A general framework to construct stationary mobility models for the simulation of mobile networks
ABSTRACT Simulation has become an indispensable tool in the design and evaluation of mobile systems. By using mobility models that describe constituent movement, one can explore large systems, producing repeatable results for comparison between alternatives. In this paper, we show that a large class of mobility models  including all those in which nodal speed and distance or destination are chosen independently  have a transient period in which the average node speed decreases until converging to some longterm average. This speed decay provides an unsound basis for simulation studies that collect results averaged over time, complicating the experimental process. In this paper, we derive a general framework for describing this decay and apply it to a number of cases. Furthermore, this framework allows us to transform a given mobility model into a stationary one by initializing the simulation using the steadystate speed distribution and using the original speed distribution subsequently. This transformation completely eliminates the transient period and the decay in average node speed and, thus, provides sound models for the simulation of mobile systems.

Conference Paper: Performance Evaluation of DYMO Protocol in Different VANET Scenarios
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ABSTRACT: In this work, we evaluate the performance of DYMO routing protocol in different VANET scenarios. We investigate the effect of density and speed in the performance of communication between nodes using different node densities and speeds for the same simulation area. As evaluation metric, we use Packet Delivery Ratio (PDR). The simulation results have shown that with the increasing of the nodes density the probability of finding a path between communicating nodes is high and the PDR is increased. When the speed of nodes is increased, DYMO protocol tents to do most of the communication with less hops.NetworkBased Information Systems (NBiS), 2012 15th International Conference on; 01/2012  SourceAvailable from: Edward CoyleIEEE Trans. Mob. Comput. 01/2007; 6:12181229.

Conference Paper: Performance Analysis of DSR and DYMO Routing Protocols for VANETs
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ABSTRACT: Wireless networks are continuing to attract attention for their potential use in several fields such as adhoc networks, sensor networks, mesh networks, and vehicular networks. Vehicular Adhoc networks (VANETs) are expected to be massively deployed in upcoming vehicles, because their use can improve the road safety and comfort. The effective implementation of vehicular communication could improve traffic management system. This effectiveness could be achieved by designing and implementing efficient vehicular network protocols. In this paper, we have implemented two routing protocols: DSR and DYMO and investigated the performance of these routing protocols using PDR and good put metrics. The simulation results shows that DYMO protocol performs better than DSR protocol.Complex, Intelligent and Software Intensive Systems (CISIS), 2012 Sixth International Conference on; 01/2012
Page 1
A General Framework to Construct
Stationary Mobility Models for
the Simulation of Mobile Networks
Jungkeun Yoon, Student Member, IEEE, Mingyan Liu, Member, IEEE, and Brian Noble, Member, IEEE
Abstract—Simulation has become an indispensable tool in the design and evaluation of mobile systems. By using mobility models that
describe constituent movement, one can explore large systems, producing repeatable results for comparison between alternatives. In
this paper, we show that a large class of mobility models—including all those in which nodal speed and distance or destination are
chosen independently—have a transient period in which the average node speed decreases until converging to some longterm
average. This speed decay provides an unsound basis for simulation studies that collect results averaged over time, complicating the
experimental process. In this paper, we derive a general framework for describing this decay and apply it to a number of cases.
Furthermore, this framework allows us to transform a given mobility model into a stationary one by initializing the simulation using the
steadystate speed distribution and using the original speed distribution subsequently. This transformation completely eliminates the
transient period and the decay in average node speed and, thus, provides sound models for the simulation of mobile systems.
Index Terms—Computing methodologies, simulation and modeling, mobility model, stationary distribution.
?
1
S
of larger scale systems than can be built practically.
Furthermore, it enables the evaluation of systems not
amenable to analysis. By carefully controlling the move
ment of nodes and wireless conditions between them,
simulations provide excellent reproducibility across experi
mental trials.
Typically, simulations of mobile systems rely upon
random mobility models. Such models are characterized by a
collection of nodes, placed within a confined space U, that
move according to certain underlying random processes.
The behavior of most mobile systems depends heavily on
the movement of constituent nodes [1]. Therefore, it is
highly desirable to have a mobility model that generates
stable nodal movement so that the mobile system maintains
a steady level of mobility over time, e.g., a fixed average
nodal speed and a fixed speed variance. This is especially
critical for simulation studies that present performance
metrics as time averages.
Our recent work [2] shows that one of the most widely
used, the random waypoint model, has a transient period in
which the average nodal speed decreases to a steadystate
level (below the initial average) as the simulation goes on.
Such speed decay can have dramatic influence on measured
performance and overhead. Consequently, one cannot
INTRODUCTION
IMULATION has become an indispensable tool in the
design and evaluation of mobile systems. It allows study
present timeaveraged metrics during this period of decay
as the underlying process is not stationary.
There are a number of ways to mitigate the negative
effect of this transient speed decay. For example, narrowing
the range from which to select speeds can reduce the degree
of decay and the time required to reach a steady state.
However, it limits the speed variation and does not remove
decay in principle. Another approach [2], [3] is to warm up
every simulation by running it until steady state is reached
and then discarding the initial data. While this is valid, it
can be cumbersome, especially because the duration of this
settling period is casedependent in general, rendering the
simulation process error prone.
The objective of this study is to develop stationary
mobility models, i.e., those that do not have such transient
speed decay, so that reliable simulation results may be
obtained via timeaverages without having to discard initial
data. In this paper, we give a general derivation of the
steadystate average speed distribution for several classes of
random mobility models and show that speed decay is not a
property exclusive to the random waypoint model, but,
rather, a much more common phenomenon. Indeed, any
random mobility model that chooses speed and destination
independently exhibits a similar transient period in which
the speed decays. The intuition is that nodes travel for
longer times at lower speeds if the destination is chosen
independently of the nodal speed.
independent of the specific distribution from which speeds
are chosen and the mechanism with which destinations are
determined. Furthermore, if pause time between successive
trips is set to zero, the distribution governing the steady
state average speed is independent of the mechanisms used
to determine destination; it depends only on the distribu
tion from which speeds are chosen.
This result is true
IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL. 5, NO. 7, JULY 20061
. The authors are with the Electrical Engineering and Computer Science
Department, University of Michigan, 1301 Beal Avenue, Ann Arbor, MI
481092122. Email: {jkyoon, mingyan, bnoble}@eecs.umich.edu.
Manuscript received 14 Apr. 2004; revised 23 Nov. 2004; accepted 24 Feb.
