Area Throughput for CSMA based Wireless Sensor Networks

Abstract

In this paper we present a mathematical approach to evaluate the area throughput of a multi-sink wireless sensor network (WSN), where nodes transmit their packets to a sink, selected among many. Sensors and sinks are both Poisson distributed in a bounded domain. A carrier sensing multiple access (CSMA) based protocol is used by nodes to access the channel. We denote as area throughput the amount of samples per second successfully transmitted to the sinks. This performance metric is strictly related to both connectivity and MAC issues: it depends, in fact, on the probability that a given sensor node is not isolated and that it succeeds in transmitting its packet (i.e., the packet does not collide). The aim of this work is to devise a mathematical model that takes CSMA and connectivity issues into account under a joint approach. Through this model some network optimisation strategies could be derived. As an example, sensors could perform an aggregation procedure, responding sporadically to queries with a single packet composed of all samples taken since the previous transmission. Our model allows the evaluation of the optimum size of the packet that should be transmitted, so that the area throughput is maximised. Finally, the effects of the connectivity on the area throughput are evaluated.

Full text

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Area Throughput for
CSMA Based Wireless Sensor Networks
(Invited Paper)
Roberto Verdone
WiLAB, DEIS
University of Bologna, Italy
Email: roberto.verdone@unibo.it
Flavio Fabbri
WiLAB, DEIS
University of Bologna, Italy
Email: flavio.fabbri@unibo.it
Chiara Buratti
WiLAB, DEIS
University of Bologna, Italy
Email: c.buratti@unibo.it
Abstract—In this paper we present a mathematical approach
to evaluate the Area Throughput of a multi-sink Wireless Sensor
Network (WSN), where nodes transmit their packets to a sink,
selected among many. Sensors and sinks are both Poisson dis-
tributed in a bounded domain. A Carrier Sensing Multiple Access
(CSMA) based protocol is used by nodes to access the channel.
We denote as Area Throughput the amount of samples per second
successfully transmitted to the sinks. This performance metric is
strictly related to both connectivity and MAC issues: it depends,
in fact, on the probability that a given sensor node is not isolated
and that it succeeds in transmitting its packet (i.e., the packet does
not collide). The aim of this work is to devise a mathematical
model that takes CSMA and connectivity issues into account
under a joint approach. Through this model some network
optimisation strategies could be derived. As an example, sensors
could perform an aggregation procedure, responding sporadically
to queries with a single packet composed of all samples taken
since the previous transmission. Our model allows the evaluation
of the optimum size of the packet that should be transmitted,
so that the Area Throughput is maximised. Finally, the effects of
the connectivity on the Area Throughput are evaluated.
I. INTRODUCTION
Wireless Sensor Networks (WSNs) [1] are about collecting
data from the environmentthrough the sampling of some phys-
ical entities (like temperature, humidity, etc), and sending them
to a user, usually through some infrastructure networks such
as the Internet, or a mobile radio system (e.g. GPRS, UMTS).
Applications of WSNs can be split into two major categories,
namely event-triggered sampling and spatial/temporal process
estimation [2]. In the latter case, the environment is observed
through queries/respond mechanisms: queries are periodically
generated by the user, and sensor nodes respond by sampling
and sending data. The user, by collecting samples taken from
different locations, and observing their temporal variations,
can estimate the realisation of the observed process [3]. Good
estimates require sufficient data taken from the environment.
If few sensor nodes are deployed over a small area, a single
network coordinator (denoted as sink in the following) can be
used as collector of the data sampled, and as gateway towards
the infrastructure. When the number of sensor nodes is large,
they are often organised in clusters, where one sink per cluster
spreads the queries to the sensors, collects the responses and
manages data transmissions.
Access to the radio channel is often based on random mul-
tiple access techniques [4], [5] as scheduling of transmission
is usually impossible for several reasons (sleep/active node
cycles, frequent change in the network topologies, etc). Carrier
Sense Multiple Access (CSMA) methods are very often used
for WSNs: the most widely adopted standard air interface
technique is IEEE802.15.4 [6], whose two operational modes
(beacon and non-beacon enabled) are based on CSMA.
Sinks are sometimes specifically designed as WSN nodes,
and deployed in optimised and planned locations with respect
to sensors. However, opportunistic exploitation of the presence
of mobile sinks, connected to the infrastructure through some
mobile radio interface is an alternative in some cases [7].
