#  Joint Power and Blocklength Optimization for URLLC in a Factory Automation
Scenario

Hong Ren, Cunhua Pan, Yansha Deng, Maged Elkashlan, and Arumugam Nallanathan
H. Ren, C. Pan, M. Elkashlan, and A. Nallanathan are with School of Electronic
Engineering and Computer Science, Queen Mary University of London, London, E1
4NS, U.K. (Email: h.ren, c.pan, maged.elkashlan, a.nallanathan@qmul.ac.uk). Y.
Deng is with the Department of Informatics, King s College London, London WC2R
2LS, U.K. (e-mail: yansha.deng@kcl.ac.uk).

###### Abstract

Ultra-reliable and low-latency communication (URLLC) is one of three pillar
applications defined in the fifth generation new radio (5G NR), and its
research is still in its infancy due to the difficulties in guaranteeing
extremely high reliability (say $10^{-9}$ packet loss probability) and low
latency (say 1 ms) simultaneously. In URLLC, short packet transmission is
adopted to reduce latency, such that conventional Shannon’s capacity formula
is no longer applicable, and the achievable data rate in finite blocklength
becomes a complex expression with respect to the decoding error probability
and the blocklength. To provide URLLC service in a factory automation
scenario, we consider that the central controller transmits different packets
to a robot and an actuator, where the actuator is located far from the
controller, and the robot can move between the controller and the actuator. In
this scenario, we consider four fundamental downlink transmission schemes,
including orthogonal multiple access (OMA), non-orthogonal multiple access
(NOMA), relay-assisted, and cooperative NOMA (C-NOMA) schemes. For all these
transmission schemes, we aim for jointly optimizing the blocklength and power
allocation to minimize the decoding error probability of the actuator subject
to the reliability requirement of the robot, the total energy constraints, as
well as the latency constraints. We further develop low-complexity algorithms
to address the optimization problems for each transmission scheme. For the
general case with more than two devices, we also develop a low-complexity
efficient algorithm for the OMA scheme. Our results show that the relay-
assisted transmission significantly outperforms the OMA scheme, while NOMA
scheme performs well when the blocklength is very limited. We further show
that the relay-assisted transmission has superior performance over the C-NOMA
scheme due to larger feasible region of the former scheme.

##  I Introduction

The fifth-generation (5G) networks are envisaged to support three pillar use
cases: enhanced mobile broadband (eMBB), massive machine type communication
(mMTC), and _mission-critical_ internet of things (IoT) [1]. Extensive
research has focused on eMBB and mMTC, but the research on mission-critical
IoT is still in its infancy [2, 3, 4, 5, 6]. The applications of mission-
critical tasks include factory automation (FA), autonomous driving, remote
surgery, smart grid automation, unmanned aerial vehicles (UAVs) control
information delivery [7], which require ultra reliable and low latency
communication (URLLC) [8, 9, 10]. For example, in Industrial 4.0 [11], wired
connection will be replaced by wireless transmission to enhance the
flexibility and reduce the infrastructure cost. This change imposes
challenging requirements on the wireless transmission in terms of latency and
reliability [12]. For mission-critical tasks in FA, the transmission duration
is expected to be lower than 100 $\mu s$ to allow processing delays during
queuing, scheduling, backhaul transmission, and propagation [13], while
guaranteeing the packet error probability of $10^{-9}$.

In conventional human-to-human (H2H) communications, the transmission delay is
relatively long (say 20-30 ms) and the packet size is large (say 1500 bytes),
thus Shannon’s capacity can be served as a tight upper bound of the achievable
data rate due to the law of large numbers [14]. In contrast, in URLLC, the
packet size should be extremely low (say 20 bytes) to support the low-latency
transmission [13]. In this case, Shannon’s capacity formula is no longer
applicable as the law of large numbers is not valid. Thus, the achievable data
rate under short blocklength needs to be retreated. In [15], the achievable
data rate in finite blocklength regime has been derived as a complicated
function of the signal-to-noise (SNR), the blocklength, and the decoding error
probability.

Recently, extensive research attention has been devoted to the short packet
transmission (SPT) design [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In
particular, the frame structure is designed in [16] for SPT, where their
results showed that it is beneficial to group multiple messages from some
users into a single packet based on approximations from finite blocklength
information theory. In [17], She _et al._ studied the network available range
maximization problem by dynamically selecting the transmission modes between
device-to-device (D2D) and cellular links. The non-asymptotic upper and lower
bounds on the coding rate for SPT over a Rician memoryless block-fading
channel were derived in [18] under a given packet error probability
requirement. The overall error probability of relay-assisted transmission
under finite blocklength was derived in [19] under the assumption of perfect
channel state information (CSI). They further extended this model to the
quasi-static Rayleigh channels where only the average CSI is available at the
source in [20], as well as to the two-way amplify-and-forward relay network in
[21]. Recently, the delay and decoding error probability were analyzed in [22]
for simultaneous wireless information and power transfer (SWIPT) relay-
assisted system, where the relay first harvests energy from the source and
then uses the harvested energy to forward the source’s information to the
destination node.

The aforementioned studies [16, 17, 18, 19, 20, 21, 22] mainly focused on the
performance analysis of finite blocklength transmission. In order to design a
practical URLLC system, it is imperative to intelligently optimize the
resource allocation including blocklength and power allocation under the given
error probability and latency requirements. Unfortunately, the achievable
coding rate expression is neither convex nor concave with respect to the
blocklength and the transmit power, which brings the difficulty in obtaining
the globally optimal solution[5]. This motivates the recent studies in
resource allocation for the SPT in [23, 24, 25, 26, 27]. Specifically, the
average throughput and the max-min throughput optimization under the latency
constraint was solved via the exhaustive search method with high complexity in
[23]. Sun _et al._ in [24] considered the SPT for a two-user downlink non-
orthogonal multiple access (NOMA) system, with an aim to maximize the
throughput of user 1 subject to the throughput requirements for user 2. Note
that the decoding error probability requirement has not been considered in
[23] and [24], and the throughput is less important in URRLC as only control
signals or measurement data with small packet size are transmitted in URLLC.
In [25], She _et al._ jointly optimized the uplink and downlink transmission
blocklengths to minimize the required total bandiwidth based on statistical
channel state information (CSI). However, the optimization is based on the
simplified expression of the rate for SPT, which cannot accurately
characterize the relationship between the decoding error probability and
blocklength. In addition, several approximations are involved in the
derivation of the decoding error probability for each user due to the fact
that only statistical CSI is available. Most recently, Hu _et al._ [26]
considered SWIPT in relay-assisted URLLC systems, where the SWIPT parameters
and blocklength are jointly optimized to maximize the reliability performance.
However, the decoding error probability at the relay cannot be guaranteed and
the power is assumed to be fixed in [26]. Most recently, in [27] we jointly
optimize the blocklength and unmanned aerial vehicle’s (UAV’s) location to
minimize the decoding error probability while guaranteeing the latency
requirement and decoding error probability target. However, the power
allocation was not considered. Furthermore, the optimization over UAV’s
location is obtained by observing the curve of the second-order derivative of
the objective function over location variable without strict proof.

In this paper, we consider a typical mission-critical scenario (i.e., a FA
scenario), where the central controller needs to transmit a certain amount of
different data to two devices within a given transmission time and under a
very low packet error probability. One device named actuator is located far
away from the controller, while the other device named robot can move between
the controller and the actuator. We consider four fundamental transmission
schemes, namely, orthogonal multiple access (OMA), NOMA, relay-assisted
transmission and cooperative NOMA (C-NOMA). In this scenario, we aim for
jointly optimizing the blocklength and the transmit power of these two devices
to minimize the decoding error probability for the actuator while guaranteeing
the decoding error probability for the robot, taking into account the energy
and blocklength constraints, which were not considered in [24, 25, 26] and new
methods needs to be developed. The main contributions of this paper are
summarized as follows:

  1. 1.

For the OMA scheme, we first prove that both the decoding error probability
and energy constraints hold with equality at the optimal point, and then
propose a novel iterative algorithm to obtain tight lower and upper bounds of
the blocklength to reduce the search complexity. A low-complexity algorithm is
proposed to find the globally optimal solution of transmit power. For the case
of more than two devices, we also develop a novel low-complexity algorithms to
find the suboptimal solution of the optimization problem.

  2. 2.

For the NOMA scheme, the search set of blocklength is first derived to reduce
the search complexity. In contrast to the OMA case, the decoding error
probability function for each given blocklength in the NOMA case is non-
continuous with respect to the transmit power, which complicates the
optimization problem. Fortunately, we rigorously proved that the decoding
error probability holds with equality at the optimal point, such that the one-
dimensional line search algorithm can be used to find the optimal solution. We
also provide a sufficient condition when the decoding error probability
function is a convex function, which facilitates the application of a low-
complexity bisection search method.

  3. 3.

For the relay-assisted scheme, we also adopt the iterative algorithm to reduce
the search complexity of blocklength. Unlike the OMA and NOMA schemes, the
decoding error probability constraint of relay-assisted transmission does not
hold with equality. To resolve this issue, we fix the blocklength, such that
the original optimization problem is reduced to a one-dimension search
optimization problem.

  4. 4.

For the C-NOMA scheme, we adapt the iterative algorithm to reduce the search
complexity of blocklength, and then one-dimension search is proposed to find
the optimal transmit power. For the special case, low-complexity bisection
search method is applied.

  5. 5.

To compare the performance of our proposed four transmission schemes, we
perform extensive simulation results, which show that the relay-assisted
scheme significantly outperforms the other three schemes for most times in
terms of both the decoding error probability and the network availability. Our
results demonstrate the effectiveness of relaying transmission in enhancing
the reliability performance in the industrial automation scenario.

The remainder of this paper is organized as follows. In Section II, the system
model and the problem formulation are provided. In Section III, the
transmission scheme is presented. The general case with more than two devices
is considered in Section IV. Simulation results and analysis are presented in
Section V. Finally, Section VI concludes the paper.

##  II System Model

###  II-A System model

Consider a downlink communication in one factory, where a central controller
serves a robot and an actuator as shown in Fig. 1. The robot is assumed to be
located in the vicinity of the controller, and the actuator is far away from
the controller. Both the robot and the actuator are equipped with a single
antenna. The controller needs to transmit two small packets to the two
devices. The packet sizes for the actuator and the robot are assumed to be the
same, and are denoted as $D$ bits.

The transmission of these two packets is subject to a latency constraint,
i.e., the transmission has to finish within $M$ symbols or channel uses. The
transmission time corresponds to $t_{\max}=MT_{s}$ seconds, where $T_{s}$ is
the symbol duration that is equal to $1/B$ with $B$ as the system bandwidth.
For the applications with URLLC requirement, short frame structure is adopted
and the end-to-end delay should be kept within 1 ms [8], which is much shorter
than the channel coherence time. Hence, the channels are quasi-static fading
and remain constant during the whole transmission. The channel fading
coefficients from the central controller to the robot and the actuator are
denoted as ${{\tilde{h}}_{1}}$ and ${{\tilde{h}}_{2}}$, respectively. The
channel fading coefficient between the robot and the actuator is denoted as
${{\tilde{h}}_{3}}$. We also assume that these channels are perfectly known at
the controller, and the total energy consumption of the system should be below
${\tilde{E}_{{\rm{tot}}}}$ Joule. Since we have assumed that the actuator is
far away from the controller, the channel power gain
$\left|{{{\tilde{h}}_{2}}}\right|^{2}$ is very small.

Figure 1: Illustration of a Factory Automation Scenario.

###  II-B Achievable data rate for a simple point-to-point system

The data rate (coding rate) $R$ of a communication system is defined as the
fraction of the number of information bits to the number of transmission
symbols. According to Shannon’s coding theorem, the Shannon capacity is
defined as the highest coding rate that there exists an encoder/decoder pair
whose decoding error probability becomes negligible when the blocklength
approaches infinity [28]. However, in URLLC, the blocklength for each frame is
limited and small, in this case, the decoding error probability at the
receiver cannot be ignored.

In URLLC scenarios, the required transmission delay is much shorter than the
channel coherence time, thus the channel is quasi-static. According to the
results in [29], for a simple point-to-point communication system transmitting
over a quasi-static Rayleigh fading channel, the channel dispersion is zero
and the achievable data rate converges to the outage capacity as the
blocklength increases. However, the closed-form expression of the outage
capacity for short-packet transmission is unavailable. In [15], the normal
approximation was adopted to approximate the coding rate $R$ at finite
blocklength, which is given by

$R\approx{\log_{2}}(1+\gamma)-\sqrt{\frac{V}{m}}\frac{{{Q^{-1}}\left(\varepsilon\right)}}{{\ln
2}},\vspace{-0.2cm}$ (1)

where $m$ is the channel blocklength, $\varepsilon$ is the decoding error
probability, $\gamma$ denotes the signal-to-noise ratio (SNR) at the receiver,
${Q^{-1}}\left(\cdot\right)$ is the inverse function
$Q(x)\\!=\\!\frac{1}{{\sqrt{2\pi}}}\int_{x}^{\infty}{{e^{-\frac{{{t^{2}}}}{2}}}dt}$,
and $V$ is given by $V\\!=\\!1-{(1+\gamma)^{-2}}$. As shown in the numerical
results in [29], this approximation is very accurate when $m$ is larger than
50, which is the case in our simulations. From (1), the decoding error
probability can be obtained as follows:

${\varepsilon}=Q\left({f\left({{{\gamma}},{m},D}\right)}\right),$ (2)

where $f\left({\gamma,m,D}\right)=\ln
2\sqrt{\frac{m}{V}}\left({{{\log}_{2}}(1+\gamma)-\frac{D}{m}}\right)$. In the
following, we aim to jointly optimize the transmission blocklength and power
to minimize the decoding error probability for four different transmission
schemes.

##  III Transmission schemes

In this section, we aim for designing efficient resource allocation algorithms
to minimize the decoding error probability of the actuator under three sets of
constraints: 1) the packets for robot and actuator need to be transmitted
within $M$ symbols; 2) the robot should satisfy its reliability requirement;
3) the total consumed energy should be kept within ${\tilde{E}_{{\rm{tot}}}}$.
The OMA, NOMA, relay-assisted transmission, and C-NOMA transmission schemes
are studied in the following subsections.

