Modelling the Behavior of a Beacon-Based Link Sensing Mechanism
with Variable Sensing Range
Geir Egeland
Department of Electrical and Computer Engineering
University of Stavanger, Norway
geir.egeland@gmail.com
Abstract
The links in an ad hoc or wireless mesh network are normally kept alive by the exchange of beacon-messages between neighboring nodes. These beacons are prone to collisions with traffic from hidden nodes. If several beacons
are lost due to overlapping transmissions, the node expecting the beacons erroneously assumes that the link is down.
This is called an apparent link-failure. This paper provides
an analytical model of apparent link-failures in a mesh network. The paper also extends the model with an algorithm
for finding the upper and lower bound for apparent linkfailures in an arbitrary mesh topology. The model is verified using simulations. The validity of the model is investigated using different link models. Since the radio sensing
and transmission range of a node influence its ability to detect ongoing transmissions, the paper also analyze how the
sensing range affects the apparent link-failure model.
1. Introduction
With the advent of the IEEE 802.11s standard [1] and
implementation in enabled devices [2], ubiquitous computing is becoming a reality, promising connection of devices and users in a truly ad hoc fashion. The usability of
such wireless ad hoc networks will depend heavily on the
routing protocol’s ability to preserve and detect failures of
links between neighboring nodes. In the majority of cases,
link-failures are present in an ad hoc network regardless of
the use of link-maintenance schemes. Some link-failures
are unavoidable, such as when a wireless node deliberately
leaves a network or is subject to a power failure. Also, a
link will cease to be operative when two nodes move outside each others radio transmission range. In addition to
these, a set of link-failures exists which are referred to as
apparent link-failures. They are primarily caused by wireless links being vulnerable to radio induced interference,
978-1-4244-9328-9/10/$26.00 ©2010 IEEE
Paal E. Engelstad
University of Oslo, Simula and
Telenor GBD&R, Norway
paal.engelstad@telenor.com
but also when a link-maintenance mechanism erroneously
assumes a link to be inoperable due to loss of beaconmessages. The purpose of beacons is to detect and keep
links alive. Beacons are normally broadcasted and are thus
not acknowledged, i.e. they are unreliable and vulnerable to
overlapping transmissions from hidden nodes [16].
Apparent link-failures are often a significant cause to
performance degradation of ad hoc networks since erroneous routing information is spread in the network. Also, it
might lead to a disconnected topology or less optimal routes
to a destination. Analysis of a real life network [12] has
demonstrated that it takes a significant amount of time to restore failed links [8]. An example of the effect of these failures is illustrated in Figure 1. Using ns2 [13] we have measured the throughput from node d8 →d7 for the topology
shown in Figure 5a. As the load from the hidden nodes [16]
increases, the link-maintenance mechanism of the routing
protocol [5] starts to loose beacons and assume that the link
between node 7 and 8 is unavailable, i.e. an apparent linkfailure. Thus, the throughput from node d8 →d7 is reduced,
either because the routing protocol forces the data packets
to traverse a longer path or simply because node 7 drops
packets when buffers are filled as a result of having no operational route to node 8. The throughput would remain relatively stable if the apparent link-failures were eliminated,
as seen from the “No apparent link failure” graph in Figure
1.
The main contribution of this paper is an analytical
model for the link-failure probability in an ad hoc network, where link-failures are a result of lost beacons caused
by overlapping transmissions from hidden nodes. With a
model in place it is possible to detect and avoid undesirable
topologies that might lead to a high frequency of apparent
failures, thus improving the performance and general user
experience. Being able to provide stable and reliable links
is prerequisite for the success of ubiquitous computing. The
model is based on prior work [7] and makes use of the probability pe , that a beacon is lost as a result of a collision with
search on ad hoc networks (e.g. cf. [11] [10]), the existence
of a link i,j depends on the Euclidean distance between the
two nodes vi and vj . If we assume a log-normal shadowing
radio propagation model [4] where the logarithmic value of
the mean power at different locations is normally distributed
with standard deviation σ around the logarithmic value of
the area mean power, the link-existence probability [10] is:
0.06
Fixed rate (d8 → d7 ) with no apparent link-failures.
