Content uploaded by Wanjiun Liao
Author content
All content in this area was uploaded by Wanjiun Liao
Content may be subject to copyright.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
Impact of Node Mobility on Link Duration in Multi-hop Mobile
Networks
Yueh-Ting Wu1, Wanjiun Liao1,2, Cheng-Lin Tsao1, and Tsung-Nan Lin1,2
1Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan
2Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan
Abstract-- In this paper, we study the impact of node mobility on link duration in multi-hop mobile networks. In multi-hop mobile
networks, each node is free to move, and each link is established between two nodes. A link between two nodes is established when one
node enters the transmission range of the other node, and the link is broken when either node leaves the transmission range of the
other. The time interval during which the link remains active is referred to as the link duration. We develop an analytical framework
for link duration in multi-hop mobile networks. We find that the link duration for two nodes is determined by the relative speed
between the two nodes and the distance during which the link is connected, which are in turn determined by the angles between the
two nodes’ velocities and the angle of one node incident to the other node’s transmission range, respectively. The analytical result is
extended to model multipoint links which appear in the existing group mobility models. The accuracy of our framework is validated by
simulations based on existing mobility models. The results show our model can describe accurately the link duration distribution for
both types of links in multi-hop mobile networks, especially when the transmission range of each node is relatively smaller than the
entire network coverage.
Index Terms—multi-hop mobile networks, link duration
I. INTRODUCTION
obility management in wireless networks has been an
active research topic for years [1-6]. Research efforts on
single hop mobile networks include the results for cellular
networks [1-4] or Mobile IP networks [5-6], i.e., only the last
hop to each mobile node is wireless, and communications
between nodes must go through the associated base stations
(BSs). Each wireless link is established only between the BS
and a node in a cell. When the node roams to another cell, the
link is handed over to the respective BS so as to retain the on-
going connection.
In a multi-hop mobile network, each node plays both roles
of a router and an end-point, and no pre-deployed
infrastructure such as BS is available for node
communications. Data are relayed by intermediate nodes if the
receiver node is beyond the transmission range of the sender
node. As a result, a wireless link is established between two
nodes. Each node is free to move arbitrarily. A link between
two nodes is activated when one node enters the transmission
range of the other, and the link is broken when either node
leaves the transmission range of the other node. When a link
on a routing path is broken, a rerouting process is initiated so
as to reduce service disruption in the network.
In this paper, we study the impact of node mobility on link
duration in multi-hop mobile networks. Specifically, we
develop an analytical framework to evaluate link duration in
such networks. In our framework, each node may move at a
different speed, pause for a while, and then move again. The
link duration here then corresponds to the time interval in
which two nodes stay within transmission range of each other.
The importance of link duration on system performance has
been identified in the literature [7-8]. For example, the link
duration may affect the lifetime of the routing path, which in
turn determines the packet delivery ratio or per-connection
throughput for an S-D pair. An analytical link duration model
can also help determine the timer setting in ad hoc routing [7]
or even on the design of better routing protocols to cope with
link breakage caused by node mobility [9].
The framework we develop is based on the relative
movement behavior of one node observed by the other node
(i.e., from the perspective of the observer node). We find that
the duration of a link between two nodes is determined by
their relative speed and the distance traversed within the
transmission range of the observer node during the link
activation, which are in turn determined by the angle between
the two nodes’ velocities and the angle (formed by the relative
velocity) incident into the observer node’s transmission range,
respectively. We also consider the link formed among a group
of nodes, referred to as the multipoint link in this paper, in
light of group mobility models (e.g., Reference Point Group
Mobility Model [10] and Reference Velocity Group Mobility
Model [11]) widely discussed in the literature for mobile ad
hoc networks. The accuracy of our model is validated by
simulations based on existing mobility models (e.g., random
waypoint models [12], the random walk [13], and group
mobility model [10-11]). We also demonstrate the usability of
the derived model for different applications.
The rest of the paper is organized as follows. In Sec. II, the
analytical framework for link duration of point-to-point links
and multipoint links in multi-hop mobile network is developed.
In Sec. III, the simulation results are provided to validate the
analytical model. Finally, the paper is concluded in Sec. IV.
M
—————————
Digital Object Identifier inserted by IEEE
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
II. ANALYTICAL MODEL OF LINK DURATION
In this section, we develop the probability distribution
function of link duration for multi-hop mobile networks. Each
node is assumed to use the same transmission power and
move independently. We do not assume any specific mobility
model for each node in this paper. Later in the simulation
section, we will validate our model based on the existing
mobility models such as the random waypoint model, random
walk model, and group mobility models.
A. Link Duration between Two Nodes
(a) Absolute viewpoint of 1
N and 2
N
(b)
Relative velocity of 1
N and 2
N
(c) Relative movement of 2
N observed by 1
N, and 12
D is the
active distance between 1
N and 2
N.
