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1
Degenerate Delay-Capacity Trade-offs
in Ad Hoc Networks with Brownian Mobility
Xiaojun Lin, Gaurav Sharma, Ravi R. Mazumdar, and Ness B. Shroff
Abstract—There has been significant recent interest within the
networking research community to characterize the impact of
mobility on the capacity and delay in mobile ad hoc networks.
In this paper, we study the fundamental trade-off between the
capacity and delay for a mobile ad hoc network under the
Brownian motion model. We show that the 2-hop relaying scheme
proposed by Grossglauser and Tse (2001), while capable of
achieving Θ(1) per-node capacity, incurs an expected packet
delay of Ω(log n/σ
2
n
), where σ
2
n
is the variance parameter of
the Brownian motion model. We then show that in order to
reduce the delay by any significant amount, one must be ready
to accept a per-node capacity close to static ad hoc networks.
In particular, we show that under a large class of scheduling
and relaying schemes, if the mean packet delay is O(n
α
/σ
2
n
), for
any α < 0, then the per-node capacity must be O(1/
√
n). This
result is in sharp contrast to other results that have recently been
reported in the literature.
I. INTRODUCTION
Since the seminal work of Gupta and Kumar [1], there has
been a lot of interest in characterizing the capacity region of ad
hoc networks. A major contribution in this direction was made
in [2], where the authors show that mobility can significantly
increase the capacity of an ad hoc network. In particular, the
authors proposed a 2-hop relaying scheme, and showed that
it can achieve Θ(1) per-node capacity
1
. However, the delay
related issues were not considered in [2]. In fact, it is pointed
out in [2] that the 2-hop relaying scheme could potentially
incur an unbounded delay.
There has been substantial recent work on the joint char-
acterization of the delay and capacity in the mobile ad hoc
networks [3], [4], [5], [6], [7], [8]. The type of node mobility
studied in the literature includes the so-called i.i.d mobility
[4], [7], [8], random way-point mobility [5], [6], Brownian
motion [3], [6], and Markovian mobility [4]. The results in
Xiaojun Lin, Gaurav Sharma, and Ness B. Shroff are with the School of
Electrical and Computer Engineering, Purdue University, West Lafayette, IN
47907, USA (email: {linx,gsharma,shroff}@ecn.purdue.edu).
Ravi R. Mazumdar is with the Department of Electrical and Computer
Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (email:
mazum@ece.uwaterloo.ca).
1
We use the following notation throughout:
f(n) = o(g(n)) ↔ lim
n→∞
f(n)
g(n)
= 0,
f(n) = O(g(n)) ↔ lim sup
n→∞
f(n)
g(n)
< ∞,
f(n) = ω(g(n)) ↔ g(n) = o(f(n)),
f(n) = Ω(g(n)) ↔ g(n) = O(f (n)),
f(n) = Θ(g(n)) ↔ f(n) = O(g(n)) and g(n) = O(f (n)).
these works are of a similar flavor. They show that it is
possible to achieve a bounded expected delay and Θ(1) per-
node capacity, using scheduling schemes that are variants of
the Grossglauser-Tse 2-hop relaying scheme. These works
then report trade-offs between the capacity and delay; i.e.,
when one is willing to sacrifice the per-node capacity, the delay
can be correspondingly reduced. The capacity-delay trade-offs
are achieved either by means of introducing some redundancy
in the 2-hop relaying scheme [4], [5], [6], or by adjusting the
cell size [3], or both [7], [8]. All the previous works report
“smooth” capacity-delay trade-offs. For instance, under the
random way-point mobility model, the authors of [5] report a
scheme that can achieve Θ(n
α−1
) per-node capacity at Θ(n
α
)
delay, for any α ∈ [1/2, 1]. Hence, by reducing α to α − ,
> 0, one can reduce the delay by a factor of Θ(n
), at
the cost of reducing the capacity by the same factor. Similar
type of smooth trade-offs have been reported under the i.i.d.
mobility model as well [4], [7], [8].
In this paper, we study the trade-off between the capacity
and delay under the Brownian motion model [3], [6]. We
note that the results of this paper can easily be extended
to other related mobility models such as the random walk
mobility model [9] and the Markovian mobility model [4].
This is because the Brownian motion model can be viewed as
a limiting case of these other mobility models. Interestingly,
our results are in complete disagreement with some recently
reported results in the literature for the Brownian motion
model or its variants [3], [6]. We show that there is virtually no
trade-off between the capacity and delay under the Brownian
motion model (see Fig. 1). In particular, we show that under
a large class of scheduling and relaying schemes, in order
to achieve a delay of Θ(n
α
/σ
2
n
) for any α < 0, where σ
2
n
is the variance parameter of the Brownian motion, the per-
node capacity must be O(1/
√
n). Further, we show that the
2-hop relaying scheme proposed by Grossglauser and Tse [2],
while capable of achieving Θ(1) per-node capacity, incurs
an expected packet delay of Ω(log n/σ
2
n
). Note that one can
achieve Θ(1/
√
n log n) per-node capacity for static wireless
networks using multi-hop transmission [1]. Thus, in order to
achieve any significant capacity gains by exploiting mobility,
one must be ready to tolerate huge delays, roughly on the
order of Θ(1/σ
2
n
), which is close to the delay at Θ(1) per-
node capacity.
We summarize the main contributions of this paper below:
• We rigorously show that, for a large class of schedul-
ing and relaying schemes, the achievable capacity-delay
trade-offunder the Brownian motion model is degenerate.
If one attempts to achieve a per-node capacity that is more
2
O(
√
n)
O(n
α
) O(n)
O(
1
√
n
)
O(1)
Capacity
Delay
Fig. 1. The degenerate delay-capacity trade-off under the Brownian motion
model (the solid line) compared with the “smooth” delay-capacity trade-off
under the random way-point mobility model (the dashed line) reported in [5].
We have chosen σ
2
n
= 1/n and ignored all logarithmic terms in the figure.
than a logarithmic factor above that of static wireless net-
works, one must be ready to incur huge delays, roughly
on the order of Θ(1/σ
2
n
).
• We consider the class of generalized 2-hop relay-
ing schemes and show that they incur a delay of
Ω(log (1/a
n
)/σ
2
n
), where a
n
is related to the concept of
the capture neighborhood and the forwarding neighbor-
hood (see Section V). Most scheduling schemes studied
in the literature fall into this class with a
n
= O(n
α
), α <
0. Hence, the delay incurred is no less than Ω(log n/σ
2
n
).
A special case is the 2-hop relaying scheme of [2] which
achieves Θ(1) per-node capacity and incurs an expected
packet delay of Ω(log n/σ
2
n
).
• In related works in the literature, a crucial step in
characterizing the delay of the 2-hop relaying scheme
is to compute the second moment of the so-called inter-
meeting time (see Section VI and the Appendix). A tech-
nical contribution of this paper is to provide a rigorous
analysis of the inter-meeting time under the Brownian
motion model, which yields more accurate results than
previous works [3].
It is interesting to compare our results for the Brownian
motion model with the results in [5] for the random way-point
mobility model. Note that both these models are continuous
mobility models (i.e., the motion of the nodes is continuous),
and both preserve the uniform distribution of the nodes at
all times, that is, an initial uniform distribution of the nodes
implies that the nodes remain uniformly distributed at all
times. However, the capacity-delay relationship under these
two models is significantly different. In particular, there exists
a smooth trade-off between the delay and capacity under the
random way-point mobility model, whereas there is virtually
no trade-off under the Brownian motion model. We believe
that this difference is a revelation of the fundamentaldifference
in the mobility pattern under these two models. In the random
way-point mobility model, nodes move “purposefully,” that
is, during each trip, a node has some target position in mind
(chosen uniformly on the sphere) and it moves along a straight-
line path, with no “wandering” at all. Thus the nodes can
cover large distances in relatively short time under the random
way-point mobility model. This is in contrast to the Brownian
motion model, where the nodes always wander around like
“drunkards,” staying in a local neighborhood for large duration
of time. It is therefore intuitive to believe that reducing the
mobility delay under the Brownian motion model would be
much more difficult.
The rest of the paper is organized as follows. In the next
section, we describe our network model and the Brownian
motion model, followed by some basic properties of the
Brownian motion model in Section III. We then derive our
main result in Section IV, showing that there is virtually no
trade-off under the Brownian motion model. In Section V, we
analyze the delay performance of generalized 2-hop relaying
schemes. We provide a discussion of the related works in
Section VI, and finally end this paper with some concluding
remarks in Section VII.
II. NETWORK AND MOBILITY MODEL
We consider an ad hoc network with n nodes moving on the
surface of a unit sphere
2
. For simplicity, we assume that each
node, say i, communicates with a single destination node, say
d(i), and that the mapping i 7→ d(i) is bijective. We assume
a uniform traffic pattern, i.e., each source generates traffic at
the same rate of λ bits per second for its destination. We
further assume that the packet arrival processes at each node
is independent of the node mobility process. The communica-
tion between any source-destination pair can possibly be via
multiple other nodes acting as relays. That is, the source could
either send a packet directly to the destination, if possible; or,
it could forward the packet to one or more relay nodes; the
relay nodes could themselves forward the packet to other relay
nodes; and finally a relay node or the source node itself could
deliver the message to the destination.