Throughput (d8 → d7 )
0.05
0.04
0.03
Fixed rate
(d8 → d7 ) with
apparent link-failures.
0.02
pd (r) = 1 −
Apparent link-failures
No apparent link-failures
0.01
0.00
0.05
0.10
0.15
0.20
Load from hidden nodes {d4 , d10 , d12 } (λc T )
1
2
h
1 + erf
10log(r/r0 )
√
2log(10)ψ
i
4
,ψ =
σ
η
(1)
0.25
Figure 1: Throughput from node d8 → d7 with and without apparent link-failures for the topology in Figure 5a. By setting an
infinite time-out value, link-failures are artificially removed.
transmissions from hidden nodes, to model a beacon-based
link-maintenance mechanism [5]. This paper also extends
this model with an algorithm for calculating pe in an arbitrary topology, thus improving the bounded pe found in [7].
The radio sensing range (rcp ) of a wireless node determines
its ability to detect whether the radio channel is busy and
will affect the loss of beacons and the apparent link-failure
probability. Consequently, the paper examines the impact
of the radio sensing range on the apparent link-failure probability model.
The rest of the paper is organized as follows: In Section 2
the link-models and the beacon-based maintenance scheme
is introduced. In Section 3 assumptions and the analytical
model is presented. In section 4 an algorithm is developed
in order to apply the model for arbitrary topologies. In Section 5 the analytical model is studied when the radio sensing
range is increased in addition to studying the effect of using
a distance-dependent link model. Related work is discussed
in Section 6 and finally conclusions are drawn in Section 7.
2. Network Link Model
2.1. Radio-link models
In radio communications, the average value of the received signal power on a link s,d decreases with an increasing geometric distance between the two nodes vs and
vd . This is often referred to as pathloss, and the area mean
power, Pa is often modeled by a power law Pa ∝r−η where
η, depends on the terrain and the environment. In the analyses we first use a pathloss model and assume that a signal
transmitted from vs is received correctly at vd if the received
signal power exceeds a minimum required threshold. Thus,
each transmitting node has a circular coverage area of radius
r0 . In reality however, the received power levels might vary
significantly in space and time around the area mean power
value of the pathloss model. For the well-known link failure
model that is distance-dependent and is being used in re-
where r0 is given by the same threshold as for the area mean
power model. Empirically ψ may vary in the range 0 to
6 [10]. In the analyses presented later, ψ will be varied
and set to either of the values {0.25,2.5}. These two values
alone gives good insight into the effect of radio propagation,
e.g. as seen in Figure 10c later in the paper.
2.2. Beacon-based link maintenance
In wireless network for ubiquitous computing, links are
usually established and maintained proactively by the use of
one-hop beacons, which are exchanged between neighboring nodes. Beacons are broadcasted in order to conserve
network resources. Thus, every link on which a beacon
is received effectively obtains maintenance from only one
transmission. Broadcast packets are not acknowledged and
beacons are therefore inherently unreliable. A node anticipates to receive a beacon from a neighbor node within a
given time-interval and can accept that some beacons occasionally will be missing due to various error conditions.
However, if a node fails to receive a number (θ+1) of consecutive beacons, it will assume that the link is inoperable.
The value of θ is typically a tradeoff between providing the
routing protocol with stable and reliable links (a large θ),
and the ability to detect link-failures in a timely and fast
manner (a small θ). Since beacons are broadcasted, they
are unable to take advantage of RTS/CTS signaling, which
adds protection against hidden nodes for the IEEE 802.11
MAC protocol’s unicast data transmissions. Although some
beacon loss is avoided using RTS/CTS for the unicast data
traffic in the network, it will only help the links of the node
that issues the CTS. Examples of routing protocols that
make use of beacons are the proactive protocol Optimized
Link State Routing (OLSR) [5] and an optional mode of operation for the reactive Ad hoc On-Demand Distance Vector
(AODV) routing protocol [15]. The typical value of θ is 2,
which we will use later in the analysis.
