#  Unified Timing Analysis for Closed-Loop Goal-Oriented Wireless
Communication ††thanks:  The work of Lintao Li and Wei Chen was supported by
the National Natural Science Foundation of China/Research Grants Council
Collaborative Research Scheme under Grant No. 62261160390. The work of A. E.
Kalør was supported in part by the Independent Research Fund Denmark (IRFD)
under Grant 1056-00006B and in part by the Horizon Europe SNS project
“6G-GOALS” (grant 101139232). The work of P. Popovski was supported in part by
the Villum Investigator Grant “WATER” from the Velux Foundation, Denmark, and
in part by the Horizon Europe SNS project “6G-GOALS” (grant 101139232). 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).
Anders E. Kalør and Petar Popovski are with the Department of Electronic
Systems, Aalborg University, 9220 Aalborg, Denmark (e-mail<EMAIL_ADDRESS>petarp@es.aau.dk).

Lintao Li, Anders E. Kalør, Petar Popovski, , and Wei Chen

###### Abstract

Goal-oriented communication has become one of the focal concepts in sixth-
generation communication systems owing to its potential to provide
intelligent, immersive, and real-time mobile services. The emerging paradigms
of goal-oriented communication constitute closed loops integrating
communication, computation, and sensing. However, challenges arise for closed-
loop timing analysis due to multiple random factors that affect the
communication/computation latency, as well as the heterogeneity of feedback
mechanisms across multi-modal sensing data. To tackle these problems, we aim
to provide a unified timing analysis framework for closed-loop goal-oriented
communication (CGC) systems over fading channels. The proposed framework is
unified as it considers computation, compression, and communication latency in
the loop with different configurations. To capture the heterogeneity across
multi-modal feedback, we categorize the sensory data into the periodic-
feedback and event-triggered, respectively. We formulate timing constraints
based on average and tail performance, covering timeliness, jitter, and
reliability of CGC systems. A method based on saddlepoint approximation is
proposed to obtain the distribution of closed-loop latency. The results show
that the modified saddlepoint approximation is capable of accurately
characterizing the latency distribution of the loop with analytically
tractable expressions. This sets the basis for low-complexity co-design of
communication and computation.

###### Index Terms:

closed-loop timing analysis, multi-modal feedback, immersive communication,
saddlepoint approximation.

##  I Introduction

###  I-A Motivation and Contributions

Driven by the rapid development of information technology and the continuous
improvement of hardware capability, the sixth-generation (6G) communication
systems are envisioned to fuse the physical and digital worlds by providing
intelligent, immersive, and real-time services [1]. This is, for instance,
reflected in the IMT-2030 draft for the 6G wireless communication technology
[2] published by the International Telecommunication Union (ITU), which in
addition to hyper reliable and low-latency communication also includes
immersive communication as central 6G usage scenarios. Against this
background, the goal-oriented communication paradigm gets significant traction
in 6G [3]. Supported by artificial intelligence (AI), goal-oriented
communication aims to provide intelligent and immersive experiences for users
by optimizing the communication for the receiver’s intention and experience,
rather than simply providing reliable, content-blind packet delivery [4, 5].
Combined with Metaverse and Digital Twin (DT) technology, this enables 6G to
break the spatial constraints to promote the achievement of the flexible
interconnection between the physical world and the digital virtual space [6].

Differently from 5G, goal-oriented paradigms require demanding _closed-loop_
interactions with strict timing requirements, starting from the commands
transmitted by the user and ending with the reception of feedback. The closed-
loop interactions are common in several of the typical 6G scenarios considered
by the Third Generation Partnership Project (3GPP) [7], including remote
control, virtual social, and gaming. However, the closed-loop timing analysis
is challenging compared to the point-to-point perspective that has dominated
the literature so far, as it needs to consider processing latency of
intermediate notes caused by, e.g., compression and computation. Thus, there
is a need to develop a unified timing-analysis framework for closed-loop goal-
oriented communication (CGC) systems over wireless channels, which can serve
as a tool to ensure that users get the desired real-time experience.

To address this problem, we aim to formulate a unified timing-analysis
framework and propose a general method to characterize the closed-loop timing
performance in CGC systems while taking into account both communication,
computation, and compression latencies. Specifically, we consider a general
CGC system with a remote-control task comprising a user (UE), a base station
(BS), and a remote agent (RA), which are assumed to operate in a closed loop.
The system is exemplified as a robotic scenario with a DT at the BS in Fig. 1.
We consider both periodic and sporadic traffic. The periodic traffic, referred
to as periodic feedback (PF), is transmitted at fixed intervals from the RA to
the UE, representing, e.g., sensor data. The sporadic traffic is initiated by
control data transmitted from the UE to the RA, which then responds with
event-triggered (ET) data. The communication goes through the BS, which is
equipped with a mobile-edge-computing (MEC) server to operate computation
tasks, such as feature extraction for AI, compression, and decompression. Note
that, although the remote agent is illustrated as a robot, it could also act
as a cloud server in a gaming scenario, or as another UE for a virtual social
scenario. A key novelty of the proposed model is that it enables closed-loop
analysis of heterogeneous multi-modal feedback comprising both event-triggered
and periodic feedback, as well as stochastic computation and compression
latency. The stochastic nature of the latter has been ignored in most existing
work, although these aspects are crucial in end-to-end latency analysis [8, 9,
10]. Besides, motivated by perceptual requirements of interactive applications
[7, 11, 12, 13], we explicitly model the asynchrony between the transmission
latency of PF and ET data, referred to as _jitter_. Further details of the
model will be given in Section II.

Note that the model in Fig. 1 is rather general, capable to cover some
commonly referred scenarios for CGC systems. For the remote control task shown
in Fig. 1, we are interested in the closed-loop latency by accounting for the
timeliness of the control data and the jitter of the multi-modal feedback. For
the authorization of avatar usage rights, specified by 3GPP in [7], the RA
acts as a cloud server. UE sends requests to the cloud server through BS
without relying on the computation process at the BS. Once the cloud server
receives the requests, it inquires and returns profiles of the avatar to the
UE. Meanwhile, authorization information will be sent to UE periodically to
authorize the use of the avatar legally. In this scenario, reducing the
asynchrony between profile reception and authorization ensures immediate
avatar utilization without additional latency. Differently from the remote
control, we focus on the closed-loop latency without a specific concern of
control link latency.

Using the proposed model, and with consideration of timeliness, jitter, and
reliability, we formulate a unified framework of timing analysis for CGC
systems,capable of addressing both average- and tail-based timing constraints
for CGC systems with different information-processing mechanisms and
configurations. To this end, we propose a general saddlepoint-approximation-
based method to obtain the analytical expression of the distribution of
closed-loop latency with high accuracy. Based on this general method, we also
give a corollary for efficiently approximating the conditional expectation in
the timing constraints. The proposed methods can be generalized to timing
analysis in other closed-loop scenarios.

Figure 1: System model for a general CGC system. This figure shows an example
of the remote control with the assistance of MEC, semantic information, and
DT. Multiple communication and computation processes are included in this
closed loop.

The main contributions are summarized as follows:

  * •

We formulate a unified timing-analysis framework for CGC systems with
different configurations, which takes the randomness of the computation,
compression, and communication latency into account. The formulated framework
integrates the timeliness, jitter, and reliability of CGC systems.

  * •

We consider two heterogeneous information flows with both PF and ET data,
representing different feedback mechanisms of multi-modal sensing data.

  * •

We propose a general saddlepoint approximation (SPA)-based method for
analytically characterizing the timing measures in CGC systems. For specific
scenarios, the central limit theorem (CLT) is also introduced to improve the
efficiency of the method while ensuring accuracy.

  * •

Based on this method, we derive an efficient approximation for calculating the
conditional expectation in timing constraints.

###  I-B Related Work

In [14], a goal-oriented communication and prediction co-design was proposed
to improve the reliability of the haptic communication system. The end-to-end
communication latency was chosen as the timing metric with a queueing model,
while the computation latency is omitted in [14]. Wen _et al._ optimized the
discriminant gain to achieve the goal of improving the accuracy of AI
inference in an integrated sensing and communication (ISAC) system [15].
Meanwhile, they considered the latency constraint in this system, which
focused on the uplink latency comprised of communication, computation, and
sensing procedures. The authors of [16] designed resource allocation and
computation offloading schemes to minimize the uplink latency of DT-enabled
Metaverse. In [15] and [16], the constant computation latency is incorporated
into the timing measurements. However, they only considered the one-way
latency instead of the closed-loop latency.

Conducting closed-loop timing analysis was proved to be effective for the
wireless networked control system (WCNS) in [17] and [18]. Thus, it is
important to analyze the timing performance of a goal-oriented communication
system from the closed-loop perspective. In [19], the authors carried out the
closed-loop timing analysis for wireless federated learning systems by taking
both random latency of local information uploading and global model
broadcasting into account. However, the computation latency was also assumed
as a deterministic term in [19], which ignored its stochastic property [8, 9,
10]. To address these problems, Suman _et al._ conducted a statistical timing
analysis for a closed-loop teleoperation system [8], in which the randomness
of computation and compression latency were accounted for. However, the
heterogeneity between the multi-modal feedback, which widely exists in various
goal-oriented communication, was not considered.

The SPA technique that we apply is a powerful method to approximate
complicated distributions with high accuracy guarantees while avoiding
intractable computations [20]. SPA has been widely used in communication
research. In [19], SPA was used to characterize the distribution of
transmission latency in wireless federated learning systems. An efficient SPA-
based method for evaluating block error probabilities was proposed in [21],
and the power-latency-throughput tradeoff of delay-constrained wireless
systems was characterized by SPA in [22].

Figure 2: Diagram of the components of the closed-loop latency. The UE
transmits commands to the RA through the control link, and the RA transmits
both periodic-feedback and event-triggered sensor data to the UE.

###  I-C Paper Organization and Notation

The paper is structured as follows. Section II presents the unified timing
framework for CGC systems considered in this paper. Section III introduces the
latency components and formally defines the overall objective, while Section
IV proposes the SPA-based timing analysis for the CGC system. Section V
validates the analysis through numerical results, and finally the paper is
concluded in Section VI.

_Notation:_ $\mathbb{E}\\{x\\}$ denotes the expectation of the random variable
$x$, and ${\rm Var}\\{x\\}$ denotes its variance. $f^{\prime}(x)$ and
$f^{\prime\prime}(x)$ denote the first-order and second-order derivative of
$f(x)$, respectively. $\mathcal{N}(0,1)$ denotes the standard Gaussian
distribution with cumulative distribution function (CDF)
$\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{y^{2}}{2}}{\rm d}y$
and probability density function (PDF)
$\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}$.
$\gamma(s,x)=\int_{0}^{x}t^{s-1}e^{-t}{\rm d}t$ is the lower incomplete gamma
function, $\\{x\\}^{+}=\max\\{x,0\\}$, and ${\rm sign}(x)$ denotes the sign of
$x$.

##  II A Unified Framework for Timing Analysis in CGC Systems with Different
Configurations

In this section, we present a unified framework for timing analysis in CGC
systems. We first give a system overview, and then introduce the components of
latency. The wireless channel model is presented in Section II-C, and Section
II-D defines different timing constraints used to formulate the proposed
framework with different configurations.

###  II-A System Overview

We study the CGC system depicted in Fig. 1, consisting of a BS, a single UE,
and a single RA equipped with a sensor. The UE and the RA communicate over a
wireless channel through the BS using dedicated channel resources, and the BS
is equipped with an MEC server to perform computation tasks, which include
feature extraction and decompression. We will refer to the channel from the UE
to the RA as the _control link_ (CL), while the link from the RA to the UE is
the _feedback link_ (FL). The communication flow is detailed in Fig. 2, where
we consider three packet types: control data, periodic-feedback (PF) data, and
event-triggered (ET) data.

Control data is transmitted sporadically by the UE as high-level (HL)
commands, and are assumed to be transmitted at relatively long intervals so
that no queuing is needed. For instance, control data could represent the
intention of UE to put the yellow box on the red one as shown in Fig. 1. The
HL commands are assumed to be much larger than the PF transmissions, and
subject to a random processing delay at the BS. Specifically, the UE transmits
an HL command to the BS, which then processes the received command to extract
the semantic information, e.g., with the help of a large language model [23]
and the DT model maintained at the BS. The HL command is assumed to consist of
$N_{\rm HL}$ packets of size $n_{\rm HL}$ bits each111There is usually one
packet containing the specific command signals. However, the analysis in this
paper does not specify the value of the number of packets for both HL and low-
level (LL) since the proposed framework and analysis are general.. At the BS,
the HL command is turned into detailed control signals, referred to as low-
level (LL) commands, which are transmitted from the BS to the RA. The LL
commands consist of $N_{\rm LL}$ packets of $n_{\rm LL}$ bits.

PF data usually has a small size, and typically represents numeric values,
e.g., the speed of the RA and the temperature of the remote environment [7].
The PF data are collected at fixed, short intervals every $T_{\rm s}$ seconds,
and can be shown on UE’s equipment without further processing. However, the
data are queued at both the RA and the BS.

Finally, ET data are generated by the RA as a response to an LL command. ET
data is usually throughput-intensive and may include multimedia data, e.g.,
video, tactile details, and audio. Thus, the ET data is first compressed by
the RA and then transmitted to the BS. The MEC server at BS decompresses the
data, extracts valuable information (VI) from the ET data, e.g., the location
and texture of the objective, and updates its DT model to prepare for the next
loop. After the processing, the valuable information is encoded into $N_{\rm
VI}$ packets with equal size $n_{\rm VI}$ bits. Finally, the valuable
information will be sent to UE. We denote the total size of the original ET
data at the RA as $n_{\rm ET}$ bits. The compressed ET data is transmitted
using $N_{\rm CD}$ packets with $n_{\rm CD}$ bits each.

