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IEEE INTERNET OF THINGS JOURNAL 1
Beacon Deployment for Unambiguous Positioning
Wei He, Pin-Han Ho, Senior Member, IEEE, and J´anos Tapolcai, Member, IEEE
Abstract—Instant and precise localization of a mobile user
is fundamental for supporting various sophisticated indoor
location-aware services. This study focuses on achieving unam-
biguous user positioning using practical Bluetooth Low Energy
(BLE) beacons with multiple discrete power levels. By receiving
the beacon coverage status from a user’s device, the cloud
server can unambiguously pinpoint the user’s location and
react correspondingly. We firstly define the problem of Beacon
Deployment for Positioning (BDP) and provide several theoretic
bounds on the number of required beacons to gain sufficient
understanding on its performance behaviour. The BDP problem
is further formulated into an integer linear program (ILP) and
solved in extensive case studies. We claim that this is the first
systematic and in-depth research on beacon deployment for
unambiguous user positioning. Our analysis and experiments
show that the proposed solution takes O(√N)to O(N
2)beacons
for NTest Positions, which is 2-8 times less beacons compared
to that by the naive approach, while the analytical bounds are
tight with the ILP results with 20% of gap.
Index Terms—Indoor Positioning, BLE beacons, Localization
I. INTRO DUC TI ON
INSTANT acquisition of precise position of each mobile
user has long been considered essential for supporting
various sophisticated indoor location-aware services. Since the
user position information may not be available via Global
Positioning System (GPS) due to the indoor shadowing effect,
many alternatives based on various technologies have been
reported, including radio (WiFi, ZigBee [1], [2], Bluetooth [3],
[4], Bluetooth Low Energy [5]–[11], FM [12], and RFID [13]),
optical (infrared, laser scanning systems [14] and camera-
based [15]), audio (sound [15], ultrasound [16]), magnetic
fields [17], barometer [18] and Inertial Navigation Systems
(INS) [19].
According to [20], these indoor positioning techniques can
be generally classified as anchor-based and non-anchor based
solutions. With the anchor-based solutions, anchor nodes are
installed in prior and periodically broadcast their SSIDs,
while the mobile nodes can determine their locations with
the received SSIDs. If incorporating with advanced position-
ing algorithms (e.g. centroid [21], triangulation [22], and/or
trilateration [23]), anchor-based solutions can achieve better
positioning granularity through manipulating various ranging
W. He is with the School of Electrical & Computer Engineering, Nantong
University, Jiangsu, China, and Zhongtian Broadband Co. Ltd., Jiangsu, China,
e-mail: heweist@gmail.com
P.-H Ho is with the Department of Electrical and Computer Engineering,
University of Waterloo, Canada, e-mail: p4ho@uwaterloo.ca.
J. Tapolcai is with MTA-BME Future Internet Research Group, Bu-
dapest University of Technology and Economics, Hungary, email: tapol-
cai@tmit.bme.hu
Manuscript received Jul. 15, 2016, accepted May 23,2017
Copyright (c) 2017 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
information: received signal strength information (RSSI), time-
of-arrival (ToA), time-difference-of-arrival (TDoA), and the
angle-of-arrival (AoA). As an example of the anchor-based
solutions, the cell-based method [11] localizes a user merely
based on the visibility of the anchor nodes. For another, Nokia
initialized the HAIP (High Accuracy Indoor Positioning) [24]
project, where directional Bluetooth Low Energy (BLE) bea-
cons can achieve 0.3mposition accuracy for indoor users, at
the expense of extra equipment and accurate installation in the
training phase.
On the other hand, non-anchor-based solutions infer a user’s
location from reading the sensors on the mobile devices.
Odometry-based and fingerprinting is the most well known
non-anchor based solutions. Odometry-based solutions use
motion sensors and a map to determine the position change
relative to the initial location. For fingerprinting, the fingerprint
(readings from sensors) of interested locations will be created
in the training phase. Subsequently, user location is identified
by comparing current fingerprint with the trained database.
An example of non-anchor-based solutions is the indoor
positioning system (IPS) by Google [25]. It is characterized
by a traditional site survey process as an offline training
phase in order to build up a radio fingerprint database. The
radio fingerprints can be by way of the sensed signal strength
from multiple local-area or near-field access points, such as
Wi-Fi, Bluetooth, and RFID. The IPS can then answer a
user’s location query through matching the measured radio
fingerprints with those in the fingerprint database.
This study investigates unambiguous user positioning via
the collaboration of a set of BLE beacons in the area of interest
(AOI). Similar to cell-based method, our solution does not
require a precise signal measurement and phase detection, but
instead the mobile device only needs to determine whether
it is in the radiation range of each beacon or not. In other
words, the server localizes each mobile device simply by
the SSIDs of the beacons successfully decoded and obtained
at the mobile device, thus applicable to any type of BLE
beacons and related protocols/SDKs, such as Apple’s iBeacon,
EddyStone beacon, Paypal Beacons, and Indoor.rs [26], etc.
Different from other literature, our solution firstly supports
multi-power-level BLE beacons and dramatically reduces the
number of beacons needed by categorizing TPs into Shared
Information Test Position Groups (SIPGs). We envision such
an approach will be a norm in the future metropolitan and
indoor positioning systems where low-cost sensors/beacons
are massively deployed and analyzed via powerful cloud
computing facilities.