2005; published online 16 May 2006.
For information on obtaining reprints of this article, please send email to:
tmc@computer.org, and reference IEEECS Log Number TMC01370404.
15361233/06/$20.00 ? 2006 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS
Page 2
Following this result, we show how this transient period
and speed decay can be completely eliminated in a
fundamental way by constructing a composite random
mobility model from any random mobility model that
exhibits speed decay. The key is to initialize the simulation
in the steady state by selecting the speed of the first trip
from the steadystate speed distribution and selecting
speeds of subsequent trips from the original speed dis
tribution. It is worth pointing out that this method is
orthogonal to any modification to a random mobility model
to obtain desired spatial distributions of nodes, e.g., uniform
distribution within the movement area. Thus, it is equally
applicable. Finally, warmup may still be needed if the
simulated mobile system starts from a “cold state.”
However, by having such stationary mobility models,
warmup is no longer needed for nodal movement, freeing
the experimenter to consider other matters.
The rest of the paper is organized as follows: Section 2
gives an overview of problems and issues. Section 3
presents a taxonomy of random mobility models and
derives their steadystate average speed distribution.
Section 4 presents the methodology of constructing a
stationary mobility model without speed decay from a
random mobility model, while Section 5 demonstrates its
effectiveness for a variety of mobility models via simula
tion. Section 6 presents related work. Section 7 discusses the
application of the results obtained in this study and
concludes the paper.
2SPEED DECAY IN MOBILITY MODELS
As mentioned earlier, the performance of a mobile ad hoc
network is highly dependent on the underlying mobility
model employed for the study, including experiment,
simulation and analysis [3], [4]. As performance measures
from experimental or simulation studies are often collected
in the form of averages over time, it is highly desirable to
have mobility models that provide a steady level of mobility
over time. The random waypoint model is one of the most
widely used models for mobile ad hoc network simulation.
A majority of work in the area is based on simulation results
with this model. However, the random waypoint model has
a transient period at the beginning of the simulation in
which the average node speed decreases before reaching the
steadystate level [2]. This poses a serious problem because,
with the decrease in average node speed, various perfor
mance measures also change over time, leading to unreli
able time averages. It was also shown that the speed decay
could last for a considerable length of time if the minimum
speed is set close to zero in the simulation; it becomes
infinitely long when the minimum speed is set to zero,
which is the default setting in ns2 [5].
Fig. 1 shows the simulation results illustrating the speed
decay phenomenon and how it affects performance of
mobile ad hoc routing protocols in simulation. Using DSR
for example, it shows that even when the minimum speed is
set to be positive and the steadystate is eventually reached,
the time it takes for this to happen may still outlast the
system warmup period. As a result, performance evaluation
is complicated not only by the system itself, but also by the
mobility model.
A natural question to ask is whether this problem is
exclusive to the random waypoint model. As will be shown
in the next section, this indeed is a problem common to a
large class of mobility models.
3MOBILITY MODELS AND STEADYSTATE SPEED
DISTRIBUTION
Classification of Mobility Models 3.1
Mobility models may be classified in many ways. In this
section, we will follow the terms used in [3] and categorize
them into entity mobility models and group mobility
models. In the former, nodes move independently of each
other, while, in the latter, nodes move in groups or in a
correlated way. Here, we will limit our attention to models
under which a node’s movement is specified by a sequence
of trips, where a trip is a miniature movement on a smaller
scale in both time and distance compared to the duration of
the simulation and the movement area.1The node’s entire
movement trajectory is formed by a sequence of such trips
and a node may pause between successive trips. A trip is
typically specified by two or more of the following random
elements: node speed, travel time, travel distance, destina
tion, and travel direction/angle. Since this study primarily
concerns the speed property of a mobility model rather than
the spatial property, we will be considering only three
elements: speed, time, and distance. This is because a
destination or direction can both be translated into travel
distance, given a starting point and choices of either speed
2 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL. 5,NO. 7,JULY 2006
Fig. 1. Speed decay and its effect on overhead packets of mobile ad hoc
routing protocols. Fifty nodes, 10 simulation runs, 30 sources, 64 bytes/
packet, 4 packets/sec, speed = [0.3, 19.7] m/s.
1. Often, the movement during a trip is on a straight line with a fixed
speed. A notable exception is a model suggested by Bettstetter [6], where
accelerations and decelerations are added to select a modal speed.
Page 3
or time. Since there are only two degrees of freedom, it
suffices to specify any two of these three elements for a trip.
Furthermore, as speed is almost always directly specified in
a mobility model, we will only consider models that are
based on the selection (speed, time) and (speed,
distance). Examples of such models include the random
waypoint model [7], [8], [9] and the random direction model
[10]. More can be found in the survey [3].
3.2 Entity Mobility Models without Pause
In this section, we consider entity mobility models with no
pause in between trips. We will study the steadystate
speed distribution of this class of mobility models given
different assumptions on the dependence of the underlying
random elements.
3.2.1 General Case, Dependence Unknown
We first consider the general case where the dependence of
the random elements are not known. The random variables,
speed, time, and distance, are denoted by V, S, and R,
respectively, and are assumed to be within finite minimum
and maximum values, denoted by Vmin, Vmax, Smin, Smax,
Rmin, and Rmax, respectively. This also implies that the
minimum speed, Vmin, is strictly positive since, otherwise,
the maximum travel time Smaxcan be unbounded.2
The cumulative distribution function (cdf) of the steady
state speed, Vss, can be obtained as follows when (speed,
time) are chosen:
PðVss? vÞ ¼ fraction of time speed falls below v
R R
¼
v0?vsfS;Vðs;v0Þ dsdv0
R R
S;VsfS;Vðs;v0Þ dsdv0;
ð1Þ
where fS;Vðs;vÞ is the joint probability density function
(pdf) of time and speed.
Similarly, when speed, distance are chosen, the
steadystate cdf of Vssis
R R
R;V
PðVss? vÞ ¼
v0?v
r
v0fR;Vðr;v0Þ drdv0
r
R R
v0fR;Vðr;v0Þ drdv0;
ð2Þ
where fR;Vðr;vÞ is the joint pdf of distance and speed.