The focus of this paper is on the closed-loop latency, $T$, of a single
transmission round under various constraints on the individual links.
Specifically, a single transmission round starts from the transmission of a HL
command by the UE and lasts until the UE has received both the corresponding
ET data _and_ the PF data generated closest to the time instant at which the
LL data is received by the RA. The proposed framework in the following
subsections takes the latency requirements for CL and the jitter between two
heterogeneous information flows in FL into account. These requirements will
affect $T$ by influencing the latency of CL and FL. For the multiple-
transmission scenario, more queueing delay is introduced. We leave the
analysis of the scenario with multiple transmissions for future work.

###  II-B Latency Components

Based on the previous description and Fig. 2, the transmission latency of the
PF data, denoted by $T_{\rm PF}$, is given by

$\displaystyle T_{\rm PF}=T^{\rm q}_{\rm PF1}+T^{\rm cm}_{\rm PF1}+T^{\rm
q}_{\rm PF2}+T^{\rm cm}_{\rm PF2},$ (1)

where $T^{\rm q}_{\rm PF1}$ and $T^{\rm cm}_{\rm PF1}$ are the queuing and
transmission delays from the RA to the BS, respectively, and $T^{\rm q}_{\rm
PF2}$ and $T^{\rm cm}_{\rm PF2}$ are the queuing and transmission delays from
the BS to the UE, respectively. For the control data, the latency from UE to
RA can be written

$T_{\rm CL}=T^{\rm cm}_{\rm HL}+T_{\rm LL}^{\rm cp}+T^{\rm cm}_{\rm
LL},\vspace{-1mm}$ (2)

where $T^{\rm cm}_{\rm HL}$ and $T^{\rm cm}_{\rm LL}$ are the transmission
delay of the HL command and the LL command, respectively, and $T_{\rm LL}^{\rm
cp}$ is the processing delay at the BS. Similarly, the latency of the ET
transmission from the RA to the UE is

$T_{\rm ET}=T^{\rm c}_{\rm ET}+T^{\rm cm}_{\rm CD}+T^{\rm d}_{\rm ET}+T^{\rm
cp}_{\rm VI}+T^{\rm cm}_{\rm VI},\vspace{-1mm}$ (3)

where $T^{\rm c}_{\rm ET}$ is the compression delay at the RA, $T^{\rm
cm}_{\rm CD}$ and $T^{\rm cm}_{\rm VI}$ are the transmission delay from the RA
to the BS and from the BS to the UE, respectively, and $T^{\rm d}_{\rm ET}$
and $T^{\rm cp}_{\rm VI}$ are the decompression delay and the processing delay
at the BS, respectively.

Recall that we are interested in the closed-loop latency from the generation
of the HL command until the reception of both the corresponding ET data and
the PT data generated closest in time to the generation time of the ET data.
Since the interval of the periodic transmission, $T_{\rm s}$, is relatively
small, its impact is negligible and the latency of the feedback link, denoted
as $T_{\rm FL}$, can thus be approximated by

$T_{\rm FL}\approx\max\left\\{T_{\rm PF},T_{\rm ET}\right\\}.\vspace{-1mm}$
(4)

Using this, we finally write the closed-loop latency as

$T=T_{\rm CL}+T_{\rm FL}.\vspace{-1mm}$ (5)

Finally, we also define the jitter of the feedback link as the absolute
difference between $T_{\rm ET}$ and $T_{\rm PF}$, i.e., $|T_{\rm ET}-T_{\rm
PF}|$.

Figure 3: Configurations considered in the proposed framework.

###  II-C Wireless Channel

We assume independent block fading channels at all links so that the channel
gain remains constant over the length of a transmitting packet. Assuming a
transmission power of $P_{i}$, at link $i\in\\{{\rm HL},{\rm LL},{\rm
PF1},{\rm PF2},{\rm CD},{\rm VI}\\}$, the signal-to-noise ratio (SNR) at the
receiver is given by

${\rm
SNR}_{i}=\frac{d_{i}^{-\ell_{i}}|h_{i}|^{2}P_{i}}{N_{0}B_{i}},\,\text{$i\in\\{{\rm
HL},{\rm LL},{\rm PF1},{\rm PF2},{\rm CD},{\rm VI}\\}$},\vspace{-1mm}$ (6)

where $|h_{i}|^{2}$ is the small-scale channel power gain, $d_{i}$ is the
distance between the transmitter and the receiver, $\ell_{i}$ is the path-loss
exponent, $N_{0}$ is the spectral density of the noise power, and $B_{i}$ is
the bandwidth. For $i\in\\{{\rm LL,CD,PF1}\\}$, $d_{i}$ refers to the distance
between RA and BS, which we denote by $d_{\rm RB}$. For $i=\\{\rm
HL,PF2,VI\\}$, $d_{i}$ refers to the distance between BS and UE, which we
denote by $d_{\rm BU}$. Besides, we let $\ell_{\rm BU}$ denote the path-loss
exponent of the link between BS and UE, while $\ell_{\rm RB}$ denote the path-
loss exponent of the link between RA and BS. Moreover, $B_{\rm PF1}=B_{\rm
PF2}=B_{\rm PF}$, while $B_{\rm CD}=B_{\rm VI}=B_{\rm ET}$. We are primarily
concerned with the case in which channel state information is known only at
the receiver side (CSIR), and the transmitter adopts the $\epsilon_{i}$-outage
rate $R_{i}$. With $|h_{i}|^{2}\sim{\rm exp}(1)$, the relationship between
$R_{i}$ and $\epsilon_{i}$ is given by [24]

$R_{i}=B_{i}\log_{2}\left(1-\frac{d^{-\ell_{i}}P_{i}}{N_{0}B_{i}}\ln(1-\epsilon_{i})\right).$
(7)

For completeness, we also consider the case where the transmitter also has
perfect channel state information for the transmission of PF packets, referred
to as CSIRT. In this case, we additionally assume that frequency or spatial
diversity (FSD) is available, e.g., through multiple antennas. This is
motivated by our previous work [25, 26], which shows that with both CSIRT and
FSD, it can be ensured that the delay violation probability is zero with
finite average power. In this case, we let $\tau_{\rm PF}$ denote the deadline
of PF data with CSIRT and FSD, which means the latency of each PF packet
cannot be larger than $\tau_{\rm PF}$. Thus, with CSIRT and FSD, we can use
$\tau_{\rm PF}$ to represent the latency of PF data.222Though the average
power is finite, the required instantaneous power may be large. If there is a
maximum power constraint, the delay violation probability of the packet is not
zero and can be characterized by large deviation theory [25] and extreme value
theory [19]. The analysis of this setting is left for future work. Moreover,
we let $\tau_{\rm PF1}$ and $\tau_{\rm PF2}$ denote the deadline for the PF
data transmission from RA to BS and BS to UE, respectively, which satisfy
$\tau_{\rm PF1}+\tau_{\rm PF2}\leq\tau_{\rm PF}$.

###  II-D A Unified Framework of Timing Constraints for CGC Systems with
Different Configurations

In this subsection, we establish a unified timing-analysis framework for CGC
systems under different configurations, shown as vertices in Fig. 3. The
proposed framework ensures the timeliness performance of CL and the jitter
performance of FL to cover general requirements for CGC systems. Meanwhile,
these constraints also affect the closed-loop latency $T$ through $T_{\rm CL}$
and $T_{\rm FL}$ to ensure the reliability of CGC systems.

As indicated on the jitter/latency constraint axis, we consider both average
and tail-based constraints on jitter and latency. The average-based
constraints are related to the expectation of timing measurements, while tail-
based constraints here are related to the violation probability of timing
measurements. For the control link (CL), which handles the transmission of
control data, we consider both with and without a latency constraint
representing a timeliness requirement on $T_{\rm CL}$. Finally, the feedback
link (FL) may or may not have CSIRT and FSD for the PF data, as represented by
the last axis. When CSIRT and FSD are available, then there is a hard delay
constraint for the PF data, $\tau_{\rm PF}$, as mentioned in Section II-C.
Therefore, under this condition, we treat the PF transmission latency $T_{\rm
PF}$ as deterministic, $T_{\rm PF}=\tau_{\rm PF}$. On the other hand, when
only CSIR is available $T_{\rm PF}$ is a random variable.

For the delay jitter, there are two different evaluating methods under
different configurations, which correspond to the FL axis in Fig. 3. The first
one refers to the case that the transmission of PF data has CSIRT and FSD. In
this case, the latency of PF data is characterized by the threshold $\tau_{\rm
PF}$. Thus, the jitter is defined as the $|T_{\rm ET}-\tau_{\rm PF}|$.

The second one refers to the case that the transmission of PF data only has
CSIR. Thus, the latency of PF data is also stochastic. Then, the jitter is the
difference between two random variables, i.e., $|T_{\rm ET}-T_{\rm PF}|$.

Based on these constraints on each axis, we can combine them to form the eight
vertices of the cube in Fig. 3.333Note that except the constraints on the
vertices of the cube, we can also form the constraints such as the combination
of tail-based FL constraint and average-based CL constraint. These constraints
can be seen as the points between vertices $i$ and $i+2$, $i\in\\{1,2,5,6\\}$.
Due to the layout limitation, we omit the detailed discussion. Next, we
elaborate on each vertex in Fig. 3. Vertices 1-4 refer to interactive systems
with control of RA under different configurations as shown in Fig. 1.

  * •

Vertex 1 – average constraints for CL and PF with CSIR:

$\displaystyle\begin{cases}\mathbb{E}\left\\{T_{\rm
CL}\right\\}\leq\varrho_{\rm CL},\\\ \mathbb{E}\left\\{\left|T_{\rm ET}-T_{\rm
PF}\right|\right\\}\leq\varrho_{\rm FLd},\end{cases}$ (8)

where $\varrho_{\rm CL}$ is the average latency constraint of CL, and
$\varrho_{\rm FLd}$ is the average jitter constraint of FL.

  * •

Vertex 2 – average constraints for CL and PF with CSIRT and FSD:

$\displaystyle\begin{cases}\mathbb{E}\left\\{T_{\rm
CL}\right\\}\leq\varrho_{\rm CL},\\\ \tau_{\rm PF1}+\tau_{\rm
PF2}\leq\tau_{\rm PF},\\\ \mathbb{E}\left\\{\left(T_{\rm ET}-\tau_{\rm
PF}\right)^{+}\right\\}\leq\varrho_{\rm FLs},\end{cases}$ (9)

where $\tau_{\rm PF}$ and $\varrho_{\rm FLs}$ are average jitter constraints
of FL. In this case, we treat the part in which $T_{\rm ET}<\tau_{\rm PF}$ as
no jitter. This is because $\tau_{\rm PF}$ is deterministic so that both ET
and PF feedback can be shown to UE until PF data is received if $T_{\rm
ET}<\tau_{\rm PF}$. Note that the third constraint in this case can be
rewritten as

$\mathbb{E}\left\\{T_{\rm ET}-\tau_{\rm PF}|T_{\rm ET}>\tau_{\rm
PF}\right\\}\leq\frac{\varrho_{\rm FLs}}{\Pr\\{T_{\rm ET}>\tau_{\rm
PF}\\}},\vspace{-1mm}$ (10)

which can be seen as a condition value-at-risk (CVaR) based constraint [27].

  * •

Vertex 3 – tail constraints for CL and PF with CSIR:

$\displaystyle\begin{cases}\Pr\left\\{T_{\rm CL}<\tau_{\rm
CL}\right\\}\geq\eta_{\rm CL},\\\ \Pr\left\\{|T_{\rm ET}-T_{\rm PF}|<\tau_{\rm
td}\right\\}\geq\eta_{\rm td},\\\ {\rm Var}\left\\{|T_{\rm ET}-T_{\rm
PF}|\right\\}\leq\rho_{\rm td},\end{cases}$ (11)

where $\eta_{\rm CL}$ and $\tau_{\rm CL}$ are tail latency constraints of CL.
$\tau_{\rm td}$, $\eta_{\rm td}$, and $\rho_{\rm td}$ are the tail jitter
constraint of FL. For the first constraint in Eq. (11), if $\eta_{\rm CL}$ is
given to find the minimum $\tau_{\rm CL}$, $\tau_{\rm CL}$ turns into value at
risk (VaR) [27], i.e.,

$\tau_{\rm CL}=\inf_{t}\left\\{\Pr\\{T_{\rm CL}<t\\}\geq\eta_{\rm
CL}\right\\}.\vspace{-1mm}$ (12)

If $\tau_{\rm CL}$ is given, solving the maximum value of $\eta_{\rm CL}$ is
equal to minimizing the delay violation probability, i.e.,

$\eta_{\rm CL}=\inf_{p}\left\\{\Pr\\{T_{\rm CL}\geq\tau_{\rm CL}\\}\leq
p\right\\}.\vspace{-1mm}$ (13)

The second and third constraints in Eq. (11) refer to the tail jitter
constraint for FL. Except for the violation probability of $|T_{\rm ET}-T_{\rm
PF}|$, the variance of the jitter is also included to cover general cases,
which is also an important measurement for tail behavior as shown in [28].

