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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/ξ)

to choose a speciﬁc preamble, and the average density of IoT

devices using a same preamble is λDp =λD/ξ, 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 q≤PACB (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}Pihikuik−α,(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[SINRo≥γthN1=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 −e−µ1

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 +σ2≥γthN1=n1

=Pho≥γth

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) = exp−2(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

ρ−2(γth)2

αb

T1λDp

λBR∞

(γth)−1

α

y

1+yαdy

(1 + γth)n1

| {z }

II

1−

exp −γthσ2

ρ−2(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 TPR,κj. 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−(TPR,κ1λDp+λBc)x(TPR,κ1λD p +cλB)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

TPR,κ1is 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) ×

(TPR,κ2λDp)n2(TPR,κ1λD p +cλB)n1+c+1

(TPR,κ1λDp +TPR,κ2λDp +λBc)n1+n2+c+1 ,(11)

where TPR,κ2is the active probability of IoT devices transmit-

ting with the power level unit κ2(i.e., TPR,κ2will 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,· · · , Nj−1=nj−1in 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

PR,κl=∞

X

n1=0

∞

X

n2=0

· · · ∞

X

nJ=0 (P[Nl=nl]

J

Y

j=1,j6=lP[Nj=nj|Nl=nl, N1=n1,· · · , Nj−1=nj−1]

| {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,· · · , Nj−1=nj−1and the typical IoT

device transmitting with the power level unit κ1is

P[Nj=nj|N1=n1, N2=n2,· · · , Nj−1=nj−1] =

Γ j

P

i=1

ni+c+ 1

Γ (nj+ 1) Γ j−1

P

i=1

ni+c+ 1×

TPR,κjλDpnjj−1

P

i=1

TPR,κiλDp +cλBj−1

P

i=1

ni+c+1

j

P

i=1

TPR,κiλDp +λBcj

P

i=1

ni+c+1

,

(12)

where TPR,κjis the active probability of IoT devices transmit-

ting with the power level unit κj(i.e., TPR,κjwill 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

PR,κl(l∈[1, J ]), where the IoT device

transmits preamble with the lth power level unit κlafter it

fails in RA for l−1times. The RA success probability of

the IoT device transmits preamble with the lth power level

unit κlin the mth time slot Pm

PR,κlis 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(J≥2)) 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

PR,κiPm

PR,κi.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

PR,κl=∞

X

n1=0

∞

X

n2=0

· · · ∞

X

nJ=0 Θ(m, l, l, ~

n)

J

Y

j=1

Ω(m, l, j, ~

n)

1−Θ(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

PR,κlλDp

λB)

nl

Γ(c+ 1)Γ(nl+ 1)(Tm

PR,κlλDp

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,· · · , Nj−1=nj−1, is derived

8

as

Ω(m, l, j, ~

n) =

Γnl+ (

j

P

i=1,i6=l

ni) + c+ 1(TPR,κjλDp)nj

Γ(nj+ 1)Γnl+

j−1

P

i=1,i6=l

ni) + c+ 1

×TPR,κl+

j−1

P

i=1,i6=l

TPR,κiλDp +cλBnl+(

j−1

P

i=1,i6=l

ni)+c+1

TPR,κl+

j

P

i=1,i6=l

TPR,κiλDp +cλBnl+(

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

κlρ−2λDp(γth )2

α

λBJ

X

i=1

(κi

κl

)2

α×

Tm

PR,κiZ∞

(γth

κi

κl)−1

α

y

1 + yαdyJ

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

κjρ−2λDp(γth )2

α

λBJ

X

i=1

(κi

κj

)2

α×

Tm

PR,κiZ∞

(γth

κi

κj)−1

α

y

1 + yαdy

(1 + γth

κl

κj

)nl+1(1 + γth )nj−1

J

Y

i=1,i6=l,j

(1 + γth

κi

κj

)ni.

(19)

Note that TPR,κiis 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

= exp−γth

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

uk∈b

Zout

EPkEhke−sκiPkhkkukk−αi

(b)

=exp −2πTm

P R,κiλDp Z∞

(P/κiρ)1

α

EPEh1−e−sκiP hx−αxdx

(c)

=exp −2πTm

P R,κiλDp(κis)2

αEP[P2

α]Z∞

(sκiρ)−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 + sκiρni(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

PR,κ1in (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 =µm−1

New +µm−1

Cum,PR −

J

X

i=1

Tm−1

PR,κiPm−1

PR,κi;

(23)

•Step 3: Calculate the active probability of each IoT device

in the mth time slot Tm

PR,all using

Tm

PR,all = 1 −e−µm

New−µm

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

PR,κiusing

Tm

PR,κi=

Tm

PR,all −

J

X

i=1

Tm−1

PR,κi1− Pm−1

PR,κi, i = 1,

1− Pm−1

PR,κi−1Tm−1

PR,κi−1, i 6= 1, i 6=J,

1− Pm−1

PR,κi−1Tm−1

PR,κi−1

+1− Pm−1

PR,κiTm−1

PR,κi, i =J,

(25)

where Pm−1

PR,κiis the RA success probability of the IoT

device transmitting with the power level unit κiin the

(m−1)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

PR,κlusing (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 + sκiρni(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

PR,κ1in (7).

