# Reliability-Latency-Rate Tradeoff in Low-Latency Communications with Finite-
Blocklength Coding

Lintao Li, Wei Chen, Petar Popovski, , and Khaled B. Letaief This work is
supported in part by the National Natural Science Foundation of China under
Grant No. 61971264, and the National Natural Science Foundation of
China/Research Grants Council Collaborative Research Scheme under Grant No.
62261160390. The preliminary work was presented at the IEEE GLOBECOM’2021 [1,
2].Lintao Li and Wei Chen are with the Department of Electronic Engineering,
Tsinghua University, Beijing 100084, China, and also with the Beijing National
Research Center for Information Science and Technology, Tsinghua University,
Beijing 100084, China (email<EMAIL_ADDRESS>wchen@tsinghua.edu.cn). Petar Popovski is with the Department of Electronic
Systems, Aalborg University, 9220 Aalborg, Denmark (e-mail: petarp@es.aau.dk).
Khaled B. Letaief is with the Department of Electronic and Computer
Engineering, Hong Kong University of Science and Technology, Clear Water Bay,
Hong Kong, and also with Peng Cheng Laboratory, Shenzhen 518066, China (email:
eekhaled@ust.hk).

###### Abstract

Low-latency communication plays an increasingly important role in delay-
sensitive applications by ensuring the real-time exchange of information.
However, due to the constraints on the maximum instantaneous power, bounded
latency is hard to be guaranteed. In this paper, we investigate the
reliability-latency-rate tradeoff in low-latency communications with finite-
blocklength coding (FBC). More specifically, we are interested in the
fundamental tradeoff between error probability, delay-violation probability
(DVP), and service rate. Based on the effective capacity (EC) and normal
approximation, we present several gain-conservation inequalities to bound the
reliability-latency-rate tradeoffs. In particular, we investigate the low-
latency transmissions over an additive white Gaussian noise (AWGN) channel,
over a Rayleigh fading channel, with frequency or spatial diversity, and over
a Nakagami-$m$ fading channel. To analytically evaluate the quality-of-
service-constrained low-latency communications with FBC, an EC-approximation
method is further conceived to derive the closed-form expression of quality-
of-service-constrained throughput. For delay-sensitive transmissions in which
the latency threshold is greater than the channel coherence time, we find an
asymptotic form of the tradeoff between the error probability and DVP over the
AWGN and Rayleigh fading channels. Our results may provide some insights into
the efficient scheduling of low-latency wireless communications in which
statistical latency and reliability metrics are adopted.

###### Index Terms:

Low-latency communications, reliability-latency-rate tradeoff, finite-
blocklength coding, effective capacity, delay-violation probability

##  I Introduction

With the tremendous development of communication technologies, the flow of
wireless data transmission has experienced an exponential growth in recent
years supported by a higher transmission rates and ubiquitous connectivity.
Under this trend, multiple delay-sensitive applications have been gaining a
lot of attention in the fifth-generation (5G) communication networks. Besides,
the ongoing works for six-generation (6G) communication networks indicate that
there will be more stringent delay and reliability requirements in 6G
applications [3]. As one of the essential requirements for the mission-
critical and more emerging applications in 5G and 6G, low-latency
communication has been a popular topic under active consideration in the
research community.

Among the research for low-latency communication, ultra-reliable and low
latency communication (URLLC) is a typical application scenario. URLLC aims to
achieve an ultra-low and bounded delay to support the above applications. For
the next generation URLLC, the requirement on latency will be more stringent
[4]. Besides, to provide real-time experience for users, the communication
delay should be controlled or deterministic [5]. However, due to the fading
nature of the wireless channels, a bounded delay is usually hard to be
guaranteed [6]. Besides, limited maximum power brings more difficulty in
achieving a bounded delay. Therefore, statistical quality-of-service (QoS)
becomes one of the most common choices as the reliability and latency metrics
for URLLC [7]. Statistical QoS is characterized by the length or delay bounds
and the corresponding thresholds of the maximal tolerable length or delay
violation probability [8]. As mentioned in [9], only focusing on the average-
based metrics without considering the tail behavior of wireless systems is
difficult to achieve the low-latency and reliability requirements. There are
some works focusing on the characterization of the statistical delay in
closed-loop control [10] and edge computing systems [11]. It is thus crucial
to pay attention to statistical QoS analysis in URLLC and other low-latency
systems to characterize the performance limit, which can in turn help us
optimize the system’s QoS performance.

To analyze the QoS of communication systems, the first procedure is to
formulate a proper model, which takes QoS metrics into account. However, using
physical-layer models for analyzing the QoS is complex and may lead to
inaccuracies resulting from the approximation errors in extracting QoS metrics
from the models [12]. In low-latency communications, we always focus on the
probability of the queue length and delay violating a certain threshold [9].
To address this issue, the effective bandwidth (EB) and effective capacity
(EC) methods are then proposed to conduct an asymptotic QoS analysis by
introducing the link-layer queueing model. With the help of EB and EC, the
queue-length-violation probability (QVP) and delay-violation probability (DVP)
has a concise expression. The EB of a source is defined as the minimum
constant service rate required by the given arrival process to satisfy the QoS
requirements [13]. Effective capacity (EC) is the dual concept of EB, which
indicates the maximum constant source rate required by the given service
process to satisfy the QoS requirement [12]. Based on the EB, EC, and Shannon
formula, there are some works focusing on the performance analysis and
resource allocation designs under statistical QoS constraints in various
communications systems with different channel and arrival models, e.g.,
independent Rayleigh fading channel [14], time-correlated Rayleigh fading
channel[15], and Nakagami-$m$ channel [16]. Specifically, the high-SNR slope
of EC was characterized in [14] with the infinite blocklength assumption.

A requirement that is often quoted in relation to 5G is a latency of less than
1 ms and a packet loss probability that is less than or equal to $10^{-5}$
[7]. With the limited bandwidth, the assumption that the blocklength shall
tend to infinity has to be changed to achieve these requirements. In
particular, the traditional Shannon formula with an infinite blocklength is
not accurate enough in this regime. Thus, finite-blocklength (FBL) coding is
expected to play a central role to solve this problem. To better characterize
the achievable coding rate in the FBL regime, many works take the influence of
FBL and error probability into account. A milestone work [17] presented the
maximum coding rate of a fixed finite-blocklength given the block error
probability. The normal approximation was also proposed in [17] to calculate
the maximum coding rate for the AWGN channel in the FBL regime. This
conclusion inspired more works studying the coding rate of various
communications scenarios in the FBL regime, e.g., Gilbert-Elliott channels
[18], quasi-static multiple-antenna fading channels [19], and coherent block
fading channels [20]. Besides, the third-order term in the normal
approximation was derived in [21], which makes the normal approximation more
accurate. With the finite blocklength assumption, the properties of EC may
have differences from the scenarios with the infinite blocklength assumption.
This problem remains open in the existing works, which also indicates that
there lacks the analysis for the fundamental tradeoff between the performance
on statistical latency, reliability, and service rate in the low-latency
systems with finite-blocklength coding (FBC). In our previous work [22], the
gain conservation equation was derived to characterize the relationships
between the reliability, latency, and service rate gains. However, in [22],
only bounded random arrival was considered, which can be seen as the
deterministic arrival with the high-SNR assumption. Thus, there still lacks an
analysis for the reliability-latency-rate tradeoff in the low-latency systems
with FBC and unbounded random arrivals.

Short packet transmission empowered by the data rate analysis of the FBC
further motivates investigations of the QoS analysis for URLLC systems from a
cross-layer perspective. However, in the general URLLC systems, the threshold
for tolerable delay is usually smaller than the coherence time of the channel.
This brings difficulty in using the large-deviation-based approximations of
QVP and DVP. To solve this problem, the related works can be divided into two
cases. The first kind of work used the LDT-based QVP and DVP approximations as
the upper bounds on the actual QVP and DVP. The authors of [23] validated that
the DVP derived from LDT is the upper bound of the actual DVP with certain
kinds of arrival processes. Based on this conclusion, a joint packet dropping
policy was proposed in [23] to minimize the transmit power for URLLC systems,
while a joint uplink and downlink resource configuration problem was
formulated in [24] to improve the spectrum usage efficiency. Another kind of
work treated EC as a metric for evaluating the QoS-constrained throughput
without considering the value of QVP or DVP. These works mainly focused on
characterizing and optimizing the EC with the given QoS exponent. The EC
maximization was considered for URLLC systems in Rayleigh fading channel [25,
26], Rician fading channels [27], and Nakagami-$m$ fading channels [28] with
different constraints and conditions. In [29], the author determined the
throughput of the simple ARQ mechanism in the FBL regime. As a follow-up to
[29], the throughput of a QoS-constrained low-latency Internet-of-Things
system was researched in [30]. However, there lacks the analysis for the
reliability-DVP tradeoff of low-latency systems with FBC in which the
tolerable delay is larger than the channel coherence time, e.g., high-mobility
and THz systems [31]. Besides, due to the introduction of the channel
dispersion term in the FBL coding rate, it is not trivial to obtain the
analytical expression of the EC. This also results in the difficulties of
evaluating the low-latency system’s QoS-constrained throughput and further
optimizing for improving the EC.

In this paper, we characterize the fundamental tradeoff between reliability,
latency, and rate in low-latency systems with FBC. Specifically, we start the
analysis from the analysis for the high-SNR slope of EC in the FBL regime. We
consider four scenarios including transmission over an AWGN channel, over a
Rayleigh fading channel, with frequency or spatial diversity, and over a
Nakagami-$m$ fading channel. First, we consider the case in which the error
probability is fixed to determine the service-rate gain, which is the
derivative of the EC to the logarithm of the SNR. Based on this result, we
further consider the case in which the error probability is a function of SNR.
The reliability gain and real-time gain are then defined as the derivatives of
functions of error probability and DVP to the logarithm of the SNR. Then, the
revised gain conservation equation is characterized in AWGN channel
immediately and in wireless systems by two inequalities in different
transmission scenarios. Moreover, a Laplace’s-method-based EC approximation
approach is proposed to derive the analytical expression of EC in the FBL
regime. Based on the discussion on the fixed-power allocation scheme, we
further propose the EC approximation for certain kinds of channel-gain-based
power allocation schemes, while its generalization to the systems with the
simple retransmission mechanism is also presented. Finally, we focus on the
low-latency systems in which the delay threshold is larger than the channel
coherence time. The closed-form expressions of the reliability-latency
tradeoff for the AWGN channel and the Rayleigh fading channel are obtained
with the help of EC and EB theories.

The rest of this paper is organized as follows. Section II-A presents the
system model, including the physical layer model and network layer model.
Section II-B shows the main results of this paper, which are summarized by
three main theorems. In Section III-A, we give the results and corresponding
for the revised gain conservation inequalities, which characterize the
reliability-latency-rate tradeoff of low-latency systems with FBC in the high-
SNR regime. In Section III-B, we consider a specific orthogonal code to give
the discussion of service-rate gain in the low-SNR regime. In Section III-C,
the EC approximation approach is proposed. Then, we characterize the
reliability-latency tradeoff in Section IV for the low-latency systems in
which the delay threshold is larger than the channel coherence time. The
simulation and numerical results are presented in Section V. The conclusion is
given in Section VI.

Throughout this paper, $u^{\prime}(x)$ denotes the first-order derivative of
$u(x)$ for $x$, while $u^{\prime\prime}(x)$ denotes the second-order
derivative. $\mathbb{N}^{+}$ denotes the set of positive natural numbers.
$\mathbb{E}\\{X\\}$ is the expectation of random variable $X$, while ${\rm
Var}\\{X\\}$ is its variance. $e$ denotes Euler’s number. $\boldsymbol{I}_{N}$
denotes a $N$-dimensional identity matrix. For two random variables $X$ and
$Y$, ${\rm Cov}(X,Y)$ denote the covariance between $X$ and $Y$.
$Q^{-1}(\cdot)$ represents the inverse $Q$-function where the $Q$-function is
$Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-\frac{t^{2}}{2}}{\rm d}t$.
$\Gamma(s)=\int_{0}^{\infty}t^{s-1}e^{-t}{\rm d}t$ denote the gamma function.
Besides, define the incomplete gamma function as
$\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}{\rm d}t$, and the exponential
integral function as ${\rm Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}{t}{\rm
d}t$. The Meijer G-function is expressed as

$G_{\rho_{3},\rho_{4}}^{\rho_{1},\rho_{2}}\Bigg{(}\begin{matrix}a_{1},a_{2},\ldots,a_{\rho_{3}}\\\
b_{1},b_{2},\ldots,b_{\rho_{4}}\end{matrix}\Bigg{|}z\Bigg{)}=\frac{1}{2\pi
i}\oint_{\mathcal{L}}\frac{\prod_{j=1}^{\rho_{1}}\Gamma(b_{j}-h)\prod_{j=2}^{\rho_{2}}\Gamma(1-a_{j}+h)}{\prod_{j=\rho_{1}+1}^{\rho_{4}}\Gamma(1-b_{j}+h)\prod_{j=\rho_{2}+1}^{\rho_{3}}\Gamma(a_{j}-h)}z^{h}dh.$

##  II System Model and Main Results

Figure 1: System Model.

In this section, we present a system model with FBC. Since we mainly focus on
the statistical delay performance of this system, the metric for evaluating
the latency performance of this system is adopted as QVP and DVP. More
specifically, we will first describe the system model in Section II-A, which
includes physical layer and queueing models. Then, we will introduce the main
results of this paper, which can be summarized by 3 main theorems in Section
II-B.

###  II-A System Model

As shown in Fig. 1, we focus on a point-to-point low-latency communication
system. The arrival data generated by the source will be transmitted through
the channel. Time is divided into time slots with equal length
$T_{0}=\frac{N}{B}$ (in seconds), where $B$ is the bandwidth of this system
and $N$ is the blocklength. Let $T_{f}$ (in seconds) denote the frame length.
For each frame, it contains $T=\frac{T_{f}}{T_{0}}$ time slots, where
$T\in\mathbb{N}^{+}$. 111Note that, for the high-mobility or THz communication
system, the channel coherence time is significantly small [31]. Thus, $T=1$
also covers the low-latency scenarios.  Let $h[n]$ denote the channel
coefficient of time slot $n$. Assume that the frame duration is equal to the
channel coherence time. Thus, the channel coefficient remains constant in a
frame, i.e., $h[iT+1]=h[iT+2]=\cdots=h[(i+1)T]$, $i=0,1,\cdots$, and varies in
an independently and identically distributed ($i.i.d.$) manner across
different frames. With blocklength $N$, the channel output $\boldsymbol{y}[n]$
at the $n$-th time slot with the input $\boldsymbol{x}[n]$ can be expressed as

$\boldsymbol{y}[n]=h[n]\boldsymbol{x}[n]+\boldsymbol{z}[n],$ (1)

where $\boldsymbol{z}[n]\sim\mathcal{CN}(0,N_{0}\boldsymbol{I}_{N})$ denote
the AWGN. For the AWGN channel, $\boldsymbol{x}[n]\in\mathbb{C}^{N}$ and
$h[n]=1$. For the fading channel, the probability density function ($p.d.f.$)
of $|h[n]|^{2}$ is denoted by $f(x)$, while its cumulative distribution
function ($c.d.f.$) is denoted by $F(x)$. For a wireless system in a Rayleigh
fading channel with $\mathbb{E}\left\\{|h[n]|^{2}\right\\}=1$, we have
$f(x)=e^{-x}$, $x>0$. For a wireless system with spatial or frequency
diversity $\kappa$ in Rayleigh fading channels, its equivalent channel power
gain by adopting maximum-ratio combining [32] is given by

$|h[n]|^{2}=\sum_{i=1}^{\kappa}|h_{i}[n]|^{2},$ (2)

where $|h_{i}[n]|$ follows the Rayleigh distribution. With the assumption that
$\mathbb{E}\\{|h_{i}[n]|^{2}\\}=1$, the $p.d.f.$ of $|h[n]|^{2}$ is given by

$f(x)=\frac{e^{-x}x^{\kappa-1}}{(\kappa-1)!},\quad x>0.$ (3)

For a wireless system in a Nakagami-$m$ fading channel with
$\mathbb{E}\left\\{|h[n]|^{2}\right\\}=\Omega$, we have

$f(x)=\left(\frac{m}{\Omega}\right)^{m}\frac{x^{m-1}}{\Gamma(m)}e^{-\frac{m}{\Omega}x},\quad
x>0,\,m\geq 0.5.$ (4)

We assume that there is a constraint on the maximum instantaneous power in
this system. Thus, to achieve the best QVP and DVP performance in this system,
we let $\|\boldsymbol{x}[n]\|^{2}=NP$, where $P$ is the maximum instantaneous
power. The transmitted signal-to-noise ratio is denoted by
$\gamma=\frac{P}{N_{0}}$.

In this paper, we assume that the channel state information (CSI) is perfectly
known at both the transmitter and the receiver through the channel estimation
as shown in Fig. 1. $\Delta N=B\Delta T$ channel uses are worked on the
channel estimation in each frame, which we neglect in this paper. The
conclusion in [17, 21] showed that for channels with capacity $C$, the maximum
coding rate (in bits per channel use) with the blocklength $N$ and block error
probability not larger than $\epsilon$, can be expressed as

$C_{\epsilon}=C-\sqrt{\frac{V}{N}}Q^{-1}(\epsilon)+\frac{\log N}{2}+O(1),$ (5)

where $V$ is the channel dispersion. Specifically, according to [21], for the
AWGN channel, the coding rate (in bits per $N$ channel uses) for a given
$\epsilon$, $N$, and SNR $\gamma$ can be approximated as

$R_{\epsilon}\approx
N\log_{2}(1+\gamma)-\sqrt{N\left(1-\frac{1}{(1+\gamma)^{2}}\right)}Q^{-1}(\epsilon)\log_{2}e+\frac{\log_{2}N}{2}.$
(6)

Moreover, given the channel gain $|h[n]|^{2}$, the coding rate (in bits per
$N$ channel uses) with $\epsilon$, $N$, and $\gamma$ for the fading channel is
approximately given by [26]

$R_{\epsilon}[n]\approx
N\log_{2}\left(1+|h[n]|^{2}\gamma\right)-\sqrt{N\left(1-\frac{1}{\left(1+|h[n]|^{2}\gamma\right)^{2}}\right)}Q^{-1}(\epsilon)\log_{2}e+\frac{\log_{2}N}{2}.$
(7)

To backlog the packets to be transmitted, an infinite-length buffer is assumed
to be available at the transmitter. Let $a[m]$ denote the size of arrival data
at the beginning of time slot $m$. In this paper, we assume that $a[m]$
follows an arbitrary stochastic distribution with finite expectation and
variance in this system, while $a[m]$ changes in an $i.i.d.$ manner across
different time slots. The average size of arrival data is denoted by
$\mu_{a}$, while the variance is denoted by $\sigma_{a}^{2}$. Accordingly, the
accumulated number of the arrival data until time slot $M$ can be expressed as
$A[M]=\sum_{m=1}^{M}a[m]$. Similarly, $s[m]$ is defined as the size of the
service data in time slot $m$. The average size of service data is denoted by
$\mu_{s}$, while the variance of $s[m]$ is denoted by $\sigma_{s}^{2}$. The
accumulated number of service data until time slot $M$ can be expressed as
$S[M]=\sum_{m=1}^{M}s[m]$.