  * •

Vertex 4 – tail constraints for CL and PF with CSIRT and FSD:

$\displaystyle\begin{cases}\Pr\left\\{T_{\rm CL}<\tau_{\rm
CL}\right\\}\geq\eta_{\rm CL},\\\ \tau_{\rm PF1}+\tau_{\rm PF2}\leq\tau_{\rm
PF},\\\ \Pr\\{T_{\rm ET}<\tau_{\rm PF}\\}\geq\eta_{\rm ts},\\\ {\rm
Var}\\{T_{\rm ET}-\tau_{\rm PF}\\}\leq\rho_{\rm ts},\end{cases}$ (14)

where $\tau_{\rm PF}$, $\eta_{\rm ts}$ and $\rho_{\rm ts}$ are the tail jitter
constraints of FL. Since $\tau_{\rm PF}$ is a constant, the fourth constraint
in this case is equivalent to

${\rm Var}\left\\{T_{\rm ET}\right\\}\leq\rho_{\rm ts}.\vspace{-1mm}$ (15)

  * •

Vertices 5-8 correspond to vertices 1-4 without the first constraint in Eqs.
(8), (9), (11), and (14), respectively. In these vertices, there are no
specific constraints for the CL. The main focus is on the performance of the
closed-loop latency and FL’s jitter, which covers the system without the
control for RA, e.g., the authorization of avatar usage rights mentioned in
Section I-A.

Note that we have to obtain the distributions of $T_{\rm PF1}^{\rm q}$ and
$T_{\rm PF2}^{\rm q}$ for modeling vertices 1, 3, 5, and 7. However, there is
no existing conclusion on the distribution of $T_{\rm PF1}^{\rm q}$ and
$T_{\rm PF2}^{\rm q}$ to the best of authors’ knowledge.444Once we have the
distribution of $T_{\rm PF1}^{\rm q}$ and $T_{\rm PF2}^{\rm q}$, vertices 1,
3, 5, 7 can be analytically expressed by the proposed method in this paper.
Thus, we mainly focus on the modeling and analysis of vertices 2, 4, 6, and 8
in the following parts, which refer to the case that the transmission of PF
data has CSIRT and FSD. Based on the proposed unified framework of timing
analysis in this paper, more timing metrics can be considered, e.g., age of
information and age of loop [17], [18]. Besides, we can also consider the
queueing process at the BS and RA sides for ET data with the aid of large
deviation theory [29, 30] to denote the asymptotic distribution of queueing
latency. We will leave these thoughts in the future work.

##  III Stochastic Latency Components and a Unified Optimization Problem in
CGC Systems

In this section, we will analyze the distribution of communication,
computation, compression, and decompression latency in CGC systems. A unified
optimization problem is formulated by combining different kinds of constraints
and stochastic latency components.

###  III-A Distributions of Latency Components in CGC Systems

We start with the introduction of the communication latency. The latency of
transmitting a packet is denoted by $t_{i}$, $i\in\\{{\rm HL},{\rm LL},{\rm
PF1},{\rm PF2},{\rm CD},{\rm VI}\\}$, which satisfies
$t_{i}=\frac{n_{i}}{R_{i}}$. Thus, the total time for successfully
transmitting one packet follows a geometric distribution. As defined in
Section II-A, $T_{i}^{\rm cm}$ denotes the transmission latency of $N_{i}$
packets. $T^{\rm cm}_{i}$ follows a negative binomial distribution
$\text{NB}(N_{i},1-\epsilon_{i})$. The PDF of $T^{\rm cm}_{i}$ is given by

$\Pr\\{T^{\rm
cm}_{i}=(N_{i}+k)t_{i}\\}={{k+N_{i}-1}\choose{N_{i}-1}}(1-\epsilon_{i})^{N_{i}}\epsilon_{i}^{k},\vspace{-1mm}$
(16)

where $k\in\\{0,1,\cdots\\}$.

The PF data is collected every $T_{\rm s}$ seconds. Thus, one packet
containing the PF data arrives at the buffer every $T_{\rm s}$ seconds, which
incurs a queueing delay. For vertices 2, 4, 6, and 8, we consider the case
that CSIRT and FSD are available for the transmission of PF data. With CSIRT
and FSD, the deadline constraint can be met with a finite average power [25].
Specifically, for $j\in\\{{\rm PF1},{\rm PF2}\\}$, the deadline-constrained
capacity is approximated by [26, 31]

$R_{j}=B_{j}\log_{2}\left(1+\frac{c_{1}d_{j}^{-\ell_{j}}P_{j}}{N_{0}B_{j}\left(c_{2}+(\tau_{j})^{-c_{3}}\right)}\right),$
(17)

where $c_{i}>0$, $i\in\\{1,2,3\\}$ are constants. Note that, for the
transmission from BS to UE, the arrival of PF data is not periodical. However,
we can transform it into periodical arrival by reintegration, i.e., let one
packet get into the transmitting buffer even more than one packet arrives in
this time slot. Therefore, the conclusion is valid. According to our previous
conclusions in [26] and [31], by transmitting the PF data through two antennas
with maximum-ratio combining, the closed-form approximation for the deadline-
constrained capacity is

$R_{j}=B_{j}\log_{2}\left(1+\frac{2.8771d_{j}^{-\ell_{j}}P_{j}}{N_{0}B_{j}\left(1.8771+\left(\tau_{j}\right)^{-3.411}\right)}\right).\vspace{-1mm}$
(18)

By this means, to ensure that all PF data is transmitted within $\tau_{j}$,
the average power $P_{j}$ can be solved from Eq. (18) by setting
$R_{j}=\frac{n_{j}}{T_{\rm s}}$.

Then we introduce the formulation of the computation time. As shown in Fig. 2,
there are two feature extraction procedures in the CGC system, which are
calculated at the MEC server. For $l\in\\{{\rm LL},{\rm VI}\\}$, the
computation time is given by

$T^{\rm cp}_{l}=\frac{\tilde{n}_{l}X^{\rm cp}_{l}}{\chi_{\rm
MEC}},\vspace{-1mm}$ (19)

where $\chi_{\rm MEC}$ is the frequency of the processing unit at the MEC
server, and $X^{\rm cp}_{l}$ is the number of processing-unit cycles required
for processing 1-bit data. $\tilde{n}_{\rm LL}=N_{\rm HL}n_{\rm HL}$ and
$\tilde{n}_{\rm VI}=n_{\rm ET}$. As shown by previous research [10, 8, 9], the
number of cycles allocated to compute 1 bit is stochastic in nature. For the
CPU case, $X^{\rm cp}_{l}$ follows a gamma distribution ${\rm
Gamma}(\alpha_{\rm MEC},\beta_{l})$ [10, 8, 9], where $\alpha_{\rm MEC}$ and
$\beta_{l}$ are the shape and rate parameters of the gamma distribution,
respectively. The shape parameter remains the same for all tasks performed by
the same processor, while the scale parameter $\beta_{l}$ will be different
for different tasks [8]. Thus, the computation time also follows a gamma
distribution ${\rm Gamma}\left(\alpha_{\rm MEC},\frac{\chi_{\rm
MEC}\beta_{l}}{\tilde{n}_{l}}\right)$. The PDF of $T^{\rm cp}_{l}$ is given by

$\\!\\!f_{l}^{\rm cp}(x)\\!=\\!\frac{\left(\chi_{\rm
MEC}\beta_{l}\right)^{\alpha_{\rm MEC}}}{\tilde{n}_{l}^{\alpha_{\rm
MEC}}\Gamma(\alpha_{\rm MEC})}x^{\alpha_{\rm
MEC}-1}\exp\left(\\!-\frac{\chi_{\rm
MEC}\beta_{l}}{\tilde{n}_{l}}x\\!\right),$ (20)

where $x>0$. The expectation of $T^{\rm cp}_{l}$ is $\mu^{\rm
cp}_{l}=\frac{\tilde{n}_{l}\alpha_{\rm MEC}}{\chi_{\rm MEC}\beta_{l}}$, while
its variance is $\xi^{\rm cp}_{l}=\frac{\tilde{n}_{l}^{2}\alpha_{\rm
MEC}}{\left(\chi_{\rm MEC}\beta_{l}\right)^{2}}$. 555Note that the analysis in
this paper is also valid for GPU scenarios if the distribution of $X^{\rm
cp}_{l}$ for GPU is given.

For the compression time, we let $\kappa\in[1,\kappa_{\rm max}]$ denote the
compression ratio, which is the ratio of the size of raw data to the size of
compressed data. $\kappa_{\rm max}$ is the maximum compression ratio. Let
$X_{\rm c}$ denote the number of CPU cycles for compressing 1-bit data, which
follows ${\rm Gamma}\left(\alpha_{\rm RA},\beta_{\rm c}\right)$, where
$\alpha_{\rm RA}$ and $\beta_{\rm c}$ denote the shape and rate parameters of
compression latency, respectively. There are two main existing conclusions to
evaluate the expected computation load with the given compression ratio. The
first one shows that

$\mathbb{E}\\{X_{\rm c}\\}=\frac{\alpha_{\rm RA}}{\beta_{\rm
c}}=\exp\left(\psi\kappa\right)-\exp\left(\psi\right),$ (21)

where $\psi>0$ is a constant. This model can be used for several lossless
compression techniques such as Zlib, Zstandard, and XZ compression [8, 9, 32].
Another conclusion shows that

$\mathbb{E}\\{X_{\rm c}\\}=\frac{\alpha_{\rm RA}}{\beta_{\rm
c}}=\omega_{1}\left(\omega_{2}\kappa^{\omega_{3}}+\omega_{4}\right),$ (22)

where $\omega_{k}$, $k=1,\cdots,4$ are the positive constants for lossless
compression algorithms GZIP and BZ2 as shown in [33]. 666Note that this model
can also be used for evaluating $\mathbb{E}\\{X_{\rm C}\\}$ of the lossy
compression [33], e.g., JPEG. Let $\zeta_{\rm c}(\kappa)=\mathbb{E}\\{X_{\rm
c}\\}$, which is an increasing function of $\kappa$.

Let $X_{\rm d}$ denote the number of CPU cycles for decompressing 1-bit data.
$X_{\rm d}$ follows ${\rm Gamma}\left(\alpha_{\rm MEC},\beta_{\rm d}\right)$,
where $\beta_{\rm d}$ is the rate parameter of decompression latency.
Accordingly, there are two main conclusions in the existing works to evaluate
$\mathbb{E}\left\\{X_{\rm d}\right\\}$. The first one shows that

$\mathbb{E}\left\\{X_{\rm d}\right\\}=\frac{\alpha_{\rm MEC}}{\beta_{\rm
d}}=\omega_{0}\left(\exp\left(\psi\kappa\right)-\exp\left(\psi\right)\right),$
(23)

where $\omega_{0}\in(0,1)$ is a constant. Another conclusion shows

$\mathbb{E}\left\\{X_{\rm d}\right\\}=\frac{\alpha_{\rm MEC}}{\beta_{\rm
d}}=\omega_{5}\left(\omega_{6}\kappa^{\omega_{7}}+\omega_{8}\right),\vspace{-1mm}$
(24)

where $\omega_{5}$, $\omega_{6}$, and $\omega_{8}$ are positive constants. The
sign of $\omega_{7}$ is determined by the specific compression algorithms
[33]. Let $\zeta_{\rm d}(\kappa)=\mathbb{E}\left\\{X_{\rm d}\right\\}$.

With the above models and definitions, $T_{\rm ET}^{\rm c}=\frac{n_{\rm
ET}X_{\rm c}}{\chi_{\rm RA}}$, and $T_{\rm ET}^{\rm d}=\frac{n_{\rm ET}X_{\rm
d}}{\kappa\chi_{\rm MEC}}$, where $\chi_{\rm RA}$ is the frequency of the
processing unit at RA. Thus, we obtain $T_{\rm ET}^{\rm c}\sim{\rm
Gamma}\left(\alpha_{\rm RA},\frac{\chi_{\rm RA}\alpha_{\rm RA}}{n_{\rm
ET}\zeta_{\rm c}(\kappa))}\right)$, with expectation $\mu_{\rm ET}^{\rm
c}=\frac{n_{\rm ET}\zeta_{\rm c}(\kappa)}{\chi_{\rm RA}}$ and variance
$\xi^{\rm c}_{\rm ET}=\frac{\left(n_{\rm ET}\zeta_{\rm
c}(\kappa)\right)^{2}}{\chi^{2}_{\rm RA}\alpha_{\rm RA}}$. $T_{\rm ET}^{\rm
d}\sim{\rm Gamma}\left(\alpha_{\rm MEC},\frac{\chi_{\rm MEC}\alpha_{\rm
MEC}\kappa}{n_{\rm ET}\zeta_{\rm d}(\kappa)}\right)$, with expectation
$\mu_{\rm ET}^{\rm d}=\frac{n_{\rm ET}\zeta_{\rm d}(\kappa)}{\chi_{\rm
MEC}\kappa}$ and variance $\xi_{\rm ET}^{\rm d}=\frac{\left(n_{\rm
ET}\zeta_{\rm d}(\kappa)\right)^{2}}{\left(\chi_{\rm
MEC}\kappa\right)^{2}\alpha_{\rm MEC}}$.

###  III-B Formulation of a Unified Optimization Problem

In this subsection, we will formulate a unified optimization problem for this
CGC system, in which average-based and tail-based optimization objectives and
constraints are considered.

