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Delay and Capacity Trade-offs in Mobile Ad Hoc
Networks: A Global Perspective
Gaurav Sharma∗, Ravi Mazumdar†, Ness Shroff∗
∗School of Electrical and Computer Engineering
Purdue University
West Lafayette, IN 47907, USA
Email: {gsharma,shroff}@ecn.purdue.edu
†Department of Electrical and Computer Engineering
University of Waterloo
Waterloo, ON N2L 3G1, Canada
Email: mazum@ece.uwaterloo.ca
Abstract— Since the original work of Grossglauser and Tse,
which showed that the mobility can increase the capacity of an ad
hoc network, there has been a lot of interest in characterizing the
delay-capacity relationship in ad hoc networks. Various mobility
models have been studied in the literature, and the delay-capacity
relationships under those models have been characterized. The
results indicate that there are trade-offs between the delay and
the capacity, and that the nature of these trade-offs is strongly
influenced by the choice of the mobility model. Some questions
that arise are: (i) How representative are these mobility models
studied in the lieterature? (ii) Can the delay-capacity relationship
be significantly different under some other “reasonable” mobility
model? (iii) What would the delay-capacity trade-off in a real
network be like? In this paper, we address these questions.
In particular, we analyze, among others, some of the mobility
models that have been used in the recent related works, under
a unified framework. We relate the nature of the delay-capacity
trade-off to the nature of the node motion, thereby providing a
better understanding of the delay-capacity relationship in ad hoc
networks than earlier works.
I. INTRODUCTION
In this paper, we study the delay and capacity trade-offs in
mobile ad hoc networks. We study the network layer notion
of capacity, first introduced by Gupta and Kumar [1]. In [1],
it was shown that the per-node capacity of a random wireless
network with static nodes scales as Θ1
√nlog n, provided
the same power level is used by all the nodes in the network.
Recent works [2], [3] have shown that a per-node capacity
of Θ(1/√n)can be achieved when the nodes are allowed to
exercise power control.
Grossglauser and Tse [4] have shown that significant gains
in the per-node capacity can be obtained by exploiting the
mobility of the nodes in a mobile ad hoc network. In particular,
they proposed a 2-hop relaying scheme, and showed that their
scheme can achieve a constant per-node capacity, which does
not vanish as the number of nodes grows arbitrarily large. The
delay related issues were not addressed in [4].
Since both capacity as well as delay are important from
an application point of view, a significant effort has recently
been devoted within the networking research community to
understand the delay-capacity relationship in ad hoc networks.
Bansal and Liu [5] were among the first to study the delay-
capacity relationship in wireless networks. They considered a
wireless network consisting of static sender-destination pairs
and mobile relays, and proposed a geographic routing scheme
that achieves a near optimal capacity and studied its delay
performance. Perevalov and Blum [6] studied the delay limited
capacity of mobile ad hoc networks. Their work was motivated
by the diversity coding approach given in [7].
Recent works of Neely and Modiano [8], ElGamal et al.
[9], Toumpis and Goldsmith [10], Sharma and Mazumdar [11],
[12], Lin and Shroff [13], and Lin et al. [14], have all studied
the delay-capacity trade-offs in mobile ad hoc networks. The
mobility models and the network settings studied in these
works differ considerably, and so do the delay-capacity trade-
offs that are reported. The mobility models studied in the
literature include the i.i.d. mobility model [8], [10], [13]; the
random way-point mobility model [11], [12]; the Brownian
mobility model [11], [14], [9]; and the random walk mobility
model [9], [15].
Since the network settings considered in these works are
quite different, it is difficult to single out the impact the nature
of the node mobility has on the delay-capacity trade-off. For
example, both [8] and [13] study the i.i.d. mobility model,
but the trade-offs reported in these works are quite different.
In particular, in [8], the authors consider a cell partitioned
network setting and show that the trade-off: λ≤ΘD
n, where
Dis the average packet delay and λis the average per-node
throughput, is both necessary as well as sufficient under their
setting. Under a less restrictive network setting in [13], the
delay-capacity trade-off is shown to be: λ3≤ΘDlog3n
n.
Thus, we see that the delay-capacity trade-off depends not only
on the nature of the node mobility, but also on the network
setting.
The main insight we provide in this paper is that there
is a critical value of delay (henceforth denoted by critical
delay), below which, the node mobility cannot be exploited for
improving the capacity. We also show that the critical delay
depends on the nature of the node mobility, but not so much
on the network setting. In terms of the notion of critical delay
(see section III for a rigorous definition), some recent results
in the literature can be summarized as follows:
•The critical delay under the Brownian motion model and
random walk model is roughly Θ(n)∗(see [14], [15]).
•The critical delay under the random way-point mobility
model is roughly Θ(√n)(see [12]).
•The critical delay under the i.i.d. mobility model is Θ(1)
(see [13]).
Observing the above results, it is natural to ask: (i) How
representative are the above mentioned mobility models? (ii)
Can the critical delay and, in general, the delay-capacity
relationship be significantly different under some other mo-
bility model? (iii) How does the critical delay scale in a real
network? This paper makes the following contributions toward
answering these fundamental questions:
•We propose and study a family of hybrid random walk
models, and show that they exhibit a continuous range
of critical delays in-between those of the i.i.d. mobility
model and the random walk mobility model. In particular,
for every βbetween 0and 1, there exists a mobility model
in the family of hybrid random walk models for which
the critical delay is roughly Θ(nβ). As expected, β= 1
corresponds to random walk mobility model, and β= 0
corresponds to the i.i.d. mobility model.
•We propose and study a family of discretized random
direction models exhibiting a continuous range of crit-
ical delays in-between those of the random way-point
mobility model and the Brownian mobility model. In
particular, for every αbetween 0and 1/2, there exists
a mobility model in the class of discretized random
direction models for which the critical delay is roughly
Θ(n1/2+α). Note that α= 0 corresponds to the random
way-point mobility model, and α= 1/2corresponds to
the Brownian mobility model. These models approximate
the motion of the nodes under the commonly used random
direction models in the literature (see, for example, [16]),
and are simpler to analyze.
