1
Supplementary for A Location-free
Semi-Directional Flooding Technique for
On-demand Routing Protocols in Low-rate
Wireless Mesh Networks
Seong Hoon Kim, Member, IEEE, Poh Kit Chong, Member, IEEE, and Daeyoung Kim, Member, IEEE,
✦
1
I NTRODUCTION TO B OUNDED -D EGREE AND -D EPTH T REE (BDDT)
1.1 A BDDT-based block-addressing scheme
A bounded-degree-and-depth tree (BDDT) [1] is a
rooted tree in which depth and degree of a tree are
bounded by three network-wide constants: Cm , Rm , and
Lm . Cm and Rm are the maximum allowable number of
children a parent may have and the maximum number
of routers a parent may have as children, respectively.
Lm is the maximum depth in the network.
Based on these network-wide constants, each parent
router device at depth l assigns network addresses of
child router devices as follows: An = Aparent + 1 + (n −
1)Cskip(l). Here, Cskip(l) indicates an offset, which is
an address space for its descendant nodes, and can be
computed as follows
m −l−1
−Cm RL
m
Cskip(l) = 1+Cm −Rm
, if Rm ≥ 2
1−Rm
The network addresses of end devices, i.e., mesh clients
that cannot have descendant, is assigned in a sequential manner with the nth address, An , given by the
following equation: An = Aparent + Cskip(l)Rm + n
where 1 ≤ n ≤ (Cm − Rm ) and Aparent stands for the
address of the parent. We exemplify a network topology
consisting of mesh routers but not mesh clients for ease
of presentation. Fig. 1 of the main text depicts a BDDTbased logical tree topology (see Fig. 1(b) of the main
text) and an address space (see Fig. 1(c) of the main
text) corresponding to the physical topology in Fig. 1(a)
of the main text where Cm , Rm and Lm are 4, 3, and
• Seong Hoon Kim and Daeyoung Kim are with the Department of
Computer Science, Korea Advanced Institute of Science and Technology
(KAIST), Daejeon, 305-701, Rep. of Korea. E-mail: shkim08@kaist.ac.kr,
kimd@kaist.ac.kr.
• Poh Kit Chong is with Faculty of Engineering and Science, Universiti
Tunku Abdul Rahman, 53300 Kuala Lumpur, Malaysia. E-mail: chongpohkit@ieee.org.
3, respectively. For instance, in Fig. 1, the node C is
the second child router of the node A (i.e., a root).
Therefore, the node C is assigned 18 = 0 + 1 + 17 where
Cskip(0) = 17.
Because of the address assignment rule of the BDDT,
when receiving a packet to destination At , a node with
address Aln can know that At belongs to its descendant
if Aln < At < Aln + Cskip(l − 1) and non-descendant otherwise where Aln+1 = Aln + Cskip(l − 1). With the BDDT,
packets from any source nodes can reach any destinations by using the network address without additional
control overheads or routing tables. That is, if Aln < At <
Aln + Cskip(l − 1), a node forwards data packets to the
next hop node with address Anh = At for end devices
A −(Aln +1)
and with address Anh = Aln +1+ tCskip(l)
Cskip(l) for
routers. Otherwise, the packet is forwarded to its parent
node Aln . For example in Figs. 1(b) and 1(c) of the main
text, when the node I with network address 24, denoted
as I-24, sends a packet to the node O-20, it first checks the
child address space to decide whether it is its descendant
or not, i.e., 24 < At < 24 + Cskip(1) = 29. Since the
node O-20 doesn’t belong to the I-24’s descendant set,
the packet is forwarded to its parent node C-18 and is
then forwarded down to the node H-19. Eventually the
node O-20 receives the packet.
1.2 The BDDT Formation
A network starts from the root node with address zero
at depth zero. Then, a node wishing to join the network
performs a scan procedure to discover candidate parents
within its transmission range and sends an association
request to a potential parent node with the lowest depth
and good link quality among discovered nodes. The potential parent replies by sending the association response
containing the node’s network address computed by the
BDDT-based distributed address assignment mechanism
as explained above.
