# Towards Lightweight Neural Animation : Exploration of Neural Network Pruning
in Mixture of Experts-based Animation Models

Antoine Maiorca1, Nathan Hubens1,2, Sohaib Laraba1 and Thierry Dutoit1
1ISIA Lab, Faculty of Engineering, University of Mons, Mons, Belgium
2Artemis, TelecomSud Paris, Paris, France
{antoine.maiorca, nathan.hubens, sohaib.laraba<EMAIL_ADDRESS>

###### Abstract

In the past few years, neural character animation has emerged and offered an
automatic method for animating virtual characters. Their motion is synthesized
by a neural network. Controlling this movement in real time with a user-
defined control signal is also an important task in video games for example.
Solutions based on fully-connected layers (MLPs) and Mixture-of-Experts (MoE)
have given impressive results in generating and controlling various movements
with close-range interactions between the environment and the virtual
character. However, a major shortcoming of fully-connected layers is their
computational and memory cost which may lead to sub-optimized solution. In
this work, we apply pruning algorithms to compress an MLP-MoE neural network
in the context of interactive character animation, which reduces its number of
parameters and accelerates its computation time with a trade-off between this
acceleration and the synthesized motion quality. This work demonstrates that,
with the same number of experts and parameters, the pruned model produces less
motion artifacts than the dense model and the learned high-level motion
features are similar for both.

##  1 INTRODUCTION

Virtual characters animation is a trending topic for video game and movie
industry. It is important that the motion generation algorithm can guarantee a
plausible and natural synthesis in order to increase the immersive factor of
the movie or the video game. Thus, the computer vision research community has
a special interest in the task of motion synthesis. Moreover, the real-time
control of the character trajectory and actions may be an important constraint
depending on the use case.

[Holden et al., 2017] showed that, with the use of a relatively simple neural
network architecture based on Multi-Layer Perceptrons (MLP), it is possible to
draw several modes of locomotion for a biped character. The motion data needs
first to be aligned on a phase vector that represents the timing of the motion
cycle. Then, a regression network whose parameters change dynamically
according to the phase vector generates the pose-by-pose motion in an
autoregressive manner. Finally, a regression network whose weights vary
dynamically according to the phase vector generates the pose-by-pose motion in
an autoregressive way. [Zhang et al., 2018] keeps the same regression network
as [Holden et al., 2017] but proposes upstream a Mixture-of-Experts (MoE) to
generalize this architecture for the motion of quadruped characters whose the
different locomotion gaits are difficult to align on one phase signal.

However, fully-connected layers are computationally expensive in terms of
number of parameters and floating point operations (FLOPs) compared to other
architectures like Convolutional Neural Network for example. Once the network
is trained, some weights in the neural network may not learn information that
significantly contributes to the quality of the generated signal. Thus, the
network capacity may be not fully exploited. Moreover, some of recent
architectures can be highly greedy in term of hardware resources and may not
be suitable for embedded deployment but reducing its number of parameters
would be too costly in terms of performance. Neural network pruning is a
research area that aims to compress a neural network while keeping its
performance intact. It is widely used when there are non-negligible
constraints on hardware resources, such as on embedded devices. The idea
behind this concept is to remove unnecessary weights and to retrain the pruned
neural network. Practically, these are not removed but nullified. This induces
that the pruned weight matrices become sparse. The proportion of zero
parameters compared to the original network size is termed as sparsity. In
this paper, we explore the impact of pruning a MLP-and-MoE based neural
network in the context of neural character animation. The benefits of this
method is to reduce the number of the network parameters and thus
theoretically accelerate the computation time while minimizing the performance
loss. To the best of our knowledge, this work is the first that applied
pruning methods in the context of neural animation.

First, the state of the art on modern data-driven animation techniques as well
as on neural network pruning is presented in section 2. Then, the experiments
performed are described in section 3 and the related results in section 4.

##  2 RELATED WORK

###  2.1 Data-driven Animation

The task of interactive motion generation is a broad field of research
focusing on synthesizing a natural motion from control signals. The
problematic of real-time pose prediction from a user-defined trajectory
belongs to this paradigm. This has been addressed by data-driven methods using
machine learning techniques [Park et al., 2002, Tilmanne and Dutoit, 2010,
Clavet, 2016]. More recent works lie in the field of autoregressive models
where, along with the controlled trajectory, the next character pose is
predicted with the previous ones. More recent models consider deep learning
algorithms with time series analysis architectures such as RNNs [Lee et al.,
2018, Pavllo et al., 2018]. This family of models combined with probabilistic
generative methods draw motions that exhibit less artefact than deterministic
models [Wang et al., 2021, Henter et al., 2020, Chen et al., 2021].

####

[Holden et al., 2017] has shown that real-time controlled locomotion of a
bipedal character can be realized thanks to a phase vector representing the
timing of the motion cycle and computed from foot contact with the ground,
like said in section 1, since MLPs do not initially extract any correlation in
sequential data. This variable is used to cyclically generates the weights of
a MLP-based regression network that synthesizes, from motion features such as
joint positions, velocities and rotations at the frame $t$, the pose as the
next frame. However, this system fails when the motion is difficult to align
to a single phase vector, such as quadruped locomotion. To tackle this issue,
[Zhang et al., 2018] proposes to use the technique of MoE [Jacobs et al.,
1991]. This technique is based on $n$ experts models. Each of them is
specialized in the realization of a particular subtask of the given problem.
The output $y_{i}$ of each expert is further weighted by $n$ coefficients
$\omega_{i}$ computed by a so-called gating network. It learns to quantify the
importance of each expert to resolve the given task. Thus, the final result
$y$ takes into account the effect of the $n$ expert models.

