On Multihop Distances in Wireless Sensor Networks with Random Node Locations

540 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 4, APRIL 2010

On Multihop Distances in Wireless Sensor
Networks with Random Node Locations
Serdar Vural, Member, IEEE, and Eylem Ekici, Member, IEEE

Abstract—Location and intersensor distance estimations are important functions for the operation of wireless sensor networks,
especially when protocols can benefit from the distance information prior to network deployment. The maximum multihop distance that
can be covered in a given number of hops in a sensor network is one such parameter related with coverage area, delay, and minimal
multihop transmission energy consumption estimations. In randomly deployed sensor networks, intersensor distances are random
variables. Hence, their evaluations require probabilistic methods, and distance models should involve investigation of distance
distribution functions. Current literature on analytical modeling of the maximum distance distribution is limited to 1D networks using the
Gaussian pdf. However, determination of the maximum multihop distance distribution in 2D networks is a quite complex problem.
Furthermore, distance distributions in 2D networks are not accurately modeled by the Gaussian pdf. Hence, we propose a greedy
method of distance maximization and evaluate the distribution of the obtained multihop distance through analytical approximations and
simulations.

Index Terms—Wireless sensor networks, multihop distance, analysis, simulation, estimation.

Ç

1 INTRODUCTION

M ETHODS providing information on the distribution of
the euclidean distance between nodes in wireless
sensor networks (WSN) are regarded as versatile tools for
[4], the distribution of the remaining distance in multihop
greedy forwarding to a destination node is derived.
Furthermore, in [5], the distribution of euclidean distances
protocol parameter tuning and performance modeling. to nth neighbor in a Poisson point process is analyzed.
Therefore, internode distance estimation is of profound Hop distance estimation between two locations is
importance for various WSN applications. For instance, studied analytically in [6] and [7]. In [6], connectivity
when urgent data are disseminated from a source node via probability in one or two hops is derived and connectivity
broadcast, it is critical to estimate the covered distance as a in multiple hops is studied with analytical bounds.
result of a sequence of data forwards. Similarly, estimation Furthermore, analytical bounds of the expected hop
of the hop distance between two network locations is distance are derived and supported by simulation results.
equivalent to estimating the minimum number of hops, In [7], the expected number n of relay nodes between two
which leads to maximization of the distance covered in randomly located sensors is analytically computed via
multihop paths. Furthermore, hop distance estimation is iteration based on expressions for connectivity in one or two
closely related with transmission delay estimation and hops. In [8], the distribution of hop distance and its
minimization of multihop energy consumption. expected value are analyzed with simulations. It is shown
The current literature on the maximization of the multi- that beamforming antennas significantly reduce the hop
hop distance is limited; however, there are several studies distance compared to omnidirectional antennas for medium
that focus on estimation of single-hop or multihop distances. and large networks with random node locations.
Methods in [1] and [2] estimate the distances to designated The maximum euclidean distance metric is first investi-
anchor nodes using optimization algorithms. There are also gated in [9] for 1D networks. The distribution of the
analytical methods that address the probabilistic evaluation maximum distance taken in a single hop is derived
of the euclidean distances such as that given in [3] and [4]. In analytically. In our previous study [10], we showed that
[3], the probability distribution of the single-hop distance the distribution of maximum multihop euclidean distance
between two randomly chosen neighbors is investigated. In in a 1D network has an increasingly Gaussian character for
increasing hop distance values. This result is used in [11] to
develop a Bayesian decision mechanism to determine the
. S. Vural is with the Department of Electrical and Computer Engineering, number of hops for a given euclidean distance. An
The Ohio State University, Columbus, OH 43210, and the Department of application of [10] to planar sensor networks is [12] which
Computer Science and Engineering, University of California, Engineering uses the Gaussian pdf to verify sensor positions. In [12], 1D
Building Unit 2, 900 University Ave., Riverside, CA 92521.
E-mail: vurals@ece.osu.edu. network results are mapped to planar networks by defining
. E. Ekici is with the Department of Electrical and Computer Engineering, a thin rectangular corridor between two locations. Since the
The Ohio State University, 205 Dreese Laboratory, 2015 Neil Avenue, definition and calculation of multihop path distances in a
Columbus, OH 43210. E-mail: ekici@ece.osu.edu. planar network are considerably complex compared to a 1D
Manuscript received 8 Jan. 2008; revised 15 Sept. 2008; accepted 4 Aug. 2009; network, the distribution model is not highly accurate.
published online 12 Aug. 2009. Hence, results of [10] are largely limited to 1D networks.
For information on obtaining reprints of this article, please send e-mail to:
tmc@computer.org, and reference IEEECS Log Number TMC-2008-01-0005. In this paper, we propose a greedy method which is
Digital Object Identifier no. 10.1109/TMC.2009.151. aimed at maximizing the multihop distance in 2D networks
1536-1233/10/$26.00 ß 2010 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS
Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:31 UTC from IEEE Xplore. Restrictions apply.

