2924 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VO L . 6, NO. 8, AUGUST 2007
REACA: An Efficient Protocol Architecture for
Large Scale Sensor Networks
Zhi Quan, Ananth Subramanian, Member, IEEE , and Ali H. Sayed, Fellow, IEEE
Abstract— The emergence of wireless sensor networks has
imposed many challenges on network design such as severe
energy constraints, limited bandwidth and computing capa-
bilities. This kind of networks necessitates network protocol
architectures that are robust, energy-efficient, scalable, an d easy
for deployment. This paper proposes a robust energy-aware
clustering architecture ( REACA) for large-scale wireless sensor
networks. We analyze the performance of the REACA network in
terms of quali ty-of-service, asymptotic throughput capacity, and
power consumption. In particular, we study h ow the throughput
capacity scales with the number of nodes and the number of
clusters. We show that by exploiting traffic locality, clustering
can achieve performance improvement both in capacity and
in power consumption over general-purpose ad hoc networks.
We also explore the fundamental trade-off b etween throughput
capacity and power consumption for single-hop and multi-
hop routing schemes in cluster-based networks. The protocol
architecture and performance analysis developed in this paper
provide useful insights for practical design and depl oyment of
large-scale wireless sen sor network.
Index Terms— Clustering, multi-hop routing, performance
analysis, throughput capacity, wireless sensor networks.
I. INTRODUCTION
W
IRELESS sen sor networks (WSNs) consist of spa-
tially d istributed sensor devices with sensing, wireless
communications, and computation capabilities. These wireless
networks have broad application s, e.g., environment monitor-
ing, target tracking, and surveillance. Unlike mobile ad hoc
networks (MANETs), WSNs are usually application -specific.
The unique char acteristics of WSNs such as limited bandwidth
and comp uting capacity, and severe energy constraints, ma ke
their design more challenging. One essential issue in the
design of WSNs is how to use bandwidth and energy resources
efficiently while pro longing the system lifetime. In this paper,
we examine the following important design considerations.
A. Throughput Capacity
Since WSNs may consist of a large number of sensor
nodes, protocols should provide good throughput capacity
as th e number of sensor nodes increases. In recent work
[1], it has been shown that the per node throughput capa c-
ity of a gene ral-purpose non-clustered wireless network
1
is
Manuscript received December 8, 2005; revised April 3, 2006; accepted
September 6, 2006. The associate editor coordinating the review of this paper
and approving it for publication was H.-H. Chen. This work was partially
supported by NSF grants ECS-0401188 and ECS-0601266.
Z. Quan and A. H. Sayed are with the Department of Electrical Engineer-
ing, University of California, Los Angeles, CA 90095-1594 USA (e-mail:
quan@ee.ucla.edu; sayed@ee.ucla.edu).
A. Subramanian is with the Institute for Infocomm Research, A-STAR,
Singapore (e-mail: msananth@ieee.org).
Digital Object Identifier 10.1109/TWC.2007.05964.
1
The notation y = Θ(f(N)) is used to signify that there exist positive
constants κ
1
and κ
2
such that κ
1
f(N ) ≤ y ≤ κ
2
f(N ).
Θ
!
R/
√
N
t
log N
t
"
, where R is the common transmission rate
of each node and N
t
is the total number of nodes in the net-
work. The result implies that the per node throughput capacity
approaches zero as the network size increases. Therefo re, it
is preferable to cluster together no des that are geographically
close and mostly communicate with each other.
B. Ene rgy Conserv ation and Awareness
Moreover, since the sensor nodes are battery operated, en-
ergy con se rvation is extre mely important. I n ord er to maximize
the system lifetime, protocols should alleviate the hot spot
problem in routing and evenly distribute the energy load
among all the node s, so th at there are no overly-u sed nodes
that will run ou t of ene rgy before the others.
C. Robustness
Still, it is quite possible that some nodes will fail or be
blocked due to lack of energy, physical damage, or environ-
mental interference. In the proposed protocol, the failure of
some sensor nodes will not prevent th e entire network from
operating.
In this paper, w e develop a robust energy-aware cluster-
ing architecture (REACA) to meet the design requirements
of WSNs. This ar chitecture supports d ata aggregation and
enables access to informatio n of interest from data collected
by spatially d istributed se nsor nodes. Applic ations include th e
average temperature of a field, an anomaly in a surveillance
network, and the location of a particular event, etc. We analyz e
the performance of the REACA network in terms of quality-of -
service (b locking p robability), throughput capacity, and power
consumpti on, and show how these performance measures scale
with the size of the network. Our results show that if the
number of clusters is medium or large, the REACA network
can achieve better performance than a general-purpose ad hoc
wireless network.