An interesting feature of the above classes of mobility models
is that all of them incur a delay of roughly Θ(n)under the 2-
hop relaying scheme; which is in line with the other mobility
models considered in the literature. Our results therefore show
that the mobility models considered in the literature are in
some sense extreme: they either exhibit the smallest critical
delays (i.i.d. mobility model and random way-point mobility
model) or the largest critical delays (Brownian motion model
and random walk mobility model), among the mobility models
in their respective classses.
The rest of the paper is organized as follows. We introduce
the hybrid random walk models and discretized random direc-
tion models, and discuss our network and transmission model
in the next section. We define the notions of critical delay and
∗Note that in [14], the results are stated in terms of the variance parameter
σ2(n). We have set σ2(n) = 1/n for the sake of easy comparison with the
other results.
2-hop delay in section III. We study the critical delay and 2-
hop delay under the hybrid random walk models in section IV,
and under the discretized random direction models in section
V. A discussion on the implications of our results for large
mobile ad hoc networks is provided in section VI. Finally, we
end this paper with some concluding remarks in section VII.
II. THE MODEL
A. The Network and Transmission Model
We consider an ad hoc network consisting of nmobile
nodes, distributed uniformly on a unit square. We consider
a homogeneous scenario in which each node generates traffic
at the same rate. Further, we assume that each node, say node
i, generates traffic for exactly one other node, say node d(i),
and that the mapping i7→ d(i)is bijective. We also assume
that the packet arrival process at each node is independent of
the node mobility process.
The communication between any source-destination pair
can possibly be carried out via multiple other nodes, acting as
relays. That is, a source node can, if possible, send a packet
directly to its destination node; or, the source node can forward
the packet to one or more relay nodes; the relay nodes can also
forward the packet to other relay nodes; and finally, a relay
node or the source node itself can deliver the packet to its
destination node.
For simplicity, we assume that the success or failure of
a transmission between a pair of nodes is governed by the
protocol model of [1]. Let Wbe the bandwidth of the system
in bits per second. Let Xi
tdenote the position of node i, for
i= 1...n, at time t. Under the protocol model, node ican
communicate directly with node jat a rate of Wbits per
second at time t, if and only if, the following interference
constraint is satisfied [1]:
d(Xk
t, Xj
t)≥(1 + ∆)d(Xi
t, Xj
t)(1)
for every other node k6=i, j that is simultaneously transmit-
ting. Here ∆is some positive number; and d(x, y)denotes
the Euclidean distance between points x, y ∈R2. Note that
for a packet to be successfully received by node j, the
above interference constraint must be satisfied over the entire
duration of the packet transmission from node ito node j.
We use the following definition of throughput: Let λi(t)
be the total number of bits delivered end-to-end for source-
destination pair iupto time t, then the throughput λ(n)of the
system is given by
λ(n) = lim inf
t→∞
n
X
i=1
λi(t)
t.
Next, we describe our mobility models, starting with the
hybrid random walk models.
B. Hybrid Random Walk Models
These models are parametrized by a single parameter β,
which takes values between 0and 1/2. The unit square is
divided into n2βsquares of area 1/n2βeach (henceforth
Cell
Subcell
Fig. 1. The figure shows the division of the unit square into cells and subcells
and also the motion of a node under the hybrid random walk model.
referred to as cells), resulting in a discrete torus of size
nβ×nβ. Each cell is then further divided into n1−2βsquare
subcells of area 1/n each†, as shown in Fig. 1. Time is
divided into slots of equal duration. At each time slot a
node is assumed to be in one of the subcells inside a cell.
Initially, each node is equally likely to be in any of the n
subcells, independent of the other nodes. At the beginning
of a time-slot, a node jumps from its current subcell to one
of the subcells in an adjacent cell, chosen in an uniformly
random fashion. By adjacent cell we mean the following: Let
(i, j) : i, j = 0,1, ..., nβ−1, be a numbering of the cells of
the 2-d torus, as shown in Fig. 2. The cells adjacent to cell
(i, j)are the cells (i+ 1, j ),(i−1, j ),(i, j +1),and (i, j −1),
where the addition and subtraction operations are performed
modulo nβ. Note that for β= 0, the above mobility model
is essentially the i.i.d. mobility model considered in [8], [13],
[10]; and for β= 1/2, it is the same as the random walk
model of [9].
Next, we describe the random direction models (see, for
example, [16]).
C. Random Direction Models
These models are parametrized by a single parameter α,
which takes values between 0and 1/2. The initial position of
each node is assumed to be uniformly distributed inside the
unit square. The motion of each node under these models is
independent and identical to the other nodes. The motion of a
node is divided into multiple trips. At the beginning of a trip,
the node chooses a direction θuniformly between [0,2π], and
moves a distance of n−αin that direction, with a speed vn,
and the process repeats itself. Note that we are assuming a
complete warp-around of the unit square (see Fig. 3).
Remark 1. In the rest of the paper, we consider vn=
Θ(1/√n), as in [12]. This particular choice of node speed is
motivated by the fact that we keep the network area fixed and
let the number of nodes increase to infinity, which means that
the average neighborhood size scales as Θ(1/√n). In order
†Throughout this paper we ignore the issues pertaining to n1−2β, n2βnot
being perfect squares.
(7,7)
(0,1) (0,7)
(1,0)
(0,0)
(7,0)
Fig. 2. A numbering of cells on a 2-d torus.
to account for this “shrinking” of the neighborhood size, we
scale the node velocity as Θ(1/√n). Note that, alternatively,
one could scale the area of the network in proportion to n,
while keeping the velocity of the nodes fixed.
Next, we describe the discretized random direction models,
that we study in this paper. As discussed before, these mod-
els approximate the motion of the nodes under the random
direction models, and are simpler to analyze.
D. Discretized Random Direction Models
These models, like the random direction models, are
parametrized by a single parameter α, which takes values
between 0and 1/2. The unit square is divided into n2αsquares
of area 1/n2αeach (henceforth referred to as cells), resulting
in a discrete torus of size nα×nα.
Time is divided into slots of equal duration. At the beginning
of a time slot, each node jumps from its current cell to an
adjacent cell, chosen uniformly from within the set of adjacent
cells. The motion of a node during the time slot is as follows:
The node chooses a start and end point uniformly from within
the current cell. During the time slot, the node moves from
the start point to the end point. In order to keep the duration
of all time slots the same, the velocity of the node is chosen
to be inversely proportional to the distance between the start
and the end point.
