþðsÞ ¼p1�

�þ sa½ �WðsÞ�j�2

12 � b½ �WðsÞ�j�222

a� bð�þsÞ�

" #;

a ¼ ½ �WðsÞ�j�221 þ ½ �WðsÞ�j�2

22

q1�

�þ s ;

b ¼ ½ �WðsÞ�j�211 þ ½ �WðsÞ�j�2

12

q1�

�þ s ;

ð18Þ

where ½ �WðsÞ�j�2ij is the ði; jÞth element of matrix ½ �WðsÞ�j�2.

The Laplace transform given by (18) can be invertednumerically by using one of the many algorithms availablein the literature [47], [48], [49].

3.4 Probability Distribution for 2D CRW MobilityModels

For special cases where the CRW on the 2D finite lattice canbe decomposed into a pair of independent 1D CRWs (asshown in Fig. 1), the transient and the steady stateprobability distributions obtained in Sections 3.1.2 and 3.2can be used to obtain the probability of the walker being atany grid point at a given point as follows: The probability ofbeing at location (x, y) at any time instant t is the product ofprobabilities of being at locations x and y for the 1D CRWsin the east-west and north-south directions, respectively, atthe time instant t.

Note that the steady state probability of a CRW on ½0; N �is 1=ðNÞ for any nonboundary state. Hence, for 2D CRWsthat can be decomposed into two independent 1D sym-metric CRW, the steady state probability of being at anynonboundary state is equal. This means that, in a simulationstudy, the 2D CRW mobility model, unlike the random waypoint


model, leads to a uniform distribution of nodes (except at theboundaries) in the steady state.

However, the results on first passage time for 1D CRWcannot be readily extended to CRWs in two dimensions.Hence, we resort to simulation studies (described in thenext section) to study various characteristics like the firstentrance time, full coverage time, and transient behaviorsfor more general cases of the proposed CRW mobilitymodel on a 2D grid.

4 SIMULATION STUDY

In this section, we describe the simulation study weconducted to compare various characteristics of theproposed CRW mobility model with widely used mobilitymodels.

4.1 Models Compared

Random Waypoint (RWP) Model. In this model, the nodestarts by choosing a new position and velocity. It thenmoves in a straight line toward the chosen destination at thechosen velocity. On reaching the destination, it chooses anew destination and velocity that are independent of allprior choices, and so on.

Correlated Random Walk (CRW) Model. We have simulateda 2D correlated random walk mobility model on a finitearea with reflecting boundaries. We have in particularconsidered CRWs in two dimensions that can be decom-posed into two 1D CRWs, one in the north-south and theother in the west-east direction. Among such walks, weconsider two variants. In the first variant, the decomposedCRWs share the same Poisson process that determines thetimes of state transitions. This results in strictly diagonalmovements, as shown in Fig. 6. Furthermore, this yields astate space where only approximately half of the states areaccessible. For a lattice with w� h states, the number ofreachable states is bwh=2c. In the second variant, thedecomposed CRWs are completely independent; they areeach driven by their own independent Poisson process. Thewalker cannot move diagonally in this variant because, for adiagonal movement, both of the Poisson processes, one foreach 1D CRW, would need to have an arrival at the sametime, an event of probability zero. Thus, in this scenario, allstates can be visited by the CRW walker.

Random Walk (RW) Model. In this model, the walkermoves in any one of the four possible directions on the 2D

grid with equal probability regardless of its previousmoves. It is therefore a special case of a 2D CRW modelwhich can be decomposed into two independent 1Dsymmetric CRWs with p ¼ :5.

Manhattan Mobility (MM) Model. In this model, thewalker moves according to the 2D CRW model describedin Section 1 with p ¼ :5, q ¼ 0, and r ¼ :25. The velocity of anode in one step is correlated to its velocity in the previousstep and restricted by the velocity of any node preceding iton the same street.

4.2 Parameter Tuning for Fair Comparison ofDifferent Mobility Models

We have simulated all models in continuous-time on a finite2D lattice of size h� v with four reflecting boundaries. Thereflecting boundaries have been implemented as illustratedby the arrows in Fig. 1.

The RW model and the MM model are special cases of

the CRW model and, hence, they can be easily compared

with the CRW model. But, in the RWP model, the walker

moves toward a chosen destination in a continuous space at

a constant chosen speed. Hence, to compare the RWP model

with the CRW model, we adapted the RWP model to a

discrete state space. We describe below the details of how

we have tuned the parameters of these models to have a fair

comparison.