This paper is organized as follows. In Section II, we
review some re la te d work. The REACA structure is described
in Se ction III. Throughout Sectio ns IV-VI, we present the
performance analysis. Section VII concludes this paper with
discussion on some future research.
II. R
ELATED WORK
The developme nt of low-energy protocols for WSNs has
attracted attention in rec ent years [2] [3]. It has been shown in
[4], for instance, that clustering enable s ban dwidth reuse and
scalability, and thus improves system capacity. In [5 ], a linked
cluster architecture was proposed and in [6] three effective
solutions for jo int clustering and power control are de scribed.
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" 2007 IE EE
QUAN et al.: REACA: AN EFFICIENT PROTOCOL ARCHITEC TURE FOR LARGE SCALE SENSOR NETWORKS 2925
Moreover, in [7], a joint rate and power control scheme is
presented that achieves a desired level of signal-to-interference
and noise ratio (SINR) at the cluster head node.
Likewise, e nergy-aware routing protocols [8]–[10] have
been developed to p rolong the network lifetime. The basic
strategy of these protocols is to select routes based on the
energy at each n ode on the route. Recently, a low-energy adap-
tive clustering hierarchy (LEACH) protocol for application-
specific WSNs was developed in [11]. LEACH achieves this
goal by randomly selecting a few sen sor nodes as cluster heads
so as to evenly distribute the energy load among nodes in the
entire network . Although LEACH can achieve a long system
lifetime, the dynamic clustering brings in pro tocol overhead
and may result in isolated nodes. Also, LEACH requir es that
transmission with in the cluster be completed through a single
hop, which is not energy efficient if the two no des are located
far away from each other.
In this paper, we develop an alternative ar chitecture
(REACA) to address these issues. This protocol architecture
differs from LEACH in the following aspects. First, REACA
uses a predetermined clustered structure and applies energy-
awareness by selecting a node with most ene rgy as the master
node (or cluster head) for each cluster. This circumvents the
unnecessary protocol overhead and possible isolated nodes in
the network. Second, to achieve energy efficiency, REACA
adopts multi-hop routing when the sour ce and destination
nodes are far apart. In a sense, REACA may be co nsidered as
a generalization of the LEACH protocol. More importantly,
this paper provides asymptotic analysis for scaling network
performance in terms of the bloc king probability, capacity,
and power consumption.
III. REACA N
ETWORK
We consider a space covered by M predetermined geo-
graphical clusters
2
as shown in Fig. 1. Each cluster contains N
terminal nodes and one master node, and thus, the total number
of nodes in the network is given by N
t
= (N + 1) × M.
A m aster node pe rforms important ta sks, such as necessary
signal processing, sending data packets to a base station (BS),
and networking information management. To maximize the
network lifetime, we let each node of a cluster take its turn to
serve as the master. Altho ugh the master node is more power-
intensive than terminal nodes, the f raction of time it functions
as a master node is only about 1/(N + 1).
The protocol opera tion is divided into cycles [12]. As shown
in Fig. 2, each cycle begins with a setup phase when the master
nodes are determined for each cluster. Following the setup
phase is the transmission phase, during which the terminal
nodes send data to their own master nodes.
A. Master Selecti on
The strategy of ene rgy-awareness is applied to maximize
the system lifetime. During the setup phase, candidates of
the master node broadcast a message through an exclusive
2
The formation of clusters depends on specific applications. For energy
and bandwidth efficiency, we cluster nodes that are close to and have strong
correlation with each other. Other clustering approaches appear in [4], [6],
[11] and the references therein.
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0n$ra12/3#$er 4-n5
Fig. 1. A generic model for REACA networks.
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+#
Fig. 2. The protocol operation shown in a time scale. Once a master node is
chosen during the setup phase, packets are transmitted from terminal nodes
to the master node during the transmission phase. In general, the duration of
transmission phases is much longer than that of setup phases, i.e., T
t
# T
s
.
frequency channel using the CSMA/CA protoc ol. The message
indicates the available ene rgy a nd the node’s unique id entifier.
Therefore , each node can automatically determine the master
node without a centralized decision maker.