VURAL AND EKICI: ON MULTIHOP DISTANCES IN WIRELESS SENSOR NETWORKS WITH RANDOM NODE LOCATIONS 541

with random node locations. Our method is based on
restricting the propagation direction outward from the
propagation source in each hop and greedily searching the
furthest neighbor each time. This is similar to the approach
in [10]; however, we demonstrate that Gaussian pdf, which
is shown to accurately model distance distributions in 1D
networks [10], cannot represent distance distributions in 2D
networks. The accuracy of the Gaussian pdf model depends
on the number of hops and the chosen parameters, which
affect the obtained error ranges. We show that a transfor-
mation of the well-known Gamma distribution effectively
approximates the obtained maximum distances in a 2D Fig. 1. Iterative definition of maximum multihop euclidean distance.
network. For the same range of number of hops, uniform (a) Definition of di and (b) multihop path of five hops.
accuracy is provided by the proposed method in this paper.
Compared to the Gaussian pdf model in [10], a much more selecting an initial propagation direction, the following
consistent and accurate representation of maximal multihop iterative definition is provided for our greedy distance
distance pdf as well as hop distance pmf in 2D networks is maximization scheme. It should be noted that our method
obtained. Using the pdf obtained by the transformation of does not locate the node with maximum distance to the
Gamma distribution, we derive the expected value and source node for a given number of hops, yet it maximizes
standard deviation of the euclidean multihop distance and the euclidean distance toward a chosen initial direction
calculate the parameters of the Gamma distribution. The greedily. For the definition of this distance, consider Fig. 1a
accuracy of the Gamma-approximated distribution is for subsequent descriptions.
demonstrated through comparisons with simulation results. Let PiÀ1 be the node with the maximum euclidean
The rest of the paper is organized as follows: In Section 2, distance, selected by the greedy selection, to a source
multihop distance in data dissemination scenarios is defined
sensor S in i À 1 hops. Furthermore, let the line liÀ1
and its similarity to well-known distributions is statistically
originating from S and passing through PiÀ1 act as a
inspected. In Section 3, multihop distance is analyzed and
direction line which defines the outward propagation from
approximation expressions of its standard deviation and
S. Then, in the ith hop of the multihop path, node PiÀ1
expected value are derived. Section 4 presents an analytical
chooses the neighbor node Pi whose euclidean distance to S
way of estimating the hop distance for a given euclidean
is the largest among all neighbors of PiÀ1 . Such a node Pi
distance using the results of Section 3. In Section 5,
may not be located on the line liÀ1 since node density is
simulation and numerical results are provided. Finally,
finite. In fact, the line segment between PiÀ1 and Pi has an
Section 6 concludes the paper.
angle of deviation from liÀ1 denoted by i . The length of this
line segment is the distance of the ith hop denoted by ri .
2 GREEDY MAXIMIZATION OF MULTIHOP Furthermore, we limit the search area of the new node Pi to a
EUCLIDEAN DISTANCE maximum angular deviation such that i is upper bounded.
2.1 Network Model Finally, the euclidean distance of Pi to the source sensor S is
WSNs considered in this study consist of sensors with denoted by di and represents the euclidean distance in a
circular communication ranges of radius R. Node locations multihop path of i hops in an outward direction from S.
are static and uniformly distributed with a planar density Fig. 1b demonstrates an example of a multihop path
. Hence, the number of points n that can be found in a generated with this definition. After locating the maximally
given area ÀA has a Poisson distribution given by
A distant node Pi to S in each hop i, the propagation direction
n
pn ðnÞ ¼ ðAÞn!e . It is assumed that nodes can receive every is changed and becomes the line li passing through S and
packet transmitted within their communication ranges. the new node Pi . For a different initial direction, other
distances can be obtained. Hence, our method maximizes
2.2 Definition of Maximum Multihop Euclidean the distance greedily while fixing the direction of propaga-
Distance tion outward from the source node. Due to the uniformly
In a 2D network of randomly located nodes, there exists distributed node locations, the distribution of di is identical
only one node with the maximum distance that can be for all propagation directions and denoted by pdi ðdi Þ.
reached in a given number of hops. Due to spatially random
node locations, the position of this node as well as its 2.3 The Probability Distribution of Multihop
distance to the source node is random. However, analytical Euclidean Distance
computation of the maximum euclidean distance in N hops In our previous study [10], the distribution of the maximum
involves considering all nodes in the previous hops and distance for a given number of hops in linear networks is
recursively reaches the source node of the multihop modeled by the Gaussian pdf. The mean r of the maximum
propagation, which becomes intractable even for small distance of a single hop is numerically found by the implicit
r
instances. Hence, we propose a greedy method of max- equation ln ð1 À RÀrÀ1Þ ¼ r, where is the linear node
imization of the euclidean multihop distance. By selecting density and R is the sensor communication range. The
locally maximally distant nodes, the multihop propagation expected value of the multihop maximum distance dN of
intends to reach further distances to the source node. After N hops is then found by E½dN Š ¼ Nr and the variance of dN



Fig. 2. Distribution estimations using 1D Model with Gaussian pdf and ML Estimation with Gamma pdf, ¼ 0:001 nodes=m2 . (a) pdi ðdi Þ estimated by
the 1D Model, (b) px ðxÞ and ML estimation, and (c) pdi ðdi Þ and ML estimation.

P 2
is calculated by 2N ﬃ E½ð N ri Þ Š À N 2 r2 , which involves
d i¼1 x ¼ NR À dN , then x is a random variable. To determine the
a recursive computation. The results in [10] illustrate that similarity of its distribution to well-known distribution
the distribution of the multihop maximum distance in 1D types, Kolmogorov-Smirnov (KS) Test is applied to a
networks can be modeled by the Gaussian distribution with simulation data set of the random variable x. The data set
high accuracy. This model is referred as the “1D Model” is collected from 10,000 different networks with uniformly
throughout the rest of this paper. distributed node locations and a sensor communication
In contrast to 1D networks, a multihop path in a planar range of R ¼ 100 m. The KS test suggests that a data set is
network is generally not located on a line. However, one similar to a given distribution type if the KS statistic is lower
can define a thin rectangular corridor with a small width W than the critical value of the test for the chosen significance
between two locations and search the multihop path within level. The critical value of the KS Test for a significance level
this corridor. Using this mapping, nodes can be assumed to of 0.05 is determined and then compared with each KS
be in a linear formation. Hence, the equivalent 1D network statistic that is obtained by x values in simulations
density becomes ¼ W , where is the actual node corresponding to different hop distance values.
density on the 2D network. With this approximation, the Fig. 3 demonstrates the test results for two node density
1D Model can be applied to planar networks to represent values, ¼ 0:001 nodes=m2 and ¼ 0:002 nodes=m2 , and for
the maximum multihop distance distribution with a various distribution types. The highest similarity results are
Gaussian pdf [12]. obtained for Gamma distribution. For ¼ 0:002 nodes=m2 ,
The simulation results for multihop maximum distance the test results for the Gamma distribution are consistently
distribution pdi ðdi Þ for i 8 and the Gaussian pdf found by below the critical value computed by the KS Test. This
the 1D Model are illustrated in Fig. 2a with W ¼ R. suggests that the distribution of x is similar to Gamma
Although Gaussian pdf accurately models 1D network distribution. The Maximum Likelihood (ML) Estimation of
distance distributions [10], it fails to represent the distribu- the distribution of x with the Gamma pdf, as illustrated in
tions in planar networks. However, since applications such Fig. 2b for a node density ¼ 0:001 nodes=m2 , also clearly
as distance estimation and localization are sensitive to even demonstrates this similarity.
small amounts of errors in intersensor distances, a more By modeling the distribution of x using Gamma
accurate model that can better represent distance distribu- distribution, the distribution of the maximum euclidean
tions is required. distance dN can be defined since di ¼ iR À x. When the
In a 2D network with random node locations, a path with distribution of x obtained by ML estimation is transformed
N hops can have a length of at most NR, with a sensor with di ¼ iR À x, the ML estimate of the maximum
communication range of R. However, due to the finite node distance distribution di is obtained, which is shown in
density , the actual maximum distance dN is smaller; hence, Fig. 2c for ¼ 0:001 nodes=m2 . This figure suggests that the
NR is an upper bound for dN . If we define this difference as pdf of di can be obtained through a simple linear

