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RACH Preamble Repetition in NB-IoT Network
Nan Jiang, Student Member, IEEE, Yansha Deng, Member, IEEE, Massimo Condoluci, Member, IEEE, Weisi Guo,
Senior Member, IEEE, Arumugam Nallanathan, Fellow, IEEE, and Mischa Dohler, Fellow, IEEE
Abstract—NarrowBand-Internet of Things (NB-IoT) is a radio
access technology recently standardized by 3GPP. To provide
reliable connections with extended coverage, a repetition trans-
mission scheme is applied in both Random Access CHannel
(RACH) procedure and data transmission. In this letter, we model
RACH in the NB-IoT network taking into account the repeated
preamble transmission and collision using stochastic geometry.
We derive the exact expression of RACH success probability
under time correlated interference, and validate the analysis with
different repetition values via independent simulations. Numerical
results have shown that the repetition scheme can efficiently
improve the RACH success probability in a light traffic scenario,
but only slightly improves that performance with very inefficient
channel resource utilization in a heavy traffic scenario.
Index Terms—NB-IoT, stochastic geometry, RACH success
probability, repetition values.
I. INTRODUCTION
To support communication of billions of miscellaneous
innovative devices, Internet of Things (IoT) has gained un-
precedented momentum and commercial interest. In view of
this, a new radio access technology was developed by the
3rd generation partnership project (3GPP) named NarrowBand-
IoT (NB-IoT), which provide reliable connections among in-
expensive IoT devices with extended coverage and low power
consumption [1, 2]. NB-IoT is built from existing Long Term
evolution (LTE) design, which aims to achieve rapid specifica-
tion and minimize development effort [3].Accordingly,it reuses
most LTE functionalities to achieve excellent co-existence perf-
ormance with legacy LTE technologies [4].
However, to meet the requirements of IoT-based applica-
tions, such as low data rate, wide coverage, and etc., a number
of LTE features are simplified, optimized, or developed [3].
One important key design difference between LTE and NB-IoT
is the Physical Random Access CHannel (PRACH) (i.e., the
channel used for uplink preamble transmission in the Random
Access CHannel procedure, a.k.a. RACH). In more detail, the
frequency hopping signal is used to transmit each preamble for
time-of-arrival estimation1, and a repetition scheme is applied
Manuscript received October 19, 2017; revised December 15, 2017; accepted
January 5, 2018. This work has been partly supported by the EC H2020
Project Data Aware Wireless Network for Internet-of-Everything (778305).
The associate editor coordinating the review of this paper and approving it for
publication was J. Ben-Othman. (Corresponding author: Yansha Deng.)
N. Jiang, and A. Nallanathan are with School of Electronic Engineering and
Computer Science, Queen Mary University of London, London, UK (e-mail:
{nan.jiang, a.nallanathan}@qmul.ac.uk).
Y. Deng, M. Condoluci and M. Dohler are with Department of Infor-
matics, King’s College London, London, UK (e-mail:{yansha.deng, mas-
simo.condoluci, mischa.dohler}@kcl.ac.uk).
W. Guo is with School of Engineering, University of Warwick, Coventry,
UK (e-mail: weisi.guo@warwick.ac.uk).
1The frequency hopping preamble is designed for calculating time-of-arrival
to facilitate distance estimation between the IoT device and its associated eNB,
which is beyond the scope of this paper [1, 2].
to improve the reliability of RACH [1,2,4]. In other words,
the preamble transmission is repeated for a dedicated number
of times according to a repetition value that decided by an
evolved NodeB (eNB) [2].
Obviously, increasing the repetition value increases the suc-
cess opportunities of preamble transmission, but occupies more
resources elements, which can bring serious burden for the
NB-IoT system leading to reduced resource elements for data
transmission. It is still unknown to what extent the repetition
scheme improves the RACH success, and how to choose repe-
tition value in different traffic scenarios to balance the RACH
success probability and data transmission channel resources.
To study the RACH in cellular-based massive IoT network, a
general model was provided to analyze preamble transmission
based on stochastic geometry [5]. Unfortunately, this model
cannot be directly adopted to analyze the RACH in NB-IoT,
due to the following three reasons: 1) new preamble format
is defined based on frequency hopping; 2) repetition scheme
significantly complicate the analysis; 3) collision events are
not addressed in [5]. Considering the three issues, this letter
provides a mathematical framework to analyze the RACH in
NB-IoT, where two independent link outage conditions are
focused: 1) the insufficient Signal-to-Interference-plus-Noise
Ratio (SINR) of the received preamble; 2) the collision event,
which occurs if two or more IoT devices choose the same
sub-carrier at the same time.
