Rex: A randomized EXclusive region based scheduling scheme for mmWave WPANs with directional antenna

TLDR: The concept of exclusive region (ER) is introduced to allow concurrent transmissions to explore the spatial multiplexing gain of wireless networks and REX, a randomized ER based scheduling scheme, is proposed to decide a set of senders that can transmit simultaneously.

Abstract: Millimeter-wave (mmWave) transmissions are promising technologies for high data rate (multi-Gbps) Wireless Personal Area Networks (WPANs). In this paper, we first introduce the concept of exclusive region (ER) to allow concurrent transmissions to explore the spatial multiplexing gain of wireless networks. Considering the unique characteristics of mmWave communications and the use of omni-directional or directional antennae, we derive the ER conditions which ensure that concurrent transmissions can always outperform serial TDMA transmissions in a mmWave WPAN. We then propose REX, a randomized ER based scheduling scheme, to decide a set of senders that can transmit simultaneously. In addition, the expected number of flows that can be scheduled for concurrent transmissions is obtained analytically. Extensive simulations are conducted to validate the analysis and demonstrate the effectiveness and efficiency of the proposed REX scheduling scheme. The results should provide important guidelines for future deployment of mmWave based WPANs.

Figures

Fig. 1. Exclusive regions for omni-directional and directional antennae.

Fig. 4. Number of concurrent transmissions vs. path loss exponent.

Fig. 3. Number of concurrent transmissions (𝛼 = 4).

Fig. 5. Spatial multiplexing gain.

Fig. 6. Number of concurrent transmissions, case 4.

Fig. 7. Throughput with different antenna efficiency, case 4.