We claim that the paper serves as the first study that
formulates the beacon deployment task for unambiguous user
positioning into a mathematical problem, namely Beacon
Deployment for Positioning (BDP), with its goal as to ensure
IEEE INTERNET OF THINGS JOURNAL 2
any two Test Positions (TPs) in different SIPGs being covered
by distinct sets of beacons. Distinguished from any reported
research, the proposed system model and formulated Integer
Linear Program (ILP) are general to practical scenarios by
considering the fact that the effective radiation range of a
BLE beacon could take any irregular shape. Based on the
formulated problem, the paper provides extensive and in-depth
analysis on the number of required beacons via geometry and
coding theories.
The rest of the paper is organized as follows. We will firstly
review the prior research of BLE based positioning systems
in Section II. Section III presents the system model and the
proposed beacon allocation problem. Section IV analyzes the
proposed problem, and Section V presents the corresponding
integer linear program (ILP), which is solved and the results
are presented in Section VI. The paper is concluded in Section
VII.
II. RELATED WORK
[11] describes a cell-based positioning method using Blue-
tooth beacons capable of modelling complex reception char-
acteristics and provides a greedy heuristic. Although the paper
defined the problem of mono-power-level beacon deployment,
it lacks any in-depth analysis and implementation. [27] lever-
ages the capture effect by assuming only the strongest signal
can be detected in a collision, in order to increase the lifetime
of mobile nodes by three orders of magnitude and be more
resilient to external interference. The position is determined
merely by the coordinates of the BLE beacons without RSSI
information. A platform using BLE beacons and ultrasound
is introduced [16] by taking advantage of the difference in
propagation characteristics between ultrasound and radio. With
a prior calibration process that automatically computes the
room geometry and the precise beacon locations, the system
can track a user’s location.
In [10], a smartphone-based indoor navigation system with
BLE beacons and visual-tags is introduced. A rough local-
ization is achieved by BLE beacons, followed by an accu-
rate navigation via decoding the visual-tags with cameras.
A Bluetooth-based IPS built upon low-cost off-the-shelf bea-
cons and mobile devices is introduced in [4], which enables
rapid prototyping of room-level applications. [7] pinpoints
the user location by a neural network fed with data from
beacons, achieving comparable accuracy to other approaches.
[8] achieves a tracking accuracy of of 0.27 meters with
iBeacon using particle filtering method.
[28] introduces a crowd-sourcing localization system based
on Wi-Fi scene analysis and Bluetooth beacons. The mobile
nodes submit their Wi-Fi fingerprints both to a map server and
the Bluetooth beacons, which further disseminate the collected
fingerprints to all the nearby mobile nodes for instantly room-
level positioning. [17] introduces a method for automatically
generating and updating the RF signal map in buildings while
localizing the mobile device. The trajectory of a smartphone is
obtained by synthesizing the sensor measurement (inertial and
RF signal) using an adaptation of the GraphSLAM technique.
The pedestrian dead reckoning estimates avoid the users to
lev pwr rssi
(dBm)(dBm)
0−30 −91
1−20 −81
2−16 −76
3−12 −74
4−8−68
5−4−66
60−62
74−60
(a) Parameters for
each power level
(b) The radiation pattern
Fig. 1. Power-levels of a beacon [34]
hold the smartphone in hand and can accommodate multiple
users.
[21] offers a low-lost, easy-deploying and reconfigurable
positioning system. User location is determined by computing
the weighted average of RSSI’s and choosing the closest
beacon. Depending on the deployment configuration, an ac-
curacy down to 0.97 meters is possible. [29] proposes two
schemes for indoor positioning by Bluetooth beacons and a
pedestrian dead reckoning (PDR) technique to provide meter-
level positioning. A multi-threshold step detection algorithm
is adopted to improve the positioning accuracy using PDR.
In addition, a heading estimation method with real-time com-
pensation is proposed, built upon a Kalman filter with map
geometry information. Moreover, this paper has implemented
two positioning approaches. In summary, Table I compares the
features of our proposed work with current literature.
III. PROB LE M FO RM ULATIO N
A 2D Positioning System is enabled in the Area of Interest
(AOI), where the Test Position (TP) at which the user can be
precisely localized is marked by “+”; and a possible position
for installing a beacon, namely Candidate Position (CP), is
marked by “×”. In practice, multiple beacons of different
power levels can be placed at a common location in the AOI.
According to the application, multiple TPs could fall into
a common Shared Information Test Position Group (SIPG) if
the same location aware information is desired at those TPs.
Next we introduce the Signal Coverage Model for beacon
deployment, which is followed by the formal definition of the
BDP problem and the solution representation.
A. Signal Coverage Model
In the proposed user positioning system the location is
identified only according to the beacons seen in the radiation
range. Without loss of generality, we assume the beacons can
be of any type and has Vdiscrete power levels1. As a part
of the input we define, whether a TP can be covered by a
beacon at CP according to 1) Radiation Pattern Specification,
1To the best of our knowledge, all current commercial BLE beacons has a
limited set of discrete power levels [26], [34].