The rest of the paper is organized as follows. The system
model and the analytical methodology are introduced in Sec-
tion II. The RACH success probability is presented in Section
III. Section IV presents the numerical results, and the paper is
concluded in Section V.
II. SYSTEM MODEL
In the NB-IoT system, the eNBs and the IoT devices
are spatially distributed in R2following two independent
homogeneous Poisson point process (HPPP), ΦBand ΦD,
with intensities λBand λD, respectively. According to [2],
each IoT device associates with its geographically nearest
eNB. Path-loss attenuation is defined as r−α, where ris the
propagation distance and αis the path-loss exponent. We
consider identically distributed (i.i.d.) Rayleigh fading channel,
where the channel power gain his assumed to be exponentially
distributed random variable with unit mean. According to [1,
2], the transmitted power of IoT devices determined by the
full path-loss inversion power control, where each IoT device
maintains the received signal power in the eNB equalling to a
same threshold ρby compensating its own path-loss.
In the uplink of NB-IoT, the Narrowband Physical Uplink
Shared CHannels (NPUSCHs) are used for data transmis-
sion and the Narrowband-PRACHs (NPRACHs) are used for
2
preamble transmission. In more detail, 180 kHz of spectrum
is occupied with 3.75 kHz tone spacing (i.e., spans over 48
sub-carriers) or 15 kHz tone spacing (i.e., spans over 12 sub-
carriers), where the NPRACH only supports 3.75 kHz tone
spacing. To fulfill the coverage requirements of different IoT
devices, NB-IoT network can configure up to 3 repetition
values from the set {1, 2, 4, 8, 16, 32, 64, 128}in a cell, and
allows flexible configuration of NPRACH resources [4]. In this
model, we consider single repetition value to provide funda-
mental insights due to repetition, where the related resources
assignment of NPRACHs only takes place in the begin of a
transmission time interval (TTI) as shown in Fig. 1.
NPUSCH
NPRACH
5.6Nτ
TTI
12 Sub-
carriers
180 kHz
5.6 ms 3.75 kHz
Sub-carriers
Repetition 1
A Preamble Symbol Group Time
Pseudo Random Hopping
Repetition 2
Repetition Nτ
Repetition 1
Repetition 2
Repetition Nτ
Fixed Size Frequency Hopping
NPUSCH
NPUSCH
Fig. 1: Structure of NPUSCH and NPRACH.
Due to the single repetition value configuration, each active
IoT device will contend on all 48 sub-carriers, and thus each
sub-carrier has an equal probability (1/48) to be chosen. As
only active IoT devices will try to request for uplink channel
resources, we define the active probability of each IoT device
pa∈[0,1] follows a Bernoulli process. According to the
thinning process [6], the density of active IoT devices choosing
the same sub-carrier can be derived
λDa =paλD/48.(1)
Recalling the repetition scheme, an active IoT device repeats
a same preamble Nτtimes (i.e., the dedicated repetition value).
In each repetition, a preamble consists of 4 symbol groups,
where the first preamble symbol group is transmitted via a
sub-carrier determined by Pseudo random hopping (i.e., the
hopping depends on the current repetition time and the Nar-
rowband physical Cell ID, a.k.a NCellID), and the following
3 preamble symbol groups are transmitted via sub-carriers
determined by the fixed size frequency hopping [1]. In other
words, if two or more IoT devices chose the same first sub-
carrier in a single RACH opportunity, the following sub-
carriers (i.e., in the same RACH opportunity) would be same,
due to that these two hopping algorithms lead to one-to-one
correspondences between the first sub-carrier and the following
sub-carriers.