Fig. 3. Kolmogorov-Smirnov Test applied to x. (a) ¼ 0:001 nodes=m2 . (b) ¼ 0:002 nodes=m2 .



Fig. 4. Angular range model and model-related approximation.
(a) Angular model and (b) model approximation.
Fig. 6. Average upper bound of model-based error, R ¼ 100 m.
(a) Different angles and (b) different density values .
transformation applied to a Gamma pdf. The Gamma pdf
a Àbx
of the random variable x is given by pR ðxÞ ¼ xaÀ1 bÀðaÞ ,
x
e
1 zÀ1 Àt Note that the region A3 can easily be defined by the
where Àð:Þ is the Gamma function ÀðzÞ ¼ 0 t e dt, and
difference between two arcs centered at P . However, this is
a and b are shape and scale parameters, respectively.
an approximation since any node in region K in Fig. 4b is
Furthermore, the mean and variance of the Gamma
more distant to S compared to the radially most distant
distribution in terms of a and b are given by E½gðx; a; bÞŠ ¼
node to PiÀ1 . In fact, the largest amount of error occurs if a
ab and varðgðx; a; bÞÞ ¼ ab2 , respectively.
point exists in location T , as shown in Fig. 4b. The distance
The maximum likelihood estimation provides the para-
of T to S is dmax ¼ dprev þ ri À , where ri is the distance of
meters a and b which uniquely define the Gamma
the radially distant node Pi to node PiÀ1 and is a negligible
distribution. However, ML estimation requires simulation
small number. However, the distance of Pi to S is
data to estimate the pdf. In contrast, our aim is to avoid the use
of simulation data, and compute the parameters a and b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dapp ¼ d2 þ ri 2 þ 2ri dprev cos ð=2Þ:
prev
analytically. Hence, the analysis in this paper aims to
analytically compute E½di Š and di and models the distribu- The maximum percent error in the approximation is then
tion of dN using the Gamma distribution with parameters a
and b that are calculated using E½di Š and di , i.e.,
max ðri ; Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2 i
d E½di Š dprev þ ri À À d2 þ ri 2 þ 2ri dprev cos ð2Þ Â 100
prev
b¼ and a ¼ : ¼ :
E½di Š 2 i
d dprev þ ri À
ð1Þ

3 ANALYSIS Denoting the probability that the maximum single-hop
distance is ri ¼ ri as pri ðri Þ, the expected value of the upper
3.1 Greedy Maximization of Multihop Distance
bound of percentage error can be calculated by evaluating
In Fig. 4a, the search area for a next hop sensor is modeled
(1) for 2 ½0; Š and ri 2 ½0; RŠ, as in (2). The derivation of
by an angular communication range with a “radial range”
pri ðri Þ is provided in Section 3.3.2
R, which is equal to the radius of the sensor communication
range, and an “angular range” . The bisector direction Z Z R
1
line represents the outward direction from the source E½
max ðri ; ÞŠ ¼ 100
max ðri ; Þpri ðri Þ dri d: ð2Þ
0 0
sensor S.
Maximization of multihop distance, as defined in Simulation results demonstrate that the amount of error
Section 2.2, is obtained by choosing the next sensor node in the approximation is negligible and upper bounded. In
within the angular communication range such that its Fig. 6, the percentage errors are all lower than 2 percent for
distance to the source sensor S is maximum. In Figs. 5a and all values of . Fig. 6a illustrates the expected value of the
5b, no nodes are found in regions A1 and A2 , and da and db upper bound of the percentage error with respect to
constitute the maximum distance to S. ra , rb , and rc denote multihop distance for different values of the angular range
the radial distance of the selected next node to P . However, in case of ¼ 0:001 nodes=m2 . Furthermore, Fig. 6b shows
the shapes of the regions created by this choice depend on the expected value of the upper bound of the percentage
the location of the node and distances da and db which error for different values of the node density with respect
complicate the analysis. to multihop distance for ¼ =3.
On the other hand, in Fig. 5c, the node with the
maximum distance to node P is chosen as the next node. 3.2 Maximum Multihop Distance Approximated as a
Sequence of Single-Hop Maximum Distances
To maximize the multihop distance, the maximum dis-
tances of individual single hops are considered within their
angular communication ranges. Note that this is an
approximation as mentioned in Section 3.1. In Fig. 7, node
PiÀ1 forwards the packet to the radially furthest node Pi .
Furthermore, it is known that no sensor node is located in
Fig. 5. Choice of next node in angular range. the shaded area Ai (hence, the area V À i) since otherwise