IEEE INTERNET OF THINGS JOURNAL 3
TABLE I
OUR W OR K VS. LI T ER ATU RE
anchor-based bcn# power-level differentiate granuality online localizing complexity
our work T low multi SIPGs coarse low
cell-based [4], [11],collocal [20] T moderate mono TPs coarse moderate
centroid [21], [30] T high mono TPs coarse moderate
trilateration [23],triangulation [22] T high mono TPs fine high
beacons + extra medium [10], [16] T moderate mono TPs fine moderate
odometry [8], [17], [29], [31] F N/A N/A TPs fine high
fingerprinting [7], [25], [28], [32], [33] F N/A N/A TPs fine moderate
2) Measurements or 3) Log Loss Radio Propagation Model. It
is stored in Qk
ij which is a binary indicator defined as follows
Qk
ij =
1,if TP iis covered by a beacon at CP j
with power-level k
0,otherwise.
See also Fig. 2c as an example for the radiation patterns of
a beacon at a CP for each power level. Ideally, the Radiation
Pattern Specification is provided by the beacon’s manufacturer,
usually obtained through experiments conducted in typical
application environments such as offices, stores etc. Note that
the radiation pattern can be different for each CP due to
shadowing by any possible barrier in the radiation range of
the beacon. To get more precise radiation patterns, case-by-
case measurements should be performed.
Alternatively, the well-known Log Loss Radio Propagation
model [35] can be adopted, where the beacon’s radiation
pattern will be a circle, as shown in (1):
P L(d) = P L(d0) + 10αlog10
d
d0
(1)
where P L(d)is the power loss in dB at dmeters away,
P L(d0)is the reference power loss at d0meters, and αis
the attenuation factor.
For BLE beacons, RSSI values are provided by the manu-
facturers [26], [34] for each power-level in related standards or
data sheets, which represents the receiver side signal strength
in dBm at d0= 1 meter away from the beacon. By definition,
rssik=pk
t−P Lk(1), where rssik,pk
tand P Lk(1) are the
RSSI, the transmission power of the beacon, and the first meter
power loss at power-level k∈ {0,1, . . . , V −1}respectively.
If the receiver sensitivity is θand the maximum distance a
beacon at power-level kcan effectively cover is dk, then the
maximum acceptable power loss from the beacon P Lk(d) =
pk
t−θ, thus P Lk(d)−P Lk(1) = pk
t−θ−P Lk(1) = rssik−θ.
From (1), dk= 10 rssik−θ
10αmeters.
Thus Qk
ij simply tests whether Dij ≤dkwhere Dij denotes
the distance between CP iand TP j. Fig. 1b illustrates the
circular radiation patterns when α= 3 for the power-levels
defined in Fig. 1a.
B. The Problem Definition
In this study, the problem of Beacon Deployment for
Positioning (BDP) is defined with its goal as minimizing the
number of beacons deployed at the CPs such that two TPs in
different SIPGs can be differentiated through beacon coverage
while the positioning delay and the power consumption of each
beacon is constrained.
The input to the BDP problem includes the set of TPs
(denoted by T), the set of SIPGs (denoted by G), the set of CPs
(denoted by C), the number of discrete power levels (denoted
by V), and the beacon ranges for each CP (denoted by Qk
ij ). As
described previously, the beacon ranges are computed based on
the attenuation factor (denoted as α), the receiver sensitivity of
user devices (denoted as θ), and the beacon specific parameters
such as power-level settings. We assume there are at least two
TPs and one CP, formally |T| ≥ 2and |C| ≥ 1.
To limit the power consumption of each beacon the input
also includes the maximum acceptable expected power for
each beacon, a time slot constant for broadcasting interval,
the transmission power for each power-level, and the range of
broadcasting interval.
The output of the problem are the positions of the required
beacons along with their power levels and broadcasting inter-
vals, which exclusively determine the set of beacons “seen”
at each TP. The solution is represented by an Area Code
Table (ACT). Each row in the ACT stores an Area Code
(AC) corresponding to a unique set of beacon ids “visible”
at the TPs listed in that row; each column is a Beacon Code
which indicates the SIPGs covered by the beacon of the
column. Specifically, a binary bit in an AC is set to 1if the
corresponding beacon covers all TPs listed in that row while
it is set to 0, otherwise. The last two rows of the ACT are
used to store the power-level and broadcasting interval settings
for each deployed beacon while the first column stands for
the SIPG that each AC belongs to. Since all SIPGs must be
differentiated, no ACs should be shared among them.
A special case of the BDP problem is the Beacon Deploy-
ment for Unambiguous Positioning (BDUP) problem where
each SIPG has a single TP. Thus the solution of BDUP can
0
1
2
3
4
0 1 2 3 4
A B
C
C
E
D
F
(a) TPs/CPs settings
SIPG TPs
A (1,4)(2,4)(0,3)
(2,3)(0,2)(1,2)
B (3,4)(4,4)
(4,3)(4,2)
C (0,0)(0,1)
(1,0)(2,1)
D (1,1)
E (4,1)
F (2,0)(3,0)
(b) SIPGs to identify (c) Radiation range of a bea-
con for each power level
Fig. 2. A BDP problem
IEEE INTERNET OF THINGS JOURNAL 4
(a) The BDP solution
SIPG CPs with beacons TPs
(0,0) (3,0) (2,3) (4,3)
A 0 0 1 0 (0,2)(1,2)(0,3)
(2,3)(1,4)(2,4)
B 0 0 0 1 (4,2)(4,3)
B 0 0 1 1 (3,4)(4,4)
C 1 0 0 0 (0,0)(1,0)(0,1)
C 0 1 1 0 (2,1)
D 1 0 1 0 (1,1)
E 0 1 0 1 (4,1)
F 0 1 0 0 (2,0)(3,0)
pwr(dBm) -4 -4 4 0
interv(sec) 0.1 0.1 0.51 0.2
(b) The Area Code Table (ACT)
(c) The naive solution (d) The BDUP solution
Fig. 3. Solutions to the problem defined by Fig. 2
serve as an upper bound to the corresponding BDP.