$y=\sum_{i}^{n}\omega_{i}y_{i}$ (1)

In the work of [Zhang et al., 2018], the gating network computes coefficients
from the skeleton leg features and simulates the effect of the phase function
in [Holden et al., 2017] for quadruped character. In this case, the experts
are sets of parameters learned during training that are then blent. Then, this
recombination is used as weights of the regression network. This technique has
shown encouraging results in the field of interactive motion generation and
constitutes the backbone of recent interactive locomotion generation
algorithms. [Starke et al., 2019] employed this method so that a character is
able to interact adaptively with objects of variable shape defined in the
virtual environment. For example, he can sit on a chair or carry a box without
having any prior knowledge of their shape. More complex movements than
locomotion can also be synthesized. [Starke et al., 2020] propose a framework
to generate basketball movements with awareness of the ball and a direct
opponent. The model must therefore learn to analyze contacts other than feet
on the ground, such as the hand-ball contact. For this, they introduce the
concept of local phase where each phase is linked to a particular subset of
the body. The movement is thus no longer aligned on a single global phase but
on several phases and they showed that this helps the model to learn multi-
contact motion. More recently, [Starke et al., 2021] have succeeded, still
with MoE, in generating with simple user-defined control signals a variety of
martial arts movements while allowing a close-character interaction. [Ling et
al., 2020] implemented the method of [Zhang et al., 2018] with a VAE
architecture. This solution allows to define the character controller through
Reinforcment Learning. Thus, the avatar can be controlled from joystick input,
learn to follow a predefined trajectory or even freely explore an environment
like a maze.

###  2.2 Pruning

Pruning methods primarily differ according to three factors: (1) the
granularity at which pruning is operated, (2) the criteria uses for selecting
parameters to remove, and (3) the schedule followed for pruning.

Granularity.  Pruning granularity is usually divided into two groups:
unstructured pruning, i.e. the weights are removed individually, without any
intent to keep structure in the weights [LeCun et al., 1989, Hassibi et al.,
1993]. This method leads to matrices able to reach a high sparsity level, but
difficult to speed-up due to the lack of regularity in the pruning patterns.
For those reasons, structured pruning, i.e. removing groups of weights, have
been introduced [He et al., 2017]. Those structures can include vector of
weights, or even kernel or entire filters when pruning convolutional
architectures.

Criteria.  Early methods proposed to use the second-order approximation of the
loss surface to find the least useful parameters [LeCun et al., 1989, Hassibi
et al., 1993]. Later work also explored the use of variational dropout
[Molchanov et al., 2017a] or even $l_{0}$ regularization for parameter removal
[Louizos et al., 2018]. However, it has been shown recently that, even though
those heuristics may provide good results under particular conditions, they
are more complex and less generalizable than computing the $l_{1}$ norm of the
weights, and using that value as a measure of the importance of the weights,
thus removing the ones with the lowest norm [Gale et al., 2019]. Moreover,
when comparing the importance of weights, one might do it locally, i.e. only
compare weights that belong to the same layer, which will provide a pruned
model that possesses the same sparsity in each layer. One also might compare
the weights globally, i.e. the weights from the whole model are compared when
the pruning is applied, resulting in a model with different sparsity levels
for each layer. Comparing the weights globally usually provides better results
but may be more expensive to compute when the model grows larger in size.

Scheduling.  There exist many ways to schedule the network pruning. Early
methods proposed to remove weights of a trained network in a single-step, the
so-called one-shot pruning [Li et al., 2017]. Such a strategy typically
required further fine-tuning of the pruned model in order to recover from the
lost performance. However, performing the pruning in several steps, i.e. the
iterative pruning, is able to provide better results and reach a higher
sparsity level [Han et al., 2015, Molchanov et al., 2017b]. Nevertheless, such
methods were usually time-consuming because of the alternation of many
iterations of pruning and fine-tuning [Li et al., 2017]. Recently, another
family of schedules has emerged, performing a pruning that is more intertwined
with the training process, allowing to obtain a pruned network in a more
reasonable time [Zhu and Gupta, 2017, Hubens et al., 2021].

##  3 EXPERIMENTS

The experimental setup consists in applying unstructured global pruning to the
model proposed by [Zhang et al., 2018] with a hidden layer size $h_{size}=512$
and 8 experts. As explained in section 2.1, it is defined by two fully
connected-based subnetworks : the gating network extracting the coefficients
$\omega_{i}$ and the motion prediction network whose its weights are the
result of the MoE processus. The chosen pruning scheduling is the one cycle
pruning [Hubens et al., 2021] with the weights l1 norm criterion [Gale et al.,
2019]. We use the same training data as [Zhang et al., 2018] which is composed
of motion capture recording of dog locomotion with various gaits.

####

First of all, we increase the network sparsity step-by-step from $10\%$ to
$90\%$ of the total amount of parameters to extract the relationship between
the network sparsity and the motion quality. To quantitatively assess the
performance of each model, we measure the foot skating artifact on the
generated motion. Foot skating is the fact that the character foot slides when
the related joint on the skeleton is considered on contact with the ground
which is practically defined if this joint foot height $h$ is under a height
threshold $H$. That effect has a bad impact on the motion naturalness and is
induced by the mean regression during the training process as explained by
[Zhang et al., 2018]. We use the equation from [Zhang et al., 2018] to
quantify the foot skating $s$ where $v$ stands for the foot horizontal
velocity. We fixed the threshold $H$ at $2.5cm$.

$s=v(2-2^{h/H})$ (2)

####

Then, we draw a comparison of the generated motion quality between the same
size dense (unpruned) and sparse models (pruned). Next, the qualitative
contribution of each expert in the generation of the movement is established
through an ablation study in order to visualize possible differences between
their roles in the case where the initial model is pruned. Finally, the
dynamic behavior of the gating network output vector $\omega$ is extracted and
subjectively compared between the dense and sparse network.