Various beacon-based schemes may differ in the method
used to determine whether or not a failed link is operational
again. Link-stability is desirable and introducing a link too
early can lead to a situation where a link oscillates between
an operational and a non-operational state. This can be
avoided by measuring the signal-to-noise ratio (SNR) of the
failed link and define the link as operational when beacons
are being received and the SNR is above a defined threshold [3]. However, if SNR measurement is not available or
not practical, a simple solution is to introduce hysteresis by
requiring a number of consecutive beacons to be received
(θh +1) before the state of the link again is set to be operational. This is the solution chosen in this analysis.
s0
B
(2)
where p0 is the probability that the packet queue of the hidden node is empty and ωb represents the average transmission time of the beacon packet.
p0 =
I:Isolated, node s1 λc = 0

1 − mρ



 1 − mρ(m+1)
m+1−ρ
III:Connected, node s1 λc = 0
1−ρps1 −ρ
II:Isolated, node s1 λc > 0
D
D
s2
s7
s0
B
s1
s5
s3
D
s6
D
s4
(a) Isolated hidden nodes
pe
The work in [7] provides an upper and a lower bound for
beacon loss probability for IEEE 802.11 based networks.
We will now briefly describe the beacon loss model using
the topologies in Figure 2.
The model depends on three main assumptions. First,
it is assumed that a beacon sent by a node has a negligible probability of colliding with a beacon from any of the
neighboring nodes since beacons are short packets that are
transmitted periodically and at a random instant at a relatively low rate. Second, it is assumed that the probability
of a beacon colliding with a data transmission from any
of the (non-hidden) neighboring nodes also is negligible,
i.e pe pcoll because the IEEE802.11 has mechanisms that
reduce such collisions to a minimum. Third, the packet
buffers of a node is modeled as an M/M/1 queue with packet
arrival rate λc and the channel access and data packet transmission times are exponential distributed with parameter
1/µ. Even though traffic in a real network will follow other
distributions, results presented later in the paper based on a
large number of random independent traffic scenarios suggest that these assumptions are fair.
The probability of a collision between a beacon from
node s1 and a data packet from the hidden node s2 in Figure
2 can be shown to be:

1−ρ




1−ρ

D
s6
s4
3.1. Beacon-loss probability pe
pe = 1 − p0 · e
s1
s3
s7
s5
(b) Connected hidden nodes
Figure 2: Sample topologies where the hidden nodes {s2 , s4 , s6 }
are isolated or connected. When the hidden nodes send data (D),
this may collide with the beacons (B) sent by node s0 .
3. Link-Failures Caused by Beacon-loss
−λc ωb /µ
D
s2
(3)
IV:Connected, node s1 λc > 0
Eq. (3) is an expression for p0 where m is the number of
hidden nodes. The hidden nodes are either isolated (Figure 2a) or connected (Figure 2b). The probability that node
s1 gains access to the channel is given by ps1 and an approximate expression is found in [7]. If the hidden nodes
are isolated (Figure 2a), i.e. outside the transmission range
of each other, the probability that one of them sends a data
1 − pe
pe
pe
s0,0
s1,0
s2,0
s2,1
1 − pe
s2,2
pe
1 − pe
(1 − pe )
Figure 3: A Markov model of a link-sensing mechanism.
Table 1: Simulation parameters.
