# Task-Oriented Prediction and Communication Co-Design for Haptic
Communications

Burak Kizilkaya, Changyang She,
Guodong Zhao, and Muhammad Ali Imran Copyright (c) 2015 IEEE. Personal use of
this material is permitted. However, permission to use this material for any
other purposes must be obtained from the IEEE by sending a request to pubs-
<EMAIL_ADDRESS>Kizilkaya, G. Zhao and M. A. Imran are with School of
Engineering, University of Glasgow, Glasgow, G12 8QQ, U.K. (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>muhammad.imran@glasgow.ac.uk).C. She is with School of Electrical and
Information Engineering, The University of Sydney, Sydney, NSW 2006,
Australia. (e-mail: shechangyang@gmail.com).

###### Abstract

Prediction has recently been considered as a promising approach to meet low-
latency and high-reliability requirements in long-distance haptic
communications. However, most of the existing methods did not take features of
tasks and the relationship between prediction and communication into account.
In this paper, we propose a task-oriented prediction and communication co-
design framework, where the reliability of the system depends on prediction
errors and packet losses in communications. The goal is to minimize the
required radio resources subject to the low-latency and high-reliability
requirements of various tasks. Specifically, we consider the _just noticeable
difference_ (JND) as a performance metric for the haptic communication system.
We collect experiment data from a real-world teleoperation testbed and use
_time-series generative adversarial networks_ (TimeGAN) to generate a large
amount of synthetic data. This allows us to obtain the relationship between
the JND threshold, prediction horizon, and the overall reliability including
communication reliability and prediction reliability. We take 5G New Radio as
an example to demonstrate the proposed framework and optimize bandwidth
allocation and data rates of devices. Our numerical and experimental results
show that the proposed framework can reduce wireless resource consumption up
to $77.80\%$ compared with a task-agnostic benchmark.

###### Index Terms:

Resource management, communication system reliability, prediction methods

##  I Introduction

Haptic communications will lie the foundation for emerging teleoperation
applications such as remote surgery and diagnosis in healthcare [1], remote
laboratory and training in education [2], remote driving in transportation
[3], and advanced manufacturing in Industry 4.0 [4]. A typical teleoperation
system with haptic communications consists of three main domains [5, 6, 7],
namely master domain, slave domain and communication domain. At master domain,
a master device equipped with a haptic interface transmits control commands
over a communication system, where the control commands are usually a series
of positions, orientations, velocities, or poses. Then, a slave device, in
slave domain, executes the received control commands to complete a task and
provides haptic feedback over a communication system. To provide high quality
user experience, haptic communications need to meet stringent requirements on
latency and reliability [6]. These requirements are well-aligned with _ultra-
reliable low-latency communications_ (URLLC) in the fifth generation (5G)
cellular networks [8]. Specifically, the _end-to-end_ (E2E) delay should be
around $1$ ms and reliability should be higher than $99.999$%. However, it is
very challenging to meet the latency and reliability requirements, especially
when the communication distance between the master and slave devices is longer
than $300$ km [7].

In the existing literature [9, 10, 11, 12, 13, 14, 15], prediction has been
considered as a promising approach to meet the stringent _quality of service_
(QoS) requirements, such as latency and reliability. In a packetized
predictive control system, predicted packets are sent to the receiver in
advance and are used in case of packet loss [9]. The authors of this work
jointly optimized prediction and communication systems to minimize wireless
resource consumption. In [10], a two-stage prediction framework was proposed
for haptic feedback prediction in remote driving to assist the driver in poor
network conditions, where prediction errors were not considered [9, 10]. In
[11], a haptic feedback prediction framework was proposed in remote surgery
use case. Gaussian mixture regression model was used to predict haptic
feedback to surgeon in a needle insertion process, where the prediction
accuracy was measured by _Root Mean Square Error_ (RMSE) between predicted
force and ground truth. In [12], authors proposed a predictive actuation
framework by predicting missing state at receiver using previously received
state. A model-based prediction algorithm was used to minimize average _age of
information_ (AoI) and transmit power. However, in both [11] and [12], human
perception capabilities were not taken into account.

Recently, user experience was considered in haptic communications, and was
characterized by _just noticeable difference_ (JND) [16]. Here, JND is defined
as the minimum difference between the master and the slave that can be
perceived by human users. Since human can only notice the difference when it
is larger than a certain JND threshold, it is reasonable to design the
predictor by considering the JND violation probability, which is defined as
the probability that the difference between the predicted value and the ground
truth is larger than a threshold. In [13], the authors used the JND violation
probability as the performance metric to measure the reliability of the
predictor and proposed a prediction and communication co-design framework to
reduce user experienced delay, where a model-based prediction algorithm was
deployed at the transmitter. It was demonstrated that prediction and
communication co-design improves the tradeoff between user experienced delay
and reliability compared to the communication system with no prediction. In
[14], JND violation probability was used as the overall performance metric to
illustrate the benefits of prediction on AoI in _Intelligent Transport
Systems_ (ITS). Similarly in [15], the tradeoff between prediction length and
AoI was analyzed by using JND violation probability in prediction design. The
results showed that the prediction algorithm used in the system can help to
improve the AoI performance.

In practice, the JND threshold is a task dependent parameter with large
variation. For example, the JND threshold value would be different when human
operator (master) controls a robotic arm (slave) for different tasks. The
large JND threshold is expected when human operator has the full arm movement.
In contrast, the small JND threshold is required in fine control. However,
current design methods cannot capture such JND threshold dynamics since the
fundamental relationship between JND threshold and QoS is not clear yet. As a
result, current design methods suffer from over-provisioning of wireless
resource that impedes their implementation in real-world systems. Recently
proposed task-oriented (or, goal-oriented/semantic) communications [17, 18,
19] advocates similar approach showing that overall system performance not
only depends on bit-level performance but also whether the intended task is
accomplished or not, given the application scenario and communication
resources available. Such a perspective envisions a paradigm shift in
communication system design from bit-level to task-level. However, this is not
an easy task since task-oriented communications is the combination of
different principles from control theory, information theory and computer
science. Effectiveness and efficiency can be enhanced by jointly considering
different task objectives and task dependent parameters. In this study, we
consider JND as a task dependent parameter and jointly design prediction and
communication systems.

In addition, it is very challenging to obtain the relationship between JND
threshold and QoS. Unlike communication systems that are built upon
fundamental theories, most of the prediction algorithms are developed via
data-driven design. This raises a huge challenge when the required error
probability is at the level of $10^{-5}$, which requires a large number of
real-world data samples to evaluate the probability of the rare event. If we
use model-based prediction, it is possible to derive the prediction error
probability, but the mismatch between the over simplified theoretical models
and practical complex systems will lead to inaccurate results. Therefore, we
need to develop innovative methods to overcome the above challenge.

In this paper, we propose a task-oriented prediction and communication co-
design framework in the context of teleoperation system, where the utilization
efficiency of the communication system is maximized subject to the
requirements of different operating tasks. In particular, in the scenario with
limited real-world data samples, we generate synthetic data via time-series
generative adversarial networks (TimeGAN). This allows us to obtain the
relationship between JND threshold and the prediction error probability that
is below $10^{-5}$. We take 5G New Radio as an example to demonstrate the
proposed framework, where we compare the performance of the systems with and
without task-oriented design. We further compare the performance difference of
deploying the predictor at transmitter and receiver sides. Specifically, the
main contributions of this paper are listed below:

  * •

We propose a task-oriented prediction and communication co-design framework
with a predictor at the receiver side. We derived an upper bound of the
overall error probability in the framework by taking packet losses and
prediction errors into account. From the upper bound, we reveal the tradeoff
between the resource utilization efficiency and the overall reliability.

  * •

To illustrate how to use this framework in practical system design, we take
remote robotic control in 5G New Radio as an example. Then, we formulate an
optimization problem to optimize bandwidth allocation and communication data
rate subject to constraints on the E2E delay and overall reliability. An
optimization algorithm is proposed to find the optimal solution.

  * •

We collect the real-world data from a teleoperation prototype. We further use
TimeGAN to generate synthetic data for predictor training and testing. With
both synthetic data and real-world data, we illustrate the tradeoff between
the prediction horizon and prediction error probability, and further evaluate
the overall reliability. Our results show that the proposed task-oriented
prediction framework can save up to $77.80\%$ bandwidth compared with a
benchmark design that is task-agnostic.

The rest of the paper is organized as follows. In Section II, we develop a
general design framework for task-oriented prediction and communication co-
design. In Section III, we illustrate packet losses in communications by
taking 5G New Radio as an example. In Section IV, we introduce prediction
algorithms. In Section V, we propose efficient resource allocation with task-
oriented prediction. In Section VI, we present simulation and numerical
results. Section VII concludes this paper. Notations used throughout the paper
are listed in Table I for clarification.