For these experiments, we use the Fasterai framework [Hubens, 2020] built on
top of Fastai [Howard et al., 2018] to implement the pruning methods. We train
150 epochs on a GTX-1080 Nvidia GPU with a batch size of $32$, a learning rate
$\eta=1e-4$ and a weight decay rate $\lambda=2.5e-3$ with a AdamWR algorithm
[Loshchilov and Hutter, 2018] warm restart with the same parameters as [Zhang
et al., 2018].

##  4 RESULTS

This section presents the experimental results of the analysis setup described
in section 3.

###  4.1 Performance/Sparsity Analysis

Figure 1 draws the average foot skating curves along the network sparsity. The
generated dog motion is evaluated when he walks and when he performs sudden
turns. The trend is that skating increases with the network sparsity. So,
there is a trade-off between the number of non-zero parameters and the quality
of the synthesized motion. This is due to the fact that the network is less
able to learn complex relationships when it has fewer parameters and the
generated motion tends towards an average pose which is reflected in the
motion by an excess of foot skating. So, following Table 1 that shows the
relationship between the sparsity imposed on the network, the number of non-
zero parameters and the floating point operations, the same trade-off exists
between the synthesized motion quality and the model size and FLOPs. A video
showing the degradation of the motion quality when the number of non-zero
parameters decreases is available at
https://www.youtube.com/watch?v=RHNLQ2Cbz3Y.

Figure 1: Average Foot Skating on all legs (cm/frame). The generated motion has been evaluated on foot skating while the dog was walking around and abruptly turning. Network sparsity | Model size (Mb) | MFLOPs
---|---|---
0% | 178 | 11.10
10% | 160 | 10.06
20% | 142 | 9.02
30% | 124 | 7.98
40% | 106 | 6.94
50% | 88 | 5.89
60% | 71 | 4.86
70% | 53 | 3.82
80% | 35 | 2.78
90% | 17.8 | 1.74
Table 1: Relationship between network size and FLOPs. Each individual
parameter is stored as a 32 bits float.

###  4.2 Comparison between same size models

This section presents a comparison of the quality of the motion generated by
two models with an equivalent number of parameters: the size of the dense
model hidden layer $h_{size}$ is set to 256, 128 and then 64. Starting from
the original network (8 experts and $h_{size}=512$), we prune the network
until we reach an equivalent number of parameters for each dense model.

Figure 2: Top : Animation generated by the dense model. - Bottom : Animation
generated by the sparse model. Both models have the same number of non-zero
parameters. In this example, we keep 10% of parameters from the initial model.
The output motion of the sparse model exhibits less motion artifact than the
dense one such as back legs stiffness while turning. The videos about these
observations can be found at https://youtu.be/AuJ0q019-nc.

The observations from the related videos show that the quality of motion
decreases with the hidden layer size. Indeed, since the network training is a
regression problem, it suffers from mean regression and the output poses may
converge to a mean pose that minimizes the mean squared error. When reducing
the number of parameters, the network has less capacity to learn complex
features and it enhances the mean pose regression issue. However, sparse
models eliminate globally rough artifacts such as the absence of motion while
turning or unnatural poses. This phenomenon is illustrated by Figure 2 showing
an example of the virtual dog animation comparing pruned model with 90% of
sparsity and the equivalent dense model. The dog motion from the dense model
is very static while turning.

Then, average foot skating is measured for each of the three dense and sparse
models while the dog is walking. The abrupt turn motion is not evaluated by
skating because the motion artifacts produced by the dense model when reducing
$h_{size}$ are too harsh and may not be correctly assessed by Equation 2. The
results of this evaluation are shown in Figure 3. It illustrates the fact that
sparse models synthesize motion with less skating than the dense models with
the same number of parameters and experts. This points out the benefits of the
application of this pruning method on this neural network : while fixing the
training budget at 150 epochs in our case, pruning a dense model into a sparse
one leads to more reasonable movements than the dense network with the same
number of non-zero parameters.

Figure 3: Comparison on average foot Skating on all legs (cm/frame) for dense
and sparse models with the same number of parameters and experts. Sparse
models exhibit less skating than dense model

###  4.3 Ablation study

$\alpha_{i}$ | MANN8 | pruned MANN8
---|---|---
$\alpha_{0}$ | fail to sit and lie | fail to sit and lie
$\alpha_{1}$ | turn badly and fail to lie | turn badly and fail to lie
$\alpha_{2}$ | cannot turn while walking |  walk badly and cannot turn while walking
$\alpha_{3}$ | cannot jump or run | cannot jump or run and turn badly
$\alpha_{4}$ | cannot run and badly walk,turn and jump | can barely move
$\alpha_{5}$ | badly right turn | badly right turn
$\alpha_{6}$ | can only jump | can only jump
$\alpha_{7}$ | cannot jump and run | cannot jump and run
Table 2: Comparison between the ablation study of the dense and the sparse
model. Both models exhibit the same experts behavior.

In order to visualize the contribution of each expert to the generation of the
movement, we make an ablation study as it is done in [Zhang et al., 2018] :
each expert is deactivated one by one and we observe the impact of this
deactivation on the synthesized movement. This study is realized for the
initial model with 8 experts and the one pruned with to a sparsity of 90% (90%
of the total amount of parameters are removed from the network) and is
presented in Table 2. For example, when removing the first expert
$\alpha_{0}$, the dog cannot properly sit or lie on the ground and this
behavior is observed on both models. Since each observation while removing the
same experts in both models is similar, this ablation study shows that, even
in a case of extreme sparsity, each expert in both models corresponds
qualitatively to the same high-level features perceived in the movement.

###  4.4 Expert activation behavior

Figure 4: Expert activation $\omega_{i}$ performing various motion. The left
column shows the results for the dense model and the right column for the 90%
sparse model. The action performed from top to bottom: walk, run, sit, jump
and turn. A similar behavior is observed for both models: the same experts
contribute to the same movements. However, the curves are slightly flatter for
the sparse model, which implies that the most significant expert in the dense
model is a little less significant in the sparse model.