We first formulate the SINR outage condition. In each
TTI, a preamble transmission success occurs if any repetition
successes, and in a single repetition, a preamble is successfully
received at the associated eNB, if all four received SINRs are
above the SINR threshold γth. Overall, the preamble transmis-
sion success probability under Nτrepetitions conditioning on
nnumber of intra-cell interfering IoT devices is expressed as
pS(Nτ, n) = 1 −
Nτ
Y
nτ=1 1−P{θnτ|ZB=n},(2)
where ZB=kZintrakis the number of active intra-cell
interfering IoT devices, and
θnτ
∆
=SINRnτ,1≥γth ,SINRnτ,2≥γth ,
SINRnτ,3≥γth ,SINRnτ,4≥γth .(3)
In (3), γth is the SINR threshold, and SINRnτ,1is the
received SINR of the 1st symbol group in the nτth repetition.
Based on Slivnyak’s theorem [6], we formulate this SINR
experienced at the typical eNB located at the origin as
SINR =ρ0h0
P
j∈Zintra
ρjhj+P
i∈Zinter
Pikuik−αhi+σ2,(4)
where k·k is the Euclidean norm, ρ0is the full path-loss
inversion power control threshold, h0is the channel power gain
from the typical IoT device to its associated eNB, Zinter is set
of the active inter-cell interfering IoT devices using the same
sub-carrier, σ2is the noise power, Piis the actual transmit
power of the ith interfering IoT device, xiand hiare distance
and channel power gain from the ith interfering IoT device to
the typical eNB, respectively.
Then, we formulate the RACH success probability under
both SINR outage and collision conditions. Recalling that a
collision occurs if an eNB receives multiple preambles from
the same set of sub-carriers at the same time. Consequently, a
RACH successes when the preamble successfully received in
the eNB, as well as no collision occurs, which is presented as
PNτ=
∞
X
n=0
P{ZB=n}pS,0(Nτ, n)
n
Y
l=1
1−pS,l (Nτ, n),(5)
where pS,0(Nτ, n)is the preamble transmission success proba-
bility of the typical IoT device, pS,l(Nτ, n)is the preamble tra-
nsmission success probability of the lth interfering IoT device
located in the typical cell, and P{ZB=n}is the probability of
nnumber of interfering IoT devices located in the typical cell.
For an expected RACH success probability, the repetition
value is written as Nτ=f−1(PNτ),(6)
where Nτis the required repetition value.
In a TTI, the repetition value determines the assigned chan-
nel resources for NPRACHs, and the least channel resources
for data transmission (i.e., NPUSCHs). A proper repetition
value can help to improve the channel efficiency, due to that
insufficient repetition may results in a low RACH success
probability, but redundant repetition may leads to deficiency
of NPUSCHs. The time vacancy ratio (i.e., the ratio between
the time of NPUSCHs and TTI) is expressed as
Ta= (TTI −5.6Nτ)/TTI.(7)
As can be seen from Eq. (7), increasing Nτdecreases the
resource elements utilization.
III. RANDOM ACCESS SUCCESS PROBABILITY
Due to that each intra-cell interfering IoT device has the
same preamble transmission success probability, the RACH
success probability presented in (5) can be simplified as
PNτ=
∞
X
n=0
P{ZB=n}pS(Nτ, n)1−pS(Nτ, n)n
,(8)
3
where P{ZB=n}is the probability of the number of interfering
IoT devices ZB=n, and pS(Nτ, n)is the preamble trans-
mission success probability of an IoT device conditioning on
{ZB=n}. The Probability Mass Function (PMF) of the number
of interfering IoT devices ZBis represented as [5, Eq.(12)]
P{ZB=n}=c(c+1)Γ(n+c+ 1)( λDa
λB)n
Γ(c+ 1)Γ(n+ 1)(λDa
λB+c)n+c+1 ,(9)
where λDa is the intensity of active IoT devices using the
same sub-carrier given in (1), λBis the intensity of eNBs,
c= 3.575 is a constant related to the approximate PMF of the
PPP Voronoi cell, and Γ{·} is the gamma function.
Without loss of generality, we assume each IoT device
remains spatially static during a TTI (i.e., we assume that the
preamble format 0 is used, where a preamble repetition only
takes 5.6 ms [1]), and thus the mutual interference among IoT
devices is temporally correlated [7]. This temporal correlation
complicate the derivation of the preamble transmission success
probability, which is the main challenge of RACH success
analysis. The preamble transmission success probability with
Nτrepetitions pS(Nτ, n)is derived in the following Theorem.