TABLE I: Descriptions of Notations Notation |  Description
---|---
$K_{m}(t)$ |  control command send by $m$-th transmitter at time slot $t$
$n$ |  number of features in a control command
$D^{\rm c}_{m}$ |  E2E communication delay for the $m$-th link
$D^{\rm e}_{m}$ |  user experienced delay for the $m$-th link
$D^{\rm r}_{m}$ |  core network and backhaul delay for the $m$-th link
$D^{\rm q}_{m}$ |  queuing delay for the $m$-th link
$D^{\rm t}_{m}$ |  transmission delay for the $m$-th link
$D^{\rm ch}_{m}$ |  coherence time for the $m$-th link
$D^{\rm th}_{m}$ |  queuing delay threshold for the $m$-th link
$T^{\text{p}}_{m}$ |  prediction horizon for the $m$-th task
$T_{\rm th}$ |  prediction horizon threshold
$D^{\max}_{m}$ |  delay requirement for the $m$-th task
$\epsilon^{\text{q}}_{m}$ |  queuing delay bound violation probability for the $m$-th task
$\epsilon^{\text{d}}_{m}$ |  decoding error probability for the $m$-th task
$\epsilon^{\text{p}}_{m}$ |  prediction error probability for the $m$-th task
$\epsilon^{\rm o}_{m}$ |  overall error probability for the $m$-th task
$\epsilon_{m}^{\max}$ |  maximum tolerable error probability for the $m$-th task
$\delta_{m}$ |  JND threshold for the $m$-th task
$\lambda_{m}$ |  average packet arrival rate for the $m$-th link
$E^{\text{B}}_{m}$ |  effective bandwidth for the $m$-th link
$b_{m}$ |  number of bits per packet for the $m$-th link
$W_{m}$ |  bandwidth for the $m$-th link
$C_{m}$ |  capacity for the $m$-th link
$R_{m}$ |  achievable rate for the $m$-th link
$V_{m}$ |  channel dispersion for the $m$-th link
$\gamma_{m}$ |  SNR for the $m$-th link
$\alpha_{m}$ |  large-scale gain for the $m$-th link
$g_{m}$ |  small-scale gain for the $m$-th link
$P_{m}$ |  maximum transmit power for the $m$-th link
$N_{0}$ |  single sided noise spectral density
$l_{m}$ |  blocklength for the $m$-th link
$A$ |  number of critical tasks
$B$ |  number of non-critical tasks
$Q(.)$ |  Q-function
$Q^{-1}(.)$ |  inverse of the Q-function
$\text{W}_{-1}(.)$ |  -1 branch of Lambert W-function
$f_{\epsilon_{m}^{\rm p}}(.)$ |  function of prediction error probability for the $m$-th task
$f_{\epsilon_{m}^{\rm q}}(.)$ |  function of queuing delay bound violation probability for the $m$-th task

##  II A General Design Framework

Figure 1: Proposed task-oriented prediction and communication co-design
framework system model with considered haptic communication system, where a
human can remotely control robots via a communication system. The predictors
are deployed at receivers to reduce user experienced delays.

We consider a haptic communication system shown in Fig. 1, where a human can
remotely control robots via a communication system. $N$ pairs of transmitters
and receivers are considered over a shared wireless channel, where orthogonal
subchannels are assigned to different transceiver pairs to avoid interference.
The predictors are deployed at receivers to reduce user experienced delays. It
is worth noting that the transmitter and receiver in the proposed design
framework might be any device depending on the application scenario. As an
example of a mission-critical application, a teleoperation scenario in haptic
communications is considered in this study. In a tele-driving [20] scenario,
the receiver can be a remote vehicle, while the transmitter is the remote
controller. Similarly, in an edge-assisted autonomous driving application
[21], the receiver can be an autonomous vehicle with the control unit on the
edge as a transmitter.

###  II-A User Experienced Delay and Delay Requirement

Time is discretized into slots. At the $t$-th time slot, the $m$-th
transmitter sends the control command
$K_{m}(t)=[k^{1}_{m}(t),k^{2}_{m}(t),...,k^{n}_{m}(t)]^{T}$ to the receiver,
where $n$ is the number of features in the command (e.g., joint angles,
angular velocity, forces, and torque). Then, $K_{m}(t)$ arrives at the $m$-th
receiver at $(t+D^{\rm c}_{m})$-th time slot, where $D^{\rm c}_{m}$ is E2E
communication delay. If the communication system has no prediction capability,
the time delay experienced by the user, denoted as $D^{\rm e}_{m}$, is the
same as the communication delay, i.e., $D^{\rm e}_{m}=D^{\rm c}_{m}$111There
are different delay components such as computing delay and control delay.
However, we focus on communication system design in this work and ignore other
delay components.. Here, the communication delay $D^{\rm c}_{m}$ consists of
the delay in core network and backhauls $D^{\rm r}_{m}$, the queuing delay in
the buffer of the base station (BS) $D^{\rm q}_{m}$, and the transmission
delay in the radio access network $D^{\rm t}_{m}$. Then, we have $D^{\rm
c}_{m}=D^{\rm r}_{m}+D^{\rm q}_{m}+D^{\rm t}_{m}$.

In this work, we consider a block fading channel. Given the 3GPP
specifications[8], $D^{\rm r}_{m}$ and $D^{\rm t}_{m}$ are assumed to be
bounded by a constant, but $D^{\rm q}_{m}$ is a random variable. As a result,
the communication delay $D^{\rm c}_{m}$ is a random variable that could be
longer than the maximum tolerable delay bound $D^{\max}_{m}$. To reduce the
user experienced delay, $D^{\rm e}_{m}$, which is defined as the time
difference between the communication delay, $D^{\rm c}_{m}$, and the
prediction horizon, $T^{\rm p}_{m}$, i.e., $D^{\rm e}_{m}=D^{\rm c}_{m}-T^{\rm
p}_{m}$, a predictor is equipped at each receiver to predict the delayed or
lost trajectories. Given the definition of the user experienced delay,
predicting lost or delayed packets reduces user experienced delay by the
amount of the prediction horizon, $T^{\rm p}_{m}$. Here, lost trajectories are
the consecutive packets that cannot be decoded by the receiver due to deep
fading in a block fading channel. We denote the maximum prediction horizon
$T_{\rm th}$ which is the upper bound of the prediction horizon $T^{\rm
p}_{m}$. This means that the predictor cannot predict the trajectory beyond
the maximum prediction horizon $T_{\rm th}$ because the temporal correlation
of data becomes very weak.

Since the user experience delay is determined by the relationship among the
communication delay $D_{m}^{c}$ and the prediction horizon $T^{\rm p}_{m}$, we
need to consider the following three cases:

  * •

Case 1: $D^{\rm c}_{m}\in(0,D^{\max}_{m}]$

The receiver only needs the predicted trajectory when some packets are lost.
The prediction horizon depends on the number of consecutive packet losses. We
consider a block fading channel. The channel gain remains constant within
duration $D^{\rm ch}_{m}$, and varies independently from one duration to
another. $D^{\rm ch}_{m}$ is referred to as the channel coherence time. When
the wireless channel is in deep fading, most of the packets cannot be decoded
by the receiver. Given this fact, it is reasonable to assume that $D^{\rm
ch}_{m}$ is upper bound of the time horizon with consecutive packet
losses.222The probability that the channel stays in deep fading in multiple
consecutive blocks is extremely small, i.e., much smaller than $10^{-5}$, and
hence is not considered. With the block fading channel, the prediction horizon
is bounded by $T^{\rm p}_{m}\leq D^{\rm ch}_{m}$. In this case, the user
experienced delay is equal to the communication delay, i.e.,

$D^{\text{e}}_{m}=D^{\text{c}}_{m}\leq D^{\max}_{m}.$

  * •

Case 2: $D^{\rm c}_{m}\in(D^{\max}_{m},T_{\rm th}+D^{\max}_{m}]$

To satisfy the delay requirement, the prediction horizon is
$T^{\text{p}}_{m}=D^{\text{c}}_{m}-D^{\max}_{m}$. With the help of the
predictor, the user experienced delay becomes

$D^{\rm e}_{m}=D^{\rm c}_{m}-T^{\rm p}_{m}=D^{\max}_{m}.$

  * •

Case 3: $D^{\rm c}_{m}\in(T_{\rm th}+D^{\max}_{m},\infty)$

In this case, it is not possible to meet the delay requirement even with the
maximum prediction horizon. Then, the user experienced delay becomes

$D^{\rm e}_{m}=D^{\rm c}_{m}-T^{\rm p}_{m}>D^{\max}_{m}.$

For haptic communications, the delay experienced by a user should not exceed a
delay bound. The maximum tolerable delay bound of the $m$-th user is denoted
by $D^{\max}_{m}$. Then, the user experienced delay, $D^{\rm e}_{m}$, should
satisfy the following requirement

$D^{\rm e}_{m}\leq D^{\max}_{m}.$ (1)

It is worth noting that the above constraint cannot be satisfied with
probability one since the queuing delay is stochastic in most of communication
systems. In the next subsection, we will analyze the delay bound violation
probability.

###  II-B Reliability Components and Reliability Requirement

Let’s denote the overall error probability and the maximum tolerable error
probability of the $m$-th task by $\epsilon^{\rm o}_{m}$ and
$\epsilon_{m}^{\max}$, respectively. The overall error probability consists of
decoding error probability, $\epsilon^{\text{d}}_{m}$, queuing delay bound
violation probability, $\epsilon^{\text{q}}_{m}$, and prediction error
probability, $\epsilon^{\text{p}}_{m}$.

The decoding error probability of the $m$-th user depends on the channel
fading, the resource allocation policy, and the modulation and coding scheme
in wireless communications. We need to optimize the communication system to
obtain a satisfactory decoding error probability. In the next section, we will
provide the expression of $\epsilon^{\text{d}}_{m}$ in a specific
communication system.