Since section 4.3 shows that the high-level motion features learned by the
experts are not affected by the employed pruning paradigm, the same behavior
of the $\omega_{i}$ (values weighting each expert to compute pose regression
network parameters) is thus expected. So, the experts activations are compared
between the initial and the sparse model for the same movement. These
activations are shown in Figure 4. When the quadruped performs a specific
action, the same weighting profile is applied to experts for both models.
However, the activations in the sparse model are slightly more distributed
among the different experts compared to the original model. We believe that
this is due to the fact that, as the number of non-zero parameters in the
pruned model experts is lower than in the dense one, the sparse network
exploits those of other experts in order to synthesize similar movements.

##  5 LIMITATIONS AND PERSPECTIVES

In this work, we applied an unstructured parameter-by-parameter pruning
method, i.e. with a granularity at the weight level. Using such a method
leads, from a dense matrix, to a sparse matrix where no constraint is a priori
imposed on the structure of the sparsity of the matrix. This means that, in
order to take advantage of the optimization that pruning theoretically brings,
it is necessary to have an adequate hardware that can handle sparse matrices.
The most evident way to overcome this limitation would be to perform a
structured pruning, which would remove blocks of weights, allowing to change
the initial architecture, instead of keeping sparse matrices.

Then, The model we considered in our experiments constitutes the backbone of
recent models as shown in section 2.1. One of the perspectives of this work is
to be able to prune these models in order to visualize and measure the impact
on more complex movements.

Finally, other methods can be employed to optimize a neural network such as
knowledge distillation [Hinton et al., 2015] in which a lightweight student
model learns the output of a large master model and tries to reach its
performance or fully-connected layers decomposition by singular value
decomposition technique [Golub and Reinsch, 1970] which replaces them by an
approximation of two smaller layers.

##  6 CONCLUSION

In this work, we explored the influence of pruning on MLP-MoE based
architectures, measured and visualized the impact on the generated motion.
These analyses showed that:

  * •

There is a trade-off between the naturalness of the generated motion and the
size of the network and its computational cost.

  * •

For an equivalent number of parameters, a pruned network performs better at
generating natural motion than a dense network.

  * •

Even in the case of extreme sparsity (90% of the network parameters have been
removed), the high level motion features that each expert learns are very
similar between the dense and sparse models, which is consistent with the
similarity of the activation profiles between these models.

As far as we know, this work is the first to make use of pruning techniques in
the context of neural network animation and we hope that this work will open
the way to further research in the context of lightweight neural animation.

## AKNOWLEDGMENTS

Unity materials and dog motion capture files we used here is given by
Sebastian Starke 111https://github.com/sebastianstarke/AI4Animation. We would
particularly like to thank him for his precious help.

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# Spatial Structure of the 12C Nucleus in a 3$\alpha$ Model with Deep
Potentials Containing Forbidden States

E. M. Tursunov<EMAIL_ADDRESS>Institute of Nuclear Physics, Academy of
Sciences, 100214, Ulugbek, Tashkent, Uzbekistan  National University of
Uzbekistan, 100174, Tashkent, Uzbekistan  M. Z. Saidov
<EMAIL_ADDRESS>Institute of Nuclear Physics, Academy of Sciences,
100214, Ulugbek, Tashkent, Uzbekistan  M. M. Begijonov
<EMAIL_ADDRESS>Institute of Nuclear Physics, Academy of Sciences,
100214, Ulugbek, Tashkent, Uzbekistan

###### Abstract

The spatial structure of the lowest 0${}_{1}^{+}$, 0${}_{2}^{+}$,
2${}_{1}^{+}$ and 2${}_{2}^{+}$ states of the 12C nucleus is studied within
the 3$\alpha$ model with the Buck, Friedrich and Wheatley $\alpha\alpha$
potential with Pauli forbidden states in the $S$ and $D$ waves. The Pauli
forbidden states in the three-body system are treated by the exact
orthogonalization method. The largest contributions to the ground and excited
2${}_{1}^{+}$ bound states energies come from the partial waves
$(\lambda,\ell)=(2,2)$ and $(\lambda,\ell)=(4,4)$. As was found earlier, these
bound states are created by the critical eigen states of the three-body Pauli
projector in the 0+ and 2+ functional spaces, respectively. These special
eigen states of the Pauli projector are responsible for the quantum phase
transitions from a weakly bound ”gas-like” phase to a deep ”quantum liquid”
phase. In contrast to the bound states, for the Hoyle resonance 0${}_{2}^{+}$
and its analog state 2${}_{2}^{+}$, dominant contributions come from the
$(\lambda,\ell)=(0,0)$ and $(\lambda,\ell)=(2,2)$ configurations,
respectively. The estimated probability density functions for the
12C(0${}_{1}^{+}$) ground and 2${}_{1}^{+}$ excited bound states show mostly a
triangular structure, where the $\alpha$ particles move at a distance of about
2.5 fm from each other. However, the spatial structure of the Hoyle resonance
and its analog state have a strongly different structure, like 8Be + $\alpha$.
In the Hoyle state the last $\alpha$ particle moves far from the doublet at
the distance between $R=3.0$ fm and $R=5.0$ fm. In the Hoyle analog
2${}_{2}^{+}$ state the two alpha particles move at a distance of about 15 fm,
but the last $\alpha$ particle can move far from the doublet at the distanse
up to $R=30.0$ fm.

3$\alpha$ model; quantum phase transition; Pauli forbidden states.