The objective of the optimization can be set from various aspects. If we aim
to reduce the closed-loop latency to improve reliability and provide a better
experience for users, the objective function is related to the closed-loop
latency $T$. If we aim to save resources, e.g., power and bandwidth, to ensure
the performance of the CGC system, the objective function is related to the
average power and bandwidth.

For the first kind of objective functions, we can also consider two cases,
i.e., the average-based and tail-based objectives. The average-based objective
is related to $\mathbb{E}\\{T\\}$. For the tail-based objective, let $T_{\rm
th}$ denote the threshold for the closed-loop latency, which is determined by
the specific applications and human’s requirements. Thus, the tail-based
objective is related to $\Pr\\{T<T_{\rm th}\\}$. We let $d(T)$ denote the
objective functions that are related to $T$. Besides, we let $g(P_{\rm
avg},B)$ denote the objective functions related to resources. We also assume
that $d(T)$ and $g(P_{\rm avg},B)$ are normalized.

A unified optimization problem for this CGC system is then formulated by
combining different kinds of objectives and constraints.

$\displaystyle\min\quad$ $\displaystyle\varepsilon
d(T)+(1-\varepsilon)g\left(P_{\rm avg},B\right)$ (25a) s.t. Constraints of a
vertex in Fig. 3, (25b) $\displaystyle P_{\rm PF1}+P_{\rm CD}\leq P_{\rm
RA}^{\rm th},$ (25c) $\displaystyle P_{\rm PF2}+P_{\rm VI}\leq P_{\rm BS}^{\rm
th},$ (25d) $\displaystyle B_{\rm PF}+B_{\rm ET}=B,$ (25e)

where $\varepsilon\in[0,1]$, $P_{\rm RA}^{\rm th}$ and $P_{\rm BS}^{\rm th}$
are the constraints of the average power on BS and RA, $B_{\rm PF1}=B_{\rm
PF2}=B_{\rm PF}$, and $B$ (in Hz) is the maximum available bandwidth. For
$\varepsilon=0$, we are interested in designing a resource-efficient
scheduling. For $\varepsilon=1$, we focus on optimizing the performance of
closed-loop latency. The optimization variables in this problem may include
$\left\\{\kappa,B_{\rm PF},B_{\rm ET},P_{\rm PF1},P_{\rm PF2},\tau_{\rm
PF1},\tau_{\rm PF2},P_{\rm CD},P_{\rm VI}\right\\}$.

It is not trivial work to solve this optimization problem in CGC systems. As
shown in Section III-A, we know the expectation and variance of each latency
component in this system, which can be used to characterize some constraints
related to the expectation and variance of $T_{\rm ET}$. However, no matter
which vertex’s constraints are adopted, we have to obtain the distribution of
closed-loop and FL’s latency. For vertices 2 and 4, we also need to obtain the
CDF of CL latency. Since the latency of CL, FL, and closed loop are sums of
gamma and negative binomial random variables, it is hard to obtain analytical
distributions of the CL, FL, and closed-loop latency. Thus, it is crucial to
derive the analytical expression for the distribution of CL, FL, and closed-
loop latency for general scenarios to provide instructions for efficient
performance evaluation or optimization.

##  IV Analytical Expression of the Latency Distribution in CGC Systems

In this section, we will present a general method for approximating the
distribution of CL, FL, and closed-loop latency. Based on this general method,
we also present methods for efficiently evaluating the terms in different
constraints. Since there are similarities in the latency distribution and
constraints in CL, FL, and closed-loop latency, we will take FL for example to
show how to derive the analytical expression for the terms in the proposed
constraints for FL. Then we will also give the results of CL and closed-loop
latency without detailed proofs. For simplicity, we define $T_{1}=T_{\rm
CL}+T_{\rm ET}$. Without loss of generality, we let $t_{\rm CD}=t_{\rm
VI}=t_{\rm u}$.

$\displaystyle L(s)=-\alpha_{\rm RA}\ln\left(1-\frac{n_{\rm ET}\zeta_{\rm
c}(\kappa)s}{\chi_{\rm RA}\alpha_{\rm RA}}\right)-$ $\displaystyle\alpha_{\rm
MEC}\ln\left(1-\frac{n_{\rm ET}\zeta_{\rm d}(\kappa)s}{\chi_{\rm
MEC}\alpha_{\rm MEC}\kappa}\right)-\alpha_{\rm MEC}\ln\left(1-\frac{n_{\rm
ET}s}{\chi_{\rm MEC}\beta_{\rm VI}}\right).$ (32)

$J(s)\\!=\\!N_{\rm CD}\left(s+\ln(1-\epsilon_{\rm CD})-\ln(1-\epsilon_{\rm
CD}e^{s})\right)\\!+\\!N_{\rm VI}\left(s+\ln(1-\epsilon_{\rm
VI})-\ln\left(1-\epsilon_{\rm
VI}e^{s}\right)\right),\,s\\!<\\!\min\left\\{-\ln\epsilon_{\rm
CD},-\ln\epsilon_{\rm VI}\right\\}.\vspace{-1mm}$ (36)

###  IV-A A General Method for Analytical Approximation for the CDF of Latency

From Section III, we find that there are discrete and continuous random
variables in the components of CL, FL, and closed-loop latency. We have to
treat them separately since they have different properties. We will first
present a general method to approximate the CDF of $T_{\rm ET}$, which can be
applied to the CDF approximation of $T_{\rm CL}$, $T_{\rm FL}$, and $T$.

In Lemma 1, we obtain the analytical expression for the CDF of $T^{\rm c}_{\rm
ET}+T^{\rm d}_{\rm ET}+T^{\rm cp}_{\rm VI}$ with the help of SPA, which is a
sum of continuous random variables. For simplicity, let $F_{1}(x)$ and
$f_{1}(x)$ denote the CDF and PDF of $T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm
ET}+T^{\rm cp}_{\rm VI}$, respectively. Besides, we let

$\iota_{1}=\left(\frac{n_{\rm ET}^{2}\zeta_{\rm c}^{2}(\kappa)}{\chi_{\rm
RA}^{2}\alpha_{\rm RA}}+\frac{n_{\rm ET}^{2}\zeta^{2}_{\rm
d}(\kappa)}{\chi_{\rm MEC}^{2}\kappa^{2}\alpha_{\rm MEC}}+\frac{n_{\rm
ET}^{2}\alpha_{\rm MEC}}{\chi^{2}_{\rm MEC}\beta_{\rm
VI}^{2}}\right)^{\frac{3}{2}},$ (26) $\iota_{2}=\frac{2n_{\rm
ET}^{3}\zeta_{\rm c}^{3}(\kappa)}{\chi_{\rm RA}^{3}\alpha_{\rm
RA}^{2}}+\frac{2n_{\rm ET}^{3}\zeta_{\rm d}^{3}(\kappa)}{\kappa^{3}\chi_{\rm
MEC}^{3}\alpha_{\rm MEC}^{2}}+\frac{2\alpha_{\rm MEC}n_{\rm ET}^{3}}{\chi_{\rm
MEC}^{3}\beta_{\rm VI}^{3}},\,\text{and}\vspace{-1mm}$ (27)
$\displaystyle\theta=\frac{n_{\rm ET}\alpha_{\rm MEC}}{\chi_{\rm
MEC}\beta_{\rm VI}}+\frac{n_{\rm ET}\zeta_{\rm c}(\kappa)}{\chi_{\rm RA}}$
$\displaystyle+\frac{n_{\rm ET}\zeta_{\rm d}(\kappa)}{\chi_{\rm MEC}\kappa}.$
(28)

Lemma 1. With $\theta$ defined in Eq. (28),

  * •

the CDF of $T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm ET}+T^{\rm cp}_{\rm VI}$ is
approximated by

$\displaystyle
F_{1}(x)=\begin{cases}\Phi(v_{x})+\phi(v_{x})\left(\frac{1}{v_{x}}-\frac{1}{u_{x}}\right),\quad&x\neq\theta,\\\
\frac{1}{2}+\frac{\iota_{2}}{6\sqrt{2\pi}\iota_{1}},\quad&x=\theta,\end{cases}$
(29)

where

$\displaystyle v_{x}$ $\displaystyle={\rm
sign}\left(\Xi_{1}\left(x\right)\right)\sqrt{2x\Xi_{1}\left(x\right)-2L(\Xi_{1}(x))},\,\text{and}$
(30) $\displaystyle u_{x}$
$\displaystyle=\Xi_{1}(x)\sqrt{L^{\prime\prime}(\Xi_{1}(x))}.$

$\Xi_{1}(\cdot)$ is the inverse function of $L^{\prime}(\cdot)$. $\iota_{1}$
and $\iota_{2}$ are given in Eqs. (26) and (27), respectively. $L(\cdot)$ is
given in Eq. (32), where $s<\min\left\\{\frac{\chi_{\rm RA}\alpha_{\rm
RA}}{n_{\rm ET}\zeta_{\rm c}(\kappa)},\frac{\chi_{\rm MEC}\alpha_{\rm
MEC}\kappa}{n_{\rm ET}\zeta_{\rm d}(\kappa)},\frac{\chi_{\rm MEC}\beta_{\rm
VI}}{n_{\rm ET}}\right\\}$.

  * •

the PDF of $T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm ET}+T^{\rm cp}_{\rm VI}$ is
approximated by

$f_{1}(x)=\frac{\exp\left(L(\Xi_{1}(x))-x\Xi_{1}(x)\right)}{\sqrt{2\pi
L^{\prime\prime}(\Xi_{1}(x))}},\,x>0.$ (33)

###### Proof:

See Appendix A. ∎

In Lemma 1, we derive the CDF and PDF of $T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm
ET}+T^{\rm cp}_{\rm VI}$, which is the sum of continuous components of $T_{\rm
ET}$. For $T_{1}$, the sum of continuous components is $T_{\rm LL}^{\rm
cp}+T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm ET}+T^{\rm cp}_{\rm VI}$. Thus, we
define $\tilde{L}(s)$ as

$\tilde{L}(s)=L(s)-\alpha_{\rm MEC}\ln\left(1-\frac{N_{\rm HL}n_{\rm
HL}s}{\chi_{\rm MEC}\beta_{\rm LL}}\right),\vspace{-1mm}$ (34)

where $s<\min\left\\{\frac{\chi_{\rm MEC}\beta_{\rm LL}}{N_{\rm HL}n_{\rm
HL}},\frac{\chi_{\rm RA}\alpha_{\rm RA}}{n_{\rm ET}\zeta_{\rm
c}(\kappa)},\frac{\chi_{\rm MEC}\alpha_{\rm MEC}\kappa}{n_{\rm ET}\zeta_{\rm
d}(\kappa)},\frac{\chi_{\rm MEC}\beta_{\rm VI}}{n_{\rm ET}}\right\\}$.
Besides, we let $\tilde{\iota}_{1}=\left(\iota_{1}^{\frac{2}{3}}+\frac{N_{\rm
HL}^{2}n_{\rm HL}^{2}\alpha_{\rm MEC}}{\chi^{2}_{\rm MEC}\beta_{\rm
LL}^{2}}\right)^{\frac{3}{2}}$,
$\tilde{\iota_{2}}=\iota_{2}+\frac{2\alpha_{\rm MEC}N^{3}_{\rm HL}n_{\rm
HL}^{3}}{\chi^{3}_{\rm MEC}\beta^{3}_{\rm LL}}$, and
$\tilde{\theta}=\theta+\frac{N_{\rm HL}n_{\rm HL}\alpha_{\rm MEC}}{\chi_{\rm
MEC}\beta_{\rm LL}}$. By replacing $\tilde{L}(s)$, $\tilde{\iota}_{1}$,
$\tilde{\iota}_{2}$, and $\tilde{\theta}$ into Eqs. (29) and (33), we obtain
the CDF and PDF of $T_{\rm LL}^{\rm cp}+T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm
ET}+T^{\rm cp}_{\rm VI}$. We let $\tilde{F}_{1}(x)$ denote the CDF of $T_{\rm
LL}^{\rm cp}+T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm ET}+T^{\rm cp}_{\rm VI}$.

In Lemma 2, we obtain the closed-form expression for the probability mass
function (PMF) of $T^{\rm cm}_{\rm CD}+T^{\rm cm}_{\rm VI}$ with the help of
saddlepoint mass functions, which is a sum of discrete components of $T_{\rm
ET}$. For simplicity, let $p_{k}$ denote $\Pr\left\\{T^{\rm cm}_{\rm
CD}+T^{\rm cm}_{\rm VI}=kt_{\rm u}\right\\}$, where $k\in\\{N_{\rm CD}+N_{\rm
VI},N_{\rm CD}+N_{\rm VI}+1,\cdots\\}$.

Lemma 2 The PMF of $T^{\rm cm}_{\rm CD}+T^{\rm cm}_{\rm VI}$ is approximated
by

$\displaystyle p_{k}=\begin{cases}(1-\epsilon_{\rm CD})^{N_{\rm
CD}}(1-\epsilon_{\rm VI})^{N_{\rm VI}},\quad k=N_{\rm CD}+N_{\rm VI},\\\
\frac{\exp\left(J(\Xi_{2}(k))-k\Xi_{2}(k)\right)}{\sqrt{2\pi\left(\frac{\epsilon_{\rm
CD}N_{\rm CD}e^{\Xi_{2}(k)}}{\left(1-\epsilon_{\rm
CD}e^{\Xi_{2}(k)}\right)^{2}}+\frac{\epsilon_{\rm VI}N_{\rm
VI}e^{\Xi_{2}(k)}}{\left(1-\epsilon_{\rm
VI}e^{\Xi_{2}(k)}\right)^{2}}\right)}},\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,k>N_{\rm
CD}+N_{\rm VI},\end{cases}$ (35)

where $\Xi_{2}(\cdot)$ is the inverse function of $J^{\prime}(\cdot)$.
$J(\cdot)$ is given in Eq. (36).