From (1) and (2) we can obtain the pdf and expectation of
the steadystate speed. Alternatively, the expectation can be
obtained through time averages. Since each node moves
independently, it suffices to consider a single node.
Denoting the longterm time average of node speed by?V V,
we have
? V V ¼ lim
t? !1
1
t
PNðtÞ
Zt
n¼1rn
PNðtÞ
0
vð?Þd? ¼ lim
t? !1
PNðtÞ
PNðTÞ
NðtÞ
n¼1vnsn
t
¼ lim
t? !1
n¼1sn
¼ lim
t? !1
1
NðtÞ
1
n¼1rn
PNðtÞ
n¼1sn
¼E½R?
E½S?;
ð3Þ
where NðtÞ is the total number of trips taken up to time t,
including the last one which may be incomplete. rn, sn, and
vnare the travel distance, time, and speed of the nth trip,
respectively. Note that frng and fsng are iid random
sequences; thus, their averages converge to the ensemble
averages as t ! 1 by the strong law of large numbers.
On the other hand, at time 0 when the first trip is
determined, the distribution of node speed is simply
fVinit¼ fVðvÞ, the distribution from which random speeds
are chosen, and the expected speed is
E½Vinit? ¼ E½V? ¼
Z
V
vfVðvÞdv:
These quantities will be used to compare with the steady
state values in subsequent sections.
3.2.2 Speed and Time, Independent
We have fS;Vðs;vÞ ¼ fSðsÞfVðvÞ. Thus, (1) reduces to the
following:
R R
Rv
Zv
Therefore, the pdf of the steadystate speed Vssis simply
PðVss? vÞ ¼
v0?vsfS;Vðs;v0Þ dsdv0
R R
RVmax
fVðv0Þ dv0:
S;VsfS;Vðs;v0Þ dsdv0
VminfVðv0Þ dv0RSmax
¼
SminsfSðsÞ ds
SminsfSðsÞ ds
VminfVðv0Þ dv0RSmax
¼
Vmin
ð4Þ
fVssðvÞ ¼ fVðvÞ;
ð5Þ
which is identical to the initial speed distribution. It
immediately follows that
E½Vss? ¼ E½V ? ¼ E½Vinit?;
ð6Þ
which means that the average speed does not change over
time. The intuition and significance of this result will be
more clearly described in the next section.
3.2.3 Speed and Distance, Independent
Since speed and distance are independent, speed and time
are necessarily dependent. Thus, (3) gives
E½Vss? ¼?V V ¼E½R?
E½S?¼E½VS?
E½S?
6¼ E½V? ¼ E½Vinit?:
ð7Þ
This indicates that the steadystate average node speed is
different from the initial average node speed. To derive this
expectation precisely, we proceed using (2). Since speed
and distance are independent, fR;Vðr;vÞ ¼ fRðrÞfVðvÞ. Plug
ging this in (2) gives
R R
R;V
Rv
Vmin
Rv
Vmin
PðVss? vÞ ¼
v0?v
r
v0fR;Vðr;v0Þ drdv0
r
v0fR;Vðr;v0Þ drdv0
v0fVðv0Þ dv0RRmax
R R
Vmin
RVmax
Vmin
RVmax
¼
1
v0fVðv0Þ dv0RRmax
RminrfRðrÞ dr
RminrfRðrÞ dr
1
¼
1
v0fVðv0Þ dv0
1
v0fVðv0Þ dv0:
ð8Þ
YOON ET AL.: A GENERAL FRAMEWORK TO CONSTRUCT STATIONARY MOBILITY MODELS...3
2. Positive minimum speed also prevents the average node speed from
asymptotically approaching zero as time goes on [2].
Page 4
Thus, we obtain the pdf of the steadystate speed
fVssðvÞ ¼
1
vfVðvÞ
1
v0fVðv0Þ dv0
RVmax
Vmin
ð9Þ
and the expectation of steadystate speed
E½Vss? ¼
ZVmax
RVmax
Vmin
v ? fVssðvÞ dv
1
v0fVðv0Þ dv0¼
¼
Vmin
1
1
E½1
V?:
ð10Þ
E½Vss? is always less than or equal to the initial average
E½V? by Jensen’s inequality [11], which states that if a
function gðXÞ is convex, then E½gðXÞ? ? gðE½X?Þ. Now, let
gðVÞ ¼1
?1
Therefore,
V(Vmin? V ? Vmax), which is convex. It follows that
?
E
V
?
1
E½V ?) E½V ? ?
1
E½1
V?:
ð11Þ
E½Vinit? ¼ E½V? ? E½Vss?;
ð12Þ
where the equality holds only when Vmin¼ Vmax. Thus, the
average speed decays with time unless the node speed is
constant.
The above results are summarized as follows:
1.
The steadystate speed distribution is different from
the initial speed distribution.
The steadystate speed distribution and expectation
of node speed are completely characterized by the
initial speed distribution fVðvÞ, which is usually
given.
The steadystate speed distribution is determined
only by the node speed distribution and not by how
distances/destinations are chosen.
The steadystate average node speed is lower than
the initial average speed. This means that if
distance/destination is chosen independently of
speed, there will always be speed decay.
Items 3 and 4 further indicate that models that only differ
in distance/destination selection are essentially indistin
guishable in terms of their speed properties.
An intuitive explanation for 4 is that when speed and
distance are chosen independently, a lower speed results in
a longer trip. Note that the steadystate speed is weighted
by travel time and, thus, is always lower than the initial
average speed. To see this more clearly, consider the
following intermediate result from (3):
2.
3.
4.
E½Vss? ¼? V V ¼ lim
t? !1
PNðtÞ
n¼1vnsn
t
:
Note that low speed vns are more likely to be weighted by
large sns, which leads to a lower longterm average node
speed. In contrast, when speed and time are selected
independently (as in Section 3.2.2), vnis not correlated with
sn. Thus, the steady state speed distribution remains the
same as the initial speed distribution. Alternatively, 4 can be
explained using the properties of harmonic mean of renewal
speed, where the steadystate average speed can be viewed
as the average rate in the system performance measure [12].
This phenomenon can also be explained via Palm calculus,
see, for example, [13].