C. Illustrative Example
Fig. 2 exemplifies the proposed BDP problem where the
CPs/TPs are shown in Fig. 2a and the AOI of 30 ·30m2is
divided into sub-areas based on the SIPGs defined in Fig. 2b.
Note that an SIPG may be the union of multiple physically
non-adjacent sub-areas. As illustrated by Fig. 2c, a beacon at
CP (0,4) has 8randomly generated polygon radiation patterns
according to specific power levels.
By solving the ILP in Section IV, an optimal solution to
the problem defined in Fig. 2 is demonstrated in Fig. 3a and
significantly outperforms the O(|T|)naive solution, commonly
adopted by iBeacon etc., where each TP is equipped with a
beacon (See Fig. 3c). Each deployed beacon is marked by
“△” while its radiation pattern is shown by the corresponding
shaded area. Fig. 3b presents the resultant ACT. In contrast,
Fig. 3d shows a corresponding BDUP solution when all
positions are TPs/CPs.
The generated ACT is kept at the server and the SIPG is
identified when a set of beacon IDs (i.e. AC) is submitted by
a mobile device. The server instantly identifies the SIPG with
which the mobile device is associated by a table look-up.
IV. PROB L EM FE A SI BIL IT Y
Next we give bounds on the number of beacons without
considering the positioning delay and the power consumption.
First we give simple general lower bounds. It is followed by a
general lower bound based on the geometry of the beacon
range. Next we formulate lower bounds depending on the
maximal number of TPs a beacon can cover. Finally, we give
several upper bounds on the number of beacons.
Definition 1: A solution is feasible to the BDP problem if
and only if
1) for any two TPs in different SIPGs, there exists a beacon
covering exactly one of them;
(a) Line topology of odd length (b) Line topology of even length
(c) 2 by ntopology, nis odd (d) 2 by ntopology, nis even
(e) mby ntopology
Fig. 4. BDUP solutions: simple beacon patterns
2) all TPs are covered.
Lemma 1: Given C⊇Tand let no more than one TP be
covered by a beacon with the minimum power level. There’s
a trivial solution for BDUP by installing a beacon at each CP
∈C.
Proof: Directly follows from Definition 1 since each TP
will be assigned a unique area code of weight 1.
Lemma 2: The minimum beacon number for BDUP is
⌈|T|+1
2⌉in a line topology of |T|TPs (i.e. the TPs are in
a row as shown on Fig. 4a). This bound is tight.
Proof: 1) At least ⌈|T|+1
2⌉are needed. Consider a line
with |T|TPs, |T|+ 1 boundaries are needed to separate
these TPs. Note that adding a beacon at most contributes 2
boundaries, hence at least ⌈|T|+1
2⌉beacons are required to fully
differentiate the TPs.
2) there exists a solution with ⌈|T|+1
2⌉beacons. A 3-hop
chaining pattern can be applied. When |T|is odd, a perfect
chain is formed (see Fig. 4a); when |T|is even, an additional
cycle is added near the end (see Fig. 4b). Note that each TP
is uniquely covered by 1−3beacons.
A. Bounds based on beacon coverage geometry
Theorem 1: The number of beacons for BDUP is at least
s2|T|
µ+1
4−2
µ+1
2
where µis the maximum number of times the boundary of
two beacons can intersect.
Proof: By viewing the AOI as a plane and the beacons
as geometric shapes, we will show that bshapes can divide
the plane into at most 1 + µb(b−1)
2regions, where µis the
maximum number of times the boundary of two shapes can
intersect.
The proof is inductive. Let r(b)denote the number of
regions into which the plane can be divided by bshapes
excluding the outside region, where a TP would have an all
zero area code. A single shape can divide the plane into an
IEEE INTERNET OF THINGS JOURNAL 5
inside and outside regions, i.e. r(1) = 1. By adding the b+ 1-
th shape, the number of regions of the plane increases by
the number of existing regions that the new shape intersects.
The new shape has at most µintersections with each of the
previous shapes, thus in total it has bµ intersections, each of
which divides one of the original regions into two. Hence
r(b+1) ≤r(b)+bµ. An explicit formula for r(b)is as follows:
r(b)≤r(b−1) + (b−1)µ≤
r(b−2) + (b−2)µ+ (b−1)µ≤ ··· ≤
r(k) + (k+...b−1)µ, ∀k∈[1, b −1) (2)
Therefore, r(b)≤r(1) + (1 + ···+b−1)µ≤1 + b(b−1)
2µ,
the minimum number of required beacons b, must satisfy:
1 + b(b−1)
2µ= 1 + µ
2(b−1
2)2−µ
8≥ |T|
⇒b≥s2|T|
µ+1
4−2
µ+1
2(3)
Corollary 1: With every beacon being in a shape of cycle,
the number of beacons for BDUP is at least q|T| − 3
4+1
2.
Proof: Two cycles can intersect at most two points, thus
µ= 2 in Theorem 1.
Corollary 2: With every beacon being in a shape of ellipse,
the number of beacons for BDUP is at least q2|T| − 1
4+1
2.