B. The Routes of Packets
In WSNs with data aggregation c apability, the sensor nodes
send processed information instead of original observations to
the base station or fusion center. A simple example of such an
application is a WSN that measures the average t emperature
of a field. In this application, each node individually measures
its surrounding temperature and sends the measureme nt to
the master node. The master node takes the average of
measureme nts over all nodes within the cluster and over a
period of time, i.e., T
m
cycles, and then sends the result to
the base station for further processing.
Without loss of generality, the intra-cl uster traffic amounts
to the majority o f the overall n etwork traffic. According to
the available energy on each node, routes of packets are es-
tablished either through a single hop or through multiple hops
via relay nodes with in the clu ster. The single-hop tran sm ission
is similar to LEACH, in which the terminal node sends data
packets directly to the destination node. Nevertheless, WSNs
are usually energy constrained and prefer multi-hop routing.
To secure efficiency, the routes should be over nearly straight-
line paths. A n example is illustrated in Fig. 3. At time cycle t,
packets generated from the source node are forwarded to the
master node if the master node is in its route to the final
destination; otherwise, the packets wait in the buffer until
the master node appears in any one of the possible routes.
At time cycle t + 1, the previous master node transfers the
relayed packets together with its own generated traffic to t h e
current master node if the current master node is in the route
to the final destination. If the de stination node becomes the
2926 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 8, AUGUST 2007
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!a#$er ()*e
=e#$-na$-)n
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Fig. 3. Example of a multi-hop route within the cluster. The cluster is divided
into many cells, each of them contains at least one node.
master at time t + 3, the p ackets will then be forwarded to
the destination. Thus, a successful multi-h op transm issio n may
take many cycles.
The master nodes process information collected from the
terminal nodes and then send the result to the BS. Th e
communication between the m aster and the BS uses a different
channel from intra-cluster communications, a nd this amount
of traffic will be negligible compared with the intra-cluster
traffic when T
m
is large. Above and thro ughout this paper,
we a ssume that each node has enough power to transmit data
packets to the BS and has a large buffer for relayed traffic.
C. Transmission Model
Consider N + 1 nodes independently and u niformly dis-
tributed in a cluster A with an area of A. When node i
communicates with its master node , it suffers interference
from the nodes in other clusters using the same c hannel and
from background noise. Let T be the set of nodes from
different clusters simultaneously tr ansmitting over the channel.
At time t, the transmission from a terminal node of cluster i
can be successfully received by it s master node if the SINR
at the master node satisfies
P
i
(t)G
ii
(t)
#
j∈T ,j"=i
P
j
(t)G
ji
(t) + σ
2
i
≥ γ (1)
where G
ji
denotes the channel gain from the j-th transmitting
node to the master no de of cluster i, σ
2
i
is thermal noise at the
receiver of the master node o f cluster i, P
i
is the transmission
power of node i, and γ is a certain thr eshold.
In a fading and shadowing channel, the c hannel gain is
modelled as
G
ji
= S
0
10
η/10
d
β
ji
(2)
where d
ji
denotes the distance between the nodes, S
0
is a
function of the carrier frequency, β denotes the path loss
exponent, and η is a zero mean Gaussian random variable
with variance σ
2
η
(i.e., 10
η/10
represents the shadowing factor
with a lognormal distribution). In practice, the values of β
and σ
η
depend on the physical environment and usually have
2 < β < 6 and 6 < σ
η
< 12.
D. Me d i a Ac cess Control (MAC)
For specific applications, the cluster needs several simul-
taneous transmissions in each cycle. We adopt time division
multiple acc e s s (TDMA) to transmit p a c kets for bandwidth
and energy-efficiency con sid erations. The master node divides
the transmission time into Q time slots and selects at most Q
terminal nodes to transmit p ackets. This scheduling informa-
tion is sent to nodes in the cluster such that no collision will
happen am ong transmitting nodes and the non -transmitting
nodes can switch to a sleeping state for energy conservation.
Throughout this paper, the analysis is carried out based on
an asymptotic regime, i.e., with a probability approaching one
as N → ∞, denoted by “with high proba bility (w.h.p.)”.
IV. B
LOCKING PROBABILITY
With the protocol described above, we define the block ing
probability in a cluster as Pr(Z
N
> Q), where Z
N
is the
number of nodes that simultaneously express a desire to
connec t to the master node. In this section, we will study the
blocking p robability in two scenarios: α-prioritized networks
[13] and distance-based prob abilistic connections.