###  III-A OMA transmission

The OMA scheme is the simplest transmission scheme, where the controller
serves the robot and the actuator in two different orthogonal channel uses or
blocklengths. In detail, the controller transmits signal $x_{1}$ to the robot
with $m_{1}$ blocklength. Due to this orthogonal property, the received signal
at the robot can be represented as

$\vspace{-0.15cm}{y_{1}}=\sqrt{{p_{1}}}{{\tilde{h}}_{1}}{x_{1}}+{n_{1}},$ (3)

where $p_{1}$ is the transmit power of the robot, $n_{1}$ is the zero-mean
additive complex white Gaussian noise (AWGN) with variance $\sigma_{1}^{2}$,
$x_{1}$ carries information knowledge for the robot with packet size $D$.
Hence, the coding rate at the robot is given by $D/m_{1}$.

From (3), the received signal to noise ratio (SNR) at the robot is given by

$\vspace{-0.1cm}{\gamma_{1}}={p_{1}}{h_{1}},$ (4)

where $h_{1}=|\tilde{h}_{1}|^{2}/{\sigma_{1}^{2}}$ denotes the normalized
channel gain from the controller to the robot. Then, according to (2), the
decoding error probability of $x_{1}$ at the robot is given by

$\vspace{-0.1cm}{\varepsilon_{1}}=Q\left({f\left({{{\gamma_{1}}},{m_{1}},D}\right)}\right).$
(5)

The controller transmits signal $x_{2}$ to the actuator with blocklength equal
to $m_{2}$. The corresponding error probability at the actuator is derived as

$\vspace{-0.1cm}{\varepsilon_{2}}=Q\left({f\left({{{\gamma_{2}}},{m_{2}},D}\right)}\right),\vspace{-0.3cm}$
(6)

where ${\gamma_{2}}={p_{2}}{h_{2}}$ with $p_{2}$ as the transmit power of the
actuator and $h_{2}=|{\tilde{h}}_{2}|^{2}/{\sigma_{2}^{2}}$ as the normalized
channel gain for the actuator. Without loss of generality (w.l.o.g.), we
assume that in this paper the robot has higher normalized channel gain than
the actuator, i.e., ${{h}_{1}}>{{h}_{2}}$.

The resource allocation problem for the OMA transmission can be formulated as:

$\displaystyle\vspace{-0.1cm}\mathop{\min}\limits_{\left\\{{{m_{1}},{m_{2}},{p_{1}},{p_{2}}}\right\\}}\;\;\;$
$\displaystyle{{\varepsilon}_{2}}$ (7a) $\displaystyle{\rm{s.t.}}\;\;\;$
$\displaystyle{{\varepsilon}_{1}}\leq\varepsilon_{1}^{\max},$ (7b)
$\displaystyle{m_{1}}{p_{1}}+{m_{2}}{p_{2}}\leq{E_{{\rm{tot}}}},$ (7c)
$\displaystyle m_{1}+m_{2}\leq M,$ (7d) $\displaystyle
m_{1},m_{2}\in\mathbb{Z},$ (7e)

where constraint in (7b) is the decoding error probability requirement of the
robot, constraint (7c) ensures the system total energy consumption is within a
budget $\tilde{E}_{\rm{tot}}=E_{\text{tot}}T_{s}$, (7d) is the constraint on
the latency constraint, and constraint (7e) ensures that the blocklength for
each transmission phase is integer with $\mathbb{Z}$ denoting the positive
integer set. The maximum decoding error probability $\varepsilon_{1}^{\max}$
is assumed to be much less than 0.1 to ensure the stringent reliability
requirement, and this assumption holds for the remaining transmission schemes.
As a result, $\varepsilon_{1}$ should be smaller than 0.1. Then, the
inequality $\frac{D}{m_{1}}<{\rm{log}}_{2}(1+\gamma_{1})$ should hold.

To solve the optimization problem in (7), we first provide the following
lemma.

_Lemma 1_ : Constraints (7b) and (7c) hold with equality at the optimum
solution.

_Proof_ : Please see Appendix A.

With $m_{1}$ and $m_{2}$ to be integers, the exhaustive search method can be
used to find the optimal solution. To reduce the search complexity when $M$ is
large, we shorten the search range of $m_{1}$ and $m_{2}$. In the following,
we aim to derive the bounds of $m_{1}$ and $m_{2}$.

####  III-A1 The upper and lower bounds of $m_{1}$ and $m_{2}$

Since $\varepsilon_{1}^{\max}$ is assumed to be a very small value that is
much smaller than $10^{-1}$, a necessary condition for constraint (7b) to hold
is that ${\log_{2}}\left({1+{p_{1}}{h_{1}}}\right)>D/{{m_{1}}}$ 111 This can
be proved as follows:
$\varepsilon_{1}=Q\left({f\left({\gamma_{1},m_{1},D}\right)}\right)<\varepsilon_{1}^{\max}<0.5=Q\left(0\right)$.
Since Q-function is a decreasing function, we have
${f\left({\gamma_{1},m_{1},D}\right)}>0$. By substituting the expression of
$f\left({\gamma_{1},m_{1},D}\right)$, the proof is complete., which leads to

$\vspace{-0.1cm}{p_{1}}>\left({{2^{\frac{D}{{{m_{1}}}}}}-1}\right)/{{h_{1}}}.$
(8)

On the other hand, based on the energy constraint (7c), we have:
$p_{1}<E_{{\rm{tot}}}/m_{1}$. Thus, the blocklength allocation of the robot
$m_{1}$ should satisfy the following inequality:

$\vspace{-0.1cm}{E_{{\rm{tot}}}}>\frac{{{m_{1}}}}{{{h_{1}}}}({2^{\frac{D}{{{m_{1}}}}}}-1)\buildrel\Delta\over{=}g({m_{1}}).$
(9)

To investigate the properties of $g(m_{1})$, the first-order and second-order
derivatives of function $g({m_{1}})$ w.r.t. $m_{1}$ are give by

$\displaystyle\vspace{-0.4cm}g^{\prime}({m_{1}})$ $\displaystyle=$
$\displaystyle{2^{\frac{D}{{{m_{1}}}}}}-1-\ln
2\cdot\frac{D}{{{m_{1}}}}{2^{\frac{D}{{{m_{1}}}}}},$ (10)
$\displaystyle\vspace{-0.4cm}g^{\prime\prime}({m_{1}})$ $\displaystyle=$
$\displaystyle{\left({\ln
2}\right)^{2}}\cdot\frac{{{D^{2}}}}{{m_{1}^{3}}}{2^{\frac{D}{{{m_{1}}}}}}\geq
0.\vspace{-0.1cm}$ (11)

Thus, $g^{\prime}({m_{1}})$ is a monotonically increasing function of $m_{1}$,
and we have

$\vspace{-0.1cm}g^{\prime}({m_{1}})\leq\mathop{\lim}\limits_{{m_{1}}\to+\infty}g^{\prime}({m_{1}})=0.\vspace{-0.2cm}$
(12)

Hence, function $g({m_{1}})$ is a monotonically decreasing function of
$m_{1}$. Then, we can find the lower bound of $m_{1}$ that satisfies the
inequality (9) which is denoted as $m_{1}^{\rm{lb}(0)}$, and $m_{1}$ should be
no smaller than $m_{1}^{\rm{lb}(0)}$, i.e., ${m_{1}}\geq
m_{1}^{{\rm{lb}}({\rm{0}})}$. Similarly, for practical applications, the
decoding error probability of the actuator ${\varepsilon}_{2}$ should be very
small, e.g., much lower than 0.5. In this case, the inequality
${\log_{2}}\left({1+{p_{2}}{h_{2}}}\right)>D/{{m_{2}}}$ should hold, which
leads to

${p_{2}}>\left({{2^{\frac{D}{{{m_{2}}}}}}-1}\right)/{{h_{2}}}.$ (13)

By using the inequality $m_{2}\leq M-m_{1}^{\rm{lb}(0)}$, we have

${m_{2}}{p_{2}}\geq\frac{{{m_{2}}}}{{{h_{2}}}}\left({{2^{\frac{D}{{{m_{2}}}}}}-1}\right)\geq\frac{{M-m_{1}^{{\rm{lb}}({\rm{0}})}}}{{{h_{2}}}}\left({{2^{\frac{D}{{M-m_{1}^{{\rm{lb}}({\rm{0}})}}}}}-1}\right)\buildrel\Delta\over{=}{A^{(0)}}.$
(14)

By using constraint (7c), we have

${E_{{\rm{tot}}}}-{A^{(0)}}>\frac{{{m_{1}}}}{{{h_{1}}}}\left({{2^{\frac{D}{{{m_{1}}}}}}-1}\right).$
(15)

By using (15), the updated lower bound of $m_{1}$ can be obtained, and denoted
as $m_{1}^{\rm{lb}(1)}$. Similar to (14), we can obtain $A^{(1)}$ by
substituting $m_{1}^{\rm{lb}(1)}$ into $m_{1}^{\rm{lb}(0)}$, and the lower
bound of $m_{1}$ can be obtained by using (15), where $A^{(0)}$ is replaced by
$A^{(1)}$, and the updated lower bound is denoted as $m_{1}^{\rm{lb}(2)}$.
Repeat the above procedure until $m_{1}^{{\rm{lb}}(n)}=m_{1}^{{\rm{lb}}(n-1)}$
or $m_{1}^{{\rm{lb}}(n)}=M-m_{2}^{{\rm{lb}}(0)}$. Then, denote the final lower
bound of $m_{1}$ as $m_{1}^{\rm{lb}}$.

This procedure is proved to converge as follows. By using ${A^{(0)}}\geq 0$
and comparing (9) and (15), we can obtain $m_{1}^{\rm{lb}(1)}\geq
m_{1}^{\rm{lb}(0)}$, and thus ${A^{(1)}}\geq A^{(0)}$, which leads to
$m_{1}^{\rm{lb}(2)}\geq m_{1}^{\rm{lb}(1)}$. Hence, the sequence of the lower
bound $m_{1}^{\rm{lb}(n)}$ is monotonically increasing. Furthermore, the
sequence is upper bounded by $M-m_{2}^{\rm{lb}(0)}$. As a result, the sequence
generated by the above iterative procedure is guaranteed to converge.

By using the similar iterative procedure, we can also obtain the lower bound
of $m_{2}$, which is denoted as $m_{2}^{\rm{lb}}$. As a result, the search
region of $m_{1}$ is given by $m_{1}^{\rm{lb}}\leq
m_{1}\leq(M-m_{2}^{\rm{lb}})\buildrel\Delta\over{=}m_{1}^{\rm{ub}}$.

For each given $m_{1}$, we need to find the search range of $m_{2}$, which is
detailed as follows. The optimal $p_{1}$ can be obtained by solving the
equation ${\varepsilon_{1}}=\varepsilon_{1}^{\max}$ with given $m_{1}$, which
is denoted as $p_{1}^{*}$. The solution can be readily obtained by using the
bisection search method due to the fact that ${\varepsilon_{1}}(p_{1})$ is a
monotonically decreasing function of $p_{1}$. Then, we have

${E_{{\rm{tot}}}}-{m_{1}}p_{1}^{*}={m_{2}}{p_{2}}\geq\frac{{{m_{2}}}}{{{h_{2}}}}\left({{2^{\frac{D}{{{m_{2}}}}}}-1}\right).$
(16)

Hence, the lower bound of $m_{2}$ with given $m_{1}$ (denoted as
$m_{2}^{\rm{lb}}(m_{1})$) can be obtained from (16), which is the minimum
integer that satisfies (16). Obviously, the upper bound of $m_{2}$ with given
$m_{1}$ is $M-m_{1}$. Hence, the search region of $m_{2}$ is given by
$m_{2}^{\rm{lb}}(m_{1})\leq m_{2}\leq(M-m_{1})$.

####  III-A2 Algorithm to solve Problem (7)

Based on the above analysis, the algorithm to solve Problem (7) is given in
Algorithm 1. The main idea can be summarized as follows. For each given
integer value of $m_{1}$ that satisfies $m_{1}^{\rm{lb}}\leq m_{1}\leq
m_{1}^{\rm{ub}}$, we calculate the value of ${{\varepsilon}_{1}}$ when $p_{1}$
is set as ${E_{{\rm{tot}}}}/m_{1}$. If
${{\varepsilon}_{1}}>\varepsilon_{1}^{\max}$, then the value of $m_{1}$ is not
feasible, and we increase the value of $m_{1}$ by one and continue to check
the updated $m_{1}$. Otherwise, we apply the bisection search method to find
the value of $p_{1}$ such that ${{\varepsilon}_{1}}=\varepsilon_{1}^{\max}$
due to the monotonically decreasing property of decoding error probability
${{\varepsilon}_{1}}$ w.r.t. $p_{1}$ [24]. By using Lemma 1, we have
$m_{2}p_{2}=E_{{\rm{tot}}}-m_{1}p_{1}$. The search range of $m_{2}$ is given
by $m_{2}^{\rm{lb}}(m_{1})\leq m_{2}\leq M-m_{1}$. For each given $m_{2}$, the
corresponding $p_{2}$ is given by $p_{2}=(E_{{\rm{tot}}}-m_{1}p_{1})/m_{2}$,
and we can calculate the value of ${\varepsilon}_{2}$. For each feasible
$m_{1}$, we can find the optimal solutions for $m_{2}$ and $p_{2}$ that yield
the minimum value of ${\varepsilon}_{2}$, respectively. At last, we check all
feasible $m_{1}$ in the range of $m_{1}^{\rm{lb}}\leq m_{1}\leq
m_{1}^{\rm{ub}}$, and choose the final globally optimal solution.

Input : $h_{1},h_{2},D,M,\varepsilon_{1}^{\max},E_{{\rm{tot}}}$

Output : $p_{1}^{\star},p_{2}^{\star},m_{1}^{\star},m_{2}^{\star}$

1 Apply the iterative procedure to calculate $m_{1}^{\rm{lb}},m_{1}^{\rm{ub}}$
and $m_{2}^{\rm{lb}}$;

2for  _ $m_{1}=m_{1}^{\rm{lb}}:m_{1}^{\rm{ub}}$ _ do

3 Set $p_{1}={E_{{\rm{tot}}}}/m_{1}$, and calculate the value of
${{\varepsilon}_{1}}$.

4 if  _${{\varepsilon}_{1}} >\varepsilon_{1}^{\max}$_ then

5 The current $m_{1}$ is not feasible, and return to the next $m_{1}$;

6 else

7 Use (16) to find the lower bound of $m_{2}$, denoted as
$m_{2}^{\rm{lb}}(m_{1})$. Apply the bisection search method to find the value
of $p_{1}$ such that ${{\varepsilon}_{1}}=\varepsilon_{1}^{\max}$;

8 for  _$m_{2}=m_{2}^{\rm{lb}}(m_{1}):M-m_{1}$_ do

9 Calculate $p_{2}=(E_{{\rm{tot}}}-m_{1}p_{1})/m_{2}$, and the value of
${\varepsilon}_{2}$, denoted as ${\varepsilon}_{2}(m_{1},m_{2})$.