Correlated Random Walk Model Parameters. We simulate

both variants of the 2D CRW. For the first variant of the 2D

CRW, the number of arrivals per unit time is, on average, �.

But it is tricky to find the distance the walker moves in each

step because it moves diagonally. We assume that this

diagonal distance is covered in two steps, one horizontal

and one vertical, in order to refer all motion to the same

lattice as the prior case. The average velocity in this case is

then given by v ¼ 2�. For the second variant of the CRW

model, each Poisson process corresponding to each decom-

posed 1D CRW has a rate of �. As they are independent, the

average number of steps per unit time is 2� and each step

covers a unit distance. The walker thus again has a speed of

v ¼ 2�.Random Waypoint Model Parameters. The Random Way-

point model is more challenging to adapt because it isactually a continuous-space model. It is apparent fromFig. 7 that there is usually no unique way of reaching aparticular destination from a particular source in a grid.Fig. 7 shows two out of 10 possible paths between twolattice points. We have considered two variants in oursimulations. In the first variant, we randomly choose oneof all the possible ways; in the second variant, we


Fig. 6. Objective area (: allowed, �: forbidden states).

Fig. 7. Two possible paths between a set of source and destination

lattice points.

minimize the distance of the discrete movement to astraight line connecting the origin and destination nodes.The velocity at which the node moves toward a newdestination is assumed to be exponentially distributed withmean 2�. The next issue to be handled is how to realizereflections at a boundary in the RWP model so as toachieve similar behavior as the CRW model. We define anextended lattice of size 2h� 2v and pick a destinationuniformly from this area. If the destination falls outside theoriginal lattice of size h� v, we map it back into the actualarea as illustrated in Fig. 8.

4.3 Simulation Results

Sample Paths. Fig. 9 shows sample paths of three mobilitymodels—random walk, asymmetric CRW (ACRW), andsymmetric CRW—on a 200� 200 grid for 100 units oftime. The random walk path appears mostly in the upperleft corner and is shown with the darkest tone, theasymmetric CRW generally runs diagonally and is shownwith the lightest tone, and the symmetric CRW is shownwith the medium tone. Each of the subfigures, Figs. 9a and9b, consist of two separate graphics, both sharing the exactsame set of parameters. This is done to show more thanone sample function for each situation. This graphicalrepresentation of the states that are visited provides anintuitive understanding of the structural differencesbetween walks generated by the different models. Thestarting position for each model is chosen in a way thatensures that each node is able to move some distance inevery direction without hitting a boundary.

One thing to note in these sample paths is that thedistance, from start to finish, covered by the random walkmodel is much smaller than that covered by the other twomodels. For the symmetric CRW model with p ¼ 0:9 inFig. 9a, we observe that the node has a tendency to move inthe same direction and travel across the lattice until it isreflected by a boundary and then trapped in the corner.This is not the case for the random walk model andsymmetric CRW with weaker correlation ðp ¼ 0:8Þ inFig. 9b. As expected, we clearly observe that, withincreasing correlation, the average length of periods with-out change of direction increases. Thus, introduction ofasymmetry in CRWs can achieve a very high degree ofdirectionality. Not only do we observe long periods withouta change in direction, but even after a reflection (a forcedchange in direction), the process resumes heading in the olddirection (toward the boundary) very rapidly.

First Entry Times. In Fig. 10a, we have plotted the time offirst entry to a 6� 6 area in the center of a 100� 100 grid forthe RW, RWP, CRW, and Manhattan mobility models as afunction of the total number of walkers wandering in thearea. We observe that, as p approaches 1, the logarithm of thefirst-entry times ðtFEÞ tends to a linear function of thelogarithm of the number of walkers ðwÞ, i.e., logðtFEðwÞÞ ¼�� logðwÞ þ � or tFEðwÞ ¼ 10�=w�. This yields the conclu-sion that, in the limit, the time of first entry is the reciprocal of the

number of walkers. With an increasing number of walkers, thebehavior converges to the (almost) deterministic case. This isintuitively reasonable because the probability of a latticepoint being occupied in the initial distribution of walkers ishigher; hence, the mean distance to the observed area issmaller and directionality is not that important anymore.

