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Analyzing Random Access Collisions in Massive IoT Networks

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The cellular-based infrastructure is regarded as one of potential solutions for massive Internet of Things (mIoT), where the Random Access (RA) procedure is used for requesting channel resources in the uplink data transmission. Due to the nature of mIoT network with the sporadic uplink transmissions of a large amount of IoT devices, massive concurrent channel resource requests lead to a high probability of RA failure. To relieve the congestion during the RA in mIoT networks, we model RA procedure, and analyze as well as evaluate the performance improvement due to different RA schemes, including power ramping (PR), back-off (BO), access class barring (ACB), hybrid ACB and back-off schemes (ACB&BO), and hybrid power ramping and back-off (PR&BO). To do so, we develop a traffic-aware spatio-temporal model for the contention-based RA analysis in the mIoT network, where the signal-to-noise-plus-interference ratio (SINR) outage and collision events jointly determine the traffic evolution and the RA success probability. Compared with existing literature only modelled collision from single cell perspective, we model both SINR outage and the collision from the network perspective. Based on this analytical model, we derive the analytical expression for the RA success probabilities to show the effectiveness of different RA schemes. We also derive the average queue lengths and the average waiting delays of each RA scheme to evaluate the packets accumulation status and packets serving efficiency. Our results show that our proposed PR&BO scheme outperforms other schemes in heavy traffic scenario in terms of the RA success probability, the average queue length, and the average waiting delay.
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1
Analyzing Random Access Collisions in Massive
IoT Networks
Nan Jiang, Student Member, IEEE, Yansha Deng, Member, IEEE, Arumugam Nallanathan, Fellow, IEEE,
Xin Kang, Member, IEEE, and Tony Q. S. Quek, Fellow, IEEE
Abstract—The cellular-based infrastructure is regarded as one
of potential solutions for massive Internet of Things (mIoT),
where the Random Access (RA) procedure is used for requesting
channel resources in the uplink data transmission. Due to the
nature of mIoT network with the sporadic uplink transmissions
of a large amount of IoT devices, massive concurrent channel
resource requests lead to a high probability of RA failure. To
relieve the congestion during the RA in mIoT networks, we
model RA procedure, and analyze as well as evaluate the per-
formance improvement due to different RA schemes, including
power ramping (PR), back-off (BO), access class barring (ACB),
hybrid ACB and back-off schemes (ACB&BO), and hybrid
power ramping and back-off (PR&BO). To do so, we develop
a traffic-aware spatio-temporal model for the contention-based
RA analysis in the mIoT network, where the signal-to-noise-
plus-interference ratio (SINR) outage and collision events jointly
determine the traffic evolution and the RA success probability.
Compared with existing literature only modelled collision from
single cell perspective, we model both SINR outage and the
collision from the network perspective. Based on this analytical
model, we derive the analytical expression for the RA success
probabilities to show the effectiveness of different RA schemes.
We also derive the average queue lengths and the average waiting
delays of each RA scheme to evaluate the packets accumulation
status and packets serving efficiency. Our results show that our
proposed PR&BO scheme outperforms other schemes in heavy
traffic scenario in terms of the RA success probability, the average
queue length, and the average waiting delay.
Index Terms—Massive IoT, Cellular Network, Random Access,
Collision, Power Ramping, Stochastic Geometry.
I. INTRODUCTION
With the rapid proliferation of innovative applications in
the paradigm of massive Internet of Things (mIoT), such as
smart city, smart home, smart industrial, and vehicular com-
munication, the demand of data traffic for wireless networks
Manuscript received Augest 2, 2017; revised October 26, 2017; revised
March 5, 2018; revised Augest 6, 2018; accepted Augest 6, 2018. This
work was supported in part by the U.K. Engineering and Physical Sciences
Research Council (EPSRC) under Grant EP/M016145/2, in part by the MOE
ARF Tier 2 under Grant MOE2015-T2-2-104, and in part by the SUTD-ZJU
Research Collaboration under Grant SUTD-ZJU/RES/01/2016. This paper was
presented in part at the IEEE International Conference on Communications,
Kansas City, MO, USA, May 2018 [1]. The associate editor coordinating
the review of this paper and approving it for publication was Brady Ruffing.
(Corresponding author: Yansha Deng.)
N. Jiang, and A. Nallanathan are with School of Electronic Engineering
and Computer Science, Queen Mary University of London, London E1 4NS,
UK (e-mail: {nan.jiang, a.nallanathan}@qmul.ac.uk).
Y. Deng is with Department of Informatics, King’s College London, London
WC2R 2LS, UK (e-mail:yansha.deng@kcl.ac.uk).
X. Kang is with Center for Intelligent Networking and Communications
(CINC), National Key Laboratory of Science and Technology on Communi-
cations, University of Electronic Science and Technology of China (UESTC),
Chengdu 610054, China.
T. Q. S. Quek is with the Information Systems Technology and Design
Pillar, Singapore University of Technology and Design, Singapore 487372
(e-mail: tonyquek@sutd.edu.sg).
is explosively grown [2, 3]. In view of this, providing reliable
wireless access for the mIoT network becomes challenging due
to its nature of massive IoT devices and diversification of data
traffic [2, 3]. Cellular-based network is deemed as a potential
solution to provide last mile connectivity for massive number
of IoT devices, due to its advantages in high scalability,
diversity, and security, as well as low cost of additional
infrastructure deployments [4, 5]. However, to provide reliable
and efficient access mechanisms for a huge number of IoT
devices is still a key challenge [5–7].
IoT devices perform Random Access (RA) procedure to re-
quest channel resources for uplink transmission in the cellular-
based mIoT network, where the massive mIoT traffic impose
enormous load at the Radio Access Network (RAN) level. To
improve the quality of service and reduce power consumption
of IoT devices, efficient RA procedure is required to enhance
the sucess RA performance. An IoT device can either perform
contention-free RA when a dedicated scheduling request re-
source is assigned by the base station (BS) (e.g., handover), or
perform contention-based RA without a dedicated scheduling
request resource (e.g., uplink data or control information
transmission). Due to the delay-tolerant and uplink preferable
characteristics of the mIoT traffic, the contention-based RA is
considered as the main access technology to request channel
resources in the uplink transmission [6, 8].
The contention-based RA is based on ALOHA-type access
(i.e., request access in the first available opportunity), where
an IoT device randomly selects a non-dedicated preamble
(i.e., orthogonal pseudo code, such as Zadoff-Chu sequence)
transmitting to its associated BS via Physical Random Access
CHannel (PRACH) in the 1st step of RA [9]. As single
preamble provides single RA opportunity, preambles con-
tention among IoT devices represents their competition of up-
link channel resources. When competing simultaneously, IoT
devices choosing the same preamble bring mutual interference
and collision risks in preamble detection, resulting in perfor-
mance degradation in terms of high RA failure probability [4,
6, 8].
A collision occurs at the step 1 of RA when a BS suc-
cessfully decodes two or more same preambles from different
IoT devices simultaneously, such that the BS cannot serve any
colliding IoT devices, and these IoT devices need to restart
the RA procedure in the next available RA time slot. The
RA opportunities are represented by the repeated PRACHs,
which are reserved in the uplink channel and defined by the
PRACH configuration index, which is selected in the BS. A
great number of possible PRACH configuration indexes are
defined in LTE [9], and the PRACH configuration index 6 is
suggested to conduct the study in the mIoT network by the
2
3GPP [5]. More specifically, the PRACH is repeated every 5
ms with 54 available preambles, in other words, this system
offer a capacity of 10800 contention-based RA oportunities
per second. However, this performance is still limited on some
applications with serious RA requirements, such as earthquake
monitoring [5], due to the facts that massive IoT devices may
create bursty traffic, and the practical system performance
might be lower than the upper bound due to the nature of
random collision.
To improve the success RA performance under limited
channel resources, efficient RA schemes need to be proposed
and analyzed, which is utilized to alleviate uplink congestion
by reducing the high interference and high collision probability
when massive IoT devices contend for the uplink channel
resources at the same time [5, 6, 8]. Accordingly, several
solutions are provided in literature to reduce congestion in
the mIoT network. For instance, Access Class Barring (ACB)
scheme had been regarded as an efficient tool to prevent
congestion when massive concurrent access occurs [5], which
was extented studied in [7, 10, 11]. In [12], a delay-estimation
based RA scheme was proposed based on the back-off (BO)
scheme, which aims at improving the collision detection and
resolution capability. In [13], the authors analyzez the success
probability, throughput, and access delay of preamble trans-
mission under three power ramping (PR) schemes with fixed,
linear, and geometric step sizes from single cell perspective.
In [14], the authors evaluate RA success with and without the
PR scheme by considering collision and Physical Downlink
Control Channel (PDCCH) deficiency. In [15], a cooperation
incentive scheme was presented which reimburses the extra
energy consumptions for the helper nodes with consideration
of signal-to-noise-plus-interference ratio (SINR) constraint and
aggregate interference. In [16], the authors suggested a tech-
nology called Distributed Queueing RA Protocol, which has
potential for handling an ideally infinite number of devices,
attaining near-optimum performance.
To characterize and analyze the performance of contention-
based RA in the mIoT network, mathematical models are
required. Previously, mathematical models mainly focused on
the SINR outage or collision problem [10, 17–20]. However,
to the best of our knowledge, most works have focused either
on studying the SINR outage from the network point of view
without considering collision, or studying the collision prob-
lem from the single cell point of view considering given fixed
value of SINR outage. In [10, 17, 18], the authors modelled
queue status by taking into account collision events with given
collision probability, but they ignored the mutual interference
between devices. In [20], the authors combine queueing theory
and stochastic geometry to analyze the stability region in a
discrete-time slotted RA network, where devices are spatially
distributed as a Poisson point process, and a infinite buffer
is modelled in each device to track the time evolution of the
queue using queueing theory. In [19], the authors extend the
model proposed in [20] to analyze the preamble transmission
in RA, where three different preamble transmission schemes
are studied and compared.
In our previous work [21], we provided a novel spatio-
temporal mathematical framework to analyze the preamble
tranmission success probability of mIoT network, where the
queue evolution of IoT devices is modelled via probability
theory (i.e., a new approach is developed to track the queue
evolution, which is different from [19,20]), and the SINR
outage of preamble transmission is studied using stochastic
geometry. Due to the page limitation, we only focused on
deriving the preamble transmission success probability as
the first work in analyzing RA procedure using stochastic
geometry and probability theory, and left the collision problem
as the future work. In this work, different from [19–21], we
take into account the SINR outage events as well as the
collision events at the BS in evaluating the success RA. The
contributions of this paper can be summarized in the following:
We propose a tractable approach to jointly model and
analyze the SINR outage and collision problem of
contention-based RA. The model is general and can be
extended to analyze different RA schemes or/and differ-
ent networks by considering different preamble transmis-
sion policy and queue evolution.