10 end for

11 Given $m_{1}$, find the blocklength $m_{2}$ with the minimum value of
${\varepsilon}_{2}(m_{1},m_{2})$:
${\left.{m_{2}^{\\#}}\right|_{{m_{1}}}}=\mathop{\arg\min}\limits_{m_{2}^{{\rm{lb}}}\leq{m_{2}}\leq
M-{m_{1}}}{\varepsilon_{2}}\left({{m_{1}},{m_{2}}}\right).$

12 end if

13

14 end for

Return
$m_{1}^{\star}=\mathop{\arg\min}\limits_{m_{1}^{{\rm{lb}}}\leq{m_{1}}\leq
m_{1}^{{\rm{ub}}}}{\varepsilon_{2}}\left({{m_{1}},{{\left.{m_{2}^{\\#}}\right|}_{{m_{1}}}}}\right),m_{2}^{\star}={\left.{m_{2}^{\\#}}\right|_{m_{1}^{\star}}}$
and the corresponding $p_{1}^{\star}$ and $p_{2}^{\star}$.

Algorithm 1 Algorithm for Problem (7)

####  III-A3 Special case of Problem (7)

In steps 8-10 of Algorithm 1, one has to calculate the value of
${\varepsilon}_{2}$ for each $m_{2}$, which may incur high complexity. In this
subsection, we consider one special case when the SNR value $\gamma$ is very
high, i.e., $\gamma\gg 1$. In this case, $V$ in (2) can be approximated as
one, i.e., $V\approx 1$ 222 In general, when $\gamma>20$ dB, the value of $V$
is larger than 0.99, which can be approximated as one.. The optimization
problem in this special case can be efficiently solved. Specifically, the
decoding error probability in (2) can be approximated as

$\tilde{\varepsilon}=Q\left({{\tilde{f}}\left({{{\gamma}},{m},D}\right)}\right),$
(17)

where ${\tilde{f}}\left({\gamma,m,D}\right)=\ln
2\sqrt{m}\left({{{\log}_{2}}(1+\gamma)-\frac{D}{m}}\right)$.

For given $m_{1}$ and $p_{1}$, the product of $m_{2}$ and $p_{2}$ should
satisfy
${m_{2}}{p_{2}}={E_{{\rm{tot}}}}-{m_{1}}{p_{1}}\buildrel\Delta\over{=}{E_{2}}$
according to Lemma 1. Then, the original problem defined in (7) can be
transformed to the following optimization problem:

$\mathop{\min}\limits_{m_{\rm{2}}^{{\rm{lb}}}\leq{m_{2}}\leq
M-{m_{1}},{m_{2}}\in\mathbb{Z}}Q\left({\tilde{f}\left({{\gamma_{2}},{m_{2}},D}\right)}\right).$
(18)

Since $Q$-function is a decreasing function, the above problem is equivalent
to the following problem by substituting $p_{2}=E_{2}/m_{2}$ into it as

$\\!\\!\mathop{\max}\limits_{m_{\rm{2}}^{{\rm{lb}}}\leq{m_{2}}\leq
M\\!-\\!{m_{1}},{m_{2}}\in\mathbb{Z}}\ln
2\sqrt{{m_{2}}}\\!\left(\\!{{{\log}_{2}}\\!\left(\\!{1\\!+\\!\frac{{{E_{2}}{h_{2}}}}{{{m_{2}}}}}\\!\right)\\!\\!-\\!\\!\frac{D}{{{m_{2}}}}}\\!\right).$
(19)

To solve the above problem, we first relax the integer variable $m_{2}$ to a
continuous variable, and define

$\tilde{g}({m_{2}})\buildrel\Delta\over{=}\sqrt{{m_{2}}}\left({{{\log}_{2}}\left({1+\frac{{{E_{2}}{h_{2}}}}{{{m_{2}}}}}\right)-\frac{D}{{{m_{2}}}}}\right).$
(20)

In the following theorem, we provide a sufficient condition for
$\tilde{g}({m_{2}})$ to be a concave function.

_Theorem 1_ : $\tilde{g}({m_{2}})$ is a concave function when
$\frac{{{E_{2}}{h_{2}}}}{{M-{m_{1}}}}\geq e-1$, where $e$ is the natural
constant.

_Proof_ : Please see Appendix B.

When the condition in Theorem 1 is satisfied, Problem (19) is a convex
optimization problem. If $\tilde{g}^{\prime}(m_{\rm{2}}^{{\rm{lb}}})\leq 0$,
the optimal $m_{2}$ is given by $m_{2}=m_{\rm{2}}^{{\rm{lb}}}$. If
$\tilde{g}^{\prime}(M-{m_{1}})\geq 0$, the optimal $m_{2}$ is
$m_{2}=M-{m_{1}}$. Otherwise, the optimal $m_{2}^{*}$ satisfies
$\tilde{g}^{\prime}(m_{\rm{2}})=0$, and the low-complexity bisection search
method can be used to find $m_{2}^{*}$. The final optimal integer $m_{2}$ is
the one with lower objective value for its two neighbor integers, i.e.,
$\left\lfloor{m_{2}^{*}}\right\rfloor$ and
$\left\lfloor{m_{2}^{*}}\right\rfloor+1$.

###  III-B NOMA transmission

In NOMA transmission, superposition coding is employed at the controller so
that the controller can transmit signals to the two devices simultaneously
with different power levels. The controller allocates higher transmit power to
the user with lower channel gains and lower power to the one with higher
channel gains. On the one hand, the robot decodes the actuator’s signal first.
If decoding correctly, the robot will subtract the actuator’s signal from its
received signals and decodes its own signal. This is the so-called successive
interference cancellation (SIC). Otherwise, it has to decode its own signal by
treating actuator’s information as interference. On the other hand, the
actuator directly decodes its own signal by treating the robot’s signal as
interference since the controller allocates higher transmit power than the
robot. To implement this scheme, it is crucial that the robot knows whether
SIC is successful or not. To this end, we assume that the controller sends the
actuator’s channel coding information along with the robot’s channel coding
information to the robot through dedicated error-free channels. The channel
coding information for both devices are different and the channel coding can
assist in detecting whether the decoded information is correct or not. Hence,
the robot knows whether the SIC is successful or not. In general, the channel
coding information changes when CSI changes, which is much longer than the
URLLC transmission. Hence, each channel coherence time can accommodate
multiple URLLC transmissions. Then, the coding information only needs to be
transmitted to the robot at the beginning of channel coherence time, which
causes negligible overhead consumption.

In NOMA, the transmission blocklength for two devices is equal to $M$.
Specifically, the received signals at the robot and the actuator are given by

$\begin{array}[]{l}{y_{1}}=\sqrt{{p_{1}}}{{\tilde{h}}_{1}}{x_{1}}+\sqrt{{p_{2}}}{{\tilde{h}}_{1}}{x_{2}}+{n_{1}},\\\
{y_{2}}=\sqrt{{p_{1}}}{{\tilde{h}}_{2}}{x_{1}}+\sqrt{{p_{2}}}{{\tilde{h}}_{2}}{x_{2}}+{n_{2}},\end{array}$
(21)

where the notations in (21) has the same meaning as those in the OMA
transmission scheme. For the robot, it first decodes the actuator’s signal,
where the decoding signal to interference plus noise ratio (SINR) is given by

$\gamma_{2}^{1}=\frac{{{p_{2}}{h_{1}}}}{{{p_{1}}{h_{1}}+1}}.$ (22)

Following (2), the decoding error probability of $x_{2}$ at the robot can be
written as
$\varepsilon_{2}^{1}=Q\left({f\left({{{\gamma_{2}^{1}}},{M},D}\right)}\right)$.
This equivalently indicates that the information $x_{2}$ can be accurately
cancelled at the robot with probability $1-\varepsilon_{2}^{1}$. Note that
this is different from the infinite blocklength case in NOMA, where perfect
decoding can be achieved by the robot. If the SIC is successful, the robot
decodes its signal $x_{1}$ by removing the decoded signal $x_{2}$. By using
the first equality in (21), the SINR of decoding the signal $x_{1}$ is given
by

${\gamma_{1}}={p_{1}}{h_{1}}.$ (23)

Thus, following (2), the decoding error probability of $x_{1}$ at the robot
under perfect SIC condition is given by
${\varepsilon_{1}}=Q\left({f\left({{{\gamma_{1}}},{M},D}\right)}\right)$.
However, if the SIC fails, the robot will decode its information $x_{1}$ while
treating $x_{2}$ as interference, and the corresponding SINR is given by

$\vspace{-0.2cm}{{\hat{\gamma}_{1}}}=\frac{{{p_{1}}{h_{1}}}}{{{p_{2}}{h_{1}}+1}}.$
(24)

Thus, the decoding error probability of $x_{1}$ at the robot is given by
${{\hat{\varepsilon}_{1}}}=Q\left({f\left({{{\hat{\gamma}_{1}}},{M},D}\right)}\right)$.
Based on the above discussion, the decoding error probability of $x_{1}$ at
the robot is Bernoulli-distributed. With probability $1-\varepsilon_{2}^{1}$,
the decoding error probability is equal to ${\varepsilon_{1}}$, and with
probability $\varepsilon_{2}^{1}$, it is equal to ${{\hat{\varepsilon}_{1}}}$.
Hence, the average decoding error probability of $x_{1}$ at the robot is
formulated as

$\vspace{-0.2cm}{{\bar{\varepsilon}}_{1}}={\varepsilon_{1}}(1-\varepsilon_{2}^{1})+{{\hat{\varepsilon}_{1}}}\varepsilon_{2}^{1}.$
(25)

Recall that the actuator directly decodes its own signal by treating the
signal from the robot as interference, and its SINR is given by

${{\gamma_{2}}}=\frac{{{p_{2}}{h_{2}}}}{{{p_{1}}{h_{2}}+1}}.$ (26)

The corresponding decoding error probability is given by
${\varepsilon_{2}}=Q\left({f\left({{{\gamma_{2}}},{M},D}\right)}\right)$.

Now, we can formulate the optimization problem under NOMA transmission as:

$\displaystyle\vspace{-0.5cm}\mathop{\min}\limits_{\left\\{{{p_{1}},{p_{2}}}\right\\}}\;\;\;$
$\displaystyle{{\varepsilon}_{2}}$ (27a) $\displaystyle{\rm{s.t.}}\;\;\;$
$\displaystyle{{\bar{\varepsilon}}_{1}}\leq\varepsilon_{1}^{\max},$ (27b)
$\displaystyle M{p_{1}}+M{p_{2}}\leq{E_{{\rm{tot}}}},$ (27c) $\displaystyle
p_{1}\leq p_{2},$ (27d)

where (27d) represents that more power should be allocated to the user with
weaker channel gains.

Similar to the proof of Lemma 1, we can show that the energy constraint in
(27c) holds with equality at the optimum point. Then, we study the feasible
range of the power allocation $p_{1}$ to facilitate the search algorithm. The
expression of ${{\bar{\varepsilon}}_{1}}$ can be reexpressed as

$\vspace{-0.1cm}{{\bar{\varepsilon}}_{1}}={\varepsilon_{1}}+({{\hat{\varepsilon}_{1}}}-{\varepsilon_{1}})\varepsilon_{2}^{1}\geq{\varepsilon_{1}}.$
(28)

By using constraints (27b) and (28), we have
${\varepsilon_{1}}\leq\varepsilon_{1}^{\max}$. By denoting
${\bar{f}}(\gamma)=f(\gamma,M,D)$, the lower bound of $p_{1}$ can be derived
as

${p_{1}}\geq\frac{{{{\bar{f}}^{-1}}\left({{Q^{-1}}(\varepsilon_{1}^{\max})}\right)}}{{{h_{1}}}}\buildrel\Delta\over{=}p_{1}^{{\rm{lb}}}.$
(29)

From constraint (27d), we know that
${p_{1}}\leq\frac{{{E_{{\rm{tot}}}}}}{{2M}}$. To guarantee the meaningfulness
of $\varepsilon_{2}^{1}$, the inequality
${\log_{2}}\left({1+\gamma_{2}^{1}}\right)\geq D/M$ should hold. Then, we have

${p_{1}}\leq\frac{{{E_{{\rm{tot}}}}{2^{-\frac{D}{M}}}}}{M}-\frac{1}{{{h_{1}}}}+\frac{{{2^{-\frac{D}{M}}}}}{{{h_{1}}}}.$
(30)

In addition, to guarantee the meaningfulness of $\varepsilon_{2}$, the
inequality ${\log_{2}}\left({1+\gamma_{2}}\right)\geq D/M$ should hold, which
yields

${p_{1}}\leq\frac{{{E_{{\rm{tot}}}}{2^{-\frac{D}{M}}}}}{M}-\frac{1}{{{h_{2}}}}+\frac{{{2^{-\frac{D}{M}}}}}{{{h_{2}}}}.$
(31)

Since $h_{1}>h_{2}$, the upper bound of $p_{1}$ is given by

${p_{1}}\leq\min\left\\{{\frac{{{E_{{\rm{tot}}}}{2^{-\frac{D}{M}}}}}{M}-\frac{1}{{{h_{2}}}}+\frac{{{2^{-\frac{D}{M}}}}}{{{h_{2}}}},\frac{{{E_{{\rm{tot}}}}}}{{2M}}}\right\\}\buildrel\Delta\over{=}p_{1}^{{\rm{ub}}}.$
(32)

To further reduce the search complexity, in the following theorem, we prove
that constraint (27b) holds with equality at the optimum point.

_Theorem 2_ : Constraint (27b) holds with equality at the optimum solution.

Proof: Please see Appendix C.

Based on Theorem 2, we can readily know that the one-dimensional line search
algorithm can be used to find the optimal $p_{1}^{\star}$.