The prediction error probability is defined as the probability that the
prediction error is larger than the required JND threshold of a task. For the
$m$-th user, the JND threshold is denoted by $\delta_{m}$. For a given
prediction algorithm, the relationship between prediction error probability,
$\epsilon^{\text{p}}_{m}$, the prediction horizon, $T^{\text{p}}_{m}$, and the
JND threshold is characterized by the following function,

$\displaystyle f_{\epsilon^{\text{p}}_{m}}(T^{\text{p}}_{m},\delta_{m})=$
$\displaystyle\Pr\\{|{\hat{K}_{m}(t+T^{\text{p}}_{m})}-K_{m}(t+T^{\text{p}}_{m})|>\delta_{m}\\},$
(2) $\displaystyle t=1,2,3,...$

where $\hat{K}_{m}(t+T^{\text{p}}_{m})$ is the predicted trajectory for the
$(t+T^{\text{p}}_{m})$-th slot and ${K_{m}(t+T^{\text{p}}_{m})}$ is the actual
trajectory in this slot. As shown in [13],
$f_{\epsilon^{\text{p}}_{m}}(T^{\text{p}}_{m},\delta_{m})$ increases with
$T^{\text{p}}_{m}$ and decreases with $\delta_{m}$.

Similarly, we denote the relationship between the queuing delay bound and the
delay bound violation probability by a function
$f_{\epsilon^{\text{q}}_{m}}(\kappa)$. It is the probability that the queuing
delay, $D^{\text{q}}_{m}$, is greater than a queuing delay bound, $\kappa$,
i.e.,

$f_{\epsilon^{\text{q}}_{m}}(\kappa)=\Pr\\{D^{\text{q}}_{m}>\kappa\\}.$ (3)

Since the delay bound violation probability decreases with the required delay
bound, $f_{\epsilon^{\text{q}}_{m}}(\kappa)$ is a monotonic decreasing
function. To meet the maximum delay bound, a threshold of queuing delay is
given by $D^{\rm th}_{m}=D^{\max}_{m}-D^{\rm t}_{m}-D^{\rm r}_{m}$.

In the sequel, we analyze the overall error probability in the three cases
discussed in the previous subsection.

  * •

Case 1: $D^{\rm c}_{m}\in(0,D^{\max}_{m}]$

In this case, we have $D^{\text{q}}_{m}\leq D^{\rm th}_{m}$. From the
definition of $f_{\epsilon_{m}^{\rm q}}(\kappa)$, the probability that the
queuing delay does not exceed $D_{m}^{\rm th}$ can be expressed by
substituting $D_{m}^{\rm th}$ into $\kappa$, so that $\kappa=D_{m}^{\rm th}$
and the probability that the queuing delay does not exceed $D_{m}^{\rm th}$
becomes

$\displaystyle\Pr\\{D_{m}^{\text{q}}\leq D_{m}^{\rm
th}\\}=1-f_{\epsilon_{m}^{\rm q}}(D_{m}^{\rm th}).$ (4)

Since the maximum prediction horizon does not exceed $D^{\rm ch}_{m}$, the
prediction error probability is bounded by $f_{\epsilon_{m}^{\rm
p}}(D_{m}^{\rm ch},\delta_{m})$. Given the decoding error probability,
$\epsilon_{m}^{\rm d}$, the error probability in case 1 can be expressed as
$\epsilon^{\text{c,1}}_{m}=f_{\epsilon^{\text{p}}_{m}}(D_{m}^{\rm
ch},\delta_{m})\epsilon^{\text{d}}_{m}$.

  * •

Case 2: $D^{\rm c}_{m}\in(D^{\max}_{m},T_{\rm th}+D^{\max}_{m}]$

In this case, $D^{\rm q}_{m}\in(D_{m}^{\rm th},D_{m}^{\rm th}+T_{\rm th}]$,
which happens with a probability of

$\displaystyle\Pr\\{D^{\rm q}_{m}\in(D_{m}^{\rm th},D_{m}^{\rm th}+T_{\rm
th}]\\}=$ $\displaystyle f_{\epsilon_{m}^{\rm q}}(D_{m}^{\rm
th})-f_{\epsilon_{m}^{\rm q}}(D_{m}^{\rm th}+T_{\rm th}).$ (5)

The prediction horizon is bounded by $T_{\rm th}$, and thus the error
probability is bounded by $\epsilon_{m}^{\rm c,2}\leq f_{\epsilon_{m}^{\rm
p}}(T_{\rm th},\delta_{m})$.

  * •

Case 3: $D^{\rm c}_{m}\in(T_{\rm th}+D^{\max}_{m},\infty)$

In this case, $D^{\text{q}}_{m}\in(T_{\rm th}+D_{m}^{\rm th},\infty)$. Case 3
happens with a probability of

$\displaystyle\Pr\\{D^{\rm q}_{m}\in(T_{\rm th}+D_{m}^{\rm
th},\infty)\\}=f_{\epsilon_{m}^{\rm q}}(D_{m}^{\rm th}+T_{\rm th}).$ (6)

Since the delay requirement is not satisfied, all the packets are lost. The
error probability in this case is $\epsilon_{m}^{\rm c,3}=1$.

The overall error probability is a combination of error probabilities in the
above three cases. It can be expressed as follows,

$\displaystyle\epsilon^{\rm o}_{m}=$
$\displaystyle\epsilon^{\text{c},1}_{m}\Pr\\{D_{m}^{\text{q}}\leq D_{m}^{\rm
th}\\}+\epsilon^{\text{c},2}_{m}\Pr\\{D^{\rm q}_{m}\in(D_{m}^{\rm
th},D_{m}^{\rm th}+T_{\rm th}]\\}$ $\displaystyle+\epsilon_{m}^{\rm
c,3}\Pr\\{D^{\rm q}_{m}\in(T_{\rm th}+D_{m}^{\rm th},\infty)\\}$
$\displaystyle\leq$ $\displaystyle f_{\epsilon^{\text{p}}_{m}}(D_{m}^{\rm
ch},\delta_{m})\epsilon^{\text{d}}_{m}(1-f_{\epsilon^{\text{q}}_{m}}(D_{m}^{\rm
th}))$ $\displaystyle+f_{\epsilon^{\text{p}}_{m}}(T_{\rm
th},\delta_{m})(f_{\epsilon_{m}^{\rm q}}(D_{m}^{\rm th})-f_{\epsilon_{m}^{\rm
q}}(D_{m}^{\rm th}+T_{\rm th}))$
$\displaystyle+f_{\epsilon^{\text{q}}_{m}}(D_{m}^{\rm th}+T_{\rm th}).$ (7)

The reliability requirement $\epsilon^{\rm o}_{m}\leq\epsilon_{m}^{\max}$ can
be satisfied if the upper bound in (7) meets the following constraint,

$\displaystyle f_{\epsilon^{\text{p}}_{m}}(D_{m}^{\rm
ch},\delta_{m})\epsilon^{\text{d}}_{m}(1-f_{\epsilon^{\text{q}}_{m}}(D_{m}^{\rm
th}))+f_{\epsilon^{\text{p}}_{m}}(T_{\rm th},\delta_{m})(f_{\epsilon_{m}^{\rm
q}}(D_{m}^{\rm th})$ $\displaystyle-f_{\epsilon_{m}^{\rm q}}(D_{m}^{\rm
th}+T_{\rm th}))+f_{\epsilon^{\text{q}}_{m}}(D_{m}^{\rm th}+T_{\rm
th})\leq\epsilon_{m}^{\max}.$ (8)

We denote the general utilization efficiency of a communication system by
$U(\bf{{x}})$, where ${\bf{x}}=[x_{1},...,x_{N}]$ is the optimization
variables of the $N$ tasks. A general task-oriented prediction and
communication co-design framework can be formulated as follows,

$\displaystyle\max_{\bf{x}}$ $\displaystyle\;{U(\bf{x})}$ (9) s.t.
$\displaystyle\;\eqref{delay_req}\;\text{and}\;\eqref{reliability_req}.$

With this framework, we can jointly optimize prediction and communication
systems to achieve better resource utilization efficiency.

##  III 5G New Radio: An Example of Communication System

To illustrate how to obtain the upper bound in (7), we consider 5G New Radio
as an example in the rest part of this work and derived the decoding error
probability, $\epsilon^{\text{d}}_{m}$, and the queuing delay bound violation
probability, $f_{\epsilon^{\text{q}}_{m}}(.)$, in this section.

###  III-A Decoding Error Probability

To achieve low transmission delay, the blocklength of channel codes is short.
In the finite blocklength regime, the maximal achievable rate can be
accurately approximated as [22]

$R_{m}\approx
C_{m}-\sqrt{\frac{V_{m}}{l_{m}}}Q^{-1}(\epsilon^{\text{d}}_{m})\;\text{(bits/s/Hz),}$
(10)

where $C_{m}=\log(1+\gamma_{m})$ is the Shannon capacity,
$\gamma_{m}=\frac{\alpha_{m}g_{m}P_{m}}{N_{0}W_{m}}$ is the received signal to
noise ratio (SNR) at BS, $\alpha_{m}$ is the large-scale channel gain, $g_{m}$
denotes the small-scale channel gain, $P_{m}$ denotes the transmit power,
$N_{0}$ is the single sided noise spectral density,
$V_{m}=\log(e)^{2}\left[1-\frac{1}{(1+\gamma_{m})^{2}}\right]$ is the channel
dispersion, $l_{m}=D^{\text{t}}_{m}W_{m}$ is the blocklength,
$D^{\text{t}}_{m}$ is the transmission duration, $W_{m}$ is the bandwidth, and
$Q^{-1}(.)$ is the inverse of the Q-function. Then, decoding error probability
can be expressed as

$\epsilon^{\text{d}}_{m}\approx
Q\left(\frac{D^{\text{t}}_{m}W_{m}C_{m}-b_{m}+\log(D^{\text{t}}_{m}W_{m})/2}{\sqrt{D^{\text{t}}_{m}W_{m}V_{m}}}\right),$
(11)

where $b_{m}=D^{\text{t}}_{m}W_{m}R_{m}$ is the number of information bits. If
SNR is higher than 5dB, channel dispersion $V_{m}$ becomes $\log(e)^{2}$ [23].
Then, $\epsilon^{\text{d}}_{m}$ becomes

$\epsilon^{\text{d}}_{m}\approx
Q\left(\frac{D^{\text{t}}_{m}W_{m}C_{m}-b_{m}+\log(D^{\text{t}}_{m}W_{m})/2}{\log(e)\sqrt{D^{\text{t}}_{m}W_{m}}}\right).$
(12)

From (12), we can obtain the following Lemma.