We assume the following Protocol Model from [1] that
governs the radio transmissions between nodes. Let W be the
bandwidth of the system in bits per second. Let X
i
t
denote the
position of the node i, for i = 1...n, at time t. Node i can
communicate directly with another node j at a rate of W bits
per second at time t, if and only if, the following interference
constraint is satisfied [1]:
d(X
k
t
, X
j
t
) ≥ (1 + ∆)d(X
i
t
, X
j
t
) (1)
for every other node k 6= i, j that is simultaneously transmit-
ting. Here, ∆ is some positive number, and d(x, y) denotes
the Euclidean distance between points x, y ∈ R
3
. Note that
when the unit of information transmitted is a packet, the
above interference constraint must be satisfied over the entire
duration of the packet transmission from node i to node j.
Let S denote the surface of the unit sphere. We assume
that nodes move independently on S according to a Brownian
motion model as in [6]. (A similar Brownian motion model on
a 2-d torus is also considered in [3].) It is easier to describe
the motion of each node using the spherical coordinates. Let
θ
t
and φ
t
denote the colatitude and longitude, respectively,
of the position of a particular node at time t (0 ≤ θ
t
≤ π
and 0 ≤ φ
t
< 2π). When a node moves according to
the Brownian motion model on the unit sphere S, the (Itˆo)
2
Note that changing the shape of the area from the surface of a sphere to
a square or a circle will not change the main results of this paper.
3
stochastic differential equations for the process (θ
t
, φ
t
) are
given by [10]:
dθ
t
= σ
n
dB
t
+
σ
2
n
2 tan θ
t
dt, (2)
and
dφ
t
=
σ
n
sin θ
t
dB
0
t
, (3)
where B
t
and B
0
t
are independent standard one-dimensional
Brownian motions (i.e., with variance 1). We call σ
2
n
the
variance of the Brownian Motion described in (2) and (3).
For analysis, it is useful to project each node’s position onto
the z-axis. Substituting Y
t
= cos θ
t
into (2), and using Itˆo’s
Lemma, we obtain
dY
t
= −σ
2
n
Y
t
dt − σ
n
q
1 − Y
2
t
dB
t
. (4)
Note that Y
t
is a diffusion process with drift coefficient −σ
2
n
Y
t
and diffusion coefficient σ
2
n
(1−Y
2
t
). We assume that the initial
positions of the nodes are i.i.d. and uniformly distributed on
the unit sphere. This implies that the positions of the nodes
will remain uniform at all times.
III. BASIC PROPERTIES OF BROWNIAN MOTION ON A
SPHERE
In this section, we summarize some basic properties of
Brownian motion on a sphere, which will be used later on.
Let us consider the motion of a single node. Let X
t
denote its
position at time t, which can be represented using the spherical
coordinates (θ
t
, φ
t
). Let Y
t
= cos θ
t
be the projection of the
node’s position on the z-axis, and recall that Y
t
is governed
by (4).
We first cite the following result from [10] concerning the
expected travel time of Y
t
:
Lemma 1: Let −1 < a < x < 1. Then, in traveling from x
to a, Y
t
takes an expected time, V
a
(x), given by:
V
a
(x) =
2
σ
2
n
log
1 + x
1 + a
.
A. The First Hitting Time
The first concept we study is the first hitting time. Let A
be an arbitrary region on the sphere. We have the following
definition:
Definition 1: The first hitting time of A, denoted by T
A
, is
the first time instant at which X
t
enters A; i.e., T
A
= inf{t ≥
0 : X
t
∈ A}.
Let Π denote the uniform distribution on the unit sphere S,
and denote by E
Π
the expectation conditioned on X
0
being
distributed according to Π. Let A = {x ∈ S : d
S
(x, y) ≤ a
n
},
where y is an arbitrary point on S, d
S
denotes the geodesic
distance on the sphere, and a
n
> 0. For a
n
↓ 0, as n → ∞,
we have the following result:
Lemma 2: E
Π
[T
A
] = Θ(log (1/a
n
)/σ
2
n
).
Proof: In view of the symmetry of the sphere, taking y
to be the south pole (i.e., the bottom most point of S that
corresponds to θ = π) entails no loss of generality. Now, for
x ∈ A, we have E[T
A
|X
0
= x] = 0. For x∈/ A, let z
x
denote
its z co-ordinate. Note that the radius of S is
1
2
√
π
. The first
time that X
t
enters A is also the first time that Y
t
travels from
z
x
to −cos(2a
n
√
π). Using Lemma 1, we obtain
E[T
A
|X
0
= x] =
2
σ
2
n
log
1 + z
x
1 − cos (2a
n
√
π)
.
Integrating over all possible positions of the point x on S, and
using the fact that x is uniformly distributed on S, we obtain
E
Π
[T
A
] =
Z
π
θ=2
√
πa
n
sin θ
σ
2
n
log
1 +
cos θ
2
√
π
1 − cos (2a
n
√
π)
!
dθ,
and the result follows after straightforward calculations.
Remark 1: If a
n
= n
α
, α < 0, then by Lemma 2, the first
hitting time is always Θ(log n/σ
2
n
), regardless of the value
of α. Even if we take a
n
=
√
π/4, which means that the
set A covers about half of the sphere, the first hitting time is
still Θ(1/σ
2
n
). Hence, the first hitting time changes very little
when the size of the set A is increased. This result reveals the
fundamental difference between the mobility pattern under the
Brownian motion model and that under other mobility models
(such as the i.i.d mobility model [8] and the random way-point
mobility model [5]). In these other models, the first hitting time
for a set A decreases substantially when the size of the set A
is increased. On the other hand, Lemma 2 is not completely
surprising given the fact that, under the Brownian motion
model, the node always wanders around like a “drunkard.”
Therefore, it is very difficult for the node to move towards
any given destination.
B. The First Exit Time
The second concept that we study is the first exit time.
Definition 2: Let A = {x ∈ S : d
S
(x, y) ≤ a
n
}. The first
exit time for the region A, denoted τ
A
, is the first instant of
time at which the Brownian motion started at y (the center of
A) exits A, i.e.,
τ
A
= inf{t ≥ 0 : X
0
= y, X
t
∈/ A}.
Assuming a
n
→ 0 as n → ∞, we have the following result:
Lemma 3: E[τ
A
] = Θ(a
2
n
/σ
2
n
).
Proof: Using the symmetry of the sphere, we can set
y to be the north pole of S (i.e., the top most point of S
that corresponds to θ = 0). It then follows that E[τ
A
] is the
expected travel time of Y
t
from 1 to cos(2a
n
√
π). Applying
Lemma 1 and performing some straightforward calculations,
the result follows.
Remark 2: From the above discussion it is clear that under
the Brownian motion model a node requires Θ(a
2
n
/σ
2
n
) time
to move a radial distance of a
n
. Thus the time a Brownian
motion process spends in a region is in proportion to the area
of the region. This also points to the well-known result that
the path of Brownian motion is nowhere differentiable [11,
p380]. Hence, it is inappropriate to define the “velocity” of a
node that is moving in accordance with the Brownian motion
model.
4
IV. THE DEGENERATE CAPACITY-DELAY TRADE-OFF
In this section, we show that there is virtually no trade-
off between the delay and the capacity under the Brownian
motion model. Specifically, we will show that whenever the
delay constraint is O(n
α
/σ
2
n
) for any α < 0, the per-node
capacity is O(1/
√
n). In order to provide the readers with the
main insight underlying this result, we use a slightly different
network model in this section. We assume that the nodes are
executing independent Brownian walks within a unit square
on a plane (instead of on a unit sphere). This change simplifies
the exposition substantially. Nonetheless, as we will see later,
our results hold for a unit sphere as well.
Consider n nodes on a unit square centered at the origin,
executing independent two-dimensional Brownian motions
within the square. As will become clear soon, our result does
not depend on how the boundary condition is handled: the
Brownian motion could either be reflected at the boundary, or
wrap around the boundary (like the 2-d torus model in [3]).
In order to prove the main result of this section, namely,
that the delay-capacity trade-off under the Brownian motion
model is degenerate, we need some supporting results. The
main idea behind the proof is that, if the delay is O(n
α
/σ
2
n
)
for α < 0, then the contribution due to node mobility in the
packet delivery is likely very small. Hence, in order to deliver
the packet to the destination, relaying over order Θ(1) distance
is required, which results in the throughput capacity being
O(1/
√
n).
We start by showing that, if the delay is O(n
α
/σ
2
n
) for
α < 0, then the contribution due to node mobility in the packet
delivery is likely very small.
Let SQ(c
n
) be the square centered at the origin with length
c
n
(see Fig. 2). Suppose there are k
n
≤ n nodes, starting at
the origin at time 0. Each node then moves according to a
two-dimensional Brownian motion with variance σ
2
n
, which
can be viewed as the composition of two independent one-
dimensional Brownian motion along the x-axis and the y-axis,
respectively, each having a variance of σ
2
n
/2. Let p
k
n
(c
n
, t
n
)
denote the probability of the event that one or more of the k
n
nodes ever exit the square SQ(c
n
) within time t
n
. We have
the following result concerning p
k
n
(c
n
, t
n
):
Lemma 4: If there exists N
0
< ∞ such that
c
2
n
t
n
≥ 8σ
2
n
log n, for n ≥ N
0
, (5)
then
lim
n→∞
p
k
n
(c
n
, t
n
) = 0.
The following corollary is an immediate consequence of
Lemma 4.