###  III-C Relay-assisted transmission

In this scheme, the robot acts as a relay that assists the transmission for
actuator, where decode-and-forward (DF) relay is assumed at the robot. The
packet ID is inserted in the packet head for each device to differentiate
their corresponding data information. The whole blocklength is divided into
two phases, the broadcast phase with blocklength $m_{1}$ and the relay phase
with blocklength $m_{2}$, which satisfy the constraint of $m_{1}+m_{2}\leq M$.

In the first phase, the controller broadcasts a large packet that is a
combination of two packets to both devices, where the combined packet size is
$2D$. The received signals at both devices are given by

$\begin{array}[]{l}{y_{1,1}}=\sqrt{{p_{s}}}{{\tilde{h}}_{1}}{\tilde{x}_{1}}+{n_{1}},\\\
{y_{1,2}}=\sqrt{{p_{s}}}{{\tilde{h}}_{2}}{\tilde{x}_{1}}+{n_{2}},\end{array}$
(33)

where $p_{s}$ denotes the power allocated to the combined packet,
$\tilde{x}_{1}$ carries the data information of the combined packet with
coding rate $2D/m_{1}$. Then, the SNR of the robot to decode the combined
packet is given by ${\gamma_{1}}={p_{s}}{h_{1}}$, and the decoding error
probability at the robot is given by
${\varepsilon_{1}}=Q\left({f\left({{{\gamma_{1}}},{m_{1}},2D}\right)}\right)$.

Since the robot acts as a relay based on the DF mode, if the robot
successfully decodes the combined packet, it will forward the actuator’s
packet to the actuator with coding rate $D/m_{2}$ in the second phase, and the
received signal at the actuator is given by

$\vspace{-0.1cm}{y_{2,2}}=\sqrt{{p_{r}}}{{\tilde{h}}_{3}}{x_{2}}+{n_{3}},$
(34)

where $p_{r}$ is the transmit power at the actuator. The received SNR is
${\gamma_{2}}={p_{r}}{h_{3}}$, where $h_{3}$ is the normalized channel gain
given by
$h_{3}={{{{\left|{{{\tilde{h}}_{3}}}\right|}^{2}}}\mathord{\left/{\vphantom{{{{\left|{{{\tilde{h}}_{3}}}\right|}^{2}}}{\sigma_{2}^{2}}}}\right.\kern-1.2pt}{\sigma_{2}^{2}}}$.
The error probability is given by
${\varepsilon_{2}}=Q\left({f\left({{{\gamma_{2}}},{m_{2}},D}\right)}\right)$.

There is a possibility that the actuator cannot decode its packet due to the
following two reasons: 1) the robot is not able to correctly decode the
combined packet and will not forward anything to the actuator with probability
$\varepsilon_{1}$; and 2) the robot correctly decodes the combined packet and
forwards the packet to the actuator with probability $1-\varepsilon_{1}$, but
with probability ${\varepsilon_{2}}$, the actuator fails to decode the packet.
In this case the actuator will have to decode the combined packet by using the
received signal from the first phase, i.e., ${y_{1,2}}$. The achieved SNR of
the actuator for decoding the combined packet is given by
${{\hat{\gamma}_{2}}}={{p_{s}}{h_{2}}}$, and the corresponding decoding error
probability is given by
${{\hat{\varepsilon}_{2}}}=Q\left({f\left({{{\hat{\gamma}_{2}}},{m_{1}},2D}\right)}\right)$.

As a result, the expected error probability of the actuator decoding its
packet in the relay-assisted transmission scheme is given by

${{\bar{\varepsilon}}_{2}}=\left({\left({1-\varepsilon_{1}}\right){\varepsilon_{2}}+\varepsilon_{1}}\right)\hat{\varepsilon}_{2}.\vspace{-0.4cm}$
(35)

Then, the resource allocation problem is formulated as

$\displaystyle\vspace{-0.5cm}\mathop{\min}\limits_{\left\\{{{m_{1}},{m_{2}},{p_{s}},{p_{r}}}\right\\}}\;\;\;$
$\displaystyle{{\bar{\varepsilon}}_{2}}$ (36a)
$\displaystyle{\rm{s.t.}}\;\;\;$
$\displaystyle{{\varepsilon}_{1}}\leq\varepsilon_{1}^{\max},$ (36b)
$\displaystyle{m_{1}}{p_{s}}+{m_{2}}{p_{r}}\leq{E_{{\rm{tot}}}},$ (36c)
$\displaystyle m_{1}+m_{2}\leq M,$ (36d) $\displaystyle
m_{1},m_{2}\in\mathbb{Z}.$ (36e)

By using the contradiction method, we can easily prove that constraint (36c)
holds with equality at the optimal solution. However, in contrast to the above
two transmission schemes, the decoding error probability constraint (36b) may
not hold with equality at the optimal solution, as the objective function may
also decrease with ${\varepsilon}_{1}$. The algorithms proposed for the OMA
and NOMA transmission schemes cannot be applied.

By using the similar iterative procedure in OMA scheme, we are able to obtain
the feasible region of $m_{1}$ as $m_{1}^{\rm{lb}}\leq m_{1}\leq
m_{1}^{\rm{ub}}$. For given $m_{1}$, the search region of $m_{2}$ can also be
obtained as $m_{2}^{\rm{lb}}(m_{1})\leq m_{2}\leq(M-m_{1})$.

In the following, we study the optimization problem of the power allocation
$p_{s}$ and $p_{r}$ under fixed $m_{1}$ and $m_{2}$. For each given $m_{2}$,
we can obtain the lower bound of $p_{r}$ to make $\varepsilon_{2}$ meaningful:
${p_{r}}\geq{{\left({{2^{{D\mathord{\left/{\vphantom{D{{m_{2}}}}}\right.\kern-1.2pt}{{m_{2}}}}}}-1}\right)}\mathord{\left/{\vphantom{{\left({{2^{{D\mathord{\left/{\vphantom{D{{m_{2}}}}}\right.\kern-1.2pt}{{m_{2}}}}}}-1}\right)}{{h_{3}}}}}\right.\kern-1.2pt}{{h_{3}}}}\buildrel\Delta\over{=}p_{r}^{{\rm{lb}}}$.
Thus, the upper bound of $p_{s}$ can be derived as

${p_{s}}\leq\frac{{{E_{{\rm{tot}}}}}}{{{m_{1}}}}-\frac{{{m_{2}}}}{{{m_{1}}}}p_{r}^{{\rm{lb}}}\buildrel\Delta\over{=}p_{s}^{{\rm{up}}}.\vspace{-0.2cm}$
(37)

Hence, the feasible region of ${p_{s}}$ is given by $p_{s}^{\rm{lb}}\leq
p_{s}\leq p_{s}^{\rm{ub}}$, where $p_{s}^{\rm{lb}}$ is the solution to
equation ${\varepsilon_{1}}({p_{s}})=\varepsilon_{1}^{\max}$ with given
$m_{1}$. When $p_{s}$ is given, $p_{r}$ can be calculated as
$p_{r}={{\left({{E_{{\rm{tot}}}}-{m_{1}}{p_{s}}}\right)}\mathord{\left/{\vphantom{{\left({{E_{{\rm{tot}}}}-{m_{1}}{p_{s}}}\right)}{{m_{2}}}}}\right.\kern-1.2pt}{{m_{2}}}}$.
Then, the original optimization problem reduces to a one-dimension
optimization problem as

$\displaystyle\mathop{\min}\limits_{{p_{s}}}\;\;\;$
$\displaystyle{{\bar{\varepsilon}}_{2}}$ (38a)
$\displaystyle{\rm{s.t.}}\;\;\;$ $\displaystyle p_{s}^{\rm{lb}}\leq p_{s}\leq
p_{s}^{\rm{ub}}.$ (38b)

The one-dimensional line search method can be used to solve Problem (38).

In summary, we provide Algorithm 2 to solve Problem (36).

Input : $h_{1},h_{2},D,M,\varepsilon_{1}^{\max},E_{{\rm{tot}}}$

Output : $p_{s}^{\star},p_{r}^{\star},m_{1}^{\star},m_{2}^{\star}$

1 Apply the iterative procedure to calculate $m_{1}^{\rm{lb}},m_{1}^{\rm{ub}}$
and $m_{2}^{\rm{lb}}$;

2for  _ $m_{1}=m_{1}^{\rm{lb}}:m_{1}^{\rm{ub}}$ _ do

3 Calculate the solution to the equation
${{\varepsilon}_{1}}=\varepsilon_{1}^{\max}$, which is denoted as
$p_{s}^{\rm{lb}}$. Calculate the lower bound of $m_{2}$ with given $m_{1}$,
denoted as $m_{2}^{\rm{lb}}(m_{1})$.

4 for  _$m_{2}=\tilde{m}_{2}^{\rm{lb}}:(M-m_{1})$_ do

5 Calculate the upper bound of $p_{s}$ as $p_{s}^{\rm{ub}}$ in (37), and solve
Problem (38). Calculate the objective value
${{\bar{\varepsilon}}_{2}}(m_{1},m_{2})$.

6 end for

7 Given $m_{1}$, find the blocklength $m_{2}$ with the minimum value of
${\varepsilon}_{2}(m_{1},m_{2})$:
${\left.{m_{2}^{\\#}}\right|_{{m_{1}}}}=\mathop{\arg\min}\limits_{\tilde{m}_{2}^{{\rm{lb}}}\leq{m_{2}}\leq
M-{m_{1}}}{\varepsilon_{2}}\left({{m_{1}},{m_{2}}}\right).$

8 end for

Return
$m_{1}^{\star}=\mathop{\arg\min}\limits_{m_{1}^{{\rm{lb}}}\leq{m_{1}}\leq
m_{1}^{{\rm{ub}}}}{\varepsilon_{2}}\left({{m_{1}},{{\left.{m_{2}^{\\#}}\right|}_{{m_{1}}}}}\right),m_{2}^{\star}={\left.{m_{2}^{\\#}}\right|_{m_{1}^{\star}}}$
and the corresponding $p_{s}^{\star}$ and $p_{r}^{\star}$.

Algorithm 2 Algorithm for Problem (36)

###  III-D C-NOMA transmission

In this part, we consider the C-NOMA transmission in [30], which is a
combination of the NOMA scheme and relay-assisted scheme. Specifically, in the
first phase, the controller transmits two signals $x_{1}$ and $x_{2}$ to the
two devices via the NOMA technique. In the second phase, the robot acts as a
relay and forwards the packet to the actuator. The blocklength for these two
phases are denoted by $m_{1}$ and $m_{2}$, which satisfies $m_{1}+m_{2}\leq
M$.

Specifically, in the first phase, the received signals at the robot and the
actuator are given by

$\begin{array}[]{l}{y_{1,1}}=\sqrt{{p_{1}}}{{\tilde{h}}_{1}}{x_{1}}+\sqrt{{p_{2}}}{{\tilde{h}}_{1}}{x_{2}}+{n_{1}},\\\
{y_{1,2}}=\sqrt{{p_{1}}}{{\tilde{h}}_{2}}{x_{1}}+\sqrt{{p_{2}}}{{\tilde{h}}_{2}}{x_{2}}+{n_{2}},\end{array}$
(39)

respectively, where $p_{1}$ and $p_{2}$ are the transmit power allocated to
the robot and the actuator, $x_{1}$ and $x_{2}$ carries different information
knowledge for different packets with size $D$. Hence, the coding rate for the
transmission to the robot and the actuator are given by $D/m_{1}$.

By using the NOMA scheme, the SIC technique is employed at the robot side to
cancel the interference from the actuator. Similar to the analysis in the NOMA
scheme, the decoding error probability of $x_{2}$ at the robot is given by

$\vspace{-0.1cm}{{\varepsilon_{2}^{1}}}=Q\left({f\left({{{\gamma_{2}^{1}}},{m_{1}},D}\right)}\right),$
(40)

where $\gamma_{2}^{1}$ is the same as that in (22). Under perfect SIC
condition, the decoding error probability of $x_{1}$ at the robot is given by

${\varepsilon_{1}}=Q\left({f\left({{{\gamma_{1}}},{m_{1}},D}\right)}\right),$
(41)

where ${\gamma_{1}}={p_{1}}{h_{1}}$. However, if SIC fails, the corresponding
decoding error probability of $x_{1}$ at the robot is given by
${\hat{\varepsilon}_{1}}=Q\left({f\left({{{\hat{\gamma}_{1}}},{m_{1}},D}\right)}\right)$,
where $\hat{\gamma}_{1}$ is given by (24). Using the same analysis as in NOMA,
the average decoding probability at the robot is given by

$\vspace{-0.1cm}{{\bar{\varepsilon}}_{1}}={\varepsilon_{1}}(1-\varepsilon_{2}^{1})+{{{\hat{\varepsilon}_{1}}}}\varepsilon_{2}^{1}.$
(42)

By using the similar analysis as in the relay-assisted scheme, the decoding
error probability of the actuator decoding $x_{2}$ under the C-NOMA scheme is
given by

$\vspace{-0.1cm}{{\bar{\varepsilon}}_{2}}=\left({\left({1-\varepsilon_{2}^{1}}\right){\varepsilon_{2}}+\varepsilon_{2}^{1}}\right)\hat{\varepsilon}_{2},$
(43)

where $\varepsilon_{2}^{1}$ and $\varepsilon_{2}$ are given in Subsection-
III-B, and $\hat{\varepsilon}_{2}$ is the decoding error probability of the
actuator when the actuator has to decode $x_{2}$ from the received signal in
the first phase. The expression of $\hat{\varepsilon}_{2}$ is given by

${\hat{\varepsilon}_{2}}=Q\left({f\left({{\hat{\gamma}_{2}},{m_{1}},D}\right)}\right),$
(44)

where ${{\hat{\gamma}_{2}}}$ is given by

${{\hat{\gamma}_{2}}}=\frac{{{p_{2}}{h_{2}}}}{{{p_{1}}{h_{2}}+1}}.$ (45)

Therefore, the optimization problem of C-NOMA transmission scheme can be
formulated as

$\displaystyle\vspace{-0.2cm}\mathop{\min}\limits_{\left\\{{{m_{1}},{m_{2}},{p_{1}},{p_{2}},{p_{r}}}\right\\}}\;\;\;$
$\displaystyle{{\bar{\varepsilon}}_{2}}$ (46a)
$\displaystyle{\rm{s.t.}}\;\;\;$
$\displaystyle{{\bar{\varepsilon}}_{1}}\leq\varepsilon_{1}^{\max},$ (46b)
$\displaystyle{m_{1}}({p_{1}}+{p_{2}})+{m_{2}}{p_{r}}\leq{E_{{\rm{tot}}}},$
(46c) $\displaystyle m_{1}+m_{2}\leq M,$ (46d) $\displaystyle
m_{1},m_{2}\in\mathbb{Z},$ (46e) $\displaystyle p_{1}\leq p_{2}.$ (46f)

Following the similar proof as Lemma 1, we can show that constraints (46b) and
(46c) hold with equality at the optimal point, thus the search method can be
used to find the optimal solution of Problem (46). To reduce the search
complexity, we need to find tight lower and upper bounds on $m_{1}$ and
$m_{2}$.