2
TABLE 1
Notations
15
ρ:0.73626
12 ρ:0.74217
min
min
the maximum allowable number of children a
parent may have
Rm
the maximum number of routers a parent may
have as children
Lm
the maximum depth in the network
n
the number of nodes in a network
N (u)
1-hop neighbors of u
depth(u) the depth of u
td(u, v)
the tree distance from node u to node v
tdN (u, d) the tree distance from node u to node v through
one of u’s neighbor such that tdN (u, d) =
td(v, d) + 1 for v ∈ N (u)
r(u)
the radius value of node u receiving a RREQ
packet
δ(u, v)
the shortest path hop distance between node u
and v
Cm
tdN (u, v)
tdN (u, v)
10
10
5
0 1
2
3
4
5
6
100
Distance (m)
Distance (m)
120 ρ:0.74615
80
60
40
20
0 2
4
5
6
7
8
9
5
min
8
9
10 11
min
(b) (4, 3, 9)
120 ρ:0.75858
120 ρ:0.75693
100
100
Distance (m)
Distance (m)
7
80
60
40
20
80
60
40
20
3
4
5
6
7
tdmin(u,
N
8
9
10 11
v)
(c) (5, 4, 7)
0 2
3
4
5
6
7
8
9
10 11
tdmin(u, v)
N
(d) (6, 5, 6)
Fig. 1. Boxplots indicating the median value and the
quartiles of Euclidean distance according to tree distance,
tdmin
N (u, v) with different address configurations: (a) (3,
2, 14), (b) (4, 3, 9), (c) (5, 4, 7), and (d) (6, 5, 6). ρ is
Pearson’s correlation coefficient between tdmin
N (u, v) and
the corresponding euclidean distance.
2
A N OTATION TABLE
For clarity, Table 1 summarizes the notations in use
throughout the paper.
3 T HE E XPLORATORY DATA A NALYSIS
T REE D ISTANCES
4
12
10
8
8
min
6
4
tdmin(u, v) = δ
N
3
4
5
6
6
7
8
9 10 11
(b) (4, 3, 9)
ρ:0.75726
2
5
7
8
9 10 11
ρ:0.74564
6
4
2
0 1
tdmin(u, v) = δ
N
2
3
4
5
6
min
tdN (u,
v)
7
8
9 10 11
v)
(d) (6, 5, 6)
Fig. 2. Boxplots indicating the median value and the
quartiles of tdmin
N (u, v) according to the shortest path discovered by AODV with different address configurations:
(a) (3, 2, 14), (b) (4, 3, 9), (c) (5, 4, 7), and (d) (6, 5, 6).
ρ is Pearson’s correlation coefficient between tdmin
N (u, v)
and the AODV’s shortest path.
tdN (u, v)
(a) (3, 2, 14)
0 2
6
3
N
(c) (5, 4, 7)
tdN (u, v)
2
tdmin(u, v)
10
2
40
4
min
tdN (u, v) = δ
0 1
tdN (u, v)
min
tdN (u, v)
12
60
3
2
9 10
(a) (3, 2, 14)
80
0 2
10
8
N
20
3
7
v) = δ
tdmin(u, v)
min
tdN (u,
100
6
4
min
(u,
N
td
0 1
120 ρ:0.74771
8
ON
Regarding the two properties above, the most important
factor is how much the tree distances are correlated with
the shortest path distance in terms of the three cases
in Fig. 3 of the main text. To confirm this, we took
exploratory data analysis (EDA) [2], [3]. For this, we
ran numerous simulations based on the configuration
in Section 7 with different address parameters where
we used 1000 random topologies, each of which has
10 communications pairs, and used AODV to find the
baseline shortest paths. Results of the experiments are
shown in Figs. 1 and 2.
As depicted in Fig. 1, the median values of physical
Euclidean distances of two arbitrary nodes tend to inincreases. The boxplots in Fig. 2 show
crease as tdmin
N
as a function of hop distance of the
quartiles of tdmin
N
corresponding shortest paths by AODV. As expected,
increases as those of shortest
the median values of tdmin
N
paths using AODV increases. To quantitatively measure
this, we use Pearson’s correlation coefficient ρ [4] to obtain the linear dependence of tree distance with physical
distance and shortest path distance by AODV, giving a
value between +1 and -1 inclusive.
In detail, the Pearson’s correlation coefficient ρ [4]
is used to measure the linear relationship of two random variables. Since the expected value of X can be
represented as μX = E[X], the Pearson’s correlation
coefficient ρX,Y of two random variables X and Y can
be expressed as below:
ρX,Y =
Cov(X, Y )
σX σY
(1)
where σX and σY are the variances of X and Y , respectively. The covariance Cov(X, Y ) is a measure of the
relationship between X and Y , which can be expressed
as below.