Figure 1. The relations between two nodes in terms of velocity and
position in the network, where a circle represents the transmission
range of the node centered at the circle.
Consider nodes i
N and j
N moving in the network. Let
i
V and j
V denote the velocities of node i
N and node j
N,
respectively; let 0
i
P denote the initial position of node i
N,
and t
i
P, the position of i
N after time t with velocity i
V. Fig.
1 shows the movements of 1
N and 2
N for time t. In Fig. 1 (a),
the dotted circles represent the transmission ranges of the
nodes centered at the circles; the solid circle represents the
relationship between nodes 1
N and 2
N. We are particularly
interested in the relative movement behavior of 2
N observed
by 1
N, as shown in Fig. 1 (b), instead of the absolute
viewpoints of both nodes as shown in Fig. 1 (a). From the
perspective of 1
N, 2
N is moving toward/away 1
N with
relative velocity 1212 VVV −= , and the relative speed is
|| 1212 VV =.
Let ij
α
denote the angle between i
V and j
V given that the
link between i
N and j
N can be established1 (as shown in Fig.
1 (b)), and ij
β
represent the incident angle, the relative
velocity ij
V, to '
i
Ns transmission range (as shown in Fig. 1
(c)). Therefore, random variable ij
α
ranges over [0, π], and
ij
β
, over [0, π/2]. Since i
N and j
N can only communicate
part of the time, the distance traversed by j
N with relative
velocity ij
V during which the link is activated is referred to as
the active distance between i
N and j
N, denoted by ij
D. Fig.
1(c) illustrates 12
D.
In Fig. 1(b), the value of angle ij
α
determines the
magnitude of ij
V, i.e.,
ijjijiij VVVVV
α
cos2
22 −+= . (1)
In Fig. 1(c), the value of angle ij
β
determines the
magnitude of ij
D, i.e.,
)cos(2 ijij rD
β
=
. (2)
Since the active distance ij
D and the relative speed ij
V are
mutually independent (which will be proved shortly in the
paper), the link duration ij
T between i
N and j
N can be
expressed by
ij
ij
ij V
D
T=. (3)
Since the probability distributions of ij
V and ij
D are
determined by those of ij
α
and ij
β
, respectively, in what
follows, we first develop the probability distributions of ij
α
and ij
β
, and then those of ij
V and ij
D, based on which the
1 ij
α
is a conditional random variable defined as the angle between velocities
i
V and j
Vgiven that the link between nodes i
N and j
N can be activated.
In other words, ij
α
is not the random variable representing the angle
between any two nodes in the network, as a link may not be realized between
any two arbitrary nodes.
Pt
1
P0
1
Pt
2
P0
2
Pt
1
P0
1
P0
1
Pt
2
Pt
2
P0
2
P0
2
N1Pt
2
P0
2
V12
12
β
D12
N1Pt
2
Pt
2
P0
2
P0
2
V12
V12
12
β
D12
D12
V1
V2
V12 = -
V2V1
12
α
V1
V1
V2
V2
V12 = -
V2V1
12
α
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
distribution of ij
T can be derived. Note that since all nodes
move independently and behave statistically identically, we
will omit the argument ij of ij
α
, ij
β
, ij
V, ij
D, and ij
T in the
subsequent analysis.
We start with the simplest case, i.e., each node is
continuously roaming and moves at the same speed (i.e.,
fixji vVV ==
→
→
). Later, we will extend it to the case with
nodes moving at different speeds, and discuss the impact of
node pause on link duration.
B. Probability Distribution of Link Duration for Point-to-
Point Links without Pause
1) fix
vVV == →→
21
Since each node moves at the same speed fix
v, the relative
speed between two nodes given that the link in between can
be established is given by
)2/sin(2
α
fix
vV =. (4)
(a) Possible locations of 2
N to reach 1
N with e
T with V
(b)
{}
tTvVb e==≤ ,|Pr
β
Figure 2. An illustration of calculating Pr{ LA
E|V,e
T}
For ease of explanation, we assume that nodes outside the
transmission range of node i
N are uniformly2 distributed in
the network. Let LA
E denote the event that the link between
i
N and j
N is activated, and e
T, the random variable
representing the elapsed time from when j
N starts to move
with relative speed V to when the transmission range of i
N is
reached. Given that j
N moves at relative speed vV
=
toward
i
N with elapsed time tTe=, the probability that the link
2 When this assumption does not hold, the derivation must be replaced with
an integral of the grey area, instead of the area coverage itself as in the
derivation.
between i
N and j
N can be activated is equal to the
probability that j
N starts moving from any point located in
the shaded area as shown in Fig. 2 (a). The area of the shaded
region can be obtained by rvt 2⋅ as illustrated in Fig. 2 (b).
