# A Downlink Puncturing Scheme for Simultaneous Transmission of URLLC and eMBB
Traffic by Exploiting Data Similarity

Mohammed AL-Mekhlafi  Mohaned Chraiti  Mohamed Amine Arfaoui
Chadi Assi  Ali Ghrayeb  and Amira Alloum M. AL-Mekhlafi, Mohamed Amine
Arfaoui and C. Assi are with the Concordia Institute for Information Systems
Engineering, Concordia University, Montreal, QC, Canada (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>assi@ciise.concordia.ca).M. Chraiti is with the LIDS lab, Massachusetts
Institute of Technology (MIT), Cambridge, Massachusetts, USA (email:
<EMAIL_ADDRESS>Ghrayeb is with Texas A & M University at Qatar, Doha,
Qatar (e-mail: ali.ghrayeb@qatar.tamu.edu).A. Alloum is with Huawei, Paris,
France (email<EMAIL_ADDRESS>
###### Abstract

Ultra Reliable and Low Latency Communications (URLLC) is deemed to be an
essential service in 5G systems and beyond (also called 6G) to accommodate a
wide range of emerging applications with stringent latency and reliability
requirements. Coexistence of URLLC alongside other service categories calls
for developing spectrally efficient multiplexing techniques. Specifically,
coupling URLLC and conventional enhanced Mobile BroadBand (eMBB) through
superposition/puncturing naturally arises as a promising option due to the
tolerance of the latter in terms of latency and reliability. The idea here is
to transmit URLLC packets (typically sporadic and of short size) over
resources occupied by ongoing eMBB transmissions while minimizing the impact
on the eMBB transmissions. In this paper, we propose a novel downlink URLLC-
eMBB multiplexing technique that exploits possible similarities among URLLC
and eMBB symbols, with the objective of reducing the size of the punctured
eMBB symbols. We propose that the base station (BS) scans the eMBB traffic’
symbol sequences and punctures those that have the highest symbol similarity
with that of the URLLC users to be served. As the eMBB and URLLC may use
different constellation sizes, we introduce the concept of symbol region
similarity to accommodate the different constellations. We assess the
performance of the proposed scheme analytically, where we derive closed-form
expressions for the symbol error rate (SER) of the eMBB and URLLC services. We
also derive an expression for the eMBB loss function due to puncturing in
terms of the eMBB SER. We demonstrate through numerical and simulation results
the efficacy of the proposed scheme where we show that 1) the eMBB spectral
efficiency is improved by puncturing fewer symbols, 2) the SER and reliability
performance of eMBB are improved, and 3) the URLLC data is accommodated within
the specified delay constraint while maintaining its reliability, 4) and the
proposed strategy has polynomial time complexity making it an efficient
solution to be used in practice.

Keywords—eMBB, multiplexing, puncturing, URLLC, 5G and beyond, 6G.

##  I Introduction

###  I-A Motivation

5G and beyond systems are anticipated to provide a variety of service classes
with different requirements in terms of latency, reliability and connectivity
[1, 2, 3]. This naturally raises concerns about their coexistence, especially
after it has been shown that allocating a dedicated bandwidth for each service
is not spectrally efficient [4]. In particular, providing a dedicated
bandwidth for ultra reliable and low latency communications (URLLC) class of
service has been shown to be poorly efficient where the effectively used
bandwidth could be less than 5% of the total allocated resource. This is
mainly due to URLLC traffic characteristics and requirements [5]. In fact, the
URLLC services come with stringent requirements in terms of latency (less than
one millisecond) and reliability (packet error less than $10^{-6}$), implying
that such services require immediate availability of spectral resources [6,
3]. Meanwhile, given the sporadic characteristic of URLLC traffic and their
short packet size, the allocated resources will only be used occasionally and
for a short period [1, 2, 7, 8]. Therefore, on-demand resource allocation for
URLLC transmissions is deemed a promising solution to make good use of
spectral resources.
Aligning with the concept of on-demand allocation of resources for URLLC
communications, the 3GPP standard has suggested using superposition/puncturing
for multiplexing URLLC and enhanced Mobile BroadBand (eMBB) services in 5G
networks [9, 10]. As time is divided into slots, and each slot consists of
several mini-slots [11], the main idea of the superposition/puncturing
framework is to transmit URLLC packets in mini-slot basis, upon their arrival,
over the resources occupied by ongoing service type transmissions.
Specifically, the eMBB traffic shares the time-frequency resources within each
slot, which can be based on the channel states of the eMBB traffic. To
accommodate the URLLC traffic of the tight latency constraints, the arriving
URLLC packets are immediately scheduled in the next mini-slot on top of the
ongoing eMBB transmissions. If the BS allocates transmission power for both
eMBB and URLLC traffic, then it is referred to as superposition. If the BS
chooses zero transmission power for the eMBB traffic, then this is referred to
as puncturing [10].

###  I-B Related Work

Superposition/puncturing is considered a promising option to allocate the
URLLC traffic due to the tolerance of the latter in terms of latency and
reliability. Hence, much work [12, 13, 14, 15, 16, 17, 18, 19] focused on
developing techniques based on coupling URLLC and conventional (eMBB) data
transmission through superposition/puncturing. In [12, 13, 14, 15, 16, 17, 18,
19], the authors investigated and developed novel superposition/puncturing
approaches aiming to minimize the impact on eMBB in terms of the contaminated
symbols. The advantages of using superposition for sharing resources in uplink
communications between eMBB, mMTC, and URLLC devices was studied in [12]. For
URLLC downlink MIMO-NOMA, network layer performance bounds and cross-layer
power control were studied in [13]. A max-matching diversity (MMD) algorithm
was proposed in [14] to allocate eMBB users, considering both heterogeneous
orthogonal and non-orthogonal multiple access network slicing strategies. A
machine learning approach for hybrid multiple access solution (HMA) was
proposed in [15]. In fact, the classical methods of NOMA, such as power-
domain, require perfect channel state information (CSI) at the BS such that
the transmitted signal can be separated at the receiver with successive
interference cancellation (SIC)[20]. A new class of NOMA, namely bits
similarity NOMA, was proposed in [20]. It was shown that, without perfect CSI,
bit-similarity NOMA can achieve better spectral efficiency and fairness among
users compared to traditional NOMA techniques. In fact, superposition
techniques (NOMA) cause severe degradation to the URLLC reliability because
the eMBB signal acts as an interference signal that increases the decoding
errors of the URLLC traffic. Moreover, the lack of the URLLC CSI at the
transmitter decreases the chances to superpose URLLC traffic on eMBB [21].
Thus, puncturing is preferred as it conserves URLLC reliability.

In order to study the impact of puncturing eMBB resources to accommodate URLLC
transmission, the authors in [16] studied the problem of joint scheduling of
eMBB and URLLC data transmission according to linear, convex and threshold
models of the eMBB rate loss associated with the eMBB resources puncturing. A
risk-sensitive approach was introduced in [17] to mitigate the risk of
puncturing eMBB resources. A resource allocation scheduler was proposed in
[18] where the formulated problem considered the overhead associated with the
URLLC load segmentation while maximizing the rate utility. A null-space-based
spatial puncturing scheduler for joint URLLC/eMBB traffic was proposed in
[22]. The authors in [23] formulated a URLLC traffic allocation problem by
adopting a superposition or puncturing scheme. Practically, when the URLLC
service is initiated in the middle of the eMBB transport block, part of eMBB
symbols are replaced by and/or superposed with the symbols of the URLLC
packet. Accordingly, the reception quality of the eMBB services could be
degraded severely.
Since eMBB tolerates delays, eMBB users can rely on long error-correction
codes in combination with re-transmission techniques to compensate for the
loss incurred by superposition/puncturing. Retransmission-based puncturing
slows the eMBB traffic, and it requires more overhead including puncturing
indicator (PI) to inform the eMBB user of the punctured resources, while the
whole information block can be re-transmitted if decoding errors occur.
Therefore, researchers have been thinking about using codes (code-based
puncturing) to correct the erroneous symbols in the eMBB message and hence
avoiding retransmissions and high overhead signal[4, 10]. Particularly, the
gain achieved by retransmission based puncturing over code-based puncturing is
moderate and less than 10% [4, 10]. Moreover, indicator-free scheme including
a transmit precoding with blind detection is proposed for resource overhead
reduction [24]. In general, the more punctured eMBB symbols there are, the
higher the number of erroneous eMBB symbols and the lower the code rate ( of
the error correction code) we get, which subsequently results in low spectral
efficiency. In this paper, we aim to reduce the number of contaminated eMBB
symbols for the uncoded system and hence the possibility to enhance the code
rate ( of the error correction code) of the eMBB and then the spectral
efficiency.

