Reliability and Efficiency Analysis of

Distributed Source Coding in

Wireless Sensor Networks

C. FISCHIONE, S. TENNINA, F. SANTUCCI, L. DI PAOLO, F. GRAZIOSI

Stockholm 2007

Automatic Control Group

School of Electrical Engineering

Kungliga Tekniska Hgskolan

IR-EE-RT 2006:036

1

Reliability and Efficiency Analysis of

Distributed Source Coding in

Wireless Sensor Networks

C. Fischione, S. Tennina, F. Santucci, L. Di Paolo, F. Graziosi

Abstract

The amount of measurements, which can be successfully gathered in tree-based wireless sensor

networks (WSNs) employing distributed source coding (DSC)algorithms, is evaluated in this paper in the

presence of different coding topologies and packet aggregation schemes (PA). The system model includes

a realistic network architecture with multi-hop communication, automatic repeat request protocol (ARQ),

packet losses due to channel impairments and collisions. Four DSC topologies and three alternatives of

PA are considered. The analysis is carried out by first computing the packet loss probability; then the

reliability is evaluated in terms of loss factor and the efficiency in terms of average energy consumption

of the network. The theoretical framework proposed in this paper is exploited to perform a numerical

evaluation of loss factor and energy consumption for some correlation profiles of the measurements.

Keywords: Wireless Sensor Networks (WSNs), Distributed Source Coding (DSC), Packet

Aggregation (PA).

C. Fischione is with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley. He

was with the School of Electrical Engineering, Royal Institute of Technology, Stockholm, Sweden, when part of this workwas

done. S. Tennina, F. Santucci, and F. Graziosi are all with the University of L’Aquila, Centre of Excellence DEWS and Department

of Electrical and Information Engineering, Poggio di Roio,67040 L’Aquila, Italy. E-mails:fischion@eecs.berkeley.edu,

stennina, santucci, graziosi@ing.univaq.it.

Work done in the framework of the HYCON Network of Excellence, contract number FP6-IST-511368. C. Fischione wishes

to acknowledge the support of the NSF ITR CHESS and the GSRC.

Part of the topics of this paper were presented at IEEE Globecom 2006, and ICTSM 2006.

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I. INTRODUCTION

Wireless Sensor Networks (WSNs) consist of sensing devicesoften deployed with limited

energy resources. Distributed Source Coding (DSC) claims to decrease the energy consumption

for lifetime extension of WSNs: to this end it exploits the richness of information usually provided

by spatial correlation of measurements taken by different sensors, to compress data without loss

of information.

Although DSC has recently found in WSNs a relevant application domain, the theoretical

foundations date back to the pioneering work by Slepian and Wolf [1]. A practical code con-

struction to achieve the theoretical bounds devised in [1] has been proposed by Pradhan et al.

in [2] and [3]. Those works have been extended by Chou et al. [4], who have proposed an

algorithm to achieve DSC over WSNs using a single codebook with variable compression rate.

Marco and Neuhoff [5] and Ganesan et al. [6] have pointed out that performance of DSC is

largely influenced by the DSC topology, i.e., the nodes’ position with respect to the spatial

correlation profile of the sensed phenomena. The problem of optimizing the DSC topology has

been investigated by Cristescu et al. [7], [8] for tree-likenetworks.

These contributions assume that DSC acts above the typical protocol stack of a sensor network

platform. However, many other mechanisms for WSNs (e.g., routing and MAC protocols) are

targeted to the same goal of achieving better energy usage. As it was observed in [9], DSC

interactions with lower layers of the protocol stack (network, data link, and physical layer)

deserve particular attention to fully exploit the claimed benefits. For instance, header overheads

that appear in any protocol data unit (PDU) format should be taken into account. A node may

aggregate packets coming from other nodes with its own data in order to reduce the impact

of packet overhead in multi-hop routing. We believe that a crucial concern is represented by a

proper (joint) combination of DSC topologies and packet aggregation (PA) in realistic multi-hop

communication scenarios.

According to our previous considerations, it can be argued that DSC has a relevant impact on

network, data link, and physical layers, and vice versa. Theinteraction between DSC and routing

has been investigated in [10]–[12]. Petrovic et al. [10] have also noted that routing with packet

aggregation allow for decreasing protocol overhead, with reduction of energy consumption. Baek

et al. [13] have dealt with the problem of finding the optimal packet aggregation alternative

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for WSNs with DSC. Classifications of aggregation protocolswhen nodes collect periodic

measurements have been done by Solis et al. [14], [15] and Roedig et al. [16]. In [17] we have

proposed an analysis of the energy consumption for various cases of DSC coding topologies

with PA. Using the same network scenario in [17], we have addressed in [18] the problem of

characterizing the percentage of measurements that cannotbe correctly decoded at the network

sink.

Although the above mentioned relevant contributions address fundamental topics of DSC, they

neglect some important aspects that relate to network design and operations in realistic scenarios,

namely: packet losses, packet aggregation and fragmentation, and overhead reduction. Indeed,

packet aggregation without packet fragmentation may be prohibitive in real multi-hop WSNs.

It is the purpose of this paper to develop a joint (comprehensive) analysis of the loss factor

and overall energy consumption. Our contribution is related to [5] because we include the four

alternatives of coding topology therein proposed. It is also related to [10], because we consider

three PA techniques. However, we simultaneously consider DSC and PA in a system scenario

including an accurate model of the physical layer, a data link layer, and multi-hop routing. In

this respect we extend considerably the analysis of DSC and PA proposed in [17] and [18].

