Mobility Ad Hoc Springer 2005

7/31/2019 Mobility Ad Hoc Springer 2005

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Connectivity Probability of Wireless Ad Hoc Networks:

Definition, Evaluation, Comparison

Tatiana K. Madsen, Frank H. P. Fitzek and Ramjee PrasadCenter for TeleInFrastructure, Deptartment of Communication Technology,Aalborg University, DenmarkEmail: [tatiana | ff | prasad]@kom.aau.dk

Gerrit SchulteTechnical University of Berlin, Berlin, GermanyEmail: [email protected]

Abstract. The paper presents a new approach investigating mobile ad hoc networkconnectivity. It is shown how to define and evaluate the connectivity probability ofa mobile network where the position of the nodes and the link quality changes overtime. The connectivity probability is a measure that can capture the impact ofthe node movement on the network connectivity. A number of mobility models isconsidered ranging from the classical Random Direction model to the Virtual Worldmodel based on the mobility measurements of a multiplayer game. We introducean Attractor model as a simple way to model nonhomogeneous node distributionby incorporating viscosity regions in the simulation area. Methods of ergodic theoryare used to show the correctness of the approach and to reduce the computationaltime. Simulation results show how the node density distribution affects the networkconnectivity.

Keywords: ad hoc network, mobility models, simulations, connectivity probability

1. Introduction

The movement pattern of users has a significant impact on perfor-mance of mobile and wireless communication systems and networks. Toincorporate user mobility in the simulation evaluation of e.g. routingprotocols, a variety of models have been proposed (see [1, 2, 3] and refer-ences therein). The mobility models define how the nodes move withinthe network. Trustworthy mobility models should satisfy some criteria.Models used for cellular systems should provide realistic values for suchparameters as the cell residence time, number of handoffs during a call,etc. In ad hoc networks the emphasis is shifted and other issues play thekey role. These issues are connectivity of a network [4], path duration(also called longevity of routes [5]) and stability of routes [3].

To achieve a connected ad hoc network, there must be a multihoppath from each node to any other node. Observing a network at a givenmoment in time, we can determine if some of the nodes are isolatedfrom other nodes or not. Assuming some initial node distribution in

c 2005 Kluwer Academic Publishers. Printed in the Netherlands.

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2 Madsen, Fitzek, Schulte, Prasad

a simulation area, we can guarantee with some probability that thenetwork is connected at that moment in time [4]. Now the questionarises what will happen when the nodes start moving around. Somelinks will be broken and new links will be established. It is desirable tohave an estimation whether or not the network will remain connectedand it would be practically convenient to have a measure of a networkbeing connected under mobility of nodes.

To estimate a network connectivity in the presence of mobility, weintroduce a measure, called connectivity probability. This parameteris defined as a measure of time intervals during which a network isconnected. It gives the probability that observing a network in ran-domly chosen point in time we will find it connected. Connectivityprobability can capture the important properties of a network whenposition of nodes and states of the links changes over time. We definea general framework for evaluating the impact of node movement on

the network connectivity. To incorporate the nodes dynamics in thetheoretical analysis of the connectivity properties of a network, we usesome results of the theory of dynamical systems. Ergodic properties ofmobility models based on deterministic modelling of node movement,allow us to derive a simple expression for the connectivity probability.For example, this can be done for Random Direction (RD) model andits extensions.

A number of different mobility models are considered in the pa-per. We illustrate how the connectivity probability can be estimatedwith the examples of Random Waypoint [2], Attractor, Virtual worldmodel [3], and Model with obstacles [14, 17]:

The Random Waypoint (RW) model is widely used for proto-col performance evaluation and is implemented in the networksimulator ns2.

We introduce the Attractor model as a simple enhancement ofany mobility model when regions of slow movements are incorpo-rated in the simulation area. This leads to nonhomogeneous nodedistribution.

The Virtual World model is based on the mobility patterns ob-tained from the measurements of the virtual world scenarios. Themobility measurements of a multiplayer game, such as Quake II,

can be used for investigation of the impact of mobility on theperformance of multihop protocols.

The mobility model with obstacles incorporates the rectangular-shaped obstructions in the simulation region to make the simula-tion scenarios more realistic. It is especially actual when simulating



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Connectivity Probability of Wireless Ad Hoc Networks 3

indoor scenarios where the influence of walls and other objects cannot be ignored.

The remainder of the paper is organized as follows. Section 2 pro-vides the motivation for our work and gives a short overview of the re-lated work. Section 3 introduces a formal definition of the connectivityprobability. Section 4 explains how the connectivity probability can beestimated for different mobility models. Section 5 presents simulationresults. Section 6 offers some concluding remarks.