###### pacs:

11.10.Ef,12.39.Fe,12.39.Ki

##  I Introduction

A discovery of quantum phase transition (QPT) in the $\alpha$-like nuclei
serdar16 , including 12C ground state, is one of the main phenomena in nuclear
structure studies of last years. The effect was found within the framework of
ab initio method based on the chiral effective field theory potentials. The
importance of the finding is connected with a role of the carbon element in
the Universe, thus allowing to conclude about ”the life near a quantum phase
transition” serdar16 . The ab initio method of serdar16  is of course a
powerful tool for studying hyperfine effects from first principles of the
quantum chromodynamics (QCD). In this sense the method has more predictive
power than other approaches (see recent review rev  and references therein).

The discovery of the QPT in 12C inspired new research interest on the
structure of this important quantum object. Very important question is,
whether it is possible to observe an effect of the QPT within the framework of
a 3$\alpha$-cluster model for practical applications. Another interesting
property of this nucleus is its special structure, associated with the
Bose–Einstein condensation review12C . On the other hand a realistic modeling
of Pauli forces in the 3$\alpha$ system is a very difficult problem. A
complicated nonlocal $\alpha\alpha$ potential derived from the resonating
group model calculations was able to reproduce the energies of the ground
state and the Hoyle ($0^{+}_{2})$ resonance suz08 . But a repulsive local
$\alpha\alpha$-potentials, both l-dependent and l-independent, strongly
underestimate the bound states of the 12C nucleus tur03 . The alternative
local deep $\alpha\alpha$\- interaction potential of Buck–Friedrich–Wheatley
(BFW) BFW  yields much more realistic description of the nuclear structure,
since it allows to treat the Pauli forbidden states (FS) exactly. However,
application of the method of orthogonalizing pseudopotentials (OPP) kuk78  for
the 3$\alpha$ quantum system met a serious problems, although it works very
well for the structure of nuclei like 6He and 6Li, which contain a single
$\alpha$ cluster in the $\alpha+N+N$ three-body model tur06 ; tur06a ; tur16 ;
bt18 ; tur18  .

Indeed, detailed studies within the OPP method have demonstrated tur01 ;
fuji06  that the energy spectrum of the ground $0_{1}^{+}$ and first excited
$2^{+}_{1}$ states is highly sensitive to the description of the
$\alpha\alpha$-Pauli forbidden states. In these studies a convergence of the
energies of the ground and excited states in respect to the projecting
constant $\lambda$ was not clear. When passing values $\lambda=10^{3}$–104 MeV
the energies of the ground $0_{1}^{+}$ and first excited $2^{+}_{1}$ states
increase sharply, which was not usual. Only the use of the direct
orthogonalization method fuji06  for the elimination of the 3$\alpha$ Pauli
forbidden states allowed to clarify the situation. It was found that the
convergence problem within the OPP method is due to the so-called almost
forbidden states (AFS) which are the special eigen states of the complete
3$\alpha$ Pauli projector with eigen values, close to zero.

Finally, recently in tur21a  it was argued that above AFS are nothing but the
critical eigen states (CES) of the three-body Pauli projector which are
responsible for the quantum phase transition in the ground and first excited
2${}_{1}^{+}$ bound states of the 12C nucleus found in serdar16 . So, when
passing the quantum critical eigen state, the energy of the lowest state of
the carbon nucleus changes sharply from -0.627 MeV (weakly bound phase) to the
deep phase with the energy value of -19.897 MeV. The same behavior was found
for the excited 12C (2${}_{1}^{+}$) bound state where the weak phase with the
energy of 1.873 MeV changes to the deep phase with the energy value of -16.572
MeV. Another interesting result is that from the left side of the quantum
critical point the lowest states present the astrophysical significant Hoyle
state 0${}_{2}^{+}$ and its excited analog state 2${}_{2}^{+}$. When passing
the above mentioned CES these Hoyle states energies almost do not change, but
they become the excited states. In other words the lowest 0${}_{1}^{+}$ and
2${}_{1}^{+}$ states in the deep phases are created by the critical eigen
states of the complete three-alpha Pauli projector.

The aim of the present work is to study the spatial structure of the ground
and lowest excited states of the 12C nucleus in the 3$\alpha$ model. In sam22
the spatial structure was studied within the local $\alpha\alpha$ potentials
with a strong repulsive core, while adding a very large three-body attractive
forces. Differently, we deal with a more realistic deep BFW $\alpha\alpha$
potential with forbidden staters in the $S$ and $D$ waves. A variational
method on symmetrized Gaussian basis is employed. For the elimination of the
3$\alpha$ Pauli forbidden states we use the same direct orthogonalization
method from fuji06 ; tur21a . We will examine a similarity of the $0^{+}$ and
$2^{+}$ spatial structure including the Hoyle band.

The theoretical model is described in Section 2. Sections 3 contains the
numerical results for the 12C(0+) and 12C(2+) states. Conclusions are given in
the last section.

##  II Theoretical model

The direct orthogonalization method fuji06 ; tur21a  is based on the
separation of the complete Hilbert functional space into two parts. The first
allowed subspace $L_{Q}$ is defined by the kernel of the 3$\alpha$ projector.
The rest subspace $L_{P}$ contains 3$\alpha$ states forbidden by the Pauli
principle. The complete Hilbert functional space of 3$\alpha$ states is
separated into the $L_{Q}$ (allowed) and $L_{P}$(forbidden) subspaces, then
the three-body Schrödinger equation is solved in $L_{Q}$.

The $\alpha\alpha$\- interaction potential of Buck–Friedrich–Wheatley BFW  of
the Gaussian form reads

$V(r)=V_{0}\exp(-\eta r^{2})+4e^{2}erf(br)/r,$ (1)

with parameters $V_{0}$ = -122.6225 MeV, $\eta=0.22$ fm-2 for the nuclear part
and $b$ = 0.75 fm-1 for the Coulomb part. This choice of the potential
parameters yields a very good description of the experimental phase shifts
$\delta_{L}(E)$ for the $\alpha\alpha$\- elastic scattering in the partial
waves $L=0,2,4$ within the energy range up to 40 MeV and the energy positions
and widths of the 8Be resonances.