In view of Remark 1, the duration of a time slot should
be Θ(n1/2−α). In order to be able to compare these models
with the Brownian motion model of [11], [14], we consider
σ2
n= 1/n, where σ2
nis the variance parameter of the Brownian
motion model, as defined in [11], [14]. Note that with the
above choice of σ2
nand vn, each node moves an average
distance of Θ(1/√n)in unit time, under all these models.
Also observe that under these settings, the discretized random
direction model with α= 1/2degenerates into the random
walk model, which is the discrete time version of the Brownian
motion model.
III. PRELIMINARIES: CRITICAL DELAY, 2-HOP DELAY,
SCHEDULING SCHEMES
In this section, we build the platform required for the study
of the critical delay in the subsequent sections. We first start
Fig. 3. The figure shows the motion of a node under the random direction
model.
by providing a rigorous definition of the critical delay and
motivating its study.
It is now well known that the per-node throughput capacity
of static ad hoc networks scales as O(1/√n), with the capacity
achieving scheme being the famous multihop relaying scheme.
Grossglauser and Tse [4] showed that a constant throughput
per-node can be achieved under any stationary and ergodic
mobility model which preserves the uniform distribution of
the nodes at all times. The delay performance of the capacity
achieving 2-hop relaying scheme of [4] has been studied in
many recent works [8], [9], [13], [11], [12], [15], and the 2-hop
relaying scheme has been shown to incur an average delay of
about Θ(n)under many different mobility models. Alternative
schemes that provide a throughput in between that of multi-
hop relaying and 2-hop relaying have also been studied in
the literature, under various mobility models. The objective
of these schemes is to provide better delay performance than
the 2-hop relaying scheme by compromising the throughput
capacity. An important quantity to study in this respect is what
we call the critical delay, which we next define:
Definition 1. Let Cbe the class of scheduling and relaying
schemes under consideration, and let Dc, λcbe the average
delay and the per-node throughput, respectively, under scheme
c∈C. The critical delay under the class of schemes C,
denoted by DC, is the minimum average delay that must be
tolerated under a given mobility model in order to achieve a
per-node capacity of ω(1/√n), that is,
DC= inf
{c∈C:λc=ω(1/√n)}Dc.(2)
Remark 2. Note that the requirement of ω(1/√n)capacity in
the above definition is to ensure that the asymptotic throughput
is above that of static ad hoc networks.
Remark 3. Note that each scheme cin the class of schemes
Cis actually sequence of schemes cn, where cnis the scheme
used with nnodes in the network. Similarly, λcand Dcare
also sequences rather than numbers. The infimum in Eq.(2)
should therefore be interpreted as a sequence of infimums,
one for each n.
Remark 4. From the above definition, it is clear that the
larger the class Cis, the smaller the critical delay would be.
Ideally, one would like the class Cto include all possible
scheduling schemes. In that case, if the delay that can be
tolerated is smaller than the critical delay for a given mobility
model, then one cannot exploit the node motion under that
mobility model to increase the throughput capacity (in order
sense) beyond its value under static conditions.
Henceforth, we will denote by 2-hop delay, the delay under
the 2-hop relaying scheme. Observe that the trade-off between
the delay and the capacity exists for delay values that are
greater than the critical delay, and smaller than the 2-hop delay.
Next, we introduce two key notions: first hitting time and the
first exit time, which will be used to study the 2-hop delay and
critical delay, respectively.
We start with some notation. Let Sdenote the unit square
centered at the origin, and let x, y be two points inside S. We
denote by [x]the set of points {x−1, x, x + 1}, and define
d([x],[y]) = min
xi∈[x],yj∈[y]d(xi, yj).
We denote by B(x, r)the set of points yinside Ssuch that
d([x],[y]) ≤r. Let Xt
idenote the position of the node iat
time t, under some mobility model. Note that the index tcould
either be continuous or discrete, depending on the mobility
model. We are now ready to define the notion of the first exit
time:
Definition 2 (First Exit Time). Let X0
i=x. The first exit
time of B(x, r), denoted by τr
E, is given by:
τr
E= inf{t≥0 : Xt
i/∈B(x, r)}.
It should be clear from the above definition that the statisti-
cal proprieties of the exit time do not depend on the choice of
the point xand node i. Also note that the notion of exit time
is well studied in the mathematics literature, under a variety
of contexts. Our interest in notion of exit time stems from the
fact that it has a close connection with the critical delay, which
will be exploited in the subsequent sections.
We will next define the first hitting time. In our case, we
find it more convinient to define it in discrete time. In order
to do so, we need some more notation. Let X(t)be a Markov
chain on state space S, with stationary distribution Π. We have
the following definition:
Definition 3 (First Hitting Time). The first hitting time for
the set of states Ais given by:
τA
H= inf{t≥0 : X(t)∈A},
with X(0) being distributed according to Π.
Again, we would like to point out that the notion of the
first hitting time has been widely studied in the mathematics
literature, under many different contexts. As will be discussed
in subsequent sections, the first hitting time has a close
connection with the 2-hop delay as well as the critical delay.
This connection has been exploited in several recent works
for estimating the 2-hop delay under various mobility models
(see, for example, [12], [14]).
Next, we recall the result concerning the first hitting time
for a single state in case of a 2-d torus of size √n×√n(for
a proof see, for example, [14]):
Lemma 1. Let Hdenote the first hitting time for a single state
on a 2-d torus of size √n×√n, then E{H}= Θ(nlog n).
The above result will be used quite often during the analysis
in subsequent sections.
We now define the class of scheduling schemes that we
will study in this paper. The only restriction we place on our
scheduling schemes is the following:
Assumption A:
•Only the source node is allowed to initiate a replication,
that is, the relay nodes holding a packet are not allowed
to initiate a replication.
Note that almost all scheduling schemes that have been consid-
ered in the literature satisfy Assumption A. To understand this,
we must elaborate on the notion of replication. By replication
we mean packet duplication, that is, creating redundant copies
of a packet. This is different from multihop relying where a
single copy of a packet exists in the network. It is important to
note that the notions of replication and relaying are different,
even though both involve forwarding packets to other relay
nodes. To understand this, suppose node idecides to replicate
a packet at node j; then node ican either transmit the
packet directly to node j; or use multihop relaying, where the
packet reaches node jthrough multiple intermediate nodes.