2. Motivation and related work

A problem of connectivity of ad hoc networks has got a lot of attentionduring last years. For multihop networks the connectivity is one of

the fundamental and important properties. The majority of researchconsiders only the static situation, when the nodes do not move. Forexample, in [4] connectivity dependence on the node density and thetransmission radius is evaluated in the static scenario. Another problemthat was tackled is defining the critical transmission range, a thresholdtransmission range that is required for connectivity (see e.g. [6, 7]).

Including the mobility of nodes complicates analysis of network char-acteristics, though many applications of ad hoc networks requires thatthe nodes are mobile but still connected. It has been realized that themobility pattern of the nodes has a big impact on the performanceof ad hoc protocols [2, 8]. But for the correct interpretation of the

results, one should be aware of the properties of the network underconsideration, including the connectivity property. Let us consider thesituation when at the initial moment of time nodes are distributeduniformly on the simulation area and the network is connected. Thenthe nodes start moving to any of the corners of the area and after sometime the network will become and remain partitioned. This example isan extreme case when an initially connected network becomes uncon-nected, but it illustrates the importance of understanding how mobilityof the nodes influences the connectivity. Under or overestimation ofnetwork connectivity can potentially lead to misinterpretation of theresults, when falls in the protocol performance would be attributed tothe specifics of a protocol design and not to the characteristics of thenetwork.

The main tools used for investigation of network connectivity comefrom the graph theory and the probability theory. For example, thetheory of geometric random graphs (GRG) deals with a set of n pointsdistributed according to some density and focuses on the properties of



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the resulting node placement. Using GRG theory some conclusions canbe made on the transmission range in a dense ad hoc network [9]. Otherresults for ad hoc network connectivity can be obtained making use ofthe percolation theory [10] (see also [11] for the connectivity of hybridad hoc networks). In [6] the problem of connectivity is approachedfrom the probabilistic angle and the transmission range is defined asa random variable allowing statistical modelling of the connectivity.Probabilistic approach can be also used for the investigation of sparsead hoc networks, namely to determine the critical transmission rangewhich generates communication graphs that are connected with highprobability [7].

Unfortunately, these instruments are only amiable for the analysis ofstatic scenarios, since they fail to capture situations when edges of theconnectivity graph changes over time. To incorporate the dynamicsof a network topology into the analysis of the connectivity problem,

we resort to the methods provided by the theory of dynamical systems.Dynamical systems are exactly the mathematical field that is developedand used for description of evolution in time of objects of differentnature. Therefore, it is also natural in our situation, for discussion ofthe qualitative estimation of the network connectivity under mobilityof nodes, to use ideas and methods of the modern theory of dynamicalsystems.

3. Definition of Connectivity Probability

We start with giving some insights in the problem of network con-nectivity in a static environment. Then we show how connectivitycan be defined when the nodes are mobile and introduce connectiv-ity probability as a measure of network connectivity in a dynamicenvironment.

Let D be a bounded domain on the Euclidean plane R2 = {x, y}with piecewise smooth borders. Inside domain D there are n nodes. Inthe initial moment t = 0 nodes are somehow placed and then they startmoving. Let ri = (xi, yi) be the radiusvector of node i. Further, weassume that every node has a transceiver with communication range r:if the distance between two nodes is larger than r, they cannot establisha direct communication link. Nodes can transmit information by usingmultihop connections.

Definition 1. A network is connected (or fully connected) if for everypair of nodes there exists a path between them.

Note that for a network to be connected we require existence ofa path from source to destination at any moment in time. As it is



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shown in [12] it might not be necessary, but it imposes design of specificrouting protocols. We do not include this special case in the scope ofour consideration.

Connectivity probability is used to quantify the network connectiv-ity. In the static environment (when the positions of nodes are fixed),the probability for a network to be connected depends on the density ofthe nodes and their transmission range. In a typical static simulationscenario, in order to evaluate the connectivity probability a number ofnodes are placed at random in the simulation area. A random variableis introduced which is equal to one if the network is found connectedand zero if it is disconnected. The average of this random variable overseveral trials gives the connectivity probability. Now we will show howto extend this approach to the case of dynamic environment.

Under motion of nodes a semiaxis of time R+ is divided into intervals

1 ,

2 ,

3 , . . .

where +k

(k

) denotes a time interval during which the network isconnected (disconnected). Lets introduce the function f+(t) such thatf+(t) = 1 ift

+k

, and f+(t) = 0 if t k

.Time intervals k

can beconsidered as randomly distributed; thus, f+(t) is a stochastic process.

Definition 2. In the dynamic environment, connectivity probabilityP+ is defined by

P+ = E[f+(t)] (1)

where E[] stands for the expected value (if the expected value exists).Analogously the function f(t) can be introduced:

f(t) =

1, t

k

0, t +k

The probability that a network is disconnected is given by P =E[f(t)]. Since f+(t) = 1 f(t), then P+ + P = 1.

