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Abstract To fully exploit the capabilities of Bluetooth for the deployment of wireless ad-hoc networks, the scatternet concept has been proposed. A scatternet is constituted by an overlapping of simple structures called piconets, each composed of up

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$*HQHUDO0HWKRGRORJ\DQG.H\0HWULFVIRU6FDWWHUQHW)RUPDWLRQLQ%OXHWRRWK
Francesca Cuomo, Tommaso Melodia
INFOCOM Department, University of Rome “La Sapienza”, Via Eudossiana 18, 00184, Rome, ITALY
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I. I
NTRODUCTION
Bluetooth (BT) is a promising short range, low power andlow cost technology for the deployment of wireless ad hocnetworks. It is packet oriented and supports about 1 Mbit/s ina so-called “piconet”, where up to 8 BT devices cansimultaneously inter-connect. The radius of a piconet is about10 meters. No network infrastructure is envisaged: self-organization and peer communication will lead to a complete“ad-hoc connectivity”. The multiple access technique isFHSS-TDD (Frequency Hopping Spread Spectrum – TimeDivision Duplexing) [1][2]. Two BT units exchangeinformation by means of a master-slave relationship. Masterand slave roles are dynamic: the device that starts thecommunication is the master, the other one is the slave.One of the key issues associated with the BT technology isthe possibility of dynamically setting-up and tearing downpiconets while interconnecting them in a “scatternet”, i.e., anoverlapping of piconets as in Fig. 1. Devices number 1, 5 and6 in Fig. 1. are masters; devices 4 and 7 are slaves in differentpiconets and on a time division basis; act as “bridge” andforward traffic to devices belonging to different piconets.
VFDWWHUQHW
PDVWHU VODYH EULGJH SLFRQHW
VFDWWHUQHW
PDVWHU VODYH EULGJH SLFRQHW
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Clearly, we can think of many alternatives to form ascatternet out of the same group of
1
devices. The way ascatternet is formed considerably affects its performance.In this paper, we define a methodology for the analysis of the scatternet formation issue. Section II briefly summarizesthe state of the art in scatternet formation. In Section III, wepresent the methodology. Section IV presents metrics that canbe used to evaluate a scatternet; related numerical results areshown in Section V. Finally, Section VI reports the mainconclusions.
II. R
ELATED WORKS
Scatternet formation has recently received a significantattention in literature. Existing works on this topicconcentrate on two main aspects: i) formation of a scatternetby means of a distributed algorithm [3][4][5][6][7][8]; ii)formation of an optimal scatternet, by using centralizedalgorithms [9], where the optimality is defined according tosuitable criteria.In the first case, the aim is to form a connected BTscatternet without
RSWLPL]LQJ
specific performance and byrelying on distributed algorithms. A second class of worksconcentrate on the
RSWLPL]DWLRQ
of the scatternet topology. Tothe best of our knowledge this issue is faced for the first timein [9] where a
FHQWUDOL]HG
optimization aims at minimizingthe load of the most congested node in the network.Finally, works exist that, even if not strictly concernedwith scatternet formation, can be of interest in ourframework. Such papers try to optimize the use of theresources of an already formed scatternet, by focusing onscheduling mechanisms that operate between adjacentpiconets [2].In this paper, we focus on two main points:
•
definition of a methodology for the scatternet formation,according to suitable metrics, based on a representation ina matrix form, which turns out to be a very simple andeffective design tool; this methodology can be used togenerate scatternets by means of both centralized anddistributed approaches;
•
identification of several metrics that can be used to formand evaluate scatternets; we emphasize the differencebetween traffic dependent and traffic independent metricsand we show some numerical results.
III. T
HE SCATTERNET FORMATION ISSUE
Before addressing the scatternet formation issue it is usefulto define a methodology for its representation.