For the PR scheme, we derive the general exact expres-
sion for the RA success probability in each time slot
with infinite number of power level units. Note that,
different from [13] considering single cell, we study
the RA success probability of the PR scheme from the
network aspect, where the analysis is more challenging
due to the difficulty in capturing both interference and
collision generated from IoT devices transmitting with
different transmit powers.
We extendedly study the ACB and back-off BO schemes
analyzed in our previous work [21] by taking into account
collision, and we also analyze two hybrid schemes,
namely, the hybrid ACB and back-off (ACB&BO)
scheme, and the hybrid power ramping and back-off
(PR&BO) scheme. We derive the exact expressions for
the RA success probability in each time slot with these
schemes, and our results show that the PR&BO scheme
outperforms all other schemes in all traffic scenarios.
We derive the average queue length and the average wait-
ing delay of each RA scheme to compensate for the fact
that the RA success probability cannot reveal the packet
accumulation status and performance of packets serving
efficiency in the time-aware network. Interestingly, our
results shown that the average queue length and the
average waiting delay of the PR&BO scheme outperforms
that of other schemes in heavy traffic scenarios.
We verify the RA success probability, the average queue
length, and the average waiting delay of each RA scheme
using our proposed realistic simulation framework, which
captures the randomness location, preamble transmission,
RA collision, as well as the real packets arrival, accumu-
lation, and departure of each IoT device in each time
slot.
The rest of the paper is organized as follows. Section II
introduces the system model. Sections III provides the single
time slot analytical model for the RA success probability.
Sections IV presents the analytical results for the RA success
probabilities in each time slot with different schemes. Section
3
V presents the analytical results for the average queue length
and the average waiting delay. Section VI provides numerical
results. Finally, Section VII concludes the paper.
II. SYSTEM MODEL
We consider a traffic-aware spatio-temporal model for the
cellular-based mIoT network: 1) the spatial model of BSs and
IoT devices are distributed in R2following two independent
homogeneous Poisson point process (HPPP), ΦBand ΦD, with
intensities λBand λD, respectively; 2) the temporal model of
the packets arrival at each IoT device in each time slot is
modelled as independent Poisson arrival process, ΛNew, with
intensities εNew. A packet can only be transmitted via the ded-
icated uplink data transmission channel, which is scheduled by
the associated BS. Before resource scheduling, the IoT device
need to execute a RA to request uplink channel resources with
the BS. We intend to analyzing the time-slotted contention-
based RA in the mIoT network, and thus assume that the
actual intended packet transmission is always successful if the
corresponding RA succeeds. Note that the data transmission
after a successful RA can be easily extended following the
analysis of preamble transmission success probability in RA.
Here, we limit ourselves to RA success to focus on the impact
of masssive access to RA procedure.
IoT devices use the contention-based RA procedure to
acquire synchronization and request uplink channel resources
with the associated BS before data transmission. Specifically,
an IoT device randomly selects a preamble from available
preamble pool for transmitting to its associated BS via PRACH
in the step 1, and exchanges control information via normal
uplink/downlink channels in the step 2, 3, and 4 [9]. In step 1,
we assume that ξnumber of available preambles are reserved
for contention-based RA in the mIoT network. Without loss
of generality, each IoT device has an equal probability (1)
to choose a specific preamble, and the average density of IoT
devices using a same preamble is λDp =λD, where λDp
is measured with unit devices/preamble/km2. The location of
each active IoT device choosing the same preamble varies in
different time slot due to that 1) IoT devices are randomly
moving such that their location are independent among differ-
ent time slots; 2) the IoT devices randomly choose a preamble
in each RA attempt, such that the set of active IoT devices
using the same preamble changes in different time slot. In
this case, the realizations of the active IoT devices that are
different is modeled as i.i.d random HPPP in each time slot.
The RA requests from massive number of IoT devices
simultaneously under limited number of available preambles
is the main challenge of mIoT network, thus we focus on the
contention of preamble in the step 1 of contention-based RA,
and we assume that the step 2,3,4 of RA are always successful
whenever the step 1 is successful. If the step 1 in RA fails,
the IoT device needs to reattempt in the next available RA
opportunity. In this case, a packet delayed in the buffer of an
IoT device causing by the access failure in the step 1 of RA
can be contributed by the following two reasons: 1) the BS
cannot decode the preamble due to the low received SINR in
the step 1 of RA; 2) the BS successfully decodes the same
preamble from two or more IoT devices in the same time,
such that the collision occurs.
It is known that collision event in the step 1 of RA can
be detected by the BS, when the collided IoT devices are
separable in terms of the power delay profile [9, 22]. Our
model follows the assumption of collision handling in [5],
where collision events are detected by BS after it decodes the
preambles in the step 1 of RA, and then no response will be
fedback from the BS to the IoT devices, such that it can not
proceed to the next step of RA [7, 23].
A. Physical Layer Description
Each IoT device is assumed to associates to its geographi-
cally nearest BS, where a Voronoi tesselation is formed, and
the BSs are uniformly distributed in the Voronoi cell [24–28].
To model the channel, a standard power-law path-loss model
is considered, where the path-loss is inversely proportional
to distance rwith the path-loss exponent α. We assume
the Rayleigh fading multi-path channel between two generic
locations x, y R2, where the channel power gains h(x, y)
is exponentially distributed random variables with unit mean.
Note that all the channel gains are independent of each other,
independent of the spatial locations, and identically distributed
(i.i.d.). For the brevity of exposition, the spatial indices (x, y)
are dropped.
Similar as [19, 21, 24, 29], we apply a full path-loss in-
version power control at all IoT devices to solve ”near-far”
problem, where each IoT device controls its transmit power
by compensating for its own path-loss to maintain the average
received signal power in the BS equalling to a same threshold
ρ. We also assume the density of BSs is high enough and no
IoT device suffers from truncation outage [21].
B. MAC Layer Description
We consider a time-slotted cellular-based mIoT network,
where the channel resources of RA are reserved in the uplink
channel and repeated in the system with a certain period
that specified by the BS. According to LTE standard [9],
most of uplink channel resources are scheduled for the data
transmission, and thus we assume that each time slot consists
of a front gap interval duration τg, which is relatively longer
than a following RA duration τc. We assume a geometric new
packets arrival process in each time slot at each IoT device,
which is modelled as independent Poisson arrival process1as
[21, 30–32]. Specifically, the number of new packets in the
mth time slot Nm
New is described by the Poisson distribution
with Nm
New Pois(µm
New), where µm
New = (τc+τg)εm
New.
More details about RA duration stracture and traffic model
(i.e., packets arriving and leaving) can be found in our previous
work [21, Section II.C]. We assume each IoT device has an
infinite buffer to store queueing packets until their successful
transmission, where none of packets will be dropped off, and
each IoT deivce transmits packets via a First Come First Serve
packets scheduling scheme2[33].
In the mIoT network, multiple RA attempts contribute to
massive concurrent signaling leading to RA fails frequently, so
1The traffic can also be modeled as the time limited Uniform Distribution
and the time limited Beta distribution [5].
2With minor modification, this model can also support multiple packets
transmitted by an IoT device within a time slot.
4
that progressively aggravates network congestion and service
degradation, in such case efficient RA transmission mecha-
nisms are required for congestion reduction [5]. In this paper,
we focus on the SINR outage and collision problem of mIoT
network with the different RA schemes, including the PR, the
ACB, the BO, the ACB&BO, and the PR&BO schemes. In
the following, we listed the five RA schemes:
PR scheme: If RA fails, the deferred packet will be
favored by stepping up the transmit power of preamble
after each unsuccessful RA attempt. Specifically, if a
RA attempt fails, the IoT device uses the full path-loss
inversion power control to maintain the average received
preamble power at a higher power level in the next RA
attempt, where κidenotes the power level unit in the
ith RA attempt by adjusting the target received preamble
power at the BS equal to κiρ[13] (i.e., κ1< κ2<· · · <
κi<· · · < κJ). Note that κJis the maximum allowable
power level unit.
ACB&BO/ACB/BO scheme: In the ACB&BO scheme,
IoT device draws a random number q[0,1] before
each RA attempt, and performs the RA attempt with its
associated BS only when qPACB (i.e., PACB is ACB
factor specified by the BS). If a RA attempt fails in the
mth time slot, the IoT device automatically defers its
following RA attempt over next tBO time slots and retry
a RA attempt for that packet in the (m+tBO + 1)th
time slot. The ACB and BO schemes correspond to the
ACB&BO scheme with the BO factor tBO = 0 and the
ACB factor PACB = 1, respectively.
PR&BO scheme: If a RA attempt fails in the mth time
slot, the IoT device will first automatically defer its
following RA attempts over next tBO time slots, and then
step up the transmit power of preamble for the deferred
packet in the RA attempt of (m+tBO + 1)th time slot.
C. SINR Expression
Different preambles represent orthogonal sub-channels,
such that only IoT devices choosing the same preamble have
correlations. The BS successfully decode a preamble when the
received SINR is above the threshold. Based on Slivnyak’s
theorem [34], we formulate the SINR of a typical BS located
at the origin in the mth time slot as
SINRm=ρh0
Im
intra +Im
inter +σ2,(1)
where ρis the full path-loss inversion power control threshold,
hois the channel power gain from the typical IoT device to
its associated BS, σ2is the noise power. Iintra and Iinter are
the aggregate intra-cell and inter-cell interference in the mth
time slot, which are represent as
Im
intra =X
uj∈Zin
1{Nm
Newj+Nm
Cumj>0}1{UR}ρhj,
Im
inter =X
ui∈Zout
1{Nm
Newi+Nm
Cumi>0}1{UR}Pihikuikα,(2)
where Zin is the set of intra-cell interfering IoT devices, Zout
is the set of inter-cell interfering IoT devices, Nm
Newjis the
numbers of new arrived packets in the mth time slot of jth
interfering IoT device, Nm
Cumjis the numbers of accumulated
packets in the mth time slot of jth interfering IoT device, 1{·}
is the indicator function that takes the value 1if the statement
1{·} is true, and zero otherwise, k·k is the Euclidean norm, ui
is the distance between the ith inter-cell IoT device and the
typical BS, Pi=ρriαis the actual transmit power of the ith
inter-cell IoT device with the distance from its associated BS.