Cov(X, Y ) = E[(X − μX )(Y − μY )] = E[XY ] − μX μY
(2)
If X and Y are statistically independent random variables, then E[XY ] = E[X]E[Y ], thus implying that the
covariance of statistically independent random variables
is zero. The correlation coefficient is bounded between
3
-1 and +1. As ρ approaches 1, it implies that the linear
dependence between two variables is strong.
As can be seen in Figs. 1 and 2, ρ is about 0.75 for
reasonably mirrors both
all cases, meaning that tdmin
N
physical and logical distance.
≥δ
It is also important to note that theoretically tdmin
N
must always hold true. However, in Fig. 2, there exist
< δ. The root reason of this is
some cases where tdmin
N
that AODV sometimes fails to find the shortest paths due
to loss of route request or reply packets during discovery,
which is a well-known inferior route selection problem
[5] of on-demand routing protocols like AODV. Indeed,
we found that less than 10% of routes discovered by
AODV was longer hop distance than those of the tdmin
N .
In summary, tree distances reasonably reflect not only
the physical Euclidean distances, but also hop distances
because of direction diversity and multiple gradients.
4
T HE A LGORITHM
TO
C OMPUTE tdmin
N
Algorithm 1 getMinTD(At , lt )
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
24:
25:
26:
27:
28:
29:
30:
A00 ← 0; A01 ← 1 + Cskip(0) ∗ Rm + (Cm − Rm );
bStop ← f alse;
for l = 0 → Lm do
it ← nblist.begin;
while it = N U LL do
if it.A = At then
tdmin
← 1;
N
return tdmin
N ;
end if
if Aln = At then
bStop ← true;
end if
if Aln ≤ it.A < Aln+1 then
td ← it.l + lt − l ∗ 2 + 1
else
td ← it.l + lt − llca ∗ 2 + 1
Remove it from nblist.
end if
it ← it.next
if td ≤ tdmin
then
N
tdmin
← td;
N
end if
end while
if bStop = true then
return tdmin
N ;
end if
At −Aln −1
Aln ← Aln + 1 + Cskip(l)
Cskip(l);
Aln+1 ← Aln + Cskip(l);
llca = l
end for
To run the O(|N |) algorithm, it is necessary for two
preconditions. The first precondition is that the neighbor
table must also contain the depth of the neighbor nodes.
This can be easily done with IEEE 802.15.4 either by
receiving the depth of the neighbor node from beacons
or by locally computing it as in [6]. Together with this,
the second one is that the neighbor table must be sorted
in a strictly increasing order of network addresses, which
can be easily done using any sorting algorithm during
neighbor discovery. Based on these two preconditions,
the algorithm runs as follows. From depth zero, the algorithm iterates all the neighbor entries in an increasing
order while checking whether each neighbor address
belongs to address spaces of target’s ascendants. Then, if
there are neighbor nodes that share a common ancestor
with the target, the algorithm goes to the next depth after
removing unnecessary neighbor nodes without computing tdmin
N . This procedure is repeated until the target
address is found.
In detail, given address At and depth lt of target t,
line 1 initializes the first and the last addresses in the
address space at depth 0. Here, the CSkip function is an
offset for descendant nodes in a BDDT, details of which
are summarized in Section 1.2 of the supplementary
file. Line 2 initializes bStop which is a boolean variable
to indicate algorithm termination. For each depth, the
iterator, it is set to the first element of a sorted neighbor
table, i.e., nblist.begin in line 4. Then, the algorithm
goes over each neighbor entry as follows. Lines 6-9 are
the case where the target is one of its neighbors, and,
is one. Lines 10-12 are the case if the
therefore, tdmin
N
target address is found and hence it does not need to go
to the next depth, thus terminating the algorithm at the
current depth.
Line 13 checks whether or not the current neighbor
entry belongs to the descendant of the subtree root, Aln
(i.e., grey or black bars in Fig. 4 of the main text). If
it is, Aln is a common ancestor and therefore the tree
distance is computed. Note that since there can be a
common ancestor at deeper depths, the neighbor entry
is still required to be computed at the next depth. In
contrast, lines 16-17 compute the tree distance with the
previous common ancestor and removes the neighbor
entry because there is no further common ancestor. In
if needed. If the iteration is
lines 29-31, it updates tdmin
N
no longer needed, the algorithm terminates in line 25.
Otherwise, it updates Aln , Aln+1 and l for the iteration at
the next depth in lines 27-29.
As a result, as presented in the following theorem we
can find tdmin
in O(|N |).
N
Theorem 1. The time complexity of getMinTD is O(|N |).