Full text

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 1, JANUARY 2010 113
REX: A Randomized EXclusive Region Based
Scheduling Scheme for mmWave WPANs with
Directional Antenna
Lin X. Cai, Student Member, IEEE , Lin Cai, Member, IEEE, Xuemin (Sherman) Shen, Fellow, IEEE,
and Jon W. Mark, Life Fellow, IEEE
Abstract—Millimeter-wave (mmWave) transmissions are
promising technologies for high data rate (multi-Gbps) Wireless
Personal Area Networks (WPANs). In this paper, we first
introduce the concept of exclusive region (ER) to allow
concurrent transmissions to explore the spatial multiplexing
gain of wireless networks. Considering the unique characteristics
of mmWave communications and the use of omni-directional or
directional antennae, we derive the ER conditions which ensure
that concurrent transmissions can always outperform serial
TDMA transmissions in a mmWave WPAN. We then propose
REX, a randomized ER based scheduling scheme, to decide a
set of senders that can transmit simultaneously. In addition, the
expected number of flows that can be scheduled for concurrent
transmissions is obtained analytically. Extensive simulations
are conducted to validate the analysis and demonstrate the
effectiveness and efficiency of the proposed REX scheduling
scheme. The results should provide important guidelines for
future deployment of mmWave based WPANs.
Index Terms—Resource management, exclusive region, service
scheduling, spatial multiplexing gain, mmWave WPAN.
I. INTRODUCTION
T
HE spectrum between 30 GHz an d 300 GHz is referred
to as th e millimeter wave (mmWave) band because
the wavelengths for these frequencies are about one to ten
millimeters. The FCC h a s recently allocated the 57-64 GHz
mmWave band for general unlicensed use, which opens a
door for very high data rate wireless applications over the
7 GHz unlicensed band. The IEEE 802.15.3c has recently
been formed to develop a mmWave-based alternative physical
layer (PHY) for the existing 802.15.3 Wireless Personal Area
Networks (WPANs) standard. The mmWave communications
have many salient features. First, it is anticipated to achieve
very high data rate (multi-Gbps), so it will enable many
killer applications such as IPTV/VoD, 3D gaming, intelligent
transportation systems, etc. These applications require not only
high data rate, but also stringent QoS, in terms of delay,
Manuscript receiv e d May 14, 2007; revised July 15, 2007; accepted July 25,
2007. The associate editor coordinating the review of this paper and approving
it for publication was X.-G. Xia.
L. X. Cai, X. Shen, and J. W. Mark are with the Centre for Wireless
Communications, Department of Electrical and Computer Engineering, Uni-
versity of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: {lcai, xshen,
jwmark}@bbcr.uwaterloo.ca).
L. Cai is with the Department of Electrical and Computer Engineering,
Univ ersity of Victoria (e-mail: cai@uvic.ca).
This work has been supported by the Natural Sciences and Engineering
Research Council (NSERC) of Canada under Grant No. RGPIN7779.
Digital Object Identifier 10.1109/TWC.2010.01.070503
jitter, and loss. Second, mmWave systems can coexist well
with existing wireless communication systems, such as WiFi
(IEEE 802.11), cellular systems, and Ultra WideBand (UWB)
systems, because of the large frequency difference. Third,
oxygen absorption peaks at 60 GHz, so the transmission and
interference ranges of mmWave communications are small,
which allows highly dense deployment of mmWave WPANs.
In addition, since the mmWave signal degrades significantly
when passing through walls and over distances, this will help
to ensu re the security of the content.
Although mmWave prototype chipsets have been emerg-
ing [1], their performance in a networked environment is still
an open area b eckoning for further investigation. To ensure
the success o f mmWave based WPANs, how to efficiently and
effectively allocate resource for co-existing mmWave devices
is a critical issue, which is the main focus of this paper.
In this paper, we first investigate the unique characteristics
of mmWave communications, the appropriate medium ac-
cess techniques, and network architecture for mmWave based
WPANs. We then identify the key opportunities and challenges
in resource management of mmWave WPANs, and propose
REX, a randomized exclusive region (ER) based scheduling
scheme to explore the spatial multiplexing gain in mmWave
WPANs. The basic concept of REX is: each flow has an ER
around the receiver, and the senders of all flows transmitting
concurrently should be outside the ERs of other flows to
ensure that concurrent transmissions are favorable.
The main contributions of the paper are four-fold. First,
to the best of our knowledge, the paper is one of the first
to systematically study the resource management issues for
mmWave b ased WPANs. Second, we propose how to allow
concurrent transmissions appropriately to explore the spa-
tial multiplexing gain in mmWave WPANs, and derive the
sufficient conditions to ensure that concurrent transmissions
are favorable in terms of per flow throughput and network
throughput, considering both omni-(directional) and direc-
tional antennae. Third, optimal scheduling for peer to peer
concurrent transmissions is known to be NP-hard [2], [3]. In
traditional scheduling problems, the utility (e.g., throughput)
obtained per unit resource (e.g., bandwidth × time slot) is
deterministic; here, utility is variable according to channel data
rate, network topology, user deployment, transmission power,
cross-correlations of interfering signals, and the scheduling
decision itself. Since the optimal scheduling problem is dif-
1536-1276/10$25.00
c
 2010 IEEE
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114 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 1, JANUARY 2010
ficult to solve in real time, we propose the REX scheme
as the first step to explore the spatial multiplexing capacity
of mmWave WPANs. Finally, given the ER condition, we
analytically investigate the network performance in terms of
the expected number of concurrent transmissions. Extensive
simulations have demonstrated the accuracy of the analysis
and the efficiency of the proposed scheduling scheme.
The remainder of the paper is organized as follows. In
Sec. II, we present the channel characteristics of mmWave