Under these circumstances, many sinks can be present in
the monitored space, but their positions are unknown and
unplanned.
In many applications of WSNs, the data must be taken
from a specific portion of space, even if the sensor nodes
are distributed over a larger area. Therefore, only a location-
driven subset of sensor nodes must respond to a query. The
aim of the query/response mechanism is to acquire the largest
possible number of samples from the target environment, per
unit of time.
However, if the sinks are randomly distributed according to
the opportunistic paradigm mentioned above, achievement of
a sufficient level of samples is not guaranteed, because the
sensor nodes might not reach any sinks according to the lim-
ited transmission range. Therefore, in such an uncoordinated
environment, network connectivity is a relevant issue, and it is
basically dominated by the randomness of radio channel and
the density of sinks.
On the other hand, the density of sensor nodes significantly
affects the ability of the CSMA mechanisms to prevent from
collisions over the air interface (i.e. simultaneous transmis-
sions from separate sensors towards the same sink); if the
number of sensor nodes per cluster is very large, collisions
and backoff procedures can make data transmission impossible
under time-constrained conditions, and the samples taken from
sensors do not reach the sinks and, consequently, the user.
If the number of sensor nodes in the target area is too large,
causing many data losses, one solution can be found in the
decimation of the sensor nodes to respond. Other improve-
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ments might be introduced by letting the sensor nodes apply a
form of aggregation procedure, responding only sporadically
to queries, with a single data packet composed of all samples
taken since the previous transmission: fewer access attempts
are performed, but with longer packets.
Such decimation process, or the aggregation strategy, must
be driven by an optimisation procedure that, by taking into
account the density of sensor nodes and sinks, the frequency
of queries, and the randomness of node locations, the radio
channel behaviour, and CSMA mechanisms, determines the
optimum number of nodes that should respond to any query,
and whether aggregating samples provides advantages.
This paper addresses such optimisation problem. We con-
sider a large area where sensors and sinks are uniformly
and randomly distributed. Then, we define a specific por-
tion of space, of finite size and given shape, as the target
environment; both the number of sensors and sinks are then
Poisson distributed in such space (see Figure 1, above part).
Without loss of generality, we assume the finite target area is
of square shape. Denoting as Area Throughput the amount of
samples per second successfully transmitted to the sinks (that
we assume to be constraintless connected to the infrastructure),
our aim is to devise a mathematical model that takes CSMA
and connectivity issues into account under a joint approach,
with particular emphasis on the IEEE802.15.4 MAC protocol
in non-beacon enable mode.
Many works in the literature are related to the modelling
of different CSMA based MAC protocols, and also to con-
nectivity models, but very few papers jointly consider the two
issues under a mathematical approach. Some analysis of the
two issues are performed through simulations: as examples, [8]
related to ad hoc networks, and [9], to WSNs. Many papers
devoted their attention to connectivity issues of wireless ad-
hoc and sensor networks in the past (e.g., [10]). Single-sink
scenarios have attracted more attention so far. However, an
example of multi-sink scenario can be found in [11]. All the
previously cited works do not account for MAC issues.
Concerning the analytical study of CSMA based MAC
protocols, in [12] the throughput for a finite population when a
persistent CSMA protocol is used, is evaluated. An analytical
model of the IEEE802.11 CSMA based MAC protocol, is
presented by Bianchi in [13]. In these works no physical layer
or channel model characteristics are accounted for. Capture
effects with CSMA in Rayleigh channels, are considered in
[14], whereas [15] addresses CSMA/CA protocols. However,
no connectivity issues are considered in these papers. In [16]
the per-node saturated throughput of an IEEE802.11b multi-
hop ad hoc network with a uniform transmission range, is
evaluated.
The model proposed here is based on some previous works:
[17], where the Authors presented a mathematical model for
the evaluation of the degree of connectivity of a multi-sink
WSN in unbounded and bounded domains; and [18], [19],
where a mathematical model to derive the success probability
for the transmission of a packet in an IEEE802.15.4 single-
sink scenario, is provided.