Proof:
1 2
1 Lm
1
+ |N |(
) . . . + |N |(
)
Rm
Rm
Rm
Lm
∞
1 l
1 l
=|N |
(
) < |N |
(
)
Rm
Rm
T (n) =|N | + |N |
d=0
d=0
Rm
≤ cN = O(|N |)
=|N |
Rm − 1
where Rm = 0 and c is a positive constant.
(3)
4
T HE A LGORITHM
TO PROCESS
RREQ
d
s(0,0)
d( ,0)
rd
Remark The SDF algorithm depends on how well tree
distances reflect the shortest path distance. However,
there is a possibility that nodes physically closer to
destinations may have greater tree distances, such that
potential shortest paths may not be discovered and thus
cannot be used. Although it happens, it rarely affects the
overall performance because it is common for multiple
shortest paths for a source-destination pair to exist [7].
Therefore, even if a few candidate nodes on the shortest
path cannot rebroadcast RREQ packets, it is very likely
that RREQ packets can be flooded over the other shortest
paths. Indeed, as will be shown in Section 6 of the main
provides
text, the proposed algorithm based on tdmin
N
significantly better or comparable performances to NWF
and EHRP, demonstrating that the existence of this problem hardly affects the overall performance of SDF.
rd
Algorithm 2 describes the pseudo code of ProcessRREQ. Given a rreq packet as a parameter, it first
determines whether it is the destination or not in line 1. If
not, it checks whether the path cost accumulated by the
rreq packet is better than the one which is stored. If it is
better and the rreq.opt is not NWF, i.e. SDF, algorithm 2
makes a decision about whether to rebroadcast the RREQ
in line 7-8. That is, if
packet or not after obtaining tdmin
N
the RREQ packet passes the forwarding condition in line
8, it is rebroadcasted after carrying out line 9-15 which
consists of an adaptive RREQ timer window in Section
5.2.1 and radius update, respectively. Subsequently, since
can be less than the radius value, it may result in
tdmin
N
further reduction of RREQ flooding.
s
rs
Algorithm 2 ProcessRREQ(rreq)
1: if This node is the target then
2:
send RREP to source after discarding rreq.
3: else
4:
if rreq has better cost than current one then
5:
update routing and route discovery tables.
6:
if rreq.opt = N W F then
7:
tdmin
← getMinTD(rreq.Ad , depth(rreq.Ad ));
N
8:
if 0 < tdmin
≤ rreq.radius then
N
9:
if rreq.radius = tdmin
then
N
10:
W = Tf ;
11:
else
tdmin
1
N
12:
W = rreq.radius−td
min rreq.radius Tf ;
N
13:
end if
14:
rreq.radius ← tdmin
N ;
15:
end if
16:
end if
17:
rebroadcast rreq with delay window W .
18:
else
19:
discard rreq;
20:
end if
21: end if
rs
5
Fig. 3. Two unit disks with overlapping area
6
AN
EXPECTED WORST CASE OVERHEAD
Theorem 2. Given γ and σ, the greatest expected number of
rebroadcasting nodes in SDF is
E[|FA∩B (s, d)|] =
δ
σ{2(td(s, d)γ)2 cos−1 ( 2td(s,d)
)
− 12 δ(s, d)γ 2 4(td(s, d))2 − δ 2 }
where γ is a maximum distance of a bidirectional communication link a node may have.
Proof: The expected number of nodes in the area A∩
B is equal to the number of nodes in the transmission
zone covered by all the nodes rebroadcasting RREQs and
is given by
E[|FA∩B (s, d)|] = σAA∩B
(4)
where rx = γtdmin
and γ is a maximum distance of
N
a bidirectional communication link a node may have.
The number of nodes rebroadcasting RREQs in SDF
is the greatest when tdmin
N (u, v) is maximum such that
tdmin
N (u, v) = td(u, v).
To find the area AA∩B , we use the formula for the
circular segment as in Fig 3. Let rx and dx be the radius
of the circle centered at node x and the height of the
isosceles triangle based on the chord length a making a
central angle θ.
For the left circle centered at node s with radius rs
such that rs = δ + ε,
As =
=
isosceles triangle
Asector
− As
s
1 2
1
2 rs θ − 2 aδs
δ
where δ = δ(s, d)γ. Since 12 θ = cos−1 ( 2r
),
As = rs2 cos−1 (
δs
1
) − δs a
rs
2
(5)
Analogously, we can derive Ad and therefore, the area
of A ∩ B is
δs
δd
1
(6)
AA∩B = rs2 cos−1 ( ) + rd2 cos−1 ( ) − aδ
rs
rd
2
We express this equation with regard to rs , rd , and δ.