One can see that in general case the probability P+ is time-dependent:P+ = P+(t). For stationary stochastic processes P+(t) = const. Ifstationary process is ergodic, then equality (1) can be substituted bythe following: with probability one

lim1

0 f+(t)dt = P+

This equality is equivalent to the following:

P+ = lim

1

mes(T+

[0, ]) (2)



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where T+ =

+k

1. That is, the probability that a network is connectedcan be defined as a density of set T+ on the semiaxis R+ = {t > 0}.

Therefore, the problem of connectivity of a network consisting ofmoving nodes is reduced to the problem of existence and estimation ofthe expected value (1). If the mobility model is stationary and ergodic,then formula (2) can be used for estimation of connectivity. Detailedanalysis of the estimation of the connectivity probability is presentedin the next section.

4. Analysis

4.1. Note on dynamical systems and stochastic processes

Dynamical systems theory can be useful for modelling the movement

of nodes in an ad hoc network. In homogeneous networks, where theproperties and capabilities of the nodes are the same, we can reasonablyassume that the movement of a single node can be described by one andthe same system of differential equations. If we introduce some form ofrandomization in the movement pattern of a node, then it would re-quire construction of stochastic differential processes. If the consideredprocess is stationary, a system of autonomous differential equations canbe used, where the rightsides of equations do not depend explicitly ontime and different nodes differ only by their initial conditions.

In the theory of dynamical systems a phase flow is introduced thatis a group of shifts along the trajectories during one and the same timeinterval. The phase flow generates a dynamical system. The system canbe described by differential equations of the following form:

x = g(x), x (3)

where is the phase space, x is a set of coordinates in (usually itis position and velocity), dot means time differentiation. Let n be anumber of nodes and x(1), . . . , x(n) are their phase coordinates. Thenthese coordinates satisfy the following differential equations:

x(k) = g(x(k)), k = 1, . . . , n (4)

In this way dynamics of n nodes is completely defined by dynamical

system (4), that is a direct product ofn copies of original dynamical sys-tem (3). Its phase space = . . . = ()n is a direct product ofncopies of initial phase space, and phase coordinates x = (x(1), . . . , x(n))

1 mes stands for measure of a domain. In this case it is the total length of theintervals T+

[0, ]



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are a set of coordinates of individual nodes. Note that if the system(3) has an invariant measure in , then the system (4) also has an

invariant measure in , that is a direct product = 1 n.

In the problem of the nodes connectivity, the phase space can bedivided into two domains D and D = \D in the following way: when

x D, then every node from n nodes x1, . . . , xn can communicatewith any other node; when x D, then some of the nodes can notreach some of the others. Following dynamical systems approach,the connectivity probability can be estimated as a fraction of time twhen x(t) D.

The problem of estimation of the connectivity measure can be sim-plified significantly if dynamical system (4) is ergodic in . By defin-ition, the system is ergodic if a measure of any invariant subdomainof the phase space is either equal to zero or is equal to the measure ofthe whole space.

The key result of the ergodic theory is Birkhoff ergodic theorem [13].Here we provide its formulation using our notations. Let f(x) be a

measurable integrable function on . Then for almost all solutions ofthe ergodic system (3) the following equality takes place:

limT

1

T

T

0f(x(t))dt =

f(x)d

mes , (5)

where

mes =

d = ()

is the measure of the whole phase space.

Let, for example, f be a characteristic function of a measurabledomain D: f(x) = 1 i f x D and f(x) = 0 i f x D. Due to the

the fact that f is limited and D is measurable, function f : Ris integrable. In this case, the leftside of (5) is equal to the fractionof time interval 0 t T when x(t) lies in the domain D. Then theconnectivity probability of a network will be equal to the rightside of(5), that in this case is

mes D/mes (6)

This approach can be also interpreted in terms of the theory ofstochastic processes. One can introduce the probabilistic measure inthe phase space . Probability for a system to be in the measurabledomain D is determined by formula (6). Let f(x) be a characteristicfunction of the domain D introduced above and x(t) be the solution ofsystem (3). Then the function f(x(t)) can be interpreted as a stochasticprocess. Its realization will be dependent on the choice of the initialconditions. Let E[f()] be the expected value of the function f(t) at



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the moment . If the rightsides of the equation (3) do not depend ontime, then the stochastic process is stationary. In particular, it meansthat E[f()] does not depend on . If, in addition to being stationary,the system is ergodic, the expected value can be calculated by usingformula (6):

limT

1

T

T

0f(x(t))dt = E[f] = mes D/mes

This is true for almost all realization of the stochastic process. There-fore, the problem of calculation of the expected value (1) is reduced tothe geometrical problem of calculation of domain volumes in a phasespace if the process is ergodic.