A
sensor
sink
Tq
D=1
D=2
t
t
Fig. 1.
aggregation strategy
Above part: The Reference Scenario considered. Below part: The
The rest of the paper is organised as follows. The following
Section introduces the scenario, the data aggregation strategy
and the link model. In Section III the Area Throughput
is evaluated, by computing the success probability for the
transmission of a packet accounting for connectivity and MAC
issues. In Section IV the particular case of the IEEE802.15.4
CSMA based protocol is considered, for the evaluation of the
success probability related to MAC. Finally, in Section V and
VI numerical results and conclusions are presented.
II. ASSUMPTIONS AND REFERENCE SCENARIO
The reference scenario considered consists of an area of
finite size and given shape, where sensors and sinks are both
distributed according to a homogeneous Poisson Point Process
(PPP). We denote as ρs[m−2] and ρ0[m−2] the sensors and
sinks densities, respectively, and with A the area of the target
domain. Denoting by k the number of sensor nodes in A, k
is Poisson distributed with mean¯k = ρs· A and p.d.f.
¯kke−¯k
gk=
k!
.
(1)
We also denote as I = ρ0· A the average number of sinks
in A.
A. The Aggregation Strategy
Sinks periodically send queries to sensors and wait for
replies. In case a sensor node receives a query from more
than one sink, it selects the one providing the largest received
power and responds to it. We assume that sensors may perform
some data aggregation before transmitting their packets. For
instance, they perform sampling from the environment upon
each query, but transmit data only when a given number of
samples have been collected. By doing so, transmissions do
not occur at each query.
We denote as T the time needed to transmit a unit of data,
that is one sample, and as TD the time needed to transmit a
packet. The frequency of the queries transmitted by the sinks
is denoted as fq = 1/Tq. Tq is the time interval between
two consecutive queries and is set to q · T; therefore, a finite
number, q, of intervals T are contained in Tq. We assume

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that sensors transmit packets composed of D samples every
D queries. At each query sensors take one sample and when
D samples are taken, data is aggregated and transmitted. We
assume that the aggregation process generates a packet whose
transmission requires a time TD = D · T, when D units of
data are aggregated. In Figure 1 (below part), the aggregation
strategies in the cases D = 1,2 are shown as examples.
B. The Link Model
The link model that we exploit accounts for the power loss
due to propagation effects including both a distance-dependent
path loss and the random channel fluctuations caused by
possible obstructions. Specifically, a direct radio link between
two nodes is said to exist if L < Lth, where L is the power
loss and Lth represents the maximum loss tolerable by the
communication system. In that case, the two nodes are said
to be ”audible”. The threshold Lthdepends on transmit power
and receiver sensitivity. The power loss in decibel scale at
distance d is expressed in the following form
L = k0+ k1lnd + s,
(2)
where k0and k1are constants, s is a Gaussian r.v. with zero
mean, variance σ2, which represents channel fluctuations. This
channel model was also adopted in [20]. By considering an
average transmission range as in [20], an average connectivity
area of the sensor can be defined as
Aσ= πe
2(Lth−k0)
k1
e
2σ2
k2
1 .
(3)
III. EVALUATION OF THE AREA THROUGHPUT
The Area Throughput is mathematically derived through
an intermediate step: we first consider the probability of
successful data transmission by an arbitrary sensor node, when
k nodes are present in the queried area. Then, the overall
Area Throughput is evaluated based on this result, given the
aggregation strategy described in Section II.
A. Joint MAC/Connectivity Probability of Success
Let us consider an arbitrary sensor node that is located
in the queried area A at a certain time instant. We aim
at computing the probability that it can connect to one of
the sinks deployed in A and successfully transmit its data
sample to the infrastructure. Such an event is clearly related to
connectivity issues (i.e., the sensor must employ an adequate
transmitting power in order to reach the sink and not be
isolated) and to MAC problems (i.e., the number of sensors
which attempt at connecting to the same sink strongly affects
the probability of successful transmission). For this reason, we
define Ps|k(x,y) as the probability of successful transmission
conditioned on the overall number, k, of sensors present in the
queried area, which also depends on the position (x,y) of the
sensor relative to a reference system with origin centered in
A. This dependence is due to the well-known border effects
in connectivity [10].