The remainder of the paper is organized as follows: in Section II, the system model is

described. Section III is dedicated to the analysis of packet loss probability, which is then used

first in Section IV to characterize the network-wide loss factor, and subsequently in Section V to

analyze the energy consumption. Our theoretical frameworkis then used in Section VI to derive

and discuss numerical results in selected scenarios. Finally, in Section VII, we summarize the

main conclusions and evidence future perspectives.

II. SYSTEM MODEL

Consider an area whereinN nodes are deployed: each of them takes some measurements and

attempts to transmit reports towards a sink. The network topology is assumed to be organized

in a tree, where each node belongs to a generic leveli (with i = 1, . . . , L), i being the distance

of a node from the sink in terms of number of hops (see Fig. 1). Tree based architecture is a

relevant network topology for the IEEE 802.15.4 standard [19, pag. 15–16]. Furthermore, the

interest for such a network topology is confirmed by a large number of relevant contributions

that can be found in the literature (see, for instance, [5] – [8], [20] – [22]), which all deal with

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tree-like networks. Denote withli the number of nodes belonging to leveli. We setl0 = 1,

which is the number of nodes of level0 (only the sink). The equality∑L

i=1 li = N holds true.

Each node is connected to a parent node and takesm measurements ofm different physical

phenomena per sensing interval. We assume that the measurements of different phenomena are

continuous i.i.d. random variables, and that the measurements of a given phenomenon by different

sensor nodes are spatially correlated. The measurements are quantized by an analog-to-digital

converter, which is assumed to have the same resolution in each sensor. Denote withYi,j, for

i = 1, . . . , L, and for j = 1, . . . , li, the quantized measurement taken from the generic nodej

of level i at a given time instant for one of them phenomena. Coherently with the purpose of

this paper, we retainYi,j as a discrete random variable taken from random process, which has

a countable discrete alphabet [4], [6], [7]. Furthermore, aset of variablesYi,js at a given time

instant are generically spatially correlated, the correlation pattern being constant in time under

the assumption of stationarity [5]. In this context we suppose that the number of bits produced

by DSC encoding of measurements is determined by the entropy(this is a common assumption

in literature, see for instance [6] – [8], [11], [23], [24], and references therein). We denote by

H(Yij) the entropy of the considered source.

Each node performs the following tasks: it encodes the measurements according to the DSC

algorithm proposed in [4], receives PDUs from the child nodes, aggregates them, and then

transmits them towards its parent node. When an ARQ protocolis employed, retransmissions

are attempted until either successful reception is attained or a maximum number of tries is

reached.

For defining and computing performance indexes, in the following we refer to a packet as a

PDU, which results from application of major headers at the various layers of the protocol stack.

Hence, a packet is composed by a frame length indicator plus apreamble and a synchronization

symbol ofP bits; a data link header ofO bits, a network header ofQ bits; a payload incorporating

coded measurements taken by nodes, andID bits for node identification;CRC bits for the

forward error detection [25]. The payload may have variablesize as a consequence of the DSC

topology and aggregation mechanism, as it will be clear in the following subsections. The packet

structure is adherent to specifications by the IEEE 802.15.4communication standard [19] for the

data link layer and physical layer of the protocol stack, while we refer to Tmote Sky sensors [26]

for the network layer.

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In the next subsections, we first describe the DSC topologiesand the aggregation mechanisms

we intend to analyze; next we define the performance indexes we intend to adopt.

A. DSC Topology and Packet Aggregation

Four prominent DSC coding schemes are here studied: No-Slepian-Wolf (NOSW), Sequential

Slepian-Wolf (SEQ), Clustered Slepian-Wolf (CL), and Slepian-Wolf Master Slave (MS). They

were firstly proposed in [5]. In the following, we briefly summarize their characteristics.

In the NOSW topology, each sensor encodes measurements independently from other nodes.

In the SEQ scheme (see Fig. 2), node(1, 1) performs encoding ofY1,1 with H1,1 = H(Y1,1)

bits; node(1, 2) encodesY1,2 with H1,2 = H(Y1,2|Y1,1) bits, provided that the decoder knows

Y1,1; node (1, j) encodesY1,j with H1,j = H(Y1,j|Y1,1 . . . Y1,j−1) bits, provided that the de-

coder knowsY1,1 . . . Y1,j−1; in general, node(i, j) encodesYi,j using the following number of

bits Hi,j = H(Yi,j|Y1,1 . . . Y1,l1 Y2,1 . . . Y2,l2 . . . Yi,1 . . . Yi,j−1), provided that the decoder knows

Y1,1 . . . Y1,l1Y2,1 . . . Y2,l2 . . . Yi,1 . . . Yi,j−1.

In the CL topology (see Fig. 3), nodes are grouped inK = l1 clusters, where each cluster

consists of a sub-tree having a node of level1 as a root. Each cluster includes a number of

nodes, which is denoted byNk. For each node of a cluster, a SEQ coding topology is adopted

independently from other clusters and starting from the root node(1, k), for k = 1, . . . , l1, of

clusterk.

In the MS topology (see Fig. 4), nodes are grouped in clustersas in the case of CL, but each

root (1, k), for k = 1, . . . , l1, of a sub-tree is elected as master node. However, in contrast to

CL, for each node of a clusterk, the DSC is performed only with respect to the master(1, k)

of the cluster the node belongs to.

We consider three alternatives for aggregation: the classical multi-hop (CMH), where nodes

relay packets without any aggregation; the aggregated multi-hop (AMH), where nodes collect

received packets, aggregate them at the MAC layer into a single frame and relay it; the Frag-

mented Aggregated Multihop (FAMH), which differs from AMH as it allows fragmentation:

when an aggregated packet reaches a size that exceeds a maximum threshold, it is fragmented

in multiple packets.