###### Proof:

See Appendix B. ∎

In Lemma 2, we derive the PMF of $T^{\rm cm}_{\rm CD}+T^{\rm cm}_{\rm VI}$,
which is the sum of discrete components of $T_{\rm ET}$. For $T_{1}$, the sum
of discrete components is $T_{\rm HL}^{\rm cm}+T_{\rm LL}^{\rm cm}+T^{\rm
cm}_{\rm CD}+T^{\rm cm}_{\rm VI}$. We let $\tilde{p}_{k}$ denote $\Pr\\{T_{\rm
HL}^{\rm cm}+T_{\rm LL}^{\rm cm}+T^{\rm cm}_{\rm CD}+T^{\rm cm}_{\rm
VI}=kt_{\rm u}\\}$. Moreover, we let $\tilde{J}(s)$ denote the CGF of $T_{\rm
HL}^{\rm cm}+T_{\rm LL}^{\rm cm}+T^{\rm cm}_{\rm CD}+T^{\rm cm}_{\rm VI}$. Due
to the layout limitation, we omit the expression of $\tilde{J}(s)$, which is
similar to Eq. (36). By substituting $\tilde{J}(s)$ into Eq. (35), we obtain
$\tilde{p}_{k}$, where $k>N_{\rm HL}+N_{\rm LL}+N_{\rm CD}+N_{\rm VI}$. For
$k=N_{\rm HL}+N_{\rm LL}+N_{\rm CD}+N_{\rm VI}$, we have

$\tilde{p}_{k}=(1-\epsilon_{\rm HL})^{N_{\rm HL}}(1-\epsilon_{\rm LL})^{N_{\rm
LL}}p_{N_{\rm CD}+N_{\rm VI}}.\vspace{-1mm}$ (36)

Moreover, we also let $p^{\rm CL}_{k}$ denote the PMF of $T_{\rm HL}^{\rm
cm}+T_{\rm LL}^{\rm cm}$, which is the sum of discrete component of $T_{\rm
CL}$. Due to the layout limitation, we omit the derivation of $p^{\rm
CL}_{k}$, which is also similar to Eq. (35).

Based on Lemmas 1 and 2, we can derive the CDF of $T_{\rm CL}$, $T_{\rm ET}$,
$T_{\rm FL}$, $T_{1}$ and $T$ in general scenarios. We let $F_{\rm CT}(x)$,
$F_{\rm ET}(x)$, $F_{\rm FL}(x)$, $F_{{\rm T}_{1}}(x)$ and $F_{\rm T}(x)$
denote the CDF of $T_{\rm CL}$, $T_{\rm ET}$, $T_{\rm FL}$, $T_{1}$, and $T$,
respectively.

Theorem 1. For vertices 2, 4, 6, and 8, the CDF of $T_{\rm CL}$ is given by

$F_{\rm CL}(x)\\!\\!=\\!\\!\\!\\!\sum_{k=N_{\rm HL}+N_{\rm
LL}}^{\left\lfloor\frac{x}{t_{\rm
u}}\right\rfloor}\\!\\!\\!\\!\frac{p_{k}^{\rm CL}}{\Gamma(\alpha_{\rm
MEC})}\gamma\left(\alpha_{\rm MEC},\frac{\chi_{\rm MEC}\beta_{\rm
LL}(x-kt_{\rm u})}{N_{\rm HL}n_{\rm HL}}\right),\vspace{-3mm}$ (37)

where $x>0$.

The CDF of $T_{\rm ET}$ is given by

$\displaystyle F_{\rm ET}(x)=\sum^{\left\lfloor\frac{x}{t_{\rm
u}}\right\rfloor}_{k=N_{\rm CD}+N_{\rm VI}}p_{k}F_{1}(x-kt_{\rm u}),\quad
x>0.$ (38)

The CDF of $T_{\rm FL}$ is given by

$F_{\rm FL}(x)=\mathbb{I}\left\\{x<\tau_{\rm PF}\right\\}\frac{F_{\rm
ET}(x)}{1-F_{\rm ET}(\tau_{\rm PF})},\quad x>0.\vspace{-1mm}$ (39)

The CDF of $T$ is given by

$F_{\rm T}(x)=\mathbb{I}\\{x>\tau_{\rm PF}\\}F_{\rm CL}(x-\tau_{\rm
PF})F_{{\rm T}_{1}}(x),\vspace{-1mm}$ (40)

where $F_{{\rm T}_{1}}(x)$ is given by

$F_{{\rm T}_{1}}(x)=\\!\\!\\!\sum^{\left\lfloor\frac{x}{t_{\rm
u}}\right\rfloor}_{k=N_{\rm HL}+N_{\rm LL}+N_{\rm CD}+N_{\rm
VI}}\\!\\!\tilde{p}_{k}\tilde{F}_{1}(x-kt_{\rm u}),\quad x>0.\vspace{-1mm}$
(41)

###### Proof:

The CDF of $T_{\rm CL}$, $T_{\rm ET}$, and $T_{1}$ can be obtained directly
according to Lemmas 1 and 2. For $T_{\rm FL}$, we can obtain its CDF based on
$F_{\rm ET}(x)$ and the definition of $T_{\rm FL}$. For the closed-loop
latency $T$, it can be expressed as $T=T_{\rm CL}+T_{\rm FL}=\max\\{T_{\rm
CL}+\tau_{\rm PF},T_{1}\\}$. Thus, the CDF of $T$ is given by

$\Pr\\{T<x\\}=\Pr\\{T_{\rm CL}<x-\tau_{\rm
PF}\\}\Pr\\{T_{1}<x\\}.\vspace{-1mm}$ (42)

Then we obtain the CDF of $T$. ∎

In Theorem 1, we derive the CDF of both CL, FL, and closed-loop latency. For
practical applications, we can reduce the complexity of obtaining the
approximation for $F_{\rm ET}(x)$ by controlling the approximation accuracy.
Based on Theorem 1, we conceive an algorithm for efficiently approximating
$F_{\rm ET}(x)$ in practical applications. The details of this algorithm are
shown in Alg. 1. Steps 7 and 10 are used for normalizing the PMF of $T^{\rm
cm}_{\rm CD}+T^{\rm cm}_{\rm VI}$. Similar procedures can be used to calculate
$F_{\rm CL}(x)$ and $F_{{\rm T}_{1}}(x)$.

$M(s)=L(s)+\vartheta t_{\rm u}s+\frac{s^{2}}{2}\left(\frac{\epsilon_{\rm
CD}N_{\rm CD}t_{\rm u}^{2}}{\left(1-\epsilon_{\rm
CD}\right)^{2}}+\frac{\epsilon_{\rm VI}N_{\rm VI}t_{\rm
u}^{2}}{\left(1-\epsilon_{\rm
VI}\right)^{2}}\right),\,s<\min\left\\{\frac{\chi_{\rm RA}\alpha_{\rm
RA}}{n_{\rm ET}\zeta_{\rm c}(\kappa)},\frac{\chi_{\rm MEC}\alpha_{\rm
MEC}\kappa}{n_{\rm ET}\zeta_{\rm d}(\kappa)},\frac{\chi_{\rm MEC}\beta_{\rm
VI}}{n_{\rm VI}}\right\\}.\vspace{-1mm}$ (48)

$\tilde{M}(s)=M(s)-\alpha_{\rm MEC}\ln\left(1-\frac{N_{\rm HL}n_{\rm
HL}s}{\chi_{\rm MEC}\beta_{\rm LL}}\right)+\left(\frac{N_{\rm HL}t_{\rm
u}}{1-\epsilon_{\rm HL}}+\frac{N_{\rm LL}t_{\rm u}}{1-\epsilon_{\rm
LL}}\right)s+\frac{s^{2}}{2}\left(\frac{\epsilon_{\rm HL}N_{\rm HL}t_{\rm
u}^{2}}{\left(1-\epsilon_{\rm HL}\right)^{2}}+\frac{\epsilon_{\rm LL}N_{\rm
LL}t_{\rm u}^{2}}{\left(1-\epsilon_{\rm
LL}\right)^{2}}\\!\right).\vspace{-2mm}$ (50)

Algorithm 1 Calculating CDF of $T_{\rm ET}$

0: parameter $\Delta\in(0,1)$ for controlling the approximation accuracy

0: $F_{\rm ET}(x)$

1: Initialization: $k=N_{\rm CD}+N_{\rm VI}-1$, $\Theta=0$

2: repeat

3: $k\leftarrow k+1$

4: Calculate $p_{k}$ according to Eq. (35)

5: until $p_{k}<\Delta$

6: $\nu\leftarrow\mathbb{I}\\{k>N_{\rm CD}+N_{\rm VI}\\}\sum\limits_{i=N_{\rm
CD}+N_{\rm VI}+1}^{k}p_{i}$

7: if $\nu>0$ then

8: for all $j=N_{\rm CD}+N_{\rm VI}+1,\cdots,k$ do

9: $p_{j}\leftarrow\left(1-p_{N_{\rm CD}+N_{\rm VI}}\right)\frac{p_{j}}{\nu}$

10: end for

11: end if

12: if $\left\lfloor\frac{x}{t_{\rm u}}\right\rfloor<N_{\rm CD}+N_{\rm VI}$
then

13: $\Theta\leftarrow 0$

14: else

15: for all $l=N_{\rm CD}+N_{\rm
VI},\cdots,\min\left\\{\left\lfloor\frac{x}{t_{\rm
u}}\right\rfloor,k\right\\}$ do

16: Calculate $F_{1}(x-lt_{\rm u})$ according to Eq. (29).

17: $\Theta\leftarrow\Theta+p_{l}F_{1}(x-lt_{\rm u})$

18: end for

19: end if

20: $F_{\rm ET}(x)\leftarrow\Theta$

###  IV-B Continuous Approximation for a Large Number of Packets or Small
Transmission Time of a Packet

In the last subsection, we present a general method to approximate the latency
distribution. Since the CL, FL, and closed-loop latency consist of sums of
continuous and discrete random variables, we adopt both the LR formula and
saddlepoint mass function to obtain a satisfying approximation performance.
However, in practical applications, the number of packets is usually large.
Besides, the transmission time of a packet can be small in short-packet
scenarios. In this subsection, we focus on these cases, in which the number of
packets is large or $t_{\rm u}$ is small. Under this condition, we can use the
central limit theorem (CLT) to approximate the sum of communication latency as
normal random variables to turn $T_{\rm FL}$ and $T$ into continuous random
variables.

We start with the analysis for the discrete components of $T_{\rm ET}$, which
can be generalized to the analysis for the discrete components of $T$.$T^{\rm
cm}_{\rm CD}$ is the sum of $N_{\rm CD}$ i.i.d. random variables following
geometric distribution, while $T^{\rm cm}_{\rm VI}$ is similar. Under the
condition that $N_{\rm CD}+N_{\rm VI}$ is large, $T^{\rm cm}_{\rm CD}+T_{\rm
CD}^{\rm cm}$ can be approximated by a normal random variable. That is,
$T^{\rm cm}_{\rm CD}+T^{\rm cm}_{\rm VI}\sim\mathcal{N}\left(\frac{N_{\rm
CD}t_{\rm u}}{1-\epsilon_{\rm CD}}+\frac{N_{\rm VI}t_{\rm u}}{1-\epsilon_{\rm
VI}},\frac{\epsilon_{\rm CD}N_{\rm CD}t_{\rm u}^{2}}{\left(1-\epsilon_{\rm
CD}\right)^{2}}+\frac{\epsilon_{\rm VI}N_{\rm VI}t_{\rm
u}^{2}}{\left(1-\epsilon_{\rm VI}\right)^{2}}\right)$. As analyzed in [8], we
are interested in the scenario in which the negative value from Gaussian
approximation has a negligible effect on the analysis. Moreover, $T_{\rm
CD}^{\rm cm}$ and $T_{\rm VI}^{\rm cm}$ are both $t_{\rm u}$-lattice random
variables. Thus, we can treat $T_{\rm CD}^{\rm cm}$ and $T_{\rm VI}^{\rm cm}$
as continuous random variables when $t_{\rm u}$ is small. There are two
possibilities for small $t_{\rm u}$ scenarios from the perspective of the
number of packets. The first possible case is that the $N_{\rm CD}+N_{\rm VI}$
is small. Since $t_{\rm u}$ is also small, the corresponding transmission time
has a small influence on the distribution of $T$. The second case is that the
$N_{\rm CD}+N_{\rm VI}$ is large. Then we can treat it as a continuous random
variable by CLT. In summary, we can also use CLT to approximate the
distribution of $T_{\rm CD}^{\rm cm}$ and $T_{\rm VI}^{\rm cm}$ with small
$t_{\rm u}$.