$6FDWWHUQHWUHSUHVHQWDWLRQ
We assume that a master can not contemporarily be a slavein any other piconet, since this case leads to inefficientbandwidth usage ([10 in fact, when the mastercommunicates as slave, no communication can occur in thepiconet where it plays the role of master. As a consequence ascatternet can be described as a
ELSDUWLWH
graph, i.e., a graphwhose nodes are divided in two disjoint sets. A link may existbetween two nodes only if they belong to different sets.A scatternet may thus be represented by a rectangular
0
x
6
binary matrix
%
, where
0
and
6
are the number of mastersand slaves respectively. Element
E
vw
in the matrix equals 1 iff slave
M
belongs to master
L
’s piconets ([10]).For instance, the scatternet of Fig. 1 may be represented by
the following matrix
%
(1a). Besides, the path between thepair of nodes (
K
,
N
may be expressed by another
0
x
6
matrix
( )
%3
N K
,
, whose element
N K LM
S
,
equals 1 iff the link betweenmaster
L
and slave
M
is part of the path between node
K
andnode
N
(1
≤
K
,
N
,
L
,
M
≤
1
)). As an example, and referring again toFig. 1, the path between node 2 and node 8 can berepresented by the matrix
3
!'
(
%
) of Eq. (1b).
%
=00115000161110123470108
%
=00115000161110123470108
(a)
3
(
%
)=00115000161010123470108
3
(
%
)=0
0115000161010123470108
(b)(1)Let us now investigate how to represent a scatternet in theabove matrix form, starting from a given set of
1
nodes. Todescribe the considered set of nodes and the relevant possiblerelationships we use the
1[1DGMDFHQF\PDWUL[
$
=[
D
vw
],whose element
D
vwÃ
equals 1 iff device
M
is in the radio range of device
L
(i.e.,
M
can directly receive the transmission of
L
).Given a set of
1
nodes
=
and an adjacency matrix
$
, wecan build the
0
x
6
matrix
%
=[
E
vw
] by associating the rows to a
=
non empty subset of
0
nodes in
=
, and the columns to a
=
non empty subset of
6
nodes in
=
, with
1
=
0
+
6
,}0{
=∩
V P
= =
and
= = =
V P
=∪
. The resulting matrix
%
may be considered representing a
“
BT-compliant
”
scatternetwith
0
masters and
6
slaves if the properties of Tab. 1 apply.
Qr
QrÃshyvhv
)
rhpuÃhrÃvÃprprqÃhÃyrhÃÃhÃyhrÃ
0 L E
6 M LM
∈∀≥
∑
=
1
1
!)
ÃrÃuhÃrrÃyhrÃirytÃÃhÃvpr
0 L E
6 M LM
∈∀≤
∑
=
7
1
")
rhpuÃyhrÃvÃprprqÃhÃyrhÃÃhÃhr
6 ME
0 L LM
∈∀≥
∑
=
1
1
#)
urÃryvtÃrxÃvÃprprq
the matrix B does not have a block structure, rows permutationnotwithstanding
UhiÃÃ±Ã7UÃrvrÃ
%$PHWKRGRORJ\IRUVFDWWHUQHWIRUPDWLRQ
By exploiting the above representation, we can introduce ageneral methodology that can be used: i) to form optimizedscatternets on the basis of suitable metrics by means of bothcentralized and distributed approaches; ii) to evaluate theensuing performance. In Fig. 2 we report a block scheme of this methodology.The upper part of the figure is concerned with a centralizedapproach:
•
block 1 randomly generates communication scenarios;
•
block 2 identifies and lists all the
“
BT-compliant
”
scatternets that may be obtained starting from thescenarios produced by the first block;
•
block 3 evaluates performance parameters of thescatternets identified by the second block by means of suitable metrics, which can be traffic-independent ortraffic-dependent;
•
block 4 chooses the optimal scatternet according to thechosen metric.As for the first block, the generation of the scenarioproceeds by complying with these constraints:
•
every node must be able to connect to at least anothernode, the radio range,
G
PD[
, is assumed to be 10 meters;
•
groups of isolated devices are not allowed (otherwisesplitted scatternets could arise).A new tentatively generated node can enter the scenario iff its distance from its closest node is less than or equal to
G
PD[
The generated scenario is then represented by an
1
x
1
adjacency matrix
$
.The function of the second block is to exhaustively identifyand list all
“
BT-compliant
”
scatternets that may be obtainedfrom the scenario represented in
$
. Let be:
•
0
the number of masters in the scatternet, with
0
PLQ
+
17
1
≤
0