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B. Performance Indexes

Two performance indexes are considered in this paper, namely: the loss factor and the energy

consumption of the network, which we denote withΛ andEN, respectively. The loss factor is

defined as the fraction of measurements generated by the network that cannot be reconstructed

at the sink. The energy consumption of the network is insteadexpressed as the average number

of bits (per reporting interval) transmitted across the network. In computingEN, we will neglect

the impact of acknowledgement packets, which is a rather common assumption in literature, and

is motivated by the fact that they have marginal impact on overall performance.1

Characterization of bothΛ and EN is founded on the packet error probability. In the next

sections we derive this probability for the various combinations of DSC topologies and PA

mechanisms.

III. PACKET LOSSPROBABILITY

Consider a wireless channel which exhibits non-selective fading behavior both in frequency and

in time, and a O-QPSK (offset quadrature phase shift keying)modulation. These assumptions are

consistent with transceivers operating in the ISM frequency band according to the IEEE 802.15.4

communication standard [19]. Therefore, the bit error probability can be expressed as [25]

Pe(d) =1

2

(

1 −

√

γ(d)

1 + γ(d)

)

, (III.1)

where γ(d) is the average value of the Signal-to-Noise Ratio (SNR) computed at distanced

from the signal source. We adopt the following model forγ(d) in dB [27]: γ(d)dB = Pt dB −

P (d)dB − Pn dB, wherePt is the transmit power,P (d) is the path loss at distanced from the

transmitter, andPn is the noise floor at the receiver. The path loss is expressed as P (d) =

P (d0) + 10 · α · log10(d/d0), whereP (d0) is the path loss computed at the reference distance

d0 (see e.g. [28]); whileα is the path-loss decay constant with typical values belonging to

the interval[2, 4]. Under the assumption that the CRC code is always able to detect perfectly

1Note that the performance indexes do not include the packet delay caused by the coding topology. Indeed, under the

assumptions that the correlation properties of the sensed phenomena are stationary, that the number of measurements per

sampling rate is the typical one provided by sensors off-the-shelf [26] (i.e. less than ten), and that the number of levels is not

large, then the delays are negligible.

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corrupted packets (see [29] for an experimental support), the average packet loss probability at

level i is defined as

Ψ(d, si) = 1 − (1 − φi) · [1 − Pe(d)]2·(P+O+Q+CRC+si) , (III.2)

whereφi is the collision probability, and the factor2 accounts for Manchester encoding. We

remark that the collision probability is computed on a bit basis in order to account for the effects

of the packet size. The termsi is defined as the average payload size among the nodes of leveli.

The average packet loss probability across one hop of leveli is defined as the average of (III.2)

with respect to the hop length:

Ψ(si) =1

dmax − dmin

∫ dmax

dmin

Ψ(ρ, si)dρ , (III.3)

wheredmin anddmax are, respectively, the minimum and maximum distances between any pair

of child-parent nodes. We call (III.3) single-hop packet loss probability. When an ARQ protocol

is implemented at levelj, with Mj maximum number of retransmissions, we can easily express

the packet loss probability of a packet generated at leveli as follows:

Ψ(si, Mj) = Ψ(si)Mj . (III.4)

The packet loss probability over a multi-hop path from nodei up to the sink can be computed

with

Ψ(si) = 1 −

i∏

j=1

[1 − Ψ(si, Mj)] . (III.5)

Remark 3.1: We can observe that in (III.3) and (III.4) the dependence on the aggregation

scheme and coding topology is included inΨ(si, Mj) through the payload size, which will be

characterized in the next subsections for the cases of aggregation CMH, AMH, and FAMH, and

coding NOSW, SEQ, CL, and MS.

We define the average connectivityCi = li+1/li, i = 0, . . . , L − 1, of a node belonging to

level i as the average number of children of that node, where we impose CL = 0. Furthermore,

for the sake of the performance analysis, we define the average coding rate of the leveli as

Hi =1

li

li∑

l=1

Hi,l .

8

A. CMH

The contribution to the average payload size for the CMH scheme at leveli, as given from

measurements generated by a node of levelj, can be expressed as

si = ID + m · Hj i = 1, . . . , L andj ≥ i , (III.6)

Therefore, the single-hop packet loss probability is givenby (III.3), wheresi is given by (III.6).

B. AMH

The following claim holds:

Claim 3.2: In the AMH scheme, the average payload size at theith level is

si =

ID ·(

1 +∑L−i

j=1

∏j

k=1 ni+k

)

+ m ·(

Hi +∑L−i

j=1 Hi+j ·∏j

k=1 ni+k

)

i = 1, . . . , L − 1 ,

ID + m · HL i = L .

(III.7)

whereni is the average number of packets of leveli that are successfully received by the parent

node of leveli − 1:

ni = [1 − Ψ(si, Mi)] · Ci−1, i = 2, . . . , L , (III.8)

Proof: In order to prove (III.7) and (III.8), consider the average number of packets that are

successfully transmitted to each parent-node of levelL − 1: nL = [1 − Ψ(sL, ML)] · CL−1 . At

level L − 1, each node aggregates the received packets and relays a unique packet having an

average payloadsL−1 = ID + m · HL−1 + sL · nL = ID + ID · nL + m · (HL−1 + HL · nL) .

At level L − 2, the average number of received packets isnL−1 = [1 − Ψ(sL−1, ML−1)] · CL−2.

Hence, the average payload size and the average number of packets coming to this level are as

follows

sL−2 =ID + ID · nL−1 · nL + ID · nL−1 + m · (HL−2 + HL−1 · nL−1 + HL · nL · nL−1) ,

nL−2 =[

1 − Ψ(sL−2, ML−2)]

· CL−3 .