###  I-C Contributions

In this work, we are motivated to satisfy QoS of both eMBB and URLLC services
in 5G and beyond 5G systems. Therefore, we seek to develop a puncturing
strategy such that the impact on the punctured eMBB symbols is minimized,
which should essentially lead to better eMBB QoS and spectral utility. In
other words, we aim at devising a puncturing strategy that can decrease the
impact of simultaneous transmissions of URLLC and eMBB traffic on the eMBB
traffic. Hence, there is no need to inform the eMBB users about punctured
resources, i.e., avoid transmitting costly and unnecessary puncturing
indicator signal. The contributions of the proposed downlink puncturing
strategy are summarized as follows:

  * •

Different from existing works, we exploit the possible similarity among the
URLLC-eMBB symbols instead of random allocation. Indeed, upon the arrival of a
URLLC packet, the BS scans the ongoing eMBB transmissions and selects the one
that maximizes the number of similar symbols between the two services. In
fact, increasing the similarity between the eMBB-URLLC symbols effectively
reduces the impacted eMBB symbols and hence the possibility to enhance the
used error correction code rate and then the spectral efficiency.

  * •

While developing the proposed technique, we consider the case where an eMBB
user could have different symbol constellations than that of URLLC users.
Accordingly, we introduce the so-called similarity region to evaluate the
similarity between the eMBB and URLLC with different constellations. We
describe in detail the encoding and decoding processes for both eMBB and URLLC
traffic.

  * •

Taking into consideration the symbol errors occurring due to the channel
impairment and the puncturing process, we derive a closed-form expression for
the symbol error rate (SER) of the eMBB traffic. The expression shows that the
SER of the eMBB traffic depends on the signal-to-noise ratio (SNR), the
average URLLC load, and the average similarity. We also consider the SER to
measure the reliability of the URLLC traffic, as conserving the SER of the
URLLC preserves the minimum packet error rate. Moreover, other reliability
improvement techniques, i.e., error control coding schemes [25], packet
duplication [26], and HARQ [27], can be used to enhance the URLLC reliability.
These enhancement techniques are outside the scope of this work.

  * •

We demonstrate through several numerical examples the efficacy of the proposed
scheme where we show that gains of up to 10 dB can be achieved in comparison
to the code-based puncturing technique. At high SNR, the eMBB SER is dominated
by error occurring due to puncturing, i.e., the impact of channel diminishes.
The opposite is true when the similarity increases, that is, the SER is
greatly affected by the channel, not the puncturing. We also show that the
proposed algorithm has low complexity computational time making it an
efficient solution to be used in practice.

###  I-D Paper Outline

The rest of the paper is organized as follows. Section II describes the
adopted system model. The proposed puncturing strategy is described in III.
Section IV provides performance analysis of the proposed strategy where
closed-form expressions for the SER for both eMBB and URLLC users are derived.
Numerical and simulations results are shown in V. We conclude the paper in
Section VI.

##  II  System Model

Figure 1: Relation between frequency resources and puncturing mechanism.

We consider a downlink wireless system consisting of one BS that serves
certain eMBB and URLLC traffics simultaneously. The system bandwidth is
partitioned into $L$ equally sized frequency resources, where each frequency
resource is referred to as a resource element (RE). Each $12$ REs constitute a
resource block (RB) that is equivalent to $180$ KHz. The time domain is
divided into slots, also known as transmission time intervals (TTIs). The
duration of each TTI is $1$ ms. To support the low latency requirement of the
URLLC traffic, each TTI is further divided into mini-slots, also known as
small TTIs (sTTIs), where the duration of each sTTI is $0.143$ ms [16]. The
REs are assigned to the eMBB traffic at the beginning of each TTI, while the
URLLC traffic arriving at each sTTI is directly transmitted in the next sTTI
by puncturing the REs belonging to the eMBB load. Each URLLC packet is divided
into blocks of $\zeta$-symbols, with $\zeta\geq 1$, and it is allocated within
one sTTI. Each eMBB receiver is assumed to decode its received data without
knowledge of the punctured resources, i.e, each eMBB receiver is assumed to be
unaware of the punctured resources at the transmission. Accordingly, the
puncturing overhead is reduced. Based on this assumption, each eMBB receiver
decodes its received data according to its decoder. Let
$m\in\\{2,4,...,{M}\\}$ denote the order of the quadrature amplitude
modulation (QAM) scheme adopted for the eMBB traffic with symbol error
probability $P_{m}(\gamma_{\rm e})$, where $\gamma_{\rm e}$ denote the
received eMBB signal-to-noise ratio (SNR). In addition, let
$n\in\\{2,4,...,{N}\\}$ denote the order of the QAM scheme adopted for the
URLLC traffic and let $\epsilon_{\rm u}$ denote its target symbol error
probability.111By definition, both $m$ and $n$ are powers of $2$. Practically,
the URLLC modulation order $n$ is low due to the following reasons: 1) the
lack of accurate channel estimation due to latency constraints; 2) the URLLC
traffic is assumed to be small, so the achievable capacity follows the short-
block regime; and 3) the high reliability constraint of the URLLC traffic,
which means very low SER. We list in Table I most of the variables used in the
analysis throughout the paper.

TABLE I:  List of Variables Used in the Analysis. Symbol | Description
---|---
$\Omega$ | similarity region
$n$ | URLLC modulation order
$m$ | eMBB modulation order
$L$ | BS downlink frequency resources
$l$ | average URLLC traffic
$\gamma_{\rm e}$ | eMBB Signal to Noise Ratio
$\gamma_{\rm u}$ | URLLC Signal to Noise Ratio
$L_{m}$ | eMBB frequency resources with modulation order $m$
$l_{n,m}$ | punctured eMBB symbols of modulation order $m$
| by URLLC traffic of modulation order $n$
$\mathcal{L}_{n,m}$ | effectively punctured eMBB symbols of modulation order $m$
| by URLLC traffic of modulation order $n$
$\overline{\mathcal{L}}_{n,m}$ | non-effectively punctured eMBB symbols of modulation order $m$
| by URLLC traffic of modulation order $n$
$p_{m}$ | the probability of encoding eMBB with modulation order $m$
$P(.)$ | eMBB traffic symbol SER
$\mathcal{P}(.)$ | URLLC traffic symbol SER
$\zeta$ | URLLC block size
$U_{n,m}(.)$ | average similar symbols
${K}$ | similarity search space
$s_{u}$ | URLLC symbol
$s_{e}$ | eMBB symbol

##  III Proposed Puncturing Scheme

###  III-A Rationale

The main idea of the proposed puncturing strategy is to exploit the similarity
between the symbols of the URLLC block and the symbols of the eMBB load such
that the punctured eMBB symbol is similar to the transmitted URLLC symbol.
Instead of puncturing the eMBB traffic randomly or greedily, we can search
through the eMBB information block to exploit the eMBB sequence that has the
highest similarity to the URLLC data block. Fig. 1 illustrates the mechanism
of the proposed scheme in terms of frequency and time resources. At each mini-
slot, one can transmit two orthogonal frequency division multiplexing (OFDM)
symbols per RE [28]. Hence, if we consider a wireless network with $100$ RBs,
then a total of $100\times 12\times 2=2400$ ODFM symbols can be transmitted in
one sTTI. The BS searches for similarity between the URLLC sequence with the
ongoing $2400$ eMBB symbols and it allocates the URLLC traffic over the eMBB
sequence that has the maximum similarity. For example, assume that the URLLC
block length is $2$ RB, i.e., $2\times 12\times 2=48$ OFDM symbols, and that
the search window (step) is one RB. Then, the proposed algorithm evaluates the
similarity between the URLLC sequence and ${K}=99$ possibilities, where ${K}$
is the search space. As a result, the BS punctures the eMBB sequence that has
the maximum similarity to the URLLC sequence.
For more elaboration, let us assume that both eMBB and URLLC services employ
binary phase shift keying (BPSK) modulation and that the transmitted URLLC
symbol is 0. Then the punctured eMBB symbol can be either 0 or 1. If the
punctured eMBB symbol is 0, then the transmitted URLLC symbol and the
punctured eMBB symbol are similar, and therefore, and therefore, the error
probability of the eMBB symbol is not affected by the puncturing scheme.
However, if the punctured eMBB symbol is 1, then the eMBB symbol will be
received erroneously with probability ($P(0)=1-P(1)=1-P_{2}(\gamma_{\rm e})$).
Therefore, it is recommended to puncture the eMBB traffic that has the maximum
similarity to the URLLC traffic. Intuitively, increasing the similarity
between the transmitted URLLC symbols and the punctured eMBB symbols will
reduce the symbol error rate at the eMBB receiver, which reduces
retransmissions and PI overhead.
In practice, the modulation schemes used by the eMBB and the URLLC traffics
can be different. In addition, the eMBB receiver, which is unaware of the
punctured part of the transmission, decodes the received signal using a
maximum likelihood receiver. Based on this, the probability of receiving the
punctured eMBB symbols in error depends on the Euclidean distance between both
the transmitted URLLC symbols and the punctured eMBB symbols. As an
illustration, let us consider the case when the URLCC traffic employs BPSK
modulation and the eMBB traffic employs 4-QAM modulation and let us suppose
that the transmitted URLLC symbol is {0}. As shown in Fig. 2, it is preferred
to puncture the eMBB symbols 00 and 01, since they have the lowest Euclidean
distance to 0 as compared to the symbols 10 and 11, and therefore, a lower
resulting error probability. Specifically, as shown in Fig. 2, we can say that
the symbols {0, 00, 10} belong to the same region which so-called the
similarity region according to the following definition:

Figure 2: Example for different modulation schemes and similarity regions.
$d^{0,01}$ is the minimum distance between the eMBB symbol 01 and the decision
line of the URLLC symbol 0.