We also observe in Fig. 10a that, given the number ofwalkers deployed, we can always find a value for the probability

p in the CRW model, which, when used, will lead to a smaller first

entry time than the RWP model; in fact, for p ¼ 0:99, the first

entry time for the CRW model is always smaller than the RWP

model. Also, we observe that the first entry times for CRWs

with p > 0:5 and p > 0:8 are much smaller than the RW (CRW

with p ¼ 0:5) and the MM models, respectively. The plots of thetime of first entry as a function of the probability p,


Fig. 8. Reflection at the boundary of the objective area.

Fig. 9. Sample paths for the random walk, asymmetric correlated random walk (ACRW), and symmetric correlated random walk (CRW). The randomwalk path appears mostly in the upper left corner and is shown with the darkest tone, the asymmetric CRW path runs diagonally and is shown withthe lightest tone, and the symmetric CRW is shown with a medium tone. (a) ACRW ðpN ¼ pE ¼ 0:9; pS ¼ pW ¼ 0:7Þ and CRW ðp ¼ 0:9Þ. (b) ACRWðpN ¼ pE ¼ 0:8; pS ¼ pW ¼ 0:6Þ and CRW ðp ¼ 0:8Þ. For a color version of this figure, please see http://www.cise.ufl.edu/~seema/papers/StochasticPropertiesTMC.pdf.

provided in 10b, are quite similar to (but not as steep as)Fig. 10a.

Full Coverage Times. In Fig. 11a, we plot the full coveragetime of the 100� 100 grid for RW, 2D CRW, RWP, andManhattan mobility models as a function of the number ofwalkers wandering in the area. We observe that, for thesame number of walkers, the time until full coverage decreasesas the correlation among movements in the CRW model increases.We also observe that the full coverage times for the RW (CRWwith p ¼ 0:5) and the RWP models is greater than the fullcoverage times for the CRW model. The full coverage time for theMM model is greater than the full coverage times for the CRWmodel with p � :8. This is as expected because the correlationamong movements in the MM model is weaker than that inthe CRW model with p � :8. We also observe that the gapbetween the full coverage times for the MM model and theCRW model with p ¼ :8 increases when the number ofwalkers is 500. This is reasonable because the MM model

puts a restriction on the velocity of a node to be less than the

nodes preceding it on a street; this means that the full

coverage times are determined by the velocity of the first

node on the streets and not all the nodes.

5 CONCLUSIONS AND FUTURE WORK

We compared the random walk, random waypoint,

Manhattan, and proposed CRW mobility models in terms

of the number of states visited in a fixed time, the time to

visit every state in a region, and the effect of the number of

wandering nodes on the time to first entrance to a set of

states. In an application where sensor-equipped vehicles

are moving in a city to detect events of interest, these results

can provide guidelines on how many such vehicles should

be deployed to achieve the required event detection rates,

given the mobility characteristics of these vehicles.


Fig. 10. (a) First entry times in a 2D CRW model as a function of the number of walkers. The first entry times for the Manhattan (MM) and RandomWaypoint (RWP) models are shown for comparison purposes. The solid curves from top to bottom in the plot are in the same order as the values ofprobability p in the legend. (b) First entry times in a 2D CRW model as a function of the probability of moving in the same direction as before. Thesolid curves from top to bottom in the plot are in the same order as the values of the number of walkers in the legend. For a color version of thisfigure, please see http://www.cise.ufl.edu/~seema/papers/StochasticPropertiesTMC.pdf.

Fig. 11. (a) Full coverage times in a 2D CRW model as a function of number of walkers. The full coverage times for the Manhattan (MM) and RandomWaypoint (RWP) models are shown for comparison purposes. The solid curves from top to bottom in the plot are in the same order as the values ofthe probability p in the legend. (b) Full coverage times in a 2D CRW model as a function of the probability of moving in the same direction as before.The solid curves from top to bottom in the plot are in the same order as the values of the number of walkers in the legend. For a color version of thisfigure, please see http://www.cise.ufl.edu/~seema/papers/StochasticPropertiesTMC.pdf.

We observe that the CRW mobility model covers moreground in a given amount of time and has a smaller firstentrance time to a small set of states than the other modelsconsidered in this study. This means that, if these vehiclesmove according to the CRW mobility model, the eventdetection times will be smaller. The CRW mobility model,unlike the random waypoint model, leads to a uniformdistribution of nodes in the steady state. We also demon-strate that, unlike most other models, tractable analyticalexpressions can be obtained for the transient behavior ofthe CRW mobility model. For future work, we intend toanalyze and simulate other mobility models including theGauss-Markov and the Smooth Random model.