Fig. 7. Multihop distance formation.
Fig. 9. Analysis of intermediate-hop maximum distance. (a) Vacant
such a node would be the furthest node and ri would be region ViÀ1 and (b) req .
larger. In the rest of the paper, areas which contain no
sensor nodes are referred to as “vacant area.” designated as A1 . Third, there should be at least one point in
In Fig. 7, the next nodes of the multihop transmission at the area S2 , as shown in Fig. 8, which is in the angular
individual hops are searched within angular communica- communication range of the second hop. The last condition
tion ranges that are pointed outward from the source node should be met as otherwise connectivity is lost.
S. The lines liÀ2 , liÀ1 , and li designate the bisectors of the Since point locations are uniformly distributed, the
angular slices at hops i À 1, i, and i þ 1, respectively. Hence, number of points that can be found within a given area
the multihop distance is established by a sequence of single-
has a Poisson distribution. Furthermore, the number of
hop distances. The next section outlines the calculation of
points that are found in nonintersecting areas are indepen-
the maximum single-hop distance.
dent random variables. Considering these facts, the prob-
3.3 Analysis of Single-Hop Maximum Distance ability that the three conditions are met can be defined as a
During the analysis of single-hop maximum distance ri , the function g = Prob(A sensor on arc a1 )ÁProb(No sensors in
following facts should be considered: A1 )ÁProb(At least one sensor in S2 ). For an infinitesimal
radial width dr1 , Prob(A sensor on arc a1 )¼ r1 dr1 .
. The vacant region ViÀ1 is known to contain no sensors. Using the definition of Poisson distribution and by the
2
. ri is a random variable whose range is upper 2
approximation AreaðS2 Þ ﬃ ðR ÀðRÀr1 Þ Þ , g becomes a func-
2
bounded by R and is independent of the angular
tion of r1 given by
deviation i .
. The angular deviation i is a random variable ðR2 Àr2 Þ
1
ðR2 ÀðRÀr1 Þ2 Þ

uniformly distributed over a range ½À ; Š and is gðr1 Þ ¼ r1 eÀ 2 1 À eÀ 2 dr1 : ð3Þ
2 2
independent of ri .
To find the probability distribution function of r1 , the
The analysis of single-hop maximum distance is differ-
function gðr1 Þ should be normalized over the range of
ent for the first hop from intermediate nodes since the first
values that r1 can take. The pdf of r1 is then given by
hop does not consider a vacant region V0 which does not
exist by definition. However, the introduction of the vacant .Z R
region ViÀ1 into the analysis of ri is inevitable for pr1 ðr1 Þ ¼ gðr1 Þ gðr1 Þdr1 :
intermediate nodes and requires some additional calcula- 0

tions due to the shape of ViÀ1 . Therefore, the single-hop Then, the expected value E½r1 Š of the first-hop maximum
maximum distance r1 is analyzed first, followed by the distance r1 can be found by the following expression:
analysis of ri for i ¼ 2; 3; . . .
Z R
3.3.1 First-Hop Maximum Distance E½r1 Š ¼ r1 pr1 ðr1 Þdr1 : ð4Þ
0
The analysis of first-hop maximum distance r1 is the
starting point of our analysis of the more general The variance of r1 can be calculated as 21 ¼ E½r1 2 Š À
r
intermediate-hop distance ri . In Fig. 8, the location of the E½r1 Š2 . Here, the second moment E½r1 2 Š of r1 is determined
source sensor S, which is known deterministically, is the using the following equation:
apex of the angular slice of the first hop. The following three Z R
conditions should be satisfied for the furthest distance r1 to Â Ã
E r2 ¼
1 r1 2 pr1 ðr1 Þdr1 : ð5Þ
be equal to a specific value r1 : First, there should be a point 0
on the circular arc a1 on which every point is r1 away from
S. Second, there should be no sensors within the region 3.3.2 Intermediate-Hop Maximum Distance
The analysis of the maximum distance ri of intermediate
hops is similar to the analysis of r1 . However, the vacant
regions Vi need also be considered.
In Fig. 9a, the lower bound of the range of values that ri
can take is determined by the vacant region ViÀ1 . However,
the shape of ViÀ1 depends on the previous distance riÀ1 and
the previous angle of deviation iÀ1 . Therefore, it is not
possible to define an exact lower bound on the range values
that ri can take for given values of riÀ1 and iÀ1 . Hence, an
Fig. 8. Analysis of first-hop maximum distance. approximate lower bound which takes the area of ViÀ1 into



account should be determined since this area contains no
sensors. This is shown in Fig. 9b, where req , called equivalent
radius, is the radius of angular slice with the same area as
ViÀ1 . The Appendix presents the calculation of req for given
values of riÀ1 and iÀ1 .
Calculation of intermediate-hop maximum distance.
Fig. 10. Calculation of E½dN Š.
req is the lower limit of the intermediate-hop maximum
distance riÀ1 and is a function given by req ðriÀ1 ; iÀ1 Þ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
Denoting riÀ1 ¼ E½riÀ1 Š, the pdf of ri is given as E½di Š ¼ E d2 þ r2 þ 2ri diÀ1 cos i ¼ E½gðr; ÞŠ;
iÀ1 i

ðR2 Àr2 Þ ðR2 ÀðRÀri Þ2 Þ

i where g is a function of r and . Hence,
gðri Þ ¼ ri eÀ 2 1 À eÀ 2 ;
Z R Z
gðri Þ ð6Þ E½di Š ¼
2
...
pri ðri Þ ¼ Z R
:
req ðriÀ1 ;iÀ1 Þ À
2
gðri Þdri Z R Z ð12Þ
2
req ðriÀ1 ;iÀ1 Þ
... f ðr; Þd1 dr1 . . . di dri ;
req ðr1 ;1 Þ À
Hence, the expected value E½ri Š and the second 2

moment E½ri 2 Š of ri are calculated using (7) and (8), where f ðr; Þ ¼ gðr; Þ Á pr; ðr; Þ and pr; ðr; Þ is the joint
respectively. The variance of ri can be calculated with probability distribution function.
2i ¼ E½ri 2 Š À E½ri Š2 , where
r The computation of the right-hand side of (12) is costly
Z Z and not scalable in number of hops. One possible approx-
1 2 R imation is the representation of E½di Š as an iterative relation
E½ri Š ¼ ri pri ðri Þdri diÀ1 ; ð7Þ
À req ðriÀ1 ;iÀ1 Þ
2 dependent on the expected value E½diÀ1 Š ¼ diÀ1 of the
previous hop. This representation is given by
Z Z
Â Ã 1 2 R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !
E r2 ¼
i r2 pri ðri Þdri diÀ1 :
i ð8Þ 2
À req ðriÀ1 ;iÀ1 Þ E½di Š ¼ E diÀ1 þ ri 2 þ 2ri diÀ1 cos i :
2