Proof: Two elips can intersect at most four points, thus
µ= 4 in Theorem 1.
Corollary 3: If every beacon has a shape of polygon with
at most xcorners, the number of beacons needed for BDUP
is at least q2|T|
x2+1
4−2
x2+1
2.
Proof: Two lines can intersect in at most one point. Thus
two polygons with at most xlines can intersect at most x2
points, thus µ=x2in Theorem 1.
B. Bounds based on the maximum number of TPs a beacon
can cover
The problem of identifying items via tests have been deeply
studied as separating systems. The version where the tests have
restrictions on their size was first raised by R ´enyi in 1961; and
the first and most important results are due to Katona (1966).
Theorem 2 ( [36]): Let Φdenote the maximum number of
TPs each beacon covers. The number of beacons for BDUP is
at least bsatisfying the following inequalities. Find the least
number b, for which there exist natural numbers j≤b−1
and a < b
j+1, such that
j
X
i=0
i·b
i+a(j+ 1) ≤Φb, (4)
j
X
i=0 b
i+a=|T|+ 1.(5)
There is no closed formula known for the above bound, but
⌈log2(|T|+1)⌉is the information theoretical lower bound and
it is known that
Theorem 3 ( [36] ): Let Φdenotes the maximum number
of TPs a beacon can cover. The number of beacons for BDUP
is at least |T|+ 1
Φ·ln(|T|+ 1)
1 + ln |T|+1
Φ
C. Constructions for beacons with circular pattern
Finally, upper bounds on BDUP for beacons with circular
radiation patterns are introduced. We assume the distance
between two adjacent TPs in the grid AOI is 1 unit, and let r
denote the maximum radius of the beacons.
Lemma 3: Consider a 2by ngrid AOI. The upper bound
on the beacon number for BDUP is nfor r= 1.
Proof: Fig. 4c and 4d demonstrate the construction pat-
terns for odd and even n. Note that each TP is uniquely
covered by 1,2or 4beacons.
Corollary 4: For a mby ngrid AOI (WLOG, m≥n),
BDUP can be achieved by ⌊m
2⌋n+ (mmod 2)⌈n
2⌉beacons
for r= 1.
Proof: A solution to the mby ntopology can be
constructed by keeping on applying the 2rows patterns (See
Fig. 4c and 4d). If there’s one row remaining, additionally ⌈n
2⌉
beacons are needed.
Fig. 4e illustrates the construction when m= 4 and
n= 7. Note that for any row (e.g. row 1), the beacons on
it and its adjacent rows (i.e. rows 0−2) separate it from the
remaining rows. Plus, by Theorem 3, the TPs in current row
are differentiated from those on its adjacent rows. Thus, any
row is uniquely coded and all TPs can be identified.
Theorem 4: For a mby ngrid AOI where m, n ∈[3,10] and
⌈m
2⌉+1 ≤n, no more than 8R+4 beacons of radius r=√2R
are needed to achieve BDUP, where R=⌈m
2⌉−(⌈m
2⌉mod 2)
2.
Proof: As shown in Fig. 5a, when m≤10,R≤2and
any beacon with radius √2Rcovers (2R+ 1)2TPs totally
as its inner square does. Thus, such a beacon can be treated
as a square which partially separates the rows and columns
simultaneously. Due to our choice of R, any beacon can cover
at least half of the TPs per row/column, thus with two such
beacons any row/column can be covered.
Based on the above observations, the pattern shown in
Fig. 5b-5c can be applied to differentiate the TPs. Basically,
the top and bottom array of beacons are used to cut the
columns while the left and right array of beacons are used
to cut the rows using the pattern introduced in Lemma 4 of
the Appendix.
Due to choice of R,4R+ 1 = 2(⌈m
2⌉−(⌈m
2⌉mod 2)) +
1≥2(⌈m
2⌉ − 1) + 1 ≥m−1, thereby at most 2R+ 2
beacons are needed for differentiating a row/column. Since
2R+ 2 = ⌈m
2⌉−(⌈m
2⌉mod 2) + 2 ≤m+1
2+ 2, when m≥3
no more than mbeacons are needed (if m= 3,4, exactly 3and
4beacons are needed while m≥5implies 2R+2 ≤m), thus
we can always find a feasible solution. Since 4beacons are
shared as shown in Fig. 5b and 5c, no more than 4(2R+2)−4
beacons are needed by Lemma 4.
Note that the above theorem 3-optimal when m=n. By
Corollary 1, at least m+ 1 beacons are required and we have
IEEE INTERNET OF THINGS JOURNAL 6
(a) TPs covered by a beacon
of radius r= 2√2
(b) 5 by 5 solution
(c) 10 by 6 solution (d) Optimal 10 by 10 solution
(e) Solution if m, n are multiples of 10.
Fig. 5. BDUP solution: mby nconstructions
4(⌈m
2⌉−(⌈m
2⌉mod 2)+1)
m+1 ≤4( m+1
2+1)
m+1 = 2 + 4
m+1 . When m=
3, the worst ratio is 3, thus the solution is 3-optimal.
Corollary 5: For a mby nAOI (m, n are multiples of 10),
BDUP is achievable with 1
5mn beacons of radius 2√2.
Proof: Note that each square area with 10 by 10 TPs can
be differentiated by 20 beacons of radius 2√2by Theorem 4,
as illustrated in Fig. 5d. Since each square area will not cover
any other TPs outside its boundary, the TPs in each square are
independently coded and can be pieced together, thus proved
the BDUP by using 1
5mn beacons. Fig. 5e shows the case
when m=n= 20.