The equations of the two circles are
x2 + y 2 = rs2
(7)
5
(x − δ)2 + y 2 = rd2
(8)
with the source-destination pairs to be involved in flooding. For all simulations, we put a PAN coordinator at the
Combining eqs. (7) and (8) and solving for x give
center of the terrain whereas other nodes are placed ran(x − δ)2 + (rs2 − x2 ) = rd2
(9) domly in given terrains. The interface queue, with two
priorities served in FIFO order, between routing layer
and MAC layer had a maximum size of 10 packets where
δ 2 + rs2 − rd2
routing packets were processed with higher priority than
(10)
x=
data packets. Link status messages were periodically
2δ
From this, we can compute the heights, i.e., δs and δd , of transmitted every 15 second to check whether the link is
two segment triangles as follows: δs = x and δd = δ − x. available or not. For SDF and EHRP, address parameters
are explained in each simulation scenario.
δ 2 + rs2 − rd2
δ 2 − rs2 + rd2
The traffic model used constant bit rate (CBR) UDP
, δd =
(11)
δs =
2δ
2δ
traffic flows with 40-byte payloads at a rate of 1 packet/s.
Since the entire chord length a is 2y, we express a with In each simulation of 250 seconds, the first 50 seconds
were used for network formation, and ten communicaregard to rs , rd , and δ as follows
tion pairs among mesh routers were selected to measure
2
2
2
δ +rs −rd 2
rs2 − (
)
y2 =
the performance of SDF and NWF. Remember that in
2δ
s +rd )(−δ+rs −rd )
a LRWMN all mesh clients wishing to transmit data
= (δ+rs +rd )(δ+rs −rd )(−δ+r
.
(12)
2
4δ
packets requests one of the mesh routers to perform
Therefore, the entire chord length a give
routing on its behalf. Thus, this simulation setup is
a = 1δ (δ + rs + rd )(δ + rs − rd )(−δ + rs + rd )(−δ + rs − rd ) reasonable for realistic scenarios in LRWMNs.
(13) For the route discovery configurations used in both
SDF and NWF, we set packets to be queued at a send
Plugging eqs. (11) and (13) into eq. (6), we have
buffer of each source when route discoveries were ini2
2
2
2
2
2
tially issued by the source. The packets were dropped if
2
−1 δ + rs − rd
2
−1 δ − rs + rd
AA∩B = rs cos (
) + rd cos (
)−
they have waited in the send buffer for more than 10s.
2δrs
2δrd
1
We used ETX [11] as the routing metric and set the size of
(δ + rs + rd )(δ + rs − rd )(−δ + rs + rd )(−δ + rs − rd ) (14)
2
maximum routing table and the size of neighbor table as
As a consequence, since δ = δ(s, d)γ and the greatest 100 (6*100 = 600 bytes) and 20, respectively. Here, the size
flooding area is when both rs and rd are equal to of routing table can be thought of as large for resource
constrained mesh routers. Nonetheless, we used these
td(s, d)γ, the above expression reduces to
settings to observe the pure flooding effect regardless of
δ(s,d)
2(td(s, d)γ)2 cos−1 ( 2td(s,d)
)
AA∩B =
memory usage in NWF.
− 21 δ(s, d)γ 2 (4(td(s, d))2 − δ(s, d)2 ). (15)
Thus, we have the following result
E[|FA∩B (s, d)|] =
δ(s,d)
σ{2(td(s, d)γ)2 cos−1 ( 2td(s,d)
)
1
2
2
− 2 δ(s, d)γ 4(td(s, d)) − δ(s, d)2 }.
Thus, theorem is proved.
7
S IMULATION C ONFIGURATION
We evaluated our algorithm in ns-2 [8] and IEEE 802.15.4
implemented by J. Zheng et al [9]. We additionally
implemented AODVjr with semi-directional flooding
(SDF), and network-wide flooding (NWF) as well as
EHRP [6].