Based on this continuous approximation, we can directly obtain the CDF and PDF
approximation of $T_{\rm ET}$ according to the LR formula. We define the
following constants as

$\Psi=\theta+\vartheta t_{\rm u},\vspace{-2mm}$ (43)

where

$\displaystyle\vartheta=\frac{N_{\rm CD}}{1-\epsilon_{\rm CD}}+\frac{N_{\rm
VI}}{1-\epsilon_{\rm VI}}.$ (44)

We also define that

$\Upsilon=\left(\iota_{1}^{\frac{2}{3}}+\frac{\epsilon_{\rm CD}N_{\rm
CD}t_{\rm u}^{2}}{\left(1-\epsilon_{\rm CD}\right)^{2}}+\frac{\epsilon_{\rm
VI}N_{\rm VI}t_{\rm u}^{2}}{\left(1-\epsilon_{\rm
VI}\right)^{2}}\right)^{\frac{3}{2}}.\vspace{-1mm}$ (45)

Lemma 3. Under the condition that $N_{\rm CD}+N_{\rm VI}$ is large or $t_{\rm
u}$ is small,

  * •

the CDF of $T_{\rm ET}$ is approximated by

$\\!\\!\\!\begin{aligned} \hat{F}_{\rm
ET}(x)=\begin{cases}\Phi(r_{x})+\phi(r_{x})\left(\frac{1}{r_{x}}-\frac{1}{z_{x}}\right),\quad&x\neq\Psi,\\\
\frac{1}{2}+\frac{\iota_{2}}{6\sqrt{2\pi}\Upsilon},\quad&x=\Psi,\end{cases}\end{aligned}\vspace{-1mm}$
(46)

where

$\\!\\!\\!\begin{aligned} r_{x}&={\rm
sign}\left(\Xi_{3}\left(x\right)\right)\sqrt{2x\Xi_{3}\left(x\right)-2M(\Xi_{3}(x))},\,\text{and}\\\
z_{x}&=\Xi_{3}(x)\sqrt{M^{\prime\prime}(\Xi_{3}(x))}.\end{aligned}\vspace{-1mm}$
(47)

$M(s)$ is given in Eq. (48). $\Xi_{3}(\cdot)$ is the inverse function of
$M^{\prime}(s)$.

  * •

the PDF of $T_{\rm ET}$ is approximated by

$\displaystyle\hat{f}_{\rm
ET}(x)=\frac{\exp\left(M\left(\Xi_{3}(x)\right)-x\Xi_{3}(x)\right)}{\sqrt{2\pi
M^{\prime\prime}(\Xi_{3}(x))}},x>0.$ (49)

###### Proof:

The proof is similar to the proof of Lemma 1. The CGF is shown in Eq. (48).
$\Psi$ is the expectation of $T_{\rm ET}$. It is obvious that $M^{\prime}(s)$
is an increasing function of $s$. Thus, the inverse function of
$M^{\prime}(s)$ exists, which we denote by $\Xi_{3}(\cdot)$. Besides, we find
that $M^{\prime\prime}(0)^{3/2}=\Upsilon$ and
$M^{\prime\prime\prime}(0)=\iota_{2}$. Thus, we obtain Eq. (46). With the help
of the saddlepoint density function, we obtain Eq. (49). ∎

By combining CLT with SPA, we can directly obtain the CDF of $T_{\rm ET}$. The
same method can be applied to obtaining the CDF of $T_{1}$ with the condition
that $N_{\rm HL}+N_{\rm LL}+N_{\rm CD}+N_{\rm VI}$ are large or $t_{\rm u}$ is
small. Let $\tilde{M}(s)$ denote the CGF of $T_{1}$, which is given in Eq.
(50), where $s<\min\left\\{\frac{\chi_{\rm MEC}\beta_{\rm LL}}{N_{\rm
HL}n_{\rm HL}},\frac{\chi_{\rm RA}\alpha_{\rm RA}}{n_{\rm ET}\zeta_{\rm
c}(\kappa)},\frac{\chi_{\rm MEC}\alpha_{\rm MEC}\kappa}{n_{\rm ET}\zeta_{\rm
d}(\kappa)},\frac{\chi_{\rm MEC}\beta_{\rm VI}}{n_{\rm ET}}\right\\}$.
Besides, we also define $\tilde{\Psi}$ and $\tilde{\Upsilon}$ as follows

$\\!\tilde{\Psi}=\tilde{\theta}+\left(\vartheta+\frac{N_{\rm
HL}}{1-\epsilon_{\rm HL}}+\frac{N_{\rm LL}}{1-\epsilon_{\rm LL}}\right)t_{\rm
u}.\vspace{-1mm}$ (51)
$\\!\\!\\!\\!\tilde{\Upsilon}\\!\\!=\\!\\!\left(\\!\Upsilon^{\frac{2}{3}}\\!+\\!\\!\frac{N_{\rm
HL}^{2}n_{\rm HL}^{2}\alpha_{\rm MEC}}{\chi^{2}_{\rm MEC}\beta_{\rm
LL}^{2}}\\!+\\!\frac{\epsilon_{\rm CD}N_{\rm CD}t_{\rm
u}^{2}}{\left(1-\epsilon_{\rm CD}\right)^{2}}\\!+\\!\frac{\epsilon_{\rm
VI}N_{\rm VI}t_{\rm u}^{2}}{\left(1-\epsilon_{\rm
VI}\right)^{2}}\\!\right)^{\frac{3}{2}}\\!\\!.\vspace{-1mm}$ (52)

$\displaystyle\mathbb{E}\left\\{\left(T_{\rm ET}-\tau_{\rm
PF}\right)^{+}\right\\}\approx\left(\theta+\vartheta t_{\rm u}-\tau_{\rm
PF}\right)\left(1-\hat{F}_{\rm ET}(\tau_{\rm PF})\right)+\frac{\tau_{\rm
PF}-\theta-\vartheta t_{\rm u}}{\Xi_{3}(\tau_{\rm PF})}\hat{f}_{\rm
ET}(\tau_{\rm PF}),\quad\text{if $\tau_{\rm PF}\neq\theta+\vartheta t_{\rm
u}$}.\vspace{-1mm}$ (54)

By substituting $\tilde{M}(s)$, $\tilde{\iota}_{2}$ (defined below Eq. (34)),
$\tilde{\Psi}$, and $\tilde{\Upsilon}$ into Eqs. (46) and (49), we can obtain
$\hat{F}_{{\rm T}_{1}}(x)$ and $\hat{f}_{{\rm T}_{1}}(x)$, where
$\hat{F}_{{\rm T}_{1}}(x)$ and $\hat{f}_{{\rm T}_{1}}(x)$ are the continuous
approximation for the CDF and PDF of $T_{1}$, respectively. We then present
the continuous approximation for the CDF of $T$ in Theorem 2.

Theorem 2. Under the condition that $N_{\rm HL}+N_{\rm LL}+N_{\rm CD}+N_{\rm
VI}$ is large or $t_{\rm u}$ is small, the CDF of the closed-loop latency is
approximated by

$\hat{F}_{\rm T}(x)=\mathbb{I}\\{x>\tau_{\rm PF}\\}F_{\rm CL}(x-\tau_{\rm
PF})\hat{F}_{{\rm T}_{1}}(x).\vspace{-1mm}$ (53)

Note that if $N_{\rm LL}+N_{\rm HL}$ is large or $t_{\rm u}$ is small, we can
also use the continuous approximation $\hat{F}_{\rm CL}(x)$ to replace $F_{\rm
CL}(x)$ in Eq. (53). However, $N_{\rm LL}$ is usually equal to 1 in practical
scenarios. Besides, $T_{\rm CL}$ only has three latency components, in which
the CDF of $T_{\rm LL}^{\rm cp}$ is known. Thus, we use the general
approximation for the CDF of $T_{\rm CL}$ in Eq. (53) to ensure high accuracy
while ensuring efficiency.

(a) CDF of $T_{\rm ET}$. (b) Approximation error of $T_{\rm ET}$.

(c) CDF of $T$.

Figure 4: $T_{\rm ET}$ and $T$ with $t_{\rm u}=5$ ms, $N_{\rm CD}^{(1.1)}=20$,
$N_{\rm VI}=3$, $\epsilon_{\rm CD}=10^{-3}$, and $\epsilon_{\rm VI}=10^{-4}$.

(a) CDF of $T_{\rm ET}$. (b) Approximation error of $T_{\rm ET}$.

(c) CDF of $T$.

Figure 5: $T_{\rm ET}$ and $T$ with $t_{\rm u}=5$ ms, $N_{\rm CD}^{(1.1)}=50$,
$N_{\rm VI}=3$, $\epsilon_{\rm CD}=10^{-3}$, and $\epsilon_{\rm VI}=10^{-4}$.

(a) CDF of $T_{\rm ET}$. (b) Approximation error of $T_{\rm ET}$.

(c) CDF of $T$.

Figure 6: $T_{\rm ET}$ and $T$ with $t_{\rm u}=5$ ms, $N_{\rm CD}^{(1.1)}=5$,
$N_{\rm VI}=1$, $\epsilon_{\rm CD}=10^{-1}$, and $\epsilon_{\rm VI}=10^{-3}$.

(a) CDF of $T_{\rm ET}$. (b) Approximation error of $T_{\rm ET}$.

(c) CDF of $T$.

Figure 7: $T_{\rm ET}$ and $T$ with $t_{\rm u}=100$ ms, $N_{\rm
CD}^{(1.1)}=5$, $N_{\rm VI}=1$, $\epsilon_{\rm CD}=10^{-1}$, and
$\epsilon_{\rm VI}=10^{-3}$.

###  IV-C Efficient Evaluation for Different Kinds of Constraints and
Properties for Optimization

In this subsection, we will propose the methods for evaluating some kinds of
constraints efficiently with the help of conclusions in Sections IV-A and
IV-B. Then, we will present some properties of the communication and
computation processes in this system, which are useful for further
optimization.

In Sections IV-A and IV-B, we have derived the CDF of $T_{\rm CL}$, $T_{\rm
ET}$, $T_{\rm FL}$, and $T$. Besides, we know the expectation and variance of
each latency component in this system. Thus, all constraints mentioned in
Section II-B can be characterized by the obtained conclusions. However, some
constraints can be evaluated more efficiently. Thus, we present the efficient
evaluating methods for these constraints by SPA-based methods.

Eq. (9) has a constraint for the conditional expectation
$\mathbb{E}\left\\{\left(T_{\rm ET}-\tau_{\rm PF}\right)^{+}\right\\}$ . If we
want to calculate it, the intuitive method is to use the PDF of $T_{\rm ET}$.
However, we can not obtain the PDF for the general case since $T_{\rm ET}$ may
not be a continuous random variable. Besides, it seems possible to obtain
$f_{\rm ET}(x)$ by Eq. (49) from the perspective of theoretical analysis when
$N_{\rm CD}+N_{\rm VI}$ are large or $t_{\rm u}$ is small. However, the
closed-form expression of $\Xi_{3}(x)$ is hard to obtain in Eq. (49), which
makes calculating $f_{\rm ET}(x)$ in a continuous interval impossible. One
possible method to approximate the PDF of $T_{\rm ET}$ is discretization as
done in our previous work [22].

Though we can design a discrete approximation for the PDF of $T_{\rm ET}$
based on Lemma 3, it is computation-intensive since we have to solve SPA
equations at a large number of points when granularity is small. Instead, we
present an efficient method to derive $\mathbb{E}\left\\{\left(T_{\rm
ET}-\tau_{\rm PF}\right)^{+}\right\\}$ in Corollary 1.

Corollary 1. Under the condition that $N_{\rm CD}+N_{\rm VI}$ is large or
$t_{\rm u}$ is small, $\mathbb{E}\left\\{\left(T_{\rm ET}-\tau_{\rm
PF}\right)^{+}\right\\}$ can be approximated by Eq. (54), where $\hat{F}_{\rm
ET}(\tau_{\rm PF})$, $\hat{f}_{\rm ET}(\tau_{\rm PF})$, and $\Xi_{3}(\tau_{\rm
PF})$ are given in Lemma 3.

###### Proof:

Corollary 1 is based on the Eq. (2) in [34]. However, this equation requires
that the random variable should be continuous. Therefore, we adopt the
continuous approximation proposed in Section VI. ∎

Based on Eq. (54) in Corollary 1, we can evaluate
$\mathbb{E}\left\\{\left(T_{\rm ET}-\tau_{\rm PF}\right)^{+}\right\\}$ by only
calculating the CDF and PDF of $T_{\rm ET}$ at one point, which is more
efficient than the discrete-approximation method.

Then, we turn our focus to Eq. (11). The communication latency of PF data is
stochastic for this constraint. To calculate $\Pr\\{|T_{\rm ET}-T_{\rm
PF}|<\tau_{\rm td}\\}$, it is equivalent to calculate

$\Pr\\{T_{\rm ET}-T_{\rm PF}<\tau_{\rm td}\\}-\Pr\\{T_{\rm ET}-T_{\rm
PF}<-\tau_{\rm td}\\}.\vspace{-1mm}$ (55)

Thus, we can use the similar method proposed in Theorem 1 to adopt the SPA-
based method by first deriving the CGF of $T_{\rm ET}-T_{\rm PF}$. Due to the
layout limitation, we omit the details.

Then, we want to provide some useful properties for optimizing the system’s
performance. For the PF data, we give a conclusion on the period allocation
for the case with CSIRT and FSD to minimize the average power consumption for
the PF data transmission. For the ET data, we present a conclusion related to
the choice of compression ratio.

Lemma 4. For the case that the transmission of PF data with CSIRT and FSD, the
period allocation to achieve a minimum $P_{\rm PF1}+P_{\rm PF2}$ is $\tau_{\rm
PF1}=\tau_{\rm PF2}=\frac{\tau_{\rm PF}}{2}$.