In particular, we assume
Ps|k(x,y) = En[PMAC(n) · PCON(x,y)]
= En[PMAC(n)] · PCON(x,y)
where we separated the impact of connectivity and MAC on
the transmission of samples. A packet will be successfully
received by a sink if the sensor node is connected to at least
one sink and if no MAC failures occur. We now analyze the
two terms that appear in (4).
PCON(x,y) represents the probability that the sensor is not
isolated (i.e., it receives a sufficiently strong signal from at
least one sink), which is computed in [17] for a scenario
analogous to the one considered here (e.g., squared and
rectangular areas). This probability decreases as the sensor
approaches the borders (border effects). Specifically, since the
position of the sensor is in general unknown, Ps|k(x,y) of (4)
can be deconditioned as follows:
(4)
Ps|k= Ex,y[Ps|k(x,y)]
= Ex,y[PCON(x,y)] · En[PMAC(n)].
It is also shown in [17] that border effects are negligible
when Aσ < 0.1A. In this case the following holds:
(5)
PCON(x,y) ? PCON= 1 − e−μsink,
where μsink= ρ0Aσ= IAσ/A is the mean number of audible
sinks on an infinite plane from any position [20].
PMAC(n), n ≥ 1, is the probability of successful transmis-
sion when n − 1 interfering sensors are present. It accounts
for MAC issues and is treated in Section IV for the particular
case of the IEEE802.15.4 standard, even though the model
is applicable to any CSMA-like protocol. For now we only
emphasize that it is a monotonic decreasing function of the
number, n, of sensors which attempt to connect to the same
serving sink. This number is in general a random variable in
the range [0,k]. In fact, note that in (4) there is no explicit
dependence on k, except for the fact that n ≤ k must hold.
Moreover in our case we assume 1 ≤ n ≤ k, as there is at
least one sensor competing for access with probability PCON
(6).
In [21], Orriss et al. showed that the number of sensors
uniformly distributed on an infinite plane that hear one par-
ticular sink as the one with the strongest signal power (i.e.,
the number of sensors competing for access to such sink) is
Poisson distributed with mean
¯ n = μs1 − e−μsink
(6)
μsink
,
(7)
with μs= ρsAσ being the mean number of sensors that are
audible by a given sink. Such a result is relevant toward our
goal even though it was derived on the infinite plane. In fact,
when border effects are negligible (i.e., Aσ < 0.1A) and k is
large, n can still be considered Poisson distributed. The only
two things that change are:
• n is upper bounded by k (i.e., the pdf is truncated)

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• the density ρsis to be computed as the ratio k/A [m−2],
thus yielding μs= kAσ
Therefore, we assume n ∼ Poisson(¯ n), with
¯ n = ¯ n(k) = kAσ
Aμsink
Finally, by making the average in (5) explicit and neglecting
border effects (see (6)), we get
A.
1 − e−μsink
= k1 − e−IAσ/A
I
.
(8)
Ps|k= (1 − e−IAσ/A) ·
1
M
k
?
n=1
PMAC(n)¯ nne−¯ n
n!
,
(9)
where
M =
k
?
n=1
¯ nne−¯ n
n!
(10)
is a normalizing factor.
B. Area Throughput
According to the aggregation strategy described in the
previous Section, the amount of data samples generated by
the network as response to a given query is equal to the
number of sensors, k, that are present and active when the
query is received. As a consequence, the average number of
data samples-per-query generated by the network is the mean
number of sensors,¯k, in the queried area.
Now denote by G the average number of data samples
generated per unit of time, given by
G =¯k · fq= ρs· A ·
1
qT[samples/sec].
(11)
From (11) we have¯k = GqT.
The average amount of data received by the infrastructure
per unit of time (Area Throughput), S, is given by:
S =
+∞
?
k=0
S(k) · gk [samples/sec],
(12)
where S(k) =
Finally, by means of (9), (10) and (11), equation (12) may
be rewritten as
S =1 − e−IAσ/A
qT
+∞
?
IV. THE IEEE802.15.4 MAC PROTOCOL
In [19] an analytical model of the IEEE802.15.4 MAC
protocol was presented, considering the non-beacon enabled
mode (see the Standard [6]). For details on the protocol we
refer to the Standard as well. Here, we just want to underline
that a maximum number of times a node can try to access
the channel and perform the backoff algorithm, NBmax, is
imposed. According to this, there will be a maximum delay
that could affect a packet transmission. We assume that a unit
of data (one sample) has a size of 10 Bytes; therefore each
node will transmit a packet of size D ·10 Bytes. In this case,
k
qTPs|k, gkas in (1) and Ps|kas in (9).