Generalization of previous expressions to obtain (III.7) and (III.8) turns out from iteration.

According to the previous Claim, the single-hop packet lossprobability is given by (III.3)

and (III.7).

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C. FAMH

In the FAMH case, the payload size is limited to a maximum length smax. When the limit is

exceeded, the payload is split intoFi fragments, withFi = ⌈ si

smax⌉, where⌈·⌉ denotes the upper

integer (ceiling). Each fragment is then mapped onto a packet and let si = (Q + si)/Fi denote

the length of the related PDU at the data link layer. Since next hop node is known at the moment

of the fragmentation, it is not necessary to transmitFi times the same network header overhead,

which can be split intoFi fragments. The fragmented packets, having payloadsi, need to have

all other headers (preamble, MAC and CRC) to be transmitted,however, they do not need to

have the network header (for next hop is known at each fragment). The loss probability of a

fragmented packet, averaged over a single-hop distance, isgiven as follows:

ΨF (si) =1

dmax − dmin

∫ dmax

dmin

1 − (1 − φi) · [1 − Pe(ρ)]2·(P+O+CRC+si) dρ . (III.9)

The packet loss probability over a single-hop can be expressed as

Ψ(si) = 1 − [1 − ΨF (si)]Fi . (III.10)

Finally, expression (III.10) can be used in (III.4) to obtain the single-hop packet loss probability.

IV. L OSSFACTOR

Here we build on the packet loss probability analysis of previous section to derive the loss

factor. First, let us introduce some definitions. Denote with Di the average number of descendants

of the generic node of leveli whose measurements are successfully received at that node.Denote

with Di,j the average number of descendants at levelj whose measurements are successfully

received at the generic node of leveli. We also imposeDi,i = 1, and thatDi,j = 0 if i > j.

Then,

Claim 4.1: At level i of the network, the average number of descendants is

Di =

L∑

j=i+1

Di,j , i = 0, . . . , (L − 1) , (IV.1)

where

Di,j =

j∏

k=i+1

[1 − Ψ(sj, Mk)] · Ck−1. (IV.2)

10

Proof: The proof is by induction. At levelL − 1, DL−1 is easily obtained byDL−1 =

DL−1,L = [1− Ψ(sL, ML)] ·CL−1 . At level L− 2, DL−2 depends on the number of packets that

are generated from levelL and levelL − 1 and that are correctly received at levelL − 2:

DL−2 = DL−2,L−1 + DL−2,L =[

1 − Ψ(sL−1, ML−1)]

· CL−2 +[

1 − Ψ(sL, ML−1)]

·[

1 − Ψ(sL, ML)]

CL−2 · CL−1 . (IV.3)

For the generic leveli, (IV.3) can be extended in a straightforward way to obtain (IV.1).

In the next subsections, we characterize the loss factor foreach case of coding topology and

aggregation mechanism.

A. NOSW

In the CMH scenario, the probability that a packet generatedat leveli does not reach the sink

is given by (III.5). Hence, after averaging over all levels and number of nodes, the loss factor

can be expressed as

Λ(a)NOSW =

1

N

L∑

i=1

1 −

i∏

j=1

[

1 − Ψ(si, Mj)]

· li , (IV.4)

where, for the sake of clarity of notation, we have denoted with the superscript(a) the CMH

case.

When aggregation procedures are adopted, derivation of theloss factor has to take into account

that loosing a packet generated from a node of leveli determines the avalanche effect of losing

measurements successfully received by that node and comingfrom lower levels of the network. It

is not difficult to see that a packet loss at leveli causes an average loss of1+Di measurements.

Therefore, the loss factor can be expressed as

Λ(b)NOSW =

1

N

L∑

i=1

1 −i

∏

j=1

[

1 − Ψ(si, Mj)]

· (1 + Di) · li , (IV.5)

where the computation ofΨ(si, Mj), which appears also in theDi, is performed by using (III.3),

(III.4), and (III.7). Note that the superscript(b) denotes AMH.

When the FAMH scheme is considered, the loss factor can be derived in the same way as in

the case of AMH:

Λ(c)NOSW =

1

N

L∑

i=1

1 −i

∏

j=1

[

1 − Ψ(si, Mj)]

· (1 + Di) · li , (IV.6)

11

where the computation ofΨ(si, Mj) is accomplished using (III.4), (III.9), and (III.10), respec-

tively. Note thatΨ(si, Mj) is also implicit in computation ofDi, although, as done before, it is

not evidenced for the sake of simple notation. The superscript (c) denotes the FAMH case.

B. SEQ

Let us firstly consider the CMH (a) case:

Claim 4.2: The loss factor can be expressed by

Λ(a)SEQ =

1

N

L∑

i=1

A(a)SEQ,i ·

li∑

j=1

BSEQ,i,j · C(a)SEQ,i,j , (IV.7)

where

A(a)SEQ,i =

1 −

i∏

m=1

[

1 − Ψ(si, Mm)]

,

BSEQ,i,j = N + 1 − j −i−1∑

m=1

lm ,

C(a)SEQ,i,j =

i−1∏

k=1

k∏

m=1

[

1 − Ψ(sk, Mm)]lk

·

i∏

m=1

[

1 − Ψ(si, Mm)]j−1

.

and whereΨ(si, Mk) is computed with (III.4).

Proof: We prove (IV.7) by induction. Denote withΦ1,1 the event of losingY1,1, so that

measurements encoded with respect toY1,1 cannot be decoded. Then, in the case of the event

Φ1,1, it causes the avalanche effect of losingN measurements, sinceY1,1 is lost along with

all measurements taken by the otherN − 1 nodes that were encoded with respect toY1,1.