Therefore,
{
}
,2,|Pr rvttTvVE eLA ==
=
where r is the
transmission range of node i
N.
The probability },|Pr{ tTvVb e==
≤
β
corresponds to the
probability that node j
N falls inside the striped area as shown
in Fig. 2 (b), i.e., the area from which j
N moving with vV
=
and tTe
=
would reach i
N’s transmission range with an
angle no more than b, where 2/0
π
≤≤ b. Therefore, the
cumulative distribution function (cdf) of
β
can be expressed
by
{
}
brvttTvVb esin2,|Pr
=
=
=
≤
β
. (5)
From (4), we have
{}
.sin
2
sin2,|Pr b
a
tvtTab fixe ⎟
⎠
⎞
⎜
⎝
⎛
===≤
αβ
Thus,
{}
∫
∫
=
=
⎟
⎠
⎞
⎜
⎝
⎛
⋅∝
⎟
⎠
⎞
⎜
⎝
⎛
⋅==≤≤
a
x
a
x
e
bdx
x
xf
bdx
x
xfttTba
0
0
.sin
2
sin)(
sin
2
sin)(|,Pr
'
'
α
α
βα
(6)
where (.)
'
α
f is the pdf of the random variable '
α
denoting
the angle between two arbitrary nodes.
Since, when the moving directions of i
N and j
N are
assumed independent and nearly uniformly distributed from 0
to 2π, the distribution of angles between two arbitrary nodes
can be approximated by a uniform distribution over [0,π], (6)
can be re-expressed by
b
a
bdx
x
a
x
sin
2
cos1
2
sin
2
sin
1
0
⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛
−=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
∫
=
ππ
. (7)
The joint probability distribution of
α
and
β
in (7) is
independent of the value taken by e
T, and can be expressed
by
{}
b
a
babaF sin
2
cos1
2
,Pr),(
,⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛
−∝≤≤=
π
βα
βα
, i.e.,
Kb
a
baF ⋅
⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛
−= sin
2
cos1
2
),(
,
π
βα
.
Since 1)
2
,(
,=
π
π
βα
F, this yields 2
π
=K. Thus,
b
a
baF sin
2
cos1),(
,⎥
⎦
⎤
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛
−=
βα
. (8)
From the joint cdf (.)
,
βα
F, we can further obtain
⎟
⎠
⎞
⎜
⎝
⎛
−= 2
cos1)( a
aF
α
, (9)
Nj
Ni
r
β
VTe
Nj
Nj
Ni
Ni
r
β
VTe
2rsinb
vt
Nj
Ni
2r
r
b
b
2rsinb
vt
Nj
Nj
Ni
Ni
2r
r
b
b
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
bbF sin)( =
β
. (10)
Thus, it follows that
⎟
⎠
⎞
⎜
⎝
⎛
== 2
sin
2
1
)(
)( a
da
adF
af
α
α
,
b
db
bdF
bf cos
)(
)( ==
β
β
.
From (8), (9) and (10), we have )()(),(
,bFaFbaF
βαβα
⋅
=.
Thus, random variables
α
and
β
are independent.
Based on the distributions of
α
and
β
in (9) and (10), we
can further derive the distributions of the active distance D
and the relative speed V. From (4) and (9), we obtain
{}
.
2
11
2
sin2
2
sin2PrPr)(
2
1⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎭
⎬
⎫
⎩
⎨
⎧≤
⎟
⎠
⎞
⎜
⎝
⎛
=≤=
−
fixfix
fixV
v
v
v
v
F
vvvVvF
α
α
(11)
2
2
42
)(
)(
vvv
v
dv
vdF
vf
fixfix
V
V−
== . (12)
From (3) and (10), we obtain
{}{ }
.
2
11
2
cos1
2
cosPr
cos2PrPr)(
2
11
⎟
⎠
⎞
⎜
⎝
⎛
−−=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⎟
⎠
⎞
⎜
⎝
⎛
−=
⎭
⎬
⎫
⎩
⎨
⎧⎟
⎠
⎞
⎜
⎝
⎛
≥=
≤=≤=
−−
r
d
r
d
F
r
d
drdDdFD
β
β
β
(13)
22
42
)(
)(
drr
d
dd
ddF
df D
D−
== . (14)
Substituting the derived distributions of D and V in (11) and
(13) into (3), we obtain the distribution of T accordingly, i.e.,
{}
.