·
k=1
?k
n=1PMAC(n)¯ nne−¯ n
?k
n!
n=1
¯ nne−¯ n
n!
·(GqT)ke−GqT
(k − 1)!
. (13)
the maximum delay is equal to (120 + D) · T [19], where T
is the time interval needed to transmit 10 Bytes, that is 320
μsec, since a bit rate of 250 kbit/sec is used. We assume that
the interval of time between two consecutive queries equals
this maximum delay: Tq= (120 + D) · T.
To evaluate PMAC(n), the scenario considered consists of n
sensors transmitting to a sink with no connectivity problems.
A finite state transition diagram is used to model sensor nodes
states. Through the analysis of this diagram the probability that
a given sensor successfully transmits its packet, PMAC(n),
is evaluated. We do not report here the expression of this
probability, owing to its complexity, but we refer to [18]
and [19], where details on formulae are given and where a
validation of the model, through comparison with simulations,
is provided for n ≤ 50. This probability PMAC(n) can be used
in (13) for the evaluation of S.
In this Section we show some results obtained through the
IEEE802.15.4 model, related to a single-sink scenario with
n sensors and no connectivity problems. These results are
interesting because they motivate the choice of the above
described aggregation strategy. It is shown indeed, that given
n, there exists an optimum value of D, Dopt, maximising the
throughput, S. Therefore, if sensors are aware of the size n
of the cluster they belong to, they could select D = Dopt,
obtained through our results, and transmit the aggregated
packet every Doptqueries.
In Figs 2 and 3 PMACand S as functions of n for different
values of D, are shown, respectively. Results are obtained by
fixing BEmin= 3, BEmax= 5, and NBmax= 4 [6]. As we
can see, PMACdecreases monotonicallyby increasing n, since
the number of sensors competing for the channel increases.
Since here we have ensured connectivity, a single sink and
a deterministically fixed number, k = n, of sensors competing
for access, we have PCON = 1 and Ps|k= PMAC. Hence,
the Area Throughput is simply S =
As seen in Figure 3, S presents a maximum. In fact, for
small n, PMACapproaches zero slower then 1/n and thus by
increasing n, S also increases. On the contrary, for large n,
PMACapproaches zero faster then 1/n and thus by increasing
n, the product n · PMAC(n) decreases, and so does S. The
physical interpretation is that too many packet losses occur
when traffic is too heavy. The maximum values of S depend
on D and are obtained for different values of n. As we can see,
for 1 < n < 12, Dopt= 7; for 12 < n < 18, Dopt= 5; for
18 < n < 68, Dopt= 2 and for n > 68 Dopt= 1. Therefore,
it clearly appears that Doptdecreases when increasing n.
n
(120+D)T· PMAC(n).
The aggregation strategy proposed here, is achievable only
in case sensors know n. This parameter could be estimated
by sensors, for example, by computing the number of times
the channel is found busy in a given interval of time. The
probability to find the channel busy, in fact, is strictly related
to n. The study of distributed protocols that provide sensors
with the knowledge of n, is left for future works.

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V. NUMERICAL RESULTS
In this Section the behavior of the Area Throughput as a
function of G (see (13)) for different connectivity levels and
for different values of D, is shown.
Let us consider a square area, having area A = 106m2,
where an average number of 10 sinks are distributed according
to a PPP (I = 10). We also set k0= 40, k1= 13.03, Lth=
120 and σ = 4. We first study the case of the IEEE802.15.4
MAC protocol; therefore T = 320μsec and q = 120 + D.
Since a typical IEEE802.15.4 air interface is considered, a
limit on the number of sensors that could be connected to
a given sink should be imposed [22], [23]. To this end, we
denote as nmax the maximum number of sensors that could
be served by a sink and define a new probability (to replace
PMAC(n) in (13)) P?
?
where PMAC(n) is obtained through the model described in
Section IV (Figure 2), and 1−nmax/n is the probability that
a sensor is not served by the sink it is connected to, owing
to the capacity constraint. Performance curves are obtained by
setting nmax = 20. Moreover, the case of negligible border
effects is considered.