Therefore, the amount of measurements lost when only the event Φ1,1 happens, is given by

ΛΦ1,1= Ψ(s1, M1) · N .

Denote withΦ1,2 the event of successfully receivingY1,1 while losingY1,2. Then, whenΦ1,2

occurs,N − 1 measurements are lost, sinceY1,2 is lost along with theN − 2 measurements that

were coded with respect toY1,2. Therefore, the amount of measurements lost when onlyΦ1,2

takes place is given byΛΦ1,2= Ψ(s1, M1)· (N − 1) · [1 − Ψ(s1, M1)] .

Denote withΦ1,l1 the event of losingY1,l1 , while Y1,1, Y1,2, . . . , Y1,l1−1 were successfully re-

ceived. IfΦ1,l1 happens, thenN−l1+1 measurements are lost, since measurementsY2,1, . . . , Y2,l2,

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Y3,1, . . . , Yl,lL were coded with respect toY1,l1. Therefore, the amount of measurements lost

in correspondence of the only eventΦ1,l1 is given by ΛΦ1,l1= Ψ(s1, M1) · (N + 1 − l1) ·

[

1 − Ψ(s1, M1)]l1−1

.

Consider now the general case of the eventΦi,j , when Yi,j is lost, while Y1,1, Y1,2, . . . ,

Y1,l1, Y2,1 . . . , Yi,1 . . . , Yi,j−1 are successfully received. WhenΦi,j occurs, thenYi,j+1, . . . , Yi,li,

Yi+1,1, . . . , YL,1, . . . , YL,ll are lost, since they were coded with respect toYi,j. Hence, the total

number of lost measurements isN + 1 − j −∑i−1

k=1 lk , and the amount of measurements lost

when the only eventΦi,j takes place isΛΦi,j= A

(a)SEQ,i · BSEQ,i,j · C

(a)SEQ,i,j . By averagingΛΦi,j

with respect to all measurements, Eq. (IV.7) is obtained.

In the cases of AMH and FAMH, expressions for the loss factorsare similar to those given

by Claim 4.2. In fact, the loss of a packet coming from node(i, j) induces the loss of all

measurements aggregated from lower layers of the network until (i, j): its average value is

given by Di in (IV.1). However, this number of losses is already included in the number of

measurements that, being coded with respect toYi,j, cannot be reconstructed. It is easy to see

from Fig. 1 that this metric can be expressed asN +1− j −∑i−1

k=1 lk. Therefore, we can readily

express the loss factor for the AMH (b) case as

Λ(b)SEQ =

1

N

L∑

i=1

A(b)SEQ,i ·

li∑

j=1

BSEQ,i,j · C(b)SEQ,i,j , (IV.8)

where the superscript(b) denotes that the probabilityΨ(si, Mj) has to be computed using (III.3),

(III.4), and (III.7).

For the case of FAMH(c), we have

Λ(c)SEQ =

1

N

L∑

i=1

A(c)SEQ,i ·

li∑

j=1

BSEQ,i,j · C(c)SEQ,i,j , (IV.9)

where the superscript(c) denotes that the computation ofΨ(si, Mj) is accomplished using (III.4),

(III.9), and (III.10).

C. CL

When the network is partitioned into clusters, for each cluster the loss factor can be computed

as in the SEQ case. Hence, the loss factor for CMH (a), AMH (b),and FAMH (c) is provided

13

by the following expression:

Λ(a,b,c)CL =

1

K

K∑

k=1

Λ(a,b,c)SEQ,k , (IV.10)

where the subscriptk denotes the loss factor of cluster numberk, and it is computed using (IV.7)

for the case(a), (IV.8) for the case(b), and (IV.9) for the case(c). When computingΛ(a,b,c)SEQ,k,

N and li have to be replaced (whenever they appear in (IV.7), (IV.8),and (IV.9)) with Nk and

l(k)i respectively, wherel(k)

i is defined as the number of nodes of leveli of clusterk. Notice that

l(k)0 = 1 is the number of nodes of level0 (only the master node) of clusterk.

D. MS

Let us consider first the CMH (a) scheme. We have the followingclaim:

Claim 4.3: The loss factor for the CMH scheme is

Λ(a)MS =

1

K

K∑

k=1

Λ(a)MS,k ,

whereΛ(a)MS,k is the loss factor for clusterk, and it is given by

Λ(a)MS,k = Ψ(s1, M1) +

1

Nk

L∑

i=2

l(k)i ·

[

1 − Ψ(s1, M1)]

·

1 −i

∏

m=1

[

1 − Ψ(si, Mm)]

,

and whereΨ(si, Mm) is computed using (III.4).

Proof: Denote withΩ1,k, for k = 1, . . . , l1, the event of losingY1,k. When Ω1,k takes

place, allNk measurements of clustersk are lost, since they were coded with respect toY1,k.

Therefore, the amount of lost measurements induced by the only eventΩ1,k is given byΛΩ1,k=

NkΨ(s1, M1) .

Consider now the eventΩi,j , for i = 2, . . . , L and j = jk, . . . , jk + l(k)i − 1, where jk is

the index of the first node of leveli of cluster k, and jk + l(k)i − 1 is the index of the last

node of the same level of the same cluster. The eventΩi,j occurs whenY1,k is successfully

received andYi,j is lost. Therefore, the loss factor induced by the only eventΩi,j is given as

ΛΩi,j=

[

1 − Ψ(s1, M1)]

·

1 −∏i

m=1

[

1 − Ψ(si, Mm)]

. By averaging with respect toΛΩi,j,

we get expression (IV.11).