In (1), 1{UR}presents that an IoT device generating in-
terference only when its RACH attempt is not restricted by
the RACH scheme (such as in the ACB scheme, generating
q > PACB leads to 1{U R}= 0), and 1{Nm
Newi+Nm
Cumi>0}
presents that only an IoT device with non-empty buffer gen-
erating interference. The queue status of an IoT device are
jointly populated by the new arrival packets (i.e., according to
Poisson arrival process ΛNew) and the accumulated packets in
the previous time slots. The evolution of queue status in each
IoT device has been detailed and analyzed in our previous
work [21, Section II.C and IV.A]. Briefly speaking, a packet
is removed from the buffer once it has been successfully
transmitted (step 1 of RA of that IoT device is successful),
otherwise, it will wait in the first place of the queue, and this
IoT device will reattempt to access the network in the next
available RA to transmit the packet. The main notations of
the proposed protocol are summarized in Table I.
TABLE I: Notation Table
Notations Physical means
λBThe intensity of BSs
λDThe intensity of IoT devices
ξThe number of available preambles *
λDp The average intensity of IoT devices using the same preamble
γth The received SINR threshold *
αThe path-loss exponent
hThe Rayleigh fading channel power gain
PThe transmit power *
ρThe full path-loss power control threshold *
σ2The noise power
rThe distance between an IoT device and its associated BS
uThe distance between an interfering IoT device and the typical
BS
Iinter The aggregate inter-cell interference
Iintra The aggregate intra-cell interference
τgThe gap interval duration
τcThe PRACH duration
εNew The intensity of new packets arrival
NThe number of interfering IoT devices in the typical cell
mThe time slot
c c = 3.575 is a constant
µm
Cum The intensity of accumulated packets in the mth time slot
µm
New The intensity of new arrival packets in the mth time slot
TmThe active probability of an IoT device in the mth time slot
BmThe non-BO probability of each IoT device in the mth time
slot with the BO scheme
κjThe power level unit in the jth RA attempt with the PR
scheme *
JThe maximum allowable power level with the PR scheme *
PACB The ACB factor with the ACB scheme *
tBO The BO factor with the BO scheme *
Remarks The variables marked with * are configurable parameters.
III. GENERAL SINGLE TIME SLOT MODEL
In this section, we provide a general single time slot
analytical model for all RA schemes. Note that in the 1st time
slot, the queue status (number of packets in buffer) of each
IoT device only depends on the new packets arrival process
Λ1
New. We perform the analysis on a BS associating with a
5
randomly chosen active IoT device in terms of the RA success
probability [23]. The RA success refers to the preamble being
successfully transmitted to the associated BS (i.e., received
SINR is greater than the SINR threshold) and no collision
occurs (i.e., no other IoT devices successfully transmits a same
preamble to the typical BS simultaneously). The probability
that the received SINR at a randomly chosen BS exceeds
a certain threshold γth has been studied in many stochastic
geometry works [24, 29, 35]. To the best of our knowledge,
there has been no work in the literature considered and
analyzed collision problem during RA via stochastic geometry
so far. The RA success probability P1is defined as
P1=
X
n1=0 P[N1=n1]
|{z }
I
P[SINRoγthN1=n1]
| {z }
II
n1
Y
i=1
P[SINRi< γthN1=n1]
| {z }
III
,(3)
where γth is the SINR threshold3,N1is the number of
intra-cell interfering IoT devices (i.e., transmitting the same
preamble as the typical IoT device simultaneously), SINRo
and SINRiare the received SINR of preamble from the
typical and the ith interfering IoT device following from (1),
I in (3) is the probability of N1number of interfering IoT
devices located in the typical BS, II in (3) represents the
preamble transmission success probability that the typical IoT
device successfully transmits the preamble to the associated
BS conditioning on N1=n1, and III in (3) represents the
preamble transmission failure probability that the preambles
transmitting from other n1intra-cell interfering IoT devices are
not successfully received by the BS conditioning on N1=n1.
The Probability Mass Function (PMF) of the number of
interfering IoT devices located in a Voronoi cell P[N1=n1]
is obtained as [35, Eq.(3)]
P[N1=n1] = c(c+1)Γ(n1+c+ 1)( b
T1λDp
λB)n1
Γ(c+ 1)Γ(n1+ 1)( b
T1λDp
λB+c)
n1+c+1 ,(4)
where c= 3.575 is a constant related to the approximate
Probability Mass function (PMF) of the PPP Voronoi cell [37],
Γ (·)is the gamma function, λDp is the density of IoT devices
using the same preamble, and
b
T1=(PACBT1,the ACB and ACB&BO scheme,
T1,the PR, BO, and PR&BO scheme.(5)
In (5), PACB is the ACB factor, T1is the active probability
of each IoT device in the 1st time slot (i.e., an IoT device has
one or more than one packets stored in the buffer waiting for
transmission), which is expressed as
T1=PN1
New >0= 1 eµ1
New .(6)
3The part III of Eq. (3) is not required for γth 0dB. Note that
determining the SINR threshold is based on characteristics of transceiver, such
as modulation scheme, coding technique, constellation size, matched filtering,
signal recovery technique, etc, and thus it is worth to study the network with
the SINR threshold in any regions [36].
Remind that in (6), µ1
New = (τc+τg)ε1
New, and N1
New is the
intensity of new arrival packets at each IoT device in the 1st
time slot.
Next, we derive the preamble transmission success prob-
ability presenting in II of (3). According to the Slivnyak’s
Theorem [34], the locations of inter-cell IoT devices follow
the Palm distribution of ΦDp, which is the same as the original
ΦDp. The probability that the received SINR at the BS from
a randomly chosen IoT device exceeds a certain threshold γth
conditioning on the given number of interfering IoT devices
in that cell n1is presented in following lemma.
Lemma 1. The probability that the received SINR at the BS
from a randomly chosen IoT device exceeds a certain threshold
γth conditioning on a given number of interfering IoT devices
in that cell n1is expressed as [21, Eq.(14)]
Pρho
Iintra +Iinter +σ2γthN1=n1
=Phoγth
ρ(Iintra +Iinter +σ2)N1=n1
(a)
=Ehexpγth
ρ(Iintra +Iinter +σ2)N1=n1i
= exp(γth
ρσ2)LIinter (γth
ρ)LIintra (γth
ρN1=n1),(7)
where the expectation in (a)is with respective to Iinter
and Iintra,LIintra (·)denotes the Laplace Transform of the
aggregate intra-cell interference Iintra, and LIinter(·)denotes
the Laplace Transform of the aggregate inter-cell interference
Iinter. In (7), the Laplace Transform of Iinter and Iintra were
derived in [21, Appendix A and B], are respectively given as
LIinter (γth
ρ) = exp2(γth)2
αb
T1λDp
λBZ
(γth)1
α
y
1 + yαdy,
LIintra (γth
ρN1=n1) = 1
(1 + γth)n1.(8)
Substituting (4) and (7) into (3), we derive the RA success
probability in the 1st time slot P1in the following theorem.
Theorem 1. In the depicted cellular-based mIoT network, the
RA success probability of a randomly chosen IoT device in the
1st time slot is derived as
P1=
X
n1=0 (c(c+1)Γ(n1+c+ 1)( b
T1λDp
λB)
n1
Γ(c+ 1)Γ(n1+ 1)( b
T1λDp
λB+c)
n1+c+1
| {z }
I
exp γthσ2
ρ2(γth)2
αb
T1λDp
λBR
(γth)1
α
y
1+yαdy
(1 + γth)n1
| {z }
II
1
exp γthσ2
ρ2(γth)2
αb
T1λDp
λBR
(γth)1
α
y
1+yαdy
(1 + γth)n1n1
| {z }
III
).
(9)
In (9), it can be shown that the preamble transmission
success probability of the typical IoT device is inversely
proportional to the received SINR threshold γth, and the
6
preamble transmission failure probabilities of other interfering
IoT devices are directly proportional to the received SINR
threshold γth, which leads to the fact that the non-collision
probability (i.e., the probability of a successful transmission
preamble does not collide with others) of the typical IoT
devices is also directly proportional to the received SINR
threshold γth. Therefore, a tradeoff between preamble trans-
mission success probability and non-collision probability is
observed. For illustration, the relationship among RA success
probability, the preamble tranmission success probability, and
the non-collision probability are shown in Fig. 1.
−20 −15 −10 −5 0 5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
gth
Probability
RA success probability (P )
Preamble transmission success
Non−collision
0
1
1
1
probability (P with III = 1)
probability (P with II = 1)
Fig. 1: Comparing RA success probability (P1), preamble transmission success proba-
bility (P1with III = 1), and non-collision probability (P1with II = 1). The parameters
are λB= 10 BS/km2,λD p = 100 IoT deivces/preamble/km2,ρ=90 dBm,
σ2=90 dBm, b
T1=T1, and the new packets arrival rate µ1
New = 0.1
packets/time slot).
IV. MULTIPLE TIME SLOTS MODEL
In this section, we analyze the RA success probability of the
cellular-based mIoT network in each time slot with different
RA schemes. Apart from the physical layer modelling in
the spatial domain based on stochastic geometry, the queue
evolution in the time domain is modelled and analyzed using
probability theory.
A. Power Ramping Scheme
Remind that the RA success probability with the PR scheme
in the 1st time slot P1
PR has been derived in (9), and the power
ramping only happens from the 2nd time slot. To derive the
RA success probability of each time slot, the main challenge
is evaluating the number of the active IoT devices transmitting
the same preamble with each power level unit in the typical
cell. Thus, we first focus on deriving the PMF of the number
of interfering IoT devices transmitting with each power level
unit.
1) PMF of the Number of Interfering IoT Devices: We first
denote the jth power level unit as κj(j[1, J], where J
is the maximum allowable power level), and the number of
interfering IoT devices transmitting the same preamble with
the power level unit κjbeing located in the same Voronoi
cell with the typical IoT device is denoted as Nj. The active
probability of IoT devices transmitting with the power level
units κjis denoted as TPRj. Note that the active probabilities
with different power level units are derived based on iteration
process, which will be represented in (25).
We assume the typical IoT device is transmitting with the
power level unit κ1with N1number of interfering IoT devices
transmitting with the same power level unit κ1(i.e., the total
number of IoT devices transmitting with the power level unit
κ1is N1+ 1 in this typical cell). To derive the PMF of
N2number of IoT devices transmitting with power level unit
κ2conditioning on N1number of interfering IoT devices
transmitting with power level unit κ1in the same cell, we
need to first obtain the Probability Density Function (PDF) of
the area size of the Voronoi cell conditioning on N1number
of interfering IoT devices transmitting with the power level
unit κ1located in such cell, which is derived in the following
Lemma.