Equation (9) is a very general result. It holds regardless
of the speed distributions used. It shows that the average
node speed of an arbitrary mobility model starts from an
initial value, decays over time, and then settles to a certain
steadystate value, as long as speed and distance are chosen
independently.
3.2.4 (Speed and Time) or (Speed and Distance),
Correlated
If speed and distance are chosen dependently, e.g., a model
that gives higher probability to higher speeds when the
distance chosen is larger, one may be able to reduce speed
decay by properly correlating the two. In [14], we showed
an example where travel time is correlated with travel
speed. In this particular example, speed decay exists.
However, it is possible to construct a joint distribution of
speed/time or speed/distance so that the resulting average
speed process is stationary, although, in this case, the
derivation of the steady state speed distribution is much
more complicated.
3.3Entity Mobility Models with Pause
3.3.1 General Case, Dependence Unknown
If pause is added between successive trips, a mobility
model can be viewed as an alternating renewal process that
has two independent renewal processes: a move process
and a pause process [15]. Since, during pause, the node has
a speed of zero, the pause process essentially has a speed
pdf of fVPðvÞ ¼ ?ðvÞ, where VP denotes the “pause speed”
(which is zero).
When (speed, time) are chosen, the cdf of Vsscan be
obtained from (1) as follows: Denoting the pause time by P
and using the fact that pause time and pause speed are
independent, we have
FVssðvÞ ¼ PðVss? vÞ
RR
RR
where fS;Vðs;v0Þ, Vmin? v0? Vmax, and fP;Vpðp;v00Þ, v00¼ 0,
are corresponding joint pdfs. E½P? is the expectation of
pause time.
Similarly, when (speed, distance) are chosen, the
steady state cdf of Vssis generalized from (2) as
¼
v0?vsfS;Vðs;v0Þdsdv0þRR
v0?vsfS;Vðs;v0Þdsdv0þ E½P?R
v00?vpfP;Vpðp;v00Þdpdv00
P;VppfP;Vpðp;v00Þdpdv00
v00?v?ðv00Þdv00
RR
S;VsfS;Vðs;v0Þdsdv0þRR
RR
¼
S;VsfS;Vðs;v0Þdsdv0þ E½P?
;
ð13Þ
FVssðvÞ ¼ PðVss? vÞ
RR
R;V
RR
¼
v0?v
RR
v0?v
r
v0fR;Vðr;v0Þdrdv0þRR
RR
v0fR;Vðr;v0Þdrdv0þRR
r
v00?vpfP;Vpðp;v00Þdpdv00
P;VppfP;Vpðp;v00Þdpdv00
v00?v?ðv00Þdv00
r
¼
v0fR;Vðr;v0Þdrdv0þE½P?R
R;V
r
v0fR;Vðr;v0Þdrdv0þ E½P?
;
ð14Þ
where fR;Vðr;v0Þ for Vmin? v0? Vmax is the joint pdf of
travel distance and speed.
With a slight modification to (3), the longterm time
average of node speed with nonzero pause time becomes
? V V ¼
E½R?
E½S? þ E½P?:
ð15Þ
4 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5,NO. 7,JULY 2006
Page 5
As in Section 3.2.1, we would like to determine the initial
average speed for comparison purposes. We will consider
the case where a node starts in either the move or pause
state with a certain probability. From the point of view of
using a mobility model for simulation, it is reasonable to
assume that these are exactly the probabilities that a node is
found to be in either state when the mobility model reaches
equilibrium, denoted by Pmove and Ppause, respectively.
Then, the initial average speed is simply
E½Vinit? ¼ E½V?Pmoveþ 0 ? Ppause¼ E½V ?Pmove:
ð16Þ
3.3.2 Speed and Time, Independent
As in Section 3.2, this independence allows (13) to reduce to
FVssðvÞ ¼ PðVss? vÞ
¼E½S?Rv
VminfVðv0Þ dv0þ E½P?R
E½S? þ E½P?
v00?v?ðv00Þdv00
:
ð17Þ
Then, the probability that a node is in a pause state is
Ppause¼ FVssðv ¼ 0Þ ¼
E½P?
E½S? þ E½P?
ð18Þ
and the probability that a node is in a move state is
Pmove¼ 1 ? Ppause¼
E½S?
E½S? þ E½P?:
ð19Þ
Therefore,
fVssðvÞ ¼
E½S?fVðvÞ
E½S?þE½P?
¼ fVðvÞ Pmove;Vmin? v ? Vmax
E½P??ðvÞ
E½S?þE½P?
¼ ?ðvÞ Ppause;v ¼ 0:
8
>
>
:
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
ð20Þ
This pdf indicates that a node either moves at a certain
speed selected from the pdf fVðvÞ with probability Pmoveor
pauses with probability Ppause. From (20), the expectation of
steadystate node speed is
E½Vss? ¼
E½S?E½V?
E½S? þ E½P?;
ð21Þ
which indicates that there is no speed decay, because E½Vss?
is the same as the initial average speed E½Vinit? ¼ E½V?Pmove
in (16).
3.3.3 Speed and Distance, Independent
As in the previous section, we can proceed from (14) and
obtain
FVssðvÞ ¼ PðVss? vÞ
¼E½R?Rv
Vmin
1
v0fVðv0Þ dv0þ E½P?R
E½R?E½1
v00?v?ðv00Þdv00
V? þ E½P?
:
ð22Þ
In the same manner as in Section 3.3.2, the probability that a
node is in a pause state is
Ppause¼ FVSSðv ¼ 0Þ ¼
E½P?
V? þ E½P?
E½R?E½1
ð23Þ
and
Pmove¼ 1 ? Pmove¼
E½R?E½1
E½R?E½1
V?
V? þ E½P?
ð24Þ
and the pdf of the steadystate speed Vssis
8
>
>
:
The steadystate pdf in (25) is interpreted in exactly the
same way as in (20): A node moves at a certain speed
according to the pdf
E½1
with probability Ppause.
From the pdf in (25), the expectation of steadystate
speed is
fVssðvÞ ¼
E½R?1
E½R?E½1
1
vfVðvÞ
E½1
E½P??ðvÞ
E½R?E½1
vfVðvÞ
V?þE½P?
¼
V?Pmove;Vmin? v ? Vmax
V?þE½P?
¼ ?ðvÞ Ppause;v ¼ 0:
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
ð25Þ
1
vfVðvÞ
V?
with probability Pmoveor pauses
E½Vss? ¼
E½R?