As we show in the following, using formula (6) is computationallymore efficient than evaluating the expected value. Thus, it should bepreferred when possible, i.e. when the system is ergodic. At this point

a natural question comes: can we expect that one or another widely-used mobility model is ergodic? The problem of ergodicity has beenintensively studied, but it is far from being completely resolved due toits complexity. There exist a number of wellknown examples of ergodicsystems from the theory of dynamical systems. For example, consider asystem of nodes within a rectangular domain moving along the straightlines. Once the boundary is reached, a new direction of movement ischosen such that the trajectory forms equal angles with the boundary.One can imagine a trajectory as a stroke of light reflected by a mirror.These kind of systems are called billiards. One can easily notice thatthey correspond to a particular case of Random Direction mobilitymodel (in RD model a new direction is chosen uniformly towards theinside of the simulation area). Billiards, restricted on the integral many-fold, are ergodic. Their ergodicity property can be established with thehelp of Weyls theorem on homogeneous distribution (see e.g. [13]). Oneshould note, though, that not every statinary process is ergodic and inrealistic scenarios ergodicity is more the exception than the rule.

If the rightside of (3) explicitly depends on time, then the processf(t) = f(x(t)) is nonstationary. In general, formula (6) is not applica-ble and the expected value depends on time. However, it turns out thatunder a large range of conditions, the expected value off(t) calculatedover a long interval of time gives sufficiently good approximation ofthe expected value E[f(t)] over semiaxis 0 t < . Thus, if E[f(t)]

changes slowly over time, then this value can be estimated as theaverage value of f(t) over long (but not infinitely-long) time interval[t, t + ]: with probability one

limT

1

T

T

0f(t)dt = lim

T

1

T

T

0E[f(t)]dt (7)



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If we consider time interval where f(t) oscillates but E[f(t)] changesslowly, then E can be approximated as a constant and the rightsideof (7) is equal to E(t). Thus, for slow stochastic processes formula (7)can be used for estimation of the connectivity probability.

4.2. Estimation of connectivity probability for differentmobility models

In [14] we present a detailed analysis and derivation of formula (6)

for Random Direction model. In this case mes is just the area of asimulation domain: mes = ab where a and b are lengths of the sidesif rectangular region is considered. mes D is area of a domain consistingof all possible positions of the nodes such that the network is connected.It seems to be difficult to find a close form expression for the measureof D as a function of communication radius r. Instead, MonteCarlo

simulations can be used: nodes are placed randomly, independently anduniformly in a simulation area and the connectivity of the network isverified. Then P+ M/N, where N is a total number of simulationtrials and M is a number of successful trials (when the network isconnected) if N is chosen sufficiently large.

Formula (6) can be used for the range of other mobility modelsthat generalize RD model: when speeds of nodes are different or whenborders of the area consist of horizontal and vertical lines, or speak-ing more generally, when rectangularshaped obstacles are placed in asimulation area. The first case is illustrated by an example of Attractormodel in Section 5.3. The realization of the second case by the model

with obstacles can be found in [15, 16]. It is discussed in Section 5.5.Using formula (6) for estimation of the connectivity probability, any

fixed moment of time can be considered, since the process is stationary.In principle, t = 0 is as good choice as any other, and by using t = 0the complexity of the simulation is of the same order as for the staticcase evaluation. However, in practise, it is recommendable to considert = t0 > 0 to make the result less correlated with the initial conditions.

Considering other stationary mobility models, it is more difficult toanswer if the limit (2) exists. If a collection of mobile nodes comprisesa dynamical system with some particular properties (namely, a systemhas an invariant measure), then for almost all initial conditions the limitexists. It can happen that for different initial conditions (i.e. differentinitial positions and velocities of the nodes) the limiting values willbe different. In this situation averaging over initial node distributionshould be done, and the obtained average connectivity probability canbe used as an estimate that a network is connected in any point intime.



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Table I. Calculation approaches for different models

RD RW Attractor VW Obst

Formula (6) Formula (1) Formula (6) Formula (7) Formula (6)

For mobility models based on RD pattern, we have two ways tocalculate the connectivity probability: either evaluating the expectedvalue (1) or by applying formula (6). Comparing these two approaches,we have done some measurements of cpu time. To obtain connectivityprobability for one set of parameters, using formula (6) it takes approx.132 sec, whereas if we calculate directly by using (1), cpu time is morethan 2000 sec. The big difference in cpu times can be explained by thefact that in the second case modelling in time is required and at each

time stamp we have to recalculate the positions of the nodes. Moreover,using MonteCarlo approach, we can estimate how many simulationruns are required in order for results to lie within a specified confidenceinterval. Applying direct calculation, it is not possible by any analyticalmeans to estimate how long in time we have to perform the simulation, it can differ in each particular case. Therefore, we have chosen tocalculate connectivity by using formula (6) when possible, since it iscomputationally less expensive.

To summarize this section, Table I is given showing what approachis used for estimation of the connectivity probability for the results inSection 5.

5. Simulation results

Different application scenarios of ad hoc networks would require dif-ferent connectivity probability. For example, in a network used forsafetycritical or lifecritical operations, the connectivity probabilityshould be more than 90%. If temporary network disconnections canbe tolerated, then the connectivity probability can go down to 50% oreven lower.