Lemma 1: Given $W_{m}$, $\epsilon^{\text{d}}_{m}$ increases with $b_{m}$.

Proof: Given $W_{m}$, the input of the Q-function in (12) decreases with
$b_{m}$. Since Q-function is a decreasing function, $\epsilon^{\text{d}}_{m}$
increases with increasing $b_{m}$.

###  III-B Queuing Delay Violation Probability

In URLLC, the transmission delay (transmission time interval could be $0.125$
ms in 5G) is much shorter than the channel coherence time. Thus, the service
rate of the queuing system is a constant. We use effective bandwidth to
characterize the minimum service rate that is required to achieve the delay
bound and delay bound violation probability [24]. To derive the closed-form
expression of queuing delay violation probability, we further assume that the
packet arrival processes are Poisson processes. Denote the average packet
arrival rate of the $m$-th user by $\lambda_{m}$ with the unit of (packets/s).
As discussed in [25, 13], the queuing delay violation probability decreases
exponentially as the delay bound increases, i.e.,

$\displaystyle
f_{\epsilon^{\text{q}}_{m}}(\kappa)=e^{\kappa\xi(E^{\text{B}}_{m},\lambda_{m})},$
(13)
$\displaystyle\xi(E^{\text{B}}_{m},\lambda_{m})=E^{\text{B}}_{m}\text{W}_{-1}\left(-\frac{\lambda_{m}}{E^{\text{B}}_{m}}e^{-\frac{\lambda_{m}}{E^{\text{B}}_{m}}}\right)+\lambda_{m},$
(14)

where $\text{W}_{-1}(.)$ is the -1 branch of Lambert W-function, which is
defined as the inverse function of $f(x)=xe^{x}$ and $E^{\text{B}}_{m}$ is the
effective bandwidth, which is the required service rate. To meet the queuing
delay bound $\kappa$ and the delay bound violation probability in (13), the
number of bits transmitted in a transmission time interval should satisfy the
following expression

$\displaystyle b_{m}/D^{\text{t}}_{m}=E^{\text{B}}_{m}.$ (15)

Upon substituting $E^{\text{B}}_{m}$ into (13) and (14), the queuing delay
bound violation probability can be expressed as

$f_{\epsilon^{\text{q}}_{m}}(\kappa)=\text{exp}\Bigg{\\{}\kappa\Bigg{[}\frac{b_{m}\text{W}_{-1}\left(-\frac{\lambda_{m}D^{\text{t}}_{m}}{b_{m}}e^{-\frac{\lambda_{m}D^{\text{t}}_{m}}{b_{m}}}\right)}{D^{\text{t}}_{m}}+\lambda_{m}\Bigg{]}\Bigg{\\}}.$
(16)

With the expression given in (16), the following property of queuing delay
bound violation probability function can be obtained.

Lemma 2: For given $\kappa$, $\lambda_{m}$, and $D^{\text{t}}_{m}$,
$f_{\epsilon^{\text{q}}_{m}}(\kappa)$ strictly decreases with $b_{m}$.

Proof: To check the monotonicity of $f_{\epsilon^{\text{q}}_{m}}(\kappa)$ in
terms of $b_{m}$, we have the partial derivative in (17).

$\begin{split}\frac{\partial f_{\epsilon^{\text{q}}_{m}}}{\partial
b_{m}}=&\kappa\,\text{exp}\left\\{{\kappa\,\left(\lambda_{m}+\frac{b_{m}\,{\mathrm{W}}_{-1}\left(-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}\,{\mathrm{e}}^{-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}}{b_{m}}}}{b_{m}}\right)}{{D^{\text{t}}_{m}}}\right)}\right\\}\times\,\left(\frac{{\mathrm{W}}_{-1}\left(-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}\,{\mathrm{e}}^{-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}}{b_{m}}}}{b_{m}}\right)}{{D^{\text{t}}_{m}}}\right.\\\
&\left.-\frac{b_{m}^{2}\,{\mathrm{e}}^{\frac{{D^{\text{t}}_{m}}\,\lambda_{m}}{b_{m}}}\,\left(\frac{{D^{\text{t}}_{m}}\,\lambda_{m}\,{\mathrm{e}}^{-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}}{b_{m}}}}{b_{m}^{2}}-\frac{{{D^{\text{t}}_{m}}}^{2}\,\lambda_{m}^{2}\,{\mathrm{e}}^{-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}}{b_{m}}}}{b_{m}^{3}}\right)\,{\mathrm{W}}_{-1}\left(-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}\,{\mathrm{e}}^{-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}}{b_{m}}}}{b_{m}}\right)}{{{D^{\text{t}}_{m}}}^{2}\,\lambda_{m}\,\left({\mathrm{W}}_{-1}\left(-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}\,{\mathrm{e}}^{-\frac{{D^{\text{t}}_{m}}\,\lambda_{m}}{b_{m}}}}{b_{m}}\right)+1\right)}\right)<0.\end{split}$
(17)

Since $W_{-1}(.)$ is always negative, and exponential function is always
positive, the partial derivative in (17) is negative. Therefore,
$f_{\epsilon^{\text{q}}_{m}}(\kappa)$ decreases with $b_{m}$ when $\kappa$,
$\lambda_{m}$, and $D^{\text{t}}_{m}$ are given.

##  IV TimeGAN Assisted Prediction: An Example of Prediction Algorithm

To characterize the tradeoff between the prediction error probability, the
prediction horizon, and the JND threshold, we require a large number of
trajectories for evaluating the prediction error probability below $10^{-5}$.
333The reliability requirement is defined by the 5G standard in [8]. However,
collecting enough trajectories in a real-world robotic platform may take
years. To address this issue, we generate synthetic data using TimeGAN. In
this section, we provide details of real-world data collection, synthetic data
generation, training and testing of different prediction algorithms.

###  IV-A Real-world Data Set Collection

The real-world trajectory data samples444The dataset is available at:
http://dx.doi.org/10.5525/gla.researchdata.1392 are collected from our
teleoperation testbed555The demonstration video: https://youtu.be/c3onK5Vh6QE.
The robotic arm is controlled by a human user to finish three types of tasks
as shown in Fig. 2.

  1. 1.

Pushing a box: Push a small box from the starting point to the end point along
a given routine.

  2. 2.

Grouping items with different colors: Move items with the same color to the
same area.

  3. 3.

Writing symbols: Write symbols by controlling the robotic arm.

Each trajectory is a time-series of observations given by
$\textbf{k}_{t:t^{{}^{\prime}}}=\\{\textbf{k}_{t},\textbf{k}_{t+1},...,\textbf{k}_{t^{{}^{\prime}}}\\}$
where each observation, $\textbf{k}_{t}=[q_{t},\dot{q_{t}}]$, consists of an
angular position, $q_{t}$, and angular velocity, $\dot{q_{t}}$, in the $t$-th
time slot. In experiments, we recorded around $1.7\times 10^{7}$ observations
with timestamps at the frequency of one thousand observations per second.

Figure 2: Tasks for data collection.

###  IV-B Synthetic Data Set Generation

To generate synthetic data, we apply TimeGAN [26], which is a framework for
time-series data generation. TimeGAN framework consists of four neural
networks namely embedding network, recovery network, sequence generator, and
sequence discriminator. The parameters of the four neural networks are denoted
by $\theta_{\rm em}$, $\theta_{\rm re}$, $\theta_{\rm g}$, and $\theta_{\rm
d}$, respectively.

Embedding and recovery networks are mappings from feature space to latent
space and vice versa. Let’s denote the latent variable and the latent space by
$\textbf{v}_{i:i^{\prime}}$ and $\mathcal{V_{T}}$, respectively. Then, the
embedding network outputs a sequence of latent variables from a given sequence
of features, i.e., $\textbf{v}_{i:i^{\prime}}=\vartheta_{\rm
em}(\textbf{k}_{i:i^{\prime}}|\theta_{\rm em})$. The recovery network
reconstructs features from latent variables according to
$\textbf{\^{k}}_{i:i^{\prime}}=\vartheta_{\rm
re}(\textbf{v}_{i:i^{\prime}}|\theta_{\rm re})$.

The generator network is denoted by
$\textbf{\^{v}}_{i:i^{\prime}}=\vartheta_{\rm
g}(\textbf{z}_{i:i^{\prime}}|\theta_{\rm g})$. It takes a sequence of random
variables $\textbf{z}_{i:i^{\prime}}$ from a known distribution as its input
and generates synthetic sequences in the latent space,
$\textbf{\^{v}}_{i:i^{\prime}}$. The discriminator is a classification network
in the latent space. Given a sequence of latent variables, a synthetic
sequence or a real sequence, the output of the discriminator is an indicator
$\tilde{y}=\vartheta_{\rm d}(\textbf{{v}}_{i:i^{\prime}}|\theta_{\rm d})$. If
$\tilde{y}=1$ the sequence is classified as a real one. Otherwise, the
sequence is classified as a synthetic one.