As shown in Fig. 1, we assume a first-in first-out (FIFO) policy for serving
the backlogged packets. Let $Q[n]$ denote the length of the queue at the end
of time slot $n$. Then, the queue length is updated as

$Q[n]=\left(Q[n-1]+a[n]-s[n]\right)^{+},$ (8)

where $(x)^{+}=\max\\{0,x\\}$. In this paper, we are interested in the
probability of the queue length or delay exceeding a certain threshold, which
characterizes the statistical performance of the FBC transmission. Thus, we
use QVP and DVP as indicators for evaluating systems’ performance on latency.
To achieve this goal, we introduce EB and EC. According to [13], the EB of the
arrival process is defined as

$\alpha_{A}(\theta)=\lim_{M\to\infty}\frac{1}{\theta
M}\ln\mathbb{E}\left\\{e^{\theta A[M]}\right\\}.$ (9)

Similarly, according to [12], the EC of the service process is defined as

$\alpha_{S}(\theta)=\lim_{M\to\infty}-\frac{1}{\theta
M}\ln\mathbb{E}\left\\{e^{-\theta S[M]}\right\\}.$ (10)

For $T\geq 1$ with a given $\gamma$, EC is given by

$\displaystyle\alpha_{S}(\theta)=$
$\displaystyle\lim_{M\to\infty}-\frac{1}{\theta
M}\ln\mathbb{E}\left\\{e^{-\theta S[M]}\right\\}$ (11) $\displaystyle=$
$\displaystyle\lim_{M\to\infty}-\frac{1}{\theta
T}\cdot\frac{T}{M}\ln\mathbb{E}\left\\{e^{-\theta\sum_{i=0}^{\frac{M}{T}-1}\sum_{j=1}^{T}s[iT+j]}\right\\}$
$\displaystyle=$ $\displaystyle-\frac{1}{\theta
T}\ln\mathbb{E}\left\\{e^{-\theta Ts[n]}\right\\}.$

To simplify the derivation, we normalize the EC by the blocklength, which is
also called effective rate [14]. According to Eq. (11), the normalized EC for
the $i.i.d.$ $s[n]$ with a given $\theta$ is given by

$\Lambda(\gamma)=-\frac{1}{\theta TN}\ln\mathbb{E}\left\\{e^{-\theta
Ts[n]}\right\\}\quad\text{bits/s/Hz}.$ (12)

Note that the effective rate defined in [14] is equal to $\Lambda(\gamma)$
when $T=1$. In this paper, we use $\Lambda(\gamma)$ to characterize the
service-rate gain of the low-latency communication systems.

Let $Q[\infty]$ denote the steady state of $Q[n]$. If $Q[\infty]$ exists, the
QVP, i.e., $\Pr\\{Q[\infty]\geq L\\}$, with a large threshold $L$ (in bits)
can then be approximated as [12]

$\chi\approx\eta e^{-\theta^{*}L},$ (13)

where $\theta^{*}$ is known as the QoS exponent, which indicates the decay
rate of the tail distribution of the queue length. $\eta$ is the probability
that the queue is not empty, which can be approximated by the ratio of
$\mu_{a}$ to $\mu_{s}$. In this paper, we focus on the high load scenario,
i.e., $\eta\to 1$. $\theta^{*}$ is the solution to the following equation
[33].

$\alpha_{A}(\theta^{*})-\alpha_{S}(\theta^{*})=0.$ (14)

Let $D[m]$ denote the delay experienced by the data arriving at the start of
time slot $m$. The steady state of $D[n]$ is denoted by $D[\infty]$. If
$D[\infty]$ exists, the DVP, i.e., $\Pr\\{D[\infty]\geq D_{\rm max}\\}$, with
a large delay bound $D_{\rm max}$ (in slots) in the high load scenario is
approximated as [12]

$\delta\approx e^{-\theta^{*}\alpha_{A}(\theta^{*})D_{\rm max}}.$ (15)

###  II-B Main Results

In this subsection, we present the main results of this paper. The main
results of this paper can be divided into 3 aspects, which refer to the
revised gain-conversation inequalities with random arrivals in the high-SNR
regime, a Laplace’s-method-based EC approximation approach, and the
reliability-latency tradeoff for low-latency systems with $T=1$.

Before presenting the main results, we first give some definitions. With the
given $\theta$, we define
$\zeta=\lim\limits_{\gamma\to\infty}\frac{\partial{\Lambda(\gamma)}}{\partial{\log_{2}\gamma}}$
as the QoS-constrained service-rate gain, which is similar to the definition
of the service-rate gain in [22]. Let
$\varpi=\sqrt{\frac{2\log_{2}e}{N}}\lim\limits_{\gamma\to\infty}\frac{\partial\sqrt{-\log_{2}\epsilon(\gamma)}}{\partial\log_{2}\gamma}$
denote the reliability gain. Besides, we define $\tau=-\frac{1}{D_{\rm
max}}\frac{\partial\log_{2}\delta}{\partial\log_{2}\gamma}$ as the real-time
gain, which is consistent with the definition in [22]. Based on the above
definitions, the high-SNR reliability-latency-rate tradeoff is shown in
Theorem 1 for low-latency communications with FBC and random arrivals.

Theorem 1. In the FBL regime with $\gamma\to\infty$,

  * •

for a SISO system in an AWGN channel, we have

$(1-\theta N\log_{2}e)\zeta+\varpi+\tau=1;$ (16)

  * •

for a SISO system in a Rayleigh fading channel, we have

$\min\left\\{1,\frac{1}{\theta TN\log_{2}e}\right\\}\leq(1-\theta
N\log_{2}e)\zeta+\varpi+\tau\leq\left\\{1,\frac{1}{\theta
TN\log_{2}e}\right\\}+c\varpi;$ (17)

  * •

for a wireless system with $f(x)$ satisfying Eq. (3), we have

$\min\left\\{1,\frac{\kappa}{\theta TN\log_{2}e}\right\\}\leq(1-\theta
N\log_{2}e)\zeta+\varpi+\tau\leq\min\left\\{1,\frac{\kappa}{\theta
TN\log_{2}e}\right\\}+c\varpi;$ (18)

  * •

for a SISO system in a Nakagami-$m$ fading channel, we have

$\min\left\\{1,\frac{m}{\theta TN\log_{2}e}\right\\}\leq(1-\theta
N\log_{2}e)\zeta+\varpi+\tau\leq\min\left\\{1,\frac{m}{\theta
TN\log_{2}e}\right\\}+c\varpi.$ (19)

In Eqs. (17), (18), and (19), $c>0$ is an arbitrary constant.

From the definition of $\zeta$, $\varpi$, and $\tau$, we find that these three
gains characterize the communication system’s performance on the service rate,
reliability, and latency with a maximum power constraint. Thus, this theorem
presents a fundamental tradeoff existing among the service rate, reliability,
and latency in the high-SNR regime for low-latency communications. Eq. (16)
gives a gain-conversation equation for the three gains in the AWGN channel,
which can be applied for the performance optimization in satellite and wired
communication systems. For wireless systems, the three gains follow an
approximate conservation relationship. We can call Eqs. (17), (18), and (19)
as gain-conservation inequalities for low-latency systems. Note that, $c>0$
can be chosen arbitrarily small. Thus, if $\varpi$ is finite, we can obtain
the following revised gain-conservation equation for the SISO system in the
Nakagami-$m$ fading channel

$(1-\theta N\log_{2}e)\zeta+\varpi+\tau=\min\left\\{1,\frac{m}{\theta
TN\log_{2}e}\right\\}.$ (20)

Similar equations can be derived for systems with $f(x)$ satisfying Eqs. (3)
and (4). The main difference between Eqs. (16) and (20) and the conclusions in
[22] results from the influence of random arrivals. Thus, Theorem 1 can be
seen as the generalization of the gain-conservation equation in [22] to the
systems with unbounded random arrivals under a large delay threshold.

EC is widely used in the researches on low-latency communications to evaluate
the QoS-constrained throughput of wireless systems. However, it is usually not
trivial to obtain the closed-form expression of EC, which brings difficulty
for the designs of resource allocation and scheduling for maximizing EC. In
this paper, we propose a Laplace’s-method-based EC approximation approach. For
simplicity of expression, let

$s(x)=\log_{2}\left(1+\Xi(x)x\right)-\sqrt{\frac{1}{N}\left(1-(1+\Xi(x)x)^{-2}\right)}Q^{-1}(\epsilon)\log_{2}e+\frac{\log_{2}N}{2N},\vspace{-1mm}$
(21)

where $\Xi(x)$ denotes the channel-power-gain-based power allocation scheme.

Theorem 2. For a low-latency system which adopts a channel-gain-based power
allocation scheme $\Xi(x)$ satisfying $\Xi[x]+\frac{{\rm d}\Xi(x)}{{\rm
d}x}x\geq 0$, EC is approximated as

$\alpha_{S}(\theta)\approx Ns(x^{*})-\frac{\ln f(x^{*})+\frac{1}{2}\ln
2\pi-\frac{1}{2}\ln|s^{\prime\prime}(x^{*})|-\frac{1}{2}\ln\theta TN}{\theta
T},$ (22)

where $x^{*}$ is given by

$\Xi(x^{*})x^{*}=\sqrt{\frac{1+\sqrt{1+4\ell}}{2}}-1.$ (23)

The approximation proposed in Theorem 2 gives us a way to obtain the closed-
from expression of the EC, which can help us to further analyze the QoS-
constrained performance of low-latency communication systems and conduct the
comparison between different scheduling schemes.

For low-latency systems with $T=1$, we further characterize the
$\epsilon-\chi$ tradeoff in AWGN channel and Rayleigh fading channel,
respectively. The results are given in Theorem 3.

Theorem 3. For a low-latency system with the $i.i.d.$ $a[n]$ in the AWGN
channel, the tradeoff between $\epsilon$ and $\chi$ in the low $\epsilon$
regime is approximated as

$\ln\epsilon\approx-\ln\left(\psi+\frac{\Phi}{L}\ln\chi\right)-\frac{1}{2}\left(\psi+\frac{\Phi}{L}\ln\chi\right)^{2}-\frac{1}{2}\ln(2\pi),$
(24)

where $\psi$ and $\Phi$ are given by

$\quad\,\,\psi=\frac{N\log(1+\gamma)-\mu_{a}+\frac{\log_{2}N}{2}}{\sqrt{N\left(1-\frac{1}{(1+\gamma)^{2}}\right)}\log_{2}e},\,\text{and}$
(25)
$\Phi=\frac{\sigma_{a}^{2}}{2\sqrt{N\left(1-\frac{1}{(1+\gamma)^{2}}\right)}\log_{2}e}.$
(26)

For a low-latency system with the $i.i.d.$ $a[n]$ over a Rayleigh fading
channel in the high SNR regime, the tradeoff between $\epsilon$ and $\chi$ is
approximated as

$\ln\epsilon\approx-\ln\left(\Upsilon+\frac{\varsigma}{L}\ln\chi\right)-\frac{1}{2}\left(\Upsilon+\frac{\varsigma}{L}\ln\chi\right)^{2}-\frac{1}{2}\ln(2\pi),$
(27)

where $\varsigma$ and $\Upsilon$ are given by

$\Upsilon=-\frac{\frac{N}{\ln 2}\cdot e^{\frac{1}{\gamma}}{\rm
Ei}\left(\frac{-1}{\gamma}\right)+\mu_{a}-\frac{\log_{2}N}{2}}{\sqrt{N}\log
e},\,\text{and}$ (28) $\varsigma=\frac{\frac{e^{\frac{1}{\gamma}}N^{2}}{(\ln
2)^{2}}\left[2G_{2,3}^{3,0}\left(\begin{array}[]{c}1,1\\\
0,0,0\end{array}\Bigg{|}\frac{1}{\gamma}\right)-e^{\frac{1}{\gamma}}{\rm
Ei}^{2}\left(\frac{-1}{\gamma}\right)\right]+\sigma_{a}^{2}}{2\sqrt{N}\log
e}.$ (29)

In Eqs. (24) and (27), $\psi$ and $\Upsilon$ are related to the mean of the
arrival data, while $\Phi$ and $\varsigma$ are related to the variance of the
arrival data. Thus, we call $\psi$ and $\Upsilon$ load-related factors, while
we call $\Phi$ and $\varsigma$ burstiness-related factors. Based on Eqs. (24)
and (27), we can determine the system’s working status, e.g., $\gamma$ and
$\epsilon$, to satisfy the statistical QoS requirement $\chi$.

##  III Reliability-latency-rate tradeoff in the finite blocklength regime

In this section, we mainly focus on characterizing the reliability-latency-
rate tradeoff in the FBL regime for low-latency communication systems. In
Section III-A, we start by deriving the high-SNR slope of EC in the FBL regime
with different channel conditions, which indicates the service-rate gain of
the QoS-constrained low-latency communication systems. Based on the results of
the high-SNR slope of EC in the FBL regime, we further generalize the
conclusions to characterize the tradeoff between the service-rate gain,
reliability gain, and real-time gain in the FBL and high-SNR regime with the
random arrival. Then, we give the discussion on the low-SNR slope of EC in the
FBL regime based on a kind of orthogonal code in Section III-B. In Section
III-C, we present an EC approximation method for low-latency systems. The
closed-form expression of the approximation characterizes the influence of the
channel distribution, error probability, and blocklength on the EC.

###  III-A High-SNR Slope of EC in the FBL regime

In this subsection, we focus on characterizing the high-SNR slope of EC in the
FBL regime and then analyzing the tradeoff between the system’s performance on
latency, reliability, and throughput. Specifically, we first derive the high-
SNR slope of EC with the give $\epsilon$ in different fading channels, which
can be seen as the generalization of conclusions in [14]. Then, we further
consider the case in which $\epsilon$ is a function of $\gamma$. The revised
gain-conservation equation is then characterized by two inequalities for
different low-latency systems. Compared to [22], the revised term in gain
conservation equation mainly results from the influence of random arrivals.

We start our discussion from the FBL-SISO system with an AWGN channel.
Specifically, the high-SNR slope of EC in the FBL regime is defined as [14]

$s_{\infty}=\lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}.$
(30)

For the AWGN channel, it is obvious that $s_{\infty}=1$ with a given
$\epsilon$ since the dispersion term in $R_{\epsilon}$ is finite. We then give
the discussion for the FBL-SISO system with a Rayleigh fading channel. In
Lemma 1.1, we present the high-SNR slope of the normalized EC for the FBL-SISO
system in a Rayleigh fading channel. For simplicity, let
$\beta=\sqrt{N}Q^{-1}(\epsilon)\log_{2}e$.

Lemma 1.1. Given $\epsilon$, the high-SNR slope of the normalized EC for the
FBL-SISO system in a Rayleigh fading channel is given by

$\displaystyle
s_{\infty}=\begin{cases}1,\quad&\theta\leq\frac{1}{TN\log_{2}e},\\\
\frac{1}{\theta N\log_{2}e},\quad&\theta>\frac{1}{TN\log_{2}e}.\end{cases}$
(31)

###### Proof:

To prove Eq. (31) holds, we try to obtain the upper and lower bounds of
$s_{\infty}$ and show that they are identical. In Theorems 5 and 6 of [14],
the author derived the value of the high-SNR slope of the normalized EC with
the infinite blocklength assumption, which can be seen as the upper
bound222Note that, [14] provided the upper bound of $s_{\infty}$ with $T=1$.
However, for $T>1$, the derivation remains the same. Thus, we omit this part
of derivation in this paper. of the $s_{\infty}$ in the FBL regime. We then
show how to derive the lower bound.