communications and the architecture design of mmWave based
WPANs. In Sec. III, we derive the exclusive regions con-
sidering both omni- and directional antennae. The proposed
REX scheme is presented in Sec. IV-A, and its performance
is analyzed in Sec. IV-B. Simulation results are presented in
Sec. V, followed by the related work in Sec. VI. Concluding
remarks and future work are given in Sec. VII.
II. S
YSTEM MODE L
A. mmWave Channel Characteristics and Multiple Access
The main characteristics of mmWave communications are
short wavelength/high frequency, large bandwidth and high
interaction with atmospheric constituents. For mmWave com-
munications with very high data rates (and thus very small
symbol duration), intersymbol interference (ISI) due to time
dispersion in multipath propagation becomes significant. Or-
thogonal frequency-division multiplexing (OFDM) signals are
relatively robust against ISI due to the reduced symbol rate
in each of the subcarriers, and thus, it is a good candidate
for mmWave communications. Although we use OFDM in
our system model, our work is independent of any particular
modulation schemes.
OFDM can b e combined with a multiple access scheme
such as Time Division Multiplex Access (TDMA) or Code Di-
vision Mu ltiple Access (CDMA) for effective multiple access
control [4]. OFDM-TDMA are straightforward: different users
share the wireless medium in different time slots. Several com-
binations of OFDM and CDMA have been discussed in [5].
For RF oscillators at mmWave spectrum, it is very difficult
to maintain a low level phase noise, which affects the signal
in the frequency conversion operations, and results in higher
bit error rate (BER) for effective communications. Different
multiple access techniques, including OFDM/TDMA, direct
sequence (DS)-CDMA, Multi-Carrier (MC)-CDMA, and MC-
DS-CDMA, have different sensitivities to phase noise. Ac-
cording to [6], MC-DS-CDMA is most robust against phase
noise and multiple access interference (MAI). Therefore, we
deploy MC-DS-CDMA as the medium access technique for
the mmWave networks.
B. Directional Antennae
Because of the unique characteristics of 60 GHz mmWave
communications, i.e., small wavelength and high path loss due
to severe oxygen absorption and atmospheric attenuation, it
is highly desired to use directional antenna to achieve much
higher antenna gain over a longer transmission range, by
radiating transmission energy to the d esired direction only [7].
There are two types of directional antennae [8]: conventional
sectored/switched antenna array and adaptive antenna array. A
sectored antenna array consists of a number of fixed beams
that provide full coverage in azimuth. An adaptive antenna
array is able to automatically adapt its radiation patterns b y
using beamforming tech nique that intelligently puts a main
beam in the direction of the wanted signal an d nulls in the
directions of the interference and noise. Since the size of the
antennae used for mmWave communications could be very
small, it is feasible to deploy multiple antenna elements in a
device to achieve directivity. In a mmWave WPAN with direc-
tional antennae, directivity and high path loss should result in
amoreefficient spectrum reuse and significant improvement
in the network throughput. In addition, directional antennae
are more energy efficient.
In the networking research community, a popular antenna
model for directional antenna is the flat-top model [9], [10]:
the antenna gain is a constant within the beamwidth and zero
outside the beamwidth. Therefore, for a beam with beamwidth
𝜃, the antenna gain of the mainlobe is 𝐺
𝑚
=2𝜋/𝜃,andthat
of sidelobe is 𝐺
𝑠
=0. A more realistic three-dimensional
cone plus sphere model is proposed in [11], taking the effects
of sidelobes into consideration. In this model, the antenna
gain consists of a mainlobe of beamwidth 𝜃 and aggregated
spherical sidelobes of beamwidth 2𝜋 − 𝜃 at the base of the
mainlobe cone. Uniform gain is also assumed for simplicity
in the cone plus sphere model. Since we consider all devices
in a WPAN to be in a plane, we employ the cone plus
circle model in a two-dimensional scenario and define the
antenna gains of the mainlobe and sidelobe as 𝐺
𝑚
= 𝜂
2𝜋
𝜃
and 𝐺
𝑠
=(1− 𝜂)
2𝜋
2𝜋 −𝜃
, respectively, where 𝜂 is the antenna
radiation efficiency.
C. WPAN Network Architecture
Since mmWave communications cannot penetrate walls, we
consider devices randomly distributed in an 𝐿 × 𝐿 square
room. IEEE 802.15.3 is the standard dedicated for high
rate WPANs. According to IEEE 802.15.3, multiple devices
form a piconet which is the basic network element. One
device is selected as the piconet controller (PNC) that collects
the global information o f the piconet. Data transmissions in
the piconet is based on the time-slotted superframe struc-
ture [12]. Considering most devices using directional antenna
in mmWave WPANs, the centralized PNC is very useful
for device/neighbor discovery. The PNC broadcasts beacons
periodically to all directions which a llow other devices to
synchronize and determine their locations. All devices send
channel time requests and their locations to the PNC, which
schedules peer-to-peer communications accordingly. However,
the scheduling algorithm is not specified in the stand a rd, and
it is our focus.
III. E
XCLUSIVE REGIONS
Let 𝑃
𝑅
denote the received signal power, 𝑅 the chan-
nel capacity (or the achievable data rate with an efficient
transceiver design), 𝑁
0
the one-sided spectral density of white
Gaussian noise, and 𝐼 the total interference power. According
to the Shannon theory, 𝑅 = 𝑊 log
2
(𝑆𝐼𝑁𝑅 +1),where
𝑆𝐼𝑁𝑅 = 𝑃
𝑅
/(𝑁
0
𝑊 + 𝐼).
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CAI et al.: REX: A RANDOMIZED EXCLUSIVE REGION BASED SCHEDULING SCHEME FOR MMWAVE WPANS WITH DIRECTIONAL ANTENNA 115
Consider a network with 𝑁 flows, {𝑓
𝑖
, ∣𝑖 ∈ 0, 1, ..., 𝑁},
requesting transmission times in a superframe with 𝑁 time
slots. The distanc e between the transmitter and receiver of
the 𝑖-th flow is 𝑑
𝑖
, and the distance between the transmit-
ter of th e 𝑗-th flow and the receiver of the 𝑖-th flow is
𝑑
𝑗,𝑖
. The average transmitting power and receiving power
of flow 𝑓
𝑖
are denoted as 𝑃
𝑇
(𝑖) and 𝑃
𝑅
(𝑖), respectively.
Using the free space path loss model, 𝑃
𝑅
(𝑖) can be calculated
as 𝑃
𝑅
(𝑖)=𝐺
𝑇
(𝑖)𝐺
𝑅
(𝑖)