Lemma 2. The PDF of the size of the Voronoi cell condition-
ing on N1number of interfering IoT devices transmitting with
the power level unit κ1is derived as
P[X=x|N1=n1] =
(x)n1+ce(TPR1λDp+λBc)x(TPR1λD p +B)n1+c+1
Γ (n1+c+ 1) ,
(10)
where xis the area size of the cell, c= 3.757 is a constant
related to the approximate PMF of the PPP Voronoi cell, and
TPR1is the active probability of IoT devices transmitting
with the power level unit κ1that will be derived in (25).
Proof. See Appendix A.
Next, we derive the PMF of N2number of IoT devices
transmitting with the power level unit κ2conditioning on
the number of interfering IoT devices transmitting with the
power level unit κ1in the typical Voronoi cell N1=n1in the
following theorem.
Theorem 2. The PMF of N2number of IoT devices transmit-
ting with the power level unit κ2in a Voronoi cell conditioning
on the number of interfering IoT devices transmitting with the
power level unit N1=n1in the same cell is derived as
P[N2=n2|N1=n1] = Γ (n1+n2+c+ 1)
Γ(n2+ 1) Γ(n1+c+ 1) ×
(TPR2λDp)n2(TPR1λD p +B)n1+c+1
(TPR1λDp +TPR2λDp +λBc)n1+n2+c+1 ,(11)
where TPR2is the active probability of IoT devices transmit-
ting with the power level unit κ2(i.e., TPR2will be derived
in (25)).
Proof. See Appendix B.
For more than two levels PR scheme (J > 2), the PMF
of Nj(j= 3,4,· · · , J ) number of active IoT devices
transmitting with the power level units κj(j= 3,4,· · · , J) in
the Voronoi cell can be derived based on the iteration process
following Lemma 2 and Theorem 2. Thus, we derive the
PMF of Njnumber of active IoT devices transmitting with
the power level unit κjconditioning on the known number
of IoT devices with other power levels N1=n1, N2=
n2,· · · , Nj1=nj1in the following proposition.
Proposition 1. The PMF of Njnumber of IoT devices
transmitting with the power level unit κjin a Voronoi cell
conditioning on number of IoT devices with other power levels
7
Pm
PRl=
X
n1=0
X
n2=0
· · ·
X
nJ=0 (P[Nl=nl]
J
Y
j=1,j6=lP[Nj=nj|Nl=nl, N1=n1,· · · , Nj1=nj1]
| {z }
I
Phκlρho
J
P
i=1 Im
interi+Im
intrai+σ2
γthN1=n1,· · · , NJ=nJi
| {z }
II
J
Y
j=1 Phκjρho
J
P
i=1 Im
interi+Im
intrai+σ2
< γthN1=n1,· · · , NJ=nJinj
| {z }
III
),(13)
N1=n1, N2=n2,· · · , Nj1=nj1and the typical IoT
device transmitting with the power level unit κ1is
P[Nj=nj|N1=n1, N2=n2,· · · , Nj1=nj1] =
Γ j
P
i=1
ni+c+ 1
Γ (nj+ 1) Γ j1
P
i=1
ni+c+ 1×
TPRjλDpnjj1
P
i=1
TPRiλDp +Bj1
P
i=1
ni+c+1
j
P
i=1
TPRiλDp +λBcj
P
i=1
ni+c+1
,
(12)
where TPRjis the active probability of IoT devices transmit-
ting with the power level unit κj(i.e., TPRjwill be derived
in (25)).
2) RA Success Probability: In the PR scheme, we assume
the maximum allowable power level unit is κJ. Based on the
PMF of the number of IoT devices transmitting with each
power level unit, we can derive the RA success probability
of the typical IoT device with the lth power level unit κlin
the mth time slot Pm
PRl(l[1, J ]), where the IoT device
transmits preamble with the lth power level unit κlafter it
fails in RA for l1times. The RA success probability of
the IoT device transmits preamble with the lth power level
unit κlin the mth time slot Pm
PRlis derived as Eq. (13).
In Eq. (13), Jis the maximum allowable power level, and
Im
interiand Im
intraidenote the aggregate inter-cell and intra-cell
interference generating by IoT devices transmitting with the
ith level power unit κi, respectively. I in (13) consists of the
probabilities that the numbers of IoT devices transmitting with
the power level units (κ1, κ2,· · · , κJ)conditioning on the
typical device transmitting with the lth power level unit κland
N1=n1, N2=n2,· · · , NJ=nJ, II in (13) represents the
preamble transmission success probability that the typical IoT
device successfully transmits the preamble to the associated
BS conditioning on N1=n1, N2=n2,· · · , NJ=nJ,
and III in (13) represents the preamble transmission success
probabilities that the preambles transmitting from all other
intra-cell interfering IoT devices are not successfully received
by the BS conditioning on N1=n1, N2=n2,· · · , NJ=nJ.
Next, we present the RA success probability of a randomly
chosen IoT device with multiple levels PR scheme (i.e., the
maximum allowable power level unit is κJ(J2)) in the
mth time slot in the next theorem.
Theorem 3. The RA success probability of a randomly chosen
IoT device (i.e. each active IoT device transmitting preamble
with any power level unit is fairly chosen) in the mth time
slot is derived as
Pm
PR,all =J
X
i=1
Tm
PRiPm
PRi.Tm
PR,all,(14)
where Jis the maximum allowable power level, the RA success
probability of IoT devices transmitting with the power level
unit κl(l[1, J ]) in the mth time slot is derived as
Pm
PRl=
X
n1=0
X
n2=0
· · ·
X
nJ=0 Θ(m, l, l, ~
n)
J
Y
j=1
Ω(m, l, j, ~
n)
1Θ(m, l, j, ~
n)nj.
(15)
In (15), ~
n={n1,· · · , nJ}, the probability that the number of
interfering IoT devices transmitting with the power level unit
κlis derived as
Ω(m, l, l, ~
n) = c(c+1)Γ(nl+c+ 1)( Tm
PRlλDp
λB)
nl
Γ(c+ 1)Γ(nl+ 1)(Tm
PRlλDp
λB+c)
nl+c+1 ,
(16)
the probability that the number of IoT devices transmitting
with the power level unit κj(when j6=l) conditioning on the
typical device transmitting with the power level unit κland
Nl=nl, N1=n1, N2=n2,· · · , Nj1=nj1, is derived
8
as
Ω(m, l, j, ~
n) =
Γnl+ (
j
P
i=1,i6=l
ni) + c+ 1(TPRjλDp)nj
Γ(nj+ 1)Γnl+
j1
P
i=1,i6=l
ni) + c+ 1
×TPRl+
j1
P
i=1,i6=l
TPRiλDp +Bnl+(
j1
P
i=1,i6=l
ni)+c+1
TPRl+
j
P
i=1,i6=l
TPRiλDp +Bnl+(
j
P
i=1,i6=l
ni)+c+1
,
(17)
the preamble transmission success probability that the received
SINR from an IoT device transmitting with the power level unit
κlexceeds the certain threshold γth are derived as
Θ(m, l, l, ~
n) = exp γth σ2
κlρ2λDp(γth )2
α
λBJ
X
i=1
(κi
κl
)2
α×
Tm
PRiZ
(γth
κi
κl)1
α
y
1 + yαdyJ
Y
i=1
(1 + γth
κi
κl
)ni,
(18)
and when j6=l, the preamble transmission success probability
of an IoT device transmitting with the power level unit κjis
Θ(m, l, j, ~
n) = exp γth σ2
κjρ2λDp(γth )2
α
λBJ
X
i=1
(κi
κj
)2
α×
Tm
PRiZ
(γth
κi
κj)1
α
y
1 + yαdy
(1 + γth
κl
κj
)nl+1(1 + γth )nj1
J
Y
i=1,i6=l,j
(1 + γth
κi
κj
)ni.
(19)
Note that TPRiis derived based on iteration process, which
will be given in (25).
Proof. The preamble transmission success probability of an
IoT device transmitting with the power level unit κjis
represented as
Θ(m, l, j, ~
n) =
Pnκjρho
J
P
i=1 Im
interi+Im
intrai+σ2
γthN1=n1,· · ·,NJ=nJo
= expγth
κjρσ2J
Y
i=1 LIm
interi(γth
κjρ)LIm
intrai(γth
κjρNi=ni),
(20)
where LIm
intrai(·)and LIm
interi(·)denote the Laplace Transform
of the PDF of the aggregate intra-cell interference Iintraiand
inter-cell interference Iinterigenerating from the IoT devices
transmitting with power level unit κi. The Laplace Transform
of aggregate inter-cell interference from IoT devices transmit-
ting with power level unit κireceived at the typical BS is
derived as
LIm
interi(s)
(a)
=Eb
Zout hY
ukb
Zout
EPkEhkeiPkhkkukkαi
(b)
=exp 2πTm
P R,κiλDp Z
(P/κiρ)1
α
EPEh1eiP hxαxdx
(c)
=exp 2πTm
P R,κiλDp(κis)2
αEP[P2
α]Z
(iρ)1
α
y
1 + yαdy,
(21)
where s=γth
κjρ,Ex[·]is the expectation with respect to the
random variable x,Tm
P R,κiis the active probability of IoT
device transmitting with ith power level unit κiin the mth
time slot, (a) follows from independence between λDp,Pk,
and hk, (b) follows from the probability generation functional
(PGFL) of the PPP, (c) obtained by changing the variables
y=x
(sP )1
η
, and the moments of the transmit power EP[·]was
presented in [21, Eq. A.2]. Substituting the moments of the
transmit power into (21), we derive the Laplace Transform of
aggregate inter-cell interference.
The Laplace Transform of aggregate intra-cell interference
from IoT devices transmitting with the power level unit κi
received at the typical BS is derived as
LIm
intrai(sNi=ni) = Ehkhexp s
ni
X
k=1
κiρhki
=1
1 + iρni(22)
where niis the number of interfering IoT devices transmitting
with the power level unit κi.