IP/MAC layer Values
Beacon/
30/
Data
1500 bytes
MAC
802.11
protocol
Queue Length
50
Physical layer Values
Propagation
Free Space
model
Data rate
11Mbps
Simulation
Values
Simulation/ 900s/25s
transient time
Traffic/
Poisson
Distribution
250m µs Replications
#50
rcp , rrx
packet which overlaps with a beacon is binomial distributed
and given by:
pe =
Pm
k=1
m
k
k
pe (1 − pe )m−k
(4)
The connected case represents an upper bound for the beacon loss probability while a lower bound will be represented
by the isolated case [7].
3.2. A model for apparent link-failures
If we assume that the event of losing a beacon is random and independent, apparent link-failures can be analyzed using a Markov model as shown in Figure 3 where
the state variable si,j describes the number of i∈[0, θ] beacons lost and j∈[0, θh ] beacons received in the hysteresis
state. Solving the state equation, the probability of apparent
link-failure (pf ) is found as the sum of the state probabiliPθh
ties j=1
pi,j given by:
pf (pe ) = p3e (2 − pe )/(p3e − pe + 1)
(5)
where pe is the probability of losing a single beacon and is
given by Eq. (2)–(4).
In order to test the model’s accuracy, we used the ns2
simulation model [13] with OLSR [5] as the ad hoc routing
protocol. The sensing range (rcp ) of the physical layer was
set to be equal to the transmission range (rrx ). This is not
the case in a real network, but simplifies our analysis and
provides some topology control. In section 5 we will study
~
Algorithm 1: H(G,
S)
Data: An undirected graph G(V, E), a directed graph S ⊆ G.
u , H l , of the vertice n that
Result: The number of neighbors, Hn,i
n,i
are hidden from vertice i and a traffic indicator Λn .
(a) Isolated hidden nodes.
(b) Connected hidden nodes.
Figure 4: The probability of link-failure (pf ) for the sample
topologies where the hidden nodes {s2 , s4 , s6 } hidden nodes send
data (D) which may collide with the beacons (B) sent by node s0 .
the effect of the sensing range on the beacon loss probability, i.e. for the more realistic case of rcp >rrx . As Figure 4
shows, the model matches well with the simulation results.
3.3. Apparent link-failure probability for
complex traffic patterns
In order to calculate the link-failure probability for links
in a given topology with an arbitrary traffic pattern, an algorithm is needed to determine the number of hidden nodes
that have an impact. A topology can be described as a directed graph G=(V, E), where the nodes in the network
serve as the vertices vj ∈V (G). Any two distinct nodes
vj and vi create an edge i,j ∈E(G) if there is a link between them. A random traffic pattern where a set of nodes
transmit data over a link i,j ∈E(G) will form a directed
graph S(V, E) that is a subset of G. It is assumed that
every node vj ∈S generates data packets at the same rate.
Alg. 1 calculates the number of neighbor nodes of n
which are hidden from a vertice i∈V (G):i,n ∈E(G) where
hu =|{j, ∀j:j∈V (G) ∧ n,j ∈E(G) ∧ ∃j→k∈V (S) ∈E(S)}|.