######  Definition 1.

Let us consider two QAM schemes with modulation orders $m$ and $n$,
respectively, and let us consider the diagram that has the superposition of
their respective constellations. The similarity region of the two above
modulation schemes is a region of the resulting constellation diagram that
contains only one constellation point from the modulation that has the lowest
order, i.e. $\min(m,n)$, and $\frac{\max(m,n)}{\min(m,n)}$ constellation
points from the modulation that has the highest order, i.e., $\max(m,n)$,
which have the minimum Euclidean distance with the constellation point of the
modulation that has the lowest order. Based on this, there exist exactly
$\min(m,n)$ similarity regions.

As an illustration for Definition 1, let us consider the case when the eMBB
traffic has a modulation order of $m=4$ and the URLLC traffic has a modulation
order of $n=2$. The superposition of the constellations diagrams of the eMBB
and URLLC modulations is shown in Fig. 2. The resulting diagram can be divided
into $\min(m,n)=2$ similarity regions, namely, similarity region 0 and
similarity region 1, where each similarity region contains only one
constellation point from the URLLC’s modulation and
$\frac{\max(m,n)}{\min(m,n)}=2$ constellation points from the eMBB modulation
that have the lowest Euclidean distance with the included constellation point
of the URLLC modulation. Moreover, according to Definition 1, the eMBB and
URLLC symbols are divided into several sets and each set consists of several
eMBB and URLLC symbols. The number of eMBB and URLLC symbols depends on the
relation between the modulation orders of the eMBB and URLLC. In practice, the
modulation schemes of the URLLC and eMBB services may have the same order
(i.e, $m=n$) or different ones ($m\neq n$). Accordingly, we classify the
relationship between the eMBB and URLLC modulation orders into the following
classes:

Figure 3: Similarity region between eMBB traffic and URLLC load with 16-QAM
(for eMBB) and BPSK (for URLLC). $d^{0,0001}$ is the minimum distance between
the eMBB symbol 0001 and the decision line of the URLLC symbol 0.

  * •

Similar-Modulation-Order: In this case, the URLLC and eMBB have the same
modulation order, i.e., $m=n$. Hence, each similarity region consists of one
eMBB symbol and one URLLC symbol. This symbol is named as the Region-index-
symbol.

  * •

Lower-URLLC-Modulation-Order: In this case, the URLLC modulation order is
lower than that of the eMBB, i.e., $m>n$. Accordingly, each similarity region
consists of one URLLC symbol and $\frac{m}{n}$ eMBB symbols. Similarly, we can
rename the URLLC symbol as the Region-index-symbol and the eMBB symbols as
mapping-symbols.

  * •

Higher-URLLC-Modulation-Order: In this case, the URLLC modulation order is
higher than that of the eMBB, i.e., $m<n$. Accordingly, each similarity region
consists of one eMBB symbol and $\frac{n}{m}$ URLLC symbols. We can rename the
eMBB symbol as the Region-index-symbol (that denotes the similarity-region)
and the URLLC symbols as mapping-symbols.

Practically, the Euclidean distance between the mapping-symbols and the
Region-index-symbol varies according to their locations on the constellation.
For clarity, as shown in Fig. 3, assume the URLLC and eMBB traffic are
modulated by the BPSK and 16-QAM, respectively. According to Definition 1, the
constellation is divided into two similarity regions. The first region has
URLLC symbol $0$ as the Region-index-symbol and the second region has the
URLLC symbols $1$ as the Region-index-symbol. Without loss of generality, let
the transmitted URLLC symbol be $0$. Then, the mapping-symbols belonging to
the same similarity region, i.e. {0000, …, 0111}, can be treated as 0 based on
the BPSK maximum-likelihood receiver. On the other hand, the eMBB receiver
receives the transmitted symbol correctly; hence the eMBB SER is not degraded.
On the other hand, the SER of the URLLC becomes worse, as the symbol energy
varies based on the eMBB constellation, which is 16-QAM in this example. We
note that symbols {0000, 0010, 0011, 0001} have the lowest Euclidean distance
with URLLC symbol $0$, hence they have lower SER at the URLLC receiver (See
Fig. 3 for more elaboration.) In light of the above discussion, we define the
symbol similarity as follows.

######  Definition 2.

The similarity relation between the Region-index-symbol $s_{x}$ and the
mapping-symbol $s_{y}$ in the same similarity region can be:

  * •

Absolute-similar: if $P(\hat{s}\neq s_{x}|s_{x}\text{ was sent})-P(\hat{s}\neq
s_{x}|s_{y}\text{ was sent})\geq 0$.

  * •

Strongly-similar: if $-\epsilon\leq P(\hat{s}\neq s_{x}|s_{x}\text{ was
sent})-P(\hat{s}\neq s_{x}|s_{y}\text{ was sent})<0$.

  * •

Weakly-similar: if $P(\hat{s}\neq s_{x}|s_{x}\text{ was sent})-P(\hat{s}\neq
s_{x}|s_{y}\text{ was sent})<-\epsilon$,

where $\epsilon\approx 0$ depends on the target URLLC SER. Accordingly, we can
call the set of symbols, which are absolute-similar and strongly-similar, as
the enhanced similarity region.

######  Definition 3.

The enhanced similarity region is a subset of the similarity region which
includes the Region-index-symbol and mapping-symbols that satisfy
$P(\hat{s}\neq s_{x}|s_{y}\text{ was sent})-P(\hat{s}\neq s_{x}|s_{x}\text{
was sent})\leq\epsilon$.

###  III-B URLLC Encoding at the BS

According to the proposed puncturing strategy, it is preferred to puncture
eMBB symbols such that the amount of symbol mismatch between the transmitted
$\zeta-$symbols of the URLLC traffic and the punctured eMBB is minimized,
i.e., smaller Hamming distance. Based on the URLLC-eMBB relationship, the
encoding at the BS is illustrated as follows.

  * •

Similar-Modulation-Order: The BS encodes the URLLC traffic according to the
desired modulation order $n$ while puncturing the eMBB symbol sequences that
has maximum similarity (Absolute-similar), i.e., maximize the similar eMBB-
URLLC OFDM symbols.

  * •

Lower-URLLC-Modulation-Order: Similar to the Higher-URLLC-Modulation-Order
case, the BS selects for puncturing the eMBB block that has a maximum number
of absolute-similar, strongly-similar, and weakly-similar symbols. To
accommodate the URLLC traffic, the BS can transmit either the encoded URLLC
symbol or the ongoing eMBB symbol, as described below.

  * •

Higher-URLLC-Modulation-Order: the BS encodes the URLLC traffic according to
the desired modulation order $n$ while puncturing the eMBB sequences that have
a maximum similarity. In other words, the BS selects the eMBB symbol sequence
that maximizes the number of absolute-similar, strong-similar, and weak-
similar symbols. Compared to the Similar-Modulation-Order case, the impact of
puncturing on the eMBB SER can not be eliminated.

When the URLLC modulation order is lower than that of the eMBB, the BS can
transmit the URLLC symbol or keep the ongoing eMBB symbol, as follows.

  * •

URLLC mapper: The BS transmits the encoded URLLC symbols. Hence, the impact of
puncturing on the eMBB resources can not be eliminated. To elaborate, let us
consider the following example. If we have the following URLLC sequence {0, 1,
1, 0}, and the punctured eMBB sequence is {00, 11, 10, 01} (See Fig. 2).
According to the similarity region definition, these URLLC symbols are in the
same similarity region of the punctured eMBB sequence. Assuming maximum-
likelihood detection, we can roughly say 50% of the punctured eMBB symbols
will be correctly received, which translates to a high SER at the eMBB
receiver (Fig. 2.)

  * •

Similarity region mapper (SRM): To overcome the high SER of eMBB using the
URLLC mapper, the BS transmits the eMBB symbol instead of the URLLC symbol, if
they belong to the same similarity region, otherwise the URLLC symbol is
transmitted. For example, assume that the modulation schemes of URLLC and eMBB
are BPSK and 16-QAM, respectively, as shown in Fig. 3. Also assume that the
transmitted URLLC symbol is 0. Then, any eMBB symbol belonging to the same
similarity region, i.e. {0000, …, 0111}, will be received as 0 at the URLLC
receiver with error probability less than 1. On the other hand, the eMBB user
receives the transmitted symbol correctly as the symbol is not affected by the
puncturing process; hence the eMBB SER will improve. On the other hand, the
SER of the URLLC becomes worse, since the eMBB symbols have different minimum
distances from the URLLC decision boundary, which is 16-QAM in this example.