Note that the above idea can be generalized when mand
nare not multiples of 10. For example, for a 12 by 12 AOI,
BDUP can be achieved with 42 beacons by placing a solution
of 10 by 10 as on Fig. 5d with 20 beacons on the top right
corner, and a solution of 12 by 2 on the left side from Lemma
3 (Fig. 4c) with 12 beacon and a solution 2 by 10 on the
bottom right corner with 10 beacons. Similarly, for a 15 by
15 AOI, BDUP can be achieved with 60 beacons by placing
a solution 10 by 10 with 20 beacons on the top right corner
surrounded by five 5 by 5 solutions of Fig. 5b each with 8
beacons.
V. ILP F ORM UL ATI ON FO R BDP
In this section, common notations used by the proposed
ILP are listed. Without confusion, all greek or upper-case
notations are given or precomputed constants, mathbold sym-
bols represent given sets, while the lower-case alphabetic
notations are the decision variables. First we introduce the
basic formulation, which ignores positioning delay and power
consumption.
A. Basic ILP formulation
Input Parameters:
T/C/G: set of TPs/CPs/SIPGs (note that each CP can be
placed with a single beacon, while a location can be mapped
by multiple CPs).
V: number of discrete power levels.
Qk
ij : indicates whether a beacon at CP iat power-level k
can cover TP j.
Ci: set of CPs can cover TP iwith maximum power-level.
Bmin: a lower bound on the number of beacons.
Decision Variables:
vij ∈ {0,1}: If it is 1install a beacon at CP iwith power
level j;0otherwise.
qij ∈ {0,1}: It takes 1when a beacon at CP icovers TP
j; it takes 0otherwise.
xk
ij ∈ {0,1}: Used to help compute the XOR value of two
binary variables, formally xk
ij =qki ·qkj .
The BDP problem can be formulated as follows:
(BDP) min X
i∈C
V−1
X
j=0
vij ,(6)
Subject to:
V−1
X
j=0
vij ≤1,∀i∈C(7)
X
i∈C
V−1
X
j=0
vij ≥Bmin (8)
qij =
V−1
X
k=0
Qk
ij vik,∀i∈C(9)
X
j∈Ci
qji ≥1,∀i∈T(10)
X
k∈Ci\Cj
qki +X
k∈Cj\Ci
qkj +X
k∈Ci∩Cj
(qki +qkj −2xk
ij )≥1,
∀i, j ∈T, /∈same SIPG
(11)
IEEE INTERNET OF THINGS JOURNAL 7
Fig. 6. Area Code Demo
2xk
ij ≤qki +qkj ≤xk
ij + 1,
∀i, j ∈T, /∈same SIPG, k ∈Ci∩Cj
(12)
The objective function (6) minimized the total number of
required beacons, where PV−1
j=0 vij is one if a beacon is placed
at CP i.
The constraint (7) illustrates: if a beacon is installed at CP i,
only one power-level can be selected2(i.e. only one vij = 1);
otherwise, vij = 0,∀j={0,1,...,V −1}. The lower bounds
of Thm. 1, 2 and 3 is added as a constraint (8) specifying that
at least Bmin beacons are in the feasible solution.
As shown in (9), to decide whether a beacon at CP ican
cover TP j, we need to compute qij based on the precomputed
values Qk
ij ’s at each power-level. When the power radiation
patterns are circular, only a portion of the TPs need to be
determined for CP idue to symmetry, which can speed up the
ILP computation. The constraint (10) ensures each TP being
covered by some beacon(s).
Any two TPs in different SIPGs should be covered by
different sets of beacons, which is achieved by the constraints
(11) and (12). The sufficient conditions of differentiating two
TPs iand jare as follows:
1) either some beacon(s) are installed at Ci\Cj, such that
there is a beacon only covers i;
2) or installed at Cj\Ci, such that there is a beacon only
covers j;
3) or installed at Ci∩Cjwithout covering both TPs.
This is also demonstrated in Fig. 6. The three conditions are
handled by the three terms on the LHS of (11), respectively.
For condition 3), a beacon installed at Ci∩Cjmay or may
not cover both TPs depending on the power-level selected. To
test whether the beacon covers both TPs, we need to compute
Pk∈Ci∩Cjqki ⊕qkj , which is translated into the third term in
(11) together with (12).
B. Considering the constraint on transmission power and
adding the positioning delay as an objective
In this subsection we extend the objective function (6) with a
target of shortening the positioning delay and add a constraint
2Alternatively, we can define the set {vij ,∀j}as a Special Ordered Set of
Type 1(SoS1) for each TP iin CPLEX. Specifically, the SoS1 weight for
vij is set to rssij−rssi0where rssijrefers to the RSSI value for the j-th
power level as shown in Fig. 1a.
on maximal transmission power. First we introduce some new
input parameters:
Pmax: maximum allowed expected power (mw) of a beacon.
τ: time slot constant for broadcasting interval (= 625µs).
ωi: transmission power3(mw) for power-level i.
Smin(max): the broadcasting interval of a beacon.
Besides we introduce new working variables:
si∈ {0,1,...Smax,}: The broadcasting interval for a
beacon at CP iin time slots. If no beacon is at i, then si= 0.
ai∈ {0,1,...,Smax}: The maximum delay at TP i.ai=
maxj∈Ci{sj}. i.e. the maximum broadcasting interval in the
surrounding area.
pi∈R+
0: The transmission power at CP i.