Mobility models based on randomly moving nodes mayprovide useful analytical approximations to the motion ofvehicles that operate in dispatch mode or delivery mode ina city. The “random” movement of such vehicles (whichcould be moving to do their normal daily activities)provides an additional detection capability beyond whatcan be provided by vehicles that follow predeterminedpaths. The unpredictable arrival of a sensor-equipped,randomly moving vehicle might very well disrupt an eventthat was planned to occur during the predictable gaps in thearrivals of vehicles following the deterministic paths. Onthe other hand, the vehicles following the deterministicpaths can be expected to visit sites more regularly, thuscompensating for the sometimes very long time betweenvisits of randomly moving nodes. The most effectivesecurity system would thus likely have a combination ofrandomly and deterministically routed vehicles. Such asystem will experience a more graceful degradation insurveillance, in event of node failures, as compared to asystem of all nodes moving on predetermined pathsbecause, in case a node moving deterministically fails, therandomly moving nodes will still monitor the coverage areaof the failed node.

The analytical results in this paper are useful to adesigner because they can be used to quickly computeperformance even when the real situation is very compli-cated. For example, the 2D CRW on squares of differentsizes and scales can be used repeatedly to construct a“tiled” and “layered” approach to a grid of city streets thathas many geographic restrictions as well as a backbone gridof higher speed corridors. The results that could be quicklyobtained in this way could provide useful insights into thenumber of vehicles that would be best to initialize suites ofsimulations of the more detailed real situation.

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Seema Bandyopadhyay received the BS de-gree in computer science and engineering fromthe Institute of Technology, Banaras HinduUniversity, India, in 1991 and the MS degree incomputer science and engineering from theIndian Institute of Technology, Delhi, India, in1997. She received the PhD degree from theSchool of Electrical and Computer Engineering,Purdue University, West Lafayette, in 2004. Sheis currently a faculty member in the Department

of Computer and Information Science and Engineering, University ofFlorida, Gainesville. Before joining the University of Florida, she was avisiting assistant professor in the School of Electrical Engineering andComputer Science, University of Central Florida, Orlando. Her researchinterests include the design and performance analysis of wirelesssensor networks, optimization of computer networks, and game theoryapplied to the design of computer networks. She is a member of theIEEE.

Edward J. Coyle received the BSEE degreefrom the University of Delaware in 1978 and thePhD degree in electrical engineering and com-puter science from Princeton University in 1982.Since 1982, he has been with Purdue University,where he is currently a professor of electricaland computer engineering and codirector of theCenter for Wireless Systems and Applications(CWSA). This year, he is a Kenan Trust VisitingProfessor for Distinguished Teaching at Prince-

ton University. Dr. Coyle’s technical research interests include wirelesssensor networks and digital signal processing. He was a corecipient ofboth the Myril B. Reed Best Paper Award from the 32nd MidwestSymposium on Circuits and Systems and the 1986 Best Paper Awardfor Authors under 30 from the IEEE Signal Processing Society. He hasserved as an associate editor of the IEEE Transactions on Circuits andSystems and as an elected member of the Board of Governors of theCircuits and Systems Society. Dr. Coyle is a fellow of the IEEE and, in1998, was named an Outstanding Engineering Alumnus of theUniversity of Delaware. Dr. Coyle’s educational research interestsinclude engineering design in a service-learning context and theintegration of undergraduate education with the research activities offaculty. He was a cofounder with Leah Jamieson of the EngineeringProjects in Community Service (EPICS) Program. For their work with theEPICS Program, Professors Coyle and Jamieson received the 1997Chester F. Carlson Award for Innovation in Engineering Education fromthe American Society for Engineering Education (ASEE). With Profes-sors Jamieson and Oakes, Dr. Coyle was a corecipient of the NationalAcademy of Engineering’s 2005 Bernard M. Gordon Prize for Innovationin Engineering and Technology Education.

Tillmann Falck is studying electrical engineeringat Ruhr University Bochum, Germany. From2006 to 2007, he was an exchange student atPurdue University, Indiana. He will receive theDipl-Ing degree in 2007. For his thesis, he workedon ultrasound contrast agent detection usingnonlinear signal processing. His current researchinterests are in nonlinear systems, detection,kernel-based methods, and optimization. He is astudent member of the IEEE.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.


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