  * •

Enhanced Similarity region mapper (ESRM). To solve the high SER of the URLLC
of the SRM, only the eMBB symbol belonging to the same enhanced similarity
region, (that have better minimum distances from the URLLC decision boundary),
i.e {0000, 0010, 0011, 0001}, are transmitted instead the URLLC symbol,
otherwise the URLLC symbol is transmitted.(See Fig. 3 for more elaboration.)

evaluate $\mathcal{P}_{1}$ ;

if  _$\mathcal{P}_{1} >\epsilon_{u}$_ then

encode the URLLC packet using URLLC mapper;

else

keep transmitting the eMBB symbols that satisfy the similarity conditions in
Definition 2;

end if

Algorithm 1 Proposed SRM/ESRM mapper

Algorithm 1 illustrates an example for the mechanism of the SRM/ESRM. Let
$\epsilon_{u}$ and $\mathcal{P}_{1}$ be the targeted SER of the URLLC traffic
and the expected SER if the SR/ESRM is used, respectively. The algorithm
starts by checking the activation condition $\mathcal{P}_{1}\leq\epsilon_{u}$,
if the activation condition is satisfied, the eMBB symbol is transmitted if
they are satisfying the similarity condition, otherwise the URLLC symbol will
be sent.

###  III-C URLLC and eMBB Decoding

As mentioned above, the eMBB receiver does not have the knowledge of the
punctured symbols and it decodes the received sequence as if no puncturing
took place. On the other hand, the URLLC receiver will perform the decoding
process normally based on their modulation scheme. For illustration, as shown
in Fig. 2, we assume that BPSK and QPSK are used to encode both the URLLC and
eMBB traffic, respectively. Let the transmitted URLLC symbol be {0}. The eMBB
decoder will translate the received symbol with probability($\approx 50\%)$ as
{00, 01}. Similarly, the URLLC receiver will decode the received symbol {00}
as {0} with probability $(1-P_{2}(0.5\gamma_{\rm e}))$. For the case when the
ESRM is used, i.e., keep transmitting the similar eMBB symbols, the URLLC
receiver will decode the received signal assuming the desired decoder, BPSK in
this example. In summary, the decoding process at either the eMBB or the URLLC
receivers is not changed.

##  IV Performance evaluation

###  IV-A eMBB SER Analysis

Although both SER and bit error rate (BER) can be used to represent the impact
of puncturing on the eMBB, the SER can represent the puncturing in the RE
(symbol) level instead of the RB level which gives more sense about the
proposed strategy. Hence, we use the SER, denoted by $P$, to measure the
impact of the proposed puncturing strategy on the eMBB traffic. As Without
loss of generality, let $L_{m}=p_{m}L$ be the average number of eMBB symbols
with modulation scheme $m$. Also, let the average number of eMBB symbols
punctured due to the URLLC traffic with modulation order $n$ be $l_{n,m}$.
According to the total probability theorem, the SER of the eMBB traffic under
the effect of both the wireless channel and the presence of URLLC load can be
expressed as

$\begin{split}P\left(\gamma_{\rm
e},l\right)=\sum_{m=2}^{M}p_{m}\times\Bigg{[}P_{m}(\gamma_{\rm
e})\times\left(1-\frac{\sum_{n=2}^{N}l_{n,m}}{L_{m}}\right)\\\
+\sum_{n=2}^{N}P_{n,m}\left(\gamma_{\rm
e},l_{n,m}\right)\times\frac{l_{n,m}}{L_{m}}\Bigg{]},\hskip
56.9055pt\end{split}$ (1)

where $P_{m}(\gamma_{\rm e})$ is the SER of the eMBB traffic with modulation
order $m$ due to the channel error only, and $P_{n,m}(\gamma_{\rm e},l_{n,m})$
is the SER due to the channel and URLLC traffic with modulation order $n$. To
quantify the actual effect on the eMBB traffic, we start with the following
definition.

Definition 4: Consider an eMBB and URLLC traffic with modulation orders $m$
and $n$, respectively. The average effectively punctured symbols,
$\mathcal{L}_{n,m}$, of eMBB traffic is a portion of the punctured eMBB
symbols, $l_{n,m}$, in which the transmitted URLLC symbol, $s_{u}$, has a
different similarity region from that of the punctured eMBB symbol, $s_{e}$,
and its range is $0\leq\mathcal{L}_{n,m}\leq l_{n,m}$.

Figure 4: Effectively punctured eMBB symbols.

Fig. 4 illustrates the relation between $\mathcal{L}_{n,m}$ and $l_{n,m}$ for
the case of similar modulation schemes, BPSK in this example. It shows that,
due to the similarity region between the punctured eMBB and URLLC symbols,
some of the punctured eMBB symbols are not affected by the puncturing process.
In general, for eMBB and URLLC traffic with modulation orders $m$ and $n$,
respectively, the expected number of the effectively eMBB punctured symbols is
$\mathcal{L}_{n,m}<=l_{n,m}$. Accordingly, (1) can be written as follows.

$\begin{split}P\left(\gamma_{\rm
e},l\right)=\sum_{m=2}^{M}p_{m}\times\Bigg{[}P_{m}\left(\gamma_{\rm
e}\right)\times\left(1-\frac{\sum_{n=2}^{N}l_{n,m}}{L_{m}}\right)+\quad\\\
\sum_{n=2}^{N}\left({P}_{n,m}(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m})\times\frac{\overline{\mathcal{L}}_{n,m}}{L_{m}}+{P}_{n,m}\left(\gamma_{\rm
e},\mathcal{L}_{n,m}\right)\times\frac{\mathcal{L}_{n,m}}{L_{m}}\right)\Bigg{]},\end{split}$
(2)

where ${P}_{n,m}(\gamma_{\rm e},{\overline{\mathcal{L}}}_{n,m})$ and
${P}_{n,m}\left(\gamma_{\rm e},\mathcal{L}_{n,m}\right)$ are the error
probabilities (which will be derived in the sequel) of the eMBB traffic in
$\overline{\mathcal{L}}_{n,m}$ and ${\mathcal{L}}_{n,m}$, respectively. Where
$\overline{\mathcal{L}}_{n,m}=l_{n,m}-{\mathcal{L}}_{n,m}$.

Equation (2) shows the SER of the eMBB traffic under the impact of the
wireless channel and the presence of URLLC load. The first term represents the
average error probability for the fraction of eMBB sequence impacted by the
channel errors only. The second term represents the average error probability
of the punctured eMBB symbols that have the same similarity region to the
URLLC symbols. The third term is the average error probability of the
effectively punctured eMBB symbols.

###  IV-B Puncturing Parameters Evaluation

In this section, we analyze the puncturing parameters ${\mathcal{L}}_{n,m},$
${P}_{n,m}(\gamma_{\rm e},{\overline{\mathcal{L}}}_{n,m})$ and
${P}_{n,m}\left(\gamma_{\rm e},\mathcal{L}_{n,m}\right)$ for the proposed
puncturing strategy. The puncturing parameters depend on the modulation
schemes of eMBB and URLLC, and how the URLLC traffic is distributed, i.e.,
$l_{n,m}$. Without loss of generality, the average punctured eMBB symbols (the
average URLLC load) is assumed to be known as it depends on the arrival rate,
$\lambda$, of the URLLC traffic. Also, the channel error, $P_{m}(\gamma_{\rm
e})$, depends on the channel and the modulation scheme. For example, the SER
of eMBB under additive white Gaussian noise (AWGN) and/or Rayleigh fading with
SNR per symbol ${\gamma}_{e}$ are [29, 30]:

$P_{m}(\gamma_{\rm
e})\approx\begin{cases}4a\,Q(\sqrt{\frac{3{\gamma}_{e}}{m-1}}),\hskip
85.27806pt\text{AWGN},\\\
2a\,(1-b)-a^{2}(1-\frac{4b}{\pi}\tan^{-1}(\frac{1}{b})),\text{Rayleigh,}\end{cases}$
(3)

where $a=(1-\frac{1}{\sqrt{m}})$ and $b=\sqrt{\frac{3{\gamma_{\rm
e}}}{2(m-1)+3{\gamma_{\rm e}}}}$.

####  IV-B1 Average Effectively Punctured Symbols

Without loss of generality, assume the distribution of the symbol similarity
between a URLLC block and a punctured eMBB sequence with modulation orders
$n\text{ and }m$ follows the Binomial distribution $B~{}(\zeta,\eta_{n,m})$.
Therefore, the average similarity between the two traffic blocks can be
defined as $U_{n,m,\zeta}\triangleq\zeta\times\eta_{n,m}$, where $\eta_{n,m}$
(probability that any two symbols are similar) is

$\eta_{n,m}=\sum_{j=0}^{n-1}\sum_{i=0}^{m-1}p_{j}\,p_{i},\forall
i,j\in\Omega,$ (4)

where $p_{j}$ and $p_{i}$ are the probabilities of sending symbol $j,i$ of the
eMBB and URLLC traffic, respectively. Accordingly, we can represent the
average effectively punctured eMBB symbols, in terms of the average URLLC load
in the following definition.