Note that, if asked by node i, the intermediate nodes may
also keep a copy of the packet with them, and in that case,
all of them are considered to receive the packet due to the
same replication decision initiated by node i. In this example,
although both node iand the other intermediate nodes forward
the packet to other nodes, their roles are different. Node iis
the one that initiates the replication, while the intermediate
nodes passively follow the instructions of node i. Thus, we
see that Assumption A only prohibits relay nodes to initiate
replication and, in particular, multihop relaying is allowed
under Assumption A.
We also allow immediate capture of the destination using
multihop relaying or long range wireless broadcast, at any time
during the replication process. Note that by capture we mean
the successful delivery of a packet to its destination node.
Although we allow for other less intuitive alternatives, in a
typical scheduling scheme 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 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 (since with more nodes holding the
packet, the likelihood of one of them capturing the destination
sooner is higher).
We use the word “immediate” to emphasize that capture can
be carried out (using multihop relaying or long range wireless
broadcast, if required) at a much faster time-scale than the
node mobility, and the same is true of the replication process
as well. The reason for this is that the packet transmission is
usually carried out at a much faster time scale than the node
mobility, and as a result the change in the position of a node
during a packet transmission is often negligible.
Note that almost all scheduling schemes that have been
studied in the literature satisfy Assumption A [4], [9], [8], [12],
[11], [10], [13], [15]. The reason for this, we believe, is that in
a distributed system, where nodes make replication decisions
and capture decisions without any knowledge of the decisions
at the other nodes, restricting the replication decisions to
the source node is a natural way to control the number
of copies of a packet. Note that higher redundancy implies
smaller throughput. The source node of a packet is in the best
position to control both the total number of replications of the
packet and the number of relay nodes getting a copy of the
packet with each replication. If the relay nodes were allowed
to initiate replication, then additional cooperation among the
relay nodes would likely be required in order to limit the
number of replicas of a packet. An interesting example of
this would be the scheme of Bansal and Liu [5], where the
relay nodes know the location of the static destination node,
and also have some knowledge of the future direction of other
relay nodes’ movement, based on which they can cooperate
to make selective and more efficient replication toward the
destination. Whether such a scheme can be devised for an ad
hoc network with mobile source-destination pairs is still an
open research challenge.
IV. CRITICAL DELAY AND 2-HOP DELAY UNDER HYBRID
RANDOM WALK MODELS
In this section, we study the critical delay and the 2-hop
delay under hybrid random walk models. We first study the
critical delay.
A. Critical Delay
Recall that the critical delay is the minimum delay that must
be tolerated in order to achieve a throughput of ω(1/√n).
Next, we develop lower bounds on the critical delay for the
hybrid random walk models. Observe that for β= 0 (which
corresponds to the i.i.d. mobility model) we have the trivial
lower bound of 1on the critical delay. Further, it has been
shown in [13] that a throughput of roughly Θ(1/n1/3) =
ω(1/√n)can be achieved with a constant average delay under
the i.i.d. mobility model, which implies that 1is also an
asymptotically tight bound (in order terms).
In the sequel, assume therefore that β > 0. The main idea is
to show that if the average delay is below a certain value, then
on average the packets travel a distance of Θ(1) using wireless
transmissions before getting delivered to their corresponding
destination nodes; and then to show that under the protocol
model, this results in a throughput of O(1/√n).
D’
S’
Disk A
Disk B
R
Replication
Capture
Fig. 4. The figure shows Disks A and B of radius 1/16 centered at
(−1/4,−1/4) and (1/4,1/4), respectively, that are used in the proof of
Lemma 3.
Recall the mobility model described in section II-B. Let
τr
E,β be the first exit time in case of a hybrid random walk
model with parameter β. We start by establishing the following
lower bound on τ1/8
E,β , that holds with probability approaching
1as n→ ∞. The proof is available in Appendix IX-A.
Lemma 2. For 0< β ≤1/2, we have
Pτ1/4
E,β ≤n2β
1024 log n≤4
n2.
Now we are ready to show that if the average delay is
smaller than fon2β
2048 log n, where fo= 1/2400, then, on average,
the packets must be relayed over Θ(1) distance.
Lemma 3. Consider a network with nnodes moving in
accordance with a hybrid random walk model with parameter
β > 0. If the average delay of the packets under a scheduling
scheme is smaller than fon2β
2048 log n, then there exists No<∞
such that for all n≥Nowe have that the packets must, on
average, be relayed over a distance no smaller than fo/10.
Proof: Consider disks A and B of radius 1/16, as shown
in Fig.4. Note that according to our assumption in section II-
A, the arrival process at each node is independent of the node
mobility process. For a given number of nodes n, let fndenote
the fraction of packets having their sources inside disk A and
the destinations inside disk B, at the time of arrival. Since the
nodes remain uniformly distributed within the subcells at all
times under our settings, we have that fn→(π(1
16 )2)2as n→
∞. Thus for all n≥No, where Nois chosen appropriately,
we have fn≥fo. Now since the average delay is smaller than
fon2β
2048 log n, the delay of at least of one-half of such packets must
be at most 2
fn·fon2β
2048 log n≤n2β
1024 log n,
for all n≥No. Consider one such packet; let its time of arrival
be t. Using Lemma 2 and the union bound, we have that with
a probability of at least
1−n·4
n2= 1 −4
n,
none of the nnodes in the network will exit the disk of radius
1/8centered around its initial position at time tbefore time t+
n2β
1024 log n. Let S′and D′be the positions of the source and the
destination, respectively, for which the distance between the
source and the destination at time tis minimized (see Fig.4).
Let the destination be captured by the relay node r, and let R
be its position when it receives the packet in consideration (see
Fig.4). Assuming that the total time spent in the replication
(which resulted in the relay rgetting a copy of the packet) and
capture is o(n2β/log n), the contribution of node movement
during the replication and capture phase in carrying the packet
toward the destination would be o(1), and can be ignored. Now
a simple geometrical argument shows that in order to reach
the destination, the packet must be relayed over a distance of
at least 1
√2−2·1
16 −2·1
8−1
8=d0>1
5,
where the term 1
√2−2·1
16 corresponds to the minimum
possible distance between the source and the destination at
time t, that is, the distance between the points S′and D′; the
term 2·1
8corresponds to the maximum distance the source and
the destination can possibly travel toward each other between
time ht, t +n2β
1024 log ni; and the term 1
8corresponds to the
maximum distance that the relay node can possibly move in
the direction of the destination after it receives the packet and
before time t+n2β
1024 log n.