Corollary 1: If
lim inf
n→∞
c
n
log n = c > 0
lim sup
n→∞
σ
2
n
t
n
n
α
= c
0
< +∞, for some α < 0 ,
then
lim
n→∞
p
k
n
(c
n
, t
n
) = 0.
n
c
n
c
origin
Fig. 2. k
n
nodes at the origin
Hence within O(n
α
/σ
2
n
) time (α < 0), none out of the
k
n
nodes can possibly travel an Θ(1/ log n) distance in any
direction.
Proof: [Proof of Lemma 4] Consider an arbitrary node.
Let X
t
be its position at time t. Let B
x
t
and B
y
t
denote its x-
coordinate and y-coordinate, respectively. Then B
x
t
and B
y
t
are
independent one-dimensional Brownian motions with variance
σ
2
n
/2. Let p(c
n
, t
n
) be the probability that this particular node
ever exits the square SQ(c
n
) within time t
n
. Let
τ
+
x
, inf{t ≥ 0 : B
x
t
= c
n
/2},
τ
−
x
, inf{t ≥ 0 : B
x
t
= −c
n
/2},
and let τ
+
y
, τ
−
y
be similarly defined with B
y
t
in place of B
x
t
.
Using the union bound, and appealing to the symmetry of the
two-dimensional Brownian motion, we obtain
p(c
n
, t
n
) ≤ P{τ
+
x
≤ t
n
or τ
−
x
≤ t
n
or τ
+
y
≤ t
n
or τ
−
y
≤ t
n
}
≤ 4P{τ
+
x
≤ t
n
}.
Further, using the Reflection Principle for one-dimensional
Brownian motion [11, p394], we have
P{τ
+
x
≤ t
n
} = 2P{B
x
t
n
≥ c
n
/2}.
Since the distribution of B
x
t
n
is Gaussian with zero mean and
variance σ
2
n
t
n
/2, we have,
P{τ
+
x
≤ t
n
} = 2
Z
∞
c
n
√
2σ
n
√
t
n
1
√
2π
exp
−
u
2
2
du.
Using the inequality,
Z
∞
x
1
√
2π
exp
−
u
2
2
du ≤
1
√
2π
Z
∞
x
u
x
exp
−
u
2
2
du
=
1
√
2πx
exp
−
x
2
2
,
we have,
P{τ
+
x
≤ t
n
} ≤ 2
s
σ
2
n
t
n
πc
2
n
exp
−
c
2
n
4σ
2
n
t
n
.
Using (5), we have
P{τ
+
x
≤ t
n
} ≤
1
√
2π log n
exp(−2 log n) =
1
n
2
√
2π log n
.
Hence,
p(c
n
, t
n
) ≤
4
n
2
√
2π log n
.
5
Finally, since there are k
n
nodes, each of them moves accord-
ing to a two-dimensional Brownian Motion, we have
p
k
n
(c
n
, t
n
) ≤ k
n
p(c
n
, t
n
) ≤
4k
n
n
2
√
2π log n
.
Noting that k
n
≤ n, the result then follows.
We now show that if each packet is relayed over Θ(1)
distance (on an average), then the throughput capacity would
be O(1/
√
n).
Consider a large enough time interval T . The total num-
ber of packets communicated end-to-end between all source-
destination pairs during the interval is c
p
λnT , where 1/c
p
is the number of bits per packet. Let h
p
be the number of
times the packet p is relayed, and let l
h
p
, for h = 1, ..., h
b
,
denote the transmission range for the h-th relaying. We have
the following result:
Lemma 5: Suppose that there exists a constant c > 0, such
that on average each packet is relayed over a total distance no
less than c, i.e.,
c
p
λnT
P
p=1
h
p
P
h=1
l
h
p
c
p
λnT
≥ c, (6)
then
λ ≤ O(1/
√
n).
Proof: We use d(x, y) to denote the Euclidean distance
between positions x and y within the unit square. Let X
i
denote the position of node i, for i = 1, ..., n. Consider nodes
i, j transmitting directly to nodes k and l, respectively, at
time t. Then, under the Protocol Model, in order for the
transmissions to be successful, the following inequalities must
hold at the time of transmission:
d(X
j
, X
k
) ≥ (1 + ∆)d(X
i
, X
k
)
d(X
i
, X
l
) ≥ (1 + ∆)d(X
j
, X
l
).
Hence,
d(X
j
, X
i
) ≥ d(X
j
, X
k
) − d(X
i
, X
k
)
≥ ∆ d(X
i
, X
k
).
Similarly,
d(X
i
, X
j
) ≥ ∆ d(X
j
, X
l
).
Therefore,
d(X
i
, X
j
) ≥
∆
2
(d(X
i
, X
k
) + d(X
j
, X
l
)).
That is, disks of radius
∆
2
times the transmission range cen-
tered at the transmitter are disjoint from each other
3
. We can
therefore measure the radio resources that each transmission
consumes by the areas of these disjoint disks. Note that the
total area of the square is 1; for each of these disks, at least
1/4 of it must lie inside the unit square; and each relaying of
a packet lasts
1
c
p
W
amount of time. Thus,
1
4
c
p
λnT
X
p=1
h
p
X
h=1
π
∆
2
l
h
p
2
≤ c
p
W T . (7)
3
A similar observation is used in [1] except that they take a receiver point
of view.
By Cauchy-Schwarz Inequality,
c
p
λnT
X
p=1
h
p
X
h=1
l
h
p
2
≤
c
p
λnT
X
p=1
h
p
X
h=1
(l
h
p
)
2
c
p
λnT
X
p=1
h
p
X
h=1
1
. (8)
Further, since there are at most n simultaneous transmissions
at any given time in the network, we have
c
p
λnT
X
p=1
h
p
≤ c
p
W T n. (9)
Therefore,
16c
p
W T
π∆
2
≥
c
p
λnT
X
p=1
h
p
X
h=1
(l
h
p
)
2
(using (7))
≥
"
c
p
λnT
P
p=1
h
p
P
h=1
l
h
p
#
2
"
c
p
λnT
P
p=1
h
p
#
(using (8))
≥
(c
p
λnT c)
2
c
p
W T n
(using (6) and (9)).
Hence,
λ ≤
r
16W
2
π∆
2
c
2
1
√
n
.
We are now ready to prove the main result of this section.
We first define a general class of scheduling polices that we
plan to study. Note that at each time and for each packet p
that has not been delivered to its destination yet, a scheduling
policy essentially needs to make the following two types of
decisions:
• Replication: The scheduler needs to decide whether to
replicate the packet p to other relay nodes that do not
have the packet yet. If yes, the scheduler needs to decide
how to schedule radio transmissions to forward the packet
p to these new relay nodes. Note that by replication we
mean packet duplication; i.e., creating redundant copies
of the packet. This is different from capture (to be defined
next) where the number of copies of the packet does not
increase.
• Capture: The scheduler needs to decide whether to de-
liver the packet p to the destination immediately, possibly
using multi-hop transmission. If yes, the scheduler needs
to choose one relay node (possibly the source) that has
a copy of packet p and schedule radio transmissions to
forward the packet to the destination. When this happens
successfully, we say that the chosen relay node has
successfully captured the destination of the packet p, or
a successful capture has occurred for the packet p.
Remark 3: Although our model does allow for other less in-
tuitive alternatives, in a typical scheduling policy a successful
capture usually occurs when a relay node holding the packet
comes within a small area around the destination node, so
that fewer resources are needed to forward the packet to the
destination. For example, a relay node could enter a disk of
6
a certain radius around the destination, or a relay node could
enter the same cell as the destination. We call such an area the
capture neighborhood. The purpose of replication is to reduce
the time before a successful capture occurs. With more nodes
holding the packet p, the likelihood of one of them capturing
the destination sooner is higher.
In this paper, we restrict our study to the class of scheduling
policies that satisfy the following assumption:
Assumption A:
• Only the source of a packet is allowed to replicate the
packet. That is, relay nodes holding a packet are not
allowed to replicate it further.
Remark 4: Note that almost all scheduling schemes that
have been proposed in the literature satisfy Assumption A [2],
[3], [4], [5], [6], [7], [8]. If the relay nodes were allowed to
replicate, then additional cooperation among the relay nodes
would most likely be required (see, for example, the scheme
in [12], where the relay nodes know the location of the
static destination, and also have some knowledge of the future
direction of other nodes’ movement, based on which they
can cooperate to make selective and more efficient replication
toward the destination) in order to limit the number of replicas
of a packet.
It is worthwhile to elaborate on Assumption A a little bit,
since it may seem restrictive at first sight. First of all, we note
that the notions of replication and relaying are different, even
though both involve forwarding packets to other relay nodes.
For example, when node i decides to replicate the packet p to
node j, node i can either transmit the packet directly to node
j, or use multi-hop transmission; i.e., node i can forward the
packet to another node k, and let node k forward the packet
to node j. (Node k may also keep the copy of the packet p,
in which case, both nodes k and j are considered to receive
the packet due to the same replication decision initiated by
node i.) In this example, although both nodes i and k forward
the packet p to other nodes, their roles are different. Node i
is the one who initiates the replication, while node k is just
passively following the instruction of node i to relay the packet
to node j. Hence, Assumption A only prohibits relay nodes to
initiate replications. Multi-hop relaying is still allowed in the
replication process. (Multi-hop relaying is also allowed for the
relay-to-destination communication, i.e., capture)
If we attempt to develop distributed scheduling policies
where nodes make replication decisions and capture decisions
without any knowledge of the decisions at other nodes, then
restricting the replication decisions to the source node is a
natural way to control the number of copies of a packet in
the system. Note that excessive redundancy will reduce the
system throughput substantially. The source node of a packet
p is in the best position to control both the total number of
replications for the packet and the number of relay nodes
getting the packet for each replication.