The RA success probabilities are derived based on the
iteration process. We assume mis a variable that denotes the
time slot from 2 to M. The iteration process for calculating
the RA success probability in the Mth time slot PM
PR,all is
shown in Fig. 2. Details of this process are described by the
following:
Step 1: Calculate the RA success probability in the 1st
time slot P1
PR1in (7) based on the known intensity
of the new arrival packets µ1
New in (6) (i.e., the power
ramping is not executed in the 1st time slot);
Step 2: Calculate the intensity of accumulated packets
µm
Cum,PR in the mth time slot via Poisson approximation
queue status analysis approach, which is given in our pre-
vious work [21, Section IV.A]. The intensity of number
of accumulated packets in the mth time slot µm
Cum,P R is
µm
Cum,PR =µm1
New +µm1
Cum,PR
J
X
i=1
Tm1
PRiPm1
PRi;
(23)
Step 3: Calculate the active probability of each IoT device
in the mth time slot Tm
PR,all using
Tm
PR,all = 1 eµm
Newµm
Cum,PR ;(24)
9
Step 4: Calculate the active probability of each IoT device
transmitting with the power level unit κi(i(1, J)) in
the mth time slot Tm
PRiusing
Tm
PRi=
Tm
PR,all
J
X
i=1
Tm1
PRi1− Pm1
PRi, i = 1,
1− Pm1
PRi1Tm1
PRi1, i 6= 1, i 6=J,
1− Pm1
PRi1Tm1
PRi1
+1− Pm1
PRiTm1
PRi, i =J,
(25)
where Pm1
PRiis the RA success probability of the IoT
device transmitting with the power level unit κiin the
(m1)th time slot given in (15);
Step 5: Calculate the RA success probabilities of IoT
devices transmitting with power level unit κl(l=
1,2,· · · , J ) in the mth time slot Pm
PRlusing (15);
Step 6: Calculate the RA success probability Pm
PR,all
using (14).
Repeating the step 2 to 6 until m=M, the RA success
probability in the Mth time slot PM
PR,all is obtained.
of the transmit power EP[·]was presented in [21, Eq. A.2]. Substituting the moments of the
transmit power into (20), we derive the Laplace Transform of aggregate inter-cell interference.
The Laplace Transform of aggregate intra-cell interference from IoT devices transmitting with
the power level unit κireceived at the typical BS is derived as
LIm
intrai(sNi=ni) = Ehkhexp s
ni
X
k=1
κiρhki=1
1 + iρni(21)
where niis the number of interfering IoT devices transmitting with the power level unit κi.
The RA success probabilities are derived based on the iteration process. We assume mis a
variable that denotes the time slot from 2 to M. The iteration process for calculating the RA
success probability in the Mth time slot PM
PR,all is shown in Fig. 1. Details of this process are
described by the following:
Step 1: Calculate P1
PR1in (7).
Step 2: Calculate µm
Cum,PR in (23).
Step 3: Calculate Tm
PR,all in (24).
Step 4: Calculate Tm
PR1,Tm
PR2,· · · ,Tm
PRJin (25).
Step 5: Calculate Pm
PR1,Pm
PR2,· · · ,Pm
PRJin (15).
Step 6: Calculate Pm
PR,all in (14).
m=M? m=m+1.
end.
m= 2
yes
no
Fig. 2: Flowchart for deriving the RA success probability in the Mth time slot with the PR scheme PM
PR,all.
Step 1: Calculate the RA success probability in the 1st time slot P1
PR1in (6) based on
the known intensity of the new arrival packets µ1
New in (5) (i.e., the power ramping is not
executed in the 1st time slot);
Step 2: Calculate the intensity of accumulated packets µm
Cum,PR in the mth time slot via
Poisson approximation queue status analysis approach, which is given in our previous work
[21, Section IV.A]. The intensity of number of accumulated packets in the mth time slot
µm
Cum,P R is µm
Cum,PR =µm1
New +µm1
Cum,PR
J
X
i=1
Tm1
PRiPm1
PRi;(22)
Fig. 2: RA success probability in each time slot with five RA schemes.
For the purpose of simplicity, we provide a special case
of the PR scheme, where each IoT device can step up the
preamble transmit power for only one time (i.e., the maximum
allowable power level J= 2), and the path-loss exponent is
set as α= 4 (i.e., close-formed expression is obtained). Next,
we present the overall RA success probability of a randomly
chosen IoT device in the mth time slot in the following
proposition.
Fig. 4 plots the RA success probabilities with the PR
scheme at the 10th time slot P10
PR,all versus the density ratio
between IoT devices transmitting the same preamble and BSs
λDpB. We study the geometric PR scheme, where the
transmit power steps up following the policy κl=gl1(i.e.,
gis a constant denoting the root of power increase, lis the
current power level, and lJ, where Jis the maximum
power level), and its effectiveness has been shown in [13].
We compare the PR schemes with the maximum power level
J= 5 and J= 2, where we set g= 2 for J= 5
(κ1,· · · , κ5= 1,2,4,8,16) and g= 2,4,8for J= 2
(κ1= 1 and κ2= 2,4,8). We observe that for J= 2,
the RA success probabilities follow P10
PR,all(J= 2, g =
8) >P10
PR,all(J= 2, g = 4) >P10
PR,all(J= 2, g = 2),
due to that increasing gresults in higher received SINR
of reattempt access and lower collision probability. We also
notice that P10
PR,all(J= 5, g = 2) performs worse than
P10
PR,all(J= 2, g = 8) before a certain density ratio, due
to that in the low density ratio region, the network condition
prefers large power gap, as this is effective in improving the
received SINR of reattempt access and reducing the collision
probability (i.e., most packets only suffer from little times of
RA fails leading to that IoT devices always use small power
level unit to transmit preambles). After that density ratio,
P10
PR,all(J= 5, g = 2) surpasses P10
PR,all(J= 2, g = 8),
due to that in the high density ratio region, the case with
J= 5 and g= 2 (κ1,· · · , κ5= 1,2,4,8,16) has relatively
smooth increase in power that decreases the high aggregate
interference.
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Dp /B
Sim.P10
PR,all (J=2,g=2)
Sim.P10
PR,all (J=2,g=4)
Sim.P10
PR,all (J=2,g=8)
Sim.P10
PR,all (J=5,g=2)
RA Success Probability
Ana.
ll
Fig. 3: RA success probability in the 10th time slot with the PR scheme. We present
4 scenarios with different parameters of the PR scheme. The simulation parameters are
λB= 10 BS/km2,γth = 1 (0dB), ρ=90 dBm, σ2=90 dBm, and µ1
New =
µ2
New =···=µm
New = 0.1packets/time slot.
Proposition 2. The RA success probability of a randomly
chosen IoT device with the PR scheme (α= 4, J = 2) in
the mth time slot is derived as
Pm
PR,all =Tm
PR1Pm
PR1+Tm
PR2Pm
PR2.Tm
PR,all.(26)
In (26), the RA success probability of a randomly chosen IoT
device transmitting with the power level unit κ1in the mth
time slot Pm
PR1is derived as
Pm
PR1=
X
n1=0
X
n2=0 Θ(m, 1,1,~
n)
2
Y
j=1
Ω(m, 1, j, ~
n)
1Θ(m, 1, j, ~
n)nj,(27)
with ~
n={n1, n2}, the RA success probability of a randomly
chosen IoT device transmitting with the power level unit κ2
in the mth time slot Pm
PR2is derived as
Pm
PR2=
X
n1=0
X
n2=0 Θ(m, 2,2,~
n)
2
Y
j=1
Ω(m, 2, j, ~
n)
1Θ(m, 2, j, ~
n)nj.(28)
10
In (27) and (28), the probabilities that the numbers of IoT de-
vices transmitting with different power level unit conditioning
on N1=n1, N2=n2are derived as
Ω(m, 1,1,~
n) = c(c+1)Γ(n1+c+ 1)( Tm
PR1λDp
λB)
n1
Γ(c+ 1)Γ(n1+ 1)(Tm
PR1λDp
λB+c)
n1+c+1 ,
(29)
Ω(m, 2,2,~
n) = c(c+1)Γ(n2+c+ 1)( Tm
PR2λDp
λB)
n2
Γ(c+ 1)Γ(n2+ 1)(Tm
PR2λDp
λB+c)
n2+c+1 ,
(30)
Ω(m, 1,2,~
n) = Γ (n1+n2+c+ 1)
Γ (n2+ 1) Γn1+c+ 1×
Tm
PR2λDpn2Tm
PR1λDP +Bn1+c+1
Tm
PR1+Tm
PR2λDP +Bn1+n2+c+1 ,
(31)
Ω(m, 2,1,~
n) = Γ (n1+n2+c+ 1)
Γ (n1+ 1) Γn2+c+ 1×
Tm
PR1λDpn1Tm
PR2λDP +Bn2+c+1
Tm
PR1+Tm
PR2λDP +Bn1+n2+c+1 ,
(32)
and the probabilities that the received SINRs at the BS exceeds
the certain threshold γth are derived as
Θ(m, 1,1,~
n) =
exp γthσ2
κ1ρ2λDpγth
λB2
P
i=1 qκi
κ1Tm
PRiarctg(qκi
κ1γth)
(1 + γth)n1(1 + γth κ2
κ1)n2,
(33)
Θ(m, 1,2,~
n) =
exp γthσ2
κ2ρ2λDpγth
λB2
P
i=1 qκi
κ2Tm
PRiarctg(qκi
κ2γth)
(1 + γth κ1
κ2)n1+1(1 + γth )n21,
(34)
Θ(m, 2,1,~
n) =
exp γthσ2
κ1ρ2λDpγth
λB2
P
i=1 qκi
κ1Tm
PRiarctg(qκi
κ1γth)
(1 + γth)n11(1 + γth κ2
κ1)n2+1 ,
(35)
Θ(m, 2,2,~
n) =
exp γthσ2
κ2ρ2λDpγth
λB2
P
i=1 qκi
κ2Tm
PRiarctg(qκi
κ2γth)
(1 + γth κ1
κ2)n1(1 + γth)n2.
(36)
Generally, the power level unit κj(when j > 1) and the
maximum allowable power level Jare the major factors in
determining the RA success probability of the PR scheme, due
to it determines the interference generated by the IoT devices
with large transmitting power. More specifically, it can be
shown in the special case of J= 2, the preamble transmission
success probabilities of IoT devices transmitting with κ2
(Θ(m, 1,2,~
n)and Θ(m, 2,2,~
n)) are directly proportional to
κ2, and these probabilities of IoT devices transmitting with
κ1(Θ(m, 1,1,~
n)and Θ(m, 2,1,~
n)) are inversely proportional
to κ2. This could be concluded that κ2introduces a tradeoff
between the performances of IoT devices transmitting with κ1
and κ2. Obviously, this special case is practical and easy to
employ to IoT devices. Furthermore, a proper κ2guarantees
large overall RA success probability Pm
PR,all, that is, less
retransmissions. However, maintaining a proper κ2is really
difficult in a complex mIoT network system with dynamic
traffic, which may result in two unexpected consequences: 1)
A relatively small power increment leads to a high outage
probability; 2) A relatively large power increment causes
serious power consumption in each retransmitting IoT device,
and large mutual interference among IoT devices.
