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Cooperative Coverage Extension in Heterogeneous Machine-to-Machine Networks

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Abstract

We study the limits of cooperative coverage extension for multicast transmission in heterogeneous machine to machine networks. Terminals are equipped with both a long range interface, through which they connect to, e.g., a cellular network or satellite system, and a short range interface, which is used to create an ad-hoc capillary network among the terminals. Such a network is intended to enhance coverage by means of properly encoded cooperative transmissions. By applying the max-flow min-cut theorem we derive an analytical lower bound on the coverage as a function of both the information rate at physical layer and the rate of innovative packets per unit-time.Our results give the tradeoff between the coverage and the rateat which the information can be injected in the network, and at the same time quantify the gain derived from cooperation and give hints on how to tune important system parameters.
Cooperative Coverage Extension in Heterogeneous
Machine-to-Machine Networks
G. Cocco, C. Ibars
Centre Tecnol`ogic de Telecomunicacions de Catalunya – CTTC
Parc Mediterrani de la Tecnologia
Av. Carl Friedrich Gauss 7 08860, Castelldefels – Spain
giuseppe.cocco@cttc.es, christian.ibars@cttc.es
N. Alagha
European Space Agency - ESTEC
Noordwijk – The Netherlands
nader.alagha@esa.int
Abstract—We study the limits of cooperative coverage exten-
sion for multicast transmission in heterogeneous machine to
machine networks. Terminals are equipped with both a long
range interface, through which they connect to, e.g., a cellular
network or satellite system, and a short range interface, which is
used to create an ad-hoc capillary network among the terminals.
Such a network is intended to enhance coverage by means of
properly encoded cooperative transmissions. By applying the
max-flow min-cut theorem we derive an analytical lower bound
on the coverage as a function of both the information rate at
physical layer and the rate of innovative packets per unit-time.
Our results give the tradeoff between the coverage and the rate
at which the information can be injected in the network, and at
the same time quantify the gain derived from cooperation and
give hints on how to tune important system parameters.
I. INTRODUCTION
Today’s wideband multimedia personal communication
plays a fundamental role in modern life and constitutes an ever
growing market. However, a new communication paradigm is
appearing, which will determine a tremendous expansion in
the already huge number of wireless terminals. According to
the World Wireless Research Forum (WWRF) forecast, by the
end of the decade around seven trillion wireless devices will be
serving a population of around seven billion people [1]. Due to
the diversity of applications and device types, M2M commu-
nications show heterogeneous needs and characteristics. As an
example, a sensor network deployed for environmental control
is likely to be characterized by low power consumption and
low data rates, while a network of on board units for road fleet
management would have virtually no energy constraint and
may support relay capabilities. Beyond the differences that
characterize M2M communications, however, an important
fraction of M2M communications are likely to have some
common characteristics, such as a certain tolerance to delay.
Delay tolerance can be exploited in order to improve network
coverage by using high level protection techniques such as
fountain codes [2]. Coverage, intended as the possibility for
all nodes to correctly receive data transmitted by a central
node (e.g., a base station or a satellite), is a main issue for
networks with a large number of terminals. In M2M networks
reliable broadcast transmission is of primary importance for
terminal software and/or firmware update, in which all the
terminals need to correctly receive all the data or, for in-
stance, navigation maps update in vehicle-mounted positioning
systems. Protocols such as ARQ, although very effective in
point-to-point communication (see, e.g., [3], section 7.1.5),
may not be applicable in multicast communications, as there
may be many retransmission requests by the terminals in
case packets are lost, which would saturate the return channel
and overwhelm the source [4]. A cooperative approach may
be applied in M2M heterogeneous capillary networks [5], in
which terminals are equipped with both a long range and a
short range communication interface.
A lot of work has been done on the use of cooperation
in multicast and broadcast communications in both terrestrial
[6][7] and satellite networks [8][9][10]. Many of these solu-
tions [4][11][12] are based on network coding [13], that can
achieve the max-flow min-cut capacity in ad-hoc networks. In
most of previous works the problem of cooperative coverage
extension has been addressed from a high-level perspective,
in which the effect of physical layer is taken into account
through a fixed packet error rate. In [10] and [14] a cooperative
architecture based on network coding has been proposed to
enhance coverage in mobile satellite networks, where simu-
lations results were presented in which the physical layer is
taken into account by using a land mobile satellite time series
generator together with a physical layer abstraction technique
adopted in the standardization process of IEEE 802.16 [15].