We are now ready to prove the main result of this section:
Proposition 1: Let
¯
D denote the expected delay averaged
over all packets and all source-destination pairs, and let λ
denote the throughput of each source-destination pair. For any
1/4
1/4
source node S
A
B
Origin
destination node D
b
b
Fig. 3. There exists a constant fraction of packets that originate from nodes
in A and are destined to nodes in B.
scheduling policy that satisfies Assumption A, if
¯
D ≤ O(n
α
/σ
2
n
), α < 0,
then
λ ≤ O(1/
√
n).
Proof: Consider squares A and B of length 1/4, centered
at (−1/4, 1/4) and (1/4, −1/4), respectively (see Fig. 3).
Since the packet arrivals are independent of the positions of
the mobile nodes, there will be a constant fraction f
0
of
the packets that have their source nodes in square A and
destinations in square B, at the time of arrival. (If the stationary
distribution of the positions of the nodes are uniform, then
f
0
=
1
4
4
= 1/256. Otherwise, f
0
is still a positive constant
independent of n.) Let Φ
AB
denote this set of packets. In order
to ensure that
¯
D ≤ O(n
α
/σ
2
n
), the delay for the packets in
Φ
AB
has to be O(n
α
/σ
2
n
). Precisely, since
¯
D ≤ O(n
α
/σ
2
n
),
there exists some N
0
> 0 and c
1
> 0, such that
¯
D ≤ c
1
n
α
/σ
2
n
, when n ≥ N
0
. (10)
Therefore, the delay of at least half of the packets in Φ
AB
must be no greater than
t
n
=
2c
1
f
0
n
α
σ
2
n
.
(Otherwise, the delay of the other half of the packets in Φ
AB
must be greater than t
n
. Because this other half contributes to
at least f
0
/2 fraction of all packets, the condition (10) will be
violated.) Let Φ
0
AB
denote the set of packets in Φ
AB
whose
delay is no greater than t
n
. Consider an arbitrary packet p
which is in Φ
0
AB
. Let S
p
and D
p
denote its source node and
destination node, respectively. Fig. 4 shows a typical packet
delivery. The source nodes S
p
moves from position S
0
to U
1
,
and replicates the packet p to a relay node, say r
1
, at position
V
1
, possibly using multi-hop transmission. The node r
1
then
moves independently of S
p
. The source node moves on to
position U
2
, where it replicates the packet p to one more relay
node, say r
2
, positioned at V
2
, and so on. It is also possible to
replicate the packet to more than one relay node at the same
time (for example, we can take U
1
= U
2
if the source node
replicates the packet to r
1
and r
2
at the same time). At time
t ≤ t
n
, a successful capture occurs, as one of the relay nodes
holding the packet p (node r
2
in the case shown in Fig. 4)
decides to forward the packet to its destination node D
p
, which
has moved from its initial position D
0
to the position D, at
7
U
1
V
1
U
2
V
2
r
2
r
1
W
1
W
2
S
0
W
0
source or destination node
relay node
replication / capture
p
S
D
0
D
p
D
Fig. 4. How a typical packet p is delivered.
S
0
source or destination node
relay node
replication / capture
V
k
W
k
D
0
U
k
S
p
D
p
D
k
r
Fig. 5. The relay node r
k
time t. Let k
n
denote the total number of relay nodes that hold
packet p in this process, and let r
k
, for k = 1, 2, ..., k
n
, denote
the k-th relay. Let U
k
and V
k
denote the position of the source
node S
p
and the position of the relay node r
k
, respectively, at
time of replication. Let W
k
denote the position of the relay
node r
k
at the time of capture (see Fig. 5). Since the direct
straight-line path is always the shortest path connecting any
two points, we have, for any k,
d(S
0
, U
k
) + d(U
k
, V
k
) + d(V
k
, W
k
) + d(W
k
, D) + d(D
0
, D)
≥ d(S
0
, D
0
).
Hence,
d(U
k
, V
k
) + d(W
k
, D)
≥ d(S
0
, D
0
) − d(D
0
, D) − d(S
0
, U
k
) − d(V
k
, W
k
). (11)
Since S
0
and D
0
are in the squares A and B, respectively,
d(S
0
, D
0
) ≥
√
2
4
.
Further, each of the terms d(D
0
, D), d(S
0
, U
k
), and
d(V
k
, W
k
), corresponds to the movement of a different node.
There are at most n nodes involved in this process. By setting
c
n
= 1/ log n in Corollary 1, we can see that, with probability
approaching 1 as n → ∞, all of the last three terms in (11)
are no greater than
√
2/ log n, for all k. Therefore,
d(U
k
, V
k
) + d(W
k
, D) ≥
√
2
4
− 3
√
2
log n
≥ 1/4
for large enough n. Finally, let W
0
denote the position of the
source node at time t. Then using a similar argument,
d(W
0
, D) ≥ d(S
0
, D
0
) − d(D, D
0
) − d(S
0
, W
0
) ≥ 1/4.
This shows that for each packet p in Φ
0
AB
, the total distance
that the packet p has to be relayed is at least 1/4. Since
Φ
0
AB
contributes to at least f
0
/2 fraction of all packets, on an
average each packet must be relayed over a distance no less
than f
0
/8 > 0. Hence, by Lemma 5, the per-node throughput
λ must be no larger than O(1/
√
n).
Remark 5: For the ease of exposition, we have shown the
above results for Brownian motion on a plane. However, it is
not difficult to see that the argument in Proposition 1 applies
to Brownian motion on a unit sphere as well. In particular, in
Lemma 4, if we choose c
n
= c/ log n, the size of the square
SQ(c
n
) diminishes to zero as n → ∞. Hence, the difference
between such a square on a plane and that on a unit sphere
vanishes. Therefore, both Corollary 1 and Proposition 1 hold
for Brownian motion on a unit sphere as well.
Proposition 1 shows that the capacity-delay trade-off under
the Brownian motion model is degenerate: For delay less than
O(n
α
/σ
2
n
), α < 0, the per-node capacity is at most O(1/
√
n).
Since one can achieve Θ(1/
√
n log n) per-node capacity for
static wireless networks using multi-hop transmission [1], our
result shows that whenever the delay is constrained to be less
than O(n
α
/σ
2
n
), α < 1, Brownian mobility cannot improve
the capacity by more than a logarithmic factor. Further, since
the packet transmissions are usually carried out at a much
faster time-scale than the node mobility, one could view the
delay under the multi-hop scheduling (see [1]) as being almost
zero. Earlier studies have shown that it is possible to achieve
Θ(1) per-node capacity at roughly Θ(1/σ
2
n
) delay under the
Brownian motion model. Obviously, Θ(1) is an upper bound
on the per-node capacity (under our network model). Hence, if
we ignore the logarithmic terms, the capacity-delay trade-off
under the Brownian motion model degenerates into two points:
one can either achieve Θ(1/
√
n log n) per-node capacity at al-
most no delay, or Θ(1) per-node capacity at roughly Θ(1/σ
2
n
)
delay, but nothing in between! Finally, although Proposition 1
is shown under the Brownian motion model, it is not difficult
to see that the result also applies to the random walk mobility
model [9], or the Markovian mobility model in [4]. This is
because as n → ∞, the difference between these mobility
models vanishes.
The result of Proposition 1 is in sharp contrast to the results
reported in the existing works [3], [6], where it is claimed that
certain schemes can provide a smooth trade-off between the
capacity and the delay. In section VI, we will point out the
most likely reasons for this discrepancy.
V. DELAY UNDER GENERALIZED TWO-HOP RELAYING
SCHEMES
In Section IV, we have established the fundamental delay-
capacity trade-off under the Brownian motion model for a
wide class of scheduling schemes. We have shown that, for
any scheduling policy that satisfies Assumption A, in order to
achieve a per-node capacity greater than Ω(1/
√
n), the delay
must be Ω(n
α
/σ
2
n
), for all α < 0. In this section, we study
the delay performance of a more restricted set of scheduling
schemes. Our interest in this class of schemes stems from
the fact that they have been used in the earlier studies for
8
achieving Ω(1/
√
n) per-node capacity under various mobility
models. Thus, we would like to understand more deeply their
delay performance under the Brownian motion model.
These schemes are similar to the 2-hop relaying scheme
of Grossglauser and Tse [2]. Hence, we refer to them as
generalized 2-hop relaying schemes. Compared with the more
general class of schemes that we considered in Section IV,
these schemes have one additional restriction: For each packet
p, the source node is only allowed to replicate the packet p to
one relay node (denoted by R(p)). Other than this restriction,
the generalized 2-hop relaying schemes still have substantial
flexibility in scheduling packet transmissions. For example,
in the replication phase, the scheduler still decides when
to replicate the packet, and how (e.g., which relay node to
replicate the packet p to, and how to schedule the packet
transmissions from the source node to the chosen relay node
R(p), possibly using multi-hop transmissions). Similarly, in
the capture phase, the scheduler decides when and how to
relay the packet p to the destination, from either the source
node or the chosen relay node R(p), possibly using multi-hop
transmissions.