𝜆
4𝜋𝑑
𝑖

2
𝑃
𝑇
(𝑖), where 𝐺
𝑇
(𝑖) and
𝐺
𝑅
(𝑖) are the antenna gains o f the transmitter and receiver,
respectively. Considering signal dispersion over distance, the
average received signal power is modeled as
𝑃
𝑅
(𝑖)=𝑘
1
𝐺
𝑇
(𝑖)𝐺
𝑅
(𝑖)𝑑
−𝛼
𝑖
𝑃
𝑇
(𝑖), (1)
where 𝑘
1
∝ (𝜆/4𝜋)
2
is a constant coefficient dependent on
the wavelength 𝜆,and𝛼 is the path loss exponent dependent
on the propagation environment and usually takes the value
between 2 to 6 [13]. Assume that 𝐺
𝑇
(𝑖), 𝐺
𝑅
(𝑖) and 𝛼 are
constant, and all devices use the same transmission power.
If only one flow is allowed to transmit at a time, i.e., flows
are transmitted in a TDMA fashion, the average data rate of
the 𝑖-th flow during the 𝑁 slots, 𝑅
𝑖
,isgivenby
𝑅
𝑖
=
𝑘
2
𝑊
𝑁
log
2
(
𝑘
1
𝐺
𝑇
(𝑖)𝐺
𝑅
(𝑖)𝑃
𝑇
(𝑖)𝑑
−𝛼
𝑖
𝑁
0
𝑊
+1) (2)
where 𝑘
2
is a coefficient related to the efficiency of the
transceiver design. If all flows can be transmitted simulta-
neously in all slots, i.e., flows are transmitted in a CDMA
fashion, the achievable data rate, 𝑅
′
𝑖
,ofthe𝑖-th flow is given
by
𝑅
′
𝑖
= 𝑘
2
𝑊 log
2
(
𝑘
1
𝐺
𝑇
(𝑖)𝐺
𝑅
(𝑖)𝑃
𝑇
(𝑖)𝑑
−𝛼
𝑖
𝑁
0
𝑊 +