In this paper we carry out an analytical study on the benefits
and limits of a cooperative approach in providing missing
coverage in single multicast M2M networks. We consider a
mathematically tractable and yet practically interesting net-
work model, in which fading and shadowing in the commu-
nication channels as well as the medium access mechanism
of cooperating nodes are taken into account. By applying
the max-flow min-cut theorem we derive an analytical lower
bound on the coverage as a function of both the transmission
rate at physical level and the rate of innovative packets per
unit-time at link level. Our results show a tradeoff between
the coverage and the rate at which the information can be
injected in the network, and at the same time quantify the gain
derived from cooperation of the nodes through the short range
interface, giving hints on how to tune important parameters
such as the medium access probability. Interestingly our results
also show that, as far as nodes are able to detect each other’s
transmissions, the supported communication rate for a given
coverage increases with the number of nodes. This is not the
case when no cooperation is used.
II. SYSTEM MODEL
An M2M network is considered in which a source (S), that
may represent either a base station or a satellite, has a set of
Ksource messages w1,...,wKof kbits each, to broadcast
to a population of terminal nodes, each of which has both long
range an short range communication capabilities. In particular,
we focus on vehicular terminals. Sprotects each message
using a channel code, in order to decrease the probability
of packet loss on the channel. No feedback is assumed from
the terminals to the source and no channel state information
(CSI) is assumed at S, which implies a non-zero probability
of packet loss. Thus, a second level of protection is also
applied by Sat packet level in order to compensate for packet
losses. Encoding at packet level takes place before the channel
encoding. NKcoded packets are created by Sapplying a
random linear network code (NC) to the Ksource messages.
We define R=K/N as the rate of the NC encoder at the
base station. Network coding operates in a finite field of size
q(GF (q)), so that each message is treated as a vector of
k/ log2(q)symbols. Source messages are linearly combined to
produce encoded packets. An encoded packet xis generated
as follows:
x=
K
X
i=1
̺iwi,
where ̺i,i= 1,...,K are random coefficients drawn ac-
cording to a uniform distribution in GF (q). The coefficients
̺i,i= 1,...,K, are appended to each message xbefore its
transmission. The set of appended coefficients represents the
coordinates of the encoded message xin GF (q)with respect
to the basis {wi},i= 1,...,K, and is called global encoding
vector.
Physical layer encoding is applied to network-encoded
packets, each consisting of of kbits. The transmitter encodes
a packet using a Gaussian codebook of size 2nr , with r=k
n
bits per channel use (bpcu), associating a codeword cmof n
i.i.d. symbols drawn at random from a Gaussian distribution
to each xm,m= 1 ...,N [3]. The time needed for the base
station to transmit a packet is called transmission slot (TS).
Terminals cooperate in order to recover the packets that are
lost in the link from the base station. We assume terminals with
high mobility, which is the case of, e.g., vehicular networks.
Thus, nodes have little time to set up connections with each
other. For this, and in order to exploit the broadcast nature
of the wireless medium, nodes act in promiscuous mode,
broadcasting packets to all terminals within reach. Similarly as
in the broadcast mode of IEEE 802.11 standards, no request to
send (RTS)/ clear to send (CTS) mechanism is assumed [16].
No CSI is assumed at the transmitter in the terminal to terminal
communication, so that there is always a non zero probability
of packet loss. Like the source, also each terminal uses two lev-
els of encoding, which are described in the following. Let Lbe
the number of packets correctly decoded at the physical level
by a terminal. The terminal selects the LLpackets which
constitute the largest set of linearly independent packets with
respect to the basis wi,i= 1,...,K. Linear independence can
be verified through the global encoding vectors of the packets.