Since $\sqrt{1-x}\leq 1$ with $x\in(0,1)$, we obtain

$\displaystyle s_{\infty}$ $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\int_{0}^{\infty}(1+\gamma x)^{-\theta
TN\log_{2}e}e^{\theta T\sqrt{N\left(1-\frac{1}{(1+\gamma
x)^{2}}\right)}Q^{-1}(\epsilon)\log_{2}e-\theta
T\frac{\log_{2}N}{2}}e^{-x}{\rm d}x}{\log_{2}\gamma}$ (32)
$\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[e^{\theta
T\beta}\int_{0}^{\infty}(1+\gamma x)^{-\theta TN\log_{2}e}e^{-x}{\rm
d}x\right]}{\log_{2}\gamma}$ $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}e^{\frac{1}{\gamma}}e^{\theta T\beta}\Gamma\left(1-\theta
TN\log_{2}e,\frac{1}{\gamma}\right)\right]}{\log_{2}\gamma}.$

For $1-\theta TN\log_{2}e<0$, we obtain

$\displaystyle s_{\infty}$ $\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-1}e^{\frac{1}{\gamma}}e^{\theta
T\beta}\right]+\ln\left[\frac{\Gamma\left(1-\theta
TN\log_{2}e,\frac{1}{\gamma}\right)}{\left(\frac{1}{\gamma}\right)^{1-\theta
TN\log_{2}e}}\right]}{\log_{2}\gamma}$ $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{-\ln\gamma}{\log_{2}\gamma}-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{1}{\gamma\log_{2}\gamma}-\frac{1}{N}\lim_{\gamma\to\infty}\frac{\beta}{\log_{2}\gamma}$
(33a) $\displaystyle=\frac{1}{\theta TN\log_{2}e},$

where Eq. (33a) holds because according to [34], for $s<0$,

$\lim_{x\to 0}\frac{\Gamma(s,x)}{x^{s}}=-\frac{1}{s}.$ (34)

Then we focus on $1-\theta TN\log_{2}e=0$. For $\theta TN\log_{2}e=1$,
$\Gamma\left(1-\theta
TN\log_{2}e,\frac{1}{\gamma}\right)=E_{1}\left(\frac{1}{\gamma}\right)$. Thus,
with $\theta TN\log_{2}e=1$, we obtain

$\displaystyle s_{\infty}$ $\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}e^{\frac{1}{\gamma}}e^{\theta
T\beta}E_{1}\left(\frac{1}{\gamma}\right)\right]}{\log_{2}\gamma}$
$\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\ln(1+\gamma)\right]}{\log_{2}\gamma}$ (35a)
$\displaystyle=-\frac{1}{\theta TN}\lim_{\gamma\to\infty}\frac{-\theta
TN\log_{2}e\ln\gamma}{\log_{2}\gamma}-\frac{1}{N}\lim_{\gamma\to\infty}\frac{\beta}{\log_{2}\gamma}-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\ln(1+\gamma)}{\log_{2}(\gamma)}$ (35b)
$\displaystyle=1.$

Eq. (35a) holds because according to [35]

$E_{1}(x)<e^{-x}\ln\left(1+\frac{1}{x}\right),\quad x>0.$ (36)

Finally, we consider $1-\theta TN\log_{2}e>0$. According to [36], for $s>0$,
$\Gamma(s,x)$ satisfies

$\Gamma(s,x)<\frac{(x+v_{s})^{s}-x^{s}}{v_{s}s}e^{-x},$ (37)

where $v_{s}=\Gamma(s+1)^{\frac{1}{s-1}}>0$. For simplicity, we let
$c_{1}=\Gamma(2-\theta TN\log_{2}e)^{-\frac{1}{\theta TN\log_{2}e}}$. Based on
Eq. (37), we obtain

$\displaystyle s_{\infty}$ $\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\frac{\left(\frac{1}{\gamma}+c_{1}\right)^{1-\theta
TN\log_{2}e}-\left(\frac{1}{\gamma}\right)^{1-\theta
TN\log_{2}e}}{c_{1}(1-\theta TN\log_{2}e)}\right]}{\log_{2}\gamma}$
$\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-1}e^{\theta
T\beta}\left((1+c_{1}\gamma)^{1-\theta
TN\log_{2}e}-1\right)\right]}{\log_{2}\gamma}$ $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{-\ln\gamma}{\log_{2}\gamma}-\frac{1}{N}\lim_{\gamma\to\infty}\frac{\beta}{\log_{2}\gamma}-\frac{1-\theta
TN\log_{2}e}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln(1+c_{1}\gamma)}{\log_{2}\gamma}$ (38a)
$\displaystyle=1.$ (38b)

Eq. (38a) holds because
$\lim_{\gamma\to\infty}\frac{\ln\left[(1+c_{1}\gamma)^{1-\theta
TN\log_{2}e}-1\right]}{(1-\theta TN\log_{2}e)\ln(1+c_{1}\gamma)}=1$ with
$c_{1}>0$ and $1-\theta TN\log_{2}e>0$. Eq. (38b) holds because
$\lim_{\gamma\to\infty}\frac{\ln(1+c_{1}\gamma)}{\ln\gamma}=1$. The derived
lower bound is equal to the upper bound proposed in [14]. Therefore, according
to squeeze theorem, we have completed the proof of Lemma 1.1. ∎

In Lemma 1.1, we obtain the value of $s_{\infty}$ in the FBL-SISO system with
a Rayleigh fading channel. According to EC theory [12], $\theta$ indicates the
statistical queueing constraints. A larger $\theta$ indicates a stricter QoS
constraint according to Eqs. (9) and (10). By comparing the results in Lemma
1.1 and the results in [14], we find that the high-SNR slope of the normalized
effective capacity is identical in the FBL-SISO and infinite-blocklength-SISO
systems. Results in Lemma 1.1 shows that with a small $\theta$, $s_{\infty}$
is irrelevant to the statistical constraints, which is consistent with the
analysis in [14] for the infinite-blocklength case.

Based on this result, we are interested in characterizing $s_{\infty}$ of the
systems with frequency or spatial diversity in Rayleigh fading channels.
Different from [14], we will consider the systems adopting maximum ratio
combing in the FBL regime in Theorem 1.1. Besides, the existing work lacks the
analysis for more general channel models, e.g., Nakagami-$m$ channel. We will
give the results for the FBL-SISO system in the Nakagami fading channel in
Theorem 1.2.

Theorem 1.1. For a wireless system with $f(x)$ satisfying Eq. (3), i.e., with
spatial or frequency diversity $\kappa$ in Rayleigh fading channels, its high-
SNR slope of the normalized EC with a given $\epsilon$ in the FBL regime is

$\displaystyle
s_{\infty}=\begin{cases}1,\quad&\theta\leq\frac{\kappa}{TN\log_{2}e},\\\
\frac{\kappa}{\theta
TN\log_{2}e},\quad&\theta>\frac{\kappa}{TN\log_{2}e}.\end{cases}$ (39)

###### Proof:

First, we give the derivation of the upper bound of $s_{\infty}$. Since
$R_{\epsilon}[n]$ is smaller than $N\log_{2}\left(1+|h[n]|^{2}\gamma\right)$,
we obtain

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\int_{0}^{\infty}(1+\gamma x)^{-\theta
TN\log_{2}e}\frac{x^{\kappa-1}e^{-x}}{(\kappa-1)!}{\rm d}x}{\log_{2}\gamma}$
(40) $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{\kappa-1}e^{-x}{\rm d}x\right]}{\log_{2}\gamma}.$

For $\theta<\frac{\kappa}{TN\log_{2}e}$, we find that

$\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{\kappa-1}e^{-x}{\rm d}x\leq\int_{0}^{\infty}x^{\kappa-\theta
TN\log_{2}e-1}e^{-x}{\rm d}x.$ (41)

According to [14], $\int_{0}^{\infty}z^{a}e^{-x}{\rm d}x<\infty$ if $a>-1$.
Since in this case $\kappa>\theta TN\log_{2}e$, we obtain $\kappa-\theta
TN\log_{2}e-1>-1$. Therefore,
$\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{\kappa-1}e^{-x}{\rm d}x$ is bounded by a finite constant. Thus,
under the condition $\theta<\frac{\kappa}{TN\log_{2}e}$, Eq. (40) can be
further expressed as

$s_{\infty}\leq-\frac{1}{\theta TN}\lim_{\gamma\to\infty}\frac{-\theta
TN\log_{2}e\ln\gamma}{\log_{2}\gamma}=1.$ (42)

For $\theta\geq\frac{\kappa}{TN\log_{2}e}$, we focus on the integral
$\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{\kappa-1}e^{-x}{\rm d}x$. We find that

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{\kappa-1}e^{-x}{\rm d}x$ (43a) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+\kappa-1}\cdot\left(\frac{x}{\frac{1}{\gamma}+x}\right)^{\kappa-1}e^{-x}{\rm
d}x$ $\displaystyle\geq$
$\displaystyle\int_{\frac{1}{\gamma}}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+\kappa-1}\left(\frac{1}{2}\right)^{\kappa-1}e^{-x}{\rm d}x$ (43b)
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)^{\kappa-1}e^{\frac{1}{\gamma}}\Gamma\left(\kappa-\theta
TN\log_{2}e,\frac{2}{\gamma}\right),$ (43c)

where Eq. (43b) holds because $\frac{x}{\frac{1}{\gamma}+x}$ an increase
function for $x\in\big{[}\frac{1}{\gamma},\infty\big{)}$. We first consider
$\theta>\frac{\kappa}{TN\log_{2}e}$, which means $\kappa-\theta
TN\log_{2}e<0$. By substituting Eq. (43c) into Eq. (40), we obtain

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\left(\frac{1}{2}\right)^{\kappa-1}e^{\frac{1}{\gamma}}\Gamma\left(\kappa-\theta
TN\log_{2}e,\frac{2}{\gamma}\right)\right]}{\log_{2}\gamma}$
$\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\kappa}\left(\frac{1}{2}\right)^{\theta
TN\log_{2}e-1}e^{\frac{1}{\gamma}}\frac{\Gamma\left(\kappa-\theta
TN\log_{2}e,\frac{2}{\gamma}\right)}{\left(\frac{2}{\gamma}\right)^{\kappa-\theta
TN\log_{2}e}}\right]}{\log_{2}\gamma}$ $\displaystyle=-\frac{1}{\theta
TN}\left(\lim_{\gamma\to\infty}\frac{-\kappa\ln\gamma}{\log_{2}\gamma}+\lim_{\gamma\to\infty}\frac{\ln\left[\frac{\Gamma\left(\kappa-\theta
TN\log_{2}e,\frac{2}{\gamma}\right)}{\left(\frac{2}{\gamma}\right)^{\kappa-\theta
TN\log_{2}e}}\right]}{\log_{2}\gamma}+\lim_{\gamma\to\infty}\frac{1}{\gamma\log_{2}\gamma}\right)$
(44a) $\displaystyle=\frac{\kappa}{\theta TN\log_{2}e}.$ (44b)

Eq. (44a) holds according to Eq. (34) with $\kappa-\theta TN\log_{2}e<0$ .

For $\kappa-\theta TN\log_{2}e=0$, $\Gamma\left(\kappa-\theta
TN\log_{2}e,\frac{2}{\gamma}\right)=E_{1}\left(\frac{2}{\gamma}\right)$. Then,
we obtain

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\left(\frac{1}{2}\right)^{\kappa-1}e^{\frac{1}{\gamma}}E_{1}\left(\frac{2}{\gamma}\right)\right]}{\log_{2}\gamma}$
$\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\left(\frac{1}{2}\right)^{\kappa}e^{-\frac{1}{\gamma}}\ln\left(1+\gamma\right)\right]}{\log_{2}\gamma}$
(45a) $\displaystyle=-\frac{1}{\theta
TN}\left(\lim_{\gamma\to\infty}\frac{-\theta
TN\log_{2}e\ln\gamma}{\log_{2}\gamma}-\lim_{\gamma\to\infty}\frac{1}{\gamma\log_{2}\gamma}+\lim_{\gamma\to\infty}\frac{\ln\ln\left(1+\gamma\right)}{\log_{2}\gamma}\right)$
$\displaystyle=1,$

where Eq. (45a) holds since according to [35],

$E_{1}(x)>\frac{1}{2}e^{-x}\ln\left(1+\frac{2}{x}\right),\quad x>0.$ (46)

Now we have proved that

$\displaystyle
s_{\infty}\leq\begin{cases}1,\quad&\theta\leq\frac{\kappa}{TN\log_{2}e},\\\
\frac{\kappa}{\theta
TN\log_{2}e},\quad&\theta>\frac{\kappa}{TN\log_{2}e}.\end{cases}$ (47)

Then, we will give the proof for the lower bound of $s_{\infty}$. Similar to
Eq. (32), since $\sqrt{1-x}\leq 1$ with $x\in(0,1)$, we obtain

$\displaystyle s_{\infty}$ $\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\int_{0}^{\infty}(1+\gamma x)^{-\theta
TN\log_{2}e}e^{\theta T\beta}\frac{x^{\kappa-1}e^{-x}}{(\kappa-1)!}{\rm
d}x}{\log_{2}\gamma}$ $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{\kappa-1}e^{-x}{\rm d}x\right]}{\log_{2}\gamma}$
$\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+\kappa-1}e^{-x}{\rm d}x\right]}{\log_{2}\gamma}$ (48a)
$\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}e^{\frac{1}{\gamma}}\Gamma\left(\kappa-\theta
TN\log_{2}e,\frac{1}{\gamma}\right)\right]}{\log_{2}\gamma}.$ (48b)

Eq. (48a) holds because $x\leq x+\frac{1}{\gamma}$ and $\kappa-1\geq 0$. By
comparing Eq. (48b) with Eq. (32), the only difference is that 1 turns into
$\kappa$. Thus, the proof of the lower bounds of $s_{\infty}$ with $\kappa$
diversity is similar to Eqs. (33), (35), and (38). We omit the detail of this
part of the proof. For $\theta>\frac{\kappa}{TN\log_{2}e}$, we obtain

$s_{\infty}\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{-\kappa\ln\gamma}{\log_{2}\gamma}-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{1}{\gamma\log_{2}\gamma}-\frac{1}{N}\lim_{\gamma\to\infty}\frac{\beta}{\log_{2}\gamma}=\frac{\kappa}{\theta
TN\log_{2}e}.$ (49)

For $\theta\leq\frac{\kappa}{TN\log_{2}e}$, we obtain

$s_{\infty}\geq-\frac{1}{\theta TN}\lim_{\gamma\to\infty}\frac{-\theta
TN\log_{2}e\ln\gamma}{\log_{2}\gamma}-\frac{1}{N}\lim_{\gamma\to\infty}\frac{\beta}{\log_{2}\gamma}=1.$
(50)

Now, we have obtained

$\displaystyle
s_{\infty}\geq\begin{cases}1,\quad&\theta\leq\frac{\kappa}{TN\log_{2}e},\\\
\frac{\kappa}{\theta
TN\log_{2}e},\quad&\theta>\frac{\kappa}{TN\log_{2}e}.\end{cases}$ (51)

Therefore, according to squeeze theorem, we have completed the proof of
Theorem 1.1. ∎

Based on Lemma 1.1 and Theorem 1.1, we can derive the $s_{\infty}$ in the
Nakagami-$m$ fading channel with $m\geq 1$ by adopting the similar methods
used in the proof of Lemma 1.1 and Theorem 1.1. For the $m\in[0.5,1)$ parts,
there are differences in the proof. However, in Theorem 1.2 we prove that the
$s_{\infty}$ in the Nakagami-$m$ fading channel has a similar expression with
the $s_{\infty}$ shown in Theorem 1.1 with $m\geq 0.5$.

Theorem 1.2. For a wireless system with $f(x)$ satisfying Eq. (4), i.e., in a
Nakagami-$m$ fading channel, its high-SNR slope of the normalized effective
capacity with a given $\epsilon$ in the FBL regime is

$\displaystyle
s_{\infty}=\begin{cases}1,\quad&\theta\leq\frac{m}{TN\log_{2}e},\\\
\frac{m}{\theta TN\log_{2}e},\quad&\theta>\frac{m}{TN\log_{2}e}.\end{cases}$
(52)

###### Proof:

Based on Eq. (4), for $m\geq 1$, we obtain

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\int_{0}^{\infty}(1+\gamma x)^{-\theta
TN\log_{2}e}\left(\frac{m}{\Omega}\right)^{m}\frac{x^{m-1}}{\Gamma(m)}e^{-\frac{m}{\Omega}x}{\rm
d}x\right]}{\log_{2}\gamma}$ (53) $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x\right]}{\log_{2}\gamma}.$

Similar to the proof of Theorem 1.1, for $\theta<\frac{m}{TN\log_{2}e}$, we
find

$\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm
d}x\leq\int_{0}^{\infty}x^{m-\theta TN\log_{2}e-1}e^{-\frac{m}{\Omega}x}{\rm
d}x.$ (54)

Since $m>\theta TN\log_{2}e$, we obtain $m-\theta TN\log_{2}e-1>-1$, which
means

$\int_{0}^{\infty}x^{m-\theta TN\log_{2}e-1}e^{-\frac{m}{\Omega}x}{\rm
d}x=\left(\frac{\Omega}{m}\right)^{m-\theta
TN\log_{2}e}\int_{0}^{\infty}t^{m-\theta TN\log_{2}e-1}e^{-t}{\rm
d}t<+\infty.$ (55)

Therefore, $\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x$ is bounded by a finite
constant. Thus, under the condition $\theta<\frac{m}{TN\log_{2}e}$, Eq. (53)
can be further expressed as

$s_{\infty}\leq-\frac{1}{\theta TN}\lim_{\gamma\to\infty}\frac{-\theta
TN\log_{2}e\ln\gamma}{\log_{2}\gamma}=1.$ (56)

For $\theta\geq\frac{m}{TN\log_{2}e}$, according to (43) we find that

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m-1}\cdot\left(\frac{x}{\frac{1}{\gamma}+x}\right)^{m-1}e^{-\frac{m}{\Omega}x}{\rm
d}x$ $\displaystyle\geq$
$\displaystyle\int_{\frac{1}{\gamma}}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m-1}\left(\frac{1}{2}\right)^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x$
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)^{m-1}\left(\frac{m}{\Omega}\right)^{\theta
TN\log_{2}e-m}e^{\frac{m}{\gamma\Omega}}\Gamma\left(m-\theta
TN\log_{2}e,\frac{2m}{\Omega\gamma}\right).$

We then obtain

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\left(\frac{1}{2}\right)^{m-1}\left(\frac{m}{\Omega}\right)^{\theta
TN\log_{2}e-m}e^{\frac{m}{\gamma\Omega}}\Gamma\left(m-\theta
TN\log_{2}e,\frac{2m}{\Omega\gamma}\right)\right]}{\log_{2}\gamma}$
$\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-m}\left(\frac{1}{2}\right)^{\theta
TN\log_{2}e-1}e^{\frac{m}{\gamma\Omega}}\frac{\Gamma\left(m-\theta
TN\log_{2}e,\frac{2m}{\Omega\gamma}\right)}{\left(\frac{2m}{\Omega\gamma}\right)^{m-\theta
TN\log_{2}e}}\right]}{\log_{2}\gamma}$ $\displaystyle=\frac{m}{\theta
TN\log_{2}e}.$

For $m-\theta TN\log_{2}e=0$, $\Gamma\left(m-\theta
TN\log_{2}e,\frac{2m}{\Omega\gamma}\right)=E_{1}\left(\frac{2m}{\Omega\gamma}\right)$.
Then, we obtain

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\left(\frac{1}{2}\right)^{m-1}\left(\frac{m}{\Omega}\right)^{\theta
TN\log_{2}e-m}e^{\frac{m}{\gamma\Omega}}E_{1}\left(\frac{2m}{\Omega\gamma}\right)\right]}{\log_{2}\gamma}$
$\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\left(\frac{1}{2}\right)^{m}e^{-\frac{m}{\Omega\gamma}}\ln\left(1+\frac{\Omega\gamma}{m}\right)\right]}{\log_{2}\gamma}$
$\displaystyle=-\frac{1}{\theta TN}\left(\lim_{\gamma\to\infty}\frac{-\theta
TN\log_{2}e\ln\gamma}{\log_{2}\gamma}-\lim_{\gamma\to\infty}\frac{m}{\Omega\gamma\log_{2}\gamma}+\lim_{\gamma\to\infty}\frac{\ln\ln\left(1+\frac{\Omega\gamma}{m}\right)}{\log_{2}\gamma}\right)$
$\displaystyle=1.$

Now we have completed the proof for the upper bound of $s_{\infty}$.