Definition 5: Consider a URLLC and eMBB traffic with modulation orders
$n\text{ and }m$, respectively. Also, let the eMBB block length and the
average length of punctured eMBB symbols by the URLLC traffic be $L_{m}$ and
$l_{n,m}$, respectively. The average effectively punctured eMBB symbols (i.e.,
modified) is given by:

${\mathcal{L}}_{n,m}=\left(1-\frac{\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})}{\zeta}\right)\,l_{n,m}\,(\text{symbols}),\vspace{-.01
in}$ (5)

where $\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})$ is the expected number of eMBB
symbols which have the same similarity region with the transmitted
$\zeta-$symbols of URLLC block. The term
$\frac{\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})}{\zeta}$ represents the ratio
(percentage) of similarity between both services. Then,
$\left(1-\frac{\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})}{\zeta}\right)\,l_{n,m}$
is the actual punctured eMBB symbols.

The definition in (5) gives an expression for the average length of the
effectively punctured (modified) eMBB symbols. However, it does not quantify
the average similarity, $\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})$, between the
URLLC and eMBB sequences. In fact, $\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})$
depends on the URLLC block size and the eMBB search space of size ${N}<L$.
Lemma 1 gives an approximated value for
$\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})$.

######  Lemma 1.

Let $L_{m}$ denote the average eMBB traffic, and let $l_{n,m}$ be the average
number of punctured eMBB symbols. Assume that the URLLC traffic is divided
into blocks with $\zeta$-symbols each. An upper bound on the expected
similarity between the URLLC and eMBB traffic is given by

$\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})=\frac{1}{\lceil\frac{l_{n,m}}{\zeta}\rceil}\sum_{1}^{\lceil\frac{l_{n,m}}{\zeta}\rceil}\sum\nolimits_{k=0}^{\zeta-1}\left[1-\\{F(k)\\}^{L_{m}-\zeta}\right],$
(6)

where, $F(k)=\sum_{{j}=0}^{k}\left(\begin{array}[]{l}{\zeta}\\\
{j}\end{array}\right)\left(\eta_{n,m}\right)_{j}(1-\eta_{n,m})^{\zeta-j}$.

###### Proof.

See Appendix A for the proof. ∎

For large $\zeta$, $\mathcal{U}_{n,m}(\zeta,l_{n,m},L_{m})$ reduces to
$\zeta\eta_{n,m}$. Hence, we can further reduce the block size, i.e., $\zeta$,
and consequently the average similarity will increase. Practically, decreasing
the size of $\zeta$ will increase the signalling overhead. Therefore, a proper
selection of $\zeta$ is important. Indeed, we examine the effect of different
values of $\zeta$ on the performance of the proposed scheme in Section V.

####  IV-B2 SER of the Effectively Punctured Symbols

The SER of the Effectively Punctured Symbols, ${P}_{n,m}\left(\gamma_{\rm
e},\mathcal{L}_{n,m}\right)$, strictly depends on the relation between the
modulation schemes of both URLLC and punctured eMBB traffic.
$\mathcal{L}_{n,m}$ is the average number of eMBB symbols that are wrongly
transmitted. In other words, the transmitted symbols belong to another region.
${P}_{n,m}\left(\gamma_{\rm e},\mathcal{L}_{n,m}\right)$ is then expressed as

${P}_{n,m}\left(\gamma_{\rm
e},\mathcal{L}_{n,m}\right)=\sum_{s_{j}\in\Omega}\sum_{s_{i}\notin\Omega}p(s_{j}|s_{i}\text{
sent})P_{err}(s_{j}|s_{i}\text{ sent}),$ (7)

where $s_{j}$ and $s_{i}$ are the effectively punctured eMBB symbol and the
transmitted URLLC symbol, respectively. We can upper bound
${P}_{n,m}\left(\gamma_{\rm e},\mathcal{L}_{n,m}\right)$ with the closed form
expression

${P}_{n,m}\left(\gamma_{\rm e},\mathcal{L}_{n,m}\right)\leq
1-P_{n}\left(\gamma_{\rm e}\right)\times\left(\frac{1}{m-1}\right)\approx 1,$
(8)

where $P_{n}\left(\gamma_{\rm e}\right)$ is the probability of error under the
URLLC modulation condition. For example, let BPSK be the modulation order used
by both URLLC and eMBB traffic and $P_{2}(\gamma_{\rm e})=10^{-2}$, then
$P_{2,2}(\gamma_{\rm e},\mathcal{L}_{n,m})=1-10^{-2}\approx 1$.

####  IV-B3 SER of the non-Effectively Punctured Symbols

The SER of the non-Effectively punctured symbols, ${P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)$ is summarized as follows.

  * •

Similar-Modulation-Order: The modulation schemes of URLLC and eMBB are
similar. Accordingly, non-effectively punctured symbols are similar to the
punctured symbol, hence ${P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)$ is expressed as:

${P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)={P}_{m}\left(\gamma_{\rm e}\right)$ (9)

  * •

Lower-URLLC-Modulation-Order: similar to Similar-Modulation-Order, in this
case, the non-effectively punctured eMBB symbols are only affected by the
channel conditions. Hence, ${P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)$ is expressed as follows:

${P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)={P}_{m}\left(\gamma_{\rm e}\right)$
(10)

  * •

Higher-URLLC-Modulation-Order: As the energy of the transmitted URLLC symbols
varies, an exact expression for the URLLC SER is not easy to obtain. Hence, we
obtain an upper bound for ${P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)$ based on the minimum distance,
${d}^{i,j}$, the transmitted URLLC symbol and the decision boundary of the
eMBB symbol. Accordingly, ${P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)$ is expressed as:

${P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)=\sum_{s_{j}\in\Omega}\sum_{s_{i}\in\Omega}p(s_{j}|s_{i}\text{
sent})P_{err}(s_{j}|s_{i}\text{ sent}),$ (11)

where $P_{err}(s_{j}|s_{i}\text{ sent})$ is expressed as

$P_{err}(s_{j}|s_{i}\text{ sent})=P_{j}(\gamma_{\rm e}{{d^{i,j}}^{2}})$ (12)

###  IV-C URLLC SER Analysis

The SER of the URLLC traffic is only affected when the URLLC traffic is
modulated using the SR/ESRM. This is because the the energy of the transmitted
symbols are different than the actual URLLC symbols. In other words, the
energy of the non-effectively punctured eMBB symbols is varies. Accordingly,
an exact expression for the URLLC SER is not easy to obtain. Hence, we drive
an upper bound expression for the URLLC SER based on the minimum distance of
the transmitted symbol and decision boundary, as

$\begin{split}\mathcal{P}_{n,m}\left(\gamma_{\rm
u}\right)=\left(1-\sum_{s_{j}}\sum_{s_{i}}p\left(s_{i}|s_{j}\text{
sent}\right)\right)\mathcal{P}_{n}\left(\gamma_{\rm u}\right)\\\
+\sum_{s_{j}}\sum_{s_{i}}p\left(s_{i}|s_{j}\text{
sent}\right)\mathcal{P}_{n}\left(\gamma_{\rm
u}{d^{i,j}}^{2}\right),\end{split}$ (13)

Equation (13) shows the SER of the URLLC traffic when the SR/ESRM is used. The
first term represents the average error probability for the fraction of URLLC
sequence impacted by the channel errors only. The second term represents the
average error probability of the URLLC symbols that have the same similarity
region to the eMBB symbols (encoded by the SRM). In fact, the SER loss is
equivalent to a power loss, $W_{dB}$ which has the following expression:

$W_{dB}\approx 10\sum_{s_{j}}\sum_{s_{i}}p\left(s_{i}|s_{j}\text{
sent}\right)\log_{10}\left({d^{i,j}}^{2}/{d^{i}}^{2}\right).$ (14)

The expression in (14) evaluates the average URLLC loss in dB. The term
$\log_{10}\left({d^{i,j}}^{2}/{d^{i}}^{2}\right)$ is the power loss for each
URLLC symbol in terms of the ratio between the distance of the URLLC and the
transmitted eMBB symbols form the decision boundary.

###  IV-D SER Scaling

In light of the above discussion, we can observe that the eMBB SER is a
function the SNR and the average similarity of the punctured eMBB symbols.
When the SNR increases, the SER improves and it is asymptotically equal to the
puncturing errors. Then, the eMBB SER based on (2) is approximated as

$P\left(l\right)\approx\sum_{m=2}^{M}p^{m}\frac{\sum_{n=2}^{N}{\mathcal{L}}_{n,m}}{L_{m}}.$
(15)

On the other hand, as the similarity increases ($L$ increases or $\zeta$
decreases), the eMBB SER reduces to only channel errors, which can be
approximated as

$\begin{split}P\left(\gamma_{\rm
e},l\right)\approx\sum_{m=2}^{M}p_{m}\times\Bigg{[}P_{m}\left(\gamma_{\rm
e}\right)\times\left(1-\frac{\sum_{n}^{N}\overline{\mathcal{L}}_{n,m}}{L_{m}}\right)\\\
+\sum_{n}^{N}{P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)\frac{\overline{\mathcal{L}}_{n,m}}{L_{m}}\Bigg{]}.\end{split}$
(16)

We observe that the eMBB performance strictly depends on the SNR and the
average similarity. As the SNR increases, the eMBB SER becomes dominated by
the puncturing errors, while increasing the similarity will reduce the SER to
errors due to the channel condition.