To solve this problem, the multi-level PR scheme (J >
2) has been studied [13], where the transmit power steps up
following specific policies. This approach offers IoT devices
finding their necessary transmitting power level by a number of
attempts. Generally, this approach avoids the two unexpected
consequences, but IoT devices may suffer from large delay as
they attempt many power increments until a success preamble
transmitting.
B. Hybrid Access Class Barring and Back-Off Scheme
In the ACB&BO scheme, the BS first broadcasts the ACB
factor PACB, then each active IoT device attempts a RA
with probability PACB or defers this RA with probability
(1PACB). If a RA fails, the back-off mechanism is executed,
where the IoT device defers its access request and waits for
tBO time slots. The RA success probability of a randomly
chosen IoT device with the ACB&BO scheme in the mth time
slot is presented in the following Theorem.
Theorem 4. The RA success probability of a randomly chosen
IoT device with the ACB&BO scheme in the mth time slot is
derived as
Pm
ACB&BO =
X
n1=0 Ω(n1, mACB&BO(n1, m)
1ΘACB&BO(n1, m)n1,(37)
where the probability of the number of interfering IoT devices
in the typical cell is derived as
Ω(n1, m) =
c(c+1)Γ(n1+c+ 1)( BmPACBTm
ACB&BOλDp
λB)n1
Γ(c+ 1)Γ(n1+ 1)(BmPACBTm
ACB&BOλDp
λB+c)n1+c+1 ,(38)
and the preamble transmission success probabilities that the
received SINR exceeds the certain threshold γth is derived as
ΘACB&BO(n1, m) =
exp γthσ2
ρ2(γth)2
αBmPACBTm
ACB&BOλDp
λBR
(γth)1
α
y
1+yαdy
(1 + γth)n1.
(39)
11
Proof. As the flowchart in Fig. 2, the detailed process of
calculating the RA success probability with the ACB&BO in
the Mth time slot PM
ACB&BO are described in the following:
Proof. As the flowchart in Fig. 2, the detailed process of
calculating the RA success probability with the ACB&BO in
the Mth time slot PM
ACB&BO are described in the following:
Step 1: Calculate the RA success probability in the 1st
time slot P1
ACB&BO,1using (7);
Step 2: Calculate the intensity of accumulated packets
µm
Cum,ACB&BO in the mth time slot using
µm
Cum,ACB&BO =µm1
New +µm1
Cum,ACB&BO
Bm1PACBPm1
ACB&BOTm1
ACB&BO; (39)
Step 3: Calculate the active probability4in the mth time
slot Tm
ACB&BO using
Tm
ACB&BO = 1 eµm
Newµm
Cum,ACB&BO; (40)
Step 4: In the back-off mechanism, each IoT device
fails to RA in the last tBO time slots will not allow to
transmit a preamble in the current mth time slot, which is
clearly introduced and analyzed in [21, Eq.(43)]. Briefly
speaking, we calculate the probability of a packet that is
not blocked in the buffer of IoT device by the back-off
mechanism in the mth time slot Bmusing
Bm=
1m1
X
j=1
(1Pmj
ACB&BO)PACBTmj
ACB&BOBmj/Tm
ACB&BO, m (tBO+1) ,
1tBO
X
j=1
(1Pmj
ACB&BO)PACBTmj
ACB&BOBmj/Tm
ACB&BO, m>tBO ;
(41)
Step 5: Calculate the RA success probability in the mth
time slot Pm
ACB&BO using (36);
Repeating the step 2 to 5 till m=M, the RA success
probability in the Mth time slot PM
ACB&BO is obtained.
of the transmit power EP[·]was presented in [21, Eq. A.2]. Substituting the moments of the
transmit power into (20), we derive the Laplace Transform of aggregate inter-cell interference.
The Laplace Transform of aggregate intra-cell interference from IoT devices transmitting with
the power level unit κireceived at the typical BS is derived as
LIm
intrai(sNi=ni) = Ehkhexp s
ni
X
k=1
κiρhki=1
1 + iρni(21)
where niis the number of interfering IoT devices transmitting with the power level unit κi.
The RA success probabilities are derived based on the iteration process. We assume mis a
variable that denotes the time slot from 2 to M. The iteration process for calculating the RA
success probability in the Mth time slot PM
PR,all is shown in Fig. 1. Details of this process are
described by the following:
Step 1: Calculate P1
PR1in (8).
Step 2: Calculate µm
Cum,PR in (22).
Step 3: Calculate Tm
PR,all in (23).
Step 4: Calculate Tm
PR1,Tm
PR2,· · · ,Tm
PRJin (24).
Step 5: Calculate Pm
PR1,Pm
PR2,· · · ,Pm
PRJin (14).
Step 6: Calculate Pm
PR,all in (13).
m=M? m=m+1.
end.
m= 2
yes
no
Fig. 2: Flowchart for deriving the RA success probability in the Mth time slot with the PR scheme PM
PR,all.
Step 1: Calculate the RA success probability in the 1st time slot P1
PR1in (6) based on
the known intensity of the new arrival packets µ1
New in (5) (i.e., the power ramping is not
executed in the 1st time slot);
Step 2: Calculate the intensity of accumulated packets µm
Cum,PR in the mth time slot via
Poisson approximation queue status analysis approach, which is given in our previous work
[21, Section IV.A]. The intensity of number of accumulated packets in the mth time slot
µm
Cum,P R is µm
Cum,PR =µm1
New +µm1
Cum,PR
J
X
i=1
Tm1
PRiPm1
PRi;(22)
Fig. 4: RA success probability in each time slot with five RA schemes.
It is important to know that the analytical results of the
ACB&BO scheme in Theorem 4 reduces to that of the ACB
scheme by setting the back-off factor tBO = 0, and reduces to
that of the BO scheme by setting the ACB factor PACB = 1.
4IoT devices may remain the radio resource control (RRC) connection with
the associated BS for a while after the RA procedure succeeded, where they do
not need to initiate RA again when new packets arrive within this duration. To
focus on studying the feature of RA schemes in the spatio-temporal model,
we assume that IoT devices release RRC connection immediately after the
packet transmission as [7, 10, 17, 19].
Step 1: Calculate P1
ACB&BO in (7).
Step 2: Calculate µm
Cum,ACB&BO in (40).
Step 3: Calculate Tm
ACB&BO in (41).
Step 4: Calculate Bm in (42).
Step 5: Calculate Pm
ACB&BO in (37).
m=M? m=m+1.
end.
m= 2
yes
no
Fig. 5: Flowchart for deriving the RA success probability in the Mth time slot with the
ACB&BO scheme PM
ACB&BO.
C. Hybrid Power Ramping and Back-Off Scheme
In the PR&BO scheme, we limit ourselves to two levels PR
policy (J= 2) with the back-off factor tBO. In detail, if the
RA fails, the IoT device defers the current RA and waits for
tBO time slots, after that the IoT device reattempt the RA by
transmitting preamble with the 2nd power level unit κ2. When
m<tBO + 2, the power ramping mechanism is not executed,
and each IoT device requests access with the BO scheme (i.e.,
IoT devices fails in RA in the 1st time slot will wait for tBO
time slot, and then reattempt RA transmitting the preamble
with power level unit κ2in the (tBO + 2)th time slot), where
the RA success probability is derived as (36) in Theorem 4
by setting the ACB factor as PACB = 1. When mtBO + 2,
the RA success probability of a randomly chosen IoT device
with the PR&BO scheme in the mth time slot is derived in
the following proposition.
Proposition 3. The RA success probability of a randomly
chosen IoT device with the PR&BO scheme (J= 2) in the
mth time slot is derived as
Pm
PR&BO,all =Tm
PR&BO1Pm
PR&BO1+Tm
PR&BO2Pm
PR&BO2
Tm
PR&BO,all
.
(42)
Proof. As the flowchart in Fig. 1, the details of the process to
calculate the RA success probability with the PR&BO scheme
(J= 2) in the mth time slot (m6= 1)Pm
PR&BO,all are
described by the following:
Step 1: Calculate the RA success probability in the 1st
time slot P1
PR&BO1using (7);
Step 2: Calculate the intensity of accumulated packets
µM
Cum,PR&BO in the mth time slot using
µm
Cum,PR&BO =
µm1
New +µm1
Cum,PR&BO − T m1
PR&BO1Pm1
PR&BO1,1< m < tBO + 2,
µm1
New +µm1
Cum,PR&BO
2
X
i=1
Tm1
PR&BOiPm1
PR&BOi, m tBO + 2;
(43)
Fig. 4: Flowchart for deriving the RA success probability in the Mth time slot with the
ACB&BO scheme PM
ACB&BO.
Step 1: Calculate the RA success probability in the 1st
time slot P1
ACB&BO,1using (7);
Step 2: Calculate the intensity of accumulated packets
µm
Cum,ACB&BO in the mth time slot using
µm
Cum,ACB&BO =µm1
New +µm1
Cum,ACB&BO
Bm1PACBPm1
ACB&BOTm1
ACB&BO;
(40)
Step 3: Calculate the active probability4in the mth time
slot Tm
ACB&BO using
Tm
ACB&BO = 1 eµm
Newµm
Cum,ACB&BO ; (41)
Step 4: In the back-off mechanism, each IoT device
fails to RA in the last tBO time slots will not allow to
transmit a preamble in the current mth time slot, which is
clearly introduced and analyzed in [21, Eq.(43)]. Briefly
speaking, we calculate the probability of a packet that is
not blocked in the buffer of IoT device by the back-off
mechanism in the mth time slot Bmusing
Bm=
1
m1
P
j=1
(1P mj
ACB&BO)PACBTmj
ACB&BOBmj
Tm
ACB&BO
,
m(tBO+1) ,
1
tBO
P
j=1
(1P mj
ACB&BO)PACBTmj
ACB&BOBmj
Tm
ACB&BO
m>tBO ;
(42)
4IoT devices may remain the radio resource control (RRC) connection with
the associated BS for a while after the RA procedure succeeded, where they do
not need to initiate RA again when new packets arrive within this duration. To
focus on studying the feature of RA schemes in the spatio-temporal model,
we assume that IoT devices release RRC connection immediately after the
packet transmission as [7, 10, 17, 19].