Applying Eq. (5) on these parameters will give the upper
bound link-failure probability pf for the link n→i. For
the calculation of the lower bound, an average value for
the number of hidden nodes is used, which is denoted hl
in Alg. 1. The justification for this is that for a set of nodes
R⊆V (S) hidden from node i, the carrier sense nature of the
MAC protocol will in the cases where two nodes k, z∈R
where ∃z6=k:z,k ∈E(G) result in that only a subset of the
nodes in R will transmit data at any given time. The parameter hl ≤hu is the average number of nodes in R that
transmit data at a given time. The upper bound for the
average apparent link-failure probability for a graph (G, E)
begin
H l ← ∅; H u ← ∅;Λ ← ∅
for i ∈ V (G) do
J ← {j, ∀j:i,j ∈E(G)};
for n ∈ J do
R ← {r, ∀r6=i:n,r ∈E(G)};
for k ∈ R do
if |{j, ∀j:k,j ∈E(S)}|>0∧k∈G
/ i then
hu ← hu + 1
N ← ∅;
for k = 0 to 2|R| do
ni ← 0; ca ← ∅;
for p = 0 to |R|] do
if k>>p & 1 ∧ n,Rp ∈ E(S) then
ca ← ca ∪ n,Rp ; ni ← ni + 1
if not
[∃z:n,z ∈ca ∧ ∃w6=z:n,w ∈ca:z,w ∈E(G)]
then
N ← N ∪ ni
P|N |
l
1
Hn,i
← |N
k=0 Nk
|
u
Hn,i
← hu
Λn ← |{j, ∀j:n,j ∈E(S)}|?0:1}
can now be calculated as:
puf
1
=
|E(G)|
X
∀i,n ∈E(G)
3
u
u
pe (Hn,i
)
2 − pe (Hn,i
)
3
u
u
)+1
) − pe (Hn,i
pe (Hn,i
(6)
where H u is provided by algorithm 1. The expression for pe
is given by Eq. (2) and p0 is calculated according to Eq. (3)
(III:Connected for Λn = 0 and IV:Connected if Λn > 0).
The calculation for the lower bound for the average apparent
link-failure probability (plf ) is analogous to Eq. (6) using
H l and Eq. (4).
For a node n, by measuring the traffic load from neighbor nodes j:n,j ∈E(G) and combining this with the 2-hop
horizon routing information gathered from the beacons, it
can estimate the probability of which neighbor nodes it is
likely to receive beacons in a timely manner. This can serve
as additional information for the link-maintenance mechanism when deciding on the status of a link.
4. Link-failures in Arbitrary Topologies
We now want to investigate how the analyses of the
topologies in Figure 2 can be applied to more complex mesh
topologies. Without loss of generality, we now focus on the
two topologies in Figure 5 as examples, observing that the
analysis can easily be generalized for any arbitrary mesh
topology. The topologies in Figure 5 do not resemble the
topologies in Figure 2, but equations Eq. (2)–(5) will together with Alg. 1 be able provide an upper and lower
bound for the link-failure probability pf .
The simplest approach to analyzing a bursty traffic pattern is to generate a snapshot of the traffic in the topology.
d10
d6
1.0
d10
d8
d12
d4
d8
d2
d7
d9
d11
d2
d7
d0
d1
d3
d5
(a) Topology A
d0
d9
d1
d11
d3
d5
(b) Topology B
Probability of apparent link failure (pf )
d4
Upper bound
Lower bound
1.0
Upper bound
Lower bound
λ = 0.9
λ = 0.5
Probability of apparent link failure (pf )
d6
λ = 0.4
0.8
λ = 0.3
0.6
0.4
λ = 0.2
0.2
λ = 0.9
0.8
λ = 0.5
0.6
λ = 0.4
0.4
λ = 0.3
0.2
λ = 0.2
Figure 5: The distribution of nodes in two example topologies.
λ = 0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Probability of traffic on a link (ptx )
(a) Results for topology A
1.0
0.0
0.1
λ = 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Probability of traffic on a link (ptx )
1.0
(b) Results for topology B
Figure 7: Analytical results for the upper/lower bound of the apparent link-failure probability for the topologies in Figure 5.
(a) Results for topology A.
(b) Results for topology B.
Figure 6: Apparent link-failure probability for Figure 5 (λc =
0.2). Simulation results are shown with a 95% confidence interval.
We assume that the time between each snapshot is sufficiently long for the traffic patterns of each snapshot to be
considered independent and that for each link in the topologies in Figure 5, a burst of data packets is transmitted with
the probability ptx /2. Each node generates data packets
within a burst according to a Poisson process with the rate
parameter λc . This snapshot will represent a possible data
transmission pattern. By generating a large number of random snapshots for a given ptx , the overall average apparent
link-failure probability for a given λc can be found.