In Figures 4 and 5, S as a function of G for different values
of D when PCON = 1 and 0.67 respectively, is shown. As
it can be observed, in both Figures there exists a value Dopt
which decreases by increasing G. Moreover, from Figure 4 we
can see that for 0 < G < 3000 samples/s (when I = 10, G =
3000 corresponds to n = 12) Dopt= 7; for 3000 < G < 4500
samples/s (G = 4500 corresponds to n = 18) Dopt= 5; and
for G > 4500 samples/s Dopt= 2. Therefore, the behavior of
Doptas a function of G is exactly the same of Figure 3.
If we compare Figs 4 and 5, we can observe the effects of
connectivity on S. As one can see, once we fix D, the values
of S reached for large offered load are approximatively the
same reached when PCON = 1. The decrease of PCON, in
fact, brings to having a lower mean number of sensors per
sink, therefore the decreasing of PCON is compensated by an
increasing of PMAC(n). However, the behavior of the curves
for low values of G is different (the curves have different
slopes). If we fix D = 5 and we want to obtain S = 1500,
when PCON = 0.67, we need to deploy on average 158
sensors, whereas, when PCON = 1, 106 sensors on average
are sufficient. Therefore, the loss of connectivity brings to a
larger cost in terms of number of sensors that must be deployed
to obtain the desired S.
To increase the values of S, instead, we need to increase I.
In fact, given a value of G, by increasing I the connectivity
improves and also the losses due to MAC decrease, since n
decreases. We do not show this result here, for the sake of
conciseness.
In Figure 6, we adopt a simpler MAC model where the
probability of success, P??
a linear function of n: P??
MAC(n) given by:
PMAC(n),
PMAC(nmax) · nmax/n,
P?
MAC(n) =
n ≤ nmax
n > nmax
(14)
MAC(n) (to be included in (13)), is
MAC(n) = m · n + 1. In fact, as
05 101520 25
n
3035 4045 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PMAC
D=1
D=2
D=5
D=7
D=10
Fig. 2.
of n, for different values of D.
The success probability of the IEEE802.15.4 protocol as a function
shown in [5], the probability of success for a non-persistent
CSMA protocol may decrease linearly with the number of
nodes. We denote by n∗the value such that P??
Figure 6 we consider three cases: m = −0.01, corresponding
to n∗= 100; m = −0.02, corresponding to n∗= 50; and
m = −0.04, corresponding to n∗= 25. As one can see, by
decreasing n∗the maximum of S is reached for lower values
of G. Therefore, for a given value of G, by increasing the
slope of P??
obtained with n∗= 50 is approximately twice as large as the
one obtained with n∗= 25, but it is reached for an offered load
that is twice over. Therefore, this increase in the maximum
value is reached at the cost of deploying more sensors.
MAC(n∗) = 0. In
MAC(n), S increases. The maximum value of S
VI. CONCLUSIONS
A multi-sink WSN where sensor nodes transmit their pack-
ets to a sink selected among many, using a CSMA based MAC
protocol, is studied. A mathematical framework is developed
to evaluate the Area Throughput, that is the amount of samples
per second successfully transmitted to the sinks. Sensors
are allowed to perform an aggregation procedure, responding
sporadically to queries with a single packet composed of all
samples taken since the previous transmission. The behavior of
the Area Throughput for different packet sizes and connectivity
levels, is shown. Results show that there exists a maximum for
the throughput and an optimum value of the packet size. This
value depends on the mean number of sensors distributed in
the network. Finally, the effects of the connectivity on the Area
Throughput are evaluated. Results show that when connectivity
decreases, the number of sensors that must de deployed to
obtain the same Area Throughput increases.
VII. ACKNOWLEDGMENT
This work has been supported by the EC-funded Network
of Excellence NEWCOM++.
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010 20 30 4050
n
6070 80 90100
0
50
100
150
200
250
300
350
S
D=1
D=2
D=5
D=7
D=10
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0750 1500 2250 3000 3750 4500 5250 6000 6750
G
0
1000
2000
3000
4000
5000
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S
n
n
n
*=100
*=50
**=25
Fig. 6.
a PMAC(n) decreasing linearly with n, for different values of n∗.
S as a function of G in the case of a CSMA based protocol, having
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