A. α-Prioritized Netwo rks
Assuming that no two nodes have the same probability to
connect to the master node a t any time, t he definition of the
α-prioritized network is given as follows.
Definition 1 (Ordered Chain
): An ord ered chain is a set of
real number s where its i-th element is less than its j-th element
if i > j (i.e., it is a set of strictly decrea sin g real nu mbers).
Definition 2 (Prioritized Cluster
): A prioritized clu ster is a
cluster in which the set of all probabilities that a termin al-
master pair becomes active forms an ordered chain.
Definition 3 (α-Prioritized Cluster
): An α-prioritized c lus-
ter is a prioritized cluster in which the set of all probabilities
that a terminal-master node pair bec omes active is uniformly
bounded above by a geometric series {α
k
, k ≥ 0} for some
known and fixed real number, 0 < α < 1.
The α-prioritized networks for m a wide c lass of networks
considering th e fact that many networks have nodes tha t
are widely and uniformly distributed and with a minimum
distance between any two nodes. The following result gives
an expression for how Q should scale with N if we desire to
maintain a fixed bound on the block ing probability, say 1/ν,
for any N.
Propo sition 1 (Time S lots and Blocking Probability
): For
an α-pr ioritized cluster with a master node and N term inal
nodes, the number of time slots sufficient to ensure a
maximum blocking probability of 1/ν (with ν ≥ 1) is
Q =
√
2N ln ν +
1 − α
N+1
1 − α
(3)
!
This implies that Q should scale with O(
√
N) to en-
sure a maximum blocking probab ility of 1/ν. Note that if
√
2N ln ν +
1−α
N +1
1−α
> N , then the blocking pr obability of
1/ν is not achievable. A r elated question of interest is how
Z
N
scales as N → ∞.
Proposition 2 (Connection Requests
): For an α-prio ritized
cluster with arbitrarily large N , there exists an N
0
dependent
QUAN et al.: REACA: AN EFFICIENT PROTOCOL ARCHITEC TURE FOR LARGE SCALE SENSOR NETWORKS 2927
on α such that for all N > N
0
, the number of nodes that
express a desire to conn ect to the m aster node in a cluster can
be bounded above w.h.p. by
Z
N
≤ min
$
N, N
%
2 ln(1/α) +
1
1 − α
&
(4)
!
For exam ple, let α = 0.95 for a cluster. Then there exists
an N
o
such that if the cluster has N > N
o
nodes, then the
number of nodes that try to connect to the m aster node is
bounded above by 0.323N + 20 w.h.p. The two p ropositions
above are established in the sequel.
Denote by B
i
the event that node i tries to connect
to the master node. Without loss of generality, we assume
that {B
i
}
N
i=1
are identically and independently distributed
(i.i.d.). Let B
N
be the sigma algebra formed by the events
{B
1
, B
2
, . . . , B
N
}. Then we have Z
N
=
#
N
i=1
I(B
i
), where
I(·) is the indic ator function. If there are k termina l no des
in the cluster, then Z
k
=
#
k
i=1
I(B
i
). Recall the following
lemma.
Lemma 1 (Azuma’s Inequality [14]
): Let {Y
0
, Y
1
, Y
2
, . . .}
be a martingale sequence such that for each k, | Y
k
−Y
k−1
|≤
c
k
, where c
k
depends on k. Then, for all k ≥ 1 and for any
µ > 0 ,
Pr(Y
k
≥ µ) ≤ exp
'
−
µ
2
2
#
k
j=1
c
2
j
(
(5)
!
Motivated by the discussion in [15], w e in troduce a mar-
tingale seque nce Y
k
and utilize the above lemma to obtain a
bound on Z
N
. Let Y
k
= Z
k
−
#
k
i=1
Pr(B
i
) with Y
0
= 1. It
can be shown tha t Y
k
is a martingale because
E[Y
k+1
|Y
k
] = E[Y
k+1
|B
k
]
= E
''
k+1
)
i=1
I(B
i
) −
k+1
)
i=1
Pr(B
i
)
(
*
*
*
*
*
B
k
(
= Y
k
+ E[I(B
k+1
|B
k
)] − E {E[I(B
k+1
)]|B
k
]}
= Y
k
Moreover, it can be shown that |Y
k
− Y
k−1
| ≤ 1. Now
applying Azuma’s inequality with µ =
√
2k ln ν, we get
Pr
+
Y
k
≥
√
2k ln ν
,
≤
1
ν
, k ≥ 1 (6)
or equivalently,
Pr
'
k
)
i=1
I(B
i
) −
k
)
i=1
Pr(B
i
) ≥
√
2k ln ν
(
≤
1
ν
, k ≥ 1 (7)
Noting that the cluster is an α-prioritized cluster, and using
the fact that {B
i
} are independent of each oth er, we obtain
k
)
i=1
Pr(B
i
) ≤
1 − α
k+1
1 − α
(8)
Substituting (8) into (7), we have
Pr
-
Z
k
≥
√
2k ln ν +
1 − α
k+1
1 − α
.