###  IV-E eMBB Loss Function

The function that represents the eMBB loss associated to the puncturing
schemes can be either a linear, convex or a threshold function [16]. The
linear loss function, i.e. $h(x)=\alpha\,x$, has been widely used to study the
impact of the superposition/puncturing scheme [16, 17]. The expected loss of
an eMBB traffic, based on the linear function, is the ratio between the
punctured eMBB symbols to total eMBB symbols:

$E[h\left(l_{m}\right)]=\frac{\sum_{n}^{N}l_{n,m}}{L_{m}},$ (17)

where $l_{m}=\sum_{n}l_{n,m}$. The loss function in (17) is widely coupled
with the eMBB rate [12, 13, 14, 15, 16, 17, 18]. Taking into consideration the
effectively punctured symbols and similar symbols, and using results in (2)
and (5), the expected loss in (17) can be generalized as follows:

$\begin{split}E[h\left(l_{m}\right)]=\frac{1}{L_{m}}\sum_{n}^{N}\bigg{(}{{P}_{n,m}\left(\gamma_{\rm
e},\overline{\mathcal{L}}_{n,m}\right)\overline{\mathcal{L}}_{n,m}}\\\
+{{P}_{n,m}\left(\gamma_{\rm
e},{\mathcal{L}}_{n,m}\right){\mathcal{L}}_{n,m}}\bigg{)}.\end{split}$ (18)

For clarity, assume a retransmission-based puncturing is adopted. Then, all
punctured eMBB symbols are lost, i.e.,
$E[h\left(l_{m}\right)]=\frac{l_{m}}{L_{m}}.$

###  IV-F Proposed Search Algorithm

$c_{k}\leftarrow 0\,\forall k\in[1,{K}]$;

Step 1: Similarity weight calculation;

for  _ $k=1\rightarrow{K}$_ do

for  _$t=1\rightarrow\zeta$_ do

count symbols in the same similarity region;

if  _${s}_{e}^{t}\,\ &\,{s}_{u}^{t}\in\Omega$ _ then

$\mathbf{c}_{k}\leftarrow\mathbf{c}_{k}+1$;

end if

end for

end for

Step 2: eMBB block selection;

$k^{*}\leftarrow\underset{k\in[1,{K}]}{\arg\max}\,c_{k}$;

Algorithm 2 Proposed search Algorithm

The latency constraint is a critical factor to maintain QoS of the URLLC
service. Therefore, we present a fast search algorithm, of time complexity
$O({K})$, that exploits the similarity between the URLLC block over multiple
eMBB traffic sequences with different modulation schemes. Initially, the
algorithm associates a counter $c_{k}$ to each eMBB in ${K}$. The subset,
${K}$, of the possible eMBB blocks for puncturing depends on the latency
requirement of the URLLC traffic. Specifically, ${K}$ should be small for the
URLLC traffic with strict latency constraint, and a relatively larger ${K}$
otherwise. Moreover, the URLLC traffic with weak latency constraint implies
that several mini-slots (URLLC-slot) is allowed for allocating the URLLC
block.
As shown in Algorithm 2, the proposed algorithm has two steps: the first step
counts the similar symbols between the eMBB blocks and the URLLC block; the
second step selects the suitable eMBB block for puncturing. The first step
contains two loops, inner and outer loop, which describe the number of eMBB
blocks for possibly punctured and the number of URLLC symbols per block. As
illustrated in Algorithm 1, the similarity region of the eMBB symbol,
$s_{e}^{t}$, is compared with the similarity region of the URLLC symbol,
$s_{u}^{t}$. Accordingly, the counter $c_{k}$ is incremented by one when both
symbols are in the same similarity region, otherwise it remains not
incremented. In the second part, the BS selects the eMBB block that has
maximum similarity with the URLLC block. In other words, the punctured eMBB
block $k^{*}$ should have a maximum count of similar symbols with the URLLC
block.
To mitigate adverse impact on eMBB traffic, the URLLC load is segmented into
smaller blocks, i.e., 1 RB [18]. Hence, Algorithm 3 guarantees that all URLLC
segments are delivered in the correct order, i.e., the URLLC symbols sequence
is correct. The algorithm initially divides the search space, ${K}$, into
ordered and equal subsets, ${K}_{1}<{K}_{2}<{K}_{3}...<{K}_{Z}$, where $Z$ is
the number of URLLC segments. Then, Algorithm 2 is applied to each segment on
the corresponding subset. To enhance the algorithm, the remaining eMBB blocks
of subset, ${K}_{k}$, which satisfy the inequality $k^{*}_{z}<k_{z}<{K}_{k}$,
are merged with the next subset, ${K}_{k+1}$, using the $merge()$ function.
Search algorithm time complexity : It can be easily shown that the proposed
algorithm has a time complexity of $O({K})$ which make it an efficient and
practical solution. For clarity, the search algorithm (Algorithm 2) consists
of two steps, i.e., Step 1 and Step 2. Step 1 consists of one outer loop and
one inner loop of ${K}$ and $\zeta$ iterations, respectively. Hence, for a
URLLC block with fixed number of symbols $\zeta$, Step 1 has a time complexity
of $O({K})$. Step 2 aims to select the maximum counter over ${K}$ elements. In
general, Step 2 performs ${K}$ comparison which means Step 2 also has a time
complexity of $O({K})$. As a result, the proposed algorithm (Algorithm 1) has
a low time complexity of $O({K})$.

${k}^{*}_{z}\leftarrow 0\,\forall z\in[1,Z]$;

for  _ $z=1\rightarrow Z$_ do

$k^{*}_{z}\leftarrow Algorithm\,1({K}_{z})$;

$N_{k+1}\leftarrow merge(k^{*}_{z}+1\rightarrow{K}_{z},{K}_{z+1})$;

end for

Algorithm 3 Search segmentation Algorithm.

##  V Simulation Results

###  V-A Simulation Setup

In this section, we carry out various simulations to evaluate the performance
of the proposed puncturing strategy. We consider a wireless network which
consists of one BS and eMBB and URLLC traffic. We assume that the eMBB traffic
belongs to 10 eMBB users and the URLLC packet arrival follows the Poisson
distribution with arrival rate, $\lambda$ and each packet size is $\zeta=96$
bits. We also assume that the BS has $L=1200$ downlink frequency resources
(RE). We assume the channel between the BS and the eMBB and URLLC receivers is
Rayleigh fading, wherein the eMBB channel gain remains constant for two time
slots (14-sTTI) and one eMBB block is transmitted within this period.
Moreover, we assume the served users are located in different distance from
the BS, hence the path loss is considered with a path loss exponent equals
$3$. The noise at the receiver is assumed to be complex AWGN
$\mathcal{CN}(N_{0},0)$, where $N_{0}=10^{-9}$ is the noise power. The BS can
use BPSK, 4-QAM, 16-QAM or 64-QAM to modulate the eMBB traffic. Particularly,
the BS adopts the modulation order $m\in\\{4,16,64\\}$ such that the channel
SER is less than or equal $0.01$, otherwise BPSK is adopted. Moreover, we
assume that the CSI of the URLLC traffic is not available at the BS. Hence,
the URLLC traffic is modulated using only BPSK to achieve the maximum
reliability.
The performance of the similarity puncturing strategy is evaluated for
different transmitting power and arrival rate, i.e., $\lambda=7$ and
$\lambda=3.5$ packets per millisecond (p/msec). we assume that the eMBB users
are unaware of the punctured resources, so we consider the code-based
puncturing proposed in [4, 10] as a baseline algorithm. Generally, when the
eMBB receiver is unaware of the punctured resources of the transmission, the
received signal is decoded as useful signal. Resource proportional (RP)
placement is used to allocate the URLLC traffic as it gives the optimal
solution for the linear loss model [16].
In the following, we start by showing the advantage of the proposed strategy
on the performance of the eMBB traffic in terms of the spectral efficiency,
SER and reliability. Second, we investigate the performance of the proposed
strategy on the URLLC traffic by considering the URLLC SER and reliability.
Finally, we evaluate the computational time of the proposed puncturing
strategy.

Figure 5: Average eMBB loss relative to the punctured symbols. Adopted
$\zeta=24~{}(1~{}RB)$ and $\gamma_{\rm e}=40~{}dB$

###  V-B eMBB Traffic Performance

####  V-B1 Spectral Efficiency

In (18), we expressed the loss function of the eMBB traffic in terms of the
SER of the punctured symbols. To measure the efficiency of the proposed
strategy and select the optimal ${K}$, we evaluate the average eMBB loss
(contaminated eMBB symbols) for both the URLLC mapper and the ESRM while
varying the size of the search space ${K}$ (see Fig. 5). The results show that
the percentages of the contaminated (lost) eMBB symbols for the ESRM are 18%
and 44% compared to 59% and 93% for the URLLC mapper for BPSK-4QAM and
BPSK-16QAM, respectively. This enhancement of the ESRM results from
transmitting the punctured eMBB symbols that fall in the same similarity
region of transmitted URLLC symbols ( i.e., punctures the eMBB symbols which
has different similarity region). Moreover, the results show that the ESRM for
(BPSK-4QAM) achieves the same loss of (BPSK-BPSK). This because the
probability of similarity in the similarity region is the same, i.e., 0.5.
However, the eMBB SER is enhanced for BPSK-4QAM using ESRM, this comes at the
expense of the URLLC SER which decreases by 2.5 dB according to equation in
(14). The results show that the proposed scheme performs better than the code-
based puncturing, i.e., ${K}=1$ for URLLC mapper. When ${K}$ is small, it
indicates that a small number of eMBB blocks can be punctured, due to the eMBB
QoS requirement. For example, the point when ${K}=300$, the search algorithm
scans only $300$ eMBB blocks (possibilities) out of ${K}=1200$ to allocate the
URLLC traffic. In other words, the QoS requirements limits ${K}$ (number of
eMBB blocks the URLLC is compared with): which means that the smaller ${K}$,
the more the eMBB traffic with strict QoS requirements.