cos
2
sinPr)(
2
sin2
cos2
Pr)(
2
sin2
cos2
PrPrPr)(
2
0
2
0
∫
∫
=
=
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
⋅
≥
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅=
⎪
⎪
⎭
⎪
⎪
⎬
⎫
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≤
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅=
⎪
⎪
⎭
⎪
⎪
⎬
⎫
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≤
⎟
⎠
⎞
⎜
⎝
⎛
=
⎭
⎬
⎫
⎩
⎨
⎧≤=≤=
π
β
π
β
α
α
α
β
bfix
bfix
fix
T
db
tv
br
bf
dbt
v
br
bf
t
v
r
t
V
D
tTtF
Since ⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⋅
≤≤
⋅
≥
⎟
⎠
⎞
⎜
⎝
⎛−
r
tv
b
v
r
t
tv
br fix
fixfix
1
cos,|
cos
2
sinPr
α
=
01
cos
2
sinPr =
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧≥
⋅
≥
⎟
⎠
⎞
⎜
⎝
⎛
tv
br
fix
α
, we can further decompose the
distribution )(tFT into two cases.
Case 1:
fix
v
r
t≤,
. ln
22
1
cos
1cos
cos
2
sinPr)()(
2
2
2
2
2
2
2
cos
2
2
cos
1
1
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
+−
−=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
−⋅=
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
⋅
≥
⎟
⎠
⎞
⎜
⎝
⎛
⋅=
∫
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⋅
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⋅
=
−
−
tvr
tVr
trv
tvr
db
tv
br
b
db
tv
br
bftF
fix
fix
fix
fix
r
tv
b
fix
r
tv
b
fix
T
fix
fix
π
π
β
α
(15)
Case 2:
fix
v
r
t>,
.ln
22
1
cos
1cos)(
22
2
2
2
2
2
0
2
2
22
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
+−
−=
⋅
−⋅= ∫
=
rtv
tvr
trv
tvr
db
tv
br
btF
fix
fix
fix
fix
bfix
T
π
(16)
From (15) and (16), we can obtain the probability distribution
of link duration for point-to-point links as follows.
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
−
+−
−=
2
2
2
2
2
2
ln
22
1
)(
tvr
tvr
trv
tvr
tF
fix
fix
fix
fix
T (17)
t
tvr
tvr
trv
tvr
dt
tdF
xf
fix
fix
fix
fix
T
T2
1
ln
2
)(
)(
2
2
2
2
2
2
2
−
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
−
++
==
2) jjii vVvV ==
→
→
and
Next, we consider the case that the velocities of i
N and
j
N are no longer fixed. Let ii vV =|| and jj vV =|| . The
relative speed between i
N and j
N can be expressed by
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
, cos
2
1cos2 22
2222
αα
⋅
+
−⋅+=−+=
ji
ji
jijiji vv
vv
vvvvvvV
which can be approximated by
.cos1 22
22
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⋅
+
−⋅+
α
ji
ji
ji vv
vv
vv (18)
Accordingly, we have
,sinsin
1
sincos1
1
, ,|,Pr
22
22
0
22
22
b
vv
vv
vv
bdx
vv
vv
vv
vVvVtTba
ji
ji
ji
a
xji
ji
ji
jjiie
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⋅
+
−⋅+=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⋅
+
−⋅+⋅∝
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧===≤≤
∫
=
→
→
αα
π
α
π
βα
and
()
,cos
1
1cos
cos2
Prcos
, |Pr)(
)
2
)(
(cos
2
)(
(cos
1
2
0
1
1
dbxxkb
t
br
Vb
vVvVtTtF
r
kktk
r
kktk
jij
jjiiT
jii
jii
∫
∫
+
−
−
→
→
−
−
⎥
⎦
⎤
⎢
⎣
⎡+−−=
⎭
⎬
⎫
⎩
⎨
⎧>⋅=
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧==≤=
π
π
where
2
1
22
22
cos2
1,,
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛−
−=
+
=+=
j
i
i
ji
ji
jjii kk
t
br
k
x
vv
vv
kvvk , and
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛−
=
ji
i
jkk
t
br
k
x
cos2
.
Denote the speed of each node by I
V, which is a random
variable distributed over [ maxmin ,II VV ]. Thus, we have
()
.cos
1
1cos
)()()(
)
2
)(
(cos
2
)(
(cos
1
1
1
max
min
max
min
ji
r
kktk
r
kktk
jij
V
V
V
V
jViVT
dvdbdvxxkb
vfvftF
jii
jii
I
I
I
I
II
∫
∫∫
+
−
−
−
−
⎥
⎦
⎤
⎢
⎣
⎡+−−
⋅⋅=
π
. (19)
C. Probability Distribution of Link Duration with Pause
We now study the impact of node pause on link duration.
Suppose that each node may be either in the state of
movement or in pause state. When either node enters the
pause state, the link duration is determined by the moving
node’s velocity.