As in tur21a ; fuji06  we use a value $\hbar^{2}/m_{\alpha}=10.4465$ MeV fm2.
This potential contains two Pauli forbidden states in the $S$ wave with
energies $E_{1}=-72.6257$ MeV and $E_{2}=-25.6186$ MeV, and a single forbidden
state in the $D$ wave with $E_{3}=-22.0005$ MeV. A realistic description of
the system requires to eliminate all FS from the solution of the three-body
Schrödinger equation.

The three-body Hamiltonian reads:

$\hat{H}=\hat{H}_{0}+V(r_{23})+V(r_{31})+V(r_{12}),$ (2)

where $\hat{H}_{0}$ is the kinetic energy operator and $V(r_{ij})$ is the
interaction potential between the $i$-th and $j$-th particles. A solution of
the Schrödinger equation

$\hat{H}\Psi_{s}^{JM}=E\Psi_{s}^{JM},\,\,\Psi_{s}^{JM}\in L_{Q}.$ (3)

should belong to the allowed subspace $L_{Q}$ of the complete 3$\alpha$
functional space.

The symmetrized wave function of the 3$\alpha$\- system is expanded in the
series of Gaussian functions tur01 :

$\Psi_{s}^{JM}=\sum_{\gamma j}c_{j}^{(\lambda,l)}\varphi_{\gamma j}^{s},$ (4)

where $\varphi_{\gamma j}^{s}=\varphi_{\gamma j}(1;2,3)+\varphi_{\gamma
j}(2;3,1)+\varphi_{\gamma j}(3;1,2),$

$\varphi_{\gamma j}(k;l,m)=N_{j}^{(\lambda
l)}x_{k}^{\lambda}y_{k}^{l}\exp(-\alpha_{\lambda
j}x_{k}^{2}-\beta_{lj}y_{k}^{2}){\cal F}_{\lambda
l}^{JM}(\widehat{\bf{x}_{k}},\widehat{\bf{y}_{k}})$ (5)

Here $(k;l,m)={(1;2,3),(2;3,1),(3;1,2)}$,
$\gamma=(\lambda,l,J,M)=(\gamma_{0},J,M);\bf{x}_{k},\bf{y}_{k}$ are the
normalized Jacobi coordinates in the $k$-set:

$\bf{x}_{k}=\frac{\sqrt{\mu}}{\hbar}(\bf{r}_{l}-\bf{r}_{m})\equiv\tau^{-1}\bf{r}_{l,m};$
$\bf{y}_{k}=\frac{2\sqrt{\mu}}{\sqrt{3}\hbar}(\frac{\bf{r}_{l}+\bf{r}_{m}}{2}-\bf{r}_{k})\equiv\tau_{1}^{-1}\bf{\rho}_{k},$
(6)

$N_{j}^{(\lambda l)}$ is a normalizing multiplier. The nonlinear variational
parameters $\alpha_{\lambda j},\beta_{lj}$ are chosen as the nodes of the
Chebyshev grid:

$\alpha_{\lambda
j}=\alpha_{0}\tan(\frac{2j-1}{2N_{\lambda}}\frac{\pi}{2}),j=1,2,...N_{\lambda},$
$\beta_{lj}=\beta_{0}\tan(\frac{2j-1}{2N_{l}}\frac{\pi}{2}),j=1,2,...N_{l},$
(7)

where $\alpha_{0}$ and $\beta_{0}$ are scale parameters for each $(\lambda l)$
partial component of the complete wave function.

When one uses the Chebyshev grid, the basis frequensies $\alpha_{\lambda
j},\beta_{lj}$ cover larger and larger intervals around the scale parameters
as the numbers $N_{\lambda}$ and $N_{l}$ increase. This allows to take into
account both short-range and long-range correlations of particles. The
extraordinary flexibility of the many-particle Gaussian basis makes it
possible to describe three-particle configurations that are formed in the
ground and excited states of multicluster systems, and which exhibit an
extremely high degree of clustering tur94 .

The angular part of the Gaussian basis is factorized as:

${\cal F}_{\lambda
l}^{JM}(\widehat{\bf{x}_{k}},\widehat{\bf{y}_{k}})=\\{Y_{\lambda}(\widehat{\bf{x}_{k}})\bigotimes
Y_{l}(\widehat{\bf{y}_{k}})\\}_{JM}\phi(1)\phi(2)\phi(3),$ (8)

where $\phi(i)$ is the internal wave functions of the $\alpha$-particles. Here
the orbital momenta $\lambda$ and $l$ are conjugate to the Jacobi coordinates
$\bf{x}_{k}$ and $\bf{y}_{k}$, respectively.

The kinetic energy operator of the Hamiltonian can be expressed in the
normalized Jacobi coordinates in a simple form as

$\hat{H}_{0}=-\frac{\partial^{2}}{\partial\bf{x}_{k}^{2}}-\frac{\partial^{2}}{\partial\bf{y}_{k}^{2}}$
(9)

within any choice of $(\bf{x}_{k},\bf{y}_{k})$, $k=1,2,3$. The detailed matrix
elements of all the operators can be found in tur94 .