In TimeGAN, the four components are trained jointly with three loss functions:
reconstruction loss, $\mathcal{L_{\text{R}}}$, unsupervised loss,
$\mathcal{L_{\text{U}}}$, and supervised loss, $\mathcal{L_{\text{S}}}$.

Reconstruction loss is used to measure the difference between the
reconstructed features, $\textbf{\^{k}}_{i:i^{\prime}}$, and the original
features, $\textbf{{k}}_{i:i^{\prime}}$ i.e.,

$\displaystyle\mathcal{L_{\text{R}}}=\mathbb{E}_{\textbf{{k}}_{i:i^{\prime}}}\left[\sum_{t}||\textbf{{k}}_{t}-\textbf{\^{k}}_{t}||_{2}\right].$
(18)

Unsupervised loss comes from the zero-sum game as in conventional GAN [27]. It
maximizes the likelihood of correct classifications for the discriminator.
Supervised loss, on the other hand, is introduced to check the discrepancy
between real and synthetic data distributions in latent space. The definitions
of $\mathcal{L_{\text{U}}}$ and $\mathcal{L_{\text{S}}}$ can be found in [26,
28].

The generator and discriminator networks are trained iteratively by solving
the following problem,

$\displaystyle\min_{\theta_{\rm em},\theta_{\rm re}}$
$\displaystyle\IEEEeqnarraymulticol{3}{l}{\alpha\mathcal{L_{\text{R}}}+\mathcal{L_{\text{S}}}},$
(19)

where $\alpha\geq 0$ is a hyperparameter.

The embedding and recovery networks are trained to minimize
$\mathcal{L_{\text{R}}}$ and $\mathcal{L_{\text{S}}}$ which yields to
following optimization

$\displaystyle\min_{\theta_{g}}$
$\displaystyle\IEEEeqnarraymulticol{3}{l}{\eta\mathcal{L_{\text{S}}}+\max_{\theta_{d}}\mathcal{L_{\text{U}}}},$
(20)

where $\eta\geq 0$ is a hyperparameter.

In our training, we have implemented the TimeGAN in Tensorflow 2.0 [29] using
original implementation in [30], and ydata-synthetic package in [31]. The
hyper-parameters are listed in Table II.

TABLE II: TimeGAN model and Hyper-parameters. Hyper-parameters | Values
---|---
Sequence length | 600
Number of features | 2
Hidden units for generator | 24
Gamma (used for discriminator loss) | 1
Noise dimension (used by generator as a starter dimension) | 32
Number of layers | 128
Batch size | 128
Learning rate |  $5$x$10^{-4}$

###  IV-C Prediction Algorithms

We design trajectory prediction algorithms with three different types of
neural networks(NNs): Recurrent Neural Networks (RNN) [32], Long Short Term
Memory (LSTM) networks [33], and Convolutional Neural Networks (CNN) [34].

####  IV-C1 RNN for Prediction

In the $t$-th time slot, the input of a RNN cell includes the feature observed
in the current slot $\textbf{k}_{t}$, and the hidden state generated by the
previous RNN cell, $\textbf{h}_{t-1}$. Then, RNN model updates the output and
hidden states from the following steps,

$\displaystyle\textbf{o}_{t}=\sigma(\textbf{W}_{\rm
o}[\textbf{h}_{t-1},\textbf{k}_{t}]+\textbf{b}_{\rm
o})\;\text{and}\;\textbf{h}_{t}=\sigma(\textbf{W}_{\rm
h}\textbf{o}_{t}+\textbf{b}_{\rm h}),$

where $\textbf{o}_{t}$ is the output of the RNN cell, $\textbf{h}_{t}$ is the
new hidden state, $\textbf{W}_{\rm o}$ and $\textbf{W}_{\rm h}$ are weight
matrices, $\textbf{b}_{\rm o}$ and $\textbf{b}_{\rm h}$ are bias terms, and
$\sigma(\cdot)$ is the activation function.

####  IV-C2 LSTM for Prediction

Each LSTM cell takes three inputs at each time slot: the feature observed in
the current slot $\textbf{k}_{t}$, the previous LSTM cell state (i.e., long-
term memory unit) $\textbf{L}_{t-1}$, and the previous hidden state (i.e., the
short-term memory unit) $\textbf{h}_{t-1}$. Then, LSTM model updates the
output, hidden state, and cell state as follows.

$\displaystyle\textbf{f}_{t}=\sigma(\textbf{W}_{\rm
f}[\textbf{h}_{t-1},\textbf{k}_{t}]+\textbf{b}_{\rm f},$ (21)
$\displaystyle\textbf{i}_{t}=\sigma(\textbf{W}_{\rm
i}[\textbf{h}_{t-1},\textbf{k}_{t}]+\textbf{b}_{\rm i}),$ (22)
$\displaystyle\textbf{\~{L}}_{t}=\tanh(\textbf{W}_{\rm
k}[\textbf{h}_{t-1},\textbf{k}_{t}]+\textbf{b}_{\rm k}),$ (23)
$\displaystyle\textbf{L}_{t}=\textbf{f}_{t}\textbf{L}_{t-1}+\textbf{i}_{t}\textbf{\~{L}}_{t},$
(24) $\displaystyle\textbf{o}_{t}=\sigma(\textbf{W}_{\rm
o}[\textbf{h}_{t-1},\textbf{k}_{t}]+\textbf{b}_{\rm o}),$ (25)
$\displaystyle\textbf{h}_{t}=\textbf{o}_{t}\tanh(\textbf{L}_{t}),$ (26)

where $\textbf{f}_{t}$ is the forget gate which decides what information will
be kept from the previous cell state, $\textbf{i}_{t}$ is the input gate which
decides what information will be added to cell state of the network,
$\textbf{o}_{t}$ is the output gate of the LSTM unit, $\textbf{h}_{t}$ is the
new hidden state of the network,
$\textbf{W}_{f},\textbf{W}_{i},\textbf{W}_{k}$, $\textbf{W}_{o}$ are the
coefficient matrices,
$\textbf{b}_{f},\textbf{b}_{i},\textbf{b}_{k},\textbf{b}_{o}$ are the bias
terms, and $\sigma(\cdot)$ is the activation function.

####  IV-C3 CNN for Prediction

CNN consists of convolution layer, pooling layer and fully connected layer. In
convolution layer the feature representations of inputs (i.e., feature maps)
are computed by applying element-wise convolution to the input with the kernel
and then applying non-linear activation function to obtain the output of the
layer. For input $\textbf{k}_{t}$ and kernel $\psi_{t}$, the resulting feature
at location $(i,j)$ can be computed from the following steps,

$\displaystyle\textbf{Z}_{t}^{i,j}=\textbf{W}_{\psi_{t}}*\textbf{k}_{t}^{i,j}+\textbf{b}_{\psi_{t}},$
(27) $\displaystyle\textbf{Y}_{t}^{i,j}=\Phi(\textbf{Z}_{t}^{i,j}),$ (28)

where $\textbf{W}_{\psi_{t}}$ and $\textbf{b}_{\psi_{t}}$ are the weights and
bias of the filter $\psi_{t}$, $\textbf{k}_{t}^{i,j}$ is the subsection of the
input centered at $(i,j)$, $\Phi(\cdot)$ is the non-linear activation
function, and ‘$*$’ is the convolution operator. Then, the pooling layer is
employed to decrease the number of features or resolution of the feature map
after the convolution layer. The most used pooling operation is max-pooling
[35] which computes a new feature map by traversing the output of convolution
layer and calculating the maximum of each patch (i.e., subsection of the
convolution output according to filter size). After computing feature maps
with several convolution and pooling layers, computed features are flatten to
be fed to fully connected layer (i.e., dense layer) as seen in the Fig. 3
which is trained to forecast the future trajectories.

Figure 3: Illustration of trajectory prediction via CNN. TABLE III: Hyper-Parameters of Prediction Algorithms | RNN | LSTM | CNN
---|---|---|---
Number of layers | 1 | 1 | 1
Number of cells in each layer | 128 RNN cells | 128 LSTM cells | 128 Convolutional cells
Batch size | 64 | 64 | 64
Optimizer | Adam | Adam | Adam
Loss function | MSE | MSE | MSE
Accuracy metric | RRMSE | RRMSE | RRMSE
Max training epochs | 1000 | 1000 | 1000
Early stopping criteria | min validation loss, patience=10 | min validation loss, patience=10 | min validation loss, patience=10
Activation function | tanh | tanh | ReLu

###  IV-D Training and Testing of Prediction Algorithms

For all the three types of NNs, we use mean squared error,
$MSE=\sum_{t}(\hat{\textbf{k}}_{t}-\textbf{k}_{t})^{2}/s$, as the loss
function since it is a differentiable function and eases the mathematical
operations in optimizations throughout training process, where $s$ is the
number of samples. In addition, we use relative root mean squared error,
$RRMSE(\%)=\frac{\sqrt{\sum_{t}(\hat{\textbf{k}}_{t}-\textbf{k}_{t})^{2}/s)}}{\bar{\textbf{k}_{t}}}\times
100$, to compare the performance of different prediction algorithms, where
$\bar{\textbf{k}_{t}}$ is the mean value of the observations. We set the
maximum number of training epochs as $1000$. To avoid over fitting, an early
stopping criteria is adopted. Specifically, the training process is terminated
and the latest model is saved if the model does not improve for 10 consecutive
epochs. The hyper-parameters of prediction algorithms are listed in Table III.
To train the predictors, each input sample consists joint position and
velocity in the past $N_{\rm in}$ time slots. Given the history window with
$N_{\rm in}$ number of steps, $\textbf{k}_{t-N_{\rm
in}:t}=\\{\textbf{k}_{t-N_{\rm in}},\textbf{k}_{t-N_{\rm
in}+1},...,\textbf{k}_{t}\\}$, predictor is trained to predict the joint
positions in the next $N_{\rm out}$ number of steps (i.e., prediction window)
as shown in the example in Fig. 3. In this study, we consider multi-step
prediction in which predictor predicts $N_{\rm out}$ steps at once since it is
more accurate than single-step prediction [36, 37].