To ensure that fewer radio resources are consumed, the
replication phase (and the capture phase, correspondingly)
typically occurs when the chosen relay node is within a small
neighborhood around the source node (and the destination
node, correspondingly). For example, a relay node could
either enter a disk of a certain radius around the source (or
destination), or a relay node could enter the same cell as the
source (or destination) in case the network is divided into cells.
We call such an area around the source or the destination
as the replication neighborhood or the capture neighborhood,
respectively. We further assume that the replication neighbor-
hood and the capture neighborhood are both contained in disks
of radius a
n
centered at the source node and the destination
node, respectively. Again, to ensure that fewer radio resources
are consumed, a
n
would typically be o(1).
Remark 6: Note that Scheme 2 and Scheme 3(b) in [3] are
both special cases of the generalized 2-hop relaying schemes
that we consider in this section.
We now give a lower bound on the average packet delay of
the generalized 2-hop relaying schemes.
Proposition 2: If the replication neighborhood and the cap-
ture neighborhood under a generalized 2-hop relaying scheme
can be contained inside a disk of radius a
n
around the source
node and the destination node, respectively, and a
n
= o(1),
then the average packet delay under the given scheduling
scheme must be Ω(log (1/a
n
)/σ
2
n
).
Proof: When a
n
= o(1), most of the packets will have
to be delivered through a relay node. Consider such a random
packet that arrives at the source node. Its delay must be no
less than the time that it takes for the relay node to move
from somewhere within distance a
n
around the source, to
somewhere within distance a
n
around the destination. Since
the packet arrivals are independent of the node mobility,
the source node and the destination node of the packet are
distributed uniformly on the unit sphere at the time of the
packet arrival. Therefore, the delay for the packet is no less
than the time that it takes for two nodes placed uniformly on
the unit sphere to come within a distance of 2a
n
from each
other. Therefore, in view of Lemma 2, the result follows.
Remark 7: Note that Proposition 2 holds even if the repli-
cation neighborhood and the capture neighborhood differ in
shape or size. Further, if a
n
= O(n
α
), for some α < 0, then
the delay of the generalized 2-hop relaying scheme is
Ω(log n/σ
2
n
). (12)
The above result provides a lower bound on the average
packet delay under any generalized 2-hop relaying scheme.
We have not provided the analysis for the upper bound on the
delay. Such an analysis could possibly be carried out using
the methodology in [3], under certain additional technical
assumptions. We will show in Section VI that, after correcting
the derivation of the variance of the so-called inter-meeting
time, the methodology in [3] will predict the packet delay
to be Θ(log n/σ
2
n
), which is consistent with our result in
Proposition 2.
VI. DISCUSSION OF RELATED WORKS
In this paper, we have investigated the capacity-delay trade-
offs of mobile ad hoc networks under the Brownian motion
model. Earlier contributions to this problem were made in [3]
and [6]. The system model in [6] is the same as in this paper.
On the other hand, the system model in [3] is not entirely
the same as in this paper, e.g., [3] considers Brownian motion
on a 2-d torus, while this paper considers Brownian motion
on a sphere. Despite these differences, we would reasonably
expect that the main results of this paper still apply for the
system model in [3]. In fact, the results in Section IV have
been shown directly for planar Brownian motion.
However, the capacity-delay trade-offs obtained in this
paper differ substantially from those reported in [3], [6]. In
particular, we show that the achievable capacity-delay trade-
off under the Brownian motion model is degenerate, while
both [3] and [6] report some sort of smooth trade-offs. We
now point out the main errors in the analysis in [3], [6] that,
we believe, have led to this difference in the results.
We first look at the results in [3]. It is our understanding that
the work in [3] also attempts to study the Brownian motion
model (see the first paragraph in [3, Section III.A]). In [3],
the nodes are assumed to move according to independent 2-d
Brownian motions on a 2-d torus. The Scheduling Scheme 2
and Scheme 3(b) in [3] are of particular interest to us because
they both belong to the generalized 2-hop relaying schemes
in Section V (and thus satisfy Assumption A in Section IV as
well). These two schemes divide the torus into
1
√
a(n)
×
1
√
a(n)
cells of equal size, where 1/n ≤ a(n) < 1. When a packet
arrives to the source, it is first replicated to a relay node in the
same cell as the source node. After the relay node moves into
the same cell as the destination node, the packet is forwarded
to the destination. In [3], the authors report that a smooth
capacity-delay trade-off can be achieved by varying the cell
size a(n). In particular, when a(n) = 1/n, Theorem 4 in [3]
reports that the delay is
Θ
√
n
v(n)
, (13)
9
where v(n) denotes the “speed” of the Brownian motion,
which we will discuss in further detail later. On the other
hand, when a(n) = ω(1/n), Theorem 6 in [3] reports that the
delay is reduced to
Θ
1
p
a(n)v(n)
!
.
The parameter v(n) in [3] is comparable to σ
n
in this
paper. In the system model in [3],
1
√
nv(n)
is approximately
the time taken by a node to move Θ(1/
√
n) distance, where
Θ(1/
√
n) is roughly the distance among neighboring nodes
(see Equation (14) in [3]). In the model in this paper, the
time to move Θ(1/
√
n) distance is Θ(
1
nσ
2
n
) (see Lemma 3).
Thus, the parameter v(n) of [3] is equivalent to
√
nσ
2
n
. If we
take this value of v(n), then the results of [3] imply that the
schemes there can achieve
Θ(1/σ
2
n
) (14)
delay when a(n) = 1/n, and
Θ
1
p
na(n)σ
2
n
!
delay when a(n) = ω(1/n). On the other hand, our result
in Section V reports that the delay must be Ω(log n/σ
2
n
),
regardless of the value of a(n). We now identify two main
reasons which, we believe, have led to the discrepancy between
the results of [3] and that of this paper.
1) The Mapping from Brownian Motion to Random Walk:
The delay analysis in [3] is based on the mapping that
approximates the Brownian motion of a single node by a 2-d
random-walk, where each state of the random walk indicates
that the node is in a corresponding cell. With such a mapping,
each change of the state of the random walk occurs when
the node moves to a neighboring cell. The authors claim that
when the area of the cell is changed from 1/n to a(n) where
a(n) = ω(1/n), the time taken by a node to move out of a
cell, say t(n), changes from O(
1
√
nv(n)
) to
t(n) = Θ
p
a(n)
v(n)
!
, (15)
where v(n) is the “speed” parameter that we discussed earlier
(see the last part of the proof of Theorem 6 in [3]). However,
if the underlying mobility model is Brownian motion, then
according to Lemma 3, the amount of time that a node takes
to move out of a cell is proportional to the area of the cell
(rather than its diameter). Hence, the correct value of t(n)
should be
t(n) = Θ
a(n)
√
n
v(n)
when the area of the cell is a(n). Using this value of t(n),
and following the rest of argument in the proof of Theorem 6
in [3], one can easily see that resizing the cells (i.e., changing
a(n) as in Scheme 3(b) of [3]) will not significantly affect the
delay. In fact, the delay of Scheme 3(b) in [3] would remain
the same as in (13) with any cell size!
An alternative way of viewing the trade-off result obtained
in [3] is that, if one forces t(n) to behave according to (15),
one must then assume that the variance of the Brownian
motion is v(n)
p
a(n) when the area of the cell is a(n). This
assumption amounts to considering a different mobility model
at each cell size a(n). Such an assumption makes the trade-
off result less useful because in practice one cannot alter the
underlying physical mobility model. Note that the main results
of this paper still apply even under this unrealistic assumption.
In particular, increasing the size of the cell (as in Scheme
3(b) of [3]) still cannot improve the order of the delay. In
fact, if the variance of the Brownian motion were increased
to v(n)
p
a(n) (as we pointed out above), then even using
a(n) = 1/n would have resulted in roughly the same reduction
in delay as that reported in [3] for Scheme 3(b). Hence, there
is no benefit to increase the cell size.
2) The Derivation of the Variance of the Inter-meeting Time:
If we take v(n) =
√
nσ
2
n
, then based on the above
discussion, the methodology in [3] would conclude that the
delay of Scheme 2 and Scheme 3(b) in [3] should be Θ(1/σ
2
n
)
for any cell size (see (14)). Thus, there is still a difference
of factor log n compared with our result in Section V (i.e.,
between (12) and (14)). As we illustrate next, this difference
is due to the incorrect calculation of the variance of the so-
called “inter-meeting time.” For simplicity, consider the case
when a(n) = 1/n. Recall that the authors of [3] map the
Brownian motion on a torus to a random walk on a
√
n ×
√
n
grid. Each state of the random walk indicates that the node is
in a corresponding cell. Each change of the state of the random
walk occurs when the node moves to a neighboring cell. The
inter-meeting time is the time that it takes for the random
walk to start from an arbitrary state and return to the same
state. Lemma 6 in [3] claims that the variance of the inter-
meeting time is Θ(n
2
). In what follows, we will show that
the variance of the inter-meeting time is actually Θ(n
2
log n),
which then accounts for the factor of log n which is missing in
Theorem 4 in [3]. Note that the random walk corresponds to an
irreducible, reversible, and temporally homogeneous Markov
chain, say s
k
, with the state space S
Z
= {(x, y) : x, y =
0, 1, ...,
√
n − 1}. Consider an arbitrary state s ∈ S
Z
. Define
the inter-meeting time to be I
s
= inf{k > 0 : s
0
= s, s
k
= s}.