In order to separate the complete 3$\alpha$ functional space into the $L_{Q}$
and $L_{P}$ one has to calculate eigen states and corresponding eigen values
of the operator fuji06

$\hat{P}=\sum_{i=1}^{3}\hat{P}_{i},$ (10)

where each $\hat{P}_{i},(i=1,2,3)$ is the sum of Pauli projectors
$\hat{\Gamma}_{i}^{(f)}$ on the partial $f$ wave forbidden states ($1S$, $2S$,
and $1D$) in the $i$-th $\alpha\alpha$ subsystem:

$\hat{P}_{i}=\sum_{f}\hat{\Gamma}_{i}^{(f)},$ (11)
$\hat{\Gamma}_{i}^{(f)}=\frac{1}{2f+1}\sum_{m_{f}}\mid\varphi_{fm_{f}}(\bf{x}_{i})><\varphi_{fm_{f}}(\bf{x^{\prime}}_{i})\mid\delta(\bf{y}_{i}-\bf{y^{\prime}}_{i}),$
(12)

with the forbidden state function expanded into the Gaussian basis:

$\varphi_{fm_{f}}({\bf
x}_{i})=x_{i}^{f}\sum_{m}N_{m}^{(f)}b_{m}^{(f)}\exp(-\frac{r_{i}^{2}}{2r_{0m}^{(f)2}})Y_{fm_{f}}(\hat{\bf{x}}_{i}).$
(13)

Here $r_{0}$ is the ”projector radius” and $N_{m}^{(f)}$ is the normalizing
multiplier:

$N_{m}^{(f)}=2^{f+7/4}\frac{\alpha_{m}^{(2f+3)/4}}{\pi^{1/4}[(2\lambda+1)!!]^{1/2}},\qquad\alpha_{m}=\tau^{2}/(2r_{0m}^{2}).$
(14)

The operator $\hat{P}$ is not a complete projector for the three-body system,
however its kernel is identical with the kernel of the complete three-body
projector tur21 . This is why one can use the operator $\hat{P}$ for the
separation of the complete Hilbert functional space into the allowed $L_{Q}$
and forbidden $L_{P}$ subspaces.

For the study of spatial structure of the 12C nucleus lowest states in the
3$\alpha$ model we have to estimate the probability density functions

$P(r,R)=\int{\Psi_{s}^{JM*}({\bf r},{\bf R})\Psi_{s}^{JM}(\bf{r},{\bf
R})}d\widehat{\bf{r}}d\widehat{\bf{R}},$ (15)

for each of the fixed bound or resonance state. In the last equation the
integral is taken only over angular part of the relative coordinates $\bf{r}$
(distance between any fixed two $\alpha$ particles) and $\bf{R}$ (distance
between the last $\alpha$ and the center of mass of already fixed two $\alpha$
particles). The normalization of the spatial density function requires the
condition

$\int{P(r,R)}drdR=1.$ (16)

##  III Numerical results

###  III.1 Partial Waves Contributions to the Lowest 0+ and 2+ States

First we estimate the contributions of different three-body partial waves into
the energies of the ground and excited states of the 12C nucleus. For the
0${}_{1}^{+}$ (ground) and 0${}_{2}^{+}$ (Hoyle) states the main contributions
come from the three-body channels $(\lambda,\ell)=(0,0),(2,2),(4,4)$ which
contain up to 280 Gaussian functions. Convergence is fast due to the use of
symmetrized basis functions. Alternatively, one can use a nonsymmetrized basis
functions for practical applications of the three-body wave function. In this
case the main contributions come from the three-body channels
$(\lambda,\ell)=(0,0),(2,2),(4,4),(6,6,),(8,8)$ requiring up to 1008 Gaussian
functions for obtaining convergent results comparable with the results of the
symmetrized basis.

Table 1: Contributions of different partial waves in $\%$ into the 0${}_{1}^{+}$ (ground) and 0${}_{2}^{+}$ (Hoyle) states $(\lambda,\ell)$ | (0, 0) | (2, 2) | (4, 4) | (6, 6) | (8, 8)
---|---|---|---|---|---
$0_{1}^{+}$ | 28.777 | 35.722 | 34.885 | 0.606 | 0.010
$0_{2}^{+}$ | 55.012 | 24.120 | 17.008 | 3.418 | 0.442
Table 2: Contributions of different partial waves in $\%$ into the 2${}_{1}^{+}$ and 2${}_{2}^{+}$ (Hoyle analog) states $(\lambda,\ell)$ | (0, 2) | (2, 0) | (2, 2) | (2, 4) | (4, 2) | (4, 4) | (6, 6) | (6, 4) | (4, 6) | (8, 8) | (8, 6) | (6, 8)
---|---|---|---|---|---|---|---|---|---|---|---|---
$2_{1}^{+}$ | 7.186 | 7.199 | 44.240 | 0.736 | 0.724 | 39.313 | 0.586 | 0.005 | 0.004 | 0.008 | 0 | 0
$2_{2}^{+}$ | 12.680 | 10.588 | 73.444 | 0.044 | 2.797 | 0.062 | 0.125 | 0.101 | 0.031 | 0.025 | 0.044 | 0.058

In Table 1 we give the contributions of different partial waves into the
0${}_{1}^{+}$ (ground) and 0${}_{2}^{+}$ (Hoyle) states. The ground state
energy in the deep phase is -19.90 MeV, which was obtained by including the
quantum critical eigen state of the Pauli projector into the functional model
space of the 0+ states tur21a . A big difference of this number from the
experimental energy value of -7.275 MeV means that the 3$\alpha$ system can be
in this deep phase with a probability much smaller than 1. Beyond the critical
point the energy of the Hoyle state increases slightly to $E=-0.458$ MeV,
which is, however, lower than the experimental energy value $E_{\rm
exp}(0_{2}^{+})$ = 0.380 MeV tur21a . Then a small three-body potential is
needed for the reproduction of the experimental energy.

As can be seen from the table, for the ground state the largest contributions
come from the partial waves $(\lambda,\ell)=(2,2)$ and $(\lambda,\ell)=(4,4)$.
The contribution of the $S$ wave is smaller due to the presence of Pauli
forbidden states. In contrast to the ground state, the main contribution to
the Hoyle state comes from $S$ waves. This result supports an idea of a
possible condensation of the $\alpha$ particles in the Hoyle state rev .