(a) eMBB SER vs transmission power in dBm for different URLLC arrival rate.

(b) eMBB SER vs transmission power in dBm for different URLLC block (segment)
size. $\lambda=7~{}p/msec$

Figure 6:  eMBB SER of the proposed puncturing strategy vs transmission power
in dBm.

####  V-B2 eMBB SER

Fig. 6 illustrates the performance of the proposed puncturing scheme in terms
of the SER of eMBB. We make the following observations from the results.

  * •

The proposed algorithm achieves better SER for the eMBB traffic compared to
the code-based baseline. In other words, the proposed scheme can achieve the
target eMBB SER with lower transmission power. Particularly, the achieved gain
increases with the SNR (transmission power) (it reaches 10 dB at high SNR), as
shown in Fig. 6(a). Also, the figure shows that the gain of our proposed
method is negligible at low transmission power (low SNR) since the channel
errors are the dominant here, however the gain improves as the SNR increases
since the error at high SNR is dominated by puncturing. Note also that the
gain saturates at high SNR according to (15), with no further improvement as
the BS allocates more transmit power for the eMBB symbols (puncturing errors
dominates).

  * •

It is intuitive that the SER of the eMBB traffic deteriorates as the URLLC
load increases, as illustrated in Fig. 6(a). For instance, the eMBB SER
saturates at $0.02$ and $0.04$ at both $\lambda=3.5$ and $\lambda=7$,
respectively. This increase in SER is due to that as the URLLC load increases,
the more eMBB resources are punctured, which leads to more SER.

  * •

Fig. 6(b) illustrates the eMBB SER for different URLLC block segment, $\zeta$.
For instance, eMBB SER is enhanced when the URLLC block size $\zeta$ becomes
smaller, as illustrated in Fig 6(b). Reducing $\zeta$ increases the
probability of similarity, which decreases the effectively punctured symbols.
In this case, the trade-off between reducing $\zeta$ and the overhead should
be optimized to achieve better spectral efficiency. In other words, as $\zeta$
decreases, the overhead increases.

(a) eMBB reliability vs the SER.

(b) eMBB users reliability. Adopted SER=.01 and $\lambda=7p/msec$

Figure 7:  eMBB traffic reliability of the proposed algorithm. $P=10~{}dBm$

####  V-B3 eMBB Reliability

In this section, we evaluate the reliability of the eMBB traffic as a function
of the achieved SER. In general, we define the reliability of both URLLC and
eMBB traffic

$\text{reliability}=\frac{\text{Number of blocks satisfying the targeted
SER}}{\text{Total transmitted blocks}}.$ (19)

Fig. 7(a) presents the eMBB reliability while varying the targeted BER, for
different URLLC arrival rates. The figure shows that the proposed puncturing
strategy achieves better reliability compared the puncturing baseline. For
instance, at $P=0.01,$ the proposed puncturing strategy achieves reliability
of 31% compared to 20% for the puncturing baseline. This means more eMBB
blocks, about 50% enhancement, are received correctly, hence less re-
transmissions and better spectral efficiency. The gain of the proposed
algorithm decreases while increasing the targeted eMBB SER, this is because
the SER becomes dominated by the channel errors at high SER. Moreover, Fig.
7(b) presents the reliability for each eMBB user. Compared to the baseline,
the figure also shows that the proposed algorithm considerably enhances the
reliability of the eMBB users, which means less retransmission for the eMBB
traffic.

###  V-C URLLC performance

In this section, the performance of the URLLC traffic is investigated in terms
of the SER and reliability.

####  V-C1 URLLC SER

Fig. 8 illustrates the URLLC SER while varying the transmitted power. As shown
in Fig. 8, the proposed puncturing scheme preserves the SER of the URLLC
traffic while enhancing the SER of the eMBB traffic. The SER loss of the
proposed strategy is negligible while taking into account the coding gain. We
emphasize here that according to the target SER or BER for both the URLLC and
eMBB, the scheduler can select either the URLLC mapper or the ESRM.

####  V-C2 URLLC Reliability

Fig. 9 illustrates the URLLC reliability (success rate) while varying the
transmitted power. The figure shows the URLLC reliability for different
$\epsilon_{u}$. The figure shows that the proposed puncturing strategy
preserves the URLLC reliability, which makes the proposed strategy a practical
method for efficient multiplexing between eMBB and URLLC traffics.

Figure 8: URLLC SER vs transmission power in dBm. Figure 9: URLLC reliability
versus the transmitting power . $\lambda=7~{}p/msec$

###  V-D Time Complexity

Figure 10: Time complexity of the proposed algorithm

Fig. 10 shows the time complexity of the proposed algorithm. The algorithm was
implemented in Matlab using a machine with the following characteristics:
System Type: x64-based PC Processor: Intel(R) i7-8700H CPU @3.20GHz. The
results show that the processing time of the proposed algorithm is less than
$1~{}ms$. As the cloud radio access network has a powerful computational
resources, the running time of the proposed algorithm will be further reduced;
Hence, the latency of the URLLC will be surely met.

In summary, the proposed puncturing scheme represents a low complexity
solution that optimizes between the SER and the spectral efficiency of the
eMBB traffic while preserving the reliability of the URLLC services. The
proposed algorithm gain, for the eMBB traffic, starts at 0 dB at low SNR and
reach up to 10 dB at high SNR, and it enhances the eMBB reliability up to 50%.

##  VI Concluding Remarks

In this paper, we proposed a downlink puncturing strategy in an effort to
reduce the impact of transmitting URLLC traffic simultaneously with eMBB
traffic. The proposed strategy mitigates the impact on the eMBB traffic by
exploiting the region similarity between the eMBB and URLLC symbols to reduce
the effectively punctured eMBB symbols. The introduced strategy covers all
relations between the eMBB and URLLC modulation schemes. Throughout the
analysis, it was shown that the eMBB SER depends on the channel gain, the
URLLC load, and the average similarity between the URLLC and eMBB traffic. At
high SNR, the eMBB SER asymptotically saturates to the errors due to
puncturing, and it is proportional to the ratio between the effectively
punctured eMBB symbols to the total eMBB load. Also, when the URLLC block is
small or the search space increases, the eMBB SER reduces to the errors due to
the channel. Numerical and simulation results demonstrated that the proposed
puncturing strategy enhances the system information rate by doubling the URLLC
load for the same SER compared to the baseline. While preserving the URLLC
quality of service requirements, the proposed puncturing scheme can achieve
gains of up to 10 dB as compared to the baseline scheme.

We believe there is still room for improving the performance of the proposed
puncturing scheme.We list here a number of possible extensions that that we
plan to undertake in future works. First, we plan to apply the proposed scheme
to a coded system in which the eMBB and URLLC streams are coded. The code rate
and the targeted block error rate should be incorporated while allocating the
URLLC load. Moreover, we plan to propose a proper receiver to extract the
effectively punctured eMBB symbols and decodes unaffected symbols. Second, the
length of the URLLC block, $\zeta$, can be optimized to achieve better
spectral efficiency by addressing the trade-off between the overhead and eMBB
SER. Finally, exploiting the eMBB block with high similarity has polynomial
time complexity, so it may be more efficient to build a learning model to
reduce the complexity of the search algorithm.