###### Proof:

By rewriting Eq. (17), we obtain that

$P_{j}=\frac{N_{0}B_{\rm
PF}}{c_{1}d_{j}^{-\ell_{j}}}\left(2^{\frac{R_{j}}{B_{\rm
PF}}}-1\right)\left(c_{2}+(\tau_{j})^{-c_{3}}\right).\vspace{-1mm}$ (56)

It is obvious that $P_{j}$ is a convex function of $\tau_{j}$. Thus, we have
the conclusion in Lemma 4 by solving a convex optimization problem. ∎

Lemma 5. For $\zeta_{\rm d}(\kappa)$ shown in Eqs. (23) and (24) with
$\omega_{7}>1+\frac{\omega_{8}}{\omega_{6}}$, $\frac{\kappa}{\zeta_{\rm
d}({\kappa})}$ is a decreasing function of $\kappa$. For $\zeta_{\rm
d}(\kappa)$ shown in Eq. (24) with $\omega_{7}\leq 1$,
$\frac{\kappa}{\zeta_{\rm d}({\kappa})}$ is an increasing function of
$\kappa$.

###### Proof:

See Appendix C. ∎

In Lemma 4, we show that it is resource-efficient to set equal deadlines for
the transmission of PF data in two hops. In Lemma 5, we present conclusions
for the monotonic properties of $\frac{\kappa}{\zeta_{\rm d}(\kappa)}$, which
is related to the distribution, expectation, and variance of $T_{\rm ET}^{\rm
d}$. This conclusion can provide instructions for the choice of $\kappa$.

##  V Simulation and Numerical Results

In this section, we validate the proposed framework through Monte Carlo
simulations and discuss the results from the perspective of the envisioned 6G
scenarios. The parameters of the simulation and numerical calculations are
mainly based on [8] and [32]. Specifically, we set $\alpha_{\rm MEC}=1.25$,
$\alpha_{\rm RA}=1.5$, $\beta_{\rm LL}=\beta_{\rm VI}=1$, $N_{\rm HL}=3$,
$N_{\rm LL}=1$, $n_{\rm HL}=128$ KB, $\epsilon_{\rm HL}=\epsilon_{\rm
LL}=10^{-5}$, $\chi_{\rm RA}=5$ GHz, $\chi_{\rm MEC}=15$ GHz,
$\Delta=10^{-5}$, and $n_{\rm ET}=1$ MB. $\zeta_{\rm c}(\kappa)$ and
$\zeta_{\rm d}(\kappa)$ follow Eq. (22) and Eq. (23), respectively, with
$\psi=3.5$ and $\omega_{0}=0.1$.

In Figs. 7-7, we present the simulation and numerical results of CDF of
$T_{\rm ET}$ and $T$. In these figures, we let $N_{\rm CD}^{(1.1)}$ denote the
number of compressed-data packets with $\kappa=1.1$. Thus, the number of
compressed-data packets with $\kappa=1.2$ is denoted by $N_{\rm
CD}^{(1.2)}=\left\lceil\frac{1.1}{1.2}N_{\rm CD}^{(1.1)}\right\rceil$. The
curves with the legend ‘Simulation’ refer to the Monte-Carlo simulation
results averaged over $10^{7}$ independent simulations, while the curves with
the legend ‘Theorem 1’, ‘Lemma 3’, and ‘Theorem 2’ represent the numerical
results of Alg. 1, Eq. (46), and Eq. (53), respectively. We considered four
scenarios in Figs. 7-7. Fig. 7 represents the general scenario. Fig. 7
represents the scenario with a large number of packets. Fig. 7 represents the
scenarios with a small $t_{\rm u}$ and a small number of packets. Fig. 7
represents the extreme scenario in which the number of packets is small and
$t_{\rm u}$ is large. By comparing the simulation and numerical results, we
find that both Theorem 1, Lemma 3, and Theorem 2 perform well in Figs. 7, 7,
and 7. However, Theorem 1 performs much better than Lemma 3 in Fig. 7(a) and
much better than Theorem 2 in Fig. 7(c). That is because, in Lemma 3 and
Theorem 2, we adopted CLT to continuously approximate the CGF of $T_{\rm ET}$
and $T$. The results in these four figures validate the accuracy of Theorem 1.
Besides, as we analyzed in Section IV, Lemma 3 can be applied when $t_{\rm u}$
is small or the number of packets is large. The results in Figs. 7, 7, and 7
show the effectiveness of Lemma 3 and Theorem 2. Note that we adopt $t_{\rm
u}=5$ ms in Fig. 7, which should have a worse performance than $t_{\rm u}<5$
ms. This result also proves the accuracy of Lemma 3 and Theorem 2 since
$t_{\rm u}=5$ ms has presented a good accuracy. The reason that Lemma 3 and
Theorem 2 perform not well in Fig. 7 is that the discrete components of
$T_{\rm ET}$ and $T$ are dominant, which can be seen from curves of simulation
results in Fig. 7. Except for the extreme scenario, we can use Lemma 3 and
Theorem 2 for the CDF approximation with satisfying accuracy.

(a) Case 1. (b) Case 2.

(c) Case 3. (d) Case 4.

Figure 8: Approximation of $\mathbb{E}\\{\left(T_{\rm ET}-\tau_{\rm
PF}\right)^{+}\\}$ versus $\tau_{\rm PF}$.

There are more interesting details. First, we find that Theorem 2 seems to
perform better than Theorem 1 in some points. The main reason may be that we
set a threshold $\Delta$ in Alg. 1, which limited the preciseness of the
results. Second, from the results shown in Fig. 7, the curves of Theorem 1
match well with those of the simulation results. This result shows the
universality of Theorem 1 even under the extreme scenario. Moreover, we see
that with the increase of $\kappa$, the CDF is not always decreasing as shown
in Figs. 7(a) and 7(c). This result shows there are difficulties in
determining the optimal compression ratio since the CDF with a given threshold
$x$ is not monotonic with the increase of $\kappa$. However, when $x$ is
large, e.g., $1-\Pr\\{T_{\rm ET}<x\\}$ is small, the CDF decreases with the
increase of $\kappa$. Since we mainly focus on the threshold with which
$1-\Pr\\{T_{\rm ET}<x\\}$ is small, it is important to find and prove the
condition that $1-\Pr\\{T_{\rm ET}<x\\}$ is decreasing with the increase of
$\kappa$ and a given $x$ in the future work.

We then present the simulation and numerical results of
$\mathbb{E}\left\\{\left(T_{\rm ET}-\tau_{\rm PF}\right)^{+}\right\\}$ in Fig.
8. The parameter settings in Figs. 8(a)-(d) are identical with those in Figs.
7-7, respectively. The curves with the legend ‘Simulation’ refer to the Monte-
Carlo simulation results averaged over $10^{7}$ independent simulations, while
the curves with the legend ‘Corollary 1’ represent the numerical results of
Eq. (54). We find that the numerical results match well with the simulation
results in Fig. 8(b). Thus, this result proves the effectiveness of Corollary
1. Moreover, the numerical results in Figs. 8(a) and 8(c) also provide
satisfying performance. The difference between the numerical and simulation
results becomes obvious in Fig. 8(d). For the extreme scenario shown in Fig.
8(d), we need a generalization of Eq. (54) to obtain an efficient and precise
approximation for the conditional expectation, which is an important future
work.

##  VI Conclusion

In this paper, we focused on formulating a unified timing-analysis framework
and proposing a general method to characterize the distribution of the closed-
loop timing performance in CGC systems. The multi-modal feedback was divided
into PF and ET data due to the heterogeneity of the feedback mechanism.
Covering both timeliness, jitter, and reliability, we formulated a unified
framework of timing analysis for CGC systems with different configurations,
which includes average-based and tail-based constraints. The randomness of
computation, compression, and communication were incorporated into the closed-
loop timing analysis. To address the difficulties resulting from the sum of
multiple random latency components, an SPA based method was conceived, which
helped us derive the analytical expression of the closed-loop latency and
other timing metrics. Important future topics include further optimization
based on this unified framework, the introduction of emerging timing metrics
into this framework, and the generalization of the approximation for the
conditional expectation of timing metrics.

##  Appendix A Proof of Lemma 1

The essence of the proof is to use the LR formula [20]. We first obtain the
cumulant generating function (CGF) of $T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm
ET}+T^{\rm cp}_{\rm VI}$. The CGF of gamma random variable
$X_{1}\sim\text{Gamma}(\alpha,\beta)$ is given by

$K_{X_{1}}(s)=-\alpha\ln\left(1-\frac{s}{\beta}\right),\quad
s<\beta.\vspace{-1mm}$ (57)

Based on Eq. (57), we obtain the CGF of $T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm
ET}+T^{\rm cp}_{\rm VI}$, which is given in Eq. (32), where
$s<\min\left\\{\frac{\chi_{\rm RA}\alpha_{\rm RA}}{n_{\rm ET}\zeta_{\rm
c}(\kappa)},\frac{\chi_{\rm MEC}\alpha_{\rm MEC}\kappa}{n_{\rm ET}\zeta_{\rm
d}(\kappa)},\frac{\chi_{\rm MEC}\beta_{\rm VI}}{n_{\rm ET}}\right\\}$. To
obtain the CDF of $T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm ET}+T^{\rm cp}_{\rm VI}$,
we need to solve the saddlepoint equation to obtain the saddlepoint $s_{x}$.
The saddlepoint equation is given by $L^{\prime}(s_{x})=x$, where
$L^{\prime}(s)$ is given by

$\displaystyle\\!\\!\\!L^{\prime}(s)\\!=\\!\frac{\alpha_{\rm
RA}}{\frac{\chi_{\rm RA}\alpha_{\rm RA}}{n_{\rm ET}\zeta_{\rm
c}(\kappa)}-s}\\!+\\!\frac{\alpha_{\rm MEC}}{\frac{\chi_{\rm MEC}\alpha_{\rm
MEC}\kappa}{n_{\rm ET}\zeta_{\rm d}(\kappa)}-s}\\!+\\!\frac{\alpha_{\rm
MEC}}{\frac{\chi_{\rm MEC}\beta_{\rm VI}}{n_{\rm ET}}-s}.$ (58)

$L^{\prime}(s_{x})$ is an increasing function of $s_{x}$. Thus, the inverse
function of $L^{\prime}:s_{x}\mapsto x$ exists, which is denoted by
$\Xi_{1}:x\mapsto s_{x}$. There is a unique solution of the saddlepoint
equation, i.e., $\Xi_{1}(x)$, which can be solved by binary search.

Then, we divide the following proof into two cases. Let
$\theta=\mathbb{E}\\{T^{\rm c}_{\rm ET}+T^{\rm d}_{\rm ET}+T^{\rm cp}_{\rm
VI}\\}$ as shown in (28). The first case is that $x\neq\theta$ holds.
According to the LR formula, we obtain

$F_{1}(x)=\Phi(v_{x})+\phi(v_{x})\left(\frac{1}{v_{x}}-\frac{1}{u_{x}}\right),\vspace{-1mm}$
(59)

where $v_{x}={\rm sign}(s_{x})\sqrt{2(xs_{x}-L(s_{x}))}$, and
$u_{x}=s_{x}\sqrt{L^{\prime\prime}(s_{x})}$. The second-order derivative of
$L(s)$ is given by

$\displaystyle L^{\prime\prime}(s)=\frac{\alpha_{\rm
RA}}{\left(\frac{\chi_{\rm RA}\alpha_{\rm RA}}{n_{\rm ET}\zeta_{\rm
c}(\kappa)}-s\right)^{2}}$ $\displaystyle+\frac{\alpha_{\rm
MEC}}{\left(\frac{\chi_{\rm MEC}\alpha_{\rm MEC}\kappa}{n_{\rm ET}\zeta_{\rm
d}(\kappa)}-s\right)^{2}}$ (60) $\displaystyle+\frac{\alpha_{\rm
MEC}}{\left(\frac{\chi_{\rm MEC}\beta_{\rm VI}}{n_{\rm ET}}-s\right)^{2}}.$

Thus, by substituting Eqs. (32) and (60) into the expressions of $v_{x}$ and
$u_{x}$, we obtain the first expression in Eq. (29).

The second case is that $x=\theta$ holds. Under this condition, $\Pr\\{T_{\rm
ET}<x\\}$ is given by

$\displaystyle F_{1}(x)$
$\displaystyle=\frac{1}{2}+\frac{L^{\prime\prime\prime}(0)}{6\sqrt{2\pi}L^{\prime\prime}(0)^{3/2}}.$
(61)

We let $\iota_{1}=L^{\prime\prime}(0)^{3/2}$ and
$\iota_{2}=L^{\prime\prime\prime}(0)$, which are given in Eqs. (26) and (27),
respectively. By substituting Eqs. (26) and (27) into Eq. (61), we obtain Eq.
(29).

For the PDF, we adopt the saddlepoint density functions in [20]. By
substituting Eqs. (32) and (60) into the expression of the saddlepoint density
function, we obtain Eq. (33). Due to the layout limitation, we omit the proof.