Figure 6 shows the analytical average upper and lower
bound for the apparent link-failure probability for λc =0.2
for Topology A and B (Figure 5) using Alg. 1 and Eq. (2)–
(5) on the randomly generated traffic patterns. The figure
also shows simulation results for the average apparent linkfailure. As the simulation results show, the analytical upper and lower bound provide a good indicator of the average link-failure probability even though it can be seen that
the gap between the upper and lower bound increases as
ptx →1. This is a result of complex traffic patterns and interaction between the nodes that the simple model does not
incorporate. At low values for ptx , the model’s upper and
lower bound is as expected, more accurate.
In Figure 7 the upper and lower bound link-failure prob-
ability for different values of λc is shown. As can be seen
from the figure, for small and large values of λc , the gap
between lower and upper bound is negligible. The reason
for this is that when λc '0, the sum of the packets awaiting
transmission in the buffers of the hidden nodes is almost
zero in both the isolated and the connected case. Therefore,
the apparent link-failure probabilities are almost identical.
For the case when λc /1, the sum of packets awaiting transmission in the buffers of the hidden nodes is always greater
that zero, i.e. there is always a packets waiting to be transmitted. Hence, the difference in apparent link-failure probability is almost negligible. For 0.2<λc <0.6, the various
combinations of empty and non-empty buffers for the isolated and the connected case is large, thus it is expected that
there will be a difference in the upper and lower bound.
5. Factors That Impact the Number of Hidden
Nodes
5.1. The radio sensing range
Until now it has been assumed that the radio transmission range (rrx ) and the radio sensing range (rcp ) are equal.
However, in a real network, the sensing range is normally
larger than the transmission range, i.e. rcp >rrx . In this
section we investigate in more detail how the sensing range
affects the apparent-link failure probability. Before doing
so, we note that at modest traffic intensities in Eq. (2),
it is primarily the number of actively transmitting hidden
nodes (given by the term p0 ) that determines the apparent
link-failure probability. At high traffic intensities, on the
contrary, p0 does not play a role, since the p0 -term is suppressed in Eq. (2), and pe =1 for links with hidden nodes.
In this stage, it is the links with no hidden nodes that contributes to an apparent link-failure probability below unity.
r cp
rrx
s0
B
s1
s2
D
s3
(a) Sample topology with rcp >rrx .
(a) rcp /rrx =1.8.
(b) rcp /rrx =1.
Figure 9: P Analytical and simulation
pf =1/|E(G)| ∀ij ∈E(G) pf (i, j).
results
for
Table 2: Simulation results for λc = 0.9.
rcp >rrx
(b) Number of hidden nodes.
(c) Links with no hidden nodes.
hu
hl
|E(S)|
h0
pf
4.3
1.3
500
97
0.806
hu
hl
|E(S)|
h0
pf
2.8
1.1
500
94
0.812
rcp =rrx
Figure 8: The effect of increasing the sensing range for rcp >rrx .
Thus, when investigating the role of the sensing range, we
are interested in how the sensing range affects the number
of hidden nodes (for the low λc case) and how it affects the
number of links with no hidden nodes (for the high λc case).
Obviously, the sensing range influences the probability
of a beacon loss, as shown for the sample topology in Figure
8a. When the sensing range is larger than the transmission
range, it is possible for a beacon from s0 to overlap/collide
with a data packet from s3 at node s1 (Fig 8a). Hence, the
transmission interferes with a larger part of the area covered
by the network topology and is thus subject to a potentially
higher number of hidden nodes. Thus, the number of hidden
nodes is expected to increase as the sensing range increases
for large networks. However, this only take place until a
certain point: As the sensing range is in the same order of
magnitude as the size of the network, a node can detect ongoing transmissions from a larger part of the topology, thus
the number of hidden nodes decreases again. This effect is
shown in Figure 8b as an average over 10000 different random topologies, each with N =70 nodes that are randomly
and uniformly distributed on the area Ω=1250×1250m2 .