Now the following Lemma gives the variance of the inter-
meeting time I
s
.
Lemma 6: V ar(I
s
) = Θ(n
2
log n).
Proof: We can show Lemma 6 using the following
standard results in the Markov chain theory. Let Π
Z
denote
the stationary distribution of s
k
. In our case, it is in fact the
uniform distribution on S
Z
. Let T
s
denote the first hitting time
to the state s, i.e.,
T
s
, inf{k ≥ 0; s
k
= s}.
We have the following relationship between the variance of I
s
and the mean of the first hitting time T
s
(see [13, Chapter 2,
pp. 21]):
V ar(I
s
) =
2E
Π
Z
[T
s
] + 1
Π
s
−
1
Π
2
s
, (16)
where the expectation E
Π
Z
is taken with respect to the station-
ary distribution of s
k
, and Π
s
is the stationary probability that
10
s
k
= s. Since the total number of states in S
Z
is n, we have
Π
s
= 1/n. It is well known that the mean of the first hitting
time is
E
Π
Z
[T
s
] = Θ(n log n). (17)
(For example, this result is listed without detailed proofs in
[13, Chapter 5, pp. 11]. We refer to the row corresponding
to 2-dimensional torus and the column corresponding to τ
0
in the table there. The definition of τ
0
in the table is given
in [13, Chapter 4, pp. 1] and is the same as E
Π
Z
[T
s
]. For
the sake of completeness, we have provided a proof for (17)
in Appendix A.) Substituting these values in (16), the result
follows.
In Lemma 6, we have shown that the variance of the inter-
meeting time I
s
is Θ(n
2
log n). If we follow the rest of the
analysis in [3], we can conclude that the delay of Scheme 2 and
Scheme 3(b) there should both be Θ(log n
√
n/v(n)), which is
then consistent with our result in Section V given the fact that
v(n) =
√
nσ
2
n
. Thus, even if one uses the same mapping from
Brownian motion to random walk as in [3], one would obtain
the same delay results as those reported in this paper. However,
this mapping from the Brownian motion to the random walk
is somewhat heuristic. In particular, it is easy to see that future
transitions of the state of the induced “random walk” are not
entirely independent of the past. We use this mapping here
mainly to point out the mistake in the proofs in [3]. It is
possible to rigorously define the notion of the inter-meeting
time for the Brownian motion and calculate the variance of
the inter-meeting time on continuous state-space, thus avoiding
the heuristic mapping to a random walk. In Appendix B, we
provide such an analysis of the variance of the inter-meeting
time for the spherical Brownian motion.
This paper also corrects the previously reported results
in [6]. In [6] it is shown that the capacity-delay trade-off
is smooth as the number of mobile relays per packet is
varied. We now point out the error in [6] that has led to
this incorrect conclusion. The derivation in [6] is based on
the empirical observation that the first meeting time nearly
follows an exponential distribution. In [6], the first meeting
time is defined to be the amount of time that it takes for a node
to move, from an initial position that is uniformly distributed
on the unit sphere, to a neighborhood of the destination. It
is then argued that when the source replicates the packet to
k
n
relays, the delay is reduced roughly by a factor of 1/k
n
,
because the mean of the minimum of k
n
i.i.d. exponentially
distributed random variables is 1/k
n
times the mean of one
of them. However, the problem with this argument is that the
paths of the relays are not independent of each other. In fact,
the paths are correlated by the fact that they all receive the
packet from the same source node. By Lemma 4, even if the
source node replicates the packet to Θ(n) relays, the delay is
not reduced much, i.e., it is still close to Θ(1/σ
2
n
). This is
in contrast to the random way-point mobility model (which is
also studied in [6]), where the correlation between the paths
of various mobile relays holding the same packet dies out in a
very short time, which then allows a smooth trade-off between
the delay and the capacity [5].
We believe that this difference in the delay-capacity trade-
off under the random way-point mobility model and the
Brownian motion model, is a revelation of the fundamental
difference in the mobility pattern under these two models. In
particular, it appears that the more “diffused” the node motion
is, the more difficult it is to reduce the mobility delay.
One possible way of characterizing the diffused nature of
the node motion could be to look at the distribution of the
inter-meeting time: We believe that the more diffused the node
motion is, the slower would be the decay of the complementary
distribution of the inter-meeting time (for the same mean), thus
resulting in larger higher order moments.
One simple measure for quantifying the diffused nature
of the node motion
4
could be to look at the coefficient of
variation of the inter-meeting time, defined as the ratio of the
inter-meeting time variance to the square of the mean inter-
meeting time, that is, V ar(I)/E
2
{I}, where I denotes the
inter-meeting time. The more diffused the mobility model, the
higher we would expect the coefficient of variation of I to be.
From the results of this paper and that of [5], it is easy to
see that the above ratio is Θ(log n)
5
in case of the Brownian
motion model and Θ(1) in case of the random way-point
mobility model. Thus, at least, in our case, the coefficient of
variation of I indeed quantifies how diffused the node motion
is under a mobility model.
From the above discussion, we see that the distribution of
the inter-meeting time has a strong influence on the delay-
capacity relationship under a mobility model. We believe that
the future studies, trying to investigate this link between the
inter-meeting time distrbution and the delay-capacity relation-
ship, can benefit from the analysis of the inter-meeting time
that we provide in Appendix B.
VII. CONCLUSION
In this paper, we have studied the fundamental trade-off
between the delay and the capacity under the Brownian motion
model. We have shown that the capacity-delay trade-off under
the Brownian motion model is degenerate: one can either
achieve a per-node capacity of Θ(1) with Ω(log n/σ
2
n
) delay
(using 2-hop relaying), or, once the delay constraint is of the
order O(n
α
/σ
2
n
), α < 0, one can at most achieve Θ(1/
√
n)
per-node capacity, which is almost the same as that can be
achieved for static wireless networks.
As we compare the results in this paper for the Brownian
motion model, with the results in [5] for the random way-point
mobility model, we find that the delay-capacity relationship
of mobile wireless networks is strongly influenced by how
directed the node motion is. In particular, it appears that
the more “diffused” the node motion is, the more difficult
it is to reduce the mobility delay. Our results also indicate
that there is some connection between the distribution of the
inter-meeting time and the delay-capacity relationship under a
mobility model. In particular, it appears that the rate of decay
of the inter-meeting time tail is fundamentally linked to the
achievable delay and capacity region under a mobility model.
4
Note that this might not be a good measure of diffusivity in some cases.
5
Here, we are considering a
n
= o(1).
11
We believe that this paper is a first step toward understand-
ing the impact, the nature of the node mobility has, on the
delay-capacity relationship in ad hoc networks.
APPENDIX
A. Average First Hitting Time Analysis for 2-D Torus
In this section, we show that the mean of the first hitting
time on 2-d torus is Θ(n log n). Recall the following defini-
tions introduced in Section VI. We have a random walk on a
2-d torus of size
√
n×
√
n. The random walk corresponds to an
irreducible, reversible, and temporally homogeneous Markov
chain, say s
k
, with the state space S
Z
= {(x, y) : x, y =
0, 1, ...,
√
n − 1}. Consider an arbitrary state s ∈ S
Z
and let
T
s
denote the first hitting time of state s, defined as follows:
T
s
, inf{k ≥ 0; s
k
= s}.
Let Π
Z
denote the uniform distribution on the torus, which is
also the stationary distribution of the Markov chain s
k
. We are
interested in finding the expectation of T
s
taken with respect
to the uniform distribution of the initial state of the Markov
chain, i.e., E
Π
Z
{T
s
}. Let Q denote the transition matrix of the
above Markov chain. It is well known (for example, see [13,
Chapter 3, pp. 19]) that
E
Π
Z
{T
s
} =
n
X
m=2
(1 − λ
m
)
−1
, (18)
where λ
m
, m = 1, 2., , , n are the eigenvalues of Q, numbered
in the descending order, i.e., 1 = λ
1
≥ λ
2
≥ ... ≥ λ
n
.
The eigenvalues in our case can easily be computed (see [13,
Chapter 5, pp. 33]) to be:
λ
i,j
=
cos
2πi
√
n
+ cos
2πj
√
n
2
, for i, j = 0, 1, ...,
√
n − 1.
Note that λ
0,0
corresponds to λ
1
= 1. Substituting this in (18),
we obtain
E
Π
Z
{T
s
} =
X
(i,j)∈S
Z
\(0,0)
2
2 − cos
2πi
√
n
− cos
2πj
√
n
=
X
(i,j)∈S
Z
\(0,0)
1
sin
2
πi
√
n
+ sin
2
πj
√
n
.