However, unlike the previous schemes that only two power allocation variables
are involved, the number of power allocation variables in C-NOMA scheme is
three. This will complicate the analysis of deriving the bounds of $m_{1}$ and
$m_{2}$. To deal with this difficulty, we regard the summation of
$p_{1}+p_{2}$ as a whole entity. To realize the functionality of the C-NOMA
scheme, $\varepsilon_{1}$ and ${{\varepsilon_{2}^{1}}}$ should be very small,
e.g., much lower than 0.5. Then, we have

$\displaystyle\vspace{-0.8cm}{p_{1}}$ $\displaystyle\geq$
$\displaystyle\frac{1}{{{h_{1}}}}\left({{2^{\frac{D}{{{m_{1}}}}}}-1}\right),$
(47) $\displaystyle{p_{2}}$ $\displaystyle\geq$
$\displaystyle\frac{1}{{{h_{1}}}}\left({{2^{\frac{D}{{{m_{1}}}}}}-1}\right)\left({1+{p_{1}}{h_{1}}}\right).\vspace{-0.2cm}$
(48)

By substituting (47) into the right hand side of (48), we have

${p_{2}}\geq\frac{1}{{{h_{1}}}}\left({{2^{\frac{D}{{{m_{1}}}}}}-1}\right)2^{\frac{D}{{{m_{1}}}}}.$
(49)

By adding (47) and (49), one can obtain

$\vspace{-0.1cm}{p_{1}}+{p_{2}}\geq\frac{1}{{{h_{1}}}}\left({{2^{\frac{{2D}}{{{m_{1}}}}}}-1}\right).$
(50)

To ensure that $\varepsilon_{2}$ is meaningful, we have

${p_{r}}\geq\frac{1}{{{h_{3}}}}\left({{2^{\frac{D}{{{m_{2}}}}}}-1}\right).$
(51)

By using the similar iterative procedure, we can also obtain the lower bounds
of $m_{1}$ and $m_{2}$, which are denoted as $m_{1}^{{\rm{lb}}}$ and
$m_{2}^{{\rm{lb}}}$, respectively. As a result, the search region of $m_{1}$
is given by $m_{1}^{{\rm{lb}}}\leq
m_{1}\leq(M-m_{2}^{{\rm{lb}}})\buildrel\Delta\over{=}m_{1}^{{\rm{ub}}}$. For
each given $m_{1}$ within the range, we need to find the search range of
$m_{2}$, which is detailed as follows. Since
${\varepsilon_{1}}<{{\bar{\varepsilon}}_{1}}\leq\varepsilon_{1}^{\max}$, the
lower bound of $p_{1}$ can be obtained by solving the equation of
${\varepsilon_{1}}({p_{1}})=\varepsilon_{1}^{\max}$ for given $m_{1}$, which
is denoted as $p_{1}^{\rm{lb}}$. By using (48), we can obtain the lower bound
of $p_{2}$ as follows:

$\vspace{-0.1cm}{p_{2}}\geq\frac{1}{{{h_{1}}}}\left({{2^{\frac{D}{{{m_{1}}}}}}-1}\right)\left({1+{p_{1}^{\rm{lb}}}{h_{1}}}\right)\buildrel\Delta\over{=}p_{2}^{\rm{lb}}.$
(52)

Based on (46c), we have

$\vspace{-0.05cm}{E_{{\rm{tot}}}}-{m_{1}}(p_{1}^{{\rm{lb}}}+p_{2}^{{\rm{lb}}})\geq{m_{2}}{p_{r}}\geq\frac{{{m_{2}}}}{{{h_{3}}}}\left({{2^{\frac{D}{{{m_{2}}}}}}-1}\right),$
(53)

where the last inequality is due to the fact that
${p_{r}}\geq\frac{1}{{{h_{3}}}}\left({{2^{\frac{D}{{{m_{2}}}}}}-1}\right)$
must hold to guarantee the meaningfulness of ${\varepsilon_{2}}$. The lower
bound of $m_{2}$ under given $m_{1}$ (denoted as $m_{2}^{{\rm{lb}}}(m_{1})$)
can be obtained from (53), which is the minimum integer that satisfies (53).
Obviously, the upper bound of $m_{2}$ with given $m_{1}$ is $M-m_{1}$. Hence,
the search region of $m_{2}$ is given by $m_{2}^{{\rm{lb}}}(m_{1})\leq
m_{2}\leq(M-m_{1})$.

Given $m_{1}$ and $m_{2}$, we need to find the optimal $p_{1}$, $p_{2}$ and
$p_{s}$. These variables are coupled and it is difficult to find the optimal
solution by using the optimization method. The one-dimensional search is
adopted to find the optimal solution. In particular, we first fix the value of
the sum of $p_{1}$ and $p_{2}$ as $t$, i.e., $t=p_{1}+p_{2}$. Since constraint
(46b) holds with equality at the optimal point, the optimal $p_{1}$ can be
obtained by solving the equation
$\bar{\varepsilon}_{1}(p_{1})=\varepsilon_{1}^{\rm{max}}$ by inserting
$p_{2}=t-p_{1}$ into this equation. By combining (48) and (46f), the upper
bound of $p_{1}$ is obtained as $p_{1}\leq\min{\left(t\cdot
2^{-\frac{D}{m_{1}}}-\frac{1}{h_{1}}+\frac{1}{h_{1}}\cdot
2^{-\frac{D}{m_{1}}},\frac{t}{2}\right)}\triangleq p_{1}^{\rm{up}}$, and
$p_{1}$ should be within the domain
$p_{1}\in(p_{1}^{\rm{lb}},p_{1}^{\rm{up}})$. This equation has only one
variable $p_{1}$ and the one-dimensional search method can be adopted to solve
the equation. As constraint (46c) holds with equality, $p_{r}$ can be directly
obtained as
$p_{r}={{({E_{{\rm{tot}}}}-t{m_{1}})}\mathord{\left/{\vphantom{{({E_{{\rm{tot}}}}-t{m_{1}})}{{m_{2}}}}}\right.\kern-1.2pt}{{m_{2}}}}$.
Calculate the objective value with given $m_{1}$, $m_{2}$, $t$ and $p_{r}$.
The remaining task is to find the tight search region $t$. Obviously, the
lower bound of $t$ is given by
$t^{\rm{lb}}=p_{1}^{{\rm{lb}}}+p_{2}^{{\rm{lb}}}$. To obtain the upper bound
of $t$, we first find the lower bound of $m_{2}p_{r}$, which is given by

$\vspace{-0.1cm}{m_{2}}{p_{r}}\geq\frac{{{m_{2}}}}{{{h_{3}}}}\left({{2^{\frac{D}{{{m_{2}}}}}}-1}\right).$
(54)

Then, the upper bound of $t$ is given by

$t\leq\frac{1}{{{m_{1}}}}\left({{E_{{\rm{tot}}}}-\frac{{{m_{2}}}}{{{h_{3}}}}\left({{2^{\frac{D}{{{m_{2}}}}}}-1}\right)}\right)={t^{{\rm{ub}}}}.$
(55)

Based on the above analysis, we provide Algorithm 3 to solve Problem (36).

Input : $h_{1},h_{2},h_{3},D,M,\varepsilon_{1}^{\max},E_{{\rm{tot}}}$

Output :
$p_{1}^{\star},p_{2}^{\star},p_{r}^{\star},m_{1}^{\star},m_{2}^{\star}$

1 Apply the iterative procedure to calculate $m_{1}^{\rm{lb}},m_{1}^{\rm{ub}}$
and $m_{2}^{\rm{lb}}$;

2for  _ $m_{1}=m_{1}^{\rm{lb}}:m_{1}^{\rm{ub}}$ _ do

3 Calculate the solution to the equation
${{\varepsilon}_{1}}=\varepsilon_{1}^{\max}$, which is denoted as
$p_{1}^{\rm{lb}}$. Use (52) to calculate the lower bound of $p_{2}$, denoted
as $p_{2}^{\rm{lb}}$. Use (53) to find the lower bound of $m_{2}$, denoted as
$m_{2}^{{\rm{lb}}}(m_{1})$.

4 if  _$m_{2}^{{\rm{lb}}}(m_{1})\leq(M-m_{1})$ _ then

5 for  _$m_{2}=m_{2}^{{\rm{lb}}}(m_{1}):(M-m_{1})$_ do

6 Calculate the lower bound of $t$ as
$t^{\rm{lb}}=p_{1}^{{\rm{lb}}}+p_{2}^{{\rm{lb}}}$ , and the upper bound of $t$
as $t^{\rm{ub}}$ from (55). Use the one-dimensional search to find the optimal
$t$ that achieves the minimum objective value. Denote the optimal objective
value ${{\bar{\varepsilon}}_{2}}(m_{1},m_{2})$.

7 end for

8 Given $m_{1}$, find the blocklength $m_{2}$ with the minimum value of
${\varepsilon}_{2}(m_{1},m_{2})$:
${\left.{m_{2}^{\\#}}\right|_{{m_{1}}}}=\mathop{\arg\min}\limits_{m_{2}^{{\rm{lb}}}(m_{1})\leq{m_{2}}\leq
M-{m_{1}}}{\varepsilon_{2}}\left({{m_{1}},{m_{2}}}\right).$

9 end if

10

11 end for

Return
$m_{1}^{\star}=\mathop{\arg\min}\limits_{m_{1}^{{\rm{lb}}}\leq{m_{1}}\leq
m_{1}^{{\rm{ub}}}}{\varepsilon_{2}}\left({{m_{1}},{{\left.{m_{2}^{\\#}}\right|}_{{m_{1}}}}}\right),m_{2}^{\star}={\left.{m_{2}^{\\#}}\right|_{m_{1}^{\star}}}$
and the corresponding $p_{1}^{\star}$ and $p_{2}^{\star}$.

Algorithm 3 Algorithm for Problem (46)

Remark: It is noted that the feasible region of C-NOMA scheme is smaller than
that of the relay-assisted transmission scheme. Specifically, if $p_{1}^{*}$
and $p_{2}^{*}$ is any one feasible solution of Problem (46), it can be
readily checked that $p_{s}=p_{1}^{*}+p_{2}^{*}$ is also a feasible solution
of Problem (36). However, if $\\{p_{1}^{*},p_{2}^{*}\\}$ is not a feasible
solution of Problem (46), $p_{s}=p_{1}^{*}+p_{2}^{*}$ may still be feasible
for Problem (36). For example, by letting

${p_{2}}=\frac{1}{{{h_{2}}}}\left({{2^{\frac{D}{{{m_{1}}}}}}-1}\right),{p_{1}}=\frac{1}{{{h_{1}}}}\left({{2^{\frac{{2D}}{{{m_{1}}}}}}-1}\right)-\frac{1}{{{h_{2}}}}\left({{2^{\frac{D}{{{m_{1}}}}}}-1}\right),$
(56)

it can be readily checked that $p_{1}$ and $p_{2}$ do not satisfy condition
(48), which is not feasible for Problem (46). However by setting
$p_{s}=p_{1}+p_{2}$, $p_{s}$ is still feasible for Problem (36). This
observation means the feasible region for Problem (36) is larger than that of
Problem (46).

##  IV Extension to More Devices for the OMA Scheme

In this section, we consider the more general case when the system has more
than two devices for the OMA scheme. The extension to other schemes will be
studied in the future work.

###  IV-A Sytem Model and Problem Formulation

Let us denote the total number of devices as $K$, and the set of all devices
as $\cal K$. We assume that the normalized channel gains of all $K$ devices
are arranged in a decreasing order, i.e., $h_{1}>h_{2}>\cdots>h_{K}$333Due to
the small-scale fading, the probability that any two or more devices have the
same channel gain is equal to zero.. Then, we aim to jointly optimize the
power and blocklength allocation to minimize the decoding error probability of
the $K$th device while guaranteeing the decoding error probability
requirements of the first $K-1$ devices. Mathematically, the optimization
problem can be formulated as follows:

$\displaystyle\min_{\left\\{m_{k},k\in\cal K\right\\},\left\\{p_{k},k\in\cal
K\right\\}}\;\;\;\;$ $\displaystyle\varepsilon_{K}$ (57a) s.t.
$\displaystyle\varepsilon_{k}\leq\varepsilon_{k}^{\max},\;\;k\in{\cal
K}\backslash K,$ (57b) $\displaystyle\sum\nolimits_{k\in\cal K}m_{k}p_{k}\leq
E_{\text{tot}},$ (57c) $\displaystyle\sum\nolimits_{k\in\cal K}m_{k}\leq M,$
(57d) $\displaystyle m_{k}\in\mathbb{Z},k\in\cal K.$ (57e)

In contrast to the case of two devices where the globally optimal solution to
Problem (7) can be obtained, the globally optimal solution to Problem (57) for
the more general case is not available. In the following, we aim to obtain a
suboptimal solution to Problem (57).

###  IV-B Problem Reformulation

To make Problem (57) tractable, we again approximate $V$ as one, i.e.,
$V\approx 1$. This approximation is very accurate when the SNR value $\gamma$
is very high, i.e., $\gamma\gg 1$. As the decoding error probability is a
decreasing function of power and blocklength, we can readily prove that
constraints (57b), (57c) and (57d) hold with equality at the optimum point by
using the contradiction method. By using the fact that
$\varepsilon_{k}=\varepsilon_{k}^{\max},k\in{\cal K}\backslash K$, $p_{k}$ can
be derived as a function of $m_{k}$, given by

$p_{k}=\frac{2^{\frac{D}{m_{k}}+\frac{Q^{-1}(\varepsilon_{k}^{\max})}{\ln
2\sqrt{m_{k}}}}-1}{h_{k}}\triangleq\chi(m_{k}),{k\in\cal K}\backslash K.$ (58)

By substituting (58) into (57), Problem (57) can be transformed as follows:

$\displaystyle\min_{\left\\{m_{k},k\in\cal K\right\\},p_{K}}\;\;\;\;$
$\displaystyle\varepsilon_{K}$ (59a) s.t. $\displaystyle\sum\limits_{k\in{\cal
K}\backslash K}m_{k}\chi(m_{k})+m_{K}p_{K}=E_{\text{tot}},$ (59b)
$\displaystyle\sum\limits_{k\in{\cal K}}m_{k}=M,m_{k}\in\mathbb{Z},k\in\cal
K.$ (59c)

Compared with the original Problem (57), the number of optimization variables
of Problem (59) is significantly reduced. However, this problem is still
difficult to solve. In the following, we first use the exhaustive search to
find $m_{K}$, and then optimize $p_{K}$. To this end, we need to find tight
lower and upper bounds of $m_{K}$ to reduce the computational complexity.