The testing results are provided in Table IV. Here, the dimension of the input
is $N_{\rm in}=500$ ms and the length of prediction window is $N_{\rm
out}=100$ ms. According to our results, CNN outperforms both LSTM and RNN.
Therefore, we use CNN in our system and will be referring to CNN as the
predictor in the rest part of the paper. It is worth noting that the
prediction accuracies of RNN, LSTM, and CNN depend on the datasets [38, 39,
40, 41]. In general, RNN and LSTM outperform CNN in time-series data. But for
some datasets, where the time-series data change suddenly, CNN can be better
than RNN and LSTM [42].

TABLE IV: Performance of Different Predictors Errors(%) | LSTM | RNN | CNN
---|---|---|---
Training RRMSE | $0.8\%$ | $0.5\%$ | $0.07\%$
Validation RRMSE | $0.8\%$ | $0.5\%$ | $0.08\%$
Test RRMSE | $0.9\%$ | $0.6\%$ | $0.2\%$

###  IV-E Tradeoff between Prediction Error Probability and Prediction Horizon

In this subsection, we illustrate how to obtain the tradeoff between the
prediction error probability and the prediction horizon. The goal is the
estimate the prediction error probability $f_{\epsilon^{\rm p}_{m}}(T_{m}^{\rm
p},\delta_{m})$, in (2), i.e., the probability that the tracking error is
greater than a required JND threshold $\delta_{m}$, when the prediction
horizon is $T^{\rm p}_{m}$.

When the probability is extremely small, e.g., $f_{\epsilon^{\rm
p}_{m}}(T_{m}^{\rm p},\delta_{m})=10^{-5}$, the real-world data set is not
enough to obtain an accurate estimation. To overcome this difficulty, the
synthetic trajectories generated by TimeGAN are used to estimate the
prediction error probability. For every input trajectory $\textbf{k}_{t-N_{\rm
in}:t}=\\{k_{t-N_{\rm in}},k_{t-N_{\rm in}+1},...,k_{t}\\}$, there is a
corresponding predicted trajectory $\textbf{\^{k}}_{t+1:t+N_{\rm
out}}=\\{\hat{k}_{t+1},\hat{k}_{t+2},...,\hat{k}_{t+N_{\rm out}}\\}$ and
ground truth $\textbf{{k}}_{t+1:t+N_{\rm
out}}=\\{k_{t+1},k_{t+2},...,k_{t+N_{\rm out}}\\}$. The prediction errors with
different prediction horizon (from $1$ time slot to $N_{\rm out}$ time slots)
are given by $\textbf{e}_{t+1:t+N_{\rm
out}}=\\{e_{t+1},e_{t+2},...,e_{t+N_{\rm out}}\\}$, where
$e_{t}=\hat{k}_{t}-k_{t}$. With both experimental data and synthetic data, we
can model the prediction error probability with different prediction horizons
and JND thresholds using feedforward neural network (FNN) that takes the
prediction horizon, $T_{m}^{\rm p}$, and the JND threshold, $\delta_{m}$, as
its inputs and outputs the prediction error probability, i.e.,

$\displaystyle f_{\epsilon^{\rm p}_{m}}(T_{m}^{\rm
p},\delta_{m})=FNN(T_{m}^{\rm p},\delta_{m}|\phi_{m}),$ (29)

where $\phi_{m}$ is the training parameters of the FNN. Hyper-parameters and
training results are provided in Table V.

TABLE V: Hyper-Parameters and Training Results of FNN Number of layers | 4
---|---
Number of cells in each layer | 64
Batch size | 64
Activation | ReLu
Max training epochs | 1000
Loss function | MSE
Accuracy metric | RRMSE
Training RRMSE | $0.6\%$
Validation RRMSE | $0.9\%$
Test RRMSE | $0.9\%$

##  V Efficient Resource Allocation with Task-oriented Prediction

Based on the communication system in Section III and the predictor in Section
IV, we illustrate how to optimize resource allocation, $W_{m}$, and data rate,
$b_{m}$, in the task-oriented prediction and communication co-design
framework. To maximize the number of users served by a BS, the optimization
problem in (9), can be re-formulated as follows,

$\displaystyle\max_{W_{m},b_{m}}$
$\displaystyle\IEEEeqnarraymulticol{3}{l}{N}$ (30) s.t.
$\displaystyle\sum_{m=1}^{N}W_{m}\leq W_{\max}$ (31a) $\displaystyle D^{\rm
e}_{m}\leq D^{\max}_{m}$ (31b)
$\displaystyle\epsilon^{\text{o}}_{m}\leq\epsilon_{\max}$
$\displaystyle\eqref{overall_error},\eqref{eps_d_formula},\eqref{eps_q_formula},\;\text{and}\;\eqref{eq:NumericalTradeoff},$

where (LABEL:Const0) is the constraint on the maximum available bandwidth,
(LABEL:Const1) and (V) are the constraints on the QoS requirement. As shown in
(7), the overall error probability is bounded by

$\displaystyle\epsilon^{\rm ub}_{m}(W_{m},b_{m})\triangleq
f_{\epsilon^{\text{p}}_{m}}(D_{m}^{\rm
ch},\delta_{m})\epsilon^{\text{d}}_{m}(1-f_{\epsilon^{\text{q}}_{m}}(D_{m}^{\rm
th}))$ $\displaystyle+f_{\epsilon^{\text{p}}_{m}}(T_{\rm
th},\delta_{m})(f_{\epsilon_{m}^{\rm q}}(D_{m}^{\rm th})-f_{\epsilon_{m}^{\rm
q}}(D_{m}^{\rm th}+T_{\rm th}))$
$\displaystyle+f_{\epsilon^{\text{q}}_{m}}(D_{m}^{\rm th}+T_{\rm th}).$

In the 5G New Radio system, the expression of the decoding error probability,
$\epsilon^{\text{d}}_{m}$, is given by (12). Given the queuing system in
Section III, the relationship between the queuing delay bound and the queuing
delay violation probability, $f_{\epsilon^{\text{q}}_{m}}(\cdot)$, is derived
in (16). With our predictor in Section IV, the tradeoff between the prediction
error probability and the prediction horizon $f_{\epsilon^{\rm
p}_{m}}(\cdot,\delta_{m})$ is obtained in (29). From (12), (16), and (29), we
can see that the upper bound is determined by the bandwidth allocation and the
data rate, and is denoted by $\epsilon^{\rm ub}_{m}(W_{m},b_{m})$.

The number of constraints and the number of optimization variables are not
deterministic in (30), i.e., they depend on the number of users, $N$. Thus, it
is hard to derive closed-form solution of this problem. To overcome this
difficulty, we decompose the problem into multiple single-user subproblems.

###  V-A Single User Subproblem

To maximize the number of users that can be served with a given bandwidth, we
turn to minimize the bandwidth that is required to guarantee the QoS of each
user. As such, the optimization problem can be decomposed into independent
single-user subproblems, i.e.,

$\displaystyle\min_{W_{m},b_{m}}$ $\displaystyle\;{W_{m}}$ (32) s.t.
$\displaystyle\;\epsilon^{\rm
ub}_{m}(W_{m},b_{m})\leq\epsilon_{\max},\eqref{delay_req},\eqref{overall_error},\eqref{eps_d_formula},\eqref{eps_q_formula},\;\text{and}\;\eqref{eq:NumericalTradeoff}.$

We develop a two dimensional binary search algorithm to find the optimal
solution of problem (32). We first fix the value of ${W_{m}}$ and find the
optimal data rate that minimizes the upper bound of the overall reliability in
(7), i.e.,

$\displaystyle\min_{b_{m}}$ $\displaystyle\;{\epsilon^{\rm
ub}_{m}(W_{m},b_{m})}$ (33) s.t.
$\displaystyle\;\eqref{delay_req},\eqref{eps_d_formula},\eqref{eps_q_formula},\;\text{and}\;\eqref{eq:NumericalTradeoff},$

The optimal solution of the problem (33) is denoted by $b_{m}^{*}(W_{m})$.
Then, we use binary search to find the value of $W_{m}$ that satisfies the
following equation,

$\displaystyle\epsilon_{m}^{\rm ub}(W_{m},b_{m}^{*}(W_{m}))=\epsilon_{\max}.$
(34)

We denote the solution of (34) by $W_{m}^{*}$.

In the rest part of this section, we prove that $W^{*}_{m}$ and
$b_{m}^{*}(W^{*}_{m})$ are the optimal solution of problem (32). To prove
this, we need two lemmas.

Lemma 3: For given $W_{m}$, $\epsilon^{\rm ub}_{m}(W_{m},b_{m})$ first
decreases and then increases with $b_{m}$.

We first validate that $\epsilon^{\rm ub}_{m}(W_{m},b_{m})$ first decreases
and then increases with $b_{m}$ in two asymptotic scenarios: 1) $b_{m}$ is
sufficiently small such that the decoding error probability is close to zero;
2) $b_{m}$ is sufficiently large such that the queuing delay violation
probability is close to zero.