##  Appendix B Proof of Lemma 2

Similar to the proof of Lemma 1, we shall first obtain the CGF of $T^{\rm
cm}_{\rm CD}+T^{\rm cm}_{\rm VI}$. The CGF of negative binomial random
variable $X_{2}\sim\text{NB}(N,p)$ is given by

$K_{X_{2}}(s)=N\left(s+\ln p-\ln\left(1-(1-p)e^{s}\right)\right),$ (62)

where $s<-\ln(1-p)$. Based on Eq. (62), we obtain the CGF of $T^{\rm cm}_{\rm
CD}+T^{\rm cm}_{\rm VI}$, which is given by Eq. (36). The first-order
derivative of $J(s)$ is thus given by

$\displaystyle J^{\prime}(s)=\frac{N_{\rm CD}}{1-\epsilon_{\rm
CD}e^{s}}+\frac{N_{\rm VI}}{1-\epsilon_{\rm VI}e^{s}},$ (63)

where $s<\min\left\\{-\ln\epsilon_{\rm CD},-\ln\epsilon_{\rm VI}\right\\}$.
The saddlepoint equation is thus given by $J^{\prime}(s_{k})=k$.
$J^{\prime}(s_{k})$ is an increasing function of $s_{k}$. Thus, the inverse
function of $J^{\prime}:s_{k}\mapsto x$ exists, which is denoted by
$\Xi_{2}:x\mapsto s_{k}$. The solution of the saddlepoint equation is thus
given by $\Xi_{2}(k)$.

Based on Eq. (63), we can also obtain

$\displaystyle J^{\prime\prime}(s)=\frac{\epsilon_{\rm CD}N_{\rm
CD}e^{s}}{\left(1-\epsilon_{\rm CD}e^{s}\right)^{2}}+\frac{\epsilon_{\rm
VI}N_{\rm VI}e^{s}}{\left(1-\epsilon_{\rm VI}e^{s}\right)^{2}},$ (64)

where $s<\min\left\\{-\ln\epsilon_{\rm CD},-\ln\epsilon_{\rm VI}\right\\}.$
According to the saddlepoint mass function [20], we obtain the PMF for
$k>N_{\rm CD}+N_{\rm VI}$, which is the interior of the span of the support of
$T^{\rm cm}_{\rm CD}+T^{\rm cm}_{\rm VI}$. For $k=N_{\rm CD}+N_{\rm VI}$, the
probability can be simply obtained as $(1-\epsilon_{\rm CD})^{N_{\rm
CD}}(1-\epsilon_{\rm VI})^{N_{\rm VI}}$.

##  Appendix C Proof of Lemma 5

Let $G(\kappa)=\frac{\kappa}{\zeta_{\rm d}({\kappa})}$. For $\zeta_{\rm
d}(\kappa)$ shown in Eq. (23), we obtain

$\frac{{\rm d}G(\kappa)}{{\rm
d}\kappa}=\frac{e^{\psi\kappa}(1-\kappa)-e^{\psi}}{\omega_{0}\left(e^{\psi\kappa}-e^{\psi}\right)}.$
(65)

Since $\kappa\geq 1$, $\frac{{\rm d}G(\kappa)}{{\rm d}\kappa}<0$. For
$\zeta_{\rm d}(\kappa)$ shown in Eq. (24), we have

$\frac{{\rm d}G(\kappa)}{{\rm
d}\kappa}=\frac{\omega_{5}\omega_{6}(1-\omega_{7})\kappa^{\omega_{7}}+\omega_{5}\omega_{8}}{\left(\omega_{5}(\omega_{6}\kappa^{\omega_{7}}+\omega_{8})\right)^{2}}.$
(66)

From this equation, we can find that
$\omega_{5}\omega_{6}(1-\omega_{7})\kappa^{\omega_{7}}$ is decreasing with
$\kappa$ when $\omega_{7}>1$. Thus, the maximum value of
$\omega_{5}\omega_{6}(1-\omega_{7})\kappa^{\omega_{7}}$ is achieved at
$\kappa=1$ since $\kappa\geq 1$, which is
$\omega_{5}\omega_{6}(1-\omega_{7})$. Thus, if
$\omega_{5}\omega_{6}(1-\omega_{7})\kappa^{\omega_{7}}+\omega_{5}\omega_{8}<0$,
i.e., $\omega_{7}>1+\frac{\omega_{8}}{\omega_{6}}$, the maximum value of
$\frac{{\rm d}G(\kappa)}{{\rm d}\kappa}<0$.

For $\omega_{7}\leq 1$,
$\omega_{5}\omega_{6}(1-\omega_{7})\kappa^{\omega_{7}}\geq 0$. Thus,
$\frac{{\rm d}G(\kappa)}{{\rm d}\kappa}>0$.

## References

  * [1] P. Popovski, F. Chiariotti, K. Huang, A. E. Kalør, M. Kountouris, N. Pappas, and B. Soret, “A perspective on time toward wireless 6G,” _Proc. IEEE_ , vol. 110, no. 8, pp. 1116–1146, Aug. 2022.
  * [2] “Framework and overall objectives of the future development of IMT for 2030 and beyond,” Recommendation ITU-R M.2160-0, Nov. 2023.
  * [3] D. Gündüz, Z. Qin, I. E. Aguerri, H. S. Dhillon, Z. Yang, A. Yener, K. K. Wong, and C.-B. Chae, “Beyond transmitting bits: Context, semantics, and task-oriented communications,” _IEEE J. Sel. Areas Commun._ , vol. 41, no. 1, pp. 5–41, Jan. 2023.
  * [4] D. Gündüz, F. Chiariotti, K. Huang, A. E. Kalør, S. Kobus, and P. Popovski, “Timely and massive communication in 6G: Pragmatics, learning, and inference,” _IEEE BITS Inf. Theory Mag._ , vol. 3, no. 1, pp. 27–40, Mar. 2023.
  * [5] E. C. Strinati, P. Di Lorenzo, V. Sciancalepore, A. Aijaz, M. Kountouris, D. Gündüz, P. Popovski, M. Sana, P. A. Stavrou, B. Soret _et al_., “Goal-Oriented and Semantic Communication in 6G AI-Native Networks: The 6G-GOALS Approach,” 2024, _arXiv:2402.07573_. [Online]. Available: https://arxiv.org/abs/2312.09007.
  * [6] M. Xu, W. C. Ng, W. Y. B. Lim, J. Kang, Z. Xiong, D. Niyato, Q. Yang, X. Shen, and C. Miao, “A full dive into realizing the edge-enabled metaverse: Visions, enabling technologies, and challenges,” _IEEE Commun. Surveys Tuts._ , vol. 25, no. 1, pp. 656–700, 1st Quart., 2023.
  * [7] “Feasibility Study on Localized Mobile Metaverse Services,” V19.1.0, document 3GPP TR 22.856, Sep. 2023.
  * [8] S. Suman, F. Chiariotti, Č. Stefanović, S. Došen, and P. Popovski, “Statistical characterization of closed-loop latency at the mobile edge,” _IEEE Trans. Commun._ , vol. 71, no. 7, pp. 4391–4405, Jul. 2023.
  * [9] D. Han, W. Chen, B. Bai, and Y. Fang, “Offloading optimization and bottleneck analysis for mobile cloud computing,” _IEEE Transactions on Communications_ , vol. 67, no. 9, pp. 6153–6167, Sept. 2019.
  * [10] J. R. Lorch and A. J. Smith, “Improving dynamic voltage scaling algorithms with pace,” _SIGMETRICS Perform. Eval. Rev._ , vol. 29, no. 1, p. 50–61, Jun. 2001.
  * [11] E. Steinbach, S. Hirche, M. Ernst, F. Brandi, R. Chaudhari, J. Kammerl, and I. Vittorias, “Haptic communications,” _Proc. IEEE_ , vol. 100, no. 4, pp. 937–956, Apr. 2012.
  * [12] K. Antonakoglou, X. Xu, E. Steinbach, T. Mahmoodi, and M. Dohler, “Toward haptic communications over the 5G tactile internet,” _IEEE Commun. Surveys Tuts._ , vol. 20, no. 4, pp. 3034–3059, 4th Quart., 2018.
  * [13] P. Popovski, “Time, simultaneity, and causality in wireless networks with sensing and communications,” _IEEE Open J. Commun. Soc._ , vol. 5, pp. 1693–1709, Mar. 2024.
  * [14] B. Kizilkaya, C. She, G. Zhao, and M. A. Imran, “Task-oriented prediction and communication co-design for haptic communications,” _IEEE Trans. Veh. Tech._ , vol. 72, no. 7, pp. 8987–9001, Jul. 2023.
  * [15] D. Wen, P. Liu, G. Zhu, Y. Shi, J. Xu, Y. C. Eldar, and S. Cui, “Task-oriented sensing, computation, and communication integration for multi-device edge AI,” _IEEE Trans. Wireless Commun._ , vol. 23, no. 3, pp. 2486–2502, Mar. 2024.
  * [16] D. Van Huynh, S. R. Khosravirad, A. Masaracchia, O. A. Dobre, and T. Q. Duong, “Edge intelligence-based ultra-reliable and low-latency communications for digital twin-enabled metaverse,” _IEEE Wireless Commun. Lett._ , vol. 11, no. 8, pp. 1733–1737, Aug. 2022.
  * [17] P. M. de Sant Ana, N. Marchenko, P. Popovski, and B. Soret, “Age of loop for wireless networked control systems optimization,” in _Proc. IEEE Int. Symp. Pers. Indoor Mob. Radio Commun. (PIMRC)_ , Sept. 2021.
  * [18] J. Cao, X. Zhu, S. Sun, P. Popovski, S. Feng, and Y. Jiang, “Age of loop for wireless networked control system in the finite blocklength regime: Average, variance and outage probability,” _IEEE Trans. Wireless Commun._ , vol. 22, no. 8, pp. 5306–5320, Aug. 2023.
  * [19] L. Li, L. Yang, X. Guo, Y. Shi, H. Wang, W. Chen, and K. B. Letaief, “Delay analysis of wireless federated learning based on saddle point approximation and large deviation theory,” _IEEE J. Sel. Areas Commun._ , vol. 39, no. 12, pp. 3772–3789, Dec. 2021.
  * [20] R. W. Butler, _Saddlepoint approximations with applications_. Cambridge, U.K.: Cambridge Univ. Press, 2007, vol. 22.
  * [21] A. O. Kislal, A. Lancho, G. Durisi, and E. G. Ström, “Efficient evaluation of the error probability for pilot-assisted URLLC with massive MIMO,” _IEEE J. Sel. Areas Commun._ , vol. 41, no. 7, pp. 1969–1981, Jul. 2023.
  * [22] L. Li, W. Chen, and K. B. Letaief, “On power-latency-throughput tradeoff of diversity enabled delay-bounded communications,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , Jun. 2023, pp. 2141–2146.
  * [23] H. Cui, Y. Du, Q. Yang, Y. Shao, and S. C. Liew, “LLMind: Orchestrating AI and IoT with LLMs for Complex Task Execution,” 2023, _arXiv:2312.09007_. [Online]. Available: https://arxiv.org/abs/2312.09007.
  * [24] T. Kallehauge, P. Ramirez-Espinosa, A. E. Kalør, and P. Popovski, _Statistical Characterization of URLLC : Frequentist and Bayesian Approaches_. New York, NY, USA: John Wiley & Sons, Ltd, 2023, ch. 2, pp. 15–59.
  * [25] L. Li, W. Chen, and K. B. Letaief, “Simple bounds on delay-constrained capacity and delay-violation probability of joint queue and channel-aware wireless transmissions,” _IEEE Trans. Wireless Commun._ , vol. 22, no. 4, pp. 2744–2759, Apr. 2023.
  * [26] L. Li, W. Chen, and K. B. Letaief, “Fitting Empowered Cross-Layer Scheduling for Real-Time Wireless Communications,” accepted by _Proc. IEEE Int. Conf. Commun. (ICC)_ , 2024.
  * [27] M. Bennis, M. Debbah, and H. V. Poor, “Ultrareliable and low-latency wireless communication: Tail, risk, and scale,” _Proc. IEEE_ , vol. 106, no. 10, pp. 1834–1853, Oct. 2018.
  * [28] C.-F. Liu, M. Bennis, M. Debbah, and H. V. Poor, “Dynamic task offloading and resource allocation for ultra-reliable low-latency edge computing,” _IEEE Trans. Commun._ , vol. 67, no. 6, pp. 4132–4150, Jun. 2019.
  * [29] L. Li, W. Chen, P. Popovski, and K. B. Letaief, “Reliability-Latency-Rate Tradeoff in Low-Latency Communications with Finite-Blocklength Coding,” 2023, _arXiv:2309.06769_. [Online]. Available: https://arxiv.org/abs/2309.06769.
  * [30] L. Li, W. Chen, and K. B. Letaief, “Ultra-reliable and low latency wireless communications with burst traffics: A large deviation method,” in _Proc. IEEE Global Commun. Conf. (GLOBECOM)_ , Dec. 2021.
  * [31] L. Li, W. Chen, and K. B. Letaief, “Wireless Communications with Hard Delay Constraints: Cross-Layer Scheduling with Its Performance Analysis,” _IEEE Internet Things J._ , major revision, 2024.
  * [32] X. Li, C. You, S. Andreev, Y. Gong, and K. Huang, “Wirelessly powered crowd sensing: Joint power transfer, sensing, compression, and transmission,” _IEEE J. Sel. Areas Commun._ , vol. 37, no. 2, pp. 391–406, Feb. 2019.
  * [33] T. T. Nguyen, V. N. Ha, L. B. Le, and R. Schober, “Joint data compression and computation offloading in hierarchical fog-cloud systems,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 1, pp. 293–309, Jan. 2020.
  * [34] Y. Zhang and Y. K. Kwok, “Saddlepoint approximations to tail expectations under non-gaussian base distributions: option pricing applications,” _J. Appl. Stat._ , vol. 47, no. 11, pp. 1936–1956, 2020.