3.3.3 Last-Hop Maximum Distance Denoting
Since there is no condition that the next hop should exist for qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
the last hop N, the pdf of the last-hop maximum distance rN sðdiÀ1 ; ri ; i Þ ¼ pri ðri Þ diÀ1 þ ri 2 þ 2ri diÀ1 cos i ;
is obtained as follows:
the computation of E½dN Š is outlined iteratively as follows:
ðR2 ÀrN 2 Þ
À
gðrN Þ ¼ rN e 2 ; 1. d1 ¼ r1 ¼ E½r1 Š, as given by (4).
gðrN Þ : ð9Þ 2. E½ri Š ¼ ri by (7) for 1 i N.
prN ðrN Þ ¼ Z R 3. For 1 i N,
gðrN ÞdrN
req ðrNÀ1 ;NÀ1 Þ Z =2 Z R Z =2
1
di ¼ sðdiÀ1 ; ri ; i Þ di dri diÀ1 :
Using (9), the expected value and the second moment of 2 À=2 req ðriÀ1 ;iÀ1 Þ À=2
the last hop are found, as given by (10) and (11). Note that the
variance of the last-hop distance is 2N ¼ E½r2 Š À E½rN Š2 ,
r N 4. Expected multihop distance is then found by
where
Z =2 Z R Z =2
1
Z Z R dN ¼ sðdNÀ1 ; rN ; N Þ
1 2 2 À=2 req ðrNÀ1 ;NÀ1 Þ À=2
E½rN Š ¼ rN prN ðrN ÞdrN dNÀ1 ; ð10Þ
À
2 req ðrNÀ1 ;NÀ1 Þ dN drN dNÀ1 :

Z Z 3.4.2 Standard Deviation dN of Multihop Maximum
Â Ã 1 2 R
E r2
N ¼ rN 2 prN ðrN Þ drN dNÀ1 : ð11Þ Distance
À
2 req ðrNÀ1 ;NÀ1 Þ
To compute the variance 2 i ¼ E½di 2 Š À E½di Š2 , the moment
d
3.4 Analysis of Multihop Maximum Distance E½di 2 Š can be found by
This section outlines how the mean E½dN Š and standard Â Ã Â Ã Â Ã
deviation dN of the maximum multihop distance distribu- E d2 ¼ E d2 þ E r2 þ 2E½ri diÀ1 cos i Š:
i iÀ1 i

tion pdN ðdN Þ are calculated. Since i is independent of diÀ1 and ri , E½ri diÀ1 cos i Š ¼
E½ri diÀ1 Šcos i , where cos i is the expected value of cos i
3.4.1 Expected Multihop Maximum Distance E½dN Š calculated over the range i ½À=2; =2Š and is equal to
Fig. 10 illustrates the relation between the single-hop 2
cos i ¼ sin . Hence, we obtain
2
distance ri and the multihop distance di in hop i and the
Â Ã Â Ã Â Ã 2
multihop distance diÀ1 of the previous hop i À 1. Denoting E d2 ¼ E d2 þ E r2 þ 2E½ri diÀ1 Š sin ;
i iÀ1 i ð13Þ
¼ {1 ; 2 ; 3 ; . . .g and r ¼ {r1 ; r2 ; r3 ; . . . }, we have 2



their transmission and sensing range patterns. The unit disk
model has been widely used in the current literature for
modeling wireless sensor networks and designing commu-
nication protocols [4], [6], [3], [16]. The unit disk model
defines the communication range as the minimum radius of
a circular reception area within which all transmissions are
successfully received if no interference or packet collisions
exist. In the event that the wireless medium is subject to the
effects of fading, the reception power at receiver nodes is
affected by the distance to the transmitter and decays
Fig. 11. Average percent error in approximation of E½dN Š and dN . exponentially with distance. Furthermore, with the pre-
(a) Percent error in dN and (b) percent error in dN . sence of Gaussian noise, the received power becomes a
random variable. This makes the reception of a packet a
where E½ri diÀ1 Š ¼ Covðri ; diÀ1 Þ þ E½ri ŠE½diÀ1 Š. Since the probabilistic event dependent on the distance to the
correlation of ri with rj for j i À 1 is negligible compared transmitter node, the statistical characteristics of the
to the correlation of ri , we have channel noise, transmission power, and the threshold of
Â Ã Â Ã Â Ã reception power. Using the Shadow Fading Model with n
E d2 ¼ E d2 þ E r2
i iÀ1 i denoting the path loss component, the path loss P LðdÞ at a
2 ð14Þ distance d from the transmitter is given by [15]
þ 2ðCovðri ; riÀ1 Þ þ E½ri ŠE½diÀ1 ŠÞ sin :
2
d
Finally, the iterative computation of E½d2 Š is outlined as P LðdÞ ¼ P Lðd0 Þ þ 10:n:log þ X ; ð15Þ
N d0
follows:
where d0 is the reference distance and X is a zero-mean
1. E½r2 Š ¼ E½d2 Š by (5).
1 1 Gaussian random variable. Considering this path loss and
2. E½r2 Š is found by (8).
i denoting the received power at distance d as PR ðdÞ, the
3. E½d2 Š is calculated using (14).
i probability that a packet is successfully received by a sensor
4. E½r2 Š is found by (11).
N at a distance d away from the transmitter node, P rðdÞ, is
5. E½d2 Š is computed by substituting E½r2 Š and
N N given by
E½d2 Š in (14).
NÀ1 Z
1 1 Àx2
3.5 Approximation Accuracy for Different Angles QðzÞ ¼ exp dx;
2 z 2 ð16Þ
Before moving to the performance analysis of the derived
approximation expressions, the effect of on the accuracy QðzÞ ¼ 1 À QðÀzÞ;
of the derived expressions is evaluated. Fig. 11a illustrates !
the average percent error in E½dN Š. For each value of , the
À P rðdÞ
error of E½dN Š approximation decreases with increasing P rðdÞ ¼ P r½PR ðdÞ
Š ¼ Q ; ð17Þ