The lower bound of $s_{\infty}$ in the Nakagami-$m$ fading channel is given by

$\displaystyle s_{\infty}$ $\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\int_{0}^{\infty}(1+\gamma x)^{-\theta
TN\log_{2}e}e^{\theta T\beta-\theta
T\frac{\log_{2}N}{2}}\left(\frac{m}{\Omega}\right)^{m}\frac{x^{m-1}}{\Gamma(m)}e^{-\frac{m}{\Omega}x}{\rm
d}x\right]}{\log_{2}\gamma}$ (57) $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x\right]}{\log_{2}\gamma}$
$\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m-1}e^{-\frac{m}{\Omega}x}{\rm d}x\right]}{\log_{2}\gamma}$
$\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\left(\frac{m}{\Omega}\right)^{\theta
TN\log_{2}e-m}e^{\frac{m}{\gamma\Omega}}\Gamma\left(m-\theta
TN\log_{2}e,\frac{m}{\Omega\gamma}\right)\right]}{\log_{2}\gamma}.$

Since $\frac{m}{\Omega}$ is a finite constant,
$\lim_{\gamma\to\infty}\frac{m}{\Omega\gamma}=0$. Therefore, the remaining
part of proof is similar to Eqs. (48) and (44). We omit the detail of this
part of proof.

For $0.5\leq m<1$, there are some differences in the proof, which mainly lie
in the proof of the upper bound with $\theta>\frac{m}{TN\log_{2}e}$ and the
lower bound. For the upper bound with $\theta>\frac{m}{TN\log_{2}e}$, Eq.
(43b) is changed into

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x$ (58) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m-1}\left(\frac{1}{\gamma
x}+1\right)^{1-m}e^{-\frac{m}{\Omega}x}{\rm d}x$ $\displaystyle\geq$
$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m-1}e^{-\frac{m}{\Omega}x}{\rm d}x$ $\displaystyle=$
$\displaystyle\left(\frac{m}{\Omega}\right)^{\theta
TN\log_{2}e-m}e^{\frac{m}{\Omega\gamma}}\Gamma\left(m-\theta
TN\log_{2}e,\frac{m}{\Omega\gamma}\right).$

Then, based on Eq. (58) and Eq. (44a), we obtain

$\displaystyle s_{\infty}\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta
TN\log_{2}e}\left(\frac{m}{\Omega}\right)^{\theta
TN\log_{2}e-m}e^{\frac{m}{\Omega\gamma}}\Gamma\left(m-\theta
TN\log_{2}e,\frac{m}{\Omega\gamma}\right)\right]}{\log_{2}\gamma}=\frac{m}{\theta
TN\log_{2}e}.$ (59)

Then, we focus on the lower bound of $s_{\infty}$ with $0.5\leq m<1$. The
essence of this part of the proof is still the scaling of
$\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x$. According to Eq. (57), the
lower bound of $s_{\infty}$ can be expressed as

$\displaystyle s_{\infty}\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x\right]}{\log_{2}\gamma}.$
(60)

For $\theta<\frac{m}{TN\log_{2}e}$, we transform Eq. (60) into

$s_{\infty}\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}x^{-\theta TN\log_{2}e+m-1}e^{-\frac{m}{\Omega}x}{\rm
d}x\right]}{\log_{2}\gamma}.$ (61)

Eq. (61) holds because $-\theta TN\log_{2}e<0$ and $\frac{1}{\gamma}+x>x$. As
we mentioned before, $\int_{0}^{\infty}z^{a}e^{-x}{\rm d}x<\infty$ holds when
$a>-1$. Since in this case $\theta<\frac{m}{TN\log_{2}e}$, $-\theta
TN\log_{2}e+m-1>-1$. Thus, in this case $\int_{0}^{\infty}x^{-\theta
TN\log_{2}e+m-1}e^{-\frac{m}{\Omega}x}{\rm
d}x=\left(\frac{\Omega}{m}\right)^{m-\theta
TN\log_{2}e}\int_{0}^{\infty}t^{m-\theta TN\log_{2}e-1}e^{-t}{\rm d}t$ is a
finite constant. Then, we obtain

$s_{\infty}\geq-\frac{1}{\theta TN}\lim_{\gamma\to\infty}\frac{-\theta
TN\log_{2}e\ln\gamma}{\log_{2}\gamma}-\frac{1}{N}\lim_{\gamma\to\infty}\frac{\beta}{\log_{2}\gamma}=1.$
(62)

For $\theta\geq\frac{m}{TN\log_{2}e}$, we find

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x$
$\displaystyle=\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m}\left(\frac{x}{\frac{1}{\gamma}+x}\right)^{m-\frac{1}{2}}\frac{x^{-\frac{1}{2}}}{\left(\frac{1}{\gamma}+x\right)^{\frac{1}{2}}}e^{-\frac{m}{\Omega}x}{\rm
d}x$ $\displaystyle\leq\left(\frac{1}{\gamma}\right)^{-\theta
TN\log_{2}e+m}\int_{0}^{\infty}\frac{e^{-\frac{m}{\Omega}x}}{\sqrt{x\left(\frac{1}{\gamma}+x\right)}}{\rm
d}x$ (63a) $\displaystyle=\gamma^{\theta
TN\log_{2}e-m}e^{\frac{m}{2\Omega\gamma}}K_{0}\left(\frac{m}{2\Omega\gamma}\right),$

where $K_{v}(z)$ is the modified Bessel function of the second kind [35]. Eq.
(63a) holds because in this case $m-\frac{1}{2}\geq 0$ and
$\frac{x}{\frac{1}{\gamma}+x}\leq 1$. According to [35], for the modified
Bessel function of the second kind $K_{v}(z)$ with $v=0$, we have

$\lim_{z\to 0}\frac{K_{0}(z)}{-\ln(z)}=c,$ (64)

where $c>0$ is a constant. Based on this conclusion, we transformed Eq. (63)
into

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}x^{m-1}e^{-\frac{m}{\Omega}x}{\rm d}x\leq\gamma^{\theta
TN\log_{2}e-m}e^{\frac{m}{2\Omega\gamma}}\ln\left(\frac{2\Omega\gamma}{m}\right)\frac{K_{0}\left(\frac{m}{2\Omega\gamma}\right)}{-\ln\left(\frac{m}{2\Omega\gamma}\right)}.$
(65)

By substituting Eq. (65) into Eq. (60), we obtain

$\displaystyle s_{\infty}$ $\displaystyle\geq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-m}e^{\theta
T\beta}e^{\frac{m}{2\Omega\gamma}}\ln\left(\frac{2\Omega\gamma}{m}\right)\frac{K_{0}\left(\frac{m}{2\Omega\gamma}\right)}{-\ln\left(\frac{m}{2\Omega\gamma}\right)}\right]}{\log_{2}\gamma}$
$\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\left(\frac{-m\ln\gamma}{\log_{2}\gamma}+\frac{\theta
T\beta}{\log_{2}\gamma}+\frac{\ln\ln\gamma}{\log_{2}\gamma}+\frac{\ln\frac{K_{0}\left(\frac{m}{2\Omega\gamma}\right)}{-\ln\left(\frac{m}{2\Omega\gamma}\right)}}{\log_{2}\gamma}\right)$
(66a) $\displaystyle=\frac{m}{\theta TN\log_{2}e},$ (66b)

where Eq. (66b) holds according to Eq. (64). Now we have completed the proof.
∎

In Lemma 1.1, Theorem 1.1, and Theorem 1.2, we characterize the high-SNR slope
of the normalized effective capacity in the FBL regime with a given
$\epsilon$. By comparing the results in Theorems 1.1 and 1.2 with Lemma 1.1,
we find that in the FBL regime, $s_{\infty}$ of the systems with the spatial
or frequency diversity is not lower than that of the SISO system in a Rayleigh
fading channel. For the SISO system in a Nakagami-$m$ fading channel, its
$s_{\infty}$ is not lower than that of the SISO system in a Rayleigh fading
channel with $m\geq 1$, while its $s_{\infty}$ is not larger than that of the
SISO system in a Rayleigh fading channel with $m\in[0.5,1)$. Besides, we also
find that, with a larger $\kappa$ or $m$, the length of the interval in which
$s_{\infty}$ is irrelevant of $\theta$ is larger. According to [32], $m$
measures the ratio of line-of-sight signal power to multipath power, which
means $m<1$ corresponds to a worse fading than the Rayleigh fading and $m>1$
corresponds to a less severe fading than the Rayleigh fading. Thus, from the
above results, we find that the severity of the fading and the spatial or
frequency diversity has an influence on the $s_{\infty}$. Less serve fading
and the larger spatial or frequency diversity help us obtain a larger
$s_{\infty}$ with a given QoS constraint and ensure $s_{\infty}$ irrelevant of
QoS constraints in a larger $\theta$ set.

We then let $\epsilon$ become a decreasing function of $\gamma$. Based on this
assumption, we will refine the conclusions in Lemma 1.1, Theorem 1.1, and
Theorem 1.2. The revised results are summarized in Corollary 1.1.

Corollary 1.1. Assume $\epsilon\in(0,0.5]$ is a decreasing function of
$\gamma$ and $\lim\limits_{\gamma\to\infty}\epsilon(\gamma)=0$.

  * •

For a FBL-SISO system in an AWGN channel, we have

$\lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}+\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}=1.$
(67)

  * •

For a FBL-SISO system in a Rayleigh fading channel, we have

$\lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}+\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}\geq\min\left\\{1,\frac{1}{\theta
TN\log_{2}e}\right\\},\text{ and }$ (68)
$\lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}+(1-c)\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}\leq\min\left\\{1,\frac{1}{\theta
TN\log_{2}e}\right\\}.$ (69)

  * •

For a system with $f(x)$ satisfying Eq. (3) in the FBL regime, we have

$\lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}+\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}\geq\min\left\\{1,\frac{\kappa}{\theta
TN\log_{2}e}\right\\},\text{ and }$ (70)
$\lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}+(1-c)\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}\leq\min\left\\{1,\frac{\kappa}{\theta
TN\log_{2}e}\right\\}.$ (71)

  * •

For a FBL-SISO system in a Nakagami-$m$ fading channel, we have

$\lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}+\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}\geq\min\left\\{1,\frac{m}{\theta
TN\log_{2}e}\right\\},\text{ and }$ (72)
$\lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}+(1-c)\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}\leq\min\left\\{1,\frac{m}{\theta
TN\log_{2}e}\right\\}.$ (73)

In Eqs. (69), (71), and (73), $c>0$ is an arbitrary constant.

###### Proof:

For the AWGN channel, we can easily obtain

$\displaystyle s_{\infty}$
$\displaystyle=\lim_{\gamma\to\infty}\frac{R_{\epsilon}}{N\log_{2}\gamma}$
(74)
$\displaystyle=1-\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}\left(\epsilon(\gamma)\right)}{\log_{2}\gamma}.$

The derivation of Eqs. (68), (70), and (72) is based on the derivation of the
lower bounds for $s_{\infty}$ in the proof of Lemma 1.1, Theorem 1.1, and
Theorem 1.2. For the FBL-SISO system in a Rayleigh fading channel, the details
are provided in Eqs. (33a), (35b), and (38a). For the system with $f(x)$
satisfying Eq. (3), the details are provided in Eqs. (49) and (50). For the
FBL-SISO in a Nakagami-$m$ fading channel, the details are provided in Eqs.
(62) and (66a).

Then, we focus on the derivation of the Eqs. (69), (71), and (73). We first
consider a FBL-SISO system in a Rayleigh fading channel. Different from the
proof of Lemma 1.1, we start the proof with the help of $\sqrt{1-x}\geq
1-\sqrt{x}$ with $x\in(0,1)$. We have

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\int_{0}^{\infty}e^{-\theta
T\left(N\log_{2}(1+\gamma x)-\beta\left(1-\frac{1}{1+\gamma
x}\right)\right)}e^{-x}{\rm d}x\right]}{\log_{2}\gamma}$ (75)
$\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}e^{-\left(x+\frac{\theta T\beta}{1+\gamma x}\right)}{\rm
d}x\right]}{\log_{2}\gamma}.$

Based on $\frac{x}{x+1}\leq\ln(1+x)$ for $x>0$, we have $\frac{1}{1+\gamma
x}\leq\ln\left(1+\frac{1}{\gamma x}\right)$. Thus, we transform Eq. (75) into

$s_{\infty}\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}\left(1+\frac{1}{\gamma x}\right)^{-\theta T\beta}e^{-x}{\rm
d}x\right]}{\log_{2}\gamma}.$ (76)

For the integral in Eq. (76), we have

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}\left(1+\frac{1}{\gamma x}\right)^{-\theta T\beta}e^{-x}{\rm d}x$
(77) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}\left(\frac{x}{x+\frac{1}{\gamma}}\right)^{\theta
T\beta}e^{-x}{\rm d}x$ $\displaystyle\geq$
$\displaystyle\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta}\int_{\frac{c_{2}}{\gamma}}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}e^{-x}{\rm d}x$ $\displaystyle=$
$\displaystyle\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta}e^{\frac{1}{\gamma}}\Gamma\left(1-\theta
TN\log_{2}e,\frac{1+c_{2}}{\gamma}\right),$

where $c_{2}$ is an arbitrary positive constant. By substituting Eq. (77) into
Eq. (76), we obtain

$\displaystyle s_{\infty}\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta}e^{\frac{1}{\gamma}}\Gamma\left(1-\theta
TN\log_{2}e,\frac{1+c_{2}}{\gamma}\right)\right]}{\log_{2}\gamma}.$ (78)

For $\theta>\frac{1}{TN\log_{2}e}$, based on Eq. (34), we have

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{-\ln\gamma}{\log_{2}\gamma}-\frac{1+\ln\frac{c_{2}}{c_{2}+1}}{N}\lim_{\gamma\to\infty}\frac{\beta}{\log_{2}\gamma}$
(79) $\displaystyle=\frac{1}{\theta
TN}-\left(1-\ln\left(1+\frac{1}{c_{2}}\right)\right)\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}.$

Note that, since $c_{2}$ is an arbitrary positive constant,
$\ln\left(1+\frac{1}{c_{2}}\right)$ is an arbitrary positive constant.

For $\theta\leq\frac{1}{TN\log_{2}e}$, we will utilize the conclusion that
$\Gamma(s,x)\geq e^{-x}(1+x)^{s-1}$ for $s\leq 1$. This can be easily proved
by Jensen’s inequality. $(t+x)^{s-1}$ is a convex function of $t$ for $s\leq
1$. Thus, we have

$\int_{x}^{\infty}y^{s-1}e^{-y}{\rm
d}y=e^{-x}\int_{0}^{\infty}(t+x)^{s-1}e^{-t}{\rm d}t\geq
e^{-x}\left(\int_{0}^{\infty}(t+x)e^{-t}{\rm
d}t\right)^{s-1}=e^{-x}(1+x)^{s-1}.$ (80)

Thus, we have

$\displaystyle s_{\infty}$ $\displaystyle\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta}e^{-\frac{c_{2}}{\gamma}}\left(1+\frac{1+c_{2}}{\gamma}\right)^{-\theta
TN\log_{2}e}\right]}{\log_{2}\gamma}$ (81) $\displaystyle=-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\left(\frac{-\theta
TN\log_{2}e\ln\gamma}{\log_{2}\gamma}+\left(1+\ln\frac{c_{2}}{c_{2}+1}\right)\frac{\theta
T\beta}{\log_{2}\gamma}-\theta
TN\log_{2}e\frac{1+c_{2}}{\gamma\log_{2}\gamma}\right)$
$\displaystyle=1-\left(1-\ln\left(1+\frac{1}{c_{2}}\right)\right)\frac{\log_{2}e}{\sqrt{N}}\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\log_{2}\gamma}.$

Now we have completed the proof of Eq. (69).

For a system with $f(x)$ satisfying Eq. (3) in the FBL regime, Eq. (76) is
changed into

$s_{\infty}\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}\left(1+\frac{1}{\gamma x}\right)^{-\theta
T\beta}x^{\kappa-1}e^{-x}{\rm d}x\right]}{\log_{2}\gamma}.$ (82)

For the integral in Eq. (82), we have

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}\left(1+\frac{1}{\gamma x}\right)^{-\theta
T\beta}x^{\kappa-1}e^{-x}{\rm d}x$ (83) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+\kappa-1}\left(\frac{1}{\gamma x}+1\right)^{1-\kappa-\theta
T\beta}e^{-x}{\rm d}x$ $\displaystyle\geq$
$\displaystyle\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta+\kappa-1}\int_{\frac{c_{2}}{\gamma}}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+\kappa-1}e^{-x}{\rm d}x$ $\displaystyle=$
$\displaystyle\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta+\kappa-1}e^{\frac{1}{\gamma}}\Gamma\left(\kappa-\theta
TN\log_{2}e,\frac{1+c_{2}}{\gamma}\right).$

By substituting Eq. (83) into Eq. (82), we obtain

$s_{\infty}\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta+\kappa-1}e^{\frac{1}{\gamma}}\Gamma\left(\kappa-\theta
TN\log_{2}e,\frac{1+c_{2}}{\gamma}\right)\right]}{\log_{2}\gamma}.$ (84)

The remaining part of the proof is similar to Eqs. (79) and (81).

For a FBL-SISO system in a Nakagami-$m$ fading channel, the integral in the
upper bound of $s_{\infty}$ is given by

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e}\left(1+\frac{1}{\gamma x}\right)^{-\theta
T\beta}x^{m-1}e^{-mx}{\rm d}x$ (85) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m-1}\left(\frac{1}{\gamma x}+1\right)^{1-m-\theta
T\beta}e^{-mx}{\rm d}x.$

Note that, with $\gamma\to+\infty$, $\beta\to+\infty$. Thus, $1-m-\theta
T\beta<0$. We obtain

$\displaystyle\int_{0}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m-1}\left(\frac{1}{\gamma x}+1\right)^{1-m-\theta
T\beta}e^{-mx}{\rm d}x$ (86) $\displaystyle\geq$
$\displaystyle\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta+m-1}\int_{\frac{c_{2}}{\gamma}}^{\infty}\left(\frac{1}{\gamma}+x\right)^{-\theta
TN\log_{2}e+m-1}e^{-mx}{\rm d}x$ $\displaystyle=$
$\displaystyle\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta+m-1}e^{\frac{m}{\gamma}}\Gamma\left(m-\theta
TN\log_{2}e,\frac{m(1+c_{2})}{\gamma}\right),$

where $c_{2}$ is an arbitrary positive constant.