We model link duration with two components3: both nodes
are in movement and either node is in pause state. The weights
of the two components can be decided by the percentage of
the time each node is in motion or in pause state. Let M denote
the mean movement duration and P be the duration of pause
time. Then, a node is in movement with a probability of
P
M
M
+ and in pause state with a probability of
P
M
P
+. Thus,
the pdf of the link duration with the consideration of pause can
be expressed by )()()( tf
P
M
P
tf
P
M
M
tf sP TTT ⋅
+
+⋅
+
=,
where )(tfT is the derivative of (19) (i.e., the case in which
both nodes are in movement) and )(tf s
T is the case that either
node is in pause state during the link activation. Thus, we
obtain
,
2
11))
2
(sin(cos
)}
2
(cosPr{}
cos2
Pr{)|(
2
1
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−==
≤=≤==
−
−
r
tv
r
tv
r
tv
t
v
r
vVtF
ii
i
i
iITS
β
β
leading to
i
V
V
i
iVT dv
r
tv
vftF
I
I
IS ∫⎟
⎠
⎞
⎜
⎝
⎛
−−⋅=
max
min
)
2
11()()(
2
. (20)
D. Link Duration of Multi-point Link
Finally, we extend the derivation to capture the behavior of
a multipoint link shared by multiple nodes. The derivation is
similar to that of the Reference Point Group Model described
in [10]. In this model, a multipoint link is formed among a
group, led by a group leader. All member nodes move along
with the leader according to a group mobility model, and each
node moves independently by following a node mobility
model. The link duration of a multipoint link is then referred
to as the time interval in which all member nodes have left the
transmission range of the leader node.
Suppose that there are m member nodes in a group (denoted
by 1
N,2
N,…, m
N) led by a group leader 0
N and all member
nodes in the group are initially inside the transmission range
of 0
N. We assume that all member nodes move independently
3 We do not consider both nodes are in pause states in this paper due to
space limitations. Later in the simulation, we will show that the probability of
both nodes in pause states is small especially when the transmission range of
each node is relatively smaller compared to the entire network coverage.
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
guided by their node mobility models and all nodes are
continuously moving. Thus, all nodes will eventually move
out of the transmission range of the leader nodes unless they
move toward the same direction as the leader node. The link
duration of a multipoint link corresponds to the longest link
duration between each member node and the leader node,
which can be expressed by
M
T = max { m
TTT ,,, 21 L}, (20)
where i
T is the link duration between nodes i
N and 0
N.
Let )(tFM denote the cdf of the link duration for a multi-
point link shared by m nodes, and )(tFi is the cdf of the link
duration between the leader 0
N and the member node i
N.
Since all nodes move mutually independently by their node
mobility models and the moving direction of each node is
uniformly distributed over [0, 2π]. We obtain
(21) ,))((
)()......()()(
}Pr{}.......Pr{}Pr{}Pr{
}.....,,Pr{}Pr{)(
321
321
321
m
T
m
m
mMM
tF
tFtFtFtF
tTtTtTtT
tTtTtTtTtTtF
=
⋅⋅⋅=
<⋅<⋅<⋅<=
<
<<<=<=
where )(tFi is the conditional cdf of member node i
N, given
i
V0 and i
R0.
Figure 3. Relations between member node i
N and leader node 0
N
The distribution of )(tFi is obtained as follows. In Fig. 3,
i
θ
is the angle formed by the relative velocity i
V0 and the
relative position vector i
R0, and ii Rd 0
=. The active
distance traversed by node i
N is
)cos()(sin 2
2
2
iiiii ddrD
θπθπ
−+−−= .
From (3) we have
,
cos2
)cos()(sin
Pr
,,|Pr)(
0
22
0
2
2
2
000
0
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
≤
−+
−+−−
=
⎭
⎬
⎫
⎩
⎨
⎧===≤=
t
vvvv
ddr
dRvVvVt
V
D
tF
ii
iii
iiii
i
i
i
α
θπθπ
where i
V0 denotes the relative speed between 0
N and i
N.
Let (.)
0i
R
Fbe the cdf of i
R0. Thus,
. if
,
)cos()(sinr
)cos()(sinr-1
1
; if
,)cos()(sinr1
1
)(
00 2222
2222
2
00
2222
2
0
0
0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎧
≥
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛−+−−−+
⎟
⎠
⎞
⎜
⎝
⎛−+−−
⋅
<
⎟
⎠
⎞
⎜
⎝
⎛⎟
⎠
⎞
⎜
⎝
⎛−+−−−⋅
=
∫∫
∫∫
v
r
t
dda
vttvF
vttvF
v
r
t
ddavttvF
tF
i
i
i
R
R
R
i
ϑ
ϑπϑπ
ϑπϑπ
π
ϑϑπϑπ
π
ππ
ππ
where avvvvv ii cos2 0
22
0−+= .