##  Appendix A Proof of Lemma 1

Consider the Binomial distribution with $B~{}(\zeta,\eta_{n,m})$ to exploit
the similar $\zeta-$symbols blocks between the URLLC load and the eMBB
sequence. Then, the CDF $F(k)$ is expressed as

$\displaystyle F(k)=\sum_{j=0}^{k}\left(\begin{array}[]{l}{\zeta}\\\
{j}\end{array}\right)\left(\eta_{n,m}\right)^{j}(1-\eta_{n,m})^{\zeta-j}.$
(22)

The expected number of similar symbols between both the URLLC and eMBB blocks
is $\mu=\eta_{n,m}\,\zeta$. Under the assumption that the eMBB blocks and the
URLLC packet are $i.i.d$, and by searching over the search space
($K_{m}=L_{m}-\zeta+1$), the order statistic after arranging the random
samples in an increasing order is $Y_{1}\leq Y_{2}\leq\cdots\leq Y_{K_{m}}$.
Based on the results of [31], the pmf of $Y_{z}$ becomes for all
$k=0,1,\ldots,\zeta$

$\begin{split}f_{z}(k)=\sum\nolimits_{r=z}^{K^{m}}\left(\begin{array}[]{c}{K^{m}}\\\
{r}\end{array}\right)\left[\\{F(k)\\}^{r}\\{1-F(k)\\}^{K_{m}-r}\right.\\\
\left.-\\{F(k-1)\\}^{r}\\{1-F(k-1)\\}^{K_{m}-r}\right].\end{split}\vspace{-.1
in}$ (23)

Considering the case when the largest ordered sample has at least $k$ similar
symbols. Also, considering that $L_{m}>>l_{n,m}$, the expected number of
similar symbols can be approximated as [31]

$U_{n,m,\zeta}=\sum\nolimits_{k=0}^{\zeta-1}\left[1-\\{F(k)\\}^{L_{m}-\zeta}\right].$
(24)

Averaging over the number of $\zeta$-blocks in $l_{n,m}$, we arrive at

$U_{n,m,\zeta}(l_{n,m})=\frac{1}{\lceil\frac{l_{n,m}}{\zeta}\rceil}\sum_{1}^{\lceil\frac{l_{n,m}}{\zeta}\rceil}\sum\nolimits_{k=0}^{\zeta-1}\left[1-\\{F(k)\\}^{L_{m}-\zeta}\right].$
(25)

This completes the proof.

## References

  * [1] W. Saad, M. Bennis, and M. Chen, “A Vision of 6G Wireless Systems: Applications, Trends, Technologies, and Open Research Problems,” _arXiv preprint arXiv:1902.10265_ , 2019.
  * [2] K. B. Letaief, W. Chen, Y. Shi, J. Zhang, and Y. A. Zhang, “The Roadmap to 6G: AI Empowered Wireless Networks,” _IEEE Commun. Mag._ , vol. 57, no. 8, pp. 84–90, August 2019.
  * [3] A. Anand, G. De Veciana, and S. Shakkottai, “Joint Scheduling of URLLC and eMBB Traffic in 5G Wireless Networks,” in _IEEE INFOCOM 2018 - IEEE Conference on Computer Communications_ , April 2018, pp. 1970–1978.
  * [4] H. Ji, S. Park, J. Yeo, Y. Kim, J. Lee, and B. Shim, “Ultra-Reliable and Low-Latency Communications in 5G Downlink: Physical Layer Aspects,” _IEEE Wireless Communications_ , vol. 25, no. 3, pp. 124–130, JUNE 2018.
  * [5] Z. Li, M. A. Uusitalo, H. Shariatmadari, and B. Singh, “5g urllc: Design challenges and system concepts,” in _2018 15th International Symposium on Wireless Communication Systems (ISWCS)_ , 2018, pp. 1–6.
  * [6] D. Maaz, A. Galindo-Serrano, and S. E. Elayoubi, “Urllc user plane latency performance in new radio,” in _2018 25th Int. Conf. on Telecommun. (ICT)_. IEEE, 2018, pp. 225–229.
  * [7] M. Bennis, M. Debbah, and H. V. Poor, “Ultrareliable and Low-Latency Wireless Communication: Tail, Risk, and Scale,” _Proceedings of the IEEE_ , vol. 106, no. 10, pp. 1834–1853, Oct 2018.
  * [8] J. Park, S. Samarakoon, H. Shiri, M. K. Abdel-Aziz, T. Nishio, A. Elgabli, and M. Bennis, “Extreme urllc: Vision, challenges, and key enablers,” _arXiv preprint arXiv:2001.09683_ , 2020.
  * [9] 3GPP, “TSG RAN WG1 Meeting 87,” 3rd Generation Partnership Project (3GPP), Tech. Rep., November 2016.
  * [10] K. I. Pedersen, G. Pocovi, J. Steiner, and S. R. Khosravirad, “Punctured scheduling for critical low latency data on a shared channel with mobile broadband,” in _2017 IEEE 86th Vehicular Technology Conference (VTC-Fall)_. IEEE, 2017, pp. 1–6.
  * [11] K. I. Pedersen, M. Niparko, J. Steiner, J. Oszmianski, L. Mudolo, and S. R. Khosravirad, “System Level Analysis of Dynamic User-Centric Scheduling for a Flexible 5G Design,” in _2016 IEEE Global Communications Conference (GLOBECOM)_ , Dec 2016, pp. 1–6.
  * [12] P. Popovski, K. F. Trillingsgaard, O. Simeone, and G. Durisi, “5G Wireless Network Slicing for eMBB, URLLC, and mMTC: A Communication-Theoretic View,” _IEEE Access_ , vol. 6, pp. 55 765–55 779, 2018.
  * [13] C. Xiao, J. Zeng, W. Ni, X. Su, R. P. Liu, T. Lv, and J. Wang, “Downlink MIMO-NOMA for Ultra-Reliable Low-Latency Communications,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 4, pp. 780–794, April 2019.
  * [14] E. J. dos Santos, R. D. Souza, J. L. Rebelatto, and H. Alves, “Network Slicing for URLLC and eMBB with Max-Matching Diversity Channel Allocation,” _IEEE Communications Letters_ , pp. 1–1, 2019.
  * [15] A. Azari, M. Ozger, and C. Cavdar, “Risk-Aware Resource Allocation for URLLC: Challenges and Strategies with Machine Learning,” _IEEE Communications Magazine_ , vol. 57, no. 3, pp. 42–48, March 2019.
  * [16] A. Anand, G. De Veciana, and S. Shakkottai, “Joint Scheduling of URLLC and eMBB Traffic in 5G Wireless Networks,” in _IEEE INFOCOM 2018 - IEEE Conference on Computer Communications_ , April 2018, pp. 1970–1978.
  * [17] M. Alsenwi, N. H. Tran, M. Bennis, A. Kumar Bairagi, and C. S. Hong, “eMBB-URLLC Resource Slicing: A Risk-Sensitive Approach,” _IEEE Commun. Lett._ , vol. 23, no. 4, pp. 740–743, April 2019.
  * [18] A. Karimi, K. I. Pedersen, N. H. Mahmood, G. Pocovi, and P. Mogensen, “Efficient Low Complexity Packet Scheduling Algorithm for Mixed URLLC and eMBB Traffic in 5G,” in _2019 IEEE 89th Vehicular Technology Conference (VTC2019-Spring)_ , April 2019, pp. 1–6.
  * [19] W. R. Ghanem, V. Jamali, Y. Sun, and R. Schober, “Resource allocation for multi-user downlink miso ofdma-urllc systems,” _IEEE Transactions on Communications_ , vol. 68, no. 11, pp. 7184–7200, 2020.
  * [20] M. Chraiti, A. Ghrayeb, and C. Assi, “A NOMA Scheme Exploiting Partial Similarity Among Users Bit Sequences,” _IEEE Trans. on Commun._ , vol. 66, no. 10, pp. 4923–4935, Oct 2018.
  * [21] R. Kassab, O. Simeone, P. Popovski, and T. Islam, “Non-orthogonal multiplexing of ultra-reliable and broadband services in fog-radio architectures,” _IEEE Access_ , vol. 7, pp. 13 035–13 049, 2019.
  * [22] N. B. Khalifa, V. Angilella, M. Assaad, and M. Debbah, “Low-complexity channel allocation scheme for urllc traffic,” _IEEE Trans. on Commun._ , pp. 1–1, 2020.
  * [23] A. Manzoor et al., “Contract-based scheduling of urllc packets in incumbent embb traffic,” _arXiv preprint arXiv:2003.11176_ , 2020.
  * [24] W. Wu, P. Lin, and Y. Lee, “An indicator-free embb and urllc multiplexed scheme for 5g downlink system,” in _2019 IEEE 90th Vehicular Technology Conference (VTC2019-Fall)_ , 2019, pp. 1–5.
  * [25] A. Sharma and M. Salim, “Polar code: The channel code contender for 5g scenarios,” in _2017 international conference on computer, communications and electronics (Comptelix)_. IEEE, 2017, pp. 676–682.
  * [26] J. Rao and S. Vrzic, “Packet duplication for urllc in 5g: Architectural enhancements and performance analysis,” _IEEE Network_ , vol. 32, no. 2, pp. 32–40, 2018.
  * [27] N. Strodthoff, B. Göktepe, T. Schierl, C. Hellge, and W. Samek, “Enhanced machine learning techniques for early harq feedback prediction in 5g,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 11, pp. 2573–2587, 2019.
  * [28] A. Ghosh, “5g new radio (nr): physical layer overview and performance,” in _Proc. IEEE Commun. Theory Workshop_ , 2018, pp. 1–38.
  * [29] H. Soury, F. Yilmaz, and M. Alouini, “Exact Symbol Error Probability of Square M-QAM Signaling Over Generalized Fading Channels Subject to Additive Generalized Gaussian Noise,” in _2013 IEEE International Symposium on Information Theory_ , July 2013, pp. 51–55.
  * [30] A. Goldsmith, _Wireless Communications_. Cambridge university press, 2005.
  * [31] B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, _A First Course in Order Statistics_. Siam, 1992, vol. 54.