In the case of packet aggregation, the procedure to derive the loss factor is the same as in the

CMH scheme. However, the loss of a measurement taken by a node(i, j) in clusterk, induces

to lose furtherD(k)i measurements on the average, whereD

(k)i is defined as in (IV.1), but using

14

C(k)i instead ofCi, with C

(k)i defined asC(k)

i = l(k)i+1/l

(k)i , i = 0, . . . , L− 1 , and thatC(k)

L = 0.

Therefore, the loss factor is given by

Λ(b)MS =

1

K

K∑

k=1

Λ(b)MS,k

where

Λ(b)MS,k = Ψ(s1, M1) + (IV.11)

1

Nk

L∑

i=2

l(k)i · (1 + D

(k)i ) ·

[

1 − Ψ(s1, M1)]

·

1 −

i∏

m=1

[

1 − Ψ(si, Mm)]

.

In (IV.11), Ψ(si, Mm) and D(k)i are computed by resorting to (III.3), (III.4), and (III.7).Ob-

viously, (III.7) is computed by replacingN , li, andCi with Nk, l(k)i , andC

(k)i whenever they

appear.

The loss factor in the FAMH (c) case is derived by applying thesame steps described for the

(b) case. Hence,

Λ(c)MS =

1

K

K∑

k=1

Λ(c)MS,k

where

Λ(c)MS,k = Ψ(s1, M1) +

1

Nk

L∑

i=2

l(k)i · (1 + D

(k)i ) ·

[

1 − Ψ(s1, M1)]

·

1 −i

∏

m=1

[

1 − Ψ(si, Mm)]

.

In previous expression,Ψ(si, Mm) andD(k)i are given by (III.4), (III.9), and (III.10).

V. AVERAGE ENERGY CONSUMPTION

Expressing the average number of bits transmitted by the network requires characterization

of the average number of transmitted packets at the generic level i. Recall that such a number

depends on the coding topology by the payload size. In the following, we derive the energy

consumption for the CMH, AMH and FAMH schemes.

The number of retransmissions of a packet sent from a node of level i over the hop between

level i and i − 1 is denoted asΘi. It is expressed as [25]

Θi =1 + Mi · Ψ(si)

Mi+1 − (Mi + 1) · Ψ(si)Mi

1 − Ψ(si).

15

Therefore, the average number of transmitted bits from the generic node of leveli, when an

ARQ protocol is employed, is

Bi =

L∑

j=i

(P + O + Q + sj + CRC) · Di,j · Θj ,

whereDi,j has been defined in (IV.2).

Consider the CMH (a) case. The average number of transmittedbits is

E(a)N =

L∑

i=1

Bi · li . (V.1)

Let us consider now the AMH (b) case. Each node of leveli transmits only one packet

containing both the measurements taken from local sensing and the measurements coming from

lower levels of the network and encapsulated therein. Therefore, EN is readily expressed as

follows:

E(b)N =

L∑

i=1

(P + O + Q + si + CRC) · Θi · li . (V.2)

Finally, consider the FAMH (c) case. It is easy to see that theaverage number of transmitted

bits is expressed as

E(c)N =

L∑

i=1

(P + O + si + CRC) · Fi · Θi · li , (V.3)

whereΘi is computed using the payload sizesi for the FAMH case.

VI. NUMERICAL EXAMPLES

In this section we report numerical results obtained by a Matlab implementation of the

analytical framework developed in previous sections. We consider two representative scenarios:

the first one refers to large spatial correlations among measurements, while the second one refers

to low correlation patterns. The parameters setting adopted for simulations is representative of the

Tmote Sky sensors [26] and the communication standard IEEE 802.15.4 [19], and are introduced

in the sequel.

We consider a network deployed in a square area having facet of 11m with the sink located in

the middle andN = 64 nodes distributed inL = 4 levels, where each node is randomly located

within a sub-square of1.2m. The values set for the number of nodes for each level, alongwith

average connectivity, are reported in Tab. II. Each node sensesm = 8 measurements per sensing

event. The packet frame format is as follows:P = 48, O = 184, Q = 56, ID = 32, CRC = 16,

16

andsmax = 760, where the unit is intended in bit. Transmission power has been set to−10 dBm.

The noise floor isPn = −120 dBm [26]. The collision probability has been set toφi = 10−5 (this

corresponds to a packet collision probability of about0.005, which is quite large if related to the

sampling rate, the packet duration, and the number of nodes). Smaller values for the collision

probability basically induce similar effects. Without loss of generality, and coherently with [19],

we set the maximum retransmission iterations of the ARQ protocol for levels1, ..., L = 4,

to 3, 3, 1, 1, respectively. This choice is motivated by the fact that nodes closer to the sink are

subject to larger packet losses as a consequence of larger traffic load. Other choices for the ARQ

protocol provide basically the same trends in the results asthose presented later in this section.

Without loss of generality, we make use of the differential entropy instead of the usual entropy,

as proposed in [6]– [8], and [30, pagg. 227–229]). By adopting this model, measurements are

characterized with aN dimensional multi-variate normal distributionGN(µ,K) having average

µ and covariance matrixK = [Ki,j ], whereKii = σ2i and Ki,j = σ2

i e−rdij for i 6= j, where

r is the spatial correlation decay parameter, anddij is the distance between nodesi and j.

The correlation decay parameter is defined such that it tendsto 0 for highly correlated sources,

whereas tends to1 for low correlated sources, as discussed in [7].