V? þ E½P?:
E½R?E½1
ð26Þ
In the case of nonzero pause times, it can also be shown
that the steadystate average E½Vss? is less than or equal to
the initial average E½Vinit?. Recall that, from (16),
E½Vinit? ¼ E½V?Pmove¼ E½V?
E½R?E½1
E½R?E½1
V? from (12), we have
E½V?E½1
E½R?E½1
V?
V? þ E½P?:
ð27Þ
Then, using (26) and 1 ? E½V?E½1
E½Vss? ?
V?E½R?
V? þ E½P? ¼ E½Vinit?:
ð28Þ
This means that the average node speed decays with time
no matter what pause time is unless the node speed is
constant.
3.4
Unlike in entity mobility models where nodes move
independently, in a group mobility model, nodes within
the same group move in a coordinated way. However, if the
movement of an entire group is specified (e.g., via a leader
node) in ways similar to those used in entity models, then
groups as a whole may again exhibit speed decay when the
speed and distance of the group movement are chosen
independently. In the remainder of this section, we will
examine the speed property of two commonly used group
mobility models: the pursue model and the reference point
group mobility (RPGM) model. In the following analysis, we
will not consider pause times since analysis with pause time
can be easily obtained using similar methods.
Group Mobility Models
3.4.1 Pursue Mode
The main feature of the pursue model is that there is a
target (or leader) node followed/tracked by other nodes [3],
[16], [17]. The framework was presented in [16], which
allows different implementations. We consider the follow
ing implementation of the pursue model: Mobile nodes are
divided into several groups and each group consists of a
single target node and a number of follower nodes. The
YOON ET AL.: A GENERAL FRAMEWORK TO CONSTRUCT STATIONARY MOBILITY MODELS...5
Page 6
target node moves around in a rectangular space, according
to the random waypoint model, without pause and with
speed range ½Vmin;Vmax? in m/s. Every T seconds, the
follower nodes choose their speed uniformly from
½Vmin;Vmax?, pick the current location of the target node as
their destination, and move to the destination. If a follower
node arrives at the destination before the next updating
instance (i.e., within T seconds), it stays there until the next
update. Thus, every T seconds, the follower nodes update
their speeds and destinations while continuously tracking a
target node. This repeats until the simulation ends. Since
each group moves independently of the others, it suffices to
look at a single group without loss of generality.
Suppose there are M nodes in a group with a single
target node and M ? 1 follower nodes. Since the target node
follows the random waypoint model where its speed and
distance are chosen independently, its average speed will
decrease over time before reaching the equilibrium. Its
steadystate distribution fVssðvÞ is given by (9). A follower
node also chooses its speed and destination independently,
but its speed is updated every fixed T seconds. Thus, a
follower’s speed is not timeweighted as opposed to the
case of the target node. However, there could be a “short
term” speed decay during an interval, since a node would
remain at the location when it arrives there before the next
update. So, there are two types of speed decay phenomena:
a longterm decay of a target node and a shortterm decay
of follower nodes, as shown in Fig. 2a. The longterm decay
is exactly the same as observed in the entity mobility
models due to the independent selection of speed and
distance, whereas the shortterm decay is a result of the
nature of this particular model. Fig. 2b shows the desired
result with the decay eliminated.
Since the speeds of the follower nodes are chosen from
distribution fVðvÞ, there is no speed decay caused by
follower nodes at update points. Therefore, at every update
point in the steady state, there exists a single target node
with fVssðvÞ and M ? 1 follower nodes with fVðvÞ. Since the
target node and the follower nodes select speeds indepen
dently, the steadystate average speed is simply
E½Vss?update¼1
ME½Vss? þM ? 1
M
E½V?:
ð29Þ
This shows that the longterm decay diminishes as the total
number of nodes in a group, M, increases. The steadystate
distribution is rather complicated to compute in this case.
Denoting by Z the average speed of all nodes, we have
Z ¼1
the speeds of the target node and the follower nodes,
respectively. Then, the pdf of Z can be obtained using
convolution integrals [18].
MX þ1
M
PM?1
i¼1Yi, where X and Yi, i ¼ 1;???;M ? 1 are
3.4.2 Reference Point Group Mobility (RPGM) Model
Instead of specifying a target node, the RPGM model [3],
[4], [19], [20] has an implicit and insubstantial tracking point
or reference point. Based on the current position and speed
of the reference point, each mobile node selects its own
speed, time, or destination. RPGM is also a framework
which allows different implementations. Here, we will
examine an implementation based on [19].
Suppose that a reference point is following some
predefined group motion. Without loss of generality,
suppose an update interval T is one second, as is
commonly assumed. Then, we can define a vector of group
motion GMð?Þ
of reference point at ? from the expected next position at
? þ 1, as shown in Fig. 3. Next, we define a vector of
random motion RMð? þ 1Þ
point at ? þ 1, by randomly selecting a distance between 0
and a predefined maximum value, RMmax, and an angle
between 0 and 360 degrees. Thus, the next position of
mobile node is determined by GMð?Þ
shown in Fig. 3. Since the movement distance of the mobile
node, Distð?Þ, is easily computed from the location of a
mobile node at each time instance and travel time is
assumed to be T ¼ 1 second, the speed of the mobile node
automatically becomesDistð?Þ
T
.
One may think that, since this model selects distance and
time independently and updates parameters every fixed
????!at time ? by subtracting the current position
???????!from the position of reference
????!and RMð? þ 1Þ
???????!as
6 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 7,JULY 2006
Fig. 2. Two types of speed decays in pursue model: a longterm decay
and a shortterm decay. (a) Illustration of pursue model simulation result.
The dotted line shows a longterm decay. (b) The desired result with the
longterm decay eliminated.
Fig. 3. Reference point group mobility model.
Page 7
interval, it may not have speed decay. However, if the
reference point moves according to the random waypoint
model, for example, then the average speed of the reference
point decays. This in turn will cause the average length of
GMð?Þ
correlated with GMð?Þ
speed decay occurs. The simulation result in Section 5 also
illustrates this.
????!
in Fig. 3 to gradually decrease. Since Distð?Þ is
????!, Distð?Þ also decreases while the
4ELIMINATING DECAY
Speed decay during the transient period is undesirable
because simulation results collected during this period
(before the steady state is reached) will not be reliable.