In the first scenario that $b_{m}$ is very small, the queuing delay violation
probability $f_{\epsilon^{\text{q}}_{m}}(.)$ is large and, the decoding error
probability $\epsilon^{\text{d}}_{m}$ is close to zero. Then, $\epsilon^{\rm
ub}_{m}(W_{m},b_{m})$ can be simplified as $f_{\epsilon^{\text{p}}_{m}}(T_{\rm
th},\delta_{m})(f_{\epsilon_{m}^{\rm q}}(D_{m}^{\rm th})-f_{\epsilon_{m}^{\rm
q}}(D_{m}^{\rm th}+T_{\rm th}))+f_{\epsilon^{\text{q}}_{m}}(D_{m}^{\rm
th}+T_{\rm th})$. Since $f_{\epsilon^{\text{q}}_{m}}(.)$ is a decreasing
function of $b_{m}$ according to Lemma 1, $\epsilon^{\rm ub}_{m}(W_{m},b_{m})$
decreases with $b_{m}$.

When $b_{m}$ is sufficiently large, the queuing delay violation probability
$f_{\epsilon^{\text{q}}_{m}}(.)$ is close to zero, but the decoding error
probability $\epsilon^{\text{d}}_{m}$ is large. Then, $\epsilon^{\rm
ub}_{m}(W_{m},b_{m})$ is dominated by $\epsilon^{\text{d}}_{m}$, which
increases with $b_{m}$ according to Lemma 2. In this case, $\epsilon^{\rm
ub}_{m}(W_{m},b_{m})$ increases with $b_{m}$.

In non-asymptotic scenarios, it is difficult to prove Lemma 3. We will
validate Lemma 3 with numerical results.

Lemma 4: $\epsilon_{m}^{\rm ub}(W_{m},b_{m}^{*}(W_{m}))$ decreases with
$W_{m}$.

Proof: From (7), (16), and (12), we can see that given the value of $b_{m}$,
only the decoding error probability decreases with the value of $W_{m}$, and
all the other terms remain constant. If $W_{m}<\tilde{W}_{m}$, then,
$\epsilon_{m}^{\rm ub}(W_{m},b_{m}^{*}(W_{m}))>\epsilon_{m}^{\rm
ub}(\tilde{W}_{m},b_{m}^{*}(W_{m}))$. According to the definition of
$b^{*}_{m}(\tilde{W}_{m})$, it is the solution of (32). Given bandwidth
$\tilde{W}_{m}$, the optimal value of $b_{m}$ that minimize the upper bound of
the decoding error probability is $b^{*}_{m}(\tilde{W}_{m})$. In other words,
$\epsilon_{m}^{\rm ub}(\tilde{W}_{m},b_{m}^{*}(W_{m}))>\epsilon_{m}^{\rm
ub}(\tilde{W}_{m},b^{*}_{m}(\tilde{W}_{m}))$. Then, we have $\epsilon_{m}^{\rm
ub}(W_{m},b^{*}_{m}(W_{m}))>\epsilon_{m}^{\rm
ub}(\tilde{W}_{m},b^{*}_{m}(\tilde{W}_{m}))$.

Lemma 4 indicates that the optimal solution of problem (32) should satisfy
(34) and the optimal bandwidth can be obtained by binary search. To find the
optimal $b_{m}$ for a given $W_{m}$, we can also use binary search as
indicated by Lemma 3. Therefore, the optimal solution of problem (32) can be
obtained via the two dimensional binary search given in Algorithm 1.

Algorithm 1 Algorithm for solving (32)

0: Delay requirement $D^{\max}_{m}$, reliability requirement
$\epsilon_{m}^{\max}$, threshold for queuing delay $D^{\rm th}_{m}$,
prediction horizon threshold $T_{\rm th}$, channel coherence time $D^{\rm
ch}_{m}$, average packet arrival rate $\lambda_{m}$, JND threshold
$\delta_{m}$, initial bandwidth $W_{0}$, maximum bandwidth $W_{\max}$, large-
scale channel gain $\alpha_{m}$, small-scale channel gain $g_{m}$, transmit
power $P_{m}$, single sided noise spectral density $N_{0}$, initial number of
bits $b_{0}$, maximum number of bits $b_{\max}$.

0: Optimal $W^{*}_{m}$, $b_{m}^{*}(W_{m}^{*})$ to ensure URLLC QoS
requirements for $m^{th}$ task.Initialisation : $W_{\text{L}}=W_{\text{0}}$,
$W_{\text{R}}=W_{\max}$, $W_{\rm mid}=(W_{\text{L}}+W_{\text{R}})/2$.

1: Binary search $b_{m}$ in $[b_{0},b_{\max}]$ and obtain $\epsilon_{m}^{\rm
ub}(W_{\rm mid},b_{m}^{*}(W_{\rm mid}))$

2: while $|\epsilon_{m}^{\rm ub}(W_{\rm mid},b_{m}^{*}(W_{\rm
mid}))-\epsilon_{m}^{\max}|>(\epsilon_{m}^{\max})^{2}$ and
$W_{\text{L}}<W_{\text{R}}$ do

3: if $\epsilon_{m}^{\rm ub}(W_{\rm mid},b_{m}^{*}(W_{\rm
mid}))\leq\epsilon_{m}^{\max}$ then

4: $W^{*}_{m}=W_{\rm mid}$

5: $W_{\rm R}=W_{\rm mid}$

6: $W_{\rm mid}=(W_{\rm L}+W_{\rm R})/2$

7: else

8: $W^{*}_{m}=W_{\rm mid}$

9: $W_{\rm L}=W_{\rm mid}$

10: $W_{\rm mid}=(W_{\rm L}+W_{\rm R})/2$

11: end if

12: Binary search $b_{m}$ in $[b_{0},b_{\max}]$ and obtain $\epsilon_{m}^{\rm
ub}(W_{\rm mid},b_{m}^{*}(W_{\rm mid}))$

13: end while

14: return $W_{m}^{*}$, $b_{m}^{*}(W_{m}^{*})$

##  VI Evaluation of the Proposed Framework

In this section, we evaluate the proposed task-oriented prediction and
communication co-design framework with numerical results where Table VI
provides the parameter settings. We consider the path loss model
$10\log_{10}(\alpha_{m})=-128.1-36.7\log_{10}(d_{m})$, where $d_{m}=200$ m is
the distance between the BS and the receiver [13]. The overall error
probability, $\epsilon^{o}_{m}$, is obtained from (7).

TABLE VI: Numerical Values of Parameters for Overall Error [13, 43]. Parameter | Value
---|---
$\lambda_{m}$, average packet arrival rate |  $100$ packets/s
$P_{m}$, maximum transmit power |  $23$ dBm
$N_{0}$, single sided noise spectral density |  $-144$ dBm/Hz
$D^{\rm r}_{m}$, core network and backhaul delay |  $10$ ms
$D^{\rm t}_{m}$, transmission delay |  $0.5$ ms
$D^{\rm ch}_{m}$, channel coherence time |  $10$ ms
$T_{\rm th}$, prediction horizon threshold |  $50$ ms
$D^{\rm th}_{m}$, queuing delay threshold | $\max\\{D^{\max}_{m}-D^{\rm t}_{m}-D^{\rm r}_{m},0\\}$
Figure 4: 5G-enabled Teleoperation Prototype

We develop a 5G-enabled teleopeartion prototype to collect real-world
trajectory data samples. As shown in Fig. 4, a haptic device is deployed at
the master domain as a controller, which has the capability of 6-degree-of-
freedoms (DoFs) positional sensing and 3-DoF force feedback. We use a Franka
Emika Panda robotic arm at slave domain, which is a 7-DoF serial manipulator
with $1$ kHz control and sensor sampling capabilities. We perform mapping from
controller’s local coordinate system to robotic arm’s local coordinate system
since their DoFs and workspace are not identical. In the communication domain,
we deploy a User Datagram Protocol (UDP) server at a 5G Base Station (BS)
Mobile Edge Computing (MEC) unit. Both the controller and the robotic arm are
equipped with 5G dongles which are small modems (i.e., 5G communication
modules) built in-house at the University of Glasgow to access the local BS
over 5G NR.

In the following, we first illustrate the tradeoff between prediction error
probability and prediction horizon for different JND thresholds. Then, we
consider the single-user scenario to illustrate the optimality of the
Algorithm 1 as well as we compare the two frameworks, where the predictors are
either deployed at transmitter sides or at the receiver sides. Finally, we
consider the multi-user scenario to evaluate the performance gain of the
proposed framework in terms of the required bandwidth for each user or the
maximum number of users that can be served.

###  VI-A Prediction Error Probability and Prediction Horizon Tradeoff

In Fig. 5, we compare a real-world trajectory segment with a synthetic
trajectory segment. The results show that real and synthetic trajectories
follow similar trend but with different values. This is reasonable since the
motivation of generating synthetic data is for increasing the diversity of the
dataset rather than generating the same data. Therefore, it is expected to
have some differences between synthetic data and real data. We use both
synthetic and real trajectories to obtain the tradeoff between the prediction
error probability and the prediction horizon for different JND thresholds. In
Fig. 6, we provide the tradeoff between the prediction error probability and
the prediction horizon. The results validate our assumption that the
prediction error probability increases with the prediction horizon $T^{p}_{m}$
and decreases with the required JND threshold $\delta_{m}$.

Figure 5: Real and synthetic trajectory comparison to illustrate the quality
of generated data. Figure 6: Prediction error probability versus prediction
horizon for different JND thresholds. Prediction error probability curves for
different JND thresholds are obtained from $1.7\times 10^{7}$ real-world data
samples and $2\times 10^{11}$ synthetic data samples which correspond to
$6.7\times 10^{8}$ history and prediction window pairs.