Theorem 1. The preamble transmission success probability of
a randomly chosen IoT device with Nτpreamble repetitions
pS(Nτ, n)is expressed as
pS(Nτ, n) = 1 −
Nτ
Y
nτ=1 1−P{θnτ
ZB=n}
=
Nτ
X
nτ=1
(−1)nτ+1Nτ
nτP{θ1,··· , θnτ
ZB=n},(10)
where (Nτ
nτ) = Nτ!
nτ!(Nτ−nτ)! is the binomial coefficient, and
P{θ1,··· , θnτ
ZB=n}is the probability that all of 4×nτ
(i.e., a preamble consists of 4 symbol groups) time-correlated
preamble symbol groups are successfully received in the eNB.
For ease of presentation, we assume m= 4 ×nτ, and
P{θ1,··· , θnτ
ZB=n}is expressed as
P{θ1,··· , θnτ
ZB=n}=
exp−mγthσ2
ρ−2(γth)2
αλDa
λBR∞
(γth)−
1
αh1−(1
1+y−α)miydy
(1 + γth)mn ,
(11)
where αis the path-loss parameter, γth is the received SINR
threshold, σ2is the noise, ρis the full path-loss inversion
power control threshold.
Proof. Based on the Binomial theorem, the preamble
transmission success probability of Nτrepetitions
P{θ1,··· , θnτ
ZB=n}is expressed as
P{SINR1≥γth ,··· ,SINRm≥γth
ZB=n}=
exp(−mγthσ2
ρ)E[e
−s
m
P
k=1
Iintra
k
ZB=n]E[e
−s
m
P
k=1
Iinter
k],(12)
where s=γth/ρ,Iintra
kand Iinter
kare the aggregate intra-cell
interference and the aggregate inter-cell interference when the
kth preamble symbol group is transmitted, respectively.
In (12), the Laplace Transform of the aggregate interference
from the intra-cell interfering IoT devices received at the
typical eNB conditioning on ZB=nis obtained as
E[e
−s
m
P
k=1
Iintra
k
ZB=n] = Ehexp −sX
j∈Zintra
ρ
m
X
k=1
hk
ji
=Ehn
Y
j=1
exp −sρ
m
X
k=1
hk
ji(a)
=(1
1 + sρ)mn ,(13)
where (a)follows by taking the average with respect to h1
j,
h2
j,···,hm
j.
In (12), the Laplace Transform of the aggregate interference
from the inter-cell interfering IoT devices received at the
typical eNB is obtained as
E[e
−s
m
P
k=1
Iinter
k] = Eexp −sX
i∈Zinter
Pi(
m
X
k=1
hk
i)kuik−α
(a)
=EY
i∈Zinter
(1
1 + sPixi−α)m
(b)
=exp −2πλDa Z∞
(P
ρ)
1
α
EPh1−(1
1 + sP x−α)mixdx
(c)
=exp −2πλDa s2
αE[P2
α]Z∞
(sρ)−
1
αh1−(1
1 + y−α)miydy,
(14)
where (a)obtained by taking the average with respect to
h1
i,h2
i,···,hm
i,(b)follows from the probability generation
functional (PGFL) of the HPPP, (c)follows by changing the
variables y=x/(sP )1
α, and E[P2
α] = ρ2
α/πλBis the
moments of the transmit power [5, Eq.(A.2,A.3)]. Substituting
(13) and (14) into (12), we derive the probability that all of m
preamble symbol groups are successfully transmitted.
For simplicity, we present a special case of RACH with two
preamble repetitions, and the preamble transmission success
probability pS(2, n)is expressed as
pS(2, n)=1−1−P{θ1
ZB=n}1−P{θ2
ZB=n}
(a)
=2P{θ1
ZB=n} − P{θ1, θ2
ZB=n},(15)
where (a)follows from P{θ1
ZB=n}=P{θ2
ZB=n}. Substi-
tuting (15) and (9) into (8), we obtain the RACH success
probability when Nτ= 2.
IV. NUMERICAL RESULTS
In this section, the derived analytical results are validated
via Monte Carlo simulations. The eNBs and IoT devices are
deployed in a 104km2circle area. We set λB=0.1eNBs/km2,
γth =0dB, α=4,ρ=−130 dBm, the bandwidth of a sub-
carrier is BW = 3.75 kHz, and thus the noise is σ2=−174 +
10log10(B W ) = −138.3dBm. Different from LTE with TTI =
40 ms [8], we set TTI in NB-IoT network as 640 ms following
4
the defined Narrowband Physical Broadcast Channel (NPBCH)
TTI [2]. The packet arrival periods of IoT devices are from a
few minutes to several days [9], hence we assume two traffic
scenarios, where the light traffic scenario is 1 packet/hour of
each IoT device, and the heavy traffic scenario is 1 packet/10
minutes of each IoT device. Therefore, the active probabilities
of each IoT device during 640 ms are pal= 640/3600000 =
0.00018 and pah= 640/600000 = 0.0011, respectively.