node density and settles to a value less than 0.2 percent,
except for ¼ =6 which requires a larger node density to where
is the receive threshold and (16) is the Q-function.
stabilize. One obvious observation is the decreasing amount When the packet transmissions occur in an environment
of percent error in E½dN Š approximation for larger angles, subject to random path loss, hop distance patterns are also
which shows that our approximation is more accurate when affected by this change. Since now the nodes may not
single hops cover larger angular ranges. receive a packet with insufficient reception power, the
The choice of a large angle, however, causes instability in maximum multihop distance that a packet can traverse in a
the approximation of the standard deviations shown in given number of hops is effectively reduced. Hence, the
Fig. 11b. Despite the decrease in the percent error, the greedy multihop maximization model, as explained in
percentage error of dN has a noticeable incline when ¼ Section 3, has to consider this change. This can be achieved
5=12 for node densities larger than ¼ 0:001 nodes=m2 . by including the probability of packet reception given by
The increasing error for large angles is due to the multi- (17) in (3), (6), and (9). Hence, (3) now becomes:
plication of the two approximated values of E½diÀ1 ri Š and
ðR2 Àr1 2 Þ
ðR2 ÀðRÀr1 Þ2 Þ

cosi in (13). Although these two terms are approximated gðr1 Þ ¼ P rðr1 Þ:r1 eÀ 2 1 À eÀ 2 dr1 ; ð18Þ
with high accuracies of the order of 2.5 and 0.62 percent,
respectively, their multiplication creates significant terms where P rðr1 Þ is found by (17). Note that P rðdÞ is, in fact, the
when and are large.
probability of reception as provided by any fading model
With a less than 2 percent error in approximated E½dN Š
which is a function of communication radius, not necessarily
and a reasonably low error of 5 percent in dN estimation,
we choose the slice angle as ¼ =3 for the practical range the Shadow Fading Model. Hence, our model captures the
of node density values used in our numerical studies. The effect of a given fading model by a multiplicative probability
subsequent performance analysis of the approximations are factor in distance probability functions in (3), (6), and (9).
based on the multihop distances with ¼ =3.

3.6 Multihop Distances with Shadow Fading Model 4 ESTIMATION OF HOP DISTANCE
The hop distances to a broadcast source in a planar sensor Apart from the estimation of coverage area boundaries as
network depends on the positions of the sensors as well as shown in previous section, an interesting application of the



maximum euclidean distance distribution is the estimation of
hop distance between two locations. In fact, hop distance is
usually a more popular metric in WSN research since
parameters like multihop end-to-end delay and total trans-
mission power can be estimated by hop distance. In this
section, hop distance N for a given euclidean distance D is
estimated using the Gamma pdf Model of maximum multi-
hop distance distribution di , where i ¼ 1; 2; 3; . . .
The posterior distribution of N for a given D is

P ðDjNÞ:P ðNÞ
P ðNjDÞ ¼ PNmax ; ð19Þ Fig. 12. Area A should contain a node.
i¼1 P ðDjiÞ:P ðiÞ
Hence, P ðDjNÞ is equal to
where P ðiÞ is the probability that a randomly selected node
is i hops away from the source node. P ðDjNÞ ¼ P robfd À R dNÀ1 Dg:
The maximum a posteriori (MAP) estimate of N is ð23Þ
P robf0 dj D À Rg:ð1 À eA Þ:
!
b P ðDjNÞ:P ðNÞ Area A is calculated using the following equations:
NMAP ¼ arg max PNmax : ð20Þ
N i¼1 P ðDjiÞ:P ðiÞ
D2 þ d2 À R2
iÀ1
x¼ ;
The probability P ðiÞ can be approximated by calculating 2D
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
the area where nodes with hop distance i are located and
y¼ d2 À x2 ;
iÀ1
then taking the ratio of this area to the area of the whole y
network. Assuming that a maximum hop distance of Nmax ð24Þ
1 ¼ arctan ;
is created as a result of the broadcast, we have DÀx
y

À Á 2 ¼ arctan ;
d2 À d2 x
i iÀ1
P ðiÞ ffi : ð21Þ A ¼ 1 ðR Þ À yðD À xÞ þ 2 d2 À yx:
2
dNmax 2 iÀ1

Hence, (20) is modified to obtain: Hence, P ðDjNÞ is found by
À 2 Á ! Z D
2
bMAP ¼ arg maxN P ðDjNÞ: dNÀÀ dNÀ1 Á :
N
P
ð22Þ
P ðDjNÞ ¼ pdNÀ1 ðdNÀ1 ÞddNÀ1 ;
Nmax dÀR
2 2
i¼1 P ðDjiÞ: di À diÀ1
! ð25Þ
NÀ2 Z DÀR
Y
Note that the value of Nmax depends on factors like pdj ðdj Þddj :ð1 À eA Þ:
j¼1 0
topology size, node locations, sensor communication range,
and node density. Nmax is lower bounded by bDM =Rc, where Finally, (25) is plugged in (22) to calculate the MAP
DM , say, is the maximum distance to the source sensor found estimation of hop distance for a given euclidean distance D.
in the topology, which is possible in case of infinite node
density. Since node density is finite and connectivity patterns
vary for different topologies, an exact value of Nmax is
5 NUMERICAL RESULTS
difficult to estimate. However, in (22), Nmax is canceled out In this section, the results of our analytical approximations for
when taking the ratio of P ðNÞ and P ðiÞ , yet it still appears as the expected value and standard deviation of the maximum
an upper bound of the summation. The pdf P ðdi Þ has low euclidean distance dN are presented and compared with the
tails as observed in Fig. 2c, and the proportion of the pdf simulation results. Furthermore, the distribution of dN is
terms for higher hop distances in the summation diminishes approximated using a transformed Gamma distribution for
quickly and become negligibly small as i increases. Hence, various node densities. The simulation results are the
for computational purposes, Nmax in (22) can be chosen as an averages of 10,000 independent topologies for all values of
arbitrarily large number. the node density, ¼ 0:0005; 0:00075; 0:001; 0:0015, and
Equation (22) provides the MAP estimation of the hop 0:002 nodes=m2 . In each simulation, the source sensor is
distance for a given euclidean distance D. However, the placed in the middle of a topology of size 1;600 m Â 1;600 m.
posterior probability P ðDjNÞ is unknown for individual The communication range of sensor nodes is R ¼ 100 m and
values of hop distance i. On the other hand, the distribution the angular range is ¼ =3. The simulations are carried out
of di for i ¼ 1; 2; 3; . . . is modeled by the Gamma pdf and by a broadcast from the source sensor to the network and
can be used to calculate P ðDjNÞ by meeting the following determining the hop distances of individual nodes to the
requirements [13]: source. For each hop distance, the greedy propagation locates
the node with the maximum distance to the previously
1. D À R dNÀ1 D. selected node toward the direction of propagation as outlined
2. 0 dj D À R; 8j N À 1. in Section 2.2. The separation between the source sensor and
3. There must be at least one point in the area A, as this new node is recorded as the multihop distance for that
shown in Fig. 12. particular hop distance in that simulation.