The Lpackets selected are re-encoded together using random
linear NC, and then re-encoded at the physical layer. NC
encoding at the terminals works as follows. Given the set of
received packets x1,...,x
L, the message y=PL
m=1 σmxm
is generated, σm,m= 1,...,L, being random coefficients
in GF (q). Each time a new encoded message is created, it
is appended its global encoding vector. The overhead this
incurs is negligible if messages are sufficiently long [17]. The
new global encoding vector ηcan be easily calculated by the
transmitting node as follows:
η=σΨ,
where σ= [σ1··· σL]is the local encoding vector, i.e., the
vector of random coefficients chosen by the transmitting node,
while Ψis an L×Kmatrix that has the global encoding
vector of xias row i. We assume that the transmission of
a message by a terminal is completed within one TS. The
physical layer encoding at a mobile node takes place in the
same way as at the base station, using the same average
transmission rate r.
A. Source-to-node Channel Model
The channel from the source Sto a generic terminal Ni(S-
N channel) is affected by both Rayleigh fading and lognormal
shadowing. The power of the signal received at the terminal
is modeled as the product of a unitary-mean random variable
γhaving exponential distribution and a log-normal random
variable ΓSwhich accounts for large scale fading. This model
has been largely used to model propagation in urban scenarios
[18] and, with some modifications, in land mobile satellite
systems [19]. The fading coefficient γtakes into account
fast variations of the channel due to terminal motion, and
is assumed to remain constant within a TS and changes
in an i.i.d. fashion at the end of each channel block. The
shadowing coefficient ΓSincludes the transmitted power at S
and accounts for obstruction of buildings in the line of sight
and changes much slowly with respect to γ. For mathematical
tractability we assume that Γremains constant for Nchannel
blocks, i.e., until all encoded packets relative to the Ksource
messages have been transmitted. We call the time needed to
transmit Nmessages a generation period. The fading and
shadowing processes of two different nodes are assumed to
be independent. We further assume that shadowing and fading
statistics are the same for all nodes, which is the case if nodes
have all about the same distance from S.
A message is lost in the S-N channel if the instantaneous
channel capacity Cis lower than the transmission rate at
physical layer r. Thus the packet loss probability in the S-
link for a generic node is:
PSN =P r {log2(1 + γΓS)< r},(1)
where γexp(1) while ΓS=eX
10 with X∼ N(µ, σ2).
ΓSis constant within a generation period, while γchanges
independently at the end of each channel block. Fixing the
value of ΓS, the packet loss probability PS N in the S-N link
is:
PSN = 1 e12r
ΓS.(2)
Due to shadowing, PSN changes randomly and independently
at each generation period and, within a generation, from one
node to the other.
B. Node-to-node Channel Model
We model the channels between the transmitting terminal
and each of the receiving terminals (N-N channel) as indepen-
dent block fading channels, i.e., the fading coefficient of each
channel changes in an i.i.d. fashion at the end of each channel
block. The probability of packet loss in the N-N channel PN N
is:
PNN =P r {log2(1 + γΓN)< r}= 1 e12r
ΓN,(3)
where ΓNaccounts for path loss and transmitted power, and is
assumed to remain constant for a whole generation period and
across terminals. In order not to saturate the terrestrial channel,
we assume that a node can transmit at most one packet within
one TS.
III. NON-COOPERATIVE COVERAGE
We define the coverage () as the probability that all the
nodes correctly decode the whole set of Ksource messages.
Assuming that Kis large enough, and using the results in
[4], the probability that node Nican decode all the Ksource
messages of a given generation in case no cooperation is
allowed is:
P r {PS N i <1R}=FPSN i (1 R),(4)
FPSN being the cumulative density function (c.d.f.) of PSN
and R=K/N the rate of the NC encoder at S. Plugging Eqn.
(2) into Eqn. (4) we find:
P r n1e12r
ΓS<1Ro.(5)
The coverage is the probability that each of the nodes decodes
all source messages, that is:
Ω = P r {PS N 1<1R,...,PSN M <1R},(6)
where PSN i is the packet loss rate in the link to the base
station of node Ni,i= 1,...,M. Under the assumption of
i.i.d. channels we have FPSN i =FPSN ,i∈ {1,...,M}.