1.5 2.5 3.5 4.5
D B 104
0
0.2
0.4
0.6
0.8
1
RACH Success Probability
(a) Light Traffic Scenario
Analysis
Simulation
RACH Success Probability
(b) Heavy Traffic Scenario
Repetition value = 8,4,2,1
0.5 1 2 3 4 5
/
Simulation
Analysis
Repetition value = 8,4,2,1
0
0.2
0.4
0.6
0.8
1
1.5 2.5 3.5 4.50.5 1 2 34 5
0.1
0.1
×
λλ
D B 104
/
×
λλ
Fig. 2: RACH success probability.
Fig. 2(a) and Fig. 2(b) plot the RACH success probabilities
of a randomly chosen IoT device in the light traffic scenario
and the heavy traffic scenario, respectively. The analytical
curves of the RACH success probability are plotted using
(8), which closely match with simulation points that validates
the accuracy of developed mathematical framework. We first
observe that for all curves, the RACH success probability
decreases with the increase of the density ratio between IoT
devices and eNBs (λD/λB), which is due to the following two
reasons: 1) increasing the number of IoT devices generating
more interference leads to lower received SINR at the eNB;
2) increasing the number of IoT devices results in higher
probability of collision. In both two sub-figures, increasing
repetition value increases the RACH success probability, which
is due to that it offers more opportunities to re-transmit a
preamble with the time and frequency diversity.
0
10
20
30
40
50
60
Nτ
(a) Repetition Value
Light Traffic Scenario
Heavy Traffic Scenario
Point of 37% RACH Success Probability
0.4
0.5
0.6
0.7
0.8
0.9
1
Ta
(b) Time Vacancy Ratio
Light Traffic Scenario
Heavy Traffic Scenario
Point of 37% RACH Success Probability
RACH Success Probability
Less Than 37%
1.5 2.5 3.5 4.5
D B
10
4
0.5 1 2345
/
0.1
×
λλ
1.5 2.5 3.5 4.5
D B
10
4
0.5 1 2 3 4 5
/
0.1
×
λλ
RACH Success Probability
Less Than 37%
Fig. 3: Required repetition values and related time vacancy ratio.
Access success probability of a IoT device is expected
to reach at least 99% within 10 RACH opportunities in the
cellular-based IoT network [10]. To meet this lower bound, the
RACH success probability for one single RACH opportunity
should achieve at least 1−10
√1−99% ≈37%. Fig. 3(a)
plots the minimum repetition value (Nτ) to achieve a RACH
success probability that is greater than or equal to 37%, and
Fig. 3(b) plots the time vacancy ratio (Ta) under the condition
of required repetition value. We assume Nτis configured from
set {1, 2, 4, 8, 16, 32, 64}, and when neither a value can help
IoT devices to reach 37% RACH success probability, the eNB
will choose maximum repetition value 64. We first observe
that Nτincreases with λD/λBin Fig. 3(a), which leads to
an opposite down trend of Tain Fig. 3(b). In the light traffic
scenario, RACH success probability is always larger than 37%
with relatively small repetition values, which contributes to a
relatively high time vacancy ratio. However, in the heavy traffic
scenario, the required repetition value increased rapidly with
λD/λB, and the performance cannot meet the requirement of
37% RACH success probability after a certain point despite
that the maximum repetition value is set.
V. CONCLUSION
In this letter, we developed a stochastic geometry framework
to analyze the RACH under the repetition scheme in the NB-
IoT system. To evaluate how the repetition mechanism fulfills
the requirement of improving RACH reliability, we derived the
exact expression of the RACH success probability under the
time correlated interference. Different from existing works, we
considered both SINR outage and collision from the network
point of view. The numerical results shed light on that the
RACH success probability increases with the repetition value,
and also revealed that very high repetition value will lead to a
low channel resources utilization in the heavy traffic scenario.
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