By choosing Nolarge enough, we can ensure that d0(1 −
1/n)≥1/5for n≥No, and the average distance that the
packets must be relayed over in that case would be no smaller
than fo
2·1
5=fo
10,
proving the Lemma.
Remark 5. Note that in the above proof we assumed that
the total time spent in the replication and capture phase is
o(n2β/log n). This assumption is motivated by the fact that
in most scheduling schemes the replication and capture are
performed using a wireless broadcast or multihop relaying,
and are therefore carried out at a much faster time-scale than
the node mobility. Note also that a packet might be relayed
over several hops during either the replication or capture
phase, and for technical consistency, one might then need to
scale down the packet size in order to keep the total time spent
in replication and capture phases small enough. This kind of
packet size scaling has been used quite often in the literature
(see, for example, [10], [9], [13], [15]).
We now recall the following result from [14], which shows
that if the packets are on average relayed over Θ(1) distance,
then the throughput capacity must be O(1/√n).
Lemma 4. Suppose that there exists a constant c > 0,
independent of n, such that on average the packets are relayed
over a total distance no less than c, then λ(n) = O(1/√n).
Remark 6. In [14], the above result is proved under the
protocol model discussed before in section II-A. By following
the line of analysis used for proving Theorem 4.2in [3], it is
possible to establish the above result under the generalized
physical model (see [3]), which is more realistic than the
protocol model.
The following result is an easy consequence of Lemmas 3
and 4, and the definition of critical delay.
Proposition 1. Under the class of scheduling schemes satisfy-
ing Assumption A, the critical delay for a hybrid random walk
model with parameter β > 0scales as Ω(n2β/log n).
We now derive an upper bound on the critical delay. Note
that the delay under any scheme that can provide a throuhput
of ω(1/√n)is essentially an upper bound on the critical
delay. For β= 1/2(which corresponds to the random walk
mobility model) it is claimed in [9] that a simple modification
of the 2-hop relaying scheme of Grossglauser and Tse can
provide a throughput capacity of Θ(1), incurring a delay
of Θ(nlog n). This result immediately establishes an upper
bound of Θ(nlog n)on the critical delay for β= 1/2. In
what follows, we consider hybrid random walk model with
parameter β, for 0< β < 1/2, and show that the critical
delay for the model scales as O(n2βlog n).
The idea is to develop a scheduling and relaying scheme
that can provide a throughput of ω(1/√n), while incurring
a delay of O(n2βlog n). In this paper, we only provide the
main insight behind such a scheme, leaving out the detailed
analysis for our future work.
Consider a scheme in which each packet is replicated at a
single relay node, which delivers the packet to the destination
node once it is in the same cell as the destination node,
possibly using multihop transmission. Note that such a scheme
would require an appropriate scaling of the packet size to
ensure that the packet can be delivered to the destination within
one slot, i.e., before the relay and the destination can possibly
move into different cells. We now provide an approximate
analysis of the delay and the throughput for such a scheme.
In order to keep our discussion simple and insightful, we will
ignore possible delays due to queueing at the source or the
relay nodes. More precisely, we will assume that the delay of
the packet is the time it takes for the relay node to move into
the same cell as the destination node, starting from the time
it receives the packet from the source node.
Now consider a packet arrival at the source node. Note that
since the packet arrival process at each node is independent of
the node mobility process, the source node and the destination
node are equally likely to be in any of the cells, at the time of
the packet arrival. Thus the delay of the packet is of the same
order as the expected first hitting time of a single state, in case
of a random walk on a 2-d torus of size nβ×nβ, which, by
Lemma 1 is Θ(n2βlog n).
Now for estimating the throughput, we need to account
for the following factors: (i) the loss in throughput due to
multiple relayings of the same packet, (ii) the loss in through-
put due to the interference. Using the standard multihop
scheme, where each hop carries the packet over Θ(plog n/n)
distance‡, and noting that the packet travels a distance of
no more than Θ(1/nβ)using multihop transmissions, it fol-
lows that the number of times a packet must be relayed is
O(n1/2−β/√log n). The loss in throughput due to multiple
relayings is correspondingly O(n1/2−β/√log n). Now since
each transmission is carried over Θ(plog n/n)distance, it
follows easily using the protocol model that the nodes which
are within Θ(plog n/n)distance of the sender must be
kept quite, resulting in a loss of throughput by a factor of
Θ(log n). Thus the throughput of such a scheme would be
Ω(nβ−1/2/√log n) = ω(1/√n)for β > 0.
The above discussion shows that the critical delay scales as
O(n2βlog n)for a hybrid random walk model of parameter β.
Although, the above arguments are heuristic, they can easily be
made precise. However, in order to do so, one would need to
specify the details of the scheduling scheme, which is beyond
the scope of this paper.
Remark 7. By choosing the size of the capture neighborhood
appropriately (e.g., Θ(1/log n)), one can show that the criti-
cal delay is bounded by Θ(n2βlog log n).
B. 2-Hop Delay
In this section, we analyze the 2-hop delay under the hybrid
random walk models. Recall that the original 2-hop relaying
protocol of Grossglauser and Tse [4] allows only the nearest
neighbor transmissions. Subsequent works [8], [11], [9], [15],
[12] have considered a slightly different version of the protocol
in which the transmissions are carried out between nodes that
are either in the same cell (of size Θ(1/√n)×Θ(1/√n)),
or are within a distance of Θ(1/√n)from each other. Note
that these different versions of the protocol are roughly the
same, because when nnodes are distributed uniformly within
a unit square the nearest neighbor distance is on the order of
Θ(1/√n).