≤
0
PD[
=
2
1
; as regards
0
PD[
we assumeto limit its value to
2
1
since a number of masters greaterthan half the nodes introduces inefficiencies (e.g.,interference) without bringing benefits to the scatternet;
•
$
’
a rectangular
0
x(
10
) adjacency matrix, whichrepresents the complete set of possible connectionsbetween masters and slaves deriving from a particularway of selecting the
0
masters among the
1
nodes.The number of possible different
$
’
matrices is equal to
∑
=
maxmin
0 0 0
0 1
, since there are
0 1
possible ways of selecting
0
masters among the
1
nodes.From each
$
’
matrix, a number of
%¶
matrices is derived,each representing a subset of the possible connections amongnodes in
$
’
. Those which respect properties 1-4 represent alland alone the scatternets obtainable with that choice of masters and constitute the output of the block (
%
matrices).In the third block, every
%
matrix is evaluated according toa metric that corresponds to a particular optimization targetfunction. Finally, the fourth block chooses the scatternetwhose representing matrix
%
optimizes (maximizing orminimizing) the selected metric.The output of the overall process is the scatternet with theoptimal topology (according to the metric applied). Thisprocess requires complete knowledge of the scenariocharacteristics and can be realized only by means of centralized algorithms. This approach has two importantadvantages: i) allows identifying all possible
“
BT-compliant
”
scatternets deriving from a given scenario; ii) allows to easilyapply different metrics to the scatternet formation processand to evaluate the resulting scatternet performance. Clearlythese pros have to be paid with the complexity of examiningall
“
BT-compliant
”
scatternets.Conversely, the lower part of Fig. 2 is concerned with adistributed approach, analyzed into details in [11]. Startingfrom the same scenario of the centralized approach, adistributed algorithm is applied to form a scatternet, with thesame metric of the previous case. The resulting scatternetmay be compared, on the basis of the selected metric, to theone formed in a centralized way.
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&HQWUDOL]HG 'LVWULEXWHG
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IV. M
ETRICS FOR THE SCATTERNET EVALUATION
The methodology to form scatternets exploits suitablemetrics that can be either dependent or independent on thetraffic loading the scatternet. In the Traffic Independent (TI)case, the scatternet is formed without
DSULRUL
knowledge of traffic relationships among involved devices. The scenario isdescribed only by means of the adjacency matrix
$
.If traffic relationships between nodes (e.g., flows at givendata rates) have to be taken into account, they can beconveniently described by a traffic matrix,
7
. In thefollowing, we refer to this case as Traffic Dependent (TD).In this Section, we will introduce several metrics; we areaware that such metrics have pros and cons. However, wepresent them to examine and discuss their characteristics sothat the network engineer can chose the most suitable one bycomparing them via a performance evaluation study.
$7,PHWULFVVFDWWHUQHWZLWKPD[LPXPFDSDFLW\
A first TI metric is the overall capacity of the scatternet,which has to be maximized. The capacity of each piconetdepends on adopted intra-piconet and inter-piconetscheduling policies [2]. We assume that each master allocatesthe same portion of capacity to each connection with any of its slaves, and that the same amount of capacity is allocatedto the two directions of each communication: master-to-slaveand slave-to-master. Finally, we assume that each bridgingnode spends the same time in each of the piconets it belongsto. These assumptions are adopted for the sake of simplicity;the proposed methodology does not depend on them and canbe applied for whatever scheduling policies.The scatternet capacity will be evaluated by normalizingits value to the overall capacity of a piconet (i.e.