###  IV-C Bounds of $m_{K}$

In this subsection, we attempt to obtain the bounds of $m_{K}$. We first
provide the following theorem.

_Theorem 3_ : Define
$A_{k}={{{Q^{-1}}(\varepsilon_{k}^{\max})}\mathord{\left/{\vphantom{{{Q^{-1}}(\varepsilon_{k}^{\max})}{\ln
2}}}\right.\kern-1.2pt}{\ln 2}}$ and
$g\left({{m_{k}}}\right)\buildrel\Delta\over{=}{m_{k}}\chi({m_{k}})$. Then,
$g\left({{m_{k}}}\right)$ is a monotonically decreasing and convex function
when $m_{k}$ satisfies:

$\sqrt{{m_{k}}}<\frac{{\frac{3}{4}{A_{k}}\ln
2+\sqrt{\frac{9}{{16}}{{\left({\ln 2}\right)}^{2}}A_{k}^{2}+8D\ln 2}}}{2}.$
(60)

_Proof_ : Please see Appendix D.

In general, for a typical URLLC system, the number of transmission bits is
around $100$ bits and the decoding error probability requirement is around
$10^{-9}$. Then, $A_{k}$ is 8.653, and the value of the right hand side of
(60) is given by 14.236. Then, when $m_{k}\leq 202$, the inequality (60)
holds. In short packet transmission with OMA scheme, the number of blocklength
to each device is generally smaller than 100. Hence, in our considered
scenario, $g\left({{m_{k}}}\right)$ can be regarded as a monotonically
decreasing and convex function.

In the following, we provide an iterative procedure to obtain the tight bounds
of $m_{K}$. Since $m_{k}p_{k}=g(m_{k})<E_{\rm{tot}}$ and $g(m_{k})$ is a
monotonically decreasing function, we can obtain the lower bound of $m_{k}$ by
using the bisection search method, which is denoted as
$m_{k}^{\rm{lb}(0)},k\in{\cal K}\backslash K$. To guarantee the meaningfulness
of ${\varepsilon_{K}}$, the following inequality holds

${p_{K}}>{{\left({{2^{\frac{D}{{{m_{K}}}}}}-1}\right)}\mathord{\left/{\vphantom{{\left({{2^{\frac{D}{{{m_{K}}}}}}-1}\right)}{{h_{K}}}}}\right.\kern-1.2pt}{{h_{K}}}}.$
(61)

Then, we have

${E_{{\rm{tot}}}}>{m_{K}}{p_{K}}>\frac{{{m_{K}}}}{{{h_{K}}}}\left({{2^{\frac{D}{{{m_{K}}}}}}-1}\right)\buildrel\Delta\over{=}q({m_{K}}).$
(62)

As a result, we can obtain the lower bound of $m_{K}$ from (62), which is
denoted as $m_{K}^{\rm{lb}(0)}$. Then, for each device $k$, the upper bound of
$m_{k}$ is given by $m_{k}^{\rm{ub}(0)}=M-\sum\nolimits_{i\in{\cal
K}\backslash k}{m_{i}^{{\rm{lb}}(0)}},k\in\cal K$. Since $q({m_{K}})$ defined
in (62) is a monotonically decreasing function, we have
$q({m_{K}})>q({m_{K}^{\rm{ub}(0)}})$. In addition, $g(m_{k})$ is a
monotonically decreasing function of $m_{k}$, and we have
$g(m_{k})>g(m_{k}^{\rm{ub}(0)}),k\in{\cal K}\backslash K$. Then, for each
$k\in{\cal K}\backslash K$, we have

${E_{{\rm{tot}}}}-\sum\nolimits_{i\in{\cal
K}\backslash\\{K,k\\}}{g(m_{i}^{{\rm{ub}(0)}})}-q({m_{K}^{\rm{ub}(0)}})>g(m_{k}),k\in{\cal
K}\backslash K.$ (63)

Then, the lower bound of $m_{k}$ for $k\in{\cal K}\backslash K$ can be
obtained as ${m_{k}^{{\rm{lb}}(1)}},k\in{\cal K}\backslash K$. For the $K$th
device, we have

${E_{{\rm{tot}}}}-\sum\nolimits_{k\in{\cal K}\backslash
K}{g(m_{k}^{{\rm{ub}(0)}})}>\frac{{{m_{K}}}}{{{h_{K}}}}\left({{2^{\frac{D}{{{m_{K}}}}}}-1}\right).$
(64)

Then, based on (64) we can update the lower bound of $m_{K}$ as
$m_{K}^{{\rm{lb}(1)}}$. Then, for each device $k$, the upper bound of $m_{k}$
is given by $m_{k}^{\rm{ub}(1)}=M-\sum\nolimits_{i\in{\cal K}\backslash
k}{m_{i}^{{\rm{lb}}(1)}},k\in\cal K$. Finally, repeat the above procedure
until $m_{K}^{{\rm{lb}}(n)}=m_{K}^{{\rm{lb}}(n+1)}$ and
$m_{K}^{{\rm{ub}}(n)}=m_{K}^{{\rm{ub}}(n+1)}$, where $n$ is the iteration
number. Similar to the case of two devices, the above procedure can be proved
to be convergent, and denote the final converged upper and lower bounds of
$m_{K}$ as $m_{K}^{\rm{ub}}$ and $m_{K}^{\rm{lb}}$, respectively.

###  IV-D Optimization of $p_{K}$ with Given $m_{K}$

Given $m_{K}$, $\varepsilon_{K}$ is a monotonically decreasing function of
$p_{K}$ and $p_{K}$ is given by

${p_{K}}=\frac{1}{{{m_{K}}}}\left({{E_{{\rm{tot}}}}-\sum\limits_{k\in{\cal
K}\backslash K}{{m_{k}}\chi({m_{k}})}}\right),$ (65)

Problem (59) can then be equivalently transformed as

$\displaystyle\min_{\left\\{m_{k},k\in{\cal K}\backslash K\right\\}}\;\;\;\;$
$\displaystyle\sum\limits_{k\in{\cal K}\backslash K}m_{k}\chi(m_{k})$ (66a)
s.t. $\displaystyle\sum\limits_{k\in{\cal K}\backslash K}m_{k}=M-m_{K},$ (66b)
$\displaystyle m_{k}\in\mathbb{Z},k\in{\cal K}\backslash K.$ (66c)

This problem is still difficult to solve due to the integer constraint (66c).
To resolve this issue, we relax $\\{m_{k},k\in{\cal K}\backslash K\\}$ to
continuous values. Then, Problem (66) can be relaxed as follows:

$\displaystyle\min_{\left\\{m_{k},k\in{\cal K}\backslash K\right\\}}\;\;\;\;$
$\displaystyle\sum\limits_{k\in{\cal K}\backslash K}m_{k}\chi(m_{k})$ (67a)
s.t. $\displaystyle m_{k}\geq m_{k}^{\rm{lb}},k\in{\cal K}\backslash
K,(\ref{edsawxs}),$ (67b)

where $\\{m_{k}^{\rm{lb}},k\in{\cal K}\backslash K\\}$ are given in the above
subsection. Since $m_{k}\chi(m_{k})$ is proved to be convex as shown in
Theorem 2, Problem (67) is a convex optimization problem, which can be solved
by using the Lagrangian dual decomposition method [31]. We first introduce the
Lagrange multiplier $\lambda$ associated with constraint (66b), the partial
Lagrangian function of Problem (67) is given by

${\cal L}({\bf{m}},\lambda)=\sum\limits_{k\in{\cal K}\backslash
K}m_{k}\chi(m_{k})+\lambda\left(\sum\limits_{k\in{\cal K}\backslash
K}m_{k}-M-m_{K}\right),$ (68)

where ${\bf{m}}=\\{m_{k},k\in{\cal K}\backslash K\\}$.

In the following, we aim to obtain the optimal $m_{k},k\in{\cal K}\backslash
K$ for given $\lambda$, which is denoted as $m_{k}^{\star}(\lambda),k\in{\cal
K}\backslash K$. As ${\cal L}({\bf{m}},\lambda)$ is a convex function of
$m_{k},k\in{\cal K}\backslash K$, the optimal $m_{k}$ for given $\lambda$ can
be obtained in the following. If

${\left.{\frac{{\partial{\cal
L}({\bf{m}},\lambda)}}{{\partial{m_{k}}}}}\right|_{{m_{k}}=m_{k}^{{\rm{lb}}}}}\geq
0,$ (69)

the optimal $m_{k}$ is given by $m_{k}^{\star}(\lambda)=m_{k}^{{\rm{lb}}}$.
Otherwise, $m_{k}^{\star}(\lambda)$ is the solution to the following equation:

$\frac{{\partial{\cal L}({\bf{m}},\lambda)}}{{\partial{m_{k}}}}=0,$ (70)

which can be obtained by using the bisection search method.

Upon obtaining the optimal ${m_{k}^{\star}}(\lambda),k\in{\cal K}\backslash
K$, we can obtain the value of the left hand side of (66b), which is defined
as function $F(\lambda)$

$F(\lambda)\buildrel\Delta\over{=}\sum\limits_{k\in{\cal K}\backslash
K}m_{k}^{\star}(\lambda).$ (71)

By using the similar technique as in Appendix A of [32], we can prove that
$F(\lambda)$ is a monotonically decreasing function of $\lambda$. Hence, the
bisection search method can be adopted to find the solution of $\lambda$ to
the equation $F(\lambda)=M-m_{K}$ if the original problem is feasible.

Denote the solution obtained by solving the relaxed problem (67) as
$\left\\{{\bar{m}}_{k},k\in{\cal K}\backslash K\right\\}$. In general,
$\left\\{{\bar{m}}_{k},k\in{\cal K}\backslash K\right\\}$ may violent the
integer requirement. Hence, we need to convert the continuous
$\left\\{{\bar{m}}_{k},k\in{\cal K}\backslash K\right\\}$ to integer
solutions, denoted as $\left\\{{m}_{k}^{\star},k\in{\cal K}\backslash
K\right\\}$. However, the integer conversion problem is a combinatorial
optimization problem, which is NP to solve. In the following, we apply the
greedy search method to solve the integer conversion problem. Specifically, we
first initialize the integer solution as
$m_{k}^{\star}=\left\lfloor{{{\bar{m}}_{k}}}\right\rfloor,k\in{\cal
K}\backslash K$. Note that $g(m_{k})$ is a monotonically decreasing function
of $m_{k}$. Each time we allocate one blocklength to the device with the
largest decrement of $g(m_{k})$, i.e.,
${k^{*}}=\mathop{\arg}\max\nolimits_{k\in{\cal K}\backslash
K}\left\\{{g({m_{k}})-g({m_{k}}+1)}\right\\}$. The rational behind this is
that based on (66b) more energy can be allocated to the $K$th device, thus
decreasing ${\varepsilon_{K}}$ most. For the $k^{*}$th device, we set
$m_{k^{*}}^{\star}=m_{k^{*}}^{\star}+1$. If $p_{K}^{\star}$ is smaller than
zero, set ${\varepsilon_{K}^{\star}}=1$. Repeat the above procedure until
$\sum\nolimits_{k\in{\cal K}\backslash K}m_{k}^{\star}=M-m_{K}$. Then, the
power allocated to the $K$th device can be recalculated as

$p_{K}=\frac{{{E_{{\rm{tot}}}}-\sum\limits_{k\in{\cal K}\backslash
K}{g(m_{k}^{\star})}}}{{m_{K}}}.$ (72)

Thus, we can calculate ${\varepsilon_{K}}$ based on current $m_{K}$ and
$p_{K}^{\star}$.

##  V Simulations Results

In this section, simulation results are provided to evaluate the performance
of the proposed algorithms. For simplicity, we assume that the controller, the
robot and the actuator are located on the same line, and the robot is moving
from the controller to the actuator, and the robot is served as the relay to
help the transmission of the actuator. The distance between the controller and
the actuator is set as $500$ m. Let us denote $d_{1}$, $d_{2}$ and $d_{3}$ as
the distances from the controller to the robot, the controller to the
actuator, and the robot to the actuator, respectively. The system bandwidth is
set as $B=1$ MHz. Hence, the downlink transmission delay duration is
calculated as $100\ {\rm{us}}$ that meets a criterion of industrial
standards[13]. The noise power spectral density is -173 dBm/Hz. The decoding
(packet) error probability requirement for the robot is set as $10^{-9}$. The
large-scale path loss model is $35.3+37.6\log_{10}{\rm{dB}}$ [33]. The
simulation section is divided into two subsections. In the first subsection,
we assume that the channel gain is only determined by the path loss in order
to obtain the insights of all the schemes. In the second subsection, we
consider the network availability performance [17] taking into account small-
scale fading obeying the Rayleigh distribution.

###  V-A Only Large-scale Fading

In Fig. 3, we first study the impact of distance $d_{1}$ on the decoding error
probability. We observe that relay-assisted transmission outperforms the other
three schemes. It is interesting to see that when the robot moves from the
controller to the actuator, the decoding error probability achieved by the OMA
and NOMA schemes always decreases. The main reason is that the channel gain
from controller to robot decreases with increasing the distance, so the energy
and blocklength required for the robot to guarantee its error probability
requirement increases. As a result, the available energy and blocklength for
the actuator will decrease. On the other hand, the reliability performances
achieved by the C-NOMA and relay-assisted schemes first increase and then
decrease when the robot moves in the line. This can be explained as follows.
When the robot moves from $50\ {\rm{m}}$ to $150\ {\rm{m}}$ for the C-NOMA and
$200\ {\rm{m}}$ for relay-assisted scheme, the channel gain from the robot to
the actuator becomes weak, which is the performance bottleneck that limits the
decoding error probability of the actuator. However, when the robot continues
to move towards the actuator, the transmission link from the controller to the
robot becomes the bottleneck link. Hence, the distance $d_{1}$ can be
optimized to additionally improve the system performance, which can be treated
in the future work. It is interesting to observe that the C-NOMA performs
worse than the relay-assisted scheme, which is due to the larger feasible
region for the latter scheme as explained at the end of Section III.