The graph shows how the average number of hidden nodes
(hu ) changes when the sensing range is gradually increased
in the interval [250m, 1250m]. In summary, for modest
traffic intensities, whether an increased sensing range has
a positive or a negative effect on the apparent link-failure
probability depends on the actual network topology, and especially on the size of the topology relative to the sensing
range.
For high traffic intensities, on the contrary, it is the number of links (h0 ) with no hidden nodes that determines how
much lower the average apparent link-failure probability is
below unity. Intuitively, as the sensing range increase, a
node can detect ongoing transmissions from a larger part of
the topology, thus the number of links (h0 ) with no hidden
nodes must increase. This is also illustrated in Figure 8c
which shows the fraction of the links with no hidden nodes,
i.e. h0 /|E(G)|, for the same topologies as used in Figure
8b. Thus, as the sensing range is increased and the traffic
intensity is high, the apparent link-failure probability will
decrease.
In order to verify this, we analyze a random topology
G(E, V ) where every node transmits data with a probability ptx =1 and the traffic is in the range λc ∈[0, 1]. The results in Figure 9a–9b show the analytical upper and lower
bound for the average apparent link-failure probability for
rcp /rrx ∈{1, 1.8}. Simulation results are also shown and it
can be seen that the simulation is bounded by the analytical
upper/lower bound.
The results illustrated in Figure 9a–9b show indeed that
the apparent link-failure probability converges towards a
limit as the load is increased. This limit can now be
calculated as pf =(|E(S)|−h0 )/|E(S)|, where |E(S)| is
the number of links formed by the sensing range. From
Tab.2 we find pf =0.806 for rcp /rrx =1.8 and pf =0.812 for
rcp /rrx =1, which match well with Figure 9a–9b.
For low value of λc , it is the number of active, i.e.
with data to transmit, hidden nodes that determines the
apparent link-failure. From Figure 8c we see that when
rcp /rrx =1.8, more hidden nodes are generated than for
rcp /rrx =1, which is also verified by the measurement
in Tab.2. The result is that the apparent link-failure for
rcp /rrx =1.8 is higher than for rcp /rrx =1. This is also verified in Figure 9a–9b.
Table 3: Average number of hidden nodes.
Hidden nodes
hu
hl
ψ = 0.25
Long links (4%)
4.7
1.5
Short links (96%)
2.8
1.1
Hidden nodes
hu
hl
ψ = 2.5
Long links (53%)
8.7
2.5
Short links (47%)
6.6
2.3
5.2. A distance-dependent link model
In the previous sections we have assumed that the links
follow a distance-independent model. In the following we
make use of a distance-dependent link model and investigate the effect such a model has on the apparent link-failure
probability.
The link-existence probability, pd , is shown in Figure
10c for ψ∈{0.25, 2.5}. The figure illustrates that a value
of ψ=0.25 gives a sharp threshold for r/r0 =1.0, which is
similar to a transmission range of rrx = 250m used in previous sub-section. However, a value of ψ=2.5 changes the
link-existence probability considerably. The result is that
the link-existence probability decreases for shorter links
(r/r0 < 1.0) while the link-existence probability for longer
links (r/r0 >1.0) increases. In Figure 10d the average apparent link-failure probability for different values of ψ is
calculated using Eqs.(1)–(5). This is a snapshot of the linkexistence probability and we assume that this does not vary
with time for when pf is calculated. The results are based
on the topology in Figure 10b, where rcp =rrx and is calculated according to Eq. (1). Every node in the topology
transmits data with probability ptx =1. Figure 10d establish that a considerably higher apparent link-failure probability is generated by the high value of ψ. For λc T <0.6, the
explanation for this can be found in the number of hidden
nodes that different values for ψ generate. Tables 3 shows
the average number of hidden nodes used in the calculation
for pf , both for Long links, i.e. r/r0 >1.0 and Short links
where r/r0 <1.0. Tab. 3 demonstrates that almost every
link is defined as a short link for ψ=0.25 while for ψ=2.5
the fraction of long links and short links are approximately
equal. The average number of hidden nodes for ψ=2.5 is
also almost twice as large as for ψ=0.25. This explains why
the average apparent link-failure probability is much higher
for ψ=2.5. For λc T ≥0.6, it is as in previous sub-section the
value of h0 that determines the apparent link-failure probability. For ψ=0.25 we find h0 =82 and |E(G)|=493, which
gives pf =(493−82)/493=0.83. For ψ=2.5, we find that
h0 =0, thus pf =1. This match well with the results in Figure 10d.