Fig. 15. Approximation of pdN ðdN Þ for low and high density. (a) ¼
0:0005 nodes=m2 and (b) ¼ 0:002 nodes=m2 .
Fig. 13. Approximation of E½ri ] and ri . (a) Approximation of E½ri Š and
(b) approximation of ri . Gamma distribution, as explained in Section 2.3. After
calculating E½dN Š and dN , the parameters a and b of the
5.1 Mean and Standard Deviation of ri Gamma distribution of the remaining distance x are
Fig. 13a illustrates the comparison between the simulated calculated. Then, transformation di ¼ iR À x is applied to
and analytical expected single-hop maximum distance get the analytical approximation of the distribution of dN .
E½ri Š. The increase in the accuracy of the approximations In Fig. 15, our approximation of multihop distance
with increasing node density can be observed. Further- distribution is illustrated for low and high node density
more, E½ri Š becomes larger with the increase in node values ¼ 0:0005 nodes=m2 and ¼ 0:002 nodes=m2 , re-
density. The approximated single-hop distances are close to spectively. The distribution of dN becomes more peaked
the simulation results and the accuracy is enhanced for with increasing node density and the pdf curves corre-
higher node densities. sponding to different values of hop distance N get more
The approximation result of the single-hop standard separated from each other due to the decrease in dN caused
deviation ri is shown in Fig. 13b. With larger node density, by larger node density.
the accuracy of the approximation of ri increases while the
The comparison of the 1D Model [10], [12] and the
magnitude of ri decreases.
proposed 2D Model is demonstrated in Fig. 16. In this
5.2 Mean and Standard Deviation of dN figure, the root-mean-square error (RMSE) between the
Fig. 14a illustrates the comparison between analytical and model pdf and the distribution obtained by simulations is
simulated E½dN Š in case of different node density values . shown. Different corridor widths (W ) are evaluated for the
It can be observed that E½dN Š monotonically increases for 1D Model. In most cases, the least RMSE is obtained for
increasing node density. Furthermore, the approximations W ¼ R. It is observed that the RMSE of the 2D Model is
closely match the simulation values with negligible errors less than the RMSE of the 1D Model regardless of the
which become even smaller for higher node densities. corridor widths W . Furthermore, the RMSE of the 2D
The standard deviation of dN is shown in Fig. 14b as an Model is more invariant to changing hop distance values
error bar around the expected value. In this figure, only the
when compared with the RMSE of 1D Model. In summary,
highest and the lowest density values used in the simula-
Fig. 16 clearly illustrates that our proposed 2D model is a
tions are shown for better visualization of the error bars.
The accuracy of the approximated dN is higher for the
higher node density. Furthermore, the increase in standard
deviation for increasing hop distance N and lower node
density can be observed.
5.3 Approximation of the pdf of dN
The pdf of the maximum euclidean distance dN correspond-
ing to a hop distance N is approximated by a “transformed”

Fig. 14. Approximation of E½dN Š and dN . (a) Approximation of E½dN Š Fig. 16. Root-mean-square error of 1D and 2D Models. (a) ¼
and (b) approximation of dN . 0:0005 nodes=m2 , (b) ¼ 0:001 nodes=m2 , and (c) ¼ 0:002 nodes=m2 .



Fig. 17. Topology shape and source locations.