Step 5: Calculate the RA success probability in the mth
time slot Pm
ACB&BO using (37);
Repeating the step 2 to 5 till m=M, the RA success
probability in the Mth time slot PM
ACB&BO is obtained.
It is important to know that the analytical results of the
ACB&BO scheme in Theorem 4 reduces to that of the ACB
scheme by setting the back-off factor tBO = 0, and reduces to
that of the BO scheme by setting the ACB factor PACB = 1.
C. Hybrid Power Ramping and Back-Off Scheme
In the PR&BO scheme, we limit ourselves to two levels PR
policy (J= 2) with the back-off factor tBO. In detail, if the
RA fails, the IoT device defers the current RA and waits for
tBO time slots, after that the IoT device reattempt the RA by
transmitting preamble with the 2nd power level unit κ2. When
m<tBO + 2, the power ramping mechanism is not executed,
and each IoT device requests access with the BO scheme (i.e.,
IoT devices fails in RA in the 1st time slot will wait for tBO
time slot, and then reattempt RA transmitting the preamble
with power level unit κ2in the (tBO + 2)th time slot), where
the RA success probability is derived as (37) in Theorem 4
by setting the ACB factor as PACB = 1. When mtBO + 2,
the RA success probability of a randomly chosen IoT device
with the PR&BO scheme in the mth time slot is derived in
the following proposition.
Proposition 3. The RA success probability of a randomly
chosen IoT device with the PR&BO scheme (J= 2) in the
mth time slot is derived as
Pm
PR&BO,all =
Tm
PR&BO1Pm
PR&BO1+Tm
PR&BO2Pm
PR&BO2
Tm
PR&BO,all
.(43)
Proof. As the flowchart in Fig. 1, the details of the process to
calculate the RA success probability with the PR&BO scheme
(J= 2) in the mth time slot (m6= 1)Pm
PR&BO,all are
described by the following:
Step 1: Calculate the RA success probability in the 1st
time slot P1
PR&BO1using (7);
Step 2: Calculate the intensity of accumulated packets
µM
Cum,PR&BO in the mth time slot using
µm
Cum,PR&BO =
µm1
New +µm1
Cum,PR&BO − T m1
PR&BO1Pm1
PR&BO1,
1< m < tBO + 2,
µm1
New +µm1
Cum,PR&BO
2
X
i=1
Tm1
PR&BOiPm1
PR&BOi,
mtBO + 2;
(44)
Step 3: Calculate the active probability in the mth time
slot Tm
PR&BO,all using
Tm
PR&BO,all = 1 exp(µm
New µm1
Cum,PR&BO); (45)
12
Step 4: Calculate the active probability Tm
PR&BO1and
Tm
PR&BO2using
Tm
PR&BO1=Tm
PR&BO,all
2
X
i=1
tBO +1
X
t=1
Tmt
PR&BOi1− Pmt
PR&BOi,
(46)
Tm
PR&BO2=
2
X
i=1
Tm1tBO
PR&BOi1− Pm1tBO
PR&BOi; (47)
Step 5: Calculate the RA success probability in the mth
time slot Pm
PR&BO1and Pm
PR&BO2using (27) and (28)
with Tm
PR&BOigiven in (46) and (47);
Step 6: Calculate the overall RA success probability in the
mth time slot Pm
PR&BO,all using (43) with Pm
PR&BOi.
Repeating the step 2 to 6 till m=M, the RA success
probability in the Mth time slot PM
PR&BO,all is obtained.
V. AVERAGE QUEUE LENGTH AND AVERAGE
WAITING DELAY
The works on RA has been mainly focused on minimizing
the failure probabilities and the service delays [6, 8]. The RA
success probability provides insights on the probability of
access for a random IoT device in each time slot, but does not
evaluate the packets accumulation status and the packets delay
over all the time slots. Many previous works have indicated
that the queue length and waiting delay are the good indication
of network congestion [4, 6, 38]. The queue length refers to the
number of packets that are waiting in buffer to be transmitted,
and the waiting delay is the duration of the time between when
a packet arrives and leaves the buffer, respectively [39].
Next, we evaluate the average queue length E[Qm]and
the average waiting delay E[Dm]. The average queue length5
denotes the average number of packets accumulated in the
buffer in the mth time slot, which is measured by mean
average the queue over all IoT devices in the network [39].
The average waiting delay6is defined as the average time slots
spent in the queue of each packet, which is measured by mean
average the waiting time over all transmitted packets between
the 1st and the mth time slot in the network [39]. Note that
there are always a number of packets being accumulated in
buffers in the mth time slot (i.e., fail to access, or still in
the queue and never been serviced before the mth time slot),
and we assume the waiting delay of these packets is the time
elapsed from the packets start to wait in the buffer to the mth
time slot. The average queue length and the average waiting
delay of each packet with the PR scheme over mtime slots
are derived as
E[Qm
PR] = µm
New +µm
Cum,PR
J
X
i=1
Tm
PRiPm
PRi,(48)
5The average queue length is looking at the average in space in a specific
time slot.
6The average waiting delay is looking at the average both in space and
time over a period of time slots.
and
E[Dm
PR] = m
X
t=1
E[Qt
PR].m
X
t=1
µt
New,(49)
where Jis the maximum allowable power level, µm
Cum,PR is
the intensity of number of accumulated packets in the mth
time slot given in (23), µt
New =τgεt
New is the intensity of
the new arrival packets in the tth time slot, and Tm
PRiand
Pm
PRiare the active probability and RA success probability
of each IoT device transmitting with ith power level unit κi
in the mth time slot given in (25) and (15), respectively.
The average queue length and the average waiting delay of
each packet with the ACB&BO scheme over mtime slots are
derived as
E[Qm
ACB&BO] =µm
New +µm
Cum,ACB&BO
BmPACBPm
ACB&BOTm
ACB&BO,(50)
and
E[Dm
ACB&BO] = m
X
t=1
E[Qt
ACB&BO].m
X
t=1
µt
New,(51)
where PACB is the ACB factor, Bmis the probability of a
packet is not blocked in the buffer by the back-off mechanism
in the mth time slot given in (42), µm
Cum,ACB&BO,Tm
ACB&BO,
and Pm
ACB&BO are given in (37), (41), and (40), respectively.
The average queue length and the average waiting delay of
each packet with the PR&BO scheme over mtime slots are
derived as
E[Qm
PR&BO] =µm
New +µm
Cum,PR&BO
2
X
i=1
Tm
PR&BOiPm
PR&BOi,(52)
and
E[Dm
PR&BO] = m
X
t=1
E[Qt
PR&BO].m
X
t=1
µt
New,(53)
where µm
Cum,PR&BO,Tm
PR&BO1, and Tm
PR&BO2are given in
(44), (46), and (47), respectively.
For each RA scheme, the network is considered stable if a
randomly selected queue is finite, which requires the packets
arrival rate to be less than the service rate. In other words,
the stability only occurs when the queue distribution reaches
a steady state. Therefore, the stability condition is related to
the average queue length, which is given by
lim
m+(E[Qm]E[Qm1]) 0.(54)
VI. NUMERICAL RESULTS
In this section, we validate the derived analytical results
via independent system level simulations. The BSs and IoT
devices are deployed via independent PPPs in a 400 km2area,
and each IoT device associated with its closest BS and transmit
with the channel inversion power control policy. Note that we
simulate the real buffer at each IoT device to capture the pack-
ets accumulated process evolved over time. In each time slot,
IoT devices randomly move to a new position, and the active
ones randomly choose a preamble for the current RA attempt.
In all figures of this section, “Analytical” and “Simulation”
13
are abbreviated as “Ana.” and “Sim.”, respectively. Unless
otherwise stated, we choose the same new packets arrival rate
for each time slot (µ1
New =µ2
New =· · · =µm
New = 0.1
packets/time slot), σ2=90 dBm, ρ=90 dBm, γth = 1
(0dB), α= 4,λB= 10 BS/km2. Unless otherwise stated, we
consider tBO = 1 for the schemes with the back-off policy
(i.e., BO, ACB&BO, and PR&BO scheme), PACB = 0.8for
the schemes with the ACB policy (i.e., ACB&BO, and ACB
scheme), and the power level unit κ1= 1 as well as the
maximum allowable power level unit κJ=κ2= 10 for the
schemes with the PR policy (i.e., PR and PR&BO scheme).
0 10 15 20 25 30
time slot
0.77
0.78
0.79
0.8
0.81
0.82
0.83
RA Success Probability
(a) RA Success Probability
Ana.
Sim.BO
Sim.ACB
Sim.ACB&BO
Sim.PRBO
Sim.PR
5
0 10 15 20 25 30
time slot
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Average Queue Length
(b) Average Queue Length
Ana.
Sim.BO
Sim.ACB
Sim.ACB&BO
Sim.PRBO
Sim.PR
5
Fig. 5: The RA success probability and the average queue length when γth =10 dB.
Fig. 5 and Fig. 6 plot the RA success probability and the
average queue length with five RA schemes within the 30
time slots when γth =10dB and γth = 0dB, respectively.
The density ratios between IoT devices transmitting the same
preamble and BSs is set as λDpB= 1. The analytical
curves of the PR scheme Pm
PR,ALL and the PR&BO scheme
Pm
PR&BO,ALL are plotted using (14) and (43), and the ana-
lytical curves of the ACB&BO, ACB, and BO schemes are
all plotted using (37). The close match between the analyt-
ical curves and simulation points validates the accuracy of
developed spatio-temporal mathematical framework. We first
observe that for all RA schemes, the RA success probabilities
in Fig. 5(a) outperform those in Fig. 6(a). This is due to
that the lower SINR threshold leading to higher preamble
transmission success probability. The stability condition is
given in (54). As can be seen in Fig. 5(b), all of the schemes
can reach stability. The average queue lengths follow PR>
PR&BO>BO>ACB>ACB&BO, which shed lights on the
buffer flushing capability of each scheme in this network
condition. In Fig. 6(b), we observe that the RA success
probabilities of the PR&BO and the PR schemes can reach
stability, rather than the other three schemes. This is due to that
the PR policy provides higher RA success probabilities (i.e., as
show in Fig. 6(a), and thus provides faster buffer flushing that
can maintain the average accumulated packets in an acceptable
level.
0 10 15 20 25 30
time slot
0.15
0.2
0.25
0.3
0.35
0.4
0.45
RA Success Probability
(a) RA Success Probability
Ana.