Methods of reducing such a negative effect have been
suggested and used in the literature [21]. One way is to
reduce the range of allowed speed by setting the
maximum speed and minimum speed to be within a
certain percentage of a set value, e.g., ? 10 percent of
15 miles per hour [3]. This significantly reduces both the
magnitude and the duration of speed decay, but also
heavily limits the variation of nodal speed within the
same experiment. Another method is to warm up the
simulation by discarding a certain portion of the initial
data or simply to run the simulation long enough and
collect results averaged over time so that the effect of the
initial decay is diluted. The problem with this method is
that it is not always clear how much one should discard
or how long is indeed long enough. If we do not warm
up enough, then the effect of speed decay still exists; on
the other hand, discarding too much results in waste. In
order to do this appropriately, we may need to prerun
the mobility model, which adds inconvenience and
wastes resources required for simulation studies. In short,
none of these methods eliminate the speed decay inherent
to such mobility models in a fundamental way.
In Section 3, we presented a method of deriving the
steadystate speed distribution. In [14], we have shown
examples of calculating these distributions for a variety of
mobility models. This naturally leads us to question
whether we can start the mobility model directly from the
steady state and construct a stationary process that is free of
the transient speed decay period.
It is important to note that this does not mean we can use
the distribution derived in (9) or (25) for the selection of
node speed throughout simulation. We restate the same
equation here, assuming a zero pause time for now:
fVssðvÞ ¼
fVðvÞ
constant
1
v
? ?
:
ð30Þ
The above result essentially indicates that the steadystate
speed distribution fVssðvÞ is different from any nontrivial
distribution fVðvÞ from which node speeds are chosen. The
former is the distribution observed at arbitrary points in
time, whereas the latter is the distribution observed at the
waypoints or the points between successive trips. This is
illustrated in Fig. 4, where speed distribution observed at
points A and C is fVðvÞ, and fVssðvÞ is observed at some
arbitrary point B. Therefore, to start the simulation in
steady state means to start the simulation from points like B
after the system has reached steady state rather than
waypoints A and C. This is also equivalent to resuming a
simulation which is suspended at an arbitrary point in the
steady state. This discussion naturally leads us to the
following method of constructing a stationary mobility
model: Start the simulation by using fVssðvÞ to select the
speed of the first trip of a moving node as if we are
resuming a simulation suspended at an arbitrary point in
steady state. After the first trip ends, we use fVðvÞ to select
node speed for all subsequent trips. We call this a composite
random mobility model as it consists of two different speed
distributions.
The same argument applies to the initial pause time
selection. That is, if a node starts from a pause state, the first
pause time should be selected from the steadystate
distribution of pause time, which is known to be (the
limiting distribution of forward recurrence time using
renewal theory) [15]:
fPssðpÞ ¼1 ? FPðpÞ
E½P?
;
ð31Þ
where FPðpÞ is the cdf of pause time. This enables us to start
a simulation directly from the steady state, which is again
equivalent to resuming a simulation suspended at an
arbitrary point during some pause period in steady state.
To summarize, we construct a composite stationary
random mobility model as follows:3
1.
Determine whether a node starts from a move state
or a pause state, with probability Pmove and Ppause,
respectively. These are calculated using methods
shown in Section 3.
If a node starts from a move state, use fVssðVmin?
v ? VmaxÞ to select the travel speed.
If a node starts from a pause state, use fPssðpÞ to
choose the pause time.
After the first trip (either move or pause) of a node,
use fVðvÞ and fPðpÞ to select all subsequent travel
speeds and pause times, respectively.
Technically, there are other ways to construct a sta
tionary process by modifying the initial part of the mobility
model. For example, if pause is not inserted between
successive trips, we could find and use the stationary
starting time of nodes’ first trips, which may seem more
intuitive. This method involves the derivation of the
distribution of the initial starting time. This may or may
2.
3.
4.
YOON ET AL.: A GENERAL FRAMEWORK TO CONSTRUCT STATIONARY MOBILITY MODELS...7
3. This modification has been implemented and included in the
latest ns2 version 2.27.
Fig. 4. Different points where distributions are observed in the steady
state.
Page 8
not be desirable, depending on the mobile system being
simulated in that a significant portion of the network may
not be moving for some period of time.
On the other hand, modification through the steadystate
speed distribution provides a very effective way of
eliminating speed decay and producing a stationary
process. We emphasize that the above composition meth
odology can be applied to any random mobility models that
choose speed and distance/destination independently and
that employ a single speed distribution, in order to obtain a
decayfree random mobility model. For example, as
described in Section 3.4, we can identify speeddecaying
nodes in a group mobility model and replace their first trip
distribution by the steadystate distribution. In doing so, we
can also build a stationary composite group mobility model
as well as a composite entity mobility model. The
effectiveness of this methodology is demonstrated in the
next section.
5SIMULATION RESULTS
In this section, we show via simulation the evolution of
instantaneous average node speed over time for a few entity
mobility models and a couple of group mobility models
examined in Section 3. As a metric, we use the instanta
neous average speed ? v vðtÞ, which is defined as
PN
where N is the total number of nodes in the simulation
scenario and viðtÞ is the speed of node i at time t.
5.1 Entity Mobility Models
? v vðtÞ ¼
i¼1viðtÞ
N
;
ð32Þ
Here, we adopt three examples of entity mobility models
for simulation: 1) a model using a uniform speed and a
uniform destination (i.e., the random waypoint model), 2) a
model using a uniform speed and uniform distance
distribution, and 3) a model choosing speed correlated
with travel time (see [14] for details). Fig. 5 depicts the
behavior of each of the original mobility models both with
and without pause, while Fig. 6 shows the behavior of the
composite models. As described in Section 4, each node in
these composite models chooses either its speed or pause
time only for the first trip from the computed steadystate
distributions of speed and pause time, respectively,
depending on the probabilities Pmoveand Ppause. Thereafter,
each node alternately chooses its speed and pause time
from the original distributions. Each graph also plots the
steadystate average node speed predicted by analysis.
In this set of simulation results, each curve is the average
over 10 different scenarios. Each scenario contains 50 mobile
nodes moving independently in a movement space of
1,500 m ? 500 m, according to the specified mobility model.