𝑖∕=𝑗
𝐼
𝑗,𝑖
+1) (3)
where 𝐼
𝑗,𝑖
is the interference power between the transmitter
of the 𝑗-th flow and the receiver of the 𝑖-th flow. Assume the
cross correlation between any two concurrent transmissions
is constant, 𝐺
𝑗,𝑖
= 𝐺
0
, ∀𝑗 ∕= 𝑖. The inter ference power is
𝐼
𝑗,𝑖
= 𝑘
1
𝐺
0
𝐺
𝑇
(𝑗)𝐺
𝑅
(𝑖)𝑃
𝑇
(𝑗)𝑑
−𝛼
𝑗,𝑖
.
To compare 𝑅 and 𝑅
′
, we consider two cases separately.
First, if SINR < 1, the achieved data rate can be approximated
as
𝑘
2
𝑊 log
2
(SINR + 1) ≈ 𝑘
2
𝑊 × SINR log
2
𝑒. (4)
With the approximation, from (2) and (3), a sufficient condi-
tion to ensure that 𝑅
′
𝑖
≥ 𝑅
𝑖
is 𝐼
𝑗,𝑖
≤ 𝑁
0
𝑊, ∀𝑗 ∕= 𝑖, i.e.,
the average interference level from any other flow should be
less than the background noise
1
. Thus, if we allow flows with
mutual interference less than that of the background noise to
transmit simultaneously, the throughput of each flow can be
higher than that of serial TDMA transmissions.
Second, if SINR ≥ 1 , the approximation in (4) may not
hold. Nevertheless, the previous derived sufficient condition
can still ensure that 𝑅
𝑖
≤ 𝑅
′
𝑖
. This is because log
2
(𝑥/𝑁 +
1) ≥ (1/𝑁 )log
2
(𝑥 +1), ∀𝑥 ≥ 1,𝑁≥ 1.If𝐼
𝑗,𝑖
≤ 𝑁
0
𝑊 ,
1
The necessary and sufficient condition to ensure that 𝑅
′
𝑖
≥ 𝑅
𝑖
is
∑
𝑗∕=𝑖
𝐼
𝑗,𝑖
≤ (𝑁 − 1)𝑊𝑁
0
,whereflow 𝑗 is scheduled to transmit
concurrently with flow 𝑖.Thesufficient condition given in the main text is
more conservative, but it allows to design much simpler and practically more
feasible scheduling algorithms.
r
2
r
1
r
3
r
0
r
5
r
4
r
r
r
8
(a) Omni − Omni (b) Directional − Omni
(d) Directional − Directional
(c) Omni − Directional
7
6
Fig. 1. Exclusiv e regions for omni-directional and directional antennae.
𝑅
′
𝑖
/𝑅
𝑖
≥ 1/𝑁 log
2
(SNR + 1)/ log
2
(SNR/𝑁 +1)≥ 1. Thus,
the derived sufficient condition is still applicable.
Assume that the noise power spectrum is constant. To
ensure that the interfer ence p ower is less than the noise,
we should not allow any interferer inside an ER around the
receiver. In other words, an interferer should be at least 𝑟(𝑖)
away from the receiver of the 𝑖-th flow , where 𝑟(𝑖) is given
as
𝑟(𝑖)=(
𝑘
1
𝐺
0
𝐺
𝑇
(𝑗)𝐺
𝑅
(𝑖)𝑃
𝑇
(𝑗)
𝑁
0
𝑊
)
1/𝛼
. (5)
The ERs are determined by the types of transmitting and
receiving antennae, i.e., omni- or directional. In the following,
we consider four cases in a two-dimensional plane, and the
results obtained can also be extended to three-dimensional
space.
Case 1: Omni-antenna to Omni-antenna
In this case, both the transmitters and receivers use omni-
antennae, 𝐺
𝑇
(𝑖)=𝐺
𝑅
(𝑖)=1, ∀𝑖 ∈ 1, 2, ..., 𝑁. The interfe r-
ence between flows 𝑗 and 𝑖 is 𝐼
𝑗,𝑖
= 𝑘
1
𝐺
0
𝑃
𝑇
(𝑗)𝑑
−𝛼
𝑗,𝑖
. Assume
all transm itters use th e same power 𝑃 for transmission. To
ensure that the interference from each interfere to be less than
the noise, all interfering sources should be at least 𝑟
0
away
from the receiver of the 𝑖-th flow (𝑑
𝑗,𝑖
≥ 𝑟
0
), where 𝑟
0
is
given by
𝑟
0
=(
𝑘
1
𝐺
0
𝑃
𝑁
0
𝑊
)
1/𝛼
. (6)
Therefore, the ER is a circle centered at the receiver, with
radius 𝑟
0
, as shown in Fig. 1 (a).
Case 2: Directional-antenna to Omni-antenna
In this case, the transmitter an tennae are direction al and
the receiver antennae are omni-antennae (𝐺
𝑅
(𝑖)=1). The
directional antenna pattern consists of a mainlobe of gain 𝐺
𝑇
𝑀
with beamwid th 𝜃 and a sidelobe of gain 𝐺
𝑇
𝑆
with beamwidth
2𝜋 − 𝜃.
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116 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 1, JANUARY 2010
As shown in Fig. 1(b), if a receiver is inside the radiation an-
gle of an interferer, the interference is 𝐼
𝑗,𝑖
= 𝑘
1
𝐺
0
𝐺
𝑇
𝑀
𝑃𝑑
−𝛼
𝑗,𝑖
.
Thus, an interferer should be outside the circle centered at the
receiver with radius 𝑟
1
:
𝑟
1
=(
𝑘
1
𝐺
0
𝐺
𝑇
𝑀
𝑃
𝑁
0
𝑊
)
1/𝛼
. (7)
If a receiver is outside the radiation angle of an interferer, we
have 𝐼
𝑗,𝑖
= 𝑘
1
𝐺
0
𝐺
𝑇
𝑆
𝑃𝑑
−𝛼
𝑗,𝑖
, and the ER is a circle with radius
𝑟
2
:
𝑟
2
=(
𝑘
1
𝐺
0
𝐺
𝑇
𝑆
𝑃
𝑁
0
𝑊
)
1/𝛼
. (8)
Case 3: Omni-antenna to Directional-antenna
When the receiver antennae are directional and the transmit-
ter antennae are omni-directional, the exclusive region in this
case is a sector of a circle centered at the receiver with radius
𝑟
4
plus a sector with radius 𝑟
3