Next, we analyze the delay under the 2-hop relaying pro-
tocol, assuming that the transmissions are scheduled between
nodes that are in the same subcell. As in the analysis of critical
delay, we will ignore the queueing delays, postponing their
analysis to future work. Thus we would mainly be interested
in estimating the time it takes for the relay node and the
destination node to come within the same subcell, starting
from two randomly and uniformly chosen subcells in the
network. Let us denote this random time by T. In subsequent
analysis, we will say that two nodes are in a “meeting” if they
are currently inside the same cell, and will denote the time
between successive meetings as the inter-meeting time.
Observe that
T=τ1n2β−1+...+ (τ1+... +τi)1−n2β−1i−1n2β−1+...,
(3)
‡Note that this is the minimum possible order of the communication range
for ensuring almost sure connectivity (see [1]).
where τ1is the time required for the nodes to enter the
same cell, starting from their initial random and uniformly
distributed positions, henceforth denoted by first meeting time;
and τifor i≥2are the successive inter-meeting times.
Observe that n2β−1is the probability that the nodes choose
the same subcell inside a given cell. It is easy to see that
the mean first meeting time is of the order of the mean first
hitting time of a single state, in case of a ranodm walk on a
2-d torus of size nβ×nβ. Using Lemma 1, it follows that
E{τ1}= Θ(n2βlog n). Further, the mean inter-meeting times
are of the order of the mean first return time (for definition,
see, for example, [17, Chap 2, p. 2]) of a random walk on a
2-d torus of size nβ×nβ, which is well known to be n2β.
We therefore have E{τi}= Θ(n2β)for i≥2. Taking the
expectations on both sides of Eq. (3), and performing some
simple algebraic manipulations, we obtain
E{T}=E{τ1}+E{τ2}n1−2β
= Θ(n2βlog n) + Θ(n2β)n1−2β
Thus for β < 1/2, we have E{T}= Θ(n); and for β= 1/2,
we have E{T}= Θ(nlog n).
Remark 8. Note that our results for β= 0 and β= 1/2
are in agreement with the corresponding results for the i.i.d.
mobility model in [8] and the random walk model in [15].
Both these works also account for the queueing delays: In [8],
queueing delays at the source nodes as well as relay nodes
are considered, whereas, [15] considers queueing delays at
the relay nodes only. It is interesting to see that our simplified
analysis yields exact results (in order sense) for two extreme
choices of β, i.e., β= 0,1/2.
Remark 9. Observe that all hybrid random walk models
incur roughly Θ(n)delay under the 2-hop relaying scheme,
but their critical delays vary significantly. More precisely, as
βincreases the critical delay increases as well (roughly as
Θ(n2β)), shrinking the delay-capacity trade-off region. The
two extreme cases being: (i) the i.i.d. mobility model (i.e.,
β= 0), for which ω(1/√n)capacity can be achieved even
under a constant delay constraint; and (ii) the random walk
model (i.e., β= 1/2), for which the delay on the order of
Θ(n/ log n)or more must be tolerated in order to achieve a
capacity of ω(1/√n).
V. CRITICAL DELAY AND 2-HOP DELAY UNDER
DISCRETIZED RANDOM DIRECTION MODELS
In this section, we study the critical delay and the 2-hop
delay under discretized random direction models. As in the
previous section, we first study the critical delay.
A. Critical Delay
Recall that the discretized random direction models are
characterized by a single parameter αthat takes values be-
tween 0and 1/2. As in the previous section, we will first
derive a lower bound on the critical delay by lower bounding
the first exit time for a disk of radius 1/8. Let τ1/8
E,α denote the
first exit time for such a disk in case of discretized random
direction model with parameter α. Let the duration of a time
slot be Cn1/2−α. For α= 0, one trivially obtains a lower
bound of Θ(√n)on τ1/8
E,α . For α > 0, we have the following
result, the proof of which follows mutatis mutandis from the
proof of Lemma 2:
Lemma 5. For 0< α ≤1/2, we have
Pτ1/4
E,α ≤Cn1/2+α
1024 log n≤4
n2.
It is interesting to note that a similar result can also be
proved for random direction models (see Appendix IX-B):
Lemma 6. Let T1/8
E,α denote the first exit time of a disk of
radius 1/8for a random direction model with parameter α.
For 0< α ≤1/2, we have
PT1/8
E,α ≤Cn1/2+α
768 log n≤4
n2.
The following Lemma and Proposition can now be proved in
a similar fashion to Lemma 3 and Proposition 1, respectively.
Lemma 7. Consider a network with nnodes moving in
accordance with a discretized random direction model with
parameter α > 0. If the average delay of the packets under a
scheduling scheme is smaller than Cfon1/2+α
2048 log n, then there exists
No<∞such that for all n≥Nowe have that the packets
must on an average be relayed over a distance greater than
fo/10.
Proposition 2. Under the class of scheduling schemes sat-
isfying Assumption A, the critical delay for a discretized
random direction model with parameter α > 0scales as
Ω(nα+1/2/log n).
Remark 10. Analogs of the results in Lemma 7 and Proposi-
tion 2 can easily be proved for random direction models using
Lemma 6.
Remark 11. Note that for α= 0, using the lower bound of
Θ(√n)on τ1/8
E,α , and arguing as in the proof of Lemma 7, we
can easily establish a lower bound of Θ(√n)on the critical
delay. Moreover, the same reasoning shows that a lower bound
of Θ(√n)on critical delay also holds under the random way-
point mobility model. This result was earlier shown in [12],
but under a more restricted class of scheduling and relaying
schemes than in this paper.
Next, we will establish an upper bound on the critical
delay. Consider the scheme discussed before in section IV-
A. Recall that the packet is replicated to at most one relay
node, which delivers it to the destination node on entering the
same cell as the destination node. The approximate analysis
of the throughput and delay under such a scheme can be
carried out following the line of analysis in section IV-A,
and it is straightforward to show that the delay under such
a scheme will be Θ(n1/2+αlog n)and the throughput will be
Ω(nα−1/2/√log n) = ω(1/√n)for α > 0. Thus for α > 0,
the critical delay is bounded above by Θ(n1/2+αlog n).
Remark 12. One might think that by increasing the size of the
capture neighbrohood to 1/nγwhere 0< γ < α, one might be
able reduce the delay below Θ(n1/2+αlog n), while maintain-
ing a throughput of Ω(1/√n). This is, however, not possible.