III. PERFORMANCE EVALUATION
In this section, we validate our analytical results via an
event-driven C++ simulator. In our simulations, 300 nodes are
uniformly distributed in a unit-square area. Each node moves
independently, including both speed and the moving direction,
guided by a mobility model. The result in each figure is an
average of 100 runs. We validate the link duration for both
point-to-point links and multipoint links as follows.
A. Link Duration of Point-to-Point Links
We consider two models, described as follows.
i) The random waypoint model: each node selects a target
location to move at a speed selected from a uniformly
distributed interval [ maxmin ,II VV ]. Once the target is
reached, the node pauses for a random time with
probability p and then selects another target with another
speed to move again.
ii) The random walk model: each node selects a direction
and a speed pair [θ, v] from uniformly distributed
intervals [0, 2π] and [ maxmin ,II VV ], respectively, and
then starts to move for a time t uniformly selected from
[0, tmax]. Once the node has moved for t units of time, it
pauses for a random time with probability p and then
starts a new movement again.
1) Nodes with Fixed Speed
We first simulate the case with nodes moving at fixed
speed 01.0
=
fix
vunit per second under both mobility
models (denoted by Sim-RWP and Sim-RW). The pdf of T
is plotted in Fig. 4 with transmission range r =0.15, where
RWP is for the random waypoint model, and RW, for the
random walk model. The figure shows that link duration
has the highest probability around the value fix
vr / for both
models, explained as follows. A larger T results from a
larger D and/or smaller V, which in turn results from a
Ni
N0
i
V0
r
i
θ
i
d
i
R0
i
D
Ni
Ni
N0
N0
i
V0
r
i
θ
i
d
i
R0
i
D
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
larger
β
and/or smaller
α
, and vice verse. Let us give a
more intuitive explanation with the following two facts.
First, two nodes moving in a similar direction have a
smaller probability to meet and form a link. Second, with a
given velocity, the point at which one node enters the other
node’s transmission range is uniformly distributed (denoted
by x, 2x, and 3x in Fig. 5). However, when the incident
point is near the tangential line, the active distance D is
shortened rapidly, i.e., )3()2()( xDxDxD >> . Since
T=D/V, the probability of large T is determined by the
nodes with similar moving direction, which has a smaller
chance for nodes to meet. On the other hand, the case of
small T is determined by the low probability of a short
active distance. Since the probability T being small or large
is low, the peak is at the point where both
α
and
β
have the
highest probabilities, i.e.,
fixfix vrvrT /)
2
sin(2/)0cos(2 ==
π
.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 20406080100
T(sec)
pdf
Analysis
Sim RWP
Sim RW
Figure 4. The pdf of T with fixed moving speed 01.0=
fix
v, r=0.15
x
3x
2x
D(3x)
D(x)
D(2x)
Tange ntial line
Figure 5. The active distances with different incident angles (i.e., at
points x, 2x, and 3x)
2) Nodes with Different Speeds
Next, we show the simulation results when the speed of
each node is uniformly selected over [0.005, 0.015] unit per
second, again under the two models. The curve with “Ana-
uniform” depicts the pdf of T with transmission range r=0.15
as in (19). The mismatch between the analytical curve and
simulation curve is due to the fact that the speeds of both
mobility models are not uniformly distributed [14]. To fix this
problem, we let the pdf of nodal moving speeds to be
inversely proportional to the speed itself, i.e., v
kvf I
V
1
)( ⋅= .
Since 1)(
max
min
=
∫dvvf
V
V
VI, we obtain
min
max
ln
1
)(
I
I
V
V
V
v
vf I
⋅
=. (21)
With the revised density function (21), we integrate (19)
over the range [ maxmin ,II VV ] to obtain the new probability
density function of the link duration for point-to-point links.
The curve in Fig. 6 with “Ana-non-uniform” shows the
analytical result based on (21). The analytical result has an
apparent accuracy improvement as compared to the uniformly
distributed one.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 1020304050607080
T(sec)
pdf
Ana: uniform
Ana: non-uniform
Sim: RWP
Sim: RW
Figure 6. The pdf of T, where [maxmin ,II VV ] =[0.005, 0.015], r=0.15
B. Link Duration with Pause
Then, we show the results when there are pauses between
movements. Here we run simulations with the random
waypoint model and the speed is fixed at 0.01 units per second
with different pause times. Fig. 7 shows the comparison of the
simulation and analysis results with a pause time of 50
seconds. We see that there are two peaks, i.e., at T=15 and 30
seconds, which are contributed by the two components (i.e.,
both nodes are in motion, and either is in pause), respectively.
In this scenario, since the transmission range of each node is
relatively smaller as compared to the entire network coverage,
our model can accurately capture the behavior of link duration.