and angle 2𝜋 −𝜃,asshownin
Fig. 1(c), where 𝜃 is the beamwidth of the directional antenna
of the receiver.
Let 𝐺
𝑅
𝑀
be the antenna gain of the receiver within the
beamwidth of 𝜃,and𝐺
𝑅
𝑆
the gain outside the beamwidth. If
an interferer is located within the beamwidth of a receiver’s
antenna, 𝐼
𝑗,𝑖
= 𝑘
1
𝐺
0
𝐺
𝑅
𝑀
𝑃𝑑
−𝛼
𝑗,𝑖
, and the interferer should be
at least 𝑟
3
away from the receiver:
𝑟
3
=(
𝑘
1
𝐺
0
𝐺
𝑅
𝑀
𝑃
𝑁
0
𝑊
)
1/𝛼
. (9)
Otherwise, 𝐼
𝑗,𝑖
= 𝑘
1
𝐺
0
𝐺
𝑅
𝑆
𝑃𝑑
−𝛼
𝑗,𝑖
and the interferer should
be at least 𝑟
4
away from the receiver:
𝑟
4
=(
𝑘
1
𝐺
0
𝐺
𝑅
𝑆
𝑃
𝑁
0
𝑊
)
1/𝛼
. (10)
Case 4: Directional-antenna to Directional-antenna
When both the transmitter and receiver antennae are direc-
tional, the ER contains four zones. If an interferer is located
within the beamwidth of the receiver, and the receiver is also
within the beamwidth of the interferer, the interferer should
be at least 𝑟
8
away from the receiver:
𝑟
8
=(
𝑘
1
𝐺
0
𝐺
𝑇
𝑀
𝐺
𝑅
𝑀
𝑃
𝑁
0
𝑊
)
1/𝛼
. (11)
Therefore, the first ER zone is a cone with angle 𝜃 and radius
𝑟
8
.
If an interferer is within the radiation angle of the receiver,
but the receiver is outside the radiation angle of the interferer,
the second ER zone is a cone with angle 𝜃 and radius 𝑟
6
:
𝑟
6
=(
𝑘
1
𝐺
0
𝐺
𝑇
𝑆
𝐺
𝑅
𝑀
𝑃
𝑁
0
𝑊
)
1/𝛼
. (12)
If an interferer is outside the radiation angle of the receiver
with its radiation beamwidth toward the receiver, the third ER
zone is a sector with angle 2𝜋 − 𝜃 and radius 𝑟
7
:
𝑟
7
=(
𝑘
1
𝐺
0
𝐺
𝑇
𝑀
𝐺
𝑅
𝑀
𝑃
𝑁
0
𝑊
)
1/𝛼
. (13)
If both the interferer and the receiver are outside of each
other’s radiation beamwidth, the last ER zone is a sector with
angle 2𝜋 − 𝜃 and radius 𝑟
5
:
𝑟
5
=(
𝑘
1
𝐺
0
𝐺
𝑇
𝑆
𝐺
𝑅
𝑆
𝑃
𝑁
0
𝑊
)
1/𝛼
. (14)
The four ER zones for this case are shown in Fig. 1 (d).
IV. REX S
CHEDULING SCHEME
A. REX Scheme
It is shown in Sec. III that concurrent transmissions are more
favorable than serial TDMA transmissions if all interfering de-
vices are sufficiently far apart, i.e., outside the ERs of the other
receivers. In other words, network throughput can be improved
by exploiting the spatial reuse of the wireless channel for
concurrent transmissions. With a random network topology,
the optimal scheduling problem for concurrent transmissions
is known to be NP-hard [2], [3]. Unlike the traditional
scheduling problems, each flow’s throughput per time slot
in mmWave WPANs is unknown before the scheduling de-
cision, and it depends on network topology, user deployment,
transmission power, cross-correlations of interfering signals,
and the scheduling decision itself. If 𝑃 ∕= 𝑁𝑃,thereis
no polynomial time algorithm to optimize the scheduling
decision.
In the following, we propose REX, a randomized ER based
scheduling scheme f or a centralized mmWave WPAN, with
computational complexity 𝑂(𝑁
2
log 𝑁) to allocate a time
slot. We consider a WPAN with 𝑁 active flows requesting
transmissions. The PNC has the global information of the
WPAN, e.g., the number of active flows, and the location
information of all devices, etc., based on which the PNC
schedules peer-to-peer transmissions for active flows
2
. Denote
the set of all active flows as 𝑆{𝑁} of 𝑁 elements. A subset
of flows 𝛾
𝑖
⊂ 𝑆{𝑁 } contains the flows scheduled in slot 𝑖
that satisfy the conditions favoring concurrent transmissions,
as derived in (6)-(14). Denote 𝐹𝑆 the set of scheduled flows
in 𝑆{𝑁 } and 𝑇
𝑎
(𝑗) the numbe r of slots allocated to flow 𝑗.
Initially, 𝐹𝑆 = 𝛾
𝑖
= 𝑁𝑈𝐿𝐿 and 𝑇
𝑎
=0for all flows in any
slot. The proposed REX scheme is as follows.
∙ Step 1: Randomly choose one flow with the minimum
𝑇
𝑎
and schedule it in slot 𝑖 (in itially, 𝑖 =1for the first
slot). Add this flow to the subsets 𝛾
𝑖
.Iftheflow is not
included in 𝐹𝑆, add it to 𝐹𝑆;
∙ Step 2: Check all the remaining active flows in the set
𝑆{𝑁}−𝛾
𝑖
for concurrent transmission conditions as
derived in (6)-(14), starting from the flow with the small-
est 𝑇
𝑎
.Ifanyflow satisfies the concurrent transmission
condition, i.e., the new flow and the flows in set 𝛾
𝑖
are
mutually outside each other’s exclusive regions, add it to
𝛾
𝑖
and increase 𝑇
𝑎
of the flow by one. If this flow is not
included in 𝐹𝑆, add it to 𝐹𝑆;