To obtain a lower bound on E
Π
Z
{T
s
}, we note that sin
2
x ≤
x
2
, and therefore
E
Π
Z
{T
s
} ≥
X
(i,j)∈S
Z
\(0,0)
n
π
2
(i
2
+ j
2
)
≥
ZZ
{0≤x,y≤
√
n}
T
{x≥1 or y≥1}
n
π
2
(x
2
+ y
2
)
dxdy
≥
Z
√
n
r=
√
2
πr
2
n
π
2
r
2
dr
(by changing to polar coordinates)
=
n
2π
[log
√
n − log
√
2]
= Θ(n log n). (19)
To obtain an upper bound on E
Π
Z
{T
s
}, let A
Z
denote the
set {(i, j) ∈ S
Z
\ (0, 0) : i, j ≤
√
n/4}. Using the inequality
sin x ≥ x/2 for x ≤ π/4, we have
X
A
Z
1
sin
2
πi
√
n
+ sin
2
πj
√
n
≤
X
A
Z
4n
π
2
(i
2
+ j
2
)
≤
ZZ
{0≤x,y≤
√
n/4+1}
T
{x≥1 or y≥1}
4n
π
2
(
x
2
+y
2
5
)
dxdy
≤
Z
√
n/2
r=1
20n
π
2
πr
2
1
r
2
dr
=
10n
π
log (
√
n/2)
= Θ(n log n). (20)
Similarly, let B
Z
denote the set {(i, j) ∈ S
Z
\ (0, 0) : i, j ≥
3
√
n/4}. Using sin x = sin (π − x) and the above techniques,
we can show that
X
B
Z
1
sin
2
πi
√
n
+ sin
2
πj
√
n
≤ Θ(n log n). (21)
Now consider (i, j) ∈ S
Z
\ A
Z
\ B
Z
. In this case, we have
either
sin (πi/
√
n) ≥ sin (π/4) = 1/
√
2,
or
sin (πj/
√
n) ≥ 1/
√
2.
Since the number of states in S
Z
\ A
Z
\ B
Z
is less than n,
we then have
X
S
Z
\A
Z
\B
Z
1
sin
2
πi
√
n
+ sin
2
πj
√
n
≤ Θ(n). (22)
Combining (20-22), we have E
Π
Z
{T
s
} ≤ Θ(n log n). In view
of (19), it follows that E
Π
Z
{T
s
} = Θ(n log n).
B. The Inter-meeting Time Analysis for Spherical Brownian
Motion
In this section, we provide a rigorous treatment of the notion
of the inter-meeting time. Our treatment does not rely on the
mapping from Brownian motion to random walk used in [3].
We start with two related definitions. Consider the motion of
an arbitrary node under the Brownian motion model. Let X
t
be its position on the unit sphere S at time t. Let A = {x ∈
S : d
S
(x, y) ≤ a
n
}, for some y ∈ S. Let A/2 = {x ∈ S :
d
S
(x, y) ≤ a
n
/2}, i.e., A/2 is the circle centered at y with
half the radius as A.
Definition 3 (Contact Time): The contact time, denoted by
τ
A
A/2
, is the time it takes for a Brownian motion started at
the boundary of A/2 to exit A; i.e., τ
A
A/2
= inf{t ≥ 0 :
d
S
(X
0
, y) = a
n
/2, d
S
(X
t
, y) = a
n
}.
Definition 4 (Return Time): The return time, denoted by
T
A/2
A
, is the time it takes for a Brownian motion started at
the boundary of A to enter A/2; i.e., T
A/2
A
= inf{t ≥ 0 :
d
S
(X
0
, y) = a
n
, d
S
(X
t
, y) = a
n
/2}.
12
We are now ready to define the inter-meeting time:
Definition 5 (Inter-Meeting Time): The inter-meeting time,
denoted by I
A
, is the time it takes for a Brownian motion
started at the boundary of A/2 to exit A, and come back to
A/2; i.e., I
A
= τ
A
A/2
+ T
A/2
A
.
The motivation for the above definitions is the following.
Consider two nodes executing independent Brownian walks
on the sphere. The return time defined as above, is related
to how much time it takes for the two nodes to come close
to each other. The contact time is related to how long the
two nodes remain close to each other. And the inter-meeting
time is related to how often the two nodes come in contact
with each other. Note that the choice of A/2 is not really
critical here: one could also use A/, with any > 1, instead
of A/2. The reason why we need > 1 is that we want
most contacts to be long enough so as to allow for a packet
exchange between the nodes. In the above definitions, we have
fixed the position of one of the nodes at y and let the other
node move. Alternatively, we can define these terms for the
case when both nodes are moving, i.e., we can replace y by
a random variable that represents the position of a moving
node. Note that when two nodes are executing i.i.d. Brownian
walks with variance σ
2
n
on the unit sphere, their relative motion
is a Brownian walk with variance 2σ
2
n
. Hence, for the sake
of estimating the order of these times, there will only be
a difference of a factor of two between these two types of
definitions.
We now estimate the mean and the variance of the inter-
meeting time.
Proposition 3: E[τ
A
A/2
] = Θ(a
2
n
/σ
2
n
), E[T
A/2
A
] = Θ(1/σ
2
n
),
and E[I
A
] = Θ(1/σ
2
n
).
Proof: The symmetry of the sphere implies that the choice
of y ∈ S, can be arbitrary. In order to estimate E[τ
A
A/2
], we
choose y to be the north pole (the top most point of S). It
is then clear that T
A
A/2
is the time it takes for Y
t
(see section
II) to travel from cos (a
n
√
π) to cos (2a
n
√
π). Appealing to
Lemma 1, gives the desired result.
To estimate E[T
A/2
A
], we choose y to be south pole (the bottom
most point of S)
6
. Thus, T
A/2
A
is the time it takes for Y
t
to
travel from −cos (2a
n
√
π) to −cos (a
n
√
π), and once again
Lemma 1 gives the desired result.
Now since E[I
A
] = E[T
A/2
A
] + E[T
A
A/2
], and a
n
= O(1), we
obtain E[I
A
] = Θ(1/σ
2
n
).
Now let us look at the variance of the inter-meeting time.
Assuming a
n
→ 0 as n → ∞, we have the following key
result:
Proposition 4: V ar(I
A
) = Θ(log (1/a
n
)/σ
4
n
) .
Remark 8: This result is consistent with that of Lemma 6.
In fact, Lemma 6 reports that the variance of the inter-meeting
time I
s
for the random walk is Θ(n
2
log n). Note that the inter-
meeting time I
s
for the random walk is defined as the number
of state transitions before returning to the same state. Since
each state transition takes Θ
1
nσ
2
n
time (when the unit area
is divided into
√
n ×
√
n cells), the variance of the actual
6
We are doing this in order to make sure that Y
t
travels in the negative
z-axis, so that we can make use of Lemma 1.
amount of time taken to return back to the same state is
Θ(n
2
log n)
Θ
1
nσ
2
n
2
= log n/σ
4
n
,
which matches with the result in Proposition 4 when a
n
=
n
α
, α < 0.
Proof: Recall that I
A
= τ
A
A/2
+ T
A/2
A
, where τ
A
A/2
and T
A/2
A
are as defined earlier. From the Strong Markov
property of the Brownian motion, it follows that τ
A
A/2
and
T
A/2
A
are independent random variables. Thus, V ar(I
A
) =
V ar(τ
A
A/2
)+V ar(T
A/2
A
). In order to estimate these variances,
we will first establish a more general result, regarding the
variance of the travel time for the process Y
t
, given by
Equation (4). Then, by appropriately choosing the starting and
the ending points, we will be able to estimate the variances of
τ
A
A/2
and T
A/2
A
.
1) Variance of the Travel Time for Y
t
: Let −1 < a < b < 1,
and let a ≤ x ≤ b. Recall that Y
t
is the z-coordinate of the
Brownian motion on the unit sphere S, and Y
t
is governed by
the diffusion process (4) with drift coefficient µ(y) = −σ
2
n
y
and diffusion coefficient σ
2
(y) = σ
2
n
(1 − y
2
). Define
τ
a∧b
, inf{t ≥ 0 : Y
t
= a or Y
t
= b}
to be the first time that Y
t
hits a or b. Let U
a∧b
(x) be the
second moment of the travel time for Y
t
from x to either a or
b, i.e.,
U
a∧b
(x) , E[τ
a∧b
2
|Y
0
= x]
= E
(
Z
τ
a∧b
0
1dτ
2
|Y
0
= x
)
.
Using the technique of [14, p202], U
a∧b
(x) must satisfy the
following differential equation:
1
2
σ
2
(x)
d
2
U
a∧b
(x)
dx
2
+ µ(x)
dU
a∧b
(x)
dx
+ 2V
a∧b
(x) = 0,
where
V
a∧b
(x) = E
Z
τ
a∧b
0
1dτ
a∧b
|Y
0
= x
= E[τ
a∧b
|Y
0
= x],
i.e., V
a∧b
(x) is the mean travel time for Y
t
from x to a or b.
The solution to U
a∧b
(x) is given by [14, p197]:
U
a∧b
(x) = 4
(
u(x)
Z
b
a
Z
η
a
V
a∧b
(ξ)m(ξ)dξ
dS(η)
−
Z
x
a
Z
η
a
V
a∧b
(ξ)m(ξ)dξ
dS(η)
, (23)
where S(·) and m(·) are the scale function and the speed
measure density [14, pp.194-195], respectively, defined as
follows for the diffusion process (4):
S(x) ,
Z
x
s(η)dη,
m(x) ,
1
σ
2
(x)s(x)
,
13
with
s(x) , exp
−
Z
x
[2µ(ξ)/σ
2
(ξ)]dξ
,
and u(x) is the probability that Y
t
reaches a before b, which
is equal to [14, p195]:
u(x) =
S(x) − S(a)
S(b) −S(a)
.