Thus, (6) can be written as:
Ω = (P r {PS N <1R})M=FM
PSN (1 R).(7)
We now look for FPS N (y), which can be obtained as
Ry
−∞ fY(z)dz,fY(y)being the probability density function
(p.d.f.) of PS N .fY(y)can be obtained from fX(x)by
applying the transformation:
Y= 1 e(12r)·eX
10 .
After some manipulation we find:
fY(y) = 10
(1 y) ln(1 y)2πσ2e{10 ln[12r
ln(1y)]η}2
2σ2,(8)
for y(0,1). Integrating the (8) we obtain the c.d.f. for the
probability of packet loss:
FY(y) = 1
21
2erf
10 ln h12r
ln(1y)iη
2σ2
,(9)
for y(0,1), where erf(x)is the error function, defined as
2
πRx
0et2dt. Finally, plugging Eqn. (9) into Eqn. (7), we
find the coverage in the non cooperative case:
Ω = 1
2M
1erf
10 ln h12r
ln(R)iη
2σ2
M
,(10)
for R(0,1). As said previously, this result holds for any
value of qas long as Kis large enough. Thus, Eqn. (10) can
also be interpreted as the coverage in a network of Mnodes
in presence of fading and shadowing that can be achieved (for
a fixed rate rat physical level) by a fountain code.
IV. COOPERATIVE COVERAGE
The network is modeled as a directed hypergraph H=
(N,A),Nbeing a set of nodes and Aa set of hyperarcs.
An hyperarc is a pair (i, J ), where iis the head node of the
hyperarc while Jis the subset of Nconnected to the head
through the hyperarc. Jis also called tail. An hyperarc (i, J )
can be used to model a broadcast transmission from node i
to nodes in J. Packet losses can also be taken into account.
We want to study the relationship between the coverage and
the rate at which the information is transferred to mobile
terminals, which depends on both the rate at physical layer
r, and the rate at which new messages are injected in the
network, which is the rate at packet level R. In [4] (Theorem
2) it is shown that, if Kis large, random linear network coding
achieves the network capacity in wireless multicast and unicast
connections, even in case of lossy links, if the number of
innovative packets transmitted by the source per unit of time
is lower than or equal to the flow across the minimum flow
cut between the source and each of the sink nodes, i.e.:
Rmin
Q∈Q(s,t)
X
(i,J)Γ+(Q)X
TQ
ziJT
(11)
where ziJT is the average injection rate of packets in the arcs
departing from ito the tail subset TJ,Q(s, t)is the set
of all cuts between Sand t, and Γ+(Q)denotes the set of
forward arcs of the cut Q, i.e.:
Γ+(Q) = {(i, j)A|iQ, j /Q}.(12)
In other words, Γ+(Q)denotes the set of arcs of Qfor which
the head node is on the same side as the source, while at least
one of the tail nodes of the relative hyperarc belongs to the
other side of the cut. The rate ziJK is defined as:
ziJT = lim
τ→∞
AiJT (τ)
τ,(13)
where AiJT (τ)is the counting process of the packets sent by
ithat arrive in TJin the temporal interval [0, τ ). The
existence of an average rate is a necessary condition for the
applicability of results in [4].
In the following we derive ziJ T for the considered network
setup as a function of both physical layer and media access
control (MAC) layer parameters such as transmission rate,
transmission power and medium access probability.
A. Multiple Access
Let us consider a network with Mnodes. We assume that
all nodes have independent S-N and N-N channels. We further
assume that channel statistics are i.i.d., which is the case if
the distances from node Nito node Njchange little i, j
{1,...,M},i6=jand with respect to each node’s distance
to the source.
In our setup the terminals are set in promiscuous mode so
that each node can receive the broadcast transmissions of any
other node [16]. The terminals share the wireless medium,
i.e., they transmit in the same bandwidth. We assume that a
carrier sense multiple access (CSMA) / collision avoidance
(CA) protocol is adopted by the nodes and that all nodes hear
each other, so that the medium is shared among the terminals
willing to transmit but no collision happens.