5.1. Random Direction model

In Random Direction model, nodes select a direction in which to move,uniformly between 0 and 2, and speed from a given distribution.Subsequently motion starts. Once the simulation boundary is reached,a node pauses for a specified period of time, then it chooses a new



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direction uniformly toward the inside of the simulation domain and theprocess continues. Especial interest has a case when a new direction ischosen such that the trajectory of movement forms equal angles withthe boundary. We refer to it as Billiard model.

One of the important characteristics of the Billiard model is a uni-form node density distribution. This can be shown directly [17] or byusing the following wellknown fact from the theory of dynamical sys-tems: in a billiard system for almost every initial conditions a trajectoryof a node covers a region of motion everywhere dense.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Connectivityprobability

Communication radius

5 nodes10 nodes

50 nodes

Figure 1. Connectivity probability as a function of communication radius. Simula-tion area 100x100 m2

Figure 1 presents the connectivity probability dependence on a com-munication radius where the simulation area is a square of 100x100 m2

and the number of nodes is equal to 5, 10 and 50. When the density ofnodes is high, in order to have a connected network with probability95% at any moment of time, we should require the communicationradius of 20 m. One can observe that for the dense network the line pre-senting connectivity probability is steeper than the one for the sparsenetwork. One should note that in the case when all nodes move withthe same speed, the connectivity probability would not depend on theactual value of the speed.

5.2. Random Waypoint model

Another widely used model for protocol performance evaluation is theRandom Waypoint model. A node chooses a random destination anda travelling speed and travel towards the destination along a straightline. Upon arrival, it pauses and repeats the process.

Table II shows connectivity probability for the following set of pa-rameters: simulation area is 150x150 m2, number of nodes n is 100



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Table I I. Connectivity Probability for Billiard, Random Way-point and Attractor models

B RW Att

r = 30, n = 100 0.92 0.017 0.85 0.007 0.55 0.003

r = 40, n = 80 0.99 0.006 0.97 0.03 0.90 0.018

and communication radius r is 30 m, and n = 80 and r = 40 m. ForRW model the results are obtained as the average of 50 simulationruns. For Billiard model 1000 trials are performed. The 95% confidenceinterval is given in Table II. Comparing results for Billiard and RandomWaypoint, one can observe that the connectivity probability for RWmodel is lower, though the same set of parameters were considered.

This result can be explained if we consider the node distribution inthe simulation area for these two models. As it was already noticed(see e.g. [18]), the node density distribution for RW is not uniform,but has a convex shape. Figure 2 shows a histogram obtained throughsimulation on a 150x150 m2 area with 80 nodes. The area is dividedinto 15x15 subareas and the number of nodes in the subarea at everytime step is counted. A higher node density in the middle of the areaand a significantly lower density at the area edges is due to the factthat nodes at the borders are more likely to move back to the middleby randomly choosing a new direction of movement.

The node density has a direct impact on the connectivity probabil-

ity. The existence of areas with low density leads to the decrease inthe connectivity, as we can see from Table II. If a sparse network isconsidered with low number of nodes, then the opposite effect can beobserved. Table III shows results when the number of nodes is equal to13. The probability to find a network fully connected under RandomWaypoint is higher than for Random Direction model. In this case thedensity of nodes is so low even in the high density region, that thenumber of nodes might not be enough to form a connected network.Therefore, here the fact that for RW model in the central part of thesimulation area the node density is higher plays crucial role.

5.3. Attractor model

The considered above models result in almost uniform spatial nodedistribution. Tough we can imagine situations when mobile users tendto gather in one subarea (shopping mall or traffic jam due to a roadconstruction). These scenarios can not be captured by group mobility



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02

46

810

1214

16X/10, m

02

46

810

1214

16

Y/10, m

0.00180.002

0.00220.00240.00260.00280.0030.0032

0.00340.0036

Figure 2. Node density distribution for Random Waypoint model

models when a group of users exhibit similar behavior. To reflect in-dividual trajectories of users movement that willingly or due to somecircumstances spend some time in one subarea, we introduce a mobilitymodel called Attractor model.

The main feature of this model is the presence of gravity pointsor areas that attract mobile users. One can think of many ways howattractors can be simulated. For example, one can specify points ofattraction and how the users move from one point to another by borrow-ing methods of fluid dynamics. Here we propose another, very simpleway introducing areas of viscosity. We can take any movement pat-tern as underlying mobility model and enhance it with regions of slowmovement.