In Fig. 5, the loss factor is plotted for various cases of the coding topology (NOSW, SEQ,

CL, and MS) and aggregation scheme (CMH, AMH, and FAMH) as obtained in Section IV,

for a highly correlated measurements scenario, wherer = 10−5. As a first general remark, it

can be observed that the adoption of ARQ significantly decreases the amount of measurements

that cannot be decoded at the sink node. This is particularlyevident in the MS case, where

the decrease of loss factor due to ARQ is relevant. When looking at the SEQ case, the loss

factor takes on the largest value among DSC topologies. Indeed, recalling the behavior of the

SEQ topology summarized in Section II.A, losing a measurement at node(i, j) determines the

avalanche effect of losing all measurementst, q, with t = i+1, . . . , L andq = j +1, . . . , li. The

same effect appears also in the CL topology, but clustering reduces the amount of performance

loss. Indeed, no relevant avalanche effect is present in this topology. When looking at aggregation

mechanisms, we can observe that they cause a slight increaseof the loss factor, since packets

have larger sizes if compared to the CMH case. A relevant conclusion that can be drawn from

Fig. 5 is that MS seems to be the most appealing coding topology in terms of reliability, and

that aggregation mechanisms exhibit similar performance within the same coding topology.

17

In Fig. 6, values of the energy consumptionEN for the various DSC topologies and aggregation

schemes are plotted as obtained for highly correlated sources. They have been computed through

the analysis presented in Section V. The right column is referred to the ARQ protocol, while

the left column is referred to the case where ARQ is not adopted. Although ARQ introduces

a larger number of packets per node to be relayed, we can clearly observe that it determines a

small energy increase. Nevertheless, by considering that the ARQ protocol is beneficial in terms

of loss factor, it can be concluded that the adoption of ARQ induces significant advantages. The

NOSW topology exhibits the worst performance, as no compression is adopted. It is interesting

to observe that SEQ, CL and MS topologies do not exhibit largedifferences. This is mainly due

to the large spatial correlation, so that measurements froma few nodes are enough to achieve

good compression rates. Considering the aggregation mechanisms, it is evident that they provide

significant performance improvement. Specifically, CMH exhibits the worst performance among

aggregation mechanisms, whatever DSC scheme is adopted. This is due to the fact that packet

overhead introduces extra bits in the packet transmission.

For a better understanding of energy consumption, define thefollowing coding efficiency

metric:

ηCT = 1 −EN,CT

EN,NOSW

, (VI.1)

where the subscript CT denotes one of the coding topologies SEQ, CL, or MS. Large values of

ηCT mean that the corresponding coding topology exhibits reduced energy consumptions when

compared to the NOSW case.

In Fig. 7, the coding efficiency is plotted versus the different coding topologies (x axis)

and aggregation algorithms for the case of highly correlated sources. The coding efficiency

increases remarkably when we move from CMH to FAMH, with AMH providing intermediate

performance. We can observe that the performance provided by the three topologies, SEQ, CL

and MS, are quite different for the cases of AMH and FAMH, while they are less evident in

CMH. It is interesting to remark that the MS coding topology shows the worst performance.

Indeed we recall that the compression is performed with respect to only one measurement for

each cluster.

In order to compare both efficiency and reliability, in Fig. 8the coding efficiency is reported

as a function of the loss factor, for the case of highly correlated sources. It turns out that the

18

SEQ alternative often offers the largest coding efficiency,but also induces larger loss factors. On

the contrary, the MS case exhibits poor energy savings, but good loss factor. The CL topology

presents fair joint performance. Finally, it is confirmed that aggregation mechanisms significantly

impact on the energy consumption.

Let us now consider the case of low correlated measurements,in which we setr = 10−3.

In Fig. 9, we plot the coding efficiency. As expected, we can observe that the maximum

achievable coding efficiency is smaller than the maximum values obtained for highly correlated

measurements in Fig. 7. Notice that once again the ARQ protocol does not induce significant

degradation of the energy efficiency. In Fig. 10, we reportedthe coding efficiency as a function

of the loss factor. The remarks done for Fig. 8 still hold true, confirming the good behavior of

the MS in terms of loss factor, and its poor performance in terms of energy efficiency.

VII. CONCLUSIONS AND FUTURE PERSPECTIVES

A theoretical framework to evaluate expressions for the loss factor and energy efficiency in the

presence of three alternatives of packet aggregation mechanisms (classic multi-hop, aggregated

multi-hop and aggregated and fragmented multi-hop), and four alternatives of distributed source

coding topology (no Slepian-Wolf coding, sequential, clustered and master slave) has been

proposed.

Numerical results show that, under the correlation model wehave used, SEQ coding topology

performs better in terms of energy consumption, yet showingpoor reliability performance.

On the contrary, the MS case offers better performance in terms of measurements that are

successfully decoded, but it is poorer in terms of energy efficiency. Furthermore, the ARQ

protocol significantly decreases the amount of measurements which cannot be decoded at the

sink; moreover, retransmissions do not seem to remarkably affect energy efficiency. We have

instead shown that packet overheads and packet loss probability may play a significant role in

network energy consumption.

Future study will be focused on the extension of our theoretical analysis to evaluate the

interplay of main communication system parameters, e.g. including radio frequency power

control. Furthermore, while in the present paper we have considered a tree-like network, we

plan to extend our analysis to a joint optimization of the routing, aggregation scheme, source

coding with respect to the spatio-temporal correlation pattern of the distributed source.