###  VI-B Single-user Scenarios

In single-user scenario, we consider two categories of tasks namely critical
and non-critical tasks (see Fig. 2 where task 1 and task 2 are non-critical,
and task 3 is critical), whose JND thresholds are $\delta_{m}=0.1\%$, and
$\delta_{m}=1\%$, respectively. For fair comparison, we consider both tasks
have the same value of delay requirement $D^{\max}_{m}$ as well as reliability
requirement $\epsilon_{m}^{\max}$.

Figure 7: Overall error probability versus delay requirement, where
$W_{m}=140$ kHz and $b_{m}=256$ bits.

Fig. 7 illustrates the trade-off between the required delay bound and the
overall error probability with different JND thresholds. Here, the allocated
bandwidth $W_{m}$ is $140\text{ kHz}$ and the packet size $b_{m}$ is
$256\text{ bits}$. The results show that the overall error probability is
constant and equal to prediction error probability when delay requirement
smaller than or equal to the communication delay, i.e., $D_{m}^{\max}\leq
D_{m}^{\rm c}$. Then, it decreases rapidly as the delay requirement
$D_{m}^{\max}$ grows from $10$ to about $25$ ms. This is reasonable because
the queuing delay violation probability $f_{\epsilon^{\text{q}}_{m}}(D^{\rm
th}_{m})$ is the dominant factor in the region $10<D_{m}^{\max}<25$ ms. When
the delay requirement is larger than $25$ ms, both curves are nearly constant
with $D_{m}^{\max}$. This is because $f_{\epsilon^{\text{p}}_{m}}(D^{\rm
ch}_{m},\delta_{m})\epsilon^{\text{d}}_{m}$ becomes the dominant factor in (7)
and does not change with the required delay bound. Furthermore, the error
probability of the critical task is much higher than that of the non-critical
task. This implies that the bandwidth consumption of different task will be
significantly different if they require the same error probability.

(a) Error probability versus number of bits, where $\delta_{m}~{}=~{}0.1\%$.

(b) Error probability versus number of bits, where $\delta_{m}~{}=~{}1\%$.

Figure 8: Error probabilities versus number of bits, where $D^{\max}_{m}=20$
ms and $\epsilon^{\max}_{m}=10^{-5}$.

To validate Lemmas 1-3, we provide the relationship between different error
probabilities in Fig. 8. The results show that the queuing delay bound
violation probability, $f_{\epsilon^{\text{q}}_{m}}(D^{\rm th}_{m})$,
decreases with number of bits and the decoding error probability,
$\epsilon^{\text{d}}_{m}$, increases with number of bits. These two curves
validate Lemma 1 and Lemma 2, respectively. For both critical and non-critical
tasks, the overall error probability, $\epsilon^{\text{o}}_{m},$ first
decreases and then increases with number of bits. Moreover, when the queuing
delay violations are larger than the decoding errors, the overall error is
dominated by the queuing delay violation. When the decoding errors are larger
than the queuing delay violations, the overall error is dominated by the
decoding errors. These observations are consistent with Lemma 3. Furthermore,
given the relationship between overall error, $\epsilon^{\rm o}_{m}$, and
other error components such as decoding error probability,
$\epsilon^{\text{d}}_{m}$, queuing delay violation probability,
$f_{\epsilon^{\text{q}}_{m}}(.)$, and prediction error probability,
$f_{\epsilon^{\text{p}}_{m}}(.,.)$, in equation (7) and given that
$f_{\epsilon^{\text{p}}_{m}}(.,.)\leq 10^{-1}$, overall error probability
could be lower than both decoding error and delay violations as seen in Fig.
8.

Figure 9: Error probabilities versus allocated bandwidth, where
$D^{\max}_{m}=20$ ms and $\epsilon^{\max}_{m}=10^{-5}$.

In Fig. 9, we compare error probabilities of critical and non-critical tasks
under different values of allocated bandwidth, $W_{m}$. Here, the delay
requirement is $20$ ms and the reliability requirement is $10^{-5}$. From the
figure, the overall error probability decreases with increasing bandwidth in
both critical and non-critical cases which is consistent with Lemma 4. The
results in Figs. 8 and 9 indicate that the overall error probability achieved
by the proposed framework for critical case is $\epsilon_{m}^{\rm
ub}(W^{*}_{m},b_{m}^{*}(W^{*}_{m}))=1.00\times 10^{-5}$ with
$W^{*}_{m}=145.24$ kHz, $b^{*}_{m}(W^{*}_{m})=268$ bits and for non-critical
case is $\epsilon_{m}^{\rm ub}(W^{*}_{m},b_{m}^{*}(W^{*}_{m}))=1.00\times
10^{-5}$ with $W^{*}_{m}=32.19$ kHz, $b^{*}_{m}(W^{*}_{m})=92$ bits.

In the existing literature, the predictor is either deployed at the
transmitter [13, 9, 10, 44] or at the receiver [11, 12]. Both deployment
strategies have advantages and disadvantages. If the predictor is deployed at
the transmitter, the advantage is that the historical information used in the
prediction algorithm is accurate. The disadvantage is that either a prediction
error or a packet loss in communication may result in a JND violation. If the
predictor is deployed at the receiver, it can adjust the prediction horizon
according to the communication delays of different packets. If the
communication delay is satisfactory, there is no need to do any prediction. In
this way, a JND violation happens when both the communication and the
prediction fail. The disadvantage of this framework is that the historical
information used in the prediction algorithm may not be accurate, because some
packets are lost or severely delayed. Nevertheless, the effects of deployment
strategy hasn’t been investigated and deserve further analyses.

(a) Overall error probability versus packet loss probability in
communications, where $D^{\max}=20$ ms.

(b) Overall error probability versus user experienced delay, where
$\epsilon^{\text{d}}=10^{-5}$.

Figure 10: Predictor at transmitter versus predictor at receiver where,
$W_{m}=140$ kHz, $\delta_{m}=1\%$.

Here, we provide comparison between two deployment strategies to highlight the
differences and deliver some insights for prediction and communication co-
design research. Fig. 10 compares the overall system reliability of two
deployment strategies under different values of packet loss probabilities in
communications (Fig. 10(a)) and user experienced delay (Fig. 10(b)). From Fig.
10(a), deploying predictor at receiver achieves better overall reliability
with identical communication conditions. The reason behind this result is that
if the predictor at the transmitter, then the overall system reliability
dominated with least reliable system (i.e. either prediction system or
communication system) since predicted future trajectories transmitted over
communication system. However, if predictor at receiver, then it becomes
compensation mechanism for communication system where lost packets can be
predicted with cost of prediction errors. From Fig. 10(b), deploying predictor
at receiver can achieve better delay and reliability tradeoff. Both strategies
can compensate or reduce user experienced delay. However, deploying predictor
at receiver can achieve similar user experienced delay with higher
reliability. The insight is that if the communication system reliability is
high, i.e., packet loss probability less than $10^{-5}$, both strategies are
suitable to reduce user experienced delay with accurate predictor. However,
when the communication system is not reliable, i.e. packet loss probability
greater than $10^{-5}$, we can achieve the URLLC QoS requirements by only
deploying the predictor at the receiver.

###  VI-C Multi-users Scenarios

In multi-user scenarios, we compare the proposed task-oriented prediction and
communication framework with the task-agnostic prediction and communication
benchmark. Similar to single-user scenario, we assume two types of tasks and
used the results from single-user scenario for the minimum required bandwidth
and optimal packet rate. Furthermore, we denote critical task ratio to total
number of tasks as $r=\frac{A}{A+B},A\geq 0,B\geq 0$ where $A$ is the number
of critical tasks and $B$ is the number of non-critical tasks.

(a) Total required bandwidth versus number of users.

(b) Number of supported users versus maximum available bandwidth.

Figure 11: The proposed task-oriented design versus task-agnostic design in
terms of resource utilization.

Fig. 11 compares the resource utilization of the proposed task-oriented
prediction and communication framework and the benchmark that is task-agnostic
under different values of available resources and critical task ratios. From
the figures, the proposed approach achieves more efficient use of resources
with up to $77.80\%$ resource saving. This is because the proposed approach
allocates resources according to JND thresholds of different tasks.

##  VII Conclusions

In this paper, we proposed the task-oriented prediction and communication co-
design framework to increase the wireless resource utilization efficiency for
haptic communications, where low latency and high reliability performance are
required. The basic tradeoff between resource utilization efficiency and
overall reliability are provided. For predictions, we see that real-world data
are not enough to achieve URLLC level reliability. To address this issue, we
deploy TimeGAN to generate realistic synthetic data which we use to reveal
experimental tradeoff between prediction reliability and prediction horizon
for different JND thresholds. We analysed prediction and communication systems
and studied their relationship to reveal the tradeoff between wireless
resources and reliability in proposed task-oriented communication and
prediction co-design. We considered the teleoperation scenario via 5G New
Radio and demonstrated a design example using the proposed framework. We
formulated a joint optimization problem to maximize the number of users in a
communication system by jointly optimizing communication data rate and task-
dependent JND threshold. Numerical results show that the proposed approach can
reduce the wireless resource consumption by $77.80\%$ compared with the
benchmark that is task-agnostic.

As a future work, the proposed framework can be extended to a two-way
prediction and communication framework, in which both transmitter and receiver
can be equipped with predictors to predict both lost and delayed packets as
well as network conditions to dynamically allocate resources accordingly. This
sequential decision making mechanism can be achieved by training Deep
Reinforcement Learning (DRL) agent with expert knowledge and real-time
information available such as channel state information (CSI).

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