In fact, it can be shown that with a capture neighborhood
of size r(n)the delay becomes Θ(n1/2+αlog (1/r(n))), and
the throughput becomes Θ(1/r(n)√nlog n). Thus choosing a
capture neighborhood of size 1/nγfor any γ > 0will not
change the order of the delay. Note also that by increasing
the size of the capture neighborhood to Θ(1/log n), one can
show an upper bound of Θ(n1/2+αlog log n)on the critical
delay.
Next, we consider α= 0. Note that for α= 0, the
discretized random direction model is almost the same as the
random way-point mobility model, with the only difference
being that the successive trips (moving between the chosen
pair of points) that a node makes under the discretized random
direction model are independent; whereas, there is some
dependency between successive trips in case of the random
way-point mobility model (since the next trip starts from the
point where the previous trip ends). The random way-point
mobility model has been analyzed§in [12]. In particular, a
protocol that allows one to trade-off throughput for the delay
has been developed in [12], and shown to achieve the following
delay-capacity trade-off:
D(n) = O(n/k(n) log n),and λ(n) = Ω(1/k(n) log n),
where D(n)is the average packet delay and the λ(n)is the
per-node throughput. Following the line of analysis in [12],
one can show that the same delay-capacity trade-off can also
be achieved under the discretized random direction model with
α= 0. Now by choosing k(n) = √n/a(n) log nwhere
a(n)→ ∞ as n→ ∞,(4)
it follows that the critical delay is O(a(n)√nlog2n)for all
a(n)satisfying condition (4). In particular, the critical delay
is o(nγ)for any γ > 1/2.
B. 2-Hop Delay
In this section, we analyze the 2-hop delay under the
discretized random direction models. We will assume that the
transmissions are scheduled between nodes that are within a
distance of 1/√nfrom each other, and will ignore the queue-
ing delays. Thus we would mainly be interested in estimating
the time it takes for the relay node and the destination node to
come within a distance of 1/sqrtn of each other, starting from
two randomly and uniformly chosen positions in the network.
Let us denote this random time by T. Arguing as in section
IV-B, it can be shown that E{T}= Θ(n1/2−α)Θ(E{τ1}+
E{τ2}/p), where τ1is the first meeting time; τ2is the inter-
meeting time; and pis the probability that two nodes will come
within a distance of 1/√nof each other any time during a
§Although [12] considers a slightly different version of the random way-
point mobility model on a sphere, the results in [12] can easily be extended
to a 2-d torus.
time slot, given that they are within the same cell in that time
slot. Note that the factor of Θ(n1/2−α)comes because the
duration of each time slot is now Θ(n1/2−α). As in section
IV-B, we have E{τ1}= Θ(n2αlog n)and E{τ2}= Θ(n2α).
Furthermore, it is easy to see that p= Θ(nα−1/2). Thus
E{T}= Θ(n1/2+αlog n) + Θ(n).
Hence we see that E{T}= Θ(nlog n)for α= 1/2, and Θ(n)
for 0≤α < 1/2.
Remark 13. Once again, we note that our results for α= 0
and α= 1/2are in agreement with the corresponding results
for the randon walk model in [15] and the random way-point
mobility model in [12]. Both these works also account for the
queueing delays: In [12], queueing delays at the source nodes
as well as relay nodes are considered, whereas, [15] considers
queueing delays at the relay nodes only. Again, we see that
our simplified analysis yields exact results (in order sense) for
two extreme choices of α, i.e., α= 0,1/2.
Remark 14. Observe that all discretized random direction
models incur roughly Θ(n)delay under the 2-hop relaying
scheme; however, their critical delays vary significantly. More
precisely, as αincreases the critical delay increases as well
(roughly as Θ(nα)), shrinking the delay-capacity trade-off
region. The two extreme cases being: (i) α= 0 (random way-
point mobility model), for which ω(1/√n)capacity incurring
delays of about Θ(√n); and (ii) α= 1/2(random walk
model), for which the delay on the order of Θ(n/ log n)or
more must be tolerated in order to achieve a capacity of
ω(1/√n).
VI. DISCUSSION
The main contribution of this paper is the definition and
study of the notion of critical delay, which provides us with
a platform to compare and contrast several existing mobility
models. The notion of critical delay is important as it provides
us a way of determining whether a particular form of node
mobility can be exploited to improve the throughput capacity
under a given delay constraint. We also showed that there
exists a strong connection between the notion of exit time and
critical delay, and used this connection to estimate the critical
delay under various mobility models.
The results obtained in the previous sections are summarized
in Fig.5. Clearly, the mobility models considered in the litera-
ture are in some sense extreme: they either exhibit the smallest
critical delays or the largest critical delays among all mobility
models having roughly the same 2-hop delay. Thus, on one
extreme, there is almost no delay-capacity trade-off under the
Brownian motion model and the random walk model, and,
on the other extreme, there is a smooth delay-capacity trade-
off for a wide range of delays under the random way-point
mobility model and the i.i.d. mobility model.
An interesting insight provided by our results is that the crit-
ical delay is inversely proportional to the characteristic path
length. By characteristic path length, we mean the distance
that a node travels without changing direction. (Recall that in
case of (discretized) random direction model with parameter
α, the characteristic path length is of the order of n−αand
the critical delay is roughly of the order of n1/2+α.) Thus
in terms of application support, a scenario where the nodes
move over long distances without changing directions (as in
the random way-point mobility model) is more desirable than
a scenario where nodes change directions over short distances
(as in the Brownian motion model). This is because the former
scenario provides more flexibility in terms of choosing the
point of operation on the delay-capacity trade-off curve, and
can therefore support a wider range of applications.
In a real world scenario, it is rather unlikely that the
(density) number of nodes in the network will have a strong
influence on the motion of the nodes. We therefore believe
that a mobility model like the random way-point model might
be more appropriate for determining the sclaing laws for large
mobile ad hoc networks, rather than a mobility model like the
Brownian motion model (random walk model). We therefore
expect that future mobile ad hoc networks would provide
network designers with ample flexibility in terms of choosing
the desired operational point on the delay-capacity trade-off
curve, and this oppurtunity must be fully exploited for optimal
operation of such networks, possibly using a cross-layer design
approach.