For the diffusion process (4), we can calculate these quantities
as,
s(x) = exp
−
Z
x
[2µ(ξ)/σ
2
(ξ)]dξ
= exp
Z
x
2ξ
1 − ξ
2
dξ
= exp
−log(1 − x
2
)
=
1
1 − x
2
,
S(x) =
Z
x
s(η)dη =
Z
x
1
1 − η
2
dη =
1
2
log
1 + x
1 − x
,
m(x) =
1
σ
2
(x)s(x)
=
1
σ
2
n
.
Note that b = 1 is an inaccessible point of the diffusion
process (4). Let b → 1, and define
τ
a
, inf{t ≥ 0 : Y
t
= a}
to be the first time that Y
t
hits a. Further, let
V
a
(x) , E[τ
a
|Y
t
= x],
and
U
a
(x) , E[τ
a
2
|Y
t
= x],
be the first moment and the second moment, respectively, of
the travel time for Y
t
from x to a. It is then not difficult to
see that, as b → 1,
U
a∧b
(x) → U
a
(x), and V
a∧b
(x) → V
a
(x).
Using Lemma 1, we have
V
a
(x) =
2
σ
2
n
log
1 + x
1 + a
.
We now use the techniques of [15, p422] for treating the
inaccessible point at b = 1. Note that
S(b) → ∞, as b → 1,
and
Z
1
x
V
a
(η)m(η)dη < ∞.
Letting b → 1 in (23), we obtain
U
a
(x) = 4
(S(x) − S(a))
Z
1
a
V
a
(ξ)m(ξ)dξ
−
Z
x
a
Z
η
a
V
a
(ξ)m(ξ)dξ
dS(η)
= 4
Z
x
a
Z
1
η
V
a
(ξ)m(ξ)dξ
dS(η).
Using the values of V
a
(x), m(x) and S(x) computed earlier,
we have
Z
1
η
V
a
(ξ)m(ξ)dξ
=
Z
1
η
2
σ
4
n
log
1 + ξ
1 + a
dξ
=
2
σ
4
n
[(1 + ξ) log(1 + ξ) −(1 + ξ) −ξ log(1 + a)]
1
η
=
2
σ
4
n
[2 log 2 − 2 − log(1 + a)]
−
2
σ
4
n
[(1 + η) log(1 + η) − (1 + η) − η log(1 + a)].
Hence,
U
a
(x) = 4
Z
x
a
s(η)
Z
1
η
V
a
(ξ)m(ξ)dξ
dη
=
8
σ
4
n
Z
x
a
2 log(2/e) − log(1 + a)
1 − η
2
dη
−
Z
x
a
log(1 + η)
1 − η
−
1
1 − η
−
η log(1 + a)
1 − η
2
dη
.
(24)
We can now compute the variances of τ
A
A/2
and T
A/2
A
.
2) The Variance of T
A/2
A
: Let
a = −cos (
√
πa
n
) ; x = −cos (2
√
πa
n
).
Then,
E[T
A/2
A
] = V
a
(x), and V ar(T
A/2
A
) = U
a
(x) − V
2
a
(x).
Using Lemma 1,
E[T
A/2
A
] = V
a
(x) = Θ(1/σ
2
n
).
We use (24) to compute the order of U
a
(x). Note first that
Z
x
a
1
1 − η
2
dη =
1
2
log
1 + η
1 − η
x
a
=
1
2
log
1 + x
1 + a
− log
1 − x
1 − a
,
Z
x
a
1
1 − η
dη = −log(1 − η)|
x
a
= −log
1 − x
1 − a
,
Z
x
a
η
1 − η
2
dη = −
1
2
log(1 − η
2
)
x
a
= −
1
2
log
(1 + x)(1 − x)
(1 + a)(1 − a)
.
As a
n
→ 0,
1 + x
1 + a
=
1 − cos(2
√
πa
n
)
1 − cos(
√
πa
n
)
→ 4,
1 − x
1 − a
→ 1.
14
Hence,
Z
x
a
1
1 − η
2
dη → log 2, (25)
Z
x
a
1
1 − η
dη → 0, (26)
Z
x
a
η
1 − η
2
dη → −log 2. (27)
Further, since −1 ≤ a ≤ x ≤ 0 when a
n
is small, we have,
Z
x
a
log(1 + η)dη ≤
Z
x
a
log(1 + η)
(1 − η)
dη ≤
1
2
Z
x
a
log(1 + η)dη.
Note that
Z
x
a
log(1 + η)dη = [(1 + η) log(1 + η) − (1 + η)]
x
a
= (1 + x) log(1 + x) − (1 + a) log(1 + a) − (x − a).
As a
n
→ 0, both (1 + x) and (1 + a) approaches zero. Using
the limit that
lim
x→0
x log x = 0,
we have
Z
x
a
log(1 + η)
(1 − η)
dη → 0. (28)
Substituting the above limits (25-28) into (24), we have
U
a
(x) =
8
σ
4
n
[−3 log(1 + a)(log 2 + o(1))
+2 log(2/e) log 2 + o(1)]
= Θ
1
σ
4
n
log(1/a
n
)
.
Therefore,
V ar(T
A/2
A
) = U
a
(x) − V
2
a
(x)
= Θ(log (1/a
n
)/σ
4
n
) − Θ(1/σ
4
n
)
= Θ(log(1/a
n
)/σ
4
n
).
3) The Variance of τ
A
A/2
: Let
a = cos (2
√
πa
n
) ; x = cos (
√
πa
n
).
Then,
E[τ
A
A/2
] = V
a
(x), and V ar(τ
A
A/2
) = U
a
(x) − V
2
a
(x).
Using Lemma 1, it follows that
E[τ
A
A/2
] = V
a
(x) =
2
σ
2
n
log
1 + cos(
√
πa
n
)
1 + cos(2
√
πa
n
)
.
Using
lim
x→0
log
1+cos x
1+cos 2x
3x
2
/4
= lim
x→0
log
h
1 +
2 sin(x/2) sin(3x/2)
1+cos 2x
i
3x
2
/4
= lim
x→0
log
h
1 +
2 sin(x/2) sin(3x/2)
1+cos 2x
i
2 sin(x/2) sin(3x/2)
1+cos 2x
× lim
x→0
sin(x/2) sin(3x/2)
(x/2)(3x/2)
× lim
x→0
2
1 + cos 2x
= 1,
we have
V
a
(x) =
3π
2σ
2
n
a
2
n
+ o(
a
2
n
σ
2
n
).
We use (24) again to compute the order of U
a
(x). By
expanding 1/(1+η) in its Taylor series around 1, and keeping
only the first three dominant terms, we obtain
1
1 + η
=
1
2
+
1 − η
4
+
(1 − η)
2
8
+ o((1 − η)
2
).
Similarly, we have
log(1 + η) = log 2 −
1 − η
2
−
(1 − η)
2
8
+ o((1 − η)
2
).
Using the above approximations, and simplifying, we have
2 log(2/e)
1 − η
2
−
log(1 + η)
1 − η
+
1
1 − η
=
1
2
log 2 +
log 2
4
(1 − η) −
1 − η
8
+ o(1 − η).
(29)
Further, using similar Taylor-series expansions (and noting that
o(1 − η) = o(1 − a)), we have
log (1 + a) = log 2 −
1 − a
2
+ o(1 − a),
1
1 + η
=
1
2
+
1 − η
4
+ o(1 − a),
and thus
−
log (1 + a)
1 − η
2
+
η log (1 + a)
1 − η
2
= −
log (1 + a)
1 + η
= −
1
2
log 2 +
1 − a
4
−
log 2
4
(1 − η) + o(1 − a).
(30)
Substituting the equalities (29-30) into (24), it follows that
U
a
(x) =
8
σ
4
n
Z
x
a
1 − a
4
−
1 − η
8
dη +
Z
x
a
o(1 − a)dη
=
1
σ
4
n
2(1 − a)(x − a) − (x − a)
+
x
2
− a
2
2
+ o((1 − a)(x − a))
=
1
σ
4
n
(x − a)
2
[3(1 − a) − (1 − x)]
+o((1 − a)(x − a))
.
Since,
1 − a = 1 − cos(2
√
πa
n
) = 2πa
2
n
+ o(a
2
n
),
1 − x = 1 − cos(
√
πa
n
) = πa
2
n
/2 + o(a
2
n
),
x − a = cos(
√
πa
n
) − cos(2
√
πa
n
) = 3πa
2
n
/2 + o(a
2
n
),
we have U
a
(x) = 33π
2
a
4
n
/8σ
4
n
+ o(a
4
n
/σ
4
n
). Thus,
V ar(τ
A
A/2
) = U
a
(x) − V
2
a
(x)
= 33π
2
a
4
n
/8σ
4
n
− 9π
2
a
4
n
/4σ
4
n
+ o(a
4
n
/σ
4
n
)
= 15π
2
a
4
n
/8σ
4
n
+ o(a
4
n
/σ
4
n
)
= Θ(a
4
n
/σ
4
n
).
15
4) The Variance of I
A
: Combining the variances of τ
A
A/2
and T
A/2
A
, we obtain
V ar(I
A
) = V ar(τ
A
A/2
) + V ar(T
A/2
A
)
= Θ(a
4
n
/σ
4
n
) + Θ(log (1/a
n
)/σ
4
n
)
= Θ(log (1/a
n
)/σ
4
n
),
proving the Proposition.
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