more accurate representation of the planar multihop Fig. 18. Percent error in expected multihop distance estimation due to
the edge effect. Topology size: 5R Â 5R. Different source locations.
maximum distance distribution. (a) Source node approaching an edge and (b) random source locations,
different node density values.
5.4 Edge Effects
The analysis in Section 2 reveals the relationship between analytical and a simulation result for a multihop distance
hop distance and euclidean distance in the greedy multihop value, respectively. First, the source sensor is placed away
forwarding mechanism, as outlined in Section 2.2. In this from the topology center and at increasingly closer locations
section, we study the effect of topology borders on the toward one of the edges. For a topology of size 5R Â 5R,
proposed estimation method. these locations are arbitrarily chosen to be at coordinates
Topology borders limit the distance that can be taken in a ðÀ4R, 0Þ, ðÀ3R, 0Þ, ðÀ2R, 0Þ, and ðÀR, 0Þ, where ð0, 0Þ is the
multihop path. For an n-hop path (if it exists) toward a topology center. Fig. 17 demonstrates the topology shape
border, the search area of the final hop (hop n) may not be and the chosen source node locations. Fig. 18a shows the
sufficiently large to form an angular slice, as defined in average percent error in multihop distance estimation with
Section 2.2. This is an edge effect which reduces the multihop this set of source locations at topologies of node density
path distance. ¼ 0:001 nodes=m2 . Note that for each source location
The edge effect is not observed on an n-hop path when value, a new set of 2,000 independent topologies of random
the distance to the border is sufficiently large. For a node locations are formed. For each topology, a single-
communication range of R, a distance of nR from the sample multihop path is selected for each hop distance n.
source node to the border ensures that the angular slice Second, we place the source node at randomly selected
search area of each hop of the greedy multihop forwarding locations and vary the node density. Similarly, we form
can be formed without being limited by the border. Hence, 2,000 independent topologies for each node density value.
scenarios with source node located at the center of The average percent error results are illustrated in Fig. 18b.
topologies of size 2nR Â 2nR avoid the edge effect on paths In Fig. 18a, when the source node is sufficiently close to
of length n and less. the edge, the percent error graphs make a peak. For instance,
The average multihop distance is determined via the graph for source location ðÀ2R; 0Þ has a peak at hop
simulations for each hop distance value n. To find the distance 4. Note that the shortest distance of this location to
average distance in n hops for a selected source node the edge is 3R. Paths of length 3 and less are not affected by
location, 2,000 independent topologies are formed, and for the edge and have comparable percent error with those of
each topology, a single sample multihop path of n hops is topologies with source node at the center. However, for 4-
located. This path is determined by choosing a random hop paths, we observe a peak in percent error as some of
initial direction (as outlined in Section 2.2. Note that if this these paths experience a distance reduction in their final
random initial direction does not produce a multihop path (4th) hops. On the other hand, it is less likely that paths of
of length n, we aggressively scan the whole area for an higher hop distance are formed toward the edge at this
angle that produces an n-hop path for that particular proximity to the edge; hence, longer paths can be established
simulation topology. With this setting, some of the n-hop in directions away from the edge. Therefore, the average of
paths experience distance limitation by the border since the the percent error in distance estimation is less for higher
distance of the source node to the border is not guaranteed values of n. Similar trends are observed at other source
to be sufficiently large to avoid the edge effect. Further- locations, although the peak of the graph is observed at a
more, note that with a random choice of propagation smaller hop distance value for locations closer to the edge.
direction, not all the n-hop paths experience a distance Another observation in Fig. 18a is that the average percent
reduction. However, on average, the effect is a decrease in error value of the peak location is higher when the source is
the expected multihop euclidean distance of a randomly closer to the edge. As mentioned before, the peak location is
chosen n-hop path. In the simulations, it is observed that the found at a smaller hop distance when the source node is
reduction in the multihop euclidean distance is largely closer to the edge. Furthermore, the edge effect is pre-
caused by the decrease in the distance taken in the final hop dominantly on the final hop a multihop path and reduces its
under the edge effects. span, with an amount of reduction comparable for all values
In Fig. 18, we study the effect of edges on the multihop of hop distance. Hence, for a smaller hop distance, the
distance estimation in terms of the average of percent error corresponding euclidean distance is smaller and the ratio
defined by % error ¼ 100 Â jaÀej , where a and e denote an
a between the distance reduction and the path distance gets



Fig. 19. Estimation of hop distance pmf for ¼ 0:001 nodes=m2 . Fig. 20. Estimation of hop distance pmf for ¼ 0:002 nodes=m2 .

higher, producing a higher percent error in distance greedy distance maximization model which approximates
estimation. (Recall that the estimated multihop distance is the maximum multihop euclidean distance in planar net-
close to the multihop value as obtained without the edge works. We demonstrated that the Gaussian pdf used to
effects.)
model the maximum multihop distance distribution in 1D
In Fig. 18b, three different node density values are tested
networks does not accurately represent the planar distance
to study the edge effect on distance estimation. The results
distribution. Using maximum likelihood estimation of
demonstrate that for a smaller node density, the edge effect is
less pronounced. This is an expected result since the edge distance distributions, we conjectured that the maximum
effect reduces the final hop distance of a multihop path, distance can be accurately modeled by a transformation of
which has a stronger limitation on higher densities with the Gamma distribution. Furthermore, we provided expres-
larger single-hop spans. As the node density gets smaller, the sions for the expected value and standard deviation of this
node with the maximum distance in the final hop is located distribution, which can be used to compute the parameters
closer to the most recently selected node (as outlined in of the distribution and defines it uniquely. In order to
Section 2.2) and its location is limited less frequently by the investigate the accuracy of the model, extensive simulations
topology border. The diminishing character in the average are made. Distribution results are also used to estimate hop
percent error values is caused by the decrease in the ratio distance in a planar network using an MAP estimator.
between the amount of distance in the final hop to the
multihop path distance as the hop distance increases.
APPENDIX
5.5 Approximation of Hop Distance pmf, P ðNjDÞ CALCULATION OF req
The MAP estimator of hop distance N for given euclidean
Derivation of the equivalent radius req at a hop i is carried
distance D presented in Section 4 and given by (25) is
studied for randomly selected values of D. Figs. 19 and 20 out according to Fig. 21 in Cartesian coordinates and
illustrate the probability mass function obtained by simula-
tions and (19) for randomly deployed networks with
uniform node density values ¼ 0:001 nodes=m2 and
¼ 0:002 nodes=m2 , respectively. The results show that
the most likely hop distance is estimated accurately for all
euclidean distances and for both node density values.
Furthermore, there are two major hop distances that are
observed for each euclidean distance D when node density
is ¼ 0:001 nodes=m2 , whereas only a single-hop distance
has the dominant probability value in the pmf when the
node density is increased to ¼ 0:002 nodes=m2 indicating a
more deterministic distribution.

6 CONCLUSION
Determination of the maximum multihop euclidean dis-
tance corresponding to a given hop distance in a 2D
network is a complex problem. In this paper, we propose a Fig. 21. Calculation of equivalent radius req .



and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
AreaðA2 Þ ¼ sB ðsB À RÞðsB À riÀ1 ÞðsB À lb Þ;
where sA ¼ 1 ðla þ R þ riÀ1 Þ and sB ¼ 1 ðlb þ R þ riÀ1 Þ. The
2 2
area of the angular slice defined by points PiÀ1 , A, and B in
2
Fig. 22 is equal to AreaðSliceðPiÀ2 ; A; BÞÞ ¼

R , where
2
2
R þ riÀ1 2 À la 2

On Multihop Distances in Wireless Sensor Networks with Random Node Locations

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