The speed range for all scenarios is from 1 m/s to 19 m/s,
which results in the initial average node speed of 10m/s
with zero pause time. When nonzero pause is applied,
pause time is randomly selected from the uniform distribu
tion from 0 to 60 seconds.
As shown in Fig. 5, speed decay exists in all cases. We see
from Fig. 6 that the constructed composite models success
fully eliminated such decay in all cases, including that with
speed correlated with time. Thus, this construction meth
odology is effective, regardless of the dependency between
travel speed and distance or time, as long as the steadystate
speed distribution can be characterized. Furthermore,
analysis in our previous version of this paper [14] showed
that average node speed settles to 6.1 m/s in all cases above
with the corresponding parameters. As expected, the
unmodified models converge to the predicted values, while
the composite models start and remain there. Such
composite models greatly simplify the evaluation process
in a simulation study.
8IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5,NO. 7, JULY 2006
Fig. 5. Average speed decays in a few examples of models
examined in Section 3 with and without pause. Speed = [1,19] m/s.
Pause = [0,60] sec. (a) Speed = uniform, destination = uniform (i.e.,
random waypoint model). (b) Speed = uniform, distance = uniform.
(c) Speed = uniform, time = bounded exponential.
Page 9
5.2
Figs. 7 and 8 show the pursue model and RPGM model,
respectively. As in the entity model simulations above, all
results are the average over 10 scenarios. In the RPGM
model, there are one reference point moving by the random
waypoint model and 10 mobile nodes moving around in a
space. For the pursue model described in Section 3.4.1, we
implemented it with 20 groups consisting of a single target
and two follower nodes each, a total of 60 nodes. The target
node moves around according to the random waypoint
model during the entire simulation time. In addition,
parameters are updated every five seconds. So, here, M is
3 and T is 5. The original speed distribution for both models
is the same as that in the entity models above: fVðvÞ is a
uniform distribution from 1 to 19 m/s. However, we did
not consider pause times for simplicity. Even if pause time
Group Mobility Models
is added, the main principle does not change and, thus, we
can apply the same framework to the nonzero pause cases.
Fig. 7a is the average speed over all nodes for 900 seconds
and Fig. 7b is a magnified version of Fig. 7a for the first
YOON ET AL.: A GENERAL FRAMEWORK TO CONSTRUCT STATIONARY MOBILITY MODELS...9
Fig. 6. No speed decays by using a steadystate pdf for the first trip.
Speed = [1,19] ms. Pause = [0,60] sec. (a) Speed = uniform, destination
= uniform (i.e., random waypoint model). (b) Speed = uniform, distance
= uniform. (c) Speed = uniform, time = bounded exponential.
Fig. 7. Pursue model with and without improvement. (a) Pursue model.
(b) First 100second result of (a) magnified. (c) Pursue model with
steadystate distribution. (d) First 100second result of (c) magnified.
Page 10
100 seconds. As described in Section 3.4.1, there exist both
longterm and shortterm decay and the dotted curve
represents a longterm decay of average node speed at
every update point. Since M ¼ 3 and T ¼ 5, according to
(29) in Section 3.4.1, the average speed at every update
point starts from E½Vinit? ¼ 10 m/s, decays over time, and
settles to
E½Vss?update¼1
ME½Vss? þM ? 1
¼ 8:7 m=s;
M
E½V? ¼1
3ð6:1Þ þ2
3ð10Þ
as shown in Fig. 7a. Since target nodes suffer from speed
decay, we can replace the first distribution of target nodes
by the steadystate speed distribution and, thus, can
construct a stationary model, as shown in Fig. 7c and its
magnified version Fig. 7d. Note that the fluctuation of short
term decay increases in the beginning because it should
indispensably take all follower nodes some time to come
close to a target node from the initial uniformly random
location, and to start tracking it.
Fig. 8 shows the RPGM model examined in Section 3.4.2
with update interval T ¼ 1 sec and maximum random
motion value RMmax¼ 10 m. As explained, the RPGM
model also exhibits speed decay because the speed of
mobile nodes is affected by the speed decay of the reference
point. Fig. 8a presents this result. That is, both speeds of
reference point and mobile nodes decay over time. But,
since the decay of reference point is the only source of speed
decay for all nodes, if it is fixed, other mobile nodes
consequently must not have the speed decay problem.
Thus, we can again apply the steadystate distribution to
the reference point to remove the decay, and Fig. 8b clearly
verifies this.
5.3
In addition to the average node speed, relative speed is also
widelyusedasacriterionforcomparisoninsimulations(e.g.,
[4]). Relative speed between a pair of nodes is defined as
Relative Speed of Mobile Nodes
Vrelði;j;tÞ ¼ jviðtÞ
??!are speed vectors of nodes i and j at
??!? vjðtÞ
??!j;
ð33Þ
where viðtÞ
time t, respectively. Thus, the average relative speed can be
defined as
PN
??!and vjðtÞ
VrelðtÞ ¼
i¼1
PN
j¼iþ1Vrelði;j;tÞ
N
2
? ?
;
ð34Þ
where N is the total number of nodes and
the total number of distinct node pairs. Since relative speed
is directly related to the individual node speed, if the
individual node speed decreases over time, so does relative
speed, as shown in Fig. 9. Here, we measured relative speed
of the random waypoint model with speed range from 1 to
19 m/s. As expected, speed decay in relative speed is
observed if speed decay in average speed occurs. Thus, if
we eliminate speed decay in average speed, speed decay in
relative speed also disappears, as shown in Fig. 9b.
N
2
? ?¼NðN?1Þ
2
is
6RELATED WORK
Mobility models are essential to the study of mobile systems
and, consequently, they have been extensively studied. One
10 IEEE TRANSACTIONS ON MOBILE COMPUTING,VOL. 5,NO. 7,JULY 2006
Fig. 8. RPGM model with and without improvement. (a) RPGM model.
(b) RPGM model with the speed decay eliminated.
Fig. 9. Relative speed. Speed = [1,19] m/s. (a) Relative speed in the
random waypoint model. (b) Relative speed in the composite model.
Page 11
can find a thorough and insightful survey by Camp et al.
in [3]. It includes not only a variety of entity random
mobility models used in ad hoc network simulations, but
also group mobility models such as RPGM [19], [22].