The new objective function would be:
(BDP) min X
i∈C
V−1
X
j=0
vij ,+1
Smax|T|+ 1 X
i∈T
ai(13)
Note that the second term in (6) is less than one such that
it will not affect the beacon number required in the solution.
The constraints are specified in Eq. (7)-(12) along with
pi=
V−1
X
j=0
ωjvij ,∀i∈C(14)
pi≤Pmaxτ si,∀i∈C(15)
Smin
V−1
X
j=0
vij ≤si,∀i∈C(16)
Smax
V−1
X
j=0
vij ≥si,∀i∈C(17)
sj≤ai,∀i∈T, j ∈Ci(18)
The constraint (14) computes the transmission power at CP
irespectively. The expected power consumption at any beacon
should not exceed the given threshold, which is stated in (15).
If Pexp or Smax decreases, possibly less power-levels are valid
for the given problem.
The constraint (16) and (17) specifies the valid broadcasting
interval range for a beacon. The constraint (18) computes
the estimated positioning delay at TP i. Note that ai=
max{sj}, j ∈Ci, which is formulated using the big-M
method.
VI. CA SE STU DIE S
In this section, case studies are conducted to validate the
proposed BDUP problem and the ILP. To simulate practical
BDUP applications, various scenarios with different attenua-
tion factors and TPs/CPs settings should be considered. For all
cases, the total area of the AOI is 60 ·60m2where TPs/CPs
are evenly distributed in it. The power-level settings of the
beacons are listed in Fig. 1a, which simulates the commercial
3Note that ωi= 10tpwr[i]/10 where tpwr[i]is the given transmission
power in dBm for power-level i.
IEEE INTERNET OF THINGS JOURNAL 8
TABLE II
SIM UL ATI ON D ETA IL S:60 BY 60 ME TE R S,θ=−97DBM,M A XI MU M D EL AY = 2secs,¯pmax = 5mw. TH E SAVE R E PR ES E NT S THE N UM B ER O F
BE ACO N S CO MPA RE D TO THE NAIV E S OL U TI ON ,CO M MO NLY A DO PT E D BY IBEAC O N ET C.
Input lower bound on bcn# upper bound ILP
α|T|ΦrThm.1 Thm.2 Thm.3 Cor 4 Cor 5 bcn# save gap% time(sec) rows, cols, nzrs
3 20*20 101 7.1 21 12 11 200 80 50 8x 40.41 221568 2·105,1·105,7·106
3 15*15 61 4.9 16 11 9 113 (60) 35 6.4x 31.40 31243.5 1·105,4·104,2·106
3 12*12 37 3.7 13 10 9 72 (42) 27 5.3x 23.57 24138 4·104,1·104,4·105
3 10*10 25 2.9 11 10 8 50 20 20 5x 0 48.93 1·104,6·103,1·105
4 20*20 21 2.3 21 37 29 200 89 4.5x 29.91 22198.5 7·104,3·104,6·105
4 15*15 13 2.1 16 32 25 113 51 4.4x 24.00 2723 3·104,1·104,2·105
4 12*12 9 1.7 13 29 22 72 40 3.6x 11.00 11336 9·103,5·103,5·104
4 10*10 5 1.1 11 33 24 50 39 2.6x 15.38 2115 5·103,3·103,2·104
5 20*20 9 1.7 21 80 56 200 116 3.4x 25.35 6252 3·104,1·104,1·105
5 15*15 5 1.1 16 75 51 113 89 2.5x 15.73 658.5 1·104,7·103,4·104
5 12*12 5 1.1 13 48 34 72 56 2.6x 14.29 202.5 5·103,3·103,2·104
estimote BLE beacons. The acceptable positioning delay is
from 0.1to 2secs, the receiver sensitivity of any user’s device
is set to -97dBm to simulate an ordinary mobile phone, and
the maximum acceptable expected power-consumption for a
beacon is 5mw.
In our experiments, attenuation factors from 3to 5are
chosen to model common indoor environment while sparse to
dense TPs/CPs settings are examined to study the performance
and scalability of our proposed work under various positioning
granularity and problem size. One application of the presented
case studies is the parking-spot identification, for which each
parking spot in the AOI is regarded as a TP(or SIPG) to be
identified. By adjusting the attenuation factor and the TP/CP
settings, various indoor environment (e.g. with or without
barriers), problem scales (e.g. number of spots) and positioning
accuracy (e.g. number of TPs per m2) can be simulated.
Table II summarizes the results we obtained, where the
Φcolumn represents the maximum number of TPs can be
covered by a beacon and the rcolumn denotes the correspond-
ing maximum radius of the beacon. It also shows the upper
and lower bounds introduced in Section IV and the details of
the ILP. In our experiments the solution by ILP requires 2-8
times less beacons compared to the commonly adopted naive
approach, like iBeacon, etc.
Fig. 7 examines the case that the radiation pattern is
circular, which is applicable when the beacons are installed
at the ceilings of an indoor environment with few obstacles.
Table II summaries the results in Fig. 7 along with the scenario
settings, where the LB column lists the analytical lower bounds
obtained from Theorems 1, 2 and 3 while the last column
indicates the size of the ILP problem. Each case is solved on
CPLEX 12.6with “mip cuts all” settings by using a server
with 16G RAM and a 4core 3.6GHz Intel CPU.