We now derive the communication rate ziJ T . We start by
finding out the communication rate zij between a transmitting
node iand a receiving node j. By the symmetry of the problem
all links have the same average rate. Consider the generic node
Ni. The average transmission rate from node Nito node Nj
is:
zi,j =pa·P r {No one else transmits}(1 PNN )
=pa·[P r {No one else tries to transmit}
+P r {Niwins contention}] (1 PN N ),(14)
where pais the probability that a node tries to contend for the
channel, and is fixed for all nodes. The first term in the sum
of Eqn. (14) is:
P r {No one else tries to transmit}= (1 pa)M1.(15)
The second term in the sum of Eqn. (14) is the probability
that one or more other nodes try to access the channel, but Ni
transmits first. To calculate this probability, we note note that
if kother nodes try to access the channel (for a total of k+ 1
nodes trying to access the channel), the probability for each
of them to occupy the channel before the others is 1/(k+ 1)
by the symmetry of the problem. Thus we can write:
P r {Niwins contention}=
M1
X
k=1 M1
kpk
a(1 pa)M1k
k+ 1
=1
Mpa
M1
X
k=1 M
k+ 1pk+1
a(1 pa)M1k
=1
Mpa1N
0(1 pa)MM
1pa(1 pa)M1
=1
Mpa1(1 pa)MM pa(1 pa)M1.(16)
Plugging (15) and (16) into (14) we obtain:
zi,j =1(1 pa)M
M(1 PNN ).(17)
Using the definition given in Eqn. (13) together with Eqn. (17),
we finally find
ziJT =1(1 pa)M
M(1 (PNN )|T|),(18)
where |T|is the cardinality of T, and the term 1(PN N )|T|
is the probability that at least one of the |T|nodes whose S-
link belongs to the cut receives correctly a transmission from
a node that is in the other side of the cut.
B. Coverage Analysis
We define the coverage as the probability that all nodes
receive all messages. In the following we derivethe conditions
for achieving coverage as a function of relevant network
parameters by applying the max-flow min-cut theorem. We
recall that such maximum coverage can be attained by using
the random coding scheme described in Section II.
Let us consider Eqn. (11). For each of the Mnodes we
must consider all the possible cuts of the network such that
the considered node and the satellite are on different sides of
the cut. Let us fix a receiving node Nt. We recall that a cut is
a set of edges that, if removed from a graph, separates source
and destination. Fig. 1 gives an example of a network with four
nodes where the cut QS N4(i.e., the cut such that N4and S
are on the same side) is put into evidence. In the example, the
destination node is Nt=N1. The dashed black lines represent
the edges which are to be removed to get the cut. Note that the
set of nodes that receive from S(only node N4in the figure)
are isolated by the cut from the nodes with satellite cut (nodes
N1,N2and N3in Fig. 1). We define an S-edge as an edge of
the kind (S, j), j 6=t. We further define a T-edge as one of the
kind: (j, t), j 6=t. First of all, note that in each possible cut of
tthe arc joining the node with the satellite is always present.
For the particular network topology considered, the rest of the
cuts consist in removing, for each of the M1remaining
nodes, either the S-link or the T-links between node tand
nodes connected with Nt. The number of possible cuts is thus
equal to 2M1. Two distinct cuts differ in either the number
nsof S-edges which are included in the cut or the identity of
the nodes for which the S-edge is part of the cut. For each
Nt∈ N and for each cut so that ns∈ {1,···, M 1}S-links
S
1
N4
N
2
N3
N
4
SN
Q
Fig. 1. Graph model for a network with four terminals. There are 2M1= 8
possible cuts for each of the Mnodes. The set of nodes that receive from
S(only node N4in the figure) are isolated by the cut from the nodes with
satellite cut.
are present, the average message rate Rat the source must be
lower than or equal to the capacity of the cut, i.e.:
α(ns)Y
jCs
Yj0,(19)
where Csis one of the cuts with nssatellite links relative to
the node tand we defined:
α(ns) = 1 R+ (Mns)1(1 pa)M
M[1 (PNN )ns].
The condition in Eqn. (19) must hold for any number nsof
S-edges. This is equivalent to imposing a new condition which
is the intersection of all the conditions like the (19), i.e.:
\
Cs∈S(ns,Nt)
Y
jCs
Yjα(ns)
,(20)
where S(ns,Nt)is the set of subsets of N\Ntwith ns
elements. The number of elements in S(ns,Nt)is M1
ns,
as each of them is obtained by choosing nselements from a
set with cardinality M1. As previously said, for a given
Ntto decode all messages, the condition on the flow must
be satisfied for all cuts, which is equivalent to imposing the
condition (20) for all ns. Finally, in order for all nodes to
decode all source messages, i.e., in order to achieve coverage,
the condition on the minimum flow cut must hold t∈ N.
Imposing this, we obtain the expression for the coverage that
is reported at the bottom of next page.
C. Lower Bound on Achievable Coverage
Although the expression in Eqn. (21) might be used to
evaluate numerically, a closed-form expression would give
more insight into the impact of cooperation on the considered
setup. Finding a simple closed form expression for Eqn. (21)
is a challenging issue, thus in the following we derive a
lowerbound. can be lower bounded by substituting in Eqn.
(21) the packet loss rate Yjfor each cut with the largest packet
loss rate among all the S-links in the network, i.e.:
Ω = P r
\
Nt∈N \
ns∈{1,...,M}\
S∈S(ns,Nt)
Y
jS
Yj< α(ns)
P r
\
Nt∈N \
ns∈{1,...,M}
ns
Y
j=1
Y(j)< α(ns)
P r
\
Nt∈N \
ns∈{1,...,M}hYns
(1) < α(ns)i
=P r
\
Nt∈N \
ns∈{1,...,M}hY(1) <ns
pα(ns)i
=P r Y(1) <min
ns∈{1,...,M}
ns
pα(ns)
=FM
Y(β),(22)
where Y(i)is the i-th largest packet loss rate across all S-edges
of the network, i.e. Y(i)Y(j)if i < j, i, j ∈ N, and we
defined
β= min
ns∈{1,...,M}
ns
pα(ns).
The first inequality in Eqn. (22) derives from the fact that:
Y
jS
Yj
ns
Y
j=1
Y(j),for S∈ S(ns,t),ns, t, (23)
i.e., we substitute the product of nsrandom variables, chosen
within a set of Mvariables, with the product of the nslargest
variables of the same set. The second inequality in Eqn. (22)
follows from the fact that
ns
Y
j=1
Y(j)Yns
(1) ,ns, t.
By plugging Eqn. (9) into Eqn. (22) we finally find:
LB =1
2M
1erf
10 ln h12r
ln(1β)iη
2σ2
M
.(24)
V. NUMERICAL RESULTS
Fig. 2 shows , obtained by evaluating numerically Eqn.
(21), plotted against message rate Rfor different network sizes
together with the relative lower bound and the coverage in case
of no cooperation. In the simulation we set r= 1.5bpcu,
pa= 0.2,ΓN= 10dB in the N-N channel, µ= 3 and σ= 1
in the S-N channel. It is interesting to note how increasing the
number of nodes increases the achievable message rate Rfor
a given . In other words, the higher number of nodes, the
higher the probability that all the information broadcasted by S
reaches the network, i.e., has not been lost. Once the informa-
tion has reached the network, it can be efficiently distributed
among the terminals thanks to the properties of random linear
network coding. Similarly, in Fig. 3 the coverage is plotted
0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.2
0.4
0.6
0.8
1
R
NC Monte Carlo N= 2
NC Monte Carlo N= 4
NC Monte Carlo N= 6
NC lower bound N= 2
NC lower bound N= 4
NC lower bound N= 6
No cooperation N= 2
No cooperation N= 4
No cooperation N= 6
Fig. 2. Coverage plotted against message rate Rin the cooperative case
for different values of N. The lower bound and the non cooperative case are
also shown. In the simulation we set r= 1.5bpcu, pa= 0.2,ΓN= 10dB
in the N-N channels, µ= 3 and σ= 1 in the S-N channel.