≤
1
ν
(9)
Since there are N terminal nodes in a cluster, Propositi on 1
is established.
Consider again A zuma’s inequ ality but choose µ =
k
%
2 ln(1/α), then
Pr
+
Y
k
≥ k
%
2 ln(1/α)
,
≤ α
k
, k ≥ 1 (10)
Summing the above inequality over k, we get
∞
)
k=0
Pr
+
Y
k
≥ k
%
2 ln(1/α)
,
≤
1
1 − α
< ∞ (11)
From the Borel Cantelli Lemma [15], we conclude that th e
event {Y
k
≥ k
%
2 ln(1/α)} cannot occur infinitely often.
Thus, for sufficiently large k, say k ≥ N
o
, we have Y
k
≤
k
%
2 ln(1/α) w.h.p. Hence, when the number of terminals in
a cluster is N , we get
Z
N
≤ N
%
2 ln(1/α) +
1
1 − α
w.h.p. (12)
which leads to Proposition 2.
B. Distance- Based Probabili stic Connection
We now consider a probabilistic connection scheme that
is based on th e distance between nodes. Assuming the master
node knows the locations of all terminal nodes, the probability
that node i succ essfully connects to the master node can be
modelled in terms of its distance from the master node as
Pr(B
i
) = F (d
i
) =
-
d
0
d
i
.
1/δ
, d
0
≤ d
i
≤ d
M
(13)
where d
0
and d
M
are respectively the minimum and maximum
distances
3
between the transmitting node and the master node,
and δ is a user defined parameter (0.5 < δ < 1). d
i
is the
distance between the terminal node and its intended master
node.
Denote the position of the master node at time t by X(t).
Considering the energy requirement f or many importan t tasks,
it is reasonable to assume that {X(t)}is stationary and ergodic
with uniform distribution on the disc. Since the nodes are i.i . d.
with a uniform distribution, conditional on X(t) = x, the
cumulative proba bility density fu nction of d
i
is given by
Pr (d
i
≤ r|X(t) = x) = c(x)
r
2
− d
2
0
d
2
M
− d
2
0
, d
0
≤ r ≤ d
M
where c(x) (0 < c(x) ≤ 1) is a constant factor dependent on
x and is used to account for edge effects. If the master node
is near the periphery of the cluster, then c(x) < 1; otherwise,
c(x) = 1. Differentiating the above equation, the prob a bility
distribution function (p.d.f.) o f d
i
follows
f
d
i
(r|X(t) = x) =
2rc (x)
d
2
M
− d
2
0
, d
0
≤ r ≤ d
M
(14)
Propo sition 3 (Number of Time Slots
): In the distance-
based probabilistic connection scen ario defined by (13),
Q should scale with O(N) in or der to ensure a constant
blocking probability
4
.
3
For the requirement of network connectivity [16], we usually have d
0
≥
κ
A log N
N
, which ensures that no node in the cluster is isolated w.h.p. Note
that κ is a certain constant.
4
The notation y = O(g(N )) denotes that there exits a constant κ such
that lim
N→∞
y
g(N )
≤ κ.
2928 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 8, AUGUST 2007
Proof: When Z
N
> Q, the master node chooses Q
closest nodes for tr ansmission. Since Z
N
takes nonnegative
values, the blocking probability can be bound ed above as
Pr(Z
N
> Q) ≤
E(Z
N
)
Q
=
E
+
#
N
i=1
I(B
i
)
,
Q
(15)
where (15) follows from the Markov inequality [17][18].
Using the fact t hat {B
i
}
N
i=1
are inde pendently and identically
distributed, we have
E
'
N
)
i=1
I(B
i
)
(
= NE (I(B
1
))
= N
/
d
M
d
0
F (r)f
d
i
(r|X(t) = x) dr
= O(N ) (16)
Combining (15) and (16) yields Proposition 3.