Step 2: Calculate µm

Cum,PR in (23).

Step 3: Calculate Tm

PR,all in (24).

Step 4: Calculate Tm

PR,κ1,Tm

PR,κ2,· · · ,Tm

PR,κJin (25).

Step 5: Calculate Pm

PR,κ1,Pm

PR,κ2,· · · ,Pm

PR,κJin (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

PR,κ1in (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 =µm−1

New +µm−1

Cum,PR −

J

X

i=1

Tm−1

PR,κiPm−1

PR,κi;(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

λDp/λB. We study the geometric PR scheme, where the

transmit power steps up following the policy κl=gl−1(i.e.,

gis a constant denoting the root of power increase, lis the

current power level, and l≤J, 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

PR,κ1Pm

PR,κ1+Tm

PR,κ2Pm

PR,κ2.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

PR,κ1is derived as

Pm

PR,κ1=∞

X

n1=0

∞

X

n2=0 Θ(m, 1,1,~

n)

2

Y

j=1

Ω(m, 1, j, ~

n)

1−Θ(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

PR,κ2is derived as

Pm

PR,κ2=∞

X

n1=0

∞

X

n2=0 Θ(m, 2,2,~

n)

2

Y

j=1

Ω(m, 2, j, ~

n)

1−Θ(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

PR,κ1λDp

λB)

n1

Γ(c+ 1)Γ(n1+ 1)(Tm

PR,κ1λDp

λB+c)

n1+c+1 ,

(29)

Ω(m, 2,2,~

n) = c(c+1)Γ(n2+c+ 1)( Tm

PR,κ2λDp

λB)

n2

Γ(c+ 1)Γ(n2+ 1)(Tm

PR,κ2λDp

λB+c)

n2+c+1 ,

(30)

Ω(m, 1,2,~

n) = Γ (n1+n2+c+ 1)

Γ (n2+ 1) Γn1+c+ 1×

Tm

PR,κ2λDpn2Tm

PR,κ1λDP +cλBn1+c+1

Tm

PR,κ1+Tm

PR,κ2λDP +cλBn1+n2+c+1 ,

(31)

Ω(m, 2,1,~

n) = Γ (n1+n2+c+ 1)

Γ (n1+ 1) Γn2+c+ 1×

Tm

PR,κ1λDpn1Tm

PR,κ2λDP +cλBn2+c+1

Tm

PR,κ1+Tm

PR,κ2λDP +cλBn1+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

κ1ρ−2λDp√γth

λB2

P

i=1 qκi

κ1Tm

PR,κiarctg(qκi

κ1γth)

(1 + γth)n1(1 + γth κ2

κ1)n2,

(33)

Θ(m, 1,2,~

n) =

exp −γthσ2

κ2ρ−2λDp√γth

λB2

P

i=1 qκi

κ2Tm

PR,κiarctg(qκi

κ2γth)

(1 + γth κ1

κ2)n1+1(1 + γth )n2−1,

(34)

Θ(m, 2,1,~

n) =

exp −γthσ2

κ1ρ−2λDp√γth

λB2

P

i=1 qκi

κ1Tm

PR,κiarctg(qκi

κ1γth)

(1 + γth)n1−1(1 + γth κ2

κ1)n2+1 ,

(35)

Θ(m, 2,2,~

n) =

exp −γthσ2

κ2ρ−2λDp√γth

λB2

P

i=1 qκi

κ2Tm

PR,κiarctg(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

(1−PACB). 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, m)ΘACB&BO(n1, m)

1−ΘACB&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

ρ−2(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 =µm−1

New +µm−1

Cum,ACB&BO−

Bm−1PACBPm−1

ACB&BOTm−1

ACB&BO; (39)

•Step 3: Calculate the active probability4in the mth time

slot Tm

ACB&BO using

Tm

ACB&BO = 1 −e−µm

New−µm

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=

1−m−1

X

j=1

(1−Pm−j

ACB&BO)PACBTm−j

ACB&BOBm−j/Tm

ACB&BO, m ≤(tBO+1) ,

1−tBO

X

j=1

(1−Pm−j

ACB&BO)PACBTm−j

ACB&BOBm−j/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 + sκiρni(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

PR,κ1in (8).