∙ Step 3: Increase the slot number 𝑖 by one and sort flows
according to 𝑇
𝑎
in ascending order;
∙ Step 4: Repeat Steps 1-3 until all flows are scheduled,
𝐹𝑆 = 𝑆{𝑁}.
The procedure can also be repeated until the requirements
of all active flows are fully satisfied. It is worth noting that
although sorting flows according to their 𝑇
𝑎
in step 2 w ill
increase the computational complexity by 𝑂(𝑁 log 𝑁),itis
essential for main taining fairness among flows. If we search
2
In WPANs, the mobility is typically low, e.g., ≤ 1 m/s, and the superframe
duration is less than 100 ms. Thus, the node movement is normally less than
0.1 m during the superframe duration. Such small change in location will not
significantly affect the received power and interference power level, and it is
acceptable to ignore mobility for scheduling decision.
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CAI et al.: REX: A RANDOMIZED EXCLUSIVE REGION BASED SCHEDULING SCHEME FOR MMWAVE WPANS WITH DIRECTIONAL ANTENNA 117
flows in a deterministic sequence for slot allocation, those
flows with smaller sequence number are more likely to be
scheduled in 𝛾
𝑖
. This will cause serious unfairness problem,
as shown in th e simulation results in a later section. With the
searching sequence used in Step 2, the maximum access delay
of all flows can be bounded.
The r esults of whether two flows are m utually exclusive
can be saved in a look-up table to reduce the execution time
of REX. Due to low mobility in WPANs, the frequency of
updating this table is low.
B. Average Number of Concurrent Transmissions
Given the number of active users in an area, what is the
number of flows that can transmit simultaneously under the
constraint of the ER condition? Since n etwork topology and
user deployment drastically affect the network performance,
we focus on the expected number of concurrent transmissions,
which is general and independent of network topology and
user deployment.
Consider an 𝐿 ×𝐿 square room containing 𝑁 active flows,
with 𝑁 transmitter s and 𝑁 receivers randomly deployed. De-
fine 𝑃 (𝑘, 𝑛) as the probability that only 𝑘 flows satisfy the ER
conditio n and can be scheduled for concurrent transmissions,
after checking the first 𝑛 ≤ 𝑁 flows one by one. Without loss
of generality, we check flows in ascending order 1, 2, ..., 𝑁.
The first flow 𝑓
1
will be scheduled for transmission in the
set 𝛾, and we have 𝑃 (1, 1) = 1.Flow𝑓
2
will be added
to 𝛾 if it does not conflict with flow 𝑓
1
.Define 𝑄 as the
probability o f a transmitter lying outside an ER of a receiver.
The probability that a flow does not conflict with anoth e r flow
is 𝑄
2
, because both transmitters should be outside the ERs of
the other receivers. Accordingly, the probability that two flows
do not satisfy the ER condition is 1 − 𝑄
2
. Therefore, in the
two-flow case, we have 𝑃 (2, 2) = 𝑄
2
and 𝑃 (1, 2) = 1 −𝑄
2
.
After we check the first 𝑛 flows, there are 𝑘 flows in 𝛾 if a)
there are 𝑘 −1 flows in 𝛾 when we check the first 𝑛−1 flows,
and the 𝑛-th flow does not conflict with the other 𝑘 −1 flows
in 𝛾;orb)thereare𝑘 flows in the set when we check the
first 𝑛 −1 flows, and the 𝑛-th flow conflicts with one of the 𝑘
flows in 𝛾. The probability that a flow does not conflict with
any of the other 𝑘 − 1 flows is 𝑄
2(𝑘−1)
.
𝑃 (𝑘, 𝑛)= 𝑃 (𝑘 − 1,𝑛− 1)𝑄
2(𝑘−1)
(15)
+𝑃 (𝑘, 𝑛 −1)(1 − 𝑄
2𝑘
) for 𝑘<𝑛.
If, among the 𝑛 flows, only the first flow can be added in
𝛾, implying that the following 𝑛 −1 flows do not satisfy the
ER condition, we have
𝑃 (1,𝑛)=(1− 𝑄
2
)
𝑛−1
for 𝑘 =1. (16)
Another extreme case is that all 𝑛 flows can be scheduled
concurrently, which means that none of the flows conflicts
with the remaining 𝑛 −1 flows,
𝑃 (𝑛,𝑛)=(𝑄
𝑛−1
)
𝑛
for 𝑘 =1. (17)
Given the initial values of 𝑃 (1, 1), 𝑃 (1, 2) and 𝑃 (2, 2),we
can iteratively obtain 𝑃 (𝑘, 𝑁) as a function of 𝑄 for ∀𝑘, 1 ≤
S
A0
A0
A0
f1
f2
f3
r0
A2
S
A1
f1
f3
r1
r0
A2
A1
A2
A1
f2
(a) Omni-Omni (b) Directional-Omni
(c) Omni-Directional
S
A 4
A3
A3
A4
A3
A4
r
4
r3
f2
f3
f1
(d) Directional-Directional
A6
A5
A 7
A 8
A 7
S
f3
f1
A 6
A5
A7
A 8
A6
A5
A 8
f2
Fig. 2. Concurrent transmissions in WPANs.
𝑘 ≤ 𝑁 . The expected number of concurrent transmissions is
𝐸[𝐶𝑇]=
𝑁