0
0.05
0.1
0.15
0.2
0 102030405060
T (se c)
pdf
Sim
Ana
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
Figure 7. The pdf of T with pause time 50s, r=0.15
C. Boundary Effect on Link Duration
To observe the boundary effect, we simulate the random
walk mobility model with a fixed movement interval of 100
seconds and a fixed speed of 0.01 units per sec. In this way,
we can avoid having a movement ends during the link
duration. From the simulation results in Fig. 8, we observe
that when the ratio of transmission range to network size is
below 0.2, the boundary effect is insignificant. However,
when the ratio exceeds 0.25, the probability of short T for
nodes near the boundary may have shorter link durations, so
our analysis can better describe the link duration when the
node transmission range is relatively smaller compared to the
entire network size.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 20406080
T(sec)
pdf
Ana
Sim
Figure 8. The pdf of T with fixed moving speed 01.0=
fix
v, r=0.2
D. Link Duration of Multipoint Links
In this simulation, there are one leader node and 10 member
nodes in a unit-square network. Each member node is initially
located in the transmission range of the leader node, and its
distance to the leader node is uniformly selected from the
range [0, r] unit, where r is the transmission range of the
leader node. All member nodes move with a fixed speed of
0.01 units per second, and the direction to move is uniformly
distributed from [0, 2π]. Fig. 9 shows that compared to the
distribution of point-to-point links, multi-point links have a
higher probability for long link duration. This is due to the
fact that as long as at least one of the member nodes chooses a
similar direction as the leader node, the link will exist for a
long time.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 20 40 60 80 100 120 140 160 180 200
T(sec)
pdf
Ana Sim
Figure 9. The pdf of multi-point link duration
IV. CONCLUSIONS AND FUTURE WORK
In this paper, we propose an analytical framework to model
the link duration for multi-hop mobile networks. We consider
both point-to-point links between nodes and multipoint links
shared by a group of nodes. Each node may be in movement
or in pause state. The model starts with the derivation of the
distributions of two parameters α and β, which in turn
determine the distributions of the relative speed of the two
considered nodes and the active distance between the two
nodes for an activated link. Then, the pdf of link duration for
point-to-point links can be obtained. The derived result is then
extended to model multipoint links. We validate the analytical
results via simulations with three different mobility models
widely used in the literature, namely, the random waypoint,
the random walk, and the group movement model.
In the future, we will further derive the path lifetime based
on the analytical results developed in this paper. The
probability distribution of the path lifetime in mobile ad hoc
networks is even more difficult to derive than link duration
since it is dependent on many system parameters such as
spatial distribution, node density, and path connectivity.
ACKNOWLEDGMENT
This work was supported in part by the Excellent Research
Projects of National Taiwan University, under Grant Number
97R0062-06, and in part by National Science Council (NSC),
Taiwan, under Grant Number NSC96-2628-E-002-003-MY3.
REFERENCES
[1] Y. Fang, I. Chlamtac, and Y.-B. Lin, “Portable Movement Modeling for
PCS Networks,” IEEE Transactions on Vehicular Technology, Vol. 49,
Issue 4, 2000, pp.1356-1363.
[2] J. Li, H. Kameda, K. Li, “Optimal Dynamic Mobility Management for
PCS Networks,” IEEE Transactions on Networking, Vol. 8, Issue 3,
June 2000, pp. 319-327.
[3] Y. Fang, “Movement-based Mobility Management and Tradeoff
Analysis for Wireless Mobile Networks,” IEEE Transactions on
Computers, Vol. 52, Issue 6, June 2003, pp. 791-803.
[4] Y. -B. Lin and S. -R. Yang, “A Mobility Management Strategy for
GPRS,” IEEE Transactions on Wireless Communications, Vol. 2, Issue
6, Nov. 2003, pp. 1178-1188.
[5] M. Wenchao and Y. Fang, “Dynamic Hierarchical Mobility
Management Strategy for Mobile IP Networks,” IEEE Journal on
Selected Areas in Communications, Vol. 22, Issue 4, May 2004, pp. 664-
676.
[6] J. -T. Weng, W. Liao, and J. –R. Lai, “Modeling Node Mobility for
Reliable Packet Delivery in Mobile IP Networks,” IEEE Transactions on
Wireless Communications, Aug. 2006.
[7] J. Boleng, W. Navidi, and T. Camp, “Metrics to Enable Adaptive
Protocols for Mobile Ad Hoc Networks,” Proc. ICWN 2002, Jun. 2002,
pp.293-298.
[8] C. -L Chao, Y. -T. Wu, W. Liao, and J. -C. Kuo, “Link Duration of the
Random Waypoint Model in Mobile Ad Hoc Networks,” Proc. IEEE
WCNC 2006.
[9] J. Boleng and T. Camp, “Adaptive Location Aided Mobile Ad Hoc
Network Routing,” Proc. IEEE IPCCC 2004, pp. 423-432, 2004.