For all the routing protocols, we used common settings
by following ns-2 implementations of IEEE 802.15.4 and
configurations in [9]. We set the nodes to have 20m omnidirectional transmission range and 25m carrier sensing
range with two-ray ground radio propagation model. We
also apply a 0.2% statistical packet error rate (PER) for all
our experiments as in [10]. This may not exactly reflect
physical characteristics of wireless propagation, but it
is enough to observe how well SDF provides optimal
routes while allowing only nodes directionally correlated
8
A DDITIONAL P ERFORMANCE E VALUATIONS
8.1 Forwarding Nodes Directionality
Even though we showed the number of nodes rebroadcasting RREQ packets, this metric did not reflect the
directionality of RREQ packets, and therefore, we also
measure how much the directionality is achieved by
SDF by introducing a new metric for forwarding nodes
directionality, F N D(s, d), which is defined as:
→
→ + |−
|−
su|
ud|
(16)
F N D(s, d) =
→
−
|sd|
u∈F (s,d)
where F (s, d) is the set of nodes forwarding RREQ
packets initiated by source s to discover destination d.
This is a sum of the normalized physical distance of
node u involved in flooding from s to d with respect
to the physical distance between s and d. This metric
reflects how much rebroadcasting node u is directionally
correlated with the shortest path between s and d. Fig. 5
shows the forwarding nodes directionality as a function
of hop counts. The top four boxplots in Figs. 5(a), (b),
(c), and (d) represent the FNDs in NWF and the bottom
four 5(e), (f), (g), and (h) are FNDs of the SDF cases. Note
6
50
50
50
50
40
40
40
40
30
30
30
30
20
20
20
20
10
10
10
10
0
0
0
0
−10
−10
−10
−10
−20
−20
−20
−20
−30
−30
−30
−30
−40
−40
−40
−50
−50
0
50
−50
−50
0
(a)
50
−40
−50
−50
0
(b)
50
−50
−50
0
(c)
50
(d)
Fig. 4. Examples of location-free semi-directional flooding (Cm = 3, Rm = 2 and Lm = 14) with tree distance. The (a),
(b), (c), and (d), are the topologies with randomly deployed nodes with 100 mesh routers in a 100 × 100 m2 terrain.
Nodes centered on the large circle rebroadcast RREQs where the large circle is to emphasize nodes rebroadcasting
RREQs. Each red arrow presents the direction from the source to the destination.
700
400
300
200
100
0 2
3
4
5
6
7
500
400
300
200
100
0 2
8 9 10 11 12
Hop
3
(a) NWF-(3,2,14)
6
7
500
400
300
200
100
3
4
5
6
7
8 9 10
Hop
(e) SDF-(3,2,14)
400
300
200
100
3
5
6
7
500
400
300
200
100
3
4
5
6
7
8 9 10
Hop
(f) SDF-(4,3,9)
500
400
300
200
100
0 2
8 9 10 11
Hop
3
5
6
7
8
Hop
9
10
700
600
500
400
300
200
100
0 2
4
(d) NWF-(6,5,6)
700
600
0 2
4
600
(c) NWF-(5,4,7)
Forwarding nodes directionality
600
500
0 2
8 9 10 11 12 13
Hop
700
Forwarding nodes directionality
Forwarding nodes directionality
5
600
(b) NWF-(4,3,9)
700
0 2
4
Forwarding nodes directionality
500
700
700
600
Forwarding nodes directionality
600
Forwarding nodes directionality
Forwarding nodes directionality
Forwarding nodes directionality
700
3
4
5
6
7
8 9 10
Hop
(g) SDF-(5,4,7)
600
500
400
300
200
100
0 2
3
4
5
6
7
8 9
Hop
(h) SDF-(6,5,6)
Fig. 5. Forwarding Nodes Directionality as a function of hop count discovered according to different address
configuration. The first and second rows are SDF and NWF, respectively. Columns from left to right are address
configuration (Cm , Rm , Lm ) (3,2,14), (4,3,9), (5,4,7), and (6,5,6).
that the pair of each column, i.e. Figs. 5(a) and 5(e), (b)
and (f), (c) and (g), and (d) and (h) in the boxplots ran
on the same simulation configuration for NWF and SDF.
As a result of this, the median value of FND for two
hop routes is about 350 in NWF, and distribution of
its data set is widely dispersed. As the hop count of
the shortest path increases, the NWF’s FNDs decrease
inverse proportional to it because RREQ packets in NWF
are omni-directionally flooded regardless of the direction
of the destination. This trend does not mean a reduction
of RREQ overhead. Instead, it is due to the increase of
→
−
|sd| in Eq. 16, meaning that the spatial proportions of
nodes in the shortest paths increase in the given terrain
size.
On the other hand, the SDF’s FND results tend to
gradually increase according to increasing hop counts.