Sim.BO
Sim.ACB
Sim.ACB&BO
Sim.PRBO
Sim.PR
5
0 10 15 20 25 30
time slot
0
0.2
0.4
0.6
0.8
1
1.2
Average Queue Length
(b) Average Queue Length
Ana.
Sim.BO
Sim.ACB
Sim.ACB&BO
Sim.PRBO
Sim.PR
5
Fig. 6: The RA success probability and the average queue length when γth = 0 dB.
Interestingly, in both Fig. 5(a) and Fig. 6(a), the
RA success probabilities follow the performance
PR&BOPRACB&BO>BO>ACB. The PR&BO scheme
and the PR scheme outperform the other schemes due to
that the deferred packets are favored by stepping up the
transmit power, which significantly increases the preamble
transmission success probability. The consistent performance
following ACB&BO>BO>ACB is due to that higher
probability of an RA attempt being deferred in the IoT device
site leads to less interference and collision probability (i.e.,
the RA success probabilities are lower than 50% leading to
more than half IoT devices deferring their RA attempts in the
BO and ACB&BO scheme, but the ACB scheme leads to only
about 20% deferring their RA attempts (i.e., PACB = 0.8),
and thus the probabilities of deferring RA attempt follows
ACB&BO>BO>ACB).
Fig. 7 plots the RA success probabilities of the PR, BO, and
14
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
RA Success Probability
5 10 15 20 25 30
2
1
k
lDp
=3
4
5
Ana.
Sim.
(a) The PR scheme
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
RA Success Probability
Ana.
Sim.
0 1 2 3 4
tBO
lDp
=3
4
5
(b) The BO scheme
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
RA Success Probability
Ana.
Sim.
lDp
=3
4
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
1−P
ACB
(c) The ACB scheme
Fig. 7: RA success probability in the 10th time slot with the PR, BO, and ACB scheme.
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1
Average Queue Length
5 10 15 20 25 30
2
1
k
lDp
=
Ana.
Sim.
5
4
3
(a) The PR scheme
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1
Average Queue Length
0 1 2 3 4
tBO
Ana.
Sim.
lDp
=5
4
3
(b) The BO scheme
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1
Average Queue Length
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
1−PACB
5
4
3
lDp
=
Ana.
Sim.
(c) The ACB scheme
Fig. 8: Average queue length over 10 time slots with the PR, BO, and ACB scheme.
5 10 15 20 25 30
2
1
k
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
4.8
5.0
Average Waiting Delay
5
4
3
lDp
=
Ana.
Sim.
(a) The PR scheme
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
4.8
5.0
Average Waiting Delay
0 1 2 3 4
tBO
5
4
3
lDp
=Ana.
Sim.
(b) The BO scheme
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
4.8
5.0
Average Waiting Delay
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
1−P
ACB
5
4
3
lDp
=Ana.
Sim.
(c) The ACB scheme
Fig. 9: Average Waiting delay over 10 time slots with the PR, BO, and ACB scheme.
ACB schemes in the 10th time slot versus the number of power
level unit κ2(the PR scheme with J= 2), the back-off factor
tBO, and the non-ACB probability 1PACB, respectively. In
Fig. 7(a), the RA success probabilities increase with increasing
κ2until reaching the performance ceilings, due to that the
average SINR2/SINR1in Table I are much larger/smaller than
the SINR threshold, which leads to slow increasing trend of
preamble transmission success probability and slow decreasing
trend of collision probability. Fig. 7(b) and (c) show that the
RA success probabilities increase with increasing tBO and
1PACB, due to that the increasing number of IoT devices
deferring access requests leads to the reduction in interference
and collision probability.
Fig. 8 and Fig. 9 plot the average queue length E[Q10]
and the average waiting delay E[D10]over 10 time slots of
the PR, BO, and ACB schemes using (48) and (49) (i.e., the
PR scheme), as well as (50) and (51) (i.e., the BO and ACB
schemes), respectively. As expected, in Fig. 8 (a) and Fig. 9
(a), the average queue length and the average waiting delay
decrease with increasing κ2until reaching the performance
floors. In Fig. 8 (b) and (c), and Fig. 9 (b) and (c), we can see
that the average queue length and the average waiting delay
15
RA Success Probability
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1 2 3 4 5 6 7 8 9 10
l
Dp/
l
B
Sim.PR&BO
Sim.PR
Sim.BO
Sim.ACB&BO
Sim.ACB
Ana.
(a) RA success probability
1 2 3 4 5 6 7 8 9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Average Queue Length
l
Dp/
l
B
Sim.PR&BO
Sim.PR
Sim.BO
Sim.ACB&BO
Sim.ACB
Ana.
(b) Average queue length
1 2 3 4 5 6 7 8 9 10
1
1.5
2
2.5
3
3.5
4.5
5
5.5
Average Waiting Delay
4
Sim.PR&BO
Sim.PR
Sim.BO
Sim.ACB&BO
Sim.ACB
Ana.
l
Dp/
l
B
(c) Average waiting delay
Fig. 10: RA success probability at the 10th time slot, and the average queue length as well as the average waiting delay over 10 time slots with five RA schemes
first decrease and achieve the lowest value, and then gradually
increase. The first decreasing trends of the average queue
length and waiting delay are mainly due to the increasing
number of IoT devices deferring their access requests, which
increases the RA success probabilities, and then the following
increasing trends are mainly due to that the continuously
increasing number of IoT devices deferring access requests
leads to the reduction in channel resources utilization. For
instance, the ACB scheme with a small PACB can provide
relatively high RA success probability sacrificing that a large
proportion of IoT devices blocks their packets by deferring
access requests, which leads to low packets serving rate and
large number of packets accumulated in buffers.
Fig. 10 plots the RA success probabilities at the
10th time slot, and the average queue length as well
as the average waiting delay over 10 time slots with
five RA schemes versus the density ratio λDpB. In
Fig. 10(a), we observe that the RA success prob-
abilities follow PR&BO>PR>ACB&BO>BO>ACB and
then PR&BO>ACB&BO>BO>PR>ACB before and after
λDpB= 4. As expected, the RA success probability of
the PR scheme decreases rapidly, due to that it does not
defer any access requests in any network condition, which
leads to the most rapid increasing interference and collision
probability. In Fig. 10(b) and (c), the RA success probabilities
of the PR and PR&BO schemes always outperform other
schemes, due to that the advantages of PR policy (i.e. as
explained in Fig. 5 and Fig. 6) leads to faster buffer flushing
(i.e., the speed of packets been served and removed from the
buffer) than other schemes. The average queue length and
the average waiting delay of schemes with PR policy follow
PR<PR&BO and then PR>PR&BO before and after certain
density ratios, due to that the BO policy leads to the reduction
in channel resources utilization in the low density ratio region,
however after certain density ratios, the increasing density ratio
increases traffic burden that leads to higher interference and
collision probability severely degrading those performances,
and thus the BO policy becomes efficient by deferring access
requests to control traffic. As seen from Fig. 10(a), (b), and (c),
all the performance of the schemes without PR policy follow
ACB>BO>ACB&BO and then ACB<BO<ACB&BO before
and after a density ratio, which can also be explained by the
same reason that the efficiency of traffic control improves with
increasing the density ratio.
VII. CONCLUSION
In this paper, we developed a spatio-temporal mathematical
model to analyze the contention-based RA in the mIoT net-
work by taking into account the SINR outage problem as well
as the collision problem. We derived the exact expressions for
the RA success probability, the average queue length, and the
average waiting delay in each time slot with the PR, ACB, BO,
ACB&BO, and PR&BO schemes. In the light traffic scenario,
the PR scheme outperforms other schemes in terms of the
average queue length and the average waiting delay, due to
its relatively high RA success probability and no deferring of
access requests leading to high utilization of channel resources.
In the heavy traffic scenario, the PR&BO scheme outperforms
other schemes in terms of RA success probability, the average
queue length, and the average waiting delay, due to that it can
maintain the efficiency of the PR policy by releasing the traffic
burden in the network via BO policy.
APPENDIX A
A PRO OF O F LEMMA 2
Using the Bayes’ theorem [40, Eq. 2-44], the PDF of the
area size of the Voronoi cell Xconditioning on N1=n1is
PX=xN1=n1=PN1=n1X=xPX=x
PN1=n1.(A.1)
In (A.1), P[N1=n1|X=x]is the PMF of the number of
interfering IoT devices N1in a cell conditioning on the area
size of the cell X=x, presented as
P[N1=n1|X=x] = (TPR1λDp x)n1
Γ (n1+ 1) e−TPR1λDp x,
(A.2)
P[X= x] is the PDF of the size of a voronoi cell that a
randomly chosen IoT device belongs to, given in [35, Lamma
2]
P[X=x] = λB
cc+1
Γ(c+ 1)(λBx)ce(λBcx),(A.3)
16
and P[N1=n1]is the PMF of N1number of interfering IoT
devices transmitting with the power level unit κ1in the voronoi
cell selected by the randomly chosen IoT device, given as [35,
Eq.(3)]
P{N1=n1}=c(c+1)Γ(n+c+ 1)( TPR1λDp
λB)n1
Γ(c+ 1)Γ(n1+ 1)(TPR1λDp
λB+c)n1+c+1 .
(A.4)
Substituting (A.2), (A.3), and (A.4) into (A.1), we verified
(10) in Lemma 2.
APPENDIX B
A PRO OF O F THE OREM 2
Using the law of the total probability [40, Eq. 2-80], the
PMF of N2number of IoT devices transmitting with the power
level unit κ2in a Voronoi cell conditioning on N1=n1is
expressed as
P[N2=n2|N1=n1] =
Z
0
P[N2=n2|X= x]P[X=x|N1=n1]dx. (B.1)
Substituting (10) and (A.2) into (B.1), we obtain
P[N2=n2|N1=n1]
=(TPR2λDp)n2(TPR1λD p +B)n1+c+1
Γ (n2+ 1) Γ (n1+c+ 1) ×
Z
0
x(n2+n1+c)e(TPR2λDp+TPR1λD p+λBc)xdx
=(TPR2λDp)n2(TPR1λD p +B)n1+c+1
Γ (n2+ 1) Γ (n1+c+ 1) ×
Lx(n2+n1+c)(TPR2λDp +TPR1λDp +λBc)
=Γ (n2+n1+c+ 1)
Γ (n2+ 1) Γ (n1+c+ 1) ×
(TPR2λDp)n2(TPR1λD p +B)n1+c+1
(TPR2λDp +TPR1λDp +λBc)n2+n1+c+1 .(B.2)
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