02

46

810

1214

16X/10, m 0

24

68

1012

1416

Y/10, m

0

0.02

0.04

0.06

0.08

0.1

0.12

Figure 3. Node density distribution for Attractor model

As an example, we consider 80 nodes moving in the 150x150 m2 areaaccording to Random Direction model with v = 5m/s. Two attractors



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are introduced: one viscosity region is a disk with center in (50, 50)and radius 30 m, the second is a disk with center located in (100,100)and radius 20 m. When coming to the first attractor, the users slowdown to 1 m/s and in the second region the speed of movement is 0.5m/s. This results in nonuniform density distribution (see Figure 3).High density corresponds to the attractor areas. One can observe thatthe node density is inverse proportional to the speed of movement:density in the second attractor is approx. 0.1 nodes/m2, in the first0.048 nodes/m2, in the rest of the area 0.001 nodes/m2.

0 1000 2000 3000 4000 5000 6000 70000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

Connectivity

Probability

r = 85 m

r = 70 m

r = 50 m

Figure 4. Connectivity probability for Virtual world model

02

46

810

1214

X/10, m 02

46

810

1214

16

Y/10, m

00.010.020.030.040.050.060.070.080.09

Figure 5. Node density distribution for Virtual World model

As expected, the probability to have a fully connected network forAttractor model is significantly lower than for Billiard and RandomWaypoint models if the same number of nodes are considered (Table II).For Attractor model nodes tend to spend more time in the viscosity



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Table III. Connectivity Probability for Quake, Billiard andRandom Waypoint models

Quake B RW

r = 85, n = 13 0.94 0.98 0.008 0.99 0.019

r = 70, n = 13 0.8 0.84 0.022 0.97 0.033

r = 50, n = 13 0.15 0.21 0.0025 0.61 0.095

regions where nodes are located close enough to be able to communi-cate directly or where there are enough nodes to support multihopconnections. In the rest of the area the node density will be low; thisexplains why the connectivity probability is low.

5.4. Virtual world model

Presented above models are generally used for simulations because theyprovide an easy way to simulate movements of nodes. It is a commonconcern among researchers in the field of selforganizing networks howrealistic these models are and if the results obtained with the helpof these models reflect the reallife situation. From the other hand,obtaining realistic measurements of mobility is complex and expensive.

To create a more realistic model, multiplayer games can be usedsince they can measure the mobility of the players in the games virtual

world [3]. The software for multiplayer games is no longer limited togames, but it is also used to emulate the real world. Therefore, it ispossible to create new virtual world and adopt them to specific needs.

Following the approach presented in [3], we use virtual world sce-narios to explore mobility issues. To obtain mobility patterns, a multiplayer game, Quake II, is used. The traces of 13 players moving inthe square area were recorded. The position of each player was loggedevery time the server updated a frame, that is approx. every 100 ms.The data was collected for about 12 min (for more details see [3]). Forpostprocessing the data, the area of players movement, 4000 x 4000units of the map editor, was scaled to 150x150 m2.

Using the measurement data, we can try to estimate the connectivityprobability in this case. Since nothing can be said about stationarityof the obtained process in advance, we have chosen to apply formula(7) and observe if the limiting value exists. Presented analysis is madebased on one record of the movement pattern. We plot the expres-sion of the rightside of (7) assuming different communication radius



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(Figure 4). One can observe that the limiting value exists (curvesasymptotically approaches constant lines).

The difficulty of using virtual traces is that limited material is atdispose for the analysis. We can not predict how the users will move inthe future, at the times when we stop recording traces. The conclusionto be made here is if the users would have the similar type of behaviorthat the one observed, then at any point in time we would find thenetwork fully connected with the probability 80% if communicationradius is 70 m.

The connectivity probabilities of Quake and Billiard models (Ta-ble III ) are close, but for the Billiard it is slightly higher. Though thedensity histograms looks quite different for these two models. In Quakethe players were given the mission to explore the area. In principle, itcould mean that a user would try to visit every subarea at least once,and if we would observe the users behavior for sufficient long interval of

time and record the density distribution, we could expect it to be closeto uniform. In reality, the density graph (Figure 5) differs from RDmodel. The area along the right wall was almost unvisited, whereasin the rest of the territory small density picks are almost uniformlyspread. Some player were more lazy than others and they preferredto spend more time in one place. This and may be the presence of someinteresting objects contributed to the nonuniform distribution.

5.5. Mobility models with obstacles

As an example of a mobility model with obstacles we consider a situ-ation pictured in Figure 6. Six users move in a room 30m by 20 m isdivided by a 20 meter wall (e.g. a museum or exhibition hall). For somepositions of the nodes the wall is blocking the communication and noline-of-sight (LOS) path exists.

Figure 6. Simulation area with an obstacle

Line C (Figure 7) is plotted under the assumption that when a signalis propagated between a pair of nodes and there is an obstacle obstruct-ing the direct transmission path, the signal is completely blocked by



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the obstacle. We can observe that the probability of having a connectednetwork is not higher than 0.9 regardless of the communication range/transmission power of the device. This is not very realistic. For compar-ison the connectivity probability is shown when no wall is present (lineA). This is equivalent to the situation when the wall is made of suchlight material that it does not seriously obscure the communication. Tomake the model more realistic, some assumptions should be made aboutthe propagation characteristics. The signal strength decay depends onmaterials of walls and ceiling, size of the room and the size and formof other objects. A proper number should be chosen in each particularcase. Line B is constructed under the assumption that in non-LOS casethe signal decay is 20 dB. From Figure 7 one can see that in contrastwith the line C, for the line B there exist parameters of the systemwhere the connectivity probability is almost one.