Based on this result, we obtain

$s_{\infty}\leq-\frac{1}{\theta
TN}\lim_{\gamma\to\infty}\frac{\ln\left[\gamma^{-\theta TN\log_{2}e}e^{\theta
T\beta}\left(\frac{c_{2}}{c_{2}+1}\right)^{\theta
T\beta+m-1}e^{\frac{m}{\gamma}}\Gamma\left(m-\theta
TN\log_{2}e,\frac{m(1+c_{2})}{\gamma}\right)\right]}{\log_{2}\gamma}.$ (87)

The remaining part of the proof is similar to Eqs. (79) and (81). Thus, we
omit the details of this part of the proof here. Then, we obtain Eq. (73). Now
we have completed the proof of Corollary 1.1. ∎

In Corollary 1.1, we proposed the result of the high-SNR asymptotic analysis
of the EC, which is related to the error probability. Note that, EC indicates
the maximum constant source rate required by the given service process to
satisfy the statistical queueing requirement [12]. With the given $\theta$, we
can define
$\zeta=\lim\limits_{\gamma\to\infty}\frac{\partial{\Lambda(\gamma)}}{\partial{\log_{2}\gamma}}$
as the service-rate gain, which we proposed in Section II-B. In the high-SNR
regime, we can approximate $\zeta$ by
$\lim\limits_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\log_{2}\gamma}$. Before
characterizing the tradeoff between the service-rate gain, reliability gain,
and real-time gain, we will first give an additional discussion on the
reliability gain for this system.

Lemma 2. The reliability gain $\varpi$ satisfies

$\lim\limits_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\sqrt{N}\varpi\log_{2}\gamma}=\frac{1}{\log_{2}e}.$
(88)

###### Proof:

Based on the definition of $\varpi$, we obtain

$\lim_{\gamma\to\infty}\frac{\partial\sqrt{-\log_{2}\epsilon(\gamma)}}{\partial\log_{2}\gamma}=\varpi\sqrt{\frac{N}{2\log_{2}e}}.$
(89)

Thus, we have

$\displaystyle\lim_{\gamma\to\infty}\sqrt{\frac{-\log_{2}\epsilon(\gamma)}{\log_{2}^{2}\gamma}}$
$\displaystyle=\lim_{\gamma\to\infty}\frac{\sqrt{-\log_{2}\epsilon(\gamma)}}{\log_{2}\gamma}$
(90a)
$\displaystyle=\lim_{\gamma\to\infty}\frac{\partial\sqrt{-\log_{2}\epsilon(\gamma)}}{\partial\log_{2}\gamma}$
(90b) $\displaystyle=\varpi\sqrt{\frac{N}{2\log_{2}e}}.$

Eq. (90b) holds according to the L’Hospital rule.

Then, we let $Q^{-1}(\epsilon(\gamma))=x$, which can also be written as
$\epsilon(\gamma)=Q(x)$. Based on Mills’ ratio, for $x>0$, we obtain that

$\frac{1}{\sqrt{2\pi}}\frac{x}{1+x^{2}}e^{-\frac{x^{2}}{2}}\leq\epsilon(\gamma)\leq\frac{1}{\sqrt{2\pi}}\frac{1}{x}e^{-\frac{x^{2}}{2}}.$
(91)

By reformulating
$\epsilon(\gamma)\leq\frac{1}{\sqrt{2\pi}}\frac{1}{x}e^{-\frac{x^{2}}{2}}$, we
obtain

$x^{2}+\ln x^{2}\leq-2\ln\left(\sqrt{2\pi}\epsilon(\gamma)\right).$ (92)

Let us introduce the Lambert W function $W(\cdot)$. $W(y)$ is the solution of
$ze^{z}=y$. Based on $W(\cdot)$, we can transform Eq. (92) into

$x\leq\pm\sqrt{W\left(\frac{\epsilon^{-2}(\gamma)}{2\pi}\right)}.$ (93)

Since $x>0$, we have
$0<x<\sqrt{W\left(\frac{\epsilon^{-2}(\gamma)}{2\pi}\right)}$. Similarly, by
reformulating
$\frac{1}{\sqrt{2\pi}}\frac{x}{1+x^{2}}e^{-\frac{x^{2}}{2}}\leq\epsilon(\gamma)$,
we obtain

$\left(x+\frac{1}{x}\right)^{2}+\ln\left(x+\frac{1}{x}\right)^{2}>x^{2}+\ln\left(x+\frac{1}{x}\right)^{2}\geq-2\ln\left(\sqrt{2\pi}\epsilon(\gamma)\right).$
(94)

Based on $W(\cdot)$, we obtain

$x+\frac{1}{x}\geq\pm\sqrt{W\left(\frac{\epsilon^{-2}(\gamma)}{2\pi}\right)}.$
(95)

Since $x>0$, we will utilize
$x+\frac{1}{x}\geq\sqrt{W\left(\frac{\epsilon^{-2}(\gamma)}{2\pi}\right)}$ in
the following.

Based on Eq. (93), we obtain

$\displaystyle\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\sqrt{N}\varpi\log_{2}\gamma}$
$\displaystyle\leq\lim_{\gamma\to\infty}\frac{\sqrt{W\left(\frac{\epsilon^{-2}(\gamma)}{2\pi}\right)}}{\sqrt{N}\varpi\log_{2}\gamma}$
$\displaystyle=\lim_{\gamma\to\infty}\frac{\sqrt{-2\log_{2}\epsilon(\gamma)}}{\sqrt{N}\varpi\log_{2}\gamma}\cdot\frac{\sqrt{W\left(\frac{\epsilon^{-2}(\gamma)}{2\pi}\right)}}{\sqrt{-2\log_{2}\epsilon(\gamma)}}$
$\displaystyle=\frac{1}{\sqrt{\log_{2}e}}\lim_{\gamma\to\infty}\sqrt{\frac{-2\log_{2}\epsilon(\gamma)}{\log_{2}^{2}\gamma}}\cdot\frac{\log_{2}\gamma}{\sqrt{N}\varpi\log_{2}\gamma}$
(96a) $\displaystyle=\frac{1}{\log_{2}e}.$ (96b)

Eq. (96a) holds because according to [37], the Lambert W function satisfies

$\ln y-\ln\ln y+\frac{\ln\ln y}{2\ln y}\leq W(y)\leq\ln y-\ln\ln
y+\frac{e}{e-1}\frac{\ln\ln y}{\ln y}.$ (97)

Thus, we obtain

$\lim_{y\to\infty}\frac{\sqrt{W(y)}}{\sqrt{\log_{2}y}}=\frac{1}{\sqrt{\log_{2}e}}.$
(98)

Besides, Eq. (96b) holds according to Eq. (90).

Similarly, based on Eq. (95), we obtain

$\displaystyle\lim_{\gamma\to\infty}\frac{Q^{-1}(\epsilon(\gamma))}{\sqrt{N}\varpi\log_{2}\gamma}\geq\lim_{\gamma\to\infty}\frac{\sqrt{W\left(\frac{\epsilon^{-2}(\gamma)}{2\pi}\right)}}{\sqrt{N}\varpi\log_{2}\gamma}-\lim_{\gamma\to\infty}\frac{1}{Q^{-1}(\epsilon(\gamma))\cdot\sqrt{N}\varpi\log_{2}\gamma}=\frac{1}{\log_{2}e}.$
(99)

Eq. (99) holds because with $\gamma\to\infty$,
$Q^{-1}(\epsilon(\gamma))\to\infty$. Thus, according to the squeeze theorem,
we have completed the proof. ∎

Based on Lemma 1.2 and Corollary 1.1, we can then obtain the inequality form
of the gain conversation equation [22] for the wireless systems with random
arrivals in the FBL regime. The result is summarized in Theorem 1.3, which is
identical to Theorem 1.

Theorem 1.3. In the FBL regime with $\gamma\to\infty$,

  * •

for a SISO system in an AWGN channel, we have

$(1-\theta N\log_{2}e)\zeta+\varpi+\tau=1;$ (100)

  * •

for a SISO system in a Rayleigh fading channel, we have

$(1-\theta N\log_{2}e)\zeta+\varpi+\tau\geq\min\left\\{1,\frac{1}{\theta
TN\log_{2}e}\right\\},\text{ and }$ (101) $(1-\theta
N\log_{2}e)\zeta+(1-c)\varpi+\tau\leq\min\left\\{1,\frac{1}{\theta
TN\log_{2}e}\right\\};$ (102)

  * •

for a wireless system with $f(x)$ satisfying Eq. (3), we have

$(1-\theta N\log_{2}e)\zeta+\varpi+\tau\geq\min\left\\{1,\frac{\kappa}{\theta
TN\log_{2}e}\right\\},\text{ and }$ (103) $(1-\theta
N\log_{2}e)\zeta+(1-c)\varpi+\tau\leq\min\left\\{1,\frac{\kappa}{\theta
TN\log_{2}e}\right\\};$ (104)

  * •

for a SISO system in a Nakagami-$m$ fading channel, we have

$(1-\theta N\log_{2}e)\zeta+\varpi+\tau\geq\min\left\\{1,\frac{m}{\theta
TN\log_{2}e}\right\\},\text{ and }$ (105) $(1-\theta
N\log_{2}e)\zeta+(1-c)\varpi+\tau\leq\min\left\\{1,\frac{m}{\theta
TN\log_{2}e}\right\\}.$ (106)

In Eqs. (102), (104), and (106), $c>0$ is an arbitrary constant.

###### Proof:

According to Eqs. (14) and (15), we obtain 333Note that, here $\theta$ is
still a constant. According to Eq. (14), with a given $\theta$ and $\gamma$,
there exists a corresponding $a[n]$ which makes $\alpha_{A}(\theta)$ equals to
$\alpha_{S}(\theta)$. Since $\theta\in(0,\infty)$, all kinds of $a[n]$ can be
involved in this theorem under this setting.

$\ln\delta=-\theta N\Lambda(\gamma)D_{\rm max}.$ (107)

Thus, by taking the derivative on both sides of Eq. (107), we have

$-\frac{1}{D_{\rm
max}}\lim_{\gamma\to\infty}\frac{\partial\log_{2}\delta}{\partial\log_{2}\gamma}=\theta
N\log_{2}e\lim_{\gamma\to\infty}\frac{\partial\Lambda(\gamma)}{\partial\log_{2}\gamma}=\left(\theta
N\log_{2}e\right)\zeta.$ (108)

By substituting Eqs. (88) and (108) into Eqs. (68), (70), and (72), we obtain
the results shown in Theorem 3. ∎

The inequalities given in Theorem 1.3 can be seen as the revised version of
the gain conversation equation in [22]. In contrast to the fixed arrival rate
in [22], we consider the random-arrival case in this paper. Based on Eq. (14)
and with the help of Taylor expansion, $\theta$ can be approximated as
$2(\mu_{s}-\mu_{a})/(\sigma_{s}^{2}+\sigma^{2}_{a})$. With the given $\mu_{s}$
and $\sigma_{s}^{2}$, a smaller $\theta$ indicates a larger $\mu_{a}$ or
larger $\sigma^{2}_{a}$. Therefore, $\theta$ in Eqs. (101)-(106) indicates the
influence from the randomness of the arrival. In summary, Theorem 1.3 presents
the tradeoff between the service-rate gain, reliability gain, and real-time
gain for the wireless systems with random arrivals in the FBL and high-SNR
regime.

###  III-B Low-SNR Slope of EC in the FBL regime

In this subsection, we focus on characterizing the low-SNR slope of the
effective capacity in the FBL regime, which can also be seen as the service-
rate gain in the low-SNR regime. Different from Section III-A, we mainly focus
on a kind of orthogonal code proposed in [38] (Chapter 8.5). We omit the
details of this code. An important conclusion of this code is that the bit
rate (in bits per $N$ channel uses) of this code satisfies

$s[n]\geq\frac{N|h[n]|^{2}\gamma}{2}\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1},$ (109)

with the condition that energy per bit to noise power spectral density ratio
is larger than $4\ln 2$. This lower bound can be used for analyzing the
achievable low-SNR slope in the FBL regime. Before presenting the results, we
define the low-SNR slope of the normalized effective capacity in the FBL
regime as

$s_{0}=\lim_{\gamma\to 0}\frac{\Lambda(\gamma)}{\gamma}.$ (110)

In Lemma 1.3, the achievable $s_{0}$ of the FBL-SISO system in a Rayleigh
fading channel is presented.

Lemma 1.3. Given a $\epsilon$, the low-SNR slope of the normalized effective
capacity for the FBL-SISO system in a Rayleigh fading channel satisfies

$s_{0}\geq\frac{1}{2\left(\ln 2-\frac{\ln\epsilon}{\log_{2}N}\right)}.$ (111)

###### Proof:

By substituting $f(x)=e^{-x}$, $x>0$ and Eq. (109) into Eq. (12), we obtain

$\displaystyle\Lambda(\gamma)$ $\displaystyle\geq-\frac{1}{\theta
TN}\ln\int_{0}^{\infty}e^{-\frac{N\theta T\gamma\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}x}{2}}e^{-x}{\rm d}x$ (112)
$\displaystyle=\frac{1}{\theta TN}\ln\left(1+\frac{N\theta T\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2}\gamma\right).$

We then obtain

$\displaystyle\lim_{\gamma\to 0}\frac{\Lambda(\gamma)}{\gamma}$
$\displaystyle\geq\lim_{\gamma\to 0}\frac{\frac{1}{\theta
TN}\ln\left(1+\frac{N\theta T\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2}\gamma\right)}{\gamma}$ (113)
$\displaystyle=\lim_{\gamma\to 0}\frac{\frac{\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2}\gamma}{\gamma}$
$\displaystyle=\frac{1}{2\left(\ln 2-\frac{\ln\epsilon}{\log_{2}N}\right)}.$

Now we have completed the proof. ∎

In Lemma 1.3, we analyze $s_{0}$ in the FBL-SISO system with a Rayleigh fading
channel based on the conclusions in [38]. By comparing Lemma 1.1 and Lemma
1.3, we find that the lower bound of $s_{0}$ is not related to $\theta$, while
we prove that $s_{\infty}$ is related to $\theta$. This is because with low
$\gamma$, $\ln\mathbb{E}\\{e^{-\theta Ts[n]}\\}$ acts as a linear function of
$\theta Ts[n]$. Besides, we also find that $s_{0}$ is related to $\epsilon$,
while $s_{\infty}$ is not related to $\epsilon$. This is because for a given
$\epsilon$, $s[n]$ approaches the Shannon capacity with $\gamma\to\infty$,
while $s[n]$ is significantly influenced by $\epsilon$ with $\gamma\to 0$.
Based on Lemma 1.3, we generalize the conclusion to other wireless systems in
Theorem 1.4.

Theorem 1.4. With a given $\epsilon$,

  * •

for a system with $f(x)$ satisfying Eq. (3) in the FBL regime, we have

$s_{0}\geq\frac{\kappa}{2\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)};\vspace{-1mm}$ (114)

  * •

for a FBL-SISO system in a Nakagami-$m$ fading channel, we have

$s_{0}\geq\frac{\Omega}{2\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)}.\vspace{-1mm}$ (115)

###### Proof:

By substituting Eqs. (3) and (109) into Eq. (12), we obtain

$\displaystyle\Lambda(\gamma)$ $\displaystyle\geq-\frac{1}{\theta
TN}\ln\int_{0}^{\infty}\frac{e^{-\frac{N\theta T\gamma\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}x}{2}}x^{\kappa-1}e^{-x}}{(\kappa-1)!}{\rm
d}x$ (116) $\displaystyle=-\frac{1}{\theta
TN}\ln\left[\frac{\Gamma(\kappa)}{(\kappa-1)!}\left(1+\frac{N\theta
T\gamma\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2}\right)^{-\kappa}\right]$
$\displaystyle=\frac{\kappa}{\theta TN}\ln\left(1+\frac{N\theta
T\gamma\left(\ln 2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2}\right).$

We then obtain

$\displaystyle\lim_{\gamma\to 0}\frac{\Lambda(\gamma)}{\gamma}$
$\displaystyle\geq\lim_{\gamma\to 0}\frac{\frac{\kappa}{\theta
TN}\ln\left(1+\frac{N\theta T\gamma\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2}\right)}{\gamma}$ (117)
$\displaystyle=\lim_{\gamma\to 0}\frac{\frac{\kappa\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2}\gamma}{\gamma}$
$\displaystyle=\frac{\kappa}{2\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)}.$

By substituting Eqs. (4) and (109) into Eq. (12), we obtain

$\displaystyle\Lambda(\gamma)$ $\displaystyle\geq-\frac{1}{\theta
TN}\ln\int_{0}^{\infty}\frac{e^{-\frac{N\theta T\gamma\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}x}{2}}m^{m}x^{m-1}e^{-\frac{m}{\Omega}x}}{\Omega^{m}\Gamma(m)}{\rm
d}x$ (118) $\displaystyle=\frac{m}{\theta TN}\ln\left(1+\frac{\Omega N\theta
T\gamma\left(\ln 2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2m}\right).$

We then obtain

$\displaystyle\lim_{\gamma\to 0}\frac{\Lambda(\gamma)}{\gamma}$
$\displaystyle\geq\lim_{\gamma\to 0}\frac{\frac{m}{\theta
TN}\ln\left(1+\frac{\Omega N\theta T\gamma\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)^{-1}}{2m}\right)}{\gamma}$ (119)
$\displaystyle=\frac{\Omega}{2\left(\ln
2-\frac{\ln\epsilon}{\log_{2}N}\right)}.$

We have completed the proof. ∎

Theorem 1.4 presents the low-SNR EC slope for systems with frequency or
spatial diversity and Nakagami-$m$ fading channels. From Theorem 1.4, we find
that, for this kind of orthogonal code, $\mathbb{E}\\{|h[n]|^{2}\\}$ has an
influence on the lower bound of $s_{0}$. It is important to obtain the precise
value of $s_{0}$ to provide instructions for practical applications. We will
focus on this part of the work in the future.

###  III-C Effective Capacity Approximation for Low-Latency Systems

As shown in Eqs. (13) and (15), there are some limits on the use of LDT to
determine the QVP and DVP in URLLC scenarios. However, EC is widely used in
the works for URLLC scenarios [27, 28, 25, 26] to evaluate the QoS-constrained
throughput. In these works, the value of the QoS exponent $\theta^{*}$ is
given to represent the QoS requirement, while the EC is adopted to
characterize the throughput under the QoS requirement. Some scheduling schemes
in URLLC were proposed to maximize the EC in these works. Thus, it is
important to derive the EC for further optimization. However, it is not
trivial to derive the closed-form expression of the EC, which brings
difficulty to the throughput analysis and the comparison between different
scheduling schemes. To solve this problem, we propose a Laplace’s-method-based
approximation for EC in this subsection. In this subsection, we set $T\geq 1$,
which covers the general low-latency scenarios.