Step 2: Calculate µm

Cum,PR in (22).

Step 3: Calculate Tm

PR,all in (23).

Step 4: Calculate Tm

PR,κ1,Tm

PR,κ2,· · · ,Tm

PR,κJin (24).

Step 5: Calculate Pm

PR,κ1,Pm

PR,κ2,· · · ,Pm

PR,κJin (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

PR,κ1in (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 =µm−1

New +µm−1

Cum,PR −

J

X

i=1

Tm−1

PR,κiPm−1

PR,κi;(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 m≥tBO + 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&BO,κ1Pm

PR&BO,κ1+Tm

PR&BO,κ2Pm

PR&BO,κ2

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&BO,κ1using (7);

•Step 2: Calculate the intensity of accumulated packets

µM

Cum,PR&BO in the mth time slot using

µm

Cum,PR&BO =

µm−1

New +µm−1

Cum,PR&BO − T m−1

PR&BO,κ1Pm−1

PR&BO,κ1,1< m < tBO + 2,

µm−1

New +µm−1

Cum,PR&BO −

2

X

i=1

Tm−1

PR&BO,κiPm−1

PR&BO,κi, 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 =µm−1

New +µm−1

Cum,ACB&BO−

Bm−1PACBPm−1

ACB&BOTm−1

ACB&BO;

(40)

•Step 3: Calculate the active probability4in the mth time

slot Tm

ACB&BO using

Tm

ACB&BO = 1 −e−µm

New−µm

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−

m−1

P

j=1

(1−P m−j

ACB&BO)PACBTm−j

ACB&BOBm−j

Tm

ACB&BO

,

m≤(tBO+1) ,

1−

tBO

P

j=1

(1−P m−j

ACB&BO)PACBTm−j

ACB&BOBm−j

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 m≥tBO + 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&BO,κ1Pm

PR&BO,κ1+Tm

PR&BO,κ2Pm

PR&BO,κ2

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&BO,κ1using (7);

•Step 2: Calculate the intensity of accumulated packets

µM

Cum,PR&BO in the mth time slot using

µm

Cum,PR&BO =

µm−1

New +µm−1

Cum,PR&BO − T m−1

PR&BO,κ1Pm−1

PR&BO,κ1,

1< m < tBO + 2,

µm−1

New +µm−1

Cum,PR&BO −

2

X

i=1

Tm−1

PR&BO,κiPm−1

PR&BO,κi,

m≥tBO + 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 −µm−1

Cum,PR&BO); (45)

12

•Step 4: Calculate the active probability Tm

PR&BO,κ1and

Tm

PR&BO,κ2using

Tm

PR&BO,κ1=Tm

PR&BO,all−

2

X

i=1

tBO +1

X

t=1

Tm−t

PR&BO,κi1− Pm−t

PR&BO,κi,

(46)

Tm

PR&BO,κ2=

2

X

i=1

Tm−1−tBO

PR&BO,κi1− Pm−1−tBO

PR&BO,κi; (47)

•Step 5: Calculate the RA success probability in the mth

time slot Pm

PR&BO,κ1and Pm

PR&BO,κ2using (27) and (28)

with Tm

PR&BO,κigiven 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&BO,κi.

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

PR,κiPm

PR,κi,(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

PR,κiand

Pm

PR,κiare 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&BO,κiPm

PR&BO,κi,(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&BO,κ1, and Tm

PR&BO,κ2are 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[Qm−1]) ≈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 λDp/λB= 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&BO≈PRACB&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 1−PACB, 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

1−PACB, 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 λDp/λB. 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

λDp/λB= 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] = (TPR,κ1λDp x)n1

Γ (n1+ 1) e−TPR,κ1λDp 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)( TPR,κ1λDp

λB)n1

Γ(c+ 1)Γ(n1+ 1)(TPR,κ1λDp

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]

=(TPR,κ2λDp)n2(TPR,κ1λD p +cλB)n1+c+1

Γ (n2+ 1) Γ (n1+c+ 1) ×

Z∞

0

x(n2+n1+c)e−(TPR,κ2λDp+TPR,κ1λD p+λBc)xdx

=(TPR,κ2λDp)n2(TPR,κ1λD p +cλB)n1+c+1

Γ (n2+ 1) Γ (n1+c+ 1) ×

Lx(n2+n1+c)(TPR,κ2λDp +TPR,κ1λDp +λBc)

=Γ (n2+n1+c+ 1)

Γ (n2+ 1) Γ (n1+c+ 1) ×

(TPR,κ2λDp)n2(TPR,κ1λD p +cλB)n1+c+1

(TPR,κ2λDp +TPR,κ1λDp +λBc)n2+n1+c+1 .(B.2)

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