Among commonly used mobility models, the random
waypoint model is perhaps the most extensively used [1], [7],
[17], [23], [24]. It is implemented and widely distributed
with the ns2 [5] simulator. Most of the studies on this
model have focused on its spatial properties such as node
distribution within the simulated area U. Bettstetter [6], [25]
showed by simulation that the random waypoint model
does not have a uniform spatial distribution of nodes. Chu
and Nikolaidis [26] mathematically proved it and also
showed that there is a relationship between node distribu
tion and node speed. Due to the boundary effect, nodes are
more likely to be near the center of U and, thus, the node
distribution becomes bellshaped. Royer et al. [10] pointed
out that the boundary effect not only causes a nonuniform
node distribution but also causes the node density to
fluctuate with time. To eliminate both problems, they
proposed a random direction model and showed satisfactory
results. Bettstetter et al. further obtained the steadystate
node spatial distribution of the random waypoint model by
analysis and verified by simulation in [9]. In addition, some
stochastic properties of the random waypoint model also
have been analyzed in [8]: epoch length (i.e., travel
distance), direction distribution, and cell change rate of
mobile nodes.
In [2], we studied the temporal properties of nodal
movement/speed under the random waypoint model. We
showed that the average node speed decreases with time
before reaching a steady state. The settling time it takes to
reach the steady state increases as the minimum speed
decreases. In particular, if the minimum speed is zero, this
transient period becomes infinitely long. Simulation results
showed how such speed decay affects ad hoc routing
protocols such as DSR [7] and AODV [27]. A simple
solution suggested was to use a positive minimum speed,
combined with simulation warmup or initial data deletion
to remove the negative effect of speed decay. This never
theless does not remove the speed decay in an essential
manner. In a followup study [14], we developed a general
framework to determine whether speed decay occurs in a
specific mobility model and to construct a stationary
composite model by using the steadystate distribution of
node speed. Navidi and Camp independently and concur
rently developed a method for constructing a stationary
process for the random waypoint model [28]. Our work
presented here is more general as it applies to multiple
classes of mobility models including the random waypoint
model. Lin et al. in [29] and Le Boudec in [13] recently
proposed similar methods to build a stationary version of
the random waypoint model by using renewal theory and
palm calculus, respectively.
The study presented in this paper does not concern
whether a mobility model is realistic; rather, it concerns
how to construct mobility models so that they are more
suitable for simulation studies. Significant effort has been
made in the literature toward developing more realistic
mobility models. Hong et al. [19] proposed a group mobility
model to reflect a realistic scenario of group movement and
Bettstetter [6], [25] developed an enhanced model by
avoiding the impossible changes of speed or direction in
reality. A recent paper by Jardosh et al. [30] considered
obstacles to constructing a more realistic mobility model
and showed the effect of them on the performance of ad hoc
routing protocols.
There are also studies on characterizing the effect
mobility models have on the performance of the mobile
system. Examples include Bai et al. [4], which analyzed
mobility models by factors and showed the impact of each
factor on performance of ad hoc routing protocols, and
Kwak et al. [20], which proposed a general metric to
characterize mobility models based on link change rate of
mobile nodes.
It has to be noted that the random mobility models
considered in this paper are more commonly used for the
study of mobile ad hoc networks rather than infrastructured
networks that may involve (static or mobile) base stations.
This is especially the case with entity mobility models
where each node moves independently. Certain group
mobility models may describe the movement of a network
with mobile base stations, but the two examples considered
in this paper are again designed for ad hoc networks.
7DISCUSSION AND CONCLUSION
This paper examined a number of random mobility
models that are based on the selection of node speed,
travel distance or destination, travel time, or pause time
from probability distributions. A large class of these
models—including all those that select node speed and
distance independently—exhibit a transient period in
which the average node speed decreases before before
reaching steadystate. Such decay poses potential pro
blems for simulation studies that collect results averaged
over time, complicating the experimental process. This
decay is easily explained with a general analytical
framework, which allows one to transform a given
random mobility model into a stationary one by selecting
initial speeds from the steadystate distribution and
subsequent speeds from the original speed distribution.
It has to be mentioned that the focus of this study is on
constructing mobility models that are suited for simulation
studies of mobile networks. The construction presented in
this paper does not make a mobility model more or less
realistic. Developing realistic mobility models is a challen
ging research area on its own and is out of the scope of this
paper. Rather, the study here aims at fixing certain hidden
problems in a model. Similar problems should also be
avoided in a more realistic mobility model, however
constructed. Thus, by such studies, we hope that a similar
analysis may be applied to the development or evaluation
of more realistic models, which would ultimately lead to
better, more efficient models.
ACKNOWLEDGMENTS
This work is partially supported by DARPA/AFPL Grant
FA 87500410114.
YOON ET AL.: A GENERAL FRAMEWORK TO CONSTRUCT STATIONARY MOBILITY MODELS... 11
Page 12
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Jungkeun Yoon received the BS degree in
electrical engineering in 2001 from Seoul Na
tional University, Korea, and the MS degree in
electrical engineering in 2003 from the Univer
sity of Michigan, Ann Arbor. He is currently a
PhD candidate in the Department of Electrical
Engineering and Computer Science at the
University of Michigan, Ann Arbor. He is a
student member of the IEEE.
Mingyan Liu received the BSc degree in
electrical engineering in 1995 from the Nanjing
University of Aeronautics and Astronautics,
Nanjing, China, and the MSc degree in systems
engineering and the PhD degree in electrical
engineering from the University of Maryland,
College Park, in 1997 and 2000, respectively.
She joined the Department of Electrical En
gineering and Computer Science at the Uni
versity of Michigan, Ann Arbor, in September
2000, where she is currently an assistant professor. Her research
interests are in performance modeling, analysis, energy efficiency and
resource allocation issues in wireless mobile ad hoc networks, wireless
sensor networks, and terrestrial satellite hybrid networks. She is the
recipient of the 2003 US National Science Foundation Career Award.
She is a member of the IEEE.
Brian Noble received the PhD degree in
computer science at Carnegie Mellon University
in 1998. He is an associate professor in the
Electrical Engineering and Computer Science
department at the University of Michigan. His
research centers on software supporting mobile
devices and their users, and he is a recipient of
the US National Science Foundation Career
award. He is a 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.
12IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 5, NO. 7,JULY 2006
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