≈
1 Mbit/s).Let us define two
0
x
6
matrices,
2
7,
(
%
)=[
R
LM
], and
'
7,
(
%
)
[
G
LM
], with
R
LM
=
E
LM
/
P
M
and
G
LM
=
E
LM
/
V
L
(
M
=1,
…
,
6
and
L
=1,
…
,
0
). In these matrices
s
i
denotes the number of slavesconnected to master
L
and
P
M
denotes the number of mastersconnected to slave
M
:
0 L IRUEV
6 M LM L
,...,1
1
==
∑
=
6 M IRU E P
0 L LM M
,...,1
1
==
∑
=
(2)The matrix
2
(
%
) represents portions of capacity a slavemay
“
spend
”
in the piconets it is connected to. The
'
(
%
)matrix represents portions of capacity the masters may
“
dedicate
”
to each of their slaves. The overall capacity of thescatternet is given by the sum of the capacities of all links.The capacity of link (
LM
is the minimum between thecapacity
R
LM
and the capacity
G
LM
.Let us define the
QRUPDOL]HGFDSDFLW\
F
7,
(
%
) and therelevant
0
x
6
matrix
&
7,
(
%
) as
∑∑
==
=
0 L 6 M LM LM 7,
G RF
11
),min()(
%
),min(][)(
LM LM LM 7,
G RF
==
%&
(3)The matrix
&
7,
(
%
) states the normalized capacity of eachlink of the scatternet. As an example, let us consider thescatternet of Fig. 1. The correspondent matrices
2
7,
(
%
),
'
7,
(
%
) and
&
7,
(
%
) are:
2
7,
(
%
)=001/21/250001/26111/20123470108
2
7,
(
%
)=001/21/250001/26111/20123470108
'
7,
(
%
)=001/21/250001/261/31/31/301234701/208
'
7,
(
%
)=001/21/250001/261/31/31/301234701/208
&
7,
(
%
)=001/21/250001/261/31/31/301234701/208
&
7,
(
%
)=001/21/250001/261/31/31/301234701/208
The resulting normalized capacity is
F
7,
(
%
)=3 (
≈
3 Mbit/s).The evaluation above is approximate, mainly for two orderof reasons:1. interference from co-located piconets may lower thecapacity of the scatternet;2. switching overhead, caused by bridging slaves thatchange piconets, may lower the capacity of the scatternet.The interference effect can be taken into account byapplying a multiplicative factor to Eq. (3). This factor isderived from results presented [12] and applied in theevaluation of the numerical results presented in Section V.We assume that a bridge slave switches from a piconet toanother with a mean time period equal to an IP packettransmission time (about 20 BT time slots). This results in aquantifiable value of loss of capacity (
≈
2 slots per switch),which is considered in the numerical results presented inSection V.
%7'PHWULFVVFDWWHUQHWZLWKPD[LPXPUHVLGXDOFDSDFLW\RUPLQLPXPDYHUDJHORDG
We consider two TD metrics: i) the so-called
UHVLGXDOFDSDFLW\
(i.e., the capacity that remains available in ascatternet, after that all pre-defined traffic relationships havebeen satisfied), which has to be maximized; ii)
QRGHV¶DYHUDJHORDG
, to be minimized.The evaluation of the above metrics is traffic dependent,and as such, is obviously tied to and reliant on the adoptedrouting strategy. As an example, given a traffic relationship,for instance a data flow between device
K
and device
N
(with
1 N K
≤≤
,1), the capacity that such flow requires from theoverall scatternet depends on the number of hops that makeup the path between device
K
and device
N
. In our analysis,we assume that a shortest path routing algorithm is applied.To evaluate the metrics, we start by describing the trafficrelationships with a
1
x
1
traffic matrix
7
[
W
KN
], whoseelement
W
KN
represents the capacity, normalized with respectto the 1 Mbit/s capacity of a piconet, required by the (
KN
)relationship (
1 N K
≤≤
,1). We also denote by
5
the numberof traffic relationships expressed by this matrix.It is easy to see that the capacity required on each link bythe traffic relationship between node
K
and node
N
is given by
( )
%3
N K KN
W
,
.The matrix representing the overall normalized capacityrequired on each link of a scatternet
%
, in the TD case
is
given by:
( )
∑∑∑∑
=≠==≠=
⋅==⋅=
1 K 1 K N N 1 K 1 K N N N K LM KN LM N K KN 7'
S W H W
1,11,1,,
)(
% 3 % &
(4)The traffic relationships defined by the matrix
7
can beeffectively supported by the scatternet
%
if the followingconditions, that assure steady state, are verified
0 L H
6 M LM
∈∀≤
∑
=
1
1
, and
6 MH
0 L LM
∈∀≤
∑
=
1
1
(5)Based on the above definitions, we can finally measure thecapacity that remains available in a scatternet, after all trafficrelationships expressed in the matrix
7
are satisfied.Recalling that the capacity of each link is assigned accordingto Eq. (3), the r
HVLGXDOFDSDFLW\
metric,
U
7'
(
%
), is given by:
( )
( )
∑∑
==
−=
0 L 6 M LM LM 7'
HFU
11
%
(6)According to this metric, a scatternet is optimal, when thevalue of
U
7'
(
%
) is maximized.Alternatively, we can adopt as metric the
QRGHV¶DYHUDJHORDG
. To evaluate such a measure, we first calculate the totalcapacity required by the traffic relationships of matrix
7
as:
( )
∑∑
==
=
0 L 6 M KN 7'
H I
11
%
(7)Since a capacity required on a link gives rise to anequivalent load on two nodes, we can calculate the averageload on each node, which we denote as
O
7'
(
%
), as:
( )
1 I O
7' 7'
%%
⋅=
2)(
(8)The minimization of the average load goes in the directionof a minimization of the average energy consumption.