In Fig. 3, we examine the impact of available blocklength $M$ on the decoding
error probability of the actuator. As expected, larger $M$ leads to much
better reliability performance in all schemes, and the decoding error
probability achieved by the relay scheme decreases from $1$ to $10^{-22}$ with
$M$ increasing from 50 to 100. It is interesting to find that when the
blocklength $M$ is equal to 50 and 60, the NOMA scheme has the best
reliability performance since the whole transmission blocklength can be used
for transmission in NOMA, while the whole blocklength should be divided into
two parts for the other schemes. Importantly, this provides insights for the
system designer that when the blocklength is very limited as in URLLC, relay
may not be a good option since some blocklengths needs to be reserved for the
two-stage transmission. However, further increasing $M$, the relay-assisted
transmission and the C-NOMA start to perform better than the NOMA scheme, and
the performance gain monotonically increases with $M$. However, the cross-
point associated with the relay scheme is much lower than that of the C-NOMA
scheme due to the shrinking feasible region associated with the latter scheme.
Furthermore, the curves of both schemes have the same slope with different
bias.

0.95

Figure 2: The decoding error probability of the actuator versus the distance
from the controller to the robot under four schemes, when $D=100$ bits,
$M=100$ symbols, $\tilde{E}_{\rm{tot}}=5\times 10^{-5}$ Joule.

0.95

Figure 3: The decoding error probability of the actuator versus the number of
symbols under four schemes, when $D=100$ bits, $\tilde{E}_{\rm{tot}}=5\times
10^{-5}$ Joule, $d_{1}=200$ m, $d_{2}=500$ m, and $d_{3}=300$ m.

In Fig. 5, we study the impact of the packet size $D$ on the decoding error
probability. As expected, a larger packet size leads to a higher error
probability for all schemes. The performance advantage of the relay-assisted
scheme over the OMA and NOMA schemes shrinks with the increase of $D$. It is
interesting to find that the curves associated with the OMA and NOMA schemes
have almost the same slope, while those of the relay-assisted transmission and
the C-NOMA scheme are similar. The main reason may be that the latter two
schemes apply relay to assist the transmission. Similar to the observations in
[24], the NOMA achieves better performance than the OMA scheme. When $D=125$
bits, the C-NOMA is even worse than the NOMA since some blocklengths should be
reserved for the two-stage transmission in the former scheme.

In Fig. 5, we study the impact of the total energy on the decoding error
probability. It is observed that more available energy leads to better
reliability performance as expected. It is also seen that the relay-assisted
transmission has the best performance, and the performance gain increases with
the amount of available energy. It is shown that with sufficient energy,
transmission with the aid of relay (i.e., the relay-assisted transmission and
the C-NOMA transmission) is beneficial for the system performance. When
${\tilde{E}_{{\rm{tot}}}}=5\times 10^{-5}$ Joule, the decoding error
probability achieved by the relay-assisted transmission is extremely low.

0.95

Figure 4: The decoding error probability of the actuator versus packet size
under four schemes, when $M=100$ sysmbols, $\tilde{E}_{\rm{tot}}=5\times
10^{-5}$ Joule, $d_{1}=200$ m, $d_{2}=500$ m, and $d_{3}=300$ m.

0.95

Figure 5: The decoding error probability of the actuator versus the energy
constraint under four schemes, when $D=100$ bits, $M=100$ symbols, $d_{1}=200$
m, $d_{2}=500$ m, and $d_{3}=300$ m.

0.95

Figure 6:  The decoding error probability of the actuator versus the number of
symbols for the OMA scheme the general OMA scheme, when $D=100$ bits, and
$\tilde{E}_{\rm{tot}}=5\times 10^{-5}$ Joule.

0.95

Figure 7: The network availability percentage versus the packet size $D$ under
four schemes, when $\tilde{E}_{\rm{tot}}=5\times 10^{-4}$ Joule, $M=100$
symbols.

In Fig. 7, we study the performance comparison between the OMA scheme in
Section III and the general OMA in Section IV. Denote the number of devices as
$K$. If $K=2$, both the OMA scheme and the general OMA scheme are applicable.
However, for the case with $K>2$, only the general OMA scheme is applicable.
For the first $K-1$th devices, the distance of the $k$th device to the
controller is set as $50\times k$ m, while the distance of the last device to
the controller is set as $500$ m. The other parameters are the same as the
previous figures. It is interesting to find that the decoding error
probability achieved by the OMA scheme and the general OMA scheme is almost
the same when $K=2$, which implies that the general OMA can achieve almost the
globally optimal solution in this setup. However, the general OMA scheme has
lower complexity than the OMA scheme. It is also noted from this figure that
the decoding error probability achieved by the $K$th device increases when the
number of total devices increases. This can be explained as follows. When the
number of total devices increases, the total resource such as energy and
channel blocklength allocated to the first $K-1$ devices will increase. Then,
the left resource allocated for the $K$th device decreases, leading to its
sworse decoding error probability performance.

###  V-B Network Availability Performance (Channel Generation Times=1000)

In this subsection, the small-scale fading channel is taken into consideration
in the channel gain, and we study the network availability performance, which
is defined as the ratio of the number of channel generations, where the
decoding error probability achieved by both devices is no larger than
$10^{-9}$, to the total number of channel generations [2]. In the following
simulations, the total number of channel generations is set as $1000$. The
distances are set as $d_{1}=200$ m, $d_{2}=500$ m, and $d_{3}=300$ m,
respectively.

Fig. 7 illustrates the network availability performance versus the packet size
$D$ for all schemes. As expected, the network availability performance
achieved by all schemes decreases with $D$. The relay-assisted transmission
has the best network availability performance over the whole region of $D$. It
is observed that when $D=100$ bits, the network availability percentage of the
relay-assisted scheme and the C-NOMA scheme is almost the same, as high as
98%. However, the performance gap of these two schemes increases rapidly with
$D$ due to the shrinking feasible region of the C-NOMA scheme compared to the
relay-assisted transmission. However, the network availability performance for
both the OMA scheme and the NOMA scheme are lower than that of relay-assisted
scheme and C-NOMA scheme, and the network availability percentage is as low as
87% for NOMA scheme even when $D=100$ bits.

Fig. 9 shows the network availability performance versus the number of symbols
for four schemes. As expected, the network availability performance increases
with $M$ for all schemes. The NOMA scheme performs slightly better than the
relay scheme when $M=50$. It is interesting to note that the C-NOMA scheme has
the worst performance when $M=50$, which means that this scheme is not a good
option when there is stringent latency requirement. However, the network
availability percentage of the C-NOMA increases rapidly with $M$, and finally
converges to almost the same value as that of the relay-assisted scheme, that
is equal to 97% when $M=100$. It is also noted that the OMA scheme converges
to almost the same performance as that of the NOMA scheme, and is low (86%
when $M=100$). It is interesting to find that the network availability
performance of all the schemes saturates in the high region of $M$, which
indicates that the number of available blocklength is not necessary to be very
large. This can be explained by using the result in [29]: The dispersion of
quasi-static fading channels converges to zero, which implies that the maximum
achievable data rate converges quickly to the outage capacity.

0.95

Figure 8: The network availability percentage versus the number of symbols $M$
under four schemes, when $\tilde{E}_{\rm{tot}}=5\times 10^{-4}$ Joule, $D=100$
bits.

0.95

Figure 9: The network availability percentage versus energy limit under four
schemes, when $D=100$ bits, $M=100$ symbols.

Finally, Fig. 9 depicts the network availability performance versus the energy
limit ${\tilde{E}_{{\rm{tot}}}}$ for all schemes. As expected, the network
performance achieved by all schemes increases with ${\tilde{E}_{{\rm{tot}}}}$.
It is also observed that relay-assisted scheme has the best network
availability performance. However, the performance gain over the C-NOMA scheme
decreases with ${\tilde{E}_{{\rm{tot}}}}$ and both curves coincide in the high
regime of ${\tilde{E}_{{\rm{tot}}}}$, where both schemes can achieve the
network availability percentage of 98%. On the other hand, both the NOMA
scheme and OMA scheme have very low network availability percentage, e.g., 86%
when ${\tilde{E}_{{\rm{tot}}}}=5\times{10^{-4}}$ Joule. The performance gap
between the relay-assisted scheme and NOMA is significant, up to 30%.

##  VI Conclusions

This work studied the resource allocation of short packet transmission for
mission-critical IoT to achieve low latency and high reliability under
fundamental transmission schemes, which include OMA, NOMA, relay-assisted
transmission and C-NOMA transmission. We formulated an optimization problem to
minimize the decoding error probability for the actuator with lower channel
gain while guaranteeing that the robot achieved a low error probability
target. To facilitate the optimal design of the blocklength and power
allocation, we derived the tight bounds on the blocklength and the transmit
power for all schemes. Simulation results demonstrated that relay-assisted
transmission significantly outperforms the other schemes for most cases in
terms of packet error probability as well as network availability percentage
performance. It was also noted that the NOMA scheme performs well when the
delay requirement is very stringent. For the C-NOMA and relay-assisted
schemes, there exists one optimal transmission distance between the central
controller and the robot. We also observed that the general OMA scheme can
achieve almost the same performance as the OMA scheme, while the former scheme
has a lower complexity.

Concerning our future work, we will consider a more general scenario with more
than two devices for the other three schemes.

##  Appendix A Proof of Lemma 1

We prove it by using contradiction. In the following, we first prove that
constraint (7b) holds with equality at the optimum solution. The second one
can be proved similarly.

Denote the optimal solution of Problem (7) as
${\bf{s}}^{\star}=\\{m_{1}^{\star},m_{2}^{\star},{p_{1}^{\star}},{p_{2}^{\star}}\\}$
and the corresponding ${\varepsilon}_{1}$ and ${\varepsilon}_{2}$ are denoted
as ${\varepsilon}_{1}^{\star}$ and $\varepsilon_{2}^{\star}$, respectively.
Suppose that ${\varepsilon}_{1}^{\star}$ is strictly smaller than
$\varepsilon_{1}^{\max}$, i.e.,
${\varepsilon}_{1}^{\star}<\varepsilon_{1}^{\max}$. In Proposition 1 of [24],
the author proved that
$Q\left({f\left({{{\gamma_{1}}},{m_{1}},D}\right)}\right)$ monotonically
decreases with $\gamma_{1}$. Then, we can construct a new solution
${\bf{s}}^{\\#}=\\{m_{1}^{\star},m_{2}^{\star},{p_{1}^{\\#}},{p_{2}^{\\#}}\\}$,
where ${p_{1}^{\\#}}={p_{1}^{\star}}-\Delta p$ and
${p_{2}^{\\#}}={p_{2}^{\star}}+\frac{{m_{1}^{\star}\Delta
p}}{{m_{2}^{\star}}}$ with $\Delta p>0$. It can be verified that the following
equation holds,

$\vspace{-0.1cm}m_{1}^{\star}p_{1}^{\\#}+m_{2}^{\star}p_{2}^{\\#}=m_{1}^{\star}p_{1}^{\star}+m_{2}^{\star}p_{2}^{\star}\leq{E_{{\rm{tot}}}}.$
(73)

Hence, the new constructed solution ${\bf{s}}^{\\#}$ still satisfies the
energy constraint (7c). In addition, we can always find a proper positive
$\Delta p$ such that the new ${\varepsilon}_{1}^{\\#}$ with the new solution
${\bf{s}}^{\\#}$ is equal to $\varepsilon_{1}^{\max}$, i.e.,
${\varepsilon}_{1}^{\\#}=\varepsilon_{1}^{\max}$, which satisfies constraint
(7b). Hence, the new constructed solution ${\bf{s}}^{\\#}$ is a feasible
solution of Problem (7). Since $p_{2}^{\\#}>p_{2}^{\star}$, we have
${\varepsilon}_{2}^{\\#}<{\varepsilon}_{2}^{\star}$. This contradicts with the
assumption that ${\bf{s}}^{\star}$ is an optimal solution. The same method is
applicable to the proof of the second conclusion.

##  Appendix B Proof of Theorem 1

The first and second derivative of function $\tilde{g}({m_{2}})$ w.r.t.
$m_{2}$ can be calculated as

$\displaystyle\vspace{-0.4cm}\tilde{g}^{\prime}({m_{2}})$ $\displaystyle=$
$\displaystyle\frac{1}{{2\ln
2}}\frac{1}{{\sqrt{m}_{2}}}\ln\left({1+\frac{{{E_{2}}{h_{2}}}}{{{m_{2}}}}}\right)-\frac{1}{{\ln
2}}\frac{1}{{\sqrt{m}_{2}}}\frac{{{E_{2}}{h_{2}}}}{{{m_{2}}+{E_{2}}{h_{2}}}}+\frac{D}{2}m_{2}^{-\frac{3}{2}}$
$\displaystyle\tilde{g}^{\prime\prime}({m_{2}})$ $\displaystyle=$
$\displaystyle\underbrace{-\frac{1}{4\ln
2}\frac{{\ln\left({1+\frac{{{E_{2}}{h_{2}}}}{{{m_{2}}}}}\right)}}{{{m_{2}{\sqrt{m}_{2}}}}}+\frac{{{E_{2}}{h_{2}}}}{{{\ln
2{\sqrt{m}_{2}}{\left({{m_{2}}+{E_{2}}{h_{2}}}\right)}^{2}}}}}_{?}\underbrace{-\frac{3}{4}Dm_{2}^{-\frac{5}{2}}}_{<0}.$

Obviously, the last term of $g^{\prime\prime}({m_{2}})$ is negative, we only
need to prove that the sum of the first two terms is negative under the
condition of $\frac{{{E_{2}}{h_{2}}}}{{M-{m_{1}}}}\geq e-1$.

Since $m_{\rm{2}}^{{\rm{lb}}}\leq{m_{2}}\leq M-{m_{1}}$, we have

$\frac{{{E_{2}}{h_{2}}}}{{{m_{2}}}}\geq\frac{{{E_{2}}{h_{2}}}}{{M-{m_{1}}}}\geq
e-1.$ (74)

Then, the following inequality follows:

$4\leq\left({\frac{{{E_{2}}{h_{2}}}}{{{m_{2}}}}+2+\frac{{{m_{2}}}}{{{E_{2}}{h_{2}}}}}\right)\ln\left({1+\frac{{{E_{2}}{h_{2}}}}{{{m_{2}}}}}\right).$
(75)

By rearranging the terms of the above inequality, we can prove that the sum of
the first two terms is negative, which completes the proof.