In this network scenario, in order to ensure a constant
blocking probability 1/ν, Q should scale with O(N). This
is beca use the connection probability of a terminal node
is completely determined by its distance from the intended
master node.
V. T
HROUGHPUT CAPACITY
Without loss of generali ty, the per node throughput is de-
fined as the time average of the number of bits per second that
can be transmitted f rom every source node to its destinatio n.
Specifically, we adopt the f ollowing asymptotic notion as
defined in [1] and [1 9].
Definiti on 4 (Fea sible Per Node Throughput
): Let M
i
(t)
denote the amount of infor mation originated fr om node i to
its destinations at time t. Given the random locations of the
source and destination nodes, we shall say a long-term pe r
node throughput of λ(N) bits per second is feasible if there
is a spatial and temporal scheduling policy π such that
lim inf
T →∞
1
T
T
)
t=1
M
i
(t) ≥ λ(N) (17)
For each transmi ssion, we assume that the transceiver is
able to adap tively control the transmitting power level so
that the SINR can maintain a value of at lea st γ. In the
capacity analysis below, we limit our attention to intra-c luster
communications.
A. Sing le-Hop REACA
In the single-hop REACA network, terminal nodes send
data packets to their master node. Since each c luster can
obtain a common transmission rate of R and there are M
simultaneous tra nsmissions at any moment, the aggregate
transmission rate is M R. This rate is shared by the N
t
nodes
in the network, and thus, the per node th roughput capacity is
given as Θ (MR/N
t
).
B. Multi-Ho p REACA
In the following, we analyze the multi-hop REACA net-
work.
1) Packet Routing: We first investigate the routing behavior
of packets within a cluster. Fig. 3 shows an example of routing
a packet from a source node to the destination. The packets
travel almost in the same direction toward the destination at
each step. To d erive the number of hops for each packet, we
partition the cluster into many cells with an equal area D, as
shown in Fig. 3. Before proceeding , we need the following
lemma.
Lemma 2 (Cluster Partitioning
): Consider a cluster that is
partitione d into many cells with an equal area D. The re exits a
deterministic positive constant µ, such that if D ≥ µ log N/N,
then every cell contains at least one node w.h.p.
Proof: Consider N + 1 nodes i.i.d. in a cluster A of
area A. Any cell D is a subset of A, i.e., D ⊂ A. Recall the
coverage result given in [20] . When N is large enough, the
number of nod es in a given area D is Poisson distributed with
mean ρD, where ρ = (N + 1)/A is the node density. That is,
the probability that a cell D contains k nodes is
Pr(There are k nodes located in D) =
(ρD)
k
k!
e
−ρD
Given a cell D, the probability that it contains at least one
node is given by
Pr(k ≥ 1|D) = 1 − Pr(k = 0|D) = 1 − e
−ρD
Let D = µ log N/N, then
Pr(k ≥ 1|D) ≥ 1 − e
−µ log N/A
−→ 1 as N −→ ∞
This completes the proof.
Corollary 1 (Cluster Coverage): As the density of nodes
ρ = (N + 1)/A increases, the cluster is fully covered w.h.p.
!
The above corollary says that any non-zero area around the
final destination node can be reached by a packet w.h.p. when
the density of no des is large e nough.
Consider eac h of the N terminal nodes in a cluster gener-
ating M
i
(t) bits to their destinations at time t. For these N
S-D pairs, the following theorem estimates how many routes
will pass through a certain node in the cluster.
Theorem 1 (Numbe r of Routes Through a Node
): At any
time, in a cluster with N terminal nodes, the number of
genera ted routes that will pass throu gh a certain node is
Θ
+
%
N/ log N
,
.
Proof: We first partition the cluster into many cells with
a common area D ≥ µ log N/N so that each cell contains
at least one node w.h.p. Consider an S-D pair i at any time
t. The number of hops needed fo r the packet to move from
the source to the destination is denoted by h
i
. The distanc e
between the source and destinatio n nodes of S-D pair i is l
i
.
Since the routes are nearly straight lines, the number o f hops
for S-D pair i can be written as h
i
= Θ
+
l
i
/
√
D
,
. For any
cell j, we define a Bernoulli ran dom variable Y
ij
as
Y
ij
=
$
1 if the route of S − D pair i intersects cell j
0 otherwise
where Pr(Y
ij
= 1) = 1 − Pr(Y
ij
= 0) = p
ij
. Then Y
j
=
#
N
i=1
Y
ij
is the total number of routes t hat int ersect cell j. Its