&0HWULFVDVVRFLDWHGWRWKHSDWKOHQJWK
We define two other metrics that do take into account thepath length. Let us denote the length of the path betweendevice
K
and device
N
in a scatternet represented by a matrix
%
(expressed in number of hops) as:
( )
∑∑
==
=
0 L 6 M N K LM N K
SK
11,,
%
(9)The first metric that we introduce is the
DYHUDJHSDWKOHQJWK
that shall be minimized
This metric corresponds tothe path length averaged over all possible relationshipsamong the
1
nodes in the TI case and over all
5
trafficrelationships in the TD case:
( )
∑∑
=≠=
−⋅=
1 K 1 K N N N K 7,
1 1 K K
1,1,
)1()(~
%
or
∑
≠∈
=
K N 7 N K
N K 7'
5 K K
,),(
,
)()(~
% %
(10)By minimizing these metrics we go in the direction of minimizing the end-to-end transfer delay.The second metric is the
DYHUDJHSDWKFDSDFLW\
defined asthe capacity available in each path averaged over all possiblerelationships among the
1
nodes (a TI metric which has to bemaximized). The metric is defined as:
( )( ) ( )
1~)(
−⋅⋅=
1 1 KFD
7, 7, 7,
%%%
(11)
V. N
UMERICAL
R
ESULTS
Numerical results presented in this section are obtained byapplying the centralized methodology described in SectionIII.B (upper part of Fig. 2). Each of the following figuresrepresents the area containing the scatternet; the x and y axesare measured in meters. The figures report nodes, their roles(master, slave or bridge) and radio links interconnectingthem. Piconets are not shown to improve neatness.
$7UDIILFLQGHSHQGHQWPHWULFV
This sub-section shows examples of scatternets resultingfrom the TI optimization. The scenario is constituted by 12devices distributed in an area of 100x100 meters. Thescatternet of Fig. 3a is obtained by selecting the one withmaximum
QRUPDOL]HGFDSDFLW\
. It can be immediatelynoticed that it presents a structure made up by a lineinterconnecting all nodes; i.e., every node is connected withtwo other nodes only. The value assumed by
F
7,
(
%
), evaluatedas in Eq. (3), is 5.5; by taking into account also the switchingoverhead,
F
7,
(
%
) decreases to 5.2727 and by considering alsothe interference effect it becomes 4.7151.Although this scatternet is the one with maximum
F
7,
(
%
), itpresents large values of the average paths length, which couldlead to high transfer delays; moreover, this characteristic,together with the peculiar scatternet structure, limits theeffective capability of supporting traffic. In fact, long pathsrequire more scatternet capacity to support the same trafficrelationship. As an example, a single 500 kbit/s bi-directionalflow between node 10 and node 12 in Fig. 3a would use allthe scatternet capacity: the nodes along the path would spendhalf their time receiving traffic from one of the two directionsand the remaining relaying the traffic in the oppositedirection.The peculiar structure produced by this metric is due to thefollowing reasons:1. the metric tends to favor scatternets formed by a largenumber of piconets, since each new piconet added to thescatternet increases the overall capacity;2. the interference effect is not significant since the numberof co-located piconets is low;3. when the switching overhead effect is taken into account,a bridge loses capacity as a function of the number of piconets it is connected to; thus, high performance, interms of capacity, are attained when a bridge node isconnected to only two piconets.These considerations explain why path lengths have to betaken into account when forming a scatternet. However,minimizing the path lengths without considering the capacity,could lead to undesirable scatternets too since if the nodes aredistributed in a small area, the resulting scatternet presents afully meshed topology where every slave is connected toevery master. In this case the resulting capacity is low as aconsequence of the high number of bridges connected to ahigh number of piconets.Let us now look at Fig. 3b, which shows a scatternetmaximizing the
DYHUDJHSDWKFDSDFLW\
. This metric seems to

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