##  Appendix C Proof of Theorem 2

We prove this theorem by using the method of contradiction. Denote the optimal
$p_{1}$ of Problem (27) as $p_{1}^{\star}$ and the corresponding decoding
error probability is given by ${\bar{\varepsilon}}_{1}(p_{1}^{\star})$.
Suppose that ${\bar{\varepsilon}}_{1}(p_{1}^{\star})$ is strictly less than
$\varepsilon_{1}^{\max}$, i.e.,
${\bar{\varepsilon}}_{1}(p_{1}^{\star})<\varepsilon_{1}^{\max}$. Since
$\hat{\varepsilon}_{1}(p_{1}^{{\rm{lb}}})>{\varepsilon_{1}}(p_{1}^{{\rm{lb}}})$,
we have

${{\bar{\varepsilon}}_{1}}(p_{1}^{{\rm{lb}}})={\varepsilon_{1}}(p_{1}^{{\rm{lb}}})+(\hat{\varepsilon}_{1}(p_{1}^{{\rm{lb}}})-{\varepsilon_{1}}(p_{1}^{{\rm{lb}}}))\varepsilon_{2}^{1}(p_{1}^{{\rm{lb}}})=\varepsilon_{1}^{\max}+(\hat{\varepsilon}_{1}(p_{1}^{{\rm{lb}}})-{\varepsilon_{1}}(p_{1}^{{\rm{lb}}}))\varepsilon_{2}^{1}(p_{1}^{{\rm{lb}}})>\varepsilon_{1}^{\max},$
(76)

where ${\varepsilon_{1}}(p_{1}^{{\rm{lb}}})=\varepsilon_{1}^{\max}$ is used in
the second equality. As ${{\bar{\varepsilon}}_{1}}({p_{1}})$ is a continuous
function, there must exist a value $p_{1}^{\&}$ within the range of
$p_{1}^{{\rm{lb}}}<p_{1}^{\&}<p_{1}^{\star}$ such that
${{\bar{\varepsilon}}_{1}}({p_{1}^{\&}})=\varepsilon_{1}^{\max}$. On the other
hand, the objective value $\varepsilon_{2}(p_{1})$ is a monotonically
increasing function of $p_{1}$ since $p_{2}=E_{\rm{tot}}/m-p_{1}$. Hence, we
have $\varepsilon_{2}(p_{1}^{\&})<\varepsilon_{2}(p_{1}^{\star})$, which
contradicts the assumption that $p_{1}^{\star}$ is an optimal solution.

##  Appendix D Proof of Theorem 3

We first prove its convexity. Define function

$J({m_{k}})\buildrel\Delta\over{=}{m_{k}}{2^{\frac{D}{{{m_{k}}}}+\frac{{{A_{k}}}}{{\sqrt{{m_{k}}}}}}}.$
(77)

Then, $g\left({{m_{k}}}\right)$ can be rewritten as
$g\left({{m_{k}}}\right)={{\left({J({m_{k}})-{m_{k}}}\right)}\mathord{\left/{\vphantom{{\left({J({m_{k}})-{m_{k}}}\right)}{{h_{k}}}}}\right.\kern-1.2pt}{{h_{k}}}}$.
Then, if $J({m_{k}})$ is convex, function $g\left({{m_{k}}}\right)$ is also
convex. Hence, in the following, we prove that $J({m_{k}})$ is a convex
function. Define function ${\tilde{J}({m_{k}})}$ as

$\tilde{J}({m_{k}})\buildrel\Delta\over{=}\ln\left({J({m_{k}})}\right)=\ln({m_{k}})+\left({\frac{D}{{{m_{k}}}}+\frac{{{A_{k}}}}{{\sqrt{{m_{k}}}}}}\right)\ln
2.$ (78)

The second-order derivative of $\tilde{J}({m_{k}})$ w.r.t. $m_{k}$ is given by

$\tilde{J}^{\prime\prime}({m_{k}})=\frac{1}{{m_{k}^{3}}}\left({2D\ln
2-{m_{k}}+\frac{3}{4}{A_{k}}\sqrt{{m_{k}}}\ln 2}\right).$ (79)

Note that the denominator of (79) is a quadratic function of
${\sqrt{{m_{k}}}}$. Hence, if the inequality in (60) is satisfied,
$\tilde{J}^{\prime\prime}({m_{k}})$ is always positive, which means
$\tilde{J}({m_{k}})$ is a convex function of $m_{k}$. Since
$J({m_{k}})={e^{\tilde{J}({m_{k}})}}$, according to the composition rule in
[31], we can show that $J({m_{k}})$ is also a convex function. Hence,
$g\left({{m_{k}}}\right)$ is a convex function of $m_{k}$ when the inequality
in (60) is satisfied.

Now, we proceed to prove that $g\left({{m_{k}}}\right)$ is a monotonically
decreasing function of $m_{k}$. The first-order derivative of
$g\left({{m_{k}}}\right)$ w.r.t. $m_{k}$ is given by

$g^{\prime}\left({{m_{k}}}\right)=\frac{1}{{{h_{k}}}}\left[{{2^{\frac{D}{{{m_{k}}}}+\frac{{{A_{k}}}}{{\sqrt{{m_{k}}}}}}}\left({-\frac{D}{{{m_{k}}}}\ln
2-\frac{{\ln 2}}{2}\frac{{{A_{k}}}}{{\sqrt{{m_{k}}}}}+1}\right)-1}\right].$
(80)

Since $g\left({{m_{k}}}\right)$ is a convex function, we have
$g^{\prime\prime}\left({{m_{k}}}\right)\geq 0$, which means
$g^{\prime}\left({{m_{k}}}\right)$ is a monotonically increasing function.
Hence, we have

$g^{\prime}\left({{m_{k}}}\right)<g^{\prime}\left(\infty\right)=0.$ (81)

Hence, $g\left({{m_{k}}}\right)$ is a monotonically decreasing function of
$m_{k}$ when the inequality in (60) holds.

## References

  * [1] M. Shafi, A. F. Molisch, P. J. Smith, T. Haustein, P. Zhu, P. De Silva, F. Tufvesson, A. Benjebbour, and G. Wunder, “5G: A Tutorial Overview of Standards, Trials, Challenges, Deployment, and Practice,” _IEEE J. Sel. Areas Commun._ , vol. 35, no. 6, pp. 1201–1221, June 2017.
  * [2] P. Schulz, M. Matthe, H. Klessig, M. Simsek, G. Fettweis, J. Ansari, S. A. Ashraf, B. Almeroth, J. Voigt, and I. Riedel, “Latency Critical IoT Applications in 5G: Perspective on the Design of Radio Interface and Network Architecture,” _IEEE Commun. Mag._ , vol. 55, no. 2, pp. 70–78, February 2017.
  * [3] 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, 2018.
  * [4] P. Popovski, J. J. Nielsen, C. Stefanovic, E. de Carvalho, E. G. Ström, K. F. Trillingsgaard, A. Bana, D. Kim, R. Kotaba, and J. Park, “Ultra-reliable low-latency communication (URLLC): principles and building blocks,” _arXiv preprint arXiv:1708.07862_ , 2017.
  * [5] C. She, C. Yang, and T. Q. S. Quek, “Radio Resource Management for Ultra-Reliable and Low-Latency Communications,” _IEEE Commun. Mag._ , vol. 55, no. 6, pp. 72–78, 2017.
  * [6] J. J. Nielsen, R. Liu, and P. Popovski, “Ultra-Reliable Low Latency Communication Using Interface Diversity,” _IEEE Trans. Commun._ , vol. 66, no. 3, pp. 1322–1334, March 2018.
  * [7] M. Mozaffari, W. Saad, M. Bennis, Y.-H. Nam, and M. Debbah, “A tutorial on UAVs for wireless networks: Applications, challenges, and open problems,” _IEEE Commun. Surveys & Tutorials_, 2019.
  * [8] M. Simsek, A. Aijaz, M. Dohler, J. Sachs, and G. Fettweis, “5G-enabled tactile internet,” _IEEE J. Sel. Areas Commun._ , vol. 34, no. 3, pp. 460–473, 2016.
  * [9] H. Ren, N. Liu, C. Pan, M. Elkashlan, A. Nallanathan, X. You, and L. Hanzo, “Power-and rate-adaptation improves the effective capacity of C-RAN for Nakagami-$m$ fading channels,” _IEEE Trans. Veh. Technol._ , vol. 67, no. 11, pp. 10 841–10 855, 2018.
  * [10] ——, “Low-latency C-RAN: An next-generation wireless approach,” _IEEE Veh. Technol. Mag._ , vol. 13, no. 2, pp. 48–56, 2018.
  * [11] A. Varghese and D. Tandur, “Wireless requirements and challenges in Industry 4.0,” in _2014 IC3I_ , Nov 2014, pp. 634–638.
  * [12] L. Liu and W. Yu, “A D2D-Based Protocol for Ultra-Reliable Wireless Communications for Industrial Automation,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 8, pp. 5045–5058, Aug 2018.
  * [13] O. N. Yilmaz, Y.-P. E. Wang, N. A. Johansson, N. Brahmi, S. A. Ashraf, and J. Sachs, “Analysis of ultra-reliable and low-latency 5G communication for a factory automation use case,” in _2015 IEEE ICC)_. IEEE, 2015, pp. 1190–1195.
  * [14] G. Durisi, T. Koch, and P. Popovski, “Toward Massive, Ultrareliable, and Low-Latency Wireless Communication With Short Packets,” _Proc. IEEE_ , vol. 104, no. 9, pp. 1711–1726, Sept 2016.
  * [15] Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel Coding Rate in the Finite Blocklength Regime,” _IEEE Trans. Inf. Theory_ , vol. 56, no. 5, pp. 2307–2359, May 2010.
  * [16] K. F. Trillingsgaard and P. Popovski, “Downlink Transmission of Short Packets: Framing and Control Information Revisited,” _IEEE Trans. Commun._ , vol. 65, no. 5, pp. 2048–2061, May 2017.
  * [17] C. She, Z. Chen, C. Yang, T. Q. Quek, Y. Li, and B. Vucetic, “Improving network availability of ultra-reliable and low-latency communications with multi-connectivity,” _IEEE Trans. Commun._ , vol. 66, no. 11, pp. 5482–5496, 2018.
  * [18] J. Östman, G. Durisi, E. G. Ström, M. C. Coşkun, and G. Liva, “Short packets over block-memoryless fading channels: Pilot-assisted or noncoherent transmission?” _IEEE Trans. Commun._ , vol. 67, no. 2, pp. 1521–1536, 2019.
  * [19] Y. Hu, J. Gross, and A. Schmeink, “On the Capacity of Relaying With Finite Blocklength,” _IEEE Trans. Veh. Technol._ , vol. 65, no. 3, pp. 1790–1794, March 2016.
  * [20] Y. Hu, A. Schmeink, and J. Gross, “Blocklength-Limited Performance of Relaying Under Quasi-Static Rayleigh Channels,” _IEEE Trans. Wireless Commun._ , vol. 15, no. 7, pp. 4548–4558, July 2016.
  * [21] Y. Gu, H. Chen, Y. Li, L. Song, and B. Vucetic, “Short-Packet Two-Way Amplify-and-Forward Relaying,” _IEEE Sig. Proc. Lett._ , vol. 25, no. 2, pp. 263–267, Feb 2018.
  * [22] O. L. A. López, R. D. Souza, H. Alves, and E. M. G. Fernández, “Ultra reliable short message relaying with wireless power transfer,” in _2017 ICC_ , May 2017, pp. 1–6.
  * [23] Y. Hu, M. C. Gursoy, and A. Schmeink, “Efficient transmission schemes for low-latency networks: NOMA vs. relaying,” in _2017 IEEE PIMRC_ , Oct 2017, pp. 1–6.
  * [24] X. Sun, S. Yan, N. Yang, Z. Ding, C. Shen, and Z. Zhong, “Short-Packet Downlink Transmission With Non-Orthogonal Multiple Access,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 7, pp. 4550–4564, July 2018.
  * [25] C. She, C. Yang, and T. Q. S. Quek, “Joint Uplink and Downlink Resource Configuration for Ultra-Reliable and Low-Latency Communications,” _IEEE Trans. Commun._ , vol. 66, no. 5, pp. 2266–2280, May 2018.
  * [26] Y. Hu, Y. Zhu, M. C. Gursoy, and A. Schmeink, “SWIPT-enabled relaying in IoT networks operating with finite blocklength codes,” _IEEE J. Sel. Areas Commun._ , vol. 37, no. 1, pp. 74–88, 2019.
  * [27] C. Pan, H. Ren, Y. Deng, M. Elkashlan, and A. Nallanathan, “Joint blocklength and location optimization for URLLC-enabled UAV relay systems,” _IEEE Commun. Lett._ , vol. 23, no. 3, pp. 498–501, March 2019\.
  * [28] C. Shannon, “A Mathematical Theory of Communication,” _The Bell System Technical Journal_ , vol. 27, no. 1, pp. 379–423, July 1948.
  * [29] W. Yang, G. Durisi, T. Koch, and Y. Polyanskiy, “Quasi-static multiple-antenna fading channels at finite blocklength,” _IEEE Trans. Inf. Theory_ , vol. 60, no. 7, pp. 4232–4265, July 2014.
  * [30] Y. Liu, Z. Ding, M. Elkashlan, and H. V. Poor, “Cooperative Non-orthogonal Multiple Access With Simultaneous Wireless Information and Power Transfer,” _IEEE J. Sel. Areas Commun._ , vol. 34, no. 4, pp. 938–953, April 2016.
  * [31] S. Boyd and L. Vandenberghe, “Convex optimization,” _Cambridge, U.K.: Cambridge Univ. Press_ , 2004.
  * [32] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, and L. Hanzo, “Intelligent reflecting surface enhanced MIMO broadcasting for simultaneous wireless information and power transfer,” _arXiv preprint arXiv:1908.04863_ , 2019.
  * [33] E. U. T. R. Access, “Further advancements for E-UTRA physical layer aspects,” _3GPP TR 36.814, Tech. Rep._ , 2010.