VII. CONCLUSION
We have studied the delay-capacity trade-offs in mobile
ad hoc networks. We introduced the meaningful notion of
critical delay to systematically study how much delay must
be tolerated for a given form of node mobility to result in an
improvement of the network capacity. The notion of critical
delay allowed us to look at various forms of node mobility
studied in the literature from a common perspective, and to
compare and contrast them.
We proposed two different classes of mobility models and
showed that they both exhibit critical delays that are in-
between that of the mobility models studied in the literature,
thus showing that the mobility models considered in the litera-
ture are rather extreme. More importantly, we showed that the
critical delay is inversely proportional to the characteristic path
length, which is the distance nodes travel without changing
directions. These results, among other things, provide a clear
understanding of why is it that the critical delay under the
Brownian motion model is larger than the critical delay under
the random way-point mobility model.
In a real world scenario, one would expect the number of
nodes or the density of nodes to have little, if any, influence on
the motion of nodes. Correspondingly, one would expect the
characteristic path length to have a rather weak depenence,
if at all, on the number of nodes or the node density. One
might therefore expect the critical delay in a real world
scenario to be close to Θ(√n), as in the case of the random
way-point mobility model. This result is optimistic, since it
suggests that the future mobile ad hoc networks would provide
network designers with ample flexibility in terms of choosing
the desired operational point on the delay-capacity trade-off
β
β
0 1/2
Critical Delay
n0.5
n
Discretized Random Direction Models
Hybrid Random Walk Models
Delay−Capacity Trade−off Regions
for a particular value of
Fig. 5. The figure shows the scaling of critical delay in case of the hybrid
random walk models and discretized random direction models.
curve; an oppurtunity that must be fully exploited for optimal
operation of such networks, possibly using a cross-layer design
approach.
VIII. ACKNOWLEDGEMENT
We wish to thank Xiaojun Lin for many valuable discussions
throughout the course of this work.
IX. APPENDIX
In this section, we provide proofs for Lemmas 2 and 6.
We start with the following simple result that is a version of
Hoeffding’s Inequality (see, for example, [18, Chapter 3, pg.
120]).
Lemma 8. Let X1, X2, ..., Xnbe i.i.d. random variables
taking values in [−l, l]for 0< l < ∞, and suppose E{Xi}=
0for all i. Let Sn=Pn
i=1 Xiand σ2
Snbe the variance of
Sn. Then
P(Sn≥µσSn)≤e−µ2/4,
for all 0≤µ≤2σSn/l.
We are now ready to prove Lemmas 2 and 6.
A. Proof of Lemma 2
Recall the hybrid random walk model of section II-B, and
the definition of τ1/8
E,β , given in section III. As discussed in
section III, the statistical properties of the first exit time do
not depend on the choice of y. So let ybe the origin, that is,
the point (0,0). Let (x0, y0)be the cell containing the origin.
Also, let (xt, yt)be the cell in which node ilies at time t.
Further, let
τ+
x,inf t≥0 : (xt−x0)≥nβ
16 ;
τ−
x,inf t≥0 : (xt−x0)≤ −nβ
16 ;
and τ+
y,τ−
ybe similarly defined with yt, y0in place of xtand
x0, respectively. Observe that
P(τ1/8
E,β ≤m)≤P(τ+
x≤mor τ−
x≤mor τ+
y≤mor τ−
y≤m)
for m≥0. Using the union bound and appealing to the
symmetry of node motion, we obtain
P(τ1/8
E,β < m)≤4P(τ+
x< m).
Now observe that before time τ1/8
E,β ,xthas the following
form:
xt=x0+
t
X
i=1
si,
where siare i.i.d. random variables taking values in {−1,0,1}
with probabilities {1/4,1/2,1/4}, respectively. Although, xt
is not a simple random walk, it is clear due to its symmetry
that the reflection principle for 1-d random walk holds in case
of xtas well, and we have
Pτ+
x≤k= 2Px⌊k⌋−x0≥nβ
16 .(5)
for k≥0, where ⌊·⌋ denotes the greatest integer function. Now
since each sihas mean 0and variance 1/2, a straightforward
application of Lemma 8 gives:
Pxt−x0≥nβ
16 ≤e
−n2β
512t,(6)
for t≥nβ/16. Substituting k=n2β
1024 log nin Eq. (5), and
combining with Eq. (6), we obtain
Pτ+
x≤n2β
1024 log n≤e−2 log n=1
n2,
and the result follows by noting that
P(τ1/8
E,β ≤m)≤4P(τ+
x≤m)
for m≥0.
B. Proof of Lemma 6
Recall the random direction model of section II-B, and the
definition of τ1/8
E,α , given in section III. Arguing as in the
previous proof, it suffices to consider y= (0,0). Let (xt, yt)
be the position of the node iafter ttrips. Let
τx,inf{t≥0 : |xt| ≥ 1/8√2},
and
τy,inf{t≥0 : |yt| ≥ 1/8√2}.
It is then clear that
P(τ1/8
E,α ≤m)≤P(τx≤mor τy≤m).
Appealing to the symmetry of the node motion and using the
union bound, we obtain
P(τ1/8
E,α ≤m)≤2P(τx≤m).
Let skbe the x-coordinate of the nodes’ position immediately
after completing the kth trip. Before time τ1/8
E,α ,skhas the
simple form:
sk=
k
X
i=1
zi,
where ziare i.i.d. random variables taking values in
[−n−α, n−α]. Note also that each zihas mean zero and
variance n−2α/2. Using Lemma 8, we have
P(sk≥1/8√2) ≤e−n2α
256k,
for k≥nα/8√2. Using the symmetry of the node motion
once again, we have
P(|sk| ≥ 1/8√2) ≤2e−n2α
256k.
Noting that the duration of each trip is Cn1/2−α, it follows
that
Pτx≤k·Cn1/2−α=P∪k
i=1|si| ≥ 1/8√2
≤
k
X
i=⌈nα/8√2⌉
2e−n2α
256i
≤k·2e−n2α
256k,
where ⌈n2α/8√2⌉denotes the smallest integer greater than
n2α/8√2. Now since n2α≤nfor α≤1/2, we have
Pτx≤Cn2α
768 log n·n1/2−α≤n·2e−3 log n=2
n2.
Thus
Pτ1/8
E,α ≤Cnα+1/2
768 log n≤4
n2,
as claimed.
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