For low-latency scenarios, the threshold $L$ and $D_{\rm max}$ is usually
small due to the stringent QoS requirement [23]. Thus, to achieve an extremely
small QVP or DVP, the value of the QoS exponent $\theta^{*}$ needs to be
large. By this means, $\theta TN$ should have a large value. Based on this
thought, we will start our analysis on EC from the fixed-power allocation in
Lemma 2.1. For convenience, let $\ell$ denote
$\left(\frac{Q^{-1}(\epsilon)}{\sqrt{N}}\right)^{2}$.

Lemma 2.1. For a low-latency system adopting fixed-power allocation, the EC is
approximated as

$\alpha_{S}(\theta)\approx Ns(x^{*})-\frac{\ln f(x^{*})+\frac{1}{2}\ln
2\pi-\frac{1}{2}\ln|s^{\prime\prime}(x^{*})|-\frac{1}{2}\ln\theta TN}{\theta
T},$ (120)

where $x^{*}$ is given by

$x^{*}=\frac{1}{\gamma}\left(\sqrt{\frac{1+\sqrt{1+4\ell}}{2}}-1\right).$
(121)

###### Proof:

See Appendix A. ∎

In Lemma 2.1, we give an approximation of EC for the fixed-power allocation,
which provides a closed-form expression of EC with arbitrary fading channels.
Besides, the approximation error is $O((\theta TN)^{-1})$. Thus, as we
mentioned before Lemma 2.1, in a low-latency system this approximation has a
better performance, which we will validate by numerical calculation in Section
V. Based on Lemma 2.1, we can then apply the approximation to more general
power allocation schemes. We summarize this generalization in Theorem 2.1,
which is identical to Theorem 2.

Theorem 2.1. For a low-latency system which adopts a channel-gain-based power
allocation scheme $\Xi(x)$ satisfying $\Xi[x]+\frac{{\rm d}\Xi(x)}{{\rm
d}x}x\geq 0$, EC is approximated as

$\alpha_{S}(\theta)\approx Ns(x^{*})-\frac{\ln f(x^{*})+\frac{1}{2}\ln
2\pi-\frac{1}{2}\ln|s^{\prime\prime}(x^{*})|-\frac{1}{2}\ln\theta TN}{\theta
T},$ (122)

where $x^{*}$ is given by

$\Xi(x^{*})x^{*}=\sqrt{\frac{1+\sqrt{1+4\ell}}{2}}-1.$ (123)

###### Proof:

See Appendix B. ∎

In Theorem 2.1, we propose the EC approximation for a specific kind of power
allocation schemes. For the policies satisfying $\frac{{\rm d}\Xi(x)}{{\rm
d}x}\geq 0$, $\Xi[x]+\frac{{\rm d}\Xi(x)}{{\rm d}x}x\geq 0$ holds. $\frac{{\rm
d}\Xi(x)}{{\rm d}x}\geq 0$ means that the allocated power does not decrease
with the increasing channel power gain. This condition can be met with some
power allocation schemes, among which one of the famous schemes is the water-
filling scheme [39]. Besides, with the condition that $\frac{{\rm
d}\Xi(x)}{{\rm d}x}\geq 0$, $\Xi(x)x$ is a monotonic function of $x$. Thus,
the solution of Eq. (23) can be obtained by the binary research directly.

Note that, though the approximation shown in Theorem 2.1 can be applied to the
policies satisfying $\Xi[x]+\frac{{\rm d}\Xi(x)}{{\rm d}x}x\geq 0$, the
accuracy of the approximation can be further improved by further optimization
according to a certain policy’s property. For example, in the water-filling
scheme, there is a channel-gain threshold $h_{\rm th}$. If the channel gain is
below this threshold, the allocated power is 0, which means the transmission
rate is approximately a constant $\frac{\log_{2}(N)}{2}$ in the FBL regime
when $h[n]<h_{\rm th}$ according to Eq. (7). Let $\xi(\theta)=e^{-\theta
T\frac{\log_{2}(N)}{2}}\int_{0}^{h_{\rm th}}f(x){\rm d}x$. According to the
above analysis and Theorem 2.1, EC can be approximated as

$\alpha_{S}(\theta)\approx-\frac{1}{\theta
N}\ln\left(\xi(\theta)+\sqrt{\frac{2\pi}{\theta
TN|s^{\prime\prime}(x^{*})|}}f(x^{*})e^{-\theta TNs(x^{*})}\right).$ (124)

For those policies which have the truncated properties, Eq. (124) is a better
choice for their EC approximation.

For the low-latency system with the retransmission mechanism, the EC is
discussed in [29], which is different from the above analysis without the
retransmission mechanism. In these works, a simple ARQ mechanism is assumed to
be adopted. With this mechanism, the receiver will send a negative
acknowledgment requesting the retransmission of the message in case of an
erroneous reception. In Corollary 2.1, we generalize the EC approximation to
the low-latency system with this ARQ mechanism.

Corollary 2.1. For a low-latency system with the above ARQ mechanism and a
channel-gain-based power allocation scheme $\Xi(x)$ satisfying
$\Xi[x]+\frac{{\rm d}\Xi(x)}{{\rm d}x}x\geq 0$, EC is approximated as

$\alpha_{S}(\theta)\approx-\frac{1}{\theta
T}\ln\left(\epsilon+(1-\epsilon)\sqrt{\frac{2\pi}{\theta
TN|s^{\prime\prime}(x^{*})|}}f(x^{*})e^{-\theta TNs(x^{*})}\right),$ (125)

where $x^{*}$ is given by Eq. (23).

###### Proof:

See Appendix C. ∎

##  IV Reliability-Latency tradeoff in the finite-blocklength regime with
$T=1$

In this section, we will conduct the QoS analysis for the low-latency system
over the AWGN and fading channel in the FBL regime with $T=1$. In Section
IV-A, we focus our attention on the AWGN channel in the FBL regime, which
models wired communications and many satellite communications. To get more
details of the QoS performance in fading channels, we propose to characterize
the QoS exponent in the Rayleigh fading channel by obtaining analytical
expressions of its upper and lower bound in Section IV-B. Moreover, the
relationships between $\epsilon$, $\chi$, and $\delta$ are obtained through an
asymptotic analysis.

###  IV-A Reliability-DVP tradeoff in AWGN channels

In this subsection, we focus our attention on the AWGN channel in the FBL
regime, which models wired communications and many satellite communications.
By using the EB and normal approximation, we analyze the QoS performance of
the AWGN channel model in the FBL regime. The closed-form upper and lower
bounds on the error probability are obtained. Based on the bounds, we further
obtain the asymptotic reliability-DVP tradeoff in the high SNR regime. Since
in this section we focus on the low-latency systems with $T=1$, the frame and
time slot are equivalent.

Since the service rate in the AWGN channel is fixed with the given $\epsilon$,
$\gamma$, and $N$, the QVP and DVP are equivalent in the AWGN channel. Thus in
this section, we will put our focus on characterizing the relationship between
the error probability $\epsilon$ and the QVP $\chi$, which also reflects the
tradeoff between the reliability and DVP. For the AWGN channel in the FBL
regime without the retransmission mechanism, we first present the relationship
between $\epsilon$ and $\chi$ in Lemma 3.1.

Lemma 3.1. For a low-latency system with the $i.i.d.$ $a[n]$ in the AWGN
channel, $\epsilon\in(0,0.5)$ satisfies

$\ln\\!\left(\\!\frac{\psi+\frac{\Phi}{L}\ln\chi}{\sqrt{2\pi}\left(1+(\psi+\frac{\Phi}{L}\ln\chi)^{2}\right)}\\!\right)-\frac{\left(\psi+\frac{\Phi}{L}\ln\chi\right)^{2}}{2}\\!\\!\leq\\!\ln\epsilon\\!\leq\\!-\\!\ln\\!\left(\\!\sqrt{2\pi}\left(\\!\psi\\!+\\!\frac{\Phi}{L}\\!\ln\chi\right)\\!\right)-\frac{\left(\psi\\!+\\!\frac{\Phi}{L}\ln\chi\right)^{2}}{2},$
(126)

where $\psi$ and $\Phi$ are given by

$\quad\,\,\psi=\frac{N\log(1+\gamma)-\mu_{a}+\frac{\log_{2}N}{2}}{\sqrt{N\left(1-\frac{1}{(1+\gamma)^{2}}\right)}\log_{2}e},\,\text{and}$
(127)
$\Phi=\frac{\sigma_{a}^{2}}{2\sqrt{N\left(1-\frac{1}{(1+\gamma)^{2}}\right)}\log_{2}e}.$
(128)

###### Proof:

See Appendix D. ∎

From Lemma 3.1, we can approximate the error probability by its upper and
lower bounds in Eq. (126) to achieve a certain $\chi$. Let $t(x)=-\ln
x-\frac{x^{2}}{2}$. We can find that $\frac{{\rm d}t(x)}{{\rm d}x}<0$ for
$x>0$. This result indicates that the upper bound of $\ln\epsilon$ is a
decreasing function of $\ln\chi$. Therefore, with a looser requirement on QoS,
i.e., a larger target $\chi$, the upper bound in Eq. (126) indicates that the
$\epsilon$ can be smaller. This is reasonable since a smaller $\epsilon$
results in a smaller transmission rate $R_{\epsilon}$. From this result, we
can see that there is a tradeoff between the reliability and the QoS
performance. That is, higher reliability (smaller $\epsilon$) results in
poorer QoS performance (larger $\chi$). Note that, this conclusion is only
valid in the system without the retransmission mechanism. In the next section,
we will give a comparison between the systems with and without the
retransmission mechanism on EC based on the EC approximation.

From Eqs. (25) and (26), we see that $\psi$ is related to the mean of the
arrival data, while $\Phi$ is related to the variance of the arrival data.
Thus, we call $\psi$ the load-related factor, and call $\Phi$ the burstiness-
related factor. These two factors reflect the influence of the statistical
properties of the source data on the system’s reliability-DVP tradeoff.

To further explore the asymptotic relationship between $\epsilon$ and $\chi$,
we give the conclusion in the low $\epsilon$ regime in Theorem 3.1, which is
identical to the conclusions of AWGN channel in Theorem 3.

Theorem 3.1. For a low-latency system with the $i.i.d.$ $a[n]$ in the AWGN
channel, the tradeoff between $\epsilon$ and $\chi$ in the low $\epsilon$
regime is approximated as

$\ln\epsilon\approx-\ln\left(\psi+\frac{\Phi}{L}\ln\chi\right)-\frac{\left(\psi+\frac{\Phi}{L}\ln\chi\right)^{2}}{2}-\frac{1}{2}\ln(2\pi).$
(129)

###### Proof:

See Appendix E. ∎

From Theorem 3.1, we obtain the closed-form tradeoff between $\epsilon$ and
$\chi$ over AWGN channels in the FBL regime with a small $\epsilon$. According
to Eq. (129), by deriving the second-order derivative of $\ln\epsilon$, we
find that for
$\ln\chi\in\left(-\min\left\\{\frac{L(1-\psi)}{\Phi},0\right\\},0\right)$,
$\ln\epsilon$ is a concave function of $\ln\chi$. Determining the concavity of
the functional relationship between $\ln\epsilon$ and $\ln\chi$ may bring
benefits for further optimization in this system. Besides, according to Eq.
(156), a small $\epsilon$ indicates a large $\psi+\frac{\Phi}{L}\ln\chi$.
Since $\lim_{x\to\infty}\frac{\ln x}{x^{2}}=0$,
$-\frac{\left(\psi+\frac{\Phi}{L}\ln\chi\right)^{2}}{2}$ is the dominant term
at the right side of Eq. (129) in the low $\epsilon$ regime.

By considering the right-hand term of Eq. (129) as the function $H(\ln\chi)$,
we obtain

$\frac{{\rm d}H(\ln\chi)}{{\rm
d}\ln\chi}=-\frac{\Phi}{L}\left(\psi+\frac{\Phi}{L}\ln\chi+\frac{1}{\psi+\frac{\Phi}{L}\ln\chi}\right).$
(130)

From this equation, we find that the burstiness-related factor $\Phi$ has a
main influence on the decreasing rate of $\ln\epsilon$ when $\ln\chi$
increases. With the same load-related factor, a higher variance of the arrival
data results in a larger decreasing rate of $\ln\epsilon$ with $\ln\chi$
increasing. This result shows that the higher variance of the arrival data
degrades the system’s performance on DVP with the given requirement on
reliability.

###  IV-B QoS exponent and reliability-DVP tradeoff in Rayleigh fading
channels

In this subsection, we try to generalize the analysis in AWGN channels as
shown in Section IV-A to fading channels. However, it is not trivial to obtain
the closed-form expression of $\theta^{*}$ in fading channels. Thus, in this
subsection, we first present the upper and lower bounds on $\theta^{*}$, which
are tight in the high SNR regime. Based on this result, the relationship
between reliability and QoS performance is then proposed.

In fading channels and for given $\gamma$, $N$, and $\epsilon$, the coding
rate is a random variable as shown in Eq. (7). Therefore, $s[n]$ is not a
constant in this case. Due to the dispersion term in Eq. (7), it is not
trivial to give the closed-form expression of the QoS exponent. As a result,
it is difficult to characterize the tradeoff between $\epsilon$ and $\chi$.
Lemma 3.2 gives one method to analyze the QoS exponent in Rayleigh fading
channels by characterizing its upper and lower bounds.

For convenience, we give the definition of the notations which will be used in
Lemma 1.1. Let $\varrho=N\log\left(1+|h[n]|^{2}\gamma\right)$,
$\omega=\sqrt{N}Q^{-1}(\epsilon)\log_{2}e$,
$\vartheta=1-\frac{1}{1+|h[n]|^{2}\gamma}$, and
$\Theta=\sqrt{1-\frac{1}{(1+|h[n]|^{2}\gamma)^{2}}}$. Then, let
$\lambda=\mathbb{E}\\{\vartheta^{2}\\}$, $\beta=\mathbb{E}\\{\vartheta\\}$,
$\rho=\mathbb{E}\\{\vartheta\varrho\\}$, $\iota=\mathbb{E}\\{\varrho\\}$, and
$\varphi=\mathbb{E}\\{\varrho^{2}\\}$. Then, we can obtain the following
lemma.

Lemma 3.2. For a low-latency system with $i.i.d.$ $a[n]$ in the Rayleigh
fading channel, if
$\omega\in\bigg{[}0,\min\Big{\\{}\frac{2(\rho-\iota\beta)}{\lambda-\beta^{2}},\frac{-2\big{(}{\rm
Cov}(\varrho,\vartheta)-{\rm Cov}(\varrho,\Theta)\big{)}}{{\rm
Var}(\Theta)-{\rm Var}(\vartheta)},\frac{2{\rm Cov}(\varrho,\Theta)}{{\rm
Var}(\Theta)}\Big{\\}}\bigg{]}$, the QoS exponent $\theta^{*}$ satisfies the
following expression:

$\theta_{1}^{*}\leq\theta^{*}\leq\theta_{2}^{*},$ (131)

where $\theta_{1}^{*}$ and $\theta_{2}^{*}$ are given by

$\displaystyle\theta_{1}^{*}=$ $\displaystyle\frac{-2\left(-\frac{N}{\ln
2}e^{\frac{1}{\gamma}}{\rm
E}_{1}\left(\frac{1}{\gamma}\right)+\sqrt{N}Q^{-1}(\epsilon)\log
e+\mu_{a}-\frac{\log_{2}N}{2}\right)}{\frac{e^{\frac{1}{\gamma}}N^{2}}{(\ln
2)^{2}}\left(2G_{2,3}^{3,0}\Bigg{(}\begin{array}[]{c}1,1\\\
0,0,0\end{array}\Bigg{|}\frac{1}{\gamma}\Bigg{)}-e^{\frac{1}{\gamma}}{\rm
E}_{1}^{2}\left(\frac{1}{\gamma}\right)\right)+\sigma_{a}^{2}},\,\text{and}$
(134) $\displaystyle\theta_{2}^{*}=$
$\displaystyle-\frac{2}{\varkappa}\left(-\frac{N}{\ln
2}e^{\frac{1}{\gamma}}{\rm
E}_{1}\left(\frac{1}{\gamma}\right)+\left(1-\frac{e^{\frac{1}{\gamma}}{\rm
E}_{1}\left(\frac{1}{\gamma}\right)}{\gamma}\right)\sqrt{N}Q^{-1}(\epsilon)\log
e+\mu_{a}-\frac{\log_{2}N}{2}\right),$ (135)

with $\varkappa$ given by

$\displaystyle\varkappa$
$\displaystyle=\left[1+\frac{1}{\gamma}-\frac{e^{\frac{1}{\gamma}}}{\gamma^{2}}{\rm
E}_{1}\left(\frac{1}{\gamma}\right)(1-2\gamma)\right]\left(\sqrt{N}Q^{-1}(\epsilon)\log
e\right)^{2}+\frac{2Ne^{\frac{1}{\gamma}}}{\ln 2}\Bigg{[}\gamma
G_{2,3}^{3,0}\Bigg{(}\begin{array}[]{c}3,3\\\
2,2,2\end{array}\Bigg{|}\frac{1}{\gamma}\Bigg{)}-\frac{e^{\frac{1}{\gamma}}}{\gamma}$
(138) $\displaystyle{\rm
E}_{1}^{2}\left(\frac{1}{\gamma}\right)\Bigg{]}\sqrt{N}Q^{-1}(\epsilon)\log
e+\frac{e^{\frac{1}{\gamma}}N^{2}}{(\ln
2)^{2}}\left[2G_{2,3}^{3,0}\left(\begin{array}[]{c}1,1\\\
0,0,0\end{array}\Bigg{|}\frac{1}{\gamma}\right)\\!-\\!e^{\frac{1}{\gamma}}{\rm
E}_{1}^{2}\left(\frac{1}{\gamma}\right)\right]+\sigma_{a}^{2}.$ (141)

When $\gamma\to\infty$, the ratio of $\theta_{1}^{*}$ to $\theta_{2}^{*}$
approaches 1.

###### Proof:

See Appendix F. ∎

In Lemma 3.2, we obtain the analytical expressions of the upper and lower
bounds on the QoS exponent in Rayleigh fading channels, which can help us
characterize the properties of the exact QoS exponent. Note that, according to
the numerical results shown in Fig. 12, we find that the minimum value of the
right end of the feasible interval of $\omega$ over all $\gamma$ is about
$5N$. For an extreme condition in which $w=5N$ and $N=100$, $\epsilon\approx
10^{-268}$. This means that with $N=100$, if $w>500$, $\epsilon$ needs to be
less than $10^{-268}$. Thus, in practice the relationship
$\theta_{1}^{*}\leq\theta^{*}\leq\theta_{2}^{*}$ can be considered as always
true. Based on Lemma 3.2, we present the $\epsilon-\chi$ relationship by
inequalities in Lemma 3.3.