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Communication radius

ConnectivityProbability

line A

line B

line C

Figure 7. Connectivity probability as a function of communication radios. Numberof nodes = 6. Line A: LOS always exists, Line B: signal is decayed in NLOS cases,Line C: signal is completely blocked in NLOS cases

6. Conclusion

We present a new method of describing the connectivity of an ad hocnetwork under mobility of nodes. This method is applicable to bothdense and sparse networks. We show that some tools of the theory ofdynamical systems can be used to conduct the analysis of the connec-tivity property and to speed up the connectivity evaluation when theunderlying mobility model possesses ergodic properties.

The results of the paper consist of a number of theoretical insightsand proofs, which also can be of interest to researchers in the area



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of ad hoc networking, since they highlight the important issue of mo-bile ad hoc network connectivity. Extensive simulation results showevaluation of network connectivity for a range of models. The connec-tivity probability can be also estimated if some measurement data isavailable, e.g. traces of players in Quake game. The relationship be-tween the connectivity probability and the node density distribution isunderlined.

In this study the focus was mainly on the stationary mobility models.It would be interesting to observe how the connectivity probabilitydepends on time for nonstationary processes. A nonstationary modelcan be created by e.g. moving the position of the gravity points in theAttractor model or changing continuously the lengths of the rectangularsides of the simulation area in RD model. This topic is a matter offurther investigation.

Acknowledgements

The authors would like to thank L. Badia and T. Henderson for theirhelp in collecting the measurement data of the multi player game.

Finally, we would like to thank the anonymous reviewer for insightlyand detailed comments that have helped to improve the quality of thepaper.

References

1. C. Bettstetter. Smooth is Better than Sharp: A Random Mobility Model forSimulation of Wireless Networks. In Proceedings of the 4th ACM InternationalWorkshop on Modelling, Analysis, and Simulation of Wireless and MobileSystems, Italy, July 2001.

2. T. Camp, J. Boleng and V. Davies. A Survey of Mobility Models for Ad hocNetwork Research. WCMC: Special Issue on Mobile Ad Hoc Networking, 2(5):483-502, 2002.

3. F. Fitzek, L. Badia, M. Zorzi, G. Schulte, P. Seeling, T. Henderson, Mobilityand Stability Evaluation in Wireless Multi-Hop Networks Using Multi-PlayerGames. In In Proceedings of NETGAMES03, May 2003.

4. C. Bettstetter. On the minimum node degree and connectivity of a wirelessmultihop network. In Proceedings of the 3rd ACM international symposium on

Mobile ad hoc networking & computing, June 2002.5. D. Turgut, S.K. Das, M. Chatterjee, Longevity of routes in mobile ad hoc

networks, In Proceeding of IEEE 53rd Vehicular Technology Conference, May2001.

6. H. Koskinen. Statistical model describing connectivity in ad hoc networks. InProceedings of WiOpt04, March 2004.



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7. P. Santi, M.D. Blough. The critical transmitting range for connectivity in sparsewireless ad hoc networks. IEEE Transactions on Mobile Computing, 2(1):25-39,January 2003.

8. F. H. P. Fitzek, T. K. Madsen, R. Prasad, M. Katz. Impact of Node Mobility

on the Protocol Design of Self-Organizing Networks. In Proceedings of WirelessWorld Research Forum 11, June 2004.

9. P. Panchapakesan, D. Manjunath, On the Transmission Range in Dense AdHoc Radio Networks. In Proceedings of IEEE SPCOM01. 2001.

10. P. Gupta, P.R. Kumar. Critical Power for Asymptotic Connectivity in Wire-less Networks. Stochastic Analysis, Control, Optimization and Applications.Boston: Birkhauser, pp. 547-566, 1998.

11. O. Dousse, P. Thiran, M. Hasler. Connectivity in ad hoc and hybrid networks.In Proceedings of INFOCOM 2002, 2002.

12. B. Bui-Xuan, A. Ferreira, and A. Jarry. Evolving graphs and least cost journeysin dynamic networks. In Proceedings of WiOpt03 Modeling and Opti-mization in Mobile, Ad-Hoc and Wireless Networks, Sophia Antipolis, March2003.

13. V. Arnold and A. Avec, Ergodic problems of classical mechanics. Benjamin,

New York. 1968.14. T. K. Madsen, F. Fitzek, R. Prasad. Impact of Different Mobility Models on

Connectivity Probability of a Wireless Ad Hoc Network. In Proceedings ofInternational Workshop on Wireless Ad Hoc Networks. June 2004.