[10] X. Hong, M. Gerla, G. Pei, and C.-C. Chiang, “A Group Mobility Model
for Ad Hoc Wireless Networks,” Proc. ACM international Workshop on
Modeling, Analysis and Simulation of Wireless and Mobile Systems
(MSWiM), Aug. 1999.
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
[11] K.H. Wang and B. Li. “Group Mobility and Partition Prediction in
Wireless Ad-Hoc Networks,” IEEE ICC 2002, Apr. 2002, pp. 1017-1021.
[12] D. B. Johnson and D. A. Maltz, “Dynamic Source Routing in Ad Hoc
Wireless Networks,” Mobile Computing, Kluwer Academic Publishers,
1996, pp. 153-181.
[13] A. B. McDonald and T. Znati, “A Path Availability Model for Wireless
Ad-Hoc Networks,” Proc. IEEE WCNC 1999, Sep. 1999, pp.35-40.
[14] T. Camp, J. Boleng, and V. Davies, "A Survey of Mobility Models for
Ad Hoc Network Research," WCMC, vol. 2, no. 5, 2002, pp. 483-502.
Manuscript received Nov 14, 2007, revised May 26, 2008, accepted Sept.
11, 2008. The review of this paper was coordinated by Dr. Mark Brian. The
authors are with the Department of Electrical Engineering, National Taiwan
University, Taipei, 106, Taiwan, R.O.C. Corresponding author: Wanjiun Liao
(e-mail: wjliao@ntu.edu.tw).
Yueh-Ting Wu received his BS degree and MS
degree from National Taiwan University, Taipei,
Taiwan, in 2004 and 2006, respectively, all in
Electrical Engineering. He is now an engineer in
Delta Network, Inc., Taiwan. His research
interests focus mainly on wireless networking.
Wanjiun Liao (M’97–SM’05) received her
Ph.D. degree in Electrical Engineering from the
University of Southern California, Los Angeles,
California, USA, in 1997. She joined the
Department of Electrical Engineering, National
Taiwan University (NTU), Taipei, Taiwan, as
an Assistant Professor in 1997, where she is
now a full professor. Her research interests
include wireless networks, multimedia
networks, and broadband access networks.
Dr. Liao is currently an Associate Editor of IEEE Transactions on
Wireless Communications, and was on the editorial board of IEEE
Transactions on Multimedia (2004-2007). She served as the Technical
Program Committee (TPC) chairs/co-chairs of many international
conferences, including the Tutorial Co-Chair of IEEE INFOCOM 2004, the
Technical Program Vice Chair of IEEE Globecom 2005 Symposium on
Autonomous Networks, a TPC Co-Chair of IEEE Globecom 2007 General
Symposium, and a TPC Co-Chair of IEEE ICC 2010 Next Generation
Networks and Internet Symposium. Dr. Liao has received many research
awards. Papers she co-authored with her students received the Best Student
Paper Award at the First IEEE International Conferences on Multimedia and
Expo (ICME) in 2000, and the Best Paper Award at the First IEEE
International Conferences on Communications, Circuits and Systems
(ICCCAS) in 2002. Dr. Liao was the recipient of K. T. Li Young Researcher
Award honored by ACM in 2003, and the recipient of Distinguished
Research Award from National Science Council in Taiwan in 2006. She is a
Senior member of IEEE.
Cheng-Lin Tsao received his BS degree and MS
degree in the Department of Electrical Engineering,
National Taiwan University, Taipei, Taiwan,
R.O.C., in 2002 and 2004, respectively. He is now a
PhD candidate in Electrical and Computer
Engineering, Georgia Institute of Technology,
Atlanta, Georgia, U.S.A. His research interest is in
the design of next generation network architectures
and protocols.
Tsung-Nan Lin received B.S. degree in electrical
engineering from National Taiwan University,
Taiwan, R.O.C. in 1989, and M.A. and Ph.D.
degrees from Princeton University in 1993 and
1996, respectively, both in electrical engineering
department. He was a Teaching Assistant with the
Department of Electrical Engineering from 1991 to
1992. He was with NEC Research Institute as a
Research Assistant from 1992 to 1996. He has been
with EPSON R&D Inc and Intovoice. He was
Engineering Consultant at EMC before he joined
NTUEE. Since Feb. 2002, he has been an Assistant Professor in the
Department of Electrical Engineering, National Taiwan University, Taipei,
Taiwan.
Tsung-Nan Lin is a member of PHI TAU PHI scholastic honor society and
a member of IEEE. He received outstanding paper award from IEEE Neural
Networks Society in 1998 and young author best award from IEEE Signal
Processing Society in 1999.
Authorized licensed use limited to: National Taiwan University. Downloaded on January 22, 2009 at 01:45 from IEEE Xplore. Restrictions apply.