Lemma 3.3. For a low-latency system with $i.i.d.$ $a[n]$ over a Rayleigh
fading channel, $\epsilon\in(0,0.5)$ satisfies

$\ln\epsilon\geq\ln\left(\frac{\Upsilon+\frac{\varsigma}{L}\ln\chi}{\sqrt{2\pi}\left(1+\left(\Upsilon+\frac{\varsigma}{L}\ln\chi\right)^{2}\right)}\right)-\frac{1}{2}\left(\Upsilon+\frac{\varsigma}{L}\ln\chi\right)^{2},\text{
and}$ (142)
$\ln\epsilon\leq-\ln\left(\sqrt{2\pi}\left(\Upsilon+\frac{\varsigma}{L}\ln\chi\right)\right)-\frac{1}{2}\left(\Upsilon+\frac{\varsigma}{L}\ln\chi\right)^{2},$
(143)

where $\varsigma$ and $\Upsilon$ are given by

$\Upsilon=-\frac{\frac{N}{\ln 2}\cdot e^{\frac{1}{\gamma}}{\rm
Ei}\left(\frac{-1}{\gamma}\right)+\mu_{a}-\frac{\log_{2}N}{2}}{\sqrt{N}\log
e},\,\text{and}$ (144) $\varsigma=\frac{\frac{e^{\frac{1}{\gamma}}N^{2}}{(\ln
2)^{2}}\left[2G_{2,3}^{3,0}\left(\begin{array}[]{c}1,1\\\
0,0,0\end{array}\Bigg{|}\frac{1}{\gamma}\right)-e^{\frac{1}{\gamma}}{\rm
Ei}^{2}\left(\frac{-1}{\gamma}\right)\right]+\sigma_{a}^{2}}{2\sqrt{N}\log
e}.$ (145)

###### Proof:

See Appendix G. ∎

By comparing Eqs. (126) and (143), we find that the $\epsilon-\chi$ tradeoff
has a similar expression in the AWGN and fading channels. $\Upsilon$ and
$\varsigma$ can also be regarded as the load-related and bustiness-related
parameters, respectively. Besides, both $\Phi$ and $\varsigma$ are larger than
$\frac{\sigma_{a}^{2}}{2\sqrt{N}\log_{2}e}$, which is the minimum burstiness-
related parameter without the influence of channel dispersion and randomness
of the service rate. However, there is differences between the $\epsilon-\chi$
tradeoff of AWGN channels and that of fading channels. In the AWGN channel,
with $\gamma$ increasing, $\psi$ becomes larger while $\Phi$ decreases. In the
Rayleigh fading channel, according to Eqs. (144) and (145), with a large
$\gamma$, $\varsigma$ and $\Upsilon$ both become large. This difference
results from the influence of the randomness of the service rate, while in the
AWGN channel the service rate is fixed.

Similar to Theorem 3.1, there is an asymptotic expression of the
$\epsilon-\chi$ tradeoff in the low $\epsilon$ regime, which we present in
Theorem 3.2. Note that the conclusions for the Rayleigh fading channel in
Theorem 3 are identical to Theorem 3.2.

Theorem 3.2. For a low-latency system with $i.i.d.$ $a[n]$ over a Rayleigh
fading channel in the high SNR regime, the tradeoff between $\epsilon$ and
$\chi$ is approximated as

$\ln\epsilon\approx-\ln\left(\Upsilon+\frac{\varsigma}{L}\ln\chi\right)-\frac{1}{2}\left(\Upsilon+\frac{\varsigma}{L}\ln\chi\right)^{2}-\frac{1}{2}\ln(2\pi).$
(146)

Since the transmission rate is a random variable in the fading channel, the
QVP is not equivalent to the DVP. Thus, based on Lemma 3.2, Lemma 3.3, and Eq.
(15), we further present the relationship between $\epsilon$ and $\delta$ in
Corollary 3.1.

Corollary 3.1. For a low-latency system with $i.i.d.$ $a[n]$ over a Rayleigh
fading channel in the high SNR regime, $\epsilon\in(0,0.5)$ and $\delta$
satisfy

$\ln\epsilon\leq-\ln\left(\Upsilon+\frac{\nu}{D_{\rm
max}}\ln\delta\right)-\frac{1}{2}\left(\Upsilon+\frac{\nu}{D_{\rm
max}}\ln\delta\right)^{2}-\frac{1}{2}\ln 2\pi,$ (147)

where $\nu$ is given by

$\nu=\frac{\frac{e^{\frac{1}{\gamma}}N^{2}}{(\ln
2)^{2}}\left[2G_{2,3}^{3,0}\left(\begin{array}[]{c}1,1\\\
0,0,0\end{array}\Bigg{|}\frac{1}{\gamma}\right)-e^{\frac{1}{\gamma}}{\rm
Ei}^{2}\left(\frac{-1}{\gamma}\right)\right]+\sigma_{a}^{2}}{2\mu_{a}\sqrt{N}\log
e}.$ (148)

###### Proof:

See Appendix E. ∎

_Remark 1: In Section IV, we characterize the tradeoff between the reliability
and the QoS performance of the system based on Eqs. ( 13) and (15). However,
as we mentioned in Section II, these approximations of $\chi$ and $\delta$ are
more accurate with large thresholds, which limits the use of these
approximations in low-latency scenarios with $T>1$. To solve this problem, the
authors of [23] proved Eqs. (13) and (15) are the upper bounds on the QVP and
DVP with some kinds of arrival process even if the threshold is not large and
$T>1$. This result indicates that the upper bounds on $\ln\epsilon$ in our
analysis are applicable in low-latency scenarios with $T>1$. To better
characterize the tradeoff between reliability and QoS performance in general
low-latency scenarios, more efforts have to be made to obtain accurate
approximations on QVP and DVP with small thresholds._

##  V Simulation and Numerical Results

In this section, we conduct Monte Carlo simulations and numerical calculations
to validate the theoretical results and conclusions in this paper. Moreover,
some characteristics of the statistical-delay performance of low-latency
systems in the FBL regime will be explained more visually by the simulation
and numerical results.

Figure 2: $s_{\infty}$ in the Rayleigh fading channel.

Figure 3: $s_{\infty}$ in the fading channel with $f(x)$ satisfying Eq. (3)
and $\kappa=2$.

(a) $s_{\infty}$ in the Nakagami-$m$ channel with $m=0.5$. (b) $s_{\infty}$ in
the Nakagami-$m$ channel with $m=2$.

Figure 4: $s_{\infty}$ in the Nakagami-$m$ fading channel.

First, we validate the conclusions of high-SNR slopes of EC in the FBL regime
in Figs. 3, 3, and 4. In these figures, we set $N=512$, $\epsilon=10^{-5}$,
and $T=1$. In Fig. 3, $\mathbb{E}\\{|h[n]|^{2}\\}=1$. In Fig. 3, $f(x)$
satisfies Eq. (3) with $\kappa=2$. In Fig. 4, $\Omega=1$. For the Rayleigh
fading case, $\frac{1}{TN\log_{2}e}\approx 0.0014$. From Fig. 3, we find that
the slopes of $\Lambda(\gamma)$ for $\theta\leq 0.0014$ are nearly identical
and close to 1, while the slopes of $\Lambda(\gamma)$ for $\theta>0.0014$ are
obviously smaller than 1. Besides, for those $\theta>0.0014$, a larger
$\theta$ results in a smaller slope of $\Lambda(\gamma)$. The slope of
$\Lambda(\gamma)$ for $\theta=0.004$ is 0.338 in Fig. 3, while
$\frac{1}{\theta N\log_{2}e}$ with $\theta=0.004$ is 0.3385. The above results
validate the accuracy of Lemma 1.1. Then we analyze the numerical results of
the case in which $f(x)$ satisfies Eq. (3). In Fig. 3,
$\frac{\kappa}{TN\log_{2}e}\approx 0.0028$. We can find that $\theta=0.002$ is
larger than $\frac{1}{TN\log_{2}e}$ in Fig. 3 while it is smaller than
$\frac{\kappa}{TN\log_{2}e}=0.0028$ in Fig. 3. Accordingly, the slope of
$\Lambda(\gamma)$ for $\theta=0.002$ in Fig. 3 is obviously different from the
slope of ergodic capacity, while the slope of $\Lambda(\gamma)$ for
$\theta=0.002$ is nearly identical to the slope of ergodic capacity in Fig. 3.
This result reveals the difference between the performance of $s_{\infty}$
with and without the spatial or frequency diversity in Rayleigh fading
channels, which also validates Theorem 1.1. In Fig. 4, we present the
numerical results of $s_{\infty}$ in the Nakagami-$m$ fading channel. In Fig.
4(a), $\frac{m}{TN\log_{2}e}\approx 0.00067$, while in Fig. 4(b),
$\frac{m}{TN\log_{2}e}\approx 0.0028$. The slopes of $\Lambda(\gamma)$ in
Figs. 4(a) and 4(b) validates the accuracy of Theorem 1.2. Besides, by
comparing the results in Figs. 3 and 4(b), we find that the slopes of
$\Lambda(\gamma)$ is nearly identical with a same $\theta$. For example, when
$\theta=0.004$, the slope of $\Lambda(\gamma)$ in Fig. 3 is 0.6731, while the
slope of $\Lambda(\gamma)$ in Fig. 4(b) is 0.6704. This result also validates
the accuracy of Theorems 1.1 and 1.2, since $s_{\infty}$ with $\kappa=2$ and
$m=2$ are identical as shown in Theorems 1.1 and 1.2.

Figure 5: EC approximation for the fixed-power allocation scheme.

Figure 6: EC approximation for the water-filling policy.

Furthermore, we validate the accuracy of the approximation for EC by numerical
calculations. First, we present the results of the EC approximation with the
fixed-power allocation in Fig. 5. In this figure, $\gamma=10$ and
$\epsilon=10^{-3}$. The channel model is set as the Rayleigh fading channel
with $\mathbb{E}\\{|h[n]|^{2}\\}=1$. The numerical result in this figure is
calculated based on the definition of the EC in Eq. (10), while the
approximation result is obtained from Eq. (120). From Fig. 5, we find that the
approximation results match well with the numerical results. The difference
between the numerical and approximation results with $N=100$ and $T=1$ is
slightly more evident when $\theta$ is small compared to the other three
pairs. This is reasonable since the approximation error of Laplace’s method is
$O((\theta TN)^{-1})$. From these results, we see that the approximation in
Lemma 2.1 satisfactorily approximates the EC in the low-latency scenarios.

Fig. 6 presents the result of EC approximation with the water-filling (WF)
policy [39]. To test the performance of the EC approximation in different
channel models, in Fig. 6 the channel gain satisfies
$|h[n]|^{2}=|h_{1}[n]|^{2}+|h_{2}[n]|^{2}$, where $|h_{i}[n]|^{2}$ follows an
exponential distribution with expectation 1. The approximation for the WF
policy we adopted here is based on Eq. (124). From Fig. 6, we can see that the
approximation and numerical results of the WF policy match well, which
validates our analysis in Section III-C. Note that, for the water-filling
policy with the average SNR $\mathbb{E}\\{\gamma\\}=10$,
$\mathbb{E}\\{s[n]\\}\approx 3932$, while for the fixed-power allocation
policy with $\gamma=10$, $\mathbb{E}\\{s[n]\\}\approx 3923$. This result
indicates that the transmission optimization based on the infinite blocklength
assumption can provide help for improving the average transmission rate in the
FBL regime under some parameter settings. However, the result in Fig. 6 shows
that, though the WF policy has a larger average transmission rate compared
with the fixed-power allocation scheme, it does not improve the EC. Since the
EC represents the maximum throughput under the QVP or DVP constraints, the
poorer performance of the WF policy on the EC demonstrates that there is a
difference between maximizing the average transmission rate and maximizing the
throughput under statistical QoS constraints.

Then, we verify the accuracy of the approximation on the QoS exponent and QVP
in Eqs. (13), (155), and (163). In Fig. 7, we present the simulation and
numerical results on QVP with the Poisson arrival. The numerical results are
calculated according to Eq. (14), while the approximation results are obtained
based on Eq. (155) in the AWGN channel and Eq. (163) in the Rayleigh fading
channel. By comparing the slope of the curves of the simulation and numerical
results, we find that QVP has a linear-decay-rate exponent which is
approximately equal to the QoS exponent proposed in Eq. (13). Besides, from
Fig. 7, we observe that with higher load, i.e., higher $\mu_{a}$, the
intercepts of the numerical results approach 0. This result validates that
$\eta\to 1$ with the high load. The above results show the accuracy of the
approximation on QVP in Eq. (13). Moreover, by comparing the slopes of the
curves of the simulation and approximation results, we find that the
approximation in Eqs. (155) and (163) is relatively accurate with the high
load. With the load decreasing, the approximation becomes inaccurate. Thus,
the approximation proposed in [15] can be applied in the high-load scenario,
which is consistent with the settings of our system model.

(a) QVP in the AWGN channel with different $\mu_{a}$. (b) QVP in the Rayleigh
fading channel with different $\mu_{a}$.

Figure 7: Simulation and numerical results on QVP with $T=1$, $N=100$,
$\epsilon=10^{-3}$ and $\gamma=10$.

Figure 8: $\log_{10}\epsilon$ versus $\log_{10}\chi$ with $\mu_{a}=1400$.

Figure 9: The upper and lower bounds on the QoS exponents with different
$\epsilon$.

Fig. 8 demonstrates the $\epsilon-\chi$ tradeoff in the AWGN channel. In this
figure, $N=256$, $\gamma=100$, and $L=50$. The parameter setting in this
figure refers to [17], which validated that the normal approximation has high
accuracy for $\epsilon\geq 10^{-6}$ with $N\geq 200$ and $\gamma=100$. The
arrival rate follows the Poisson distribution with $\mu_{a}=1400$. The
numerical results are calculated according to Eq. (156), while the upper and
lower bounds are obtained from Eq. (126). Note that, the points on the curves
of the numerical results represent the minimum value of $\epsilon$ for
achieving the corresponding $\chi$. If the point
$(\log_{10}\chi,\log_{10}\epsilon)$ locates below this curve, then $\chi$ can
not be achieved with the given $N$, $\gamma$, and $\epsilon$. From Fig. 8, we
find that $\ln\epsilon$ is a concave function of $\ln\chi$. With the
decreasing $\epsilon$, the difference between the upper bound and the
numerical result becomes smaller. The above results show that the closed-form
upper bound can be used to characterize the $\epsilon-\chi$ tradeoff in the
low $\epsilon$ regime, which validates Lemma 3.1 and Theorem 3.1.

Figure 10: The $\epsilon-\chi$ tradeoff in the Rayleigh fading channel with
$T=1$, $L=4000$, $N=256$, and $\gamma=100$.

In Fig. 9, we present the numerical result of the upper and lower bounds on
the QoS exponents with different $\epsilon$. The arrival rate follows the
Poisson distribution with $\mu_{a}=\sigma_{a}^{2}=40$, and the blocklength is
$N=100$. The numerical results show that the upper and lower bounds become
larger with the increasing $\gamma$. Besides, the difference between
$\theta_{1}^{*}$ and $\theta_{2}^{*}$ becomes negligible with a high $\gamma$.
Note that, this result is not equivalent to
$\lim_{\gamma\to\infty}\frac{\theta_{1}^{*}}{\theta_{2}^{*}}=1$ proposed in
Lemma 3.2. That is because Lemma 3.2 is based on the premise that the system
has a high load. In Fig. 9, as $\gamma$ increases, the load of the system
becomes low, which is not consistent with the premise of Lemma 3.2. However,
according to Eq. (163), the numerical results in this figure shows that with
large $\gamma$, $\frac{\mu_{1}/\sigma_{1}^{2}}{\mu_{2}/\sigma_{2}^{2}}\approx
1$, where $\mu_{i}$ and $\sigma_{i}^{2}$, $i=1,2$ is the mean and variance of
$R_{\epsilon}^{(i)}[n]$ defined in Eqs. (161) and (162). From this
perspective, the results in Fig. 9 indirectly validates Lemma 3.2.

Fig. 10 shows the $\epsilon-\chi$ tradeoff in the Rayleigh fading channel. The
arrival rate follows the Poisson distribution with $\mu_{a}=1000$. The
expectation of $|h[n]|^{2}$ is 1 in this figure. By comparing Fig. 8 with Fig.
10, we find that the $\epsilon-\chi$ tradeoffs in the AWGN and the Rayleigh
fading channel have similar properties. However, there are some differences.
Note that, the threshold is 50 in the AWGN channel, while $L=4000$ in the
Rayleigh fading channel. With a larger threshold, the $\epsilon-\chi$ tradeoff
in the Rayleigh fading channel has a close performance with that in the AWGN
channel. Moreover, the results in Fig. 7 also show that the QVP decays faster
in the AWGN channel than in the Rayleigh fading channel. These results show
that the randomness of the service rate results in the degradation of the
queueing performance.

##  VI Conclusion

In this paper, we mainly focused on characterizing the reliability-latency-
rate tradeoff for low-latency systems with FBC. Based on the previous
conclusions on the gain-conservation property for low-latency systems with
deterministic arrival, we defined the reliability gain, real-time gain, and
service-rate gain for the low-latency systems with random arrivals. EB and EC
were adopted to establish the connection between the gain-conservation
equation and the influence of random arrivals. By considering transmissions
over an AWGN channel, over a Rayleigh fading channel, with frequency or
spatial diversity, and over a Nakagami-$m$ fading channel, we derived the
service-rate gains in different scenarios. Then, the revised gain-conservation
inequalities were derived by taking service-rate, reliability, and real-time
gains into account. Moreover, an EC approximation method was proposed to
obtain the analytical expression of EC for some channel-gain-based scheduling
policies in the FBL regime. Finally, we put our focus on the low-latency
systems in which the delay threshold is larger than the channel coherence
time. The closed-form expressions of the reliability-latency tradeoff for the
AWGN channel and the Rayleigh fading channel were obtained. The conclusions
proposed in this paper characterize the fundamental tradeoff between the
reliability, latency, and throughput for low-latency systems with FBC, which
will provide instructions on the design of efficient scheduling schemes for
the low-latency systems. Future promising research directions include
obtaining more accurate approximations of QVP and DVP with the small
threshold, generalizing the analysis of the upper and lower bounds on QoS
exponent in other fading channels, and designing efficient scheduling schemes
based on the analytical forms of reliability-DVP tradeoff and EC approximation
for low-latency systems.