For the 2${}_{1}^{+}$ (bound) and 2${}_{2}^{+}$ (Hoyle analog) states the most
important contributions come from the three-body channels
$(\lambda,\ell)=(0,2),(2,0),(2,2),(2,4),(4,2),(4,4)$ which contain up to 437
symmetrized Gaussian functions. Or, alternatively, nonsymmetrized basis
functions can be used within the three-body channels
$(\lambda,\ell)=(0,2),(2,0),(2,2),(2,4),(4,2),(4,4),(4,6),(6,4),(6,6),(6,8),(8,6),(8,8)$.
In the last case a convergent results were obtained with 1292 Gaussian
functions.

In Table 2 we give the contributions of different partial waves to the
2${}_{1}^{+}$ (bound) and 2${}_{2}^{+}$ (Hoyle analog) states. The energy of
the 2${}_{1}^{+}$ state in the deep phase is -16.572 MeV, which was obtained
by including the quantum critical eigen state of the Pauli projector into the
functional model space of the 2+ states tur21a . As in the case of the ground
state, a strong difference of this number from the experimental energy value
of -2.836 MeV means that the 3$\alpha$ system can be in this deep phase with a
probability much smaller than 1. Beyond the critical point the energy of the
Hoyle analog state increases slightly to $E=2.279$ MeV, which is close to the
experimental energy value $E_{\rm exp}(2_{2}^{+})$ = 2.596 MeV tur21a . The
same small additional three-body potential, which was used for fitting Hoyle
state energy, can reproduce the experimental energy of its analog state tur21a
.

From Table 2 one can see that for the 2${}_{1}^{+}$ state the largest
contributions come from the partial waves $(\lambda,\ell)=(2,2)$ and (4, 4).
The contributions of the partial $S$ waves $(\lambda,\ell)=(0,2)$ and (2,0)
are comparable and smaller due to the precence of Pauli forbidden states. For
the 2${}_{2}^{+}$ Hoyle analog state the dominant contribution comes from the
partial wave $(\lambda,\ell)=(2,2)$.

###  III.2 Probability Density Functions of the Lowest 0+ and 2+ States

Now we go to estimate the probability density functions of the lowest 0+ and
2+ states of the carbon nucleus on the base of Eq. (15). These functions yield
a spatial structure or a matter distribution in each state.

Figure 1: Probability density for the 12C($0_{1}^{+}$) ground state. Figure 2:
Probability density for the 12C($0_{2}^{+}$) Hoyle resonance state. Figure 3:
Probability density for the 12C($2_{1}^{+}$) excited bound state. Figure 4:
Probability density for the 12C($2_{2}^{+}$) Hoyle resonance analog state.

In Fig. 1 we show the probability density functions for the 12C(0${}_{1}^{+}$)
ground state. As can be seen from the figure, in this state the alpha
particles mostly move in a regular triangular configuration with a maximum
probability at the distance of $r=2.7$ fm and $R=2.5$ fm.

In Fig. 2 the probability density function for the 12C( 0${}_{2}^{+}$) Hoyle
state is presented. Here one can see a very different structure of the matter
distribution. In this state the two alpha particles move at the distance of
about $r=$4 fm, and the last alpha particle moves far from the doublet at the
distance between $R=3.0$ fm and $R=5.0$ fm. There is also a probability to go
to smaller distances which corresponds to the smaller maxima in the structure
function. Here both linear and 8Be + $\alpha$ configurations are possible as
was found in sam22 .

In Fig. 3 the probability density function for the 12C(2${}_{1}^{+}$) bound
state is demonstrated. As can be seen from the figure, the spatial structure
of this state is also triangular. It is close to the matter ditribution of the
ground state, but there is a difference at small distances.

In Fig. 4 the probability density function for the 12C( 2${}_{2}^{+}$) Hoyle
analog state is presented. Here one can see a very extended structure. In this
state the two alpha particles move at the distance about $r=$15 fm, and the
last alpha particle moves far from the doublet at the distance between
$R=15.0$ fm and $R=30.0$ fm.

##  IV Conclusion

In summary, the spatial structure of the lowest 0+ and 2+ states of the 12C
nucleus has been analyzed within the 3$\alpha$ model with the BFW
$\alpha\alpha$ potential with two forbidden states in the $S$ wave and a
single forbidden state in the $D$ wave. The Pauli forbidden states in the
three-body system were treated by the exact orthogonalization method. As was
found previously, the ground and excited $2_{1}^{+}$ bound states in the deep
phases are completely defined by the so-called quantum critical eigen states
of the three-body Pauli projector for the 0+ and 2+ functional spaces,
respectively. In the weak phases these states are very close to the Hoyle and
its analog states.

The contributions of different partial waves into the 0${}_{1}^{+}$,
0${}_{2}^{+}$, 2${}_{1}^{+}$ and 2${}_{2}^{+}$ states have been estimated. For
the ground and excited 2${}_{1}^{+}$ bound states in the deep phases the
largest contributions come from the partial waves $(\lambda,\ell)=(2,2)$ and
$(\lambda,\ell)=(4,4)$. In contrast to the bound states, for the Hoyle
resonance 0${}_{2}^{+}$ and its analog state 2${}_{2}^{+}$, dominant
contributions come from the $(\lambda,\ell)=(0,0)$ and $(\lambda,\ell)=(2,2)$
configurations, respectively.

The estimated probability density functions for the 12C(0${}_{1}^{+}$) ground
and 2${}_{1}^{+}$ excited bound states show mostly a triangular structure with
a distance of about 2.5 fm between the $\alpha$ particles. However, the
spatial structures of the Hoyle resonance and its analog have a strongly
different structure, like 8Be + $\alpha$. In the Hoyle state the last $\alpha$
particle moves far from the doublet at the distance between $R=3.0$ fm and
$R=5.0$ fm. Even more, in the Hoyle analog state the two alpha particles move
at the distance of about 15 fm and the last $\alpha$ particle goes far from
the doublet at the distanse between $R=15.0$ fm and $R=30.0$ fm.

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# Age of Computