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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. N≥Kcoded 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 L′≤Lpackets 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 L′packets 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 γ∼exp(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 −e1−2r

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 −e1−2r

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 (Ω) 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 <1−R}=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 n1−e1−2r

ΓS<1−Ro.(5)

The coverage is the probability that each of the nodes decodes

all source messages, that is:

Ω = P r {PS N 1<1−R,...,PSN M <1−R},(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 <1−R})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(1−2r)·e−X

10 .

After some manipulation we ﬁnd:

fY(y) = 10

(1 −y) ln(1 −y)√2πσ2e−{10 ln[1−2r

ln(1−y)]−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

2−1

2erf

10 ln h1−2r

ln(1−y)i−η

2σ2

,(9)

for y∈(0,1), where erf(x)is the error function, deﬁned as

2

√πRx

0e−t2dt. Finally, plugging Eqn. (9) into Eqn. (7), we

ﬁnd the coverage in the non cooperative case:

Ω = 1

2M

1−erf

10 ln h1−2r

ln(R)i−η

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.:

R≤min

Q∈Q(s,t)

X

(i,J)∈Γ+(Q)X

T⊂Q

ziJT

(11)

where ziJT is the average injection rate of packets in the arcs

departing from ito the tail subset T⊂J,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|i∈Q, 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 T⊂Jin 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)M−1.(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}=

M−1

X

k=1 M−1

kpk

a(1 −pa)M−1−k

k+ 1

=1

Mpa

M−1

X

k=1 M

k+ 1pk+1

a(1 −pa)M−1−k

=1

Mpa1−N

0(1 −pa)M−M

1pa(1 −pa)M−1

=1

Mpa1−(1 −pa)M−M pa(1 −pa)M−1.(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 Ωas the probability that all nodes

receive all messages. In the following we derive the 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 M−1remaining

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 2M−1. 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 2M−1= 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

j∈Cs

Yj≥0,(19)

where Csis one of the cuts with nssatellite links relative to

the node tand we deﬁned:

α(ns) = 1 −R+ (M−ns)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

j∈Cs

Yj≤α(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 M−1

ns,

as each of them is obtained by choosing nselements from a

set with cardinality M−1. 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 Ωnumerically, 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. Ωcan 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

j∈S

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

j∈S

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:

ΩLB =1

2M

1−erf

10 ln h1−2r

ln(1−β)i−η

2σ2

M

.(24)

V. NUMERICAL RESULTS

Fig. 2 shows Ω, 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 Ω. 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

Ω

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 Ωplotted 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

Ω

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 Ωplotted 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 Ωincreases 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

Ω

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 Ωplotted 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,...,M−1}\

S∈S(ns,Nt)"Y

j∈S

Yj<1−R+ (M−ns)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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