Implicitly ψ is also an expression for the radio receiver’s
ability to correctly receive data. As the bitrate on a link
increase, the probability of correctly receiving data ap-
r0
na
nc
nb
nd
(a) Distance dependent model
(c) Link-existence probability (pd ).
(b) Random topology
(d) Link-failure probability (pf ).
Figure 10: Average apparent link-failure probability for different
link-existence probability.
proaches a step function, which is similar to ψ approaching
zero.
6. Related Work
The paper is based on prior work in [7] where a model
for beacon loss was developed. This work however did not
encompass a link-maintenance mechanism nor was it applicable for a general topology. The work in [14] aims to
improve performance by introducing a mobility detection
mechanism in order to differentiate between node mobility and congestion. This detector is similar to the linkmaintenance mechanism in [5], however, the work does not
provide any probabilistic measures of the correctness of the
detector. In [9] introduces a mechanism for detecting when
a link is about to be broken instead of waiting for the break
to happen. This approach the signal strength from neighbor nodes in order to estimate whether an active route to a
neighboring node will fail. The disadvantage with this approach is that the detector needs a constant flow of packet
on which it can measure the signal strength. In [6] the reliability and availability of a set of topologies are studied using
both a distance-dependent and a distance-independent linkexistence model, but the effects of beacon-based link maintenance and hidden nodes are ignored. Here it is assumed
that apparent link-failures are a result of radio-induced interference only. In [17] the performance of neighbor sensing in ad hoc networks is studied, but only parameters such
as the transmission frequency of the beacons and the linklayer feedback are covered.
7. Conclusion and Future Work
This paper introduces an approximate model for the
probability of apparent link-failure in beacon based link
maintenance schemes. The model can provide a rough upper and lower bound for the apparent link-failure probability. Using a simulation tool, it was observed that the model
provides acceptable accuracy for simple topologies. Also,
arbitrary mesh topologies with random traffic patterns and
bursty traffic have been studied, where the model was extended with an algorithm in order to provide an average upper and lower bound for the link-failure probability.
The number of hidden nodes is a key factor for the
calculation of the apparent link-failure probability. Since
the number of hidden nodes in an arbitrary mesh topology
will depend on parameters such as radio sensing range and
different link-existence models, the apparent link-failure
model has been thoroughly studied varying these parameters. The analyses show that our model is also valid for
networks where the sensing range is larger than the transmission range. Also, the geometry of the topology determines the number of hidden nodes, so an increase in the
sensing range can either increase or decrease the apparent
link-failure probability. However, above a certain sensing
range, the number of hidden nodes will decrease as the sensing range increases.
The paper shows that the link-existence model has a
considerable impact on the number of hidden nodes in a
topology, thus affecting the apparent link-failure probability. However, for future work it would be interesting to include the time-aspect of the link-existence model in the apparent link-failure calculations. Last, but not least, we have
analyzed how a distance-dependent model affects apparent
link-failure probability, showing that the radio transceiver’s
ability to sense/receive data influences the existence of hidden nodes.
In future work it will be interesting to study how the apparent link-failure model can be applied as an additional
parameter upon which a link-maintenance mechanism can
base its link-status decision.
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