15. T. K. Madsen, F. Fitzek, R. Prasad. Simulating Mobile Ad Hoc Networks:Estimation of Connectivity Probability. In Proceedings of WPMC04. 2004.

16. A. Jardosh, E.M. Belding-Royer, K.C. Almeroth, S. Suri, Towards realisticmobility models for mobile ad hoc networks, In Proceedings of the 9th annualinternational conference on Mobile computing and networking, September 2003.

17. D. Yu, H. Li. Influence of Mobility Models on Node Distribution in Ad HocNetworks. In Proceedings of ICCT03. 2003.

18. J. Yoon, M. Liu, B. Noble. Random Waypoint Considered Harmful. InProceedings of INFOCOM 2003, April 2003.

Authors Vitae

Tatiana K. Madsen was born in Moscow, Russia, in 1975. She is anAssistant Professor in the Department of Communication Technology,University of Aalborg, Denmark. She received her M. Sc. and Ph.D.degrees in Mathematics from Moscow State University, Russia in 1997and 2000, respectively. The area of her PhD studies was dynamicalsystems, their ergodic properties and integrability. Her current researchinterests lies within the areas of ad hoc networks with the focus onrealistic mobility models and ad hoc routing and 4G wireless commu-nications including QoS support for multimedia services and IP headercompression techniques for wireless networks.



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Frank H. P. Fitzek was born in Berlin, Germany in 1971. He isan Associate Professor in the Department of Communication Technol-ogy, University of Aalborg, Denmark heading the Future Vision group.He received his diploma (Dipl.-Ing.) degree in electrical engineeringfrom the University of Technology - Rheinisch-Westfalische TechnischeHochschule (RWTH) - Aachen, Germany, in 1997 and his Ph.D. (Dr.-Ing.) in Electrical Engineering from the Technical University Berlin,Germany in 2002. As a visiting student at the Arizona State Universityhe conducted research in the field of video services over wireless net-works. He co-founded the start-up company acticom GmbH in Berlinin 1999. In 2002 he was Adjunct Professor at the University of Fer-rara, Italy giving lectures on wireless communications and conductingresearch on multi-hop networks. His current research interests are inthe areas of 4G wireless communication, QoS support for multimedia

services, access techniques, security for wireless communication, andthe integration of multi hop networks in cellular systems. Dr. Fitzekserves on the Editorial Board of the IEEE Communications Surveys& Tutorials. He is the program chair for the International Conferenceon Advances in Computer Entertainment Technology (ACE2004) andserves in the program committee for VTC2003, VTC2004, ACE2004,and IEEE MWN2004.

Jan Gerrit Schulte is the CEO of acticom mobile networks GmbH inBerlin, Germany. He studied electrical engineering at the Technical

University Berlin from 1989 until 2001, while the main focus weretelecommunication networks, the project alpha core dealt with In-ternet Measurements and Characterists, and the Diploma thesis wasabout routing and stability in wireless ad-hoc networks. From 1996until 2000 he worked as student researcher at the TelekommunicationsNetwork Group at TU-Berlin. From there Gerrit Schulte changed toSIEMENS AG WM where he worked within Bluetooth Developmentand Design. In 2001 Gerrit Schulte began to work as CEO of acticom,a company that he also co-founded in 1999 and which is developingsoftware for wireless telecommunication networks and devices, such asmobile phones and cellular networks.

Ramjee Prasad received a B.Sc. (eng) degree from Bihar Institute ofTechnology, Sindri, India, and M.Sc. (Eng) and Ph.D degrees from BirlaInstitute of Technology (BIT), Ranchi, India, in 1968, 1970 and 1979,respectively. During February 1988 May 1999 he has been with the



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Telecommunications and Traffic-Control Systems Group of Delft Uni-versity of Technology (DUT), The Netherlands, where he was activelyinvolved in the area of wireless personal and multimedia communica-tions (WPMC). He was head of the Transmission Research Sectionof International research Centre for Telecommunications Transmissionand Radar (IRCTR) and also Program Director of the Centre for Wire-less Personal Communications (CEWPC). As from June 1999 RamjeePrasad joined as the Wireless Information Multimedia Communicationschair and co-director of Center for PersonKommunikation at AalborgUniversity, Denmark. From January 2004 he is Director of the Cen-ter for Teleinfrastruktur (CTIF). He has published over 500 technicalpapers, and authored and co-edited a series of book about WirelessMultimedia Communications (Artech House). His research interest liesin wireless networks, packet communications, multiple access protocols,adaptive equalizers, spread-spectrum CDMA systems and multimedia

communications. He is a fellow of the IEE, a fellow of IETE, a seniormember of IEEE, a member of NERG, and a member of the DanishEngineering Society (IDA).



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Mobility Ad Hoc Springer 2005

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