However, in all cases FND is less than 100. Interestingly,
the data sets of NWF show routes having more than
10 hops whereas those in SDF are at most 10. This
indicates that some of routes based on NWF fail to
discover the shortest path resulting from the inferior
route selection problem of on-demand routing protocols
[5]. These results imply that RREQ packets using SDF
are efficiently flooded towards the destination since SDF
reduces the RREQ overheads by preventing nodes with
low correlation to the shortest path from rebroadcasting
the RREQ packets.
8.2 Influence of the Network Density
To measure the performance of SDF with respect to
various node densities, we used a constant terrain size
while increasing the number of router nodes. The results
of Fig. 6 depict the influence of varying node density.
When the node density is low (i.e., 70 nodes), we
observed that SDF requires about 79.4% less routing
overhead compared to NWF. This is because with low
density each router node has less information about the
7
EHRP
NWF
SDF
60000
50000
40000
30000
20000
10000
0
60
80
100 120 140 160 180 200
The number of routers
0.98
6.5
0.96
6
0.032
EHRP
NWF
SDF
5.5
0.94
0.92
0.9
5
4.5
4
0.88 EHRP
NWF
SDF
0.86
60 80 100 120 140 160 180 200
The number of router nodes
(a)
(b)
EHRP
NWF
SDF
0.03
End-to-end delay
70000
Hop count
80000
Packet Delivery Ratio
Routing overhead (packets)
90000
3.5
0.028
0.026
0.024
0.022
0.02
0.018
3
0.016
60
80 100 120 140 160 180
The number of router nodes
200
60
(c)
80 100 120 140 160 180 200
The number of router nodes
(d)
Fig. 6. Performance results of (a) routing overhead, (b) Packet delivery ratio, (c) hop count, and (d) end-to-end delay
as a function of node density.
Route discovery delay (sec)
0.8
0.7
0.6
8.3 Influence of the Traffic Load
EHRP
NWF
SDF
0.5
0.4
0.3
0.2
0.1
0
60
80
100
120
140
160
180
200
The number of router nodes
(a)
Fig. 7. Route discovery delay.
tree distances to the others and SDF is constrained by
the limited tree distance information. However, as density becomes higher, routing overhead of NWF steeply
increases, while there is a gradual growth for SDF.
In terms of packet delivery ratio, SDF outperforms the
others in all cases. In particular, as density increases,
the packet delivery ratio of SDF increases up to about
97.3% when the number of nodes is 130, but shows
a slight decrease afterward. On the other hand, NWF
shows that its packet delivery ratio drops steeply due
to a considerable amount of routing overhead. EHRP
shows a gradual improvement and matches NWF after
130 nodes. These results are due to SDF benefitting from
increasing neighbor information, which are likely to
reduce the detour routes. As expected from our previous
observations, the results of hop count and end-to-end
delay shown in Fig 6(c) and 6(d) show a similar trend.
Remarkably, SDF shows comparable or better performance than NWF and EHRP in terms of hop count and
end-to-end delay.
We plot route discovery delay in Fig. 7. Due to the
adaptive RREQ timer reflecting the tree distance, route
discovery delays of both SDF and NWF tend to form a
’U’ shape that initially reduces as the density increases,
and curves upwards after an inflection point as the probability of collisions increases with higher node densities.
However, the delay for SDF remains relatively lower
for higher densities compared to lower densities, while
NWF experiences a sharp increase in delay as the density
of nodes increases mainly due to collisions.
Since the number of communication pairs has an effect
on the initiation of router discoveries, we also evaluated
SDF with respect to various traffic loads. The results in
Fig. 8(a) present an exponential increase of the routing
overhead of NWF whereas that of SDF slowly goes
up, resulting in 88.1, 86.5, 87.0, 84.6 and 72.6 percent
reduction of routing overhead for 10, 20, 30, 40 and
50 communication pairs. With increasing communication
pairs, many route discoveries are initiated. However, due
to limited interface queue size (i.e., 10), the results show
saturation in NWF when the number of communication
pairs is close to 50. Subsequently, as can be seen from the
packet delivery ratio of NWF in Fig. 8(b), we observed
sudden drops when the number of communication pairs
is 50, resulting from frequent route discovery failures
due to resource shortage in NWF. In contrast, SDF shows
a linear decline due to reduced routing overhead. Note
that EHRP shows comparable performance with NWF
but is always worse than SDF due to triangular detours
and the growth of collision probability resulting from
traffic concentration at nodes close to the root. Hop
count metric in Fig. 8(c) shows that SDF and NWF are
comparable, while the EHRP has about one-hop longer
routes. In Fig. 8(d), end-to-end delay tends to increase
with the increasing number of communication pairs, yet
SDF shows comparable or better performance than the
others.