##  Appendix A Proof of Lemma 2.1

We put our focus on the expectation term in the definition of EC. We define
$g(x)$ as $-s(x)$. The derivative of $g(x)$ is given by

$g^{\prime}(x)=\frac{-(\ln 2)^{-1}\left(\gamma(1+\gamma x)\sqrt{(1+\gamma
x)^{2}-1}-\sqrt{\frac{1}{N}}Q^{-1}(\epsilon)\gamma\right)}{(1+\gamma
x)^{3}\sqrt{\big{(}1-(1+\gamma x)^{-2}\big{)}}}.$ (149)

$(1+\gamma x)\sqrt{(1+\gamma x)^{2}-1}$ is an increasing function of $x$,
which is equals to 0 with $x=0$. Thus, based on Eq. (149), we find that there
exists a $x^{*}$, when $x=x^{*}$, $g^{\prime}(x)=0$, $x<x^{*}$,
$g^{\prime}(x)>0$, and $x>x^{*}$, $g^{\prime}(x)<0$. $x^{*}$ satisfies

$(1+\gamma x^{*})\sqrt{(1+\gamma
x^{*})^{2}-1}=\frac{Q^{-1}(\epsilon)}{\sqrt{N}}.$ (150)

$x^{*}$ can be solved from Eq. (150) as

$x^{*}=\frac{1}{\gamma}\left(\sqrt{\frac{1+\sqrt{1+4c}}{2}}-1\right),$ (151)

which is an interior point of $(0,\infty)$. Then, according to Laplace’s
method [40], the integral in the definition of EC can be expressed as

$\int_{0}^{\infty}f(x)e^{-\theta TNs(x)}dx=\sqrt{\frac{2\pi}{\theta
TN|s^{\prime\prime}(x^{*})|}}f(x^{*})e^{-\theta TNs(x^{*})}+O\left((\theta
TN)^{-1}\right).$ (152)

By substituting Eq. (152) into Eq. (10), we obtain Eq. (120).

Note that, the actual FBL coding rate can be expressed as $s(x)+O(1)$ to avoid
$s(x)<0$. If the $O(1)$ term can be represented by a constant $l$, then the EC
approximation needs to add $Nl$ at the right side of Eq. (120). Since $Nl$ is
a deterministic term, it does not have influence on the accuracy of the
approximation.

##  Appendix B Proof of Theorem 2.1

Similar to the proof of Lemma 4, the derivative of $g(x)$ is given by

$g^{\prime}(x)=\frac{-(\ln 2)^{-1}\left(\Xi+\Xi^{\prime}x\right)\left((1+\Xi
x)\sqrt{(1+\Xi x)^{2}-1}-\sqrt{\frac{1}{N}}Q^{-1}(\epsilon)\right)}{(1+\Xi
x)^{3}\sqrt{\big{(}1-(1+\Xi x)^{-2}\big{)}}},$ (153)

where for simplicity we omit the argument of $\Xi$. By comparing Eqs. (153)
and (149), we find that if $\Xi+\Xi^{\prime}x\geq 0$, the remaining analysis
is identical with the proof of Lemma 4.

##  Appendix C Proof of Corollary 2.1

According to [29], the EC with the simple ARQ mechanism that has an unlimited
number of retransmission is given by

$\displaystyle\alpha_{S}(\theta)$ $\displaystyle=-\frac{1}{\theta
T}\ln\mathbb{E}\left\\{\epsilon+(1-\epsilon)e^{-\theta TNs(x)}\right\\}$ (154)
$\displaystyle=-\frac{1}{\theta
T}\ln\left(\epsilon+(1-\epsilon)\int_{0}^{\infty}e^{-\theta TNs(x)}f(x){\rm
d}x\right).$

Similar to the proofs of Lemma 3 and Theorem 4, if the power allocation scheme
satisfies $\Xi(x)+\frac{{\rm d}\Xi(x)}{{\rm d}x}x\geq 0$, the integral
$\int_{0}^{\infty}e^{-\theta TNs(x)}f(x){\rm d}x$ can be approximated as shown
in Eq. (152) with the help of Laplace’s method. By substituting Eq. (152) into
Eq. (154), we obtain Eq. (125).

##  Appendix D Proof of Lemma 3.1

According to [15], for the $i.i.d.$ $a[n]$, the EB is approximated by
$\alpha_{A}(\theta^{*})\approx\mu_{a}+\frac{\sigma_{a}^{2}}{2}\theta^{*}$.
Since in the AWGN channel, $s[n]$ is a constant with the given $N$, $\gamma$,
and $\epsilon$. Thus, according to Eq. (14), the QoS exponent satisfies
$\alpha_{S}[\theta^{*}]=R_{\epsilon}$. Therefore, by combing this result with
EB approximation, $\theta^{*}$ is given by 444The accuracy of this
approximation will be validated in Section V.

$\theta^{*}\approx\frac{2(R_{\epsilon}-\mu_{a})}{\sigma^{2}_{a}}.$ (155)

By substituting Eqs. (6) and (13) into Eq. (155), we obtain that

$\epsilon=Q\left(\psi+\Phi\ln\chi\right).$ (156)

According to Mills ratio [41], if $\psi+\Phi\ln\chi>0$, $\epsilon$ satisfies

$\frac{1}{\sqrt{2\pi}}\frac{\psi+\frac{\Phi}{L}\ln\chi}{1+\left(\psi+\frac{\Phi}{L}\ln\chi\right)^{2}}e^{-\frac{\left(\psi+\frac{\Phi}{L}\ln\chi\right)^{2}}{2}}\leq\epsilon\leq\frac{1}{\sqrt{2\pi}}\frac{1}{\psi+\frac{\Phi}{L}\ln\chi}e^{-\frac{\left(\psi+\frac{\Phi}{L}\ln\chi\right)^{2}}{2}},$
(157)

where $\psi$ and $\Phi$ are given in Eqs. (25) and (26). Thus, we obtain Eq.
(126) in Lemma 3.1.

##  Appendix E Proof of Theorem 3.1

According to Mills ratio [41], $Q(x)$ satisfies

$\lim_{x\to\infty}\frac{Q(x)}{\frac{1}{\sqrt{2\pi}}\frac{1}{x}e^{-\frac{x^{2}}{2}}}=1.$
(158)

Based on this equation, we can prove that

$\displaystyle\lim_{x\to\infty}\frac{\ln
Q(x)}{\ln\left(\frac{1}{\sqrt{2\pi}}\frac{1}{x}e^{-\frac{x^{2}}{2}}\right)}$
(159) $\displaystyle=$
$\displaystyle\lim_{x\to\infty}\frac{Q^{\prime}(x)}{Q(x)\left(-x-\frac{1}{x}\right)}$
$\displaystyle=$
$\displaystyle\lim_{x\to\infty}\frac{x}{x+\frac{1}{x}}\cdot\frac{\frac{1}{\sqrt{2\pi}}\frac{1}{x}e^{-\frac{x^{2}}{2}}}{Q(x)}$
$\displaystyle=1.$

From Eqs. (25) and (156), we can see that with $\epsilon\to 0$,
$\psi+\Phi\ln\chi\to\infty$. Thus, according to Eq. (159), we obtain that

$\lim_{\psi+\frac{\Phi}{L}\ln\chi\to\infty}\frac{\ln
Q\left(\psi+\frac{\Phi}{L}\ln\chi\right)}{-\ln(\psi+\frac{\Phi}{L}\ln\chi)-\frac{(\psi+\frac{\Phi}{L}\ln\chi)^{2}}{2}-\frac{1}{2}\ln(2\pi)}=1.$
(160)

Then, we obtain the approximation in Eq. (129).

##  Appendix F Proof of Lemma 3.2

We define $R^{(1)}_{\epsilon}[n]$ as

$R^{(1)}_{\epsilon}[n]=N\log\left(1+|h[n]|^{2}\gamma\right)-\sqrt{N}Q^{-1}(\epsilon)\log
e+\frac{\log_{2}N}{2}.$ (161)

Besides, we define $R^{(2)}_{\epsilon}[n]$ as

$R^{(2)}_{\epsilon}[n]=N\log\left(1+|h[n]|^{2}\gamma\right)-\sqrt{N}Q^{-1}(\epsilon)\left(1-\frac{1}{1+|h[n]|^{2}\gamma}\right)\log
e+\frac{\log_{2}N}{2}.$ (162)

According to [15], with the $i.i.d.$ $a[n]$ and $s[n]$, the QoS exponent can
be expressed as 555Note that, the numerical results shown in Section V
indicate that this approximation is more accurate in high load scenarios,
which is identical to the settings of our system model.

$\theta^{*}\approx\frac{2\left(\mu_{s}-\mu_{a}\right)}{\sigma_{a}^{2}+\sigma_{s}^{2}}.$
(163)

Let $\theta_{1}^{*}$ and $\theta_{2}^{*}$ denote the QoS exponent of systems
with transmission rates $R^{(1)}_{\epsilon}[n]$ and $R^{(2)}_{\epsilon}[n]$,
respectively. To get the relationship
$\theta_{1}^{*}\leq\theta^{*}\leq\theta_{2}^{*}$, we need to derive the
condition in which
$\mathbb{E}\\{R^{(1)}_{\epsilon}[n]\\}\leq\mathbb{E}\\{R_{\epsilon}[n]\\}\leq\mathbb{E}\\{R^{(2)}_{\epsilon}[n]\\}$
and ${\rm Var}(R^{(1)}_{\epsilon}[n])\geq{\rm Var}(R_{\epsilon}[n])\geq{\rm
Var}(R^{(2)}_{\epsilon}[n])$. Since the constant term $\frac{\log_{2}N}{2}$ is
existed in both $R_{\epsilon}[n]$, $R^{(1)}_{\epsilon}[n]$, and
$R^{(2)}_{\epsilon}[n]$, which has no influence on the comparison of their
means and variances, we omit it in the following analysis.

Since $\sqrt{1-x}\geq 1-\sqrt{x}$ and $\sqrt{1-x}\leq 1$ when $x\in[0,1]$, we
obtain that
$\mathbb{E}\\{R^{(1)}_{\epsilon}[n]\\}\leq\mathbb{E}\\{R_{\epsilon}[n]\\}\leq\mathbb{E}\\{R^{(2)}_{\epsilon}[n]\\}.$
Then, we have to find the condition in which ${\rm
Var}(R^{(1)}_{\epsilon}[n])\geq{\rm Var}(R_{\epsilon}[n])\geq{\rm
Var}(R^{(2)}_{\epsilon}[n])$ is satisfied.

First, we present the condition under which ${\rm
Var}(R^{(1)}_{\epsilon}[n])-{\rm Var}(R^{(2)}_{\epsilon}[n])\geq 0$. For
$R^{(1)}_{\epsilon}[n]$, ${\rm Var}(R^{(1)}_{\epsilon}[n])$ is equal to ${\rm
Var}(\varrho)$. Therefore, we obtain

$\displaystyle{\rm Var}(R^{(1)}_{\epsilon}[n])-{\rm
Var}(R^{(2)}_{\epsilon}[n])$ $\displaystyle=2\cdot{\rm
Cov}(\varrho,\vartheta)-\omega^{2}\cdot{\rm Var}(\vartheta)$ (164)
$\displaystyle=-(\lambda-\beta^{2})\omega^{2}+2(\rho-\iota\beta)\omega.$

Eq. (164) can be seen as a downward quadratic function of $\omega$. The two
zero points of this function are 0 and
$\frac{2(\rho-\iota\beta)}{\lambda-\beta^{2}}$. Our aim is to get the feasible
regime of ${\rm Var}(R^{(1)}_{\epsilon}[n])-{\rm
Var}(R^{(2)}_{\epsilon}[n])>0$. Let us denote
$\frac{2(\rho-\iota\beta)}{\lambda-\beta^{2}}$ as zero point 1 in the
following derivations. The curve of zero point 1 versus $\gamma$ is shown in
Fig. 12. From Fig. 12, we obtain that ${\rm Var}(R^{(1)}_{\epsilon}[n])-{\rm
Var}(R^{(2)}_{\epsilon}[n])>0$ holds when
$\omega\in[0,\frac{2(\rho-\iota\beta)}{\lambda-\beta^{2}}]$.

Figure 11: ${\rm Var}(\Theta)-{\rm Var}(\vartheta)$ versus $\gamma$.

Figure 12: The zero points values versus $\gamma$.

Next, we present the condition under which ${\rm Var}(R_{\epsilon}[n])-{\rm
Var}(R^{(2)}_{\epsilon}[n])\geq 0$. ${\rm Var}(R_{\epsilon}[n])-{\rm
Var}(R^{(2)}_{\epsilon}[n])$ is given by

${\rm Var}(R_{\epsilon}[n])-{\rm
Var}(R^{(2)}_{\epsilon}[n])=\omega^{2}\left({\rm Var}(\Theta)-{\rm
Var}(\vartheta)\right)+2\omega\left({\rm Cov}(\varrho,\vartheta)-{\rm
Cov}(\varrho,\Theta)\right),$ (165)

where ${\rm Var}(\vartheta)=\lambda-\beta^{2}$ and ${\rm
Cov}(\varrho,\vartheta)=\rho-\iota\beta$. We find that Eq. (165) is also a
quadratic function of $\omega$. However, ${\rm Var}(\Theta)$ and ${\rm
Cov}(\varrho,\Theta)$ do not have analytical forms. Therefore, we characterize
the quadratic coefficient and the zero points numerically. Fig. 12 lists the
quadratic coefficient ${\rm Var}(\Theta)-{\rm Var}(\vartheta)$ versus
$\gamma$. From Fig. 12, we see that Eq. (165) is a downward quadratic function
of $\omega$. Besides, 0 is one zero point of this function, while another zero
point is $\frac{-2\big{(}{\rm Cov}(\varrho,\vartheta)-{\rm
Cov}(\varrho,\Theta)\big{)}}{{\rm Var}(\Theta)-{\rm Var}(\vartheta)}$. Let us
denote $\frac{-2\big{(}{\rm Cov}(\varrho,\vartheta)-{\rm
Cov}(\varrho,\Theta)\big{)}}{{\rm Var}(\Theta)-{\rm Var}(\vartheta)}$ as zero
point 2. The curve of zero point 2 versus $\gamma$ is shown in Fig. 12. From
Fig. 12 we know that $\frac{-2\big{(}{\rm Cov}(\varrho,\vartheta)-{\rm
Cov}(\varrho,\Theta)\big{)}}{{\rm Var}(\Theta)-{\rm Var}(\vartheta)}$ is
positive. Therefore, ${\rm Var}(R_{\epsilon}[n])\geq{\rm
Var}(R^{(2)}_{\epsilon}[n])$ holds when
$\omega\in\bigg{[}0,\frac{-2\big{(}{\rm Cov}(\varrho,\vartheta)-{\rm
Cov}(\varrho,\Theta)\big{)}}{{\rm Var}(\Theta)-{\rm Var}(\vartheta)}\bigg{]}$.

Finally, we present the condition under which ${\rm
Var}(R^{(1)}_{\epsilon}[n])-{\rm Var}(R_{\epsilon}[n])\geq 0$. Similar to the
above analysis, we first express ${\rm Var}(R^{(1)}_{\epsilon}[n])-{\rm
Var}(R_{\epsilon}[n])$ as a quadratic function of $\omega$ as

${\rm Var}(R^{(1)}_{\epsilon}[n])-{\rm
Var}(R_{\epsilon}[n])=-\omega^{2}\cdot{\rm Var}(\Theta)+2\omega\cdot{\rm
Cov}(\varrho,\Theta).$ (166)

Eq. (166) can be seen as a downward quadratic function of $\omega$. Let us
denote $\frac{2{\rm Cov}(\varrho,\Theta)}{{\rm Var}(\Theta)}$ as zero point 3.
The curve of zero point 3 versus $\gamma$ is shown in Fig. 12. Therefore,
${\rm Var}(R^{(1)}_{\epsilon}[n])\geq{\rm Var}(R_{\epsilon}[n])$ holds when
$\omega\in[0,\frac{2{\rm Cov}(\varrho,\Theta)}{{\rm Var}(\Theta)}]$.

From the above analysis, we find that if

$\omega\in\bigg{[}0,\min\Big{\\{}\frac{2(\rho-\iota\beta)}{\lambda-\beta^{2}},\frac{-2\big{(}{\rm
Cov}(\varrho,\vartheta)-{\rm Cov}(\varrho,\Theta)\big{)}}{{\rm
Var}(\Theta)-{\rm Var}(\vartheta)},\frac{2{\rm Cov}(\varrho,\Theta)}{{\rm
Var}(\Theta)}\Big{\\}}\bigg{]},$ (167)

then ${\rm Var}(R^{(1)}_{\epsilon}[n])\geq{\rm Var}(R_{\epsilon}[n])\geq{\rm
Var}(R^{(2)}_{\epsilon}[n])$. Therefore,
$\theta_{1}^{*}\leq\theta^{*}\leq\theta_{2}^{*}$ holds if Eq. (167) is
satisfied.

Based on Eq. (163), we can now express $\theta^{*}_{1}$ and $\theta^{*}_{2}$
in Eqs. (168) and (169), respectively.

$\theta_{1}^{*}=\frac{2\left(\iota-\omega-\mu_{a}+\frac{\log_{2}N}{2}\right)}{\varphi-\iota^{2}+\sigma_{a}^{2}}.$
(168)
$\theta_{2}^{*}=\frac{2\left(\iota-\beta\omega-\mu_{a}+\frac{\log_{2}N}{2}\right)}{(\lambda-\beta^{2})\omega^{2}-2(\rho-\iota\beta)\omega+\varphi-\iota^{2}+\sigma_{a}^{2}}.$
(169)

Besides, since $\lim_{x\to 0}\frac{1}{1-\sqrt{x}}=1$, $R_{\epsilon}^{(1)}[n]$
and $R_{\epsilon}^{(2)}[n]$ are equivalent with $\gamma\to\infty$. Therefore,
$\lim_{\gamma\to\infty}\frac{\theta^{*}_{1}}{\theta^{*}_{2}}=1$.

##  Appendix G Proof of Lemma 3.3

As we mentioned in Lemma 3.2, with $\gamma\to\infty$,
$\lim_{\gamma\to\infty}\frac{\theta^{*}_{1}}{\theta^{*}_{2}}=1$. Thus, here we
adopt $\theta^{*}_{1}$ to characterize the QoS exponent in the high SNR
regime.

By substituting Eq. (168) into Eq. (13), we obtain

$\ln\chi=-\frac{2\left(\iota-\omega-\mu_{a}+\frac{\log_{2}N}{2}\right)L}{\varphi-\iota^{2}+\sigma_{a}^{2}}.$
(170)

According to the definition of $\omega$,