19

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21

Symbol Meaning

N Number of nodes

Nk Number of nodes in the clusterk

L Number of levels of the network

li Number of nodes of leveli of the network

Ci Average connectivity of a node of leveli

Di Average number of descendants of a node of leveli

Di,j Average number of descendants at levelj whose measurements are successfully received at a node of level i

K Number of clusters of nodes

dmin Minimum distance among nodes

dmax Maximum distance among nodes

Yi,j Data measured by nodej of level i

m Number of sensed phenomena per node per sampling time

H(·) Entropy function of the source

P Frame length indicator + Preamble+Sync size (bits)

O DataLink section size (bits)

Q Network section size (bits)

si Average payload size (bits)

si Average fragmented payload size (bits)

smax Maximum payload size (bits)

ni Average number of packets of leveli that are successfully received by a parent node of leveli − 1

CRC Correction code section size (bits)

ID Identification code size (bits)

Fi Number of packet fragments at leveli

Mi Maximum number of retrasmissions at leveli

γ(·) SNR function

φi Collision probability at leveli

Pe(·) Bit error probability function

Ψ(d, si) Single-hop packet loss probability function

Ψ(si) Average single-hop packet loss probability function

Ψ(si, Mj) Single-hop packet loss probability function with ARQ

Ψ(si) Multi-hop packet loss probability function with ARQ

Θi Average number of retransmissions for a packet of leveli

Bi Average number of transmitted bits for a node of leveli

Λ Loss Factor of the network

EN Average number of transmitted bits of the network

ηCT Coding efficiency

TABLE I

SYMBOLS.

22

i 0 1 2 3 4

li 1 4 12 20 28

Ci 4 3 20/12 28/20 0

TABLE II

NUMBER OF NODES FOR EACH LEVEL AND AVERAGE CONNECTIVITY.

Fig. 1. Network topology. Nodes are organized as a tree withL levels. The root node acts as sink.

Fig. 2. Sequential coding topology (SEQ).

23

Fig. 3. Clustered coding topology (CL). In each cluster, a SEQ coding is employed. The superscript in the indexes corresponds

to the cluster the node belongs to. Notice that the level 1 of each cluster contains only a node.

Fig. 4. Master Slave coding topology (MS). It is similar to the CL, but the coding is done only with respect to a cluster’

master. The superscript in the indexes corresponds to the cluster the node belongs to. Notice that the level 1 of each cluster

contains only a node.

24

CMH AMH FAMH0

0.2

0.4

0.6

0.8

LOS

S F

AC

TO

R

NO SLEPIAN−WOLF

CMH AMH FAMH0

0.2

0.4

0.6

0.8

LOS

S F

AC

TO

R

SEQUENTIAL SLEPIAN−WOLF

CMH AMH FAMH0

0.2

0.4

0.6

0.8

LOS

S F

AC

TO

R

CLUSTERED SLEPIAN−WOLF

CMH AMH FAMH0

0.2

0.4

0.6

0.8

LOS

S F

AC

TO

R

MASTER−SLAVE

NO ARQ

ARQ

Fig. 5. Loss factor for CMH, AMH, and FAMH with and without ARQ, for the case of highly correlated measurements. The

right bars are referred to ARQ case, whereas the left ones arereferred to the case without ARQ.

CMH AMH FAMH0

2

4

6

8x 10

4

BIT

TX

/ S

NA

PS

HO

T

NO SLEPIAN−WOLF

CMH AMH FAMH0

2

4

6

8x 10

4

BIT

TX

/ S

NA

PS

HO

T

SEQUENTIAL SLEPIAN−WOLF

CMH AMH FAMH0

2

4

6

8x 10

4

BIT

TX

/ S

NA

PS

HO

T

CLUSTERED SLEPIAN−WOLF

CMH AMH FAMH0

2

4

6

8x 10

4

BIT

TX

/ S

NA

PS

HO

T

MASTER−SLAVE

NO ARQ

ARQ

Fig. 6. Average number of byte transmitted by the network forthe case of highly correlated measurements. The right bars are

referred to ARQ case, whereas the left ones are referred to the case without ARQ.

25

SEQ CL MS5

10

15

20

25

CO

DIN

G E

FF

ICIE

NC

Y (

%)

CMHCMH−ARQAMHAMH−ARQFAMHFAMH−ARQ

Fig. 7. Coding Efficiency for CMH, AMH, and FAMH with and without ARQ for the case of highly correlated measurements.

On the abscissa the cases SEQ, CL, and MS are reported.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.96

8

10

12

14

16

18

20

22

24

26

LOSS FACTOR

CO

DIN

G E

FF

ICIE

NC

Y (

%)

SEQ

CL

MS

CMHCMH−ARQAMHAMH−ARQFAMHFAMH−ARQ

Fig. 8. Coding Efficiency vs Loss factor for CMH, AMH, and FAMHwith and without ARQ, for the case of highly correlated

measurements. On each curve, the markers on the left are referred to MS, in the middle to CL, and on the right to SEQ.

26

SEQ CL MS2

4

6

8

10

12

14

16

CO

DIN

G E

FF

ICIE

NC

Y (

%)

CMHCMH−ARQAMHAMH−ARQFAMHFAMH−ARQ

Fig. 9. Coding Efficiency for CMH, AMH, and FAMH with and without ARQ, for the case of weakly correlated measurements.

On the abscissa the cases SEQ, CL, and MS are reported.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92

4

6

8

10

12

14

16

LOSS FACTOR

CO

DIN

G E

FF

ICIE

NC

Y (

%)

SEQ

CL

MS

CMHCMH−ARQAMHAMH−ARQFAMHFAMH−ARQ

Fig. 10. Coding Efficiency vs Loss factor for CMH, AMH, and FAMH with and without ARQ, for the case of weakly correlated

measurements. On each curve, the markers on the left are referred to MS, in the middle to CL, and on the right to SEQ.