𝑘=1
𝑘𝑃(𝑘, 𝑁). (18)
To obtain 𝐸[𝐶𝑇], we need to know 𝑄. Let the size o f the
ER of a receiver be 𝐴, and total area 𝑆 = 𝐿
2
.Asshown
in Fig. 2, with each device randomly deployed in the room,
an interferer of one flow is outside the ER of the receiver of
another flow with probability 𝑄 =1− 𝐴/𝑆. Since the ER
region and 𝑄 are related to the types of antennae used, in the
following, we derive 𝑄 by considering the four cases shown
in Fig. 1.
Case 1: Omni-antenna to Omni-antenna
In case 1, the ER is a circle with radius 𝑟
0
and 𝐴
0
= 𝜋𝑟
2
0
,
as shown in Fig. 2(a ). The probability that an interferer is
outside the ER of a receiver is given by
𝑄
1
=1−
𝐴
0
𝑆
=1−
𝜋𝑟
2
0
𝑆
, for 𝑟
0
<< 𝐿. (19)
Case 2: Directional-antenna to Omni-antenna
Due to the omni- receivers and directional transmitters,
the ER in case 2 contains two zones, a circle with radius
𝑟
1
and another circle with radius 𝑟
2
, as shown in Fig. 2(b).
Accordingly, the areas of the two zones are 𝐴
1
= 𝜋𝑟
2
1
and
𝐴
2
= 𝜋𝑟
2
2
. If a receiver is within the radiation angle of
an interferer with probability 𝜃/2𝜋, the interferer is outside
the first ER zone (𝐴
1
) with probability 1 − 𝐴
1
/𝑆. Similarly,
if a receiver is outside the radiation angle of an interferer
with probability 1 −𝜃/2𝜋, the interferer is outside the second
ER zone (𝐴
2
) with probability 1 − 𝐴
2
/𝑆. Therefore, the
probability that an interferer is outside the ER of a receiver is
given by
𝑄
2
=1−
𝜋𝑟
2
2
𝑆
+
𝑟
2
2
𝜃
2𝑆
−
𝑟
2
1
𝜃
2𝑆
, for 𝑟
1
,𝑟
2
<< 𝐿. (20)
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