against the physical rate rfor a fixed R. Also in this case, an
important gain in transmission rate can be observed, with an
increase of about 0.4bpcu when passing from no cooperation
to cooperation in a network with 2nodes, and about 1bpcu
in case of a network with 4nodes. In both figures the lower
bound is pretty tight for M= 2 and M= 4. An important
point is that this result is achieved without any feedback to
the base station or any packet request among nodes, as the
decision on whether to encode and transmit or not is taken
autonomously by each terminal only based on the probability
of media contention pa. In Fig. 4 the coverage is plotted
0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
r (bpcu)
NC Monte Carlo N= 2
NC Monte Carlo N= 4
NC Monte Carlo N= 6
NC lower bound N= 2
NC lower bound N= 4
NC lower bound N= 6
No cooperation N= 2
No cooperation N= 4
No cooperation N= 6
Fig. 3. Coverage plotted against rate at physical layer rin the cooperative
case for different values of N. The lower bound and the non cooperative case
are also shown. In the simulation we set R= 2/3messages/slot, pa= 0.2,
ΓN= 10dB in the N-N channels, µ= 3 and σ= 1 in the S-N channel.
against the per-node probability of transmission attempt pafor
two different N-N channel conditions. In one case the average
channel power is set to 10 dB while in the other it is set to 0
dB. Two main observations can be done. The first is that the
paneeded to achieve a given increases significantly with
the worsening of the N-N channel. This is because a bad inter
terminal communication channel determines many losses, and
thus more attempts of channel access are needed by the nodes
to compensate such losses. We see that, in order to achieve
full coverage, the required paalmost doubles when passing
from ΓN= 10 to ΓN= 0. The second observation is that,
even in bad channel conditions, relatively small values of pa
(0.2for ΓN= 0) are sufficient to achieve full coverage for
values of rand Rwhich are of practical interest (Fig. 4 was
obtained by setting R= 2/3messages/slot and r= 1 bpcu).
We further observe that the lower bound tightly approximates
the simulated curves. The coverage for the non cooperative
case in the setup considered in Fig. 4 is 0, coherently with
Fig. 3.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
0.2
0.4
0.6
0.8
1
pa
NC Monte Carlo ΓN N = 10dB
NC lower bound ΓN N = 10dB
NC Monte Carlo ΓN N = 0dB
NC lower bound ΓN N = 0dB
Fig. 4. Coverage plotted against the probability of media contention pain
the cooperative case for a network with M= 4 and nodes two different values
of average SNR in the N-N link, i.e., ΓN= 10dB and ΓN= 0dB. The
lower bound is also shown. In the simulation we set R= 2/3messages/slot,
r= 1 bpcu, µ= 3 and σ= 1 in the S-N channel.
VI. CONCLUSIONS
In this paper we investigated the performance of a coop-
erative approach in providing missing coverage for broadcast
M2M networks. We considered a mathematically tractable and
yet practically interesting network model, in which fading and
shadowing effects in communication channels as well as the
medium access mechanism of cooperating node have been
taken into account. By applying the max-flow min-cut theorem
we derived an analytical lower bound on the coverage as a
function of both the information rate at physical layer and
the rate of innovative packets per unit-time. Our results give
a tradeoff between the coverage and the rate at which the
information can be injected in the network, and at the same
time quantify the gain derived from node cooperation through
Ω = P r
\
Nt∈N \
ns∈{1,...,M1}\
S∈S(ns,Nt)"Y
jS
Yj<1R+ (Mns)1(1 pa)M
M[1 (PNN )ns]#
.(21)
the short range interface, showing that the diversity gain grows
with the number of terminals, and thus important gains in
terms of transmission rate at the base station can be achieved
through cooperation.
Currently we are investigating how to extend the results to
networks with fixed repeaters and imperfect medium access
control mechanism. We also plan to compare the analytical
results obtained thus far with the simulation results that include
state of the art propagation channel models.
ACKNOWLEDGMENT
This work was partially supported by the European
Commission under project ICT-FP7-258512 (EXALTED).
Giuseppe Cocco is partially supported by the European Space
Agency under the Networking/Partnering Initiative.
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