8.4 Influence of the Node Failure
In this experiment, we focused on the protocol behavior
in presence of frequent node failures. Node failures
introduced here are modeled as a uniformly distributed
on/off process by following the model in [12]. We increase the number of nodes that toggles their state from 0
to 50 percent with random interval from 0 to 250 second.
Note that the results shown here exclude communication
pairs if one of either source or destination becomes a
failing node.
As can be seen in Fig. 9(a), compared to a zero failed
node case, routing overhead in NWF increases with the
increasing number of failed nodes. The routing overhead
in NWF shows saturation when the number of failed
nodes is more than 20. This is because any source nodes
8
40000
30000
20000
10000
0
10
15 20 25 30 35 40 45 50
The number of communication pairs
1
5.2
0.95
5
0.03
EHRP
NWF
SDF
0.028
End-to-end delay
EHRP
NWF
SDF
4.8
0.9
Hop count
50000
Packet Delivery Ratio
Routing overhead (packets)
60000
0.85
0.8
4.6
4.4
4.2
0.75 EHRP
NWF
SDF
0.7
10 15 20 25 30 35 40 45 50
The number of communication pairs
(a)
EHRP
NWF
SDF
0.026
0.024
0.022
0.02
4
3.8
0.018
10
15 20 25 30 35 40 45 50
The number of communication pairs
(b)
10
15 20 25 30 35 40 45 50
The number of communication pairs
(c)
(d)
Fig. 8. Performance results of (a) routing overhead, (b) packet delivery ratio, (c) hop count, and (d) end-to-end delay
as a function of traffic load.
25000
20000
15000
10000
5000
0
0
10
20
30
40
The number of failed nodes
(a)
50
5.6
5.4
5.2
EHRP
NWF
SDF
End-to-end delay
30000
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55 EHRP
0.5 NWF
SDF
0.45
0
10
Hop count
EHRP
NWF
SDF
35000
40000
Packet Delivery Ratio
Routing overhead (packets)
45000
5
4.8
4.6
4.4
4.2
4
20
30
40
50
10
The number of failed nodes
15
20
25
30
35
40
The number of failed nodes
(b)
(c)
45
50
0.03
EHRP
0.029 NWF
0.028
SDF
0.027
0.026
0.025
0.024
0.023
0.022
0.021
0.02
0.019
0
10
20
30
40
50
The number of failed nodes
(d)
Fig. 9. Performance results of (a) routing overhead, (b) Packet delivery ratio, (c) hop count and (d) end-to-end delay
as a function of node failures.
failing to discover the destinations that were off at that
time kept trying to find the destination again, increasing
the number of route discoveries within a short interval.
On the other hand, the growth of routing overhead in
SDF is relatively small. These routing overheads incurred
due to the failing nodes also affect the packet delivery
ratio as depicted in Fig. 9(b). As a consequence, the
packet delivery ratio of the SDF outperforms that of the
NWF when the number of failed nodes is more than
20. Unlike SDF and NWF, the impact of failing nodes
close to the root, which may be the intermediate nodes
along the tree routes, is more severe for EHRP. The
results in Fig. 9(c) and 9(d) reveal that the hop count
of SDF is slightly longer than that of NWF, and the
end-to-end delay of SDF is lower than that of NWF.
From these results, we conclude that with even small
amount of routing overhead, SDF provides better routing
performance compared to EHRP and NWF.
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A PPENDIX
A summary of terms in use throughout this paper appears in Table 2.
9
Abbreviation
AODV
BDDT
DF
DODAG
EDA
EHRP
FND
LRWMN
NWF
RREP
RREQ
RRER
SDF
WMN
WMAN
WLAN
WPAN
Term
Ad-hoc On-demand Distance Vector
Bounded-Degree-and-Depth Tree
Directional Flooding
Destination-Oriented Directed Acyclic Graph
Exploratory Data Analysis
Enhanced Hierarchical Routing Protocol
Forwarding Nodes Directionality
Low-Rate Wireless Mesh Networks
Network-Wide Flooding
Route Reply
Route Request
Route Error
Semi-Directional Flooding
Wireless Mesh Networks
Wireless Metropolitan Area Networks
Wireless Local Area Networks
Wireless Personal Area Networks
TABLE 2
Abbreviation and terms in use