Fig. 7i plots the performance gap between the ILP under
various CP/TP and attenuation factor settings based on Ta-
ble II. The x-axis represents the number of TPs while the
y-axis represents the corresponding optimal solution for the
number of beacons required. As illustrated in Fig. 7i, the
number of beacons grows almost linearly with respect to the
number of TPs.
Fig. 7 demonstrates the optimal solutions under various
scenarios where each beacon is drawn as a circle with its center
marked by “△” and its radius corresponds to the farthest grid
distance it can cover. As shown in Fig. 7, each TP is covered
by a unique set of beacons.
VII. CO NC LUS IO N S
In this paper, the problem of beacon deployment for un-
ambiguous user positioning (BDP) was investigated. Based
on the problem definition and feasibility, we theoretically
proved a series of performance bounds on the number of
required beacons, and formulated a novel ILP that jointly
determines the beacon positions along with their power levels
and broadcast intervals. We claim that this is the first complete
research work on beacon deployment with multiple power
levels and irregular radiation patterns. The ILP was solved
and our concluded observations are given as follows:
1) the analytical bounds we derived are tight with the ILP
results with about 20% of gap, which can serve as a
viable designing guide for larger instances;
2) with a larger power constraint (thus a larger radiation
range), less beacons are required at the expense of
shorter beacon lifetime and/or longer delay;
3) the number of required beacons grows approximately
linearly with the number of TPs in an AOI;
4) In our experiments the solution by ILP requires 2-8
times less beacons compared to the commonly adopted
naive approach, like iBeacon, etc.
Our future research will be on development of effective
heuristic algorithms in solving the BDP problem in order
to gain more insights into the performance behaviour of the
proposed approach under larger and irregular topologies.
APP EN D IX A
Lemma 4: For a 1by nAOI, BDUP is achievable with
2⌊r⌋+ 1 beacons of radius rwhen n∈[2⌊r⌋+ 1,4⌊r⌋+ 1]
and n−2⌊r⌋such beacons when n > 4⌊r⌋+ 1.
Proof: By consecutively putting 2⌊r⌋+ 1 beacons of
radius r, any 1by nAOI (n∈[2⌊r⌋+ 1,4⌊r⌋+ 1]) can
be differentiated using the pattern demonstrated in Fig. 8a(if
n < 4⌊r⌋+ 1, the pattern should be shifted and truncated to
fit nwhile always keeping the 2⌊r⌋+ 1 beacons, see Fig. 8b).
IEEE INTERNET OF THINGS JOURNAL 9
(a) α= 3,225 TPs (b) α= 3,144 TPs (c) α= 3,100 TPs
(d) α= 4,225 TPs (e) α= 4,144 TPs (f) α= 4,100 TPs
(g) α= 5,225 TPs (h) α= 5,144 TPs
0
20
40
60
80
100
120
140
100 200 300 400
bcn#
TP#
a= 3
a= 4
a= 5
(i) ILP simulation Results
Fig. 7. Beacon ILP Deployment: 60 ·60m2meters, θ=−97dBm, maximum delay = 2secs,¯pmax = 5mw
When n > 4⌊r⌋+ 1, in addition to the 2⌊r⌋+ 1 beacons,
another n−(4⌊r⌋+ 1) beacons are required at the end of
the line AOI, and all TPs can be differentiated as Fig. 8c
demonstrates.
As shown in Fig. 8, the pattern is coded as a block diagonal
matrix. When n= 4⌊r⌋+ 1, a nby 2⌊r⌋+ 1 matrix is used,
as shown in Fig. 8d; when n∈[2⌊r⌋+ 1,4⌊r⌋+ 1) or n >
4⌊r⌋+ 1, the pattern should be shifted and truncated (See
Fig. 8e) or appended to fit n(See Fig. 8f).
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Wei He Dr. Wei He received his B.Sc. degree in computer science from
Fudan University and M.SE. degree from Peking University, in 2005 and 2008
respectively; and Ph.D. degree from School of Computer Science, University
of Waterloo in 2013. He is now an assistant professor in the School of
Electronics & Information Engineering, Nantong University, Jiangsu, China
and the lab director of Network Architecture at Zhongtian Broadband Co.
Ltd., Jiangsu, China. His current research interests include optical networks,
survivable network design, Cyber-physical systems and Internet of Things.
Pin-Han Ho Dr. Pin-Han Ho received his B.Sc. and M.Sc. degree from the
Electrical Engineering dept. in National Taiwan University in 1993 and 1995,
and Ph.D. degree from Queens University at Kingston at 2002. He is now a
professor in the department of Electrical & Computer Engineering, University
of Waterloo, Canada. His current research interests include survivable network
design, Fiber-Wireless (FIWI) network integration etc.
J´
anos Tapolcai Dr. J´anos Tapolcai received his M.Sc. (’00 in Technical
Informatics), Ph.D. (’05 in Computer Science) degrees from Budapest Univer-
sity of Technology and Economics (BME), Budapest, and D.Sc. (’13 in En-
gineering Science) from Hungarian Academy of Sciences (MTA). Currently
he is a Full Professor at the High-Speed Networks Laboratory at the Dept. of
Telecommunications and Media Informatics at BME. His research interests
include applied mathematics, optical networks and IP routing. He is a winner
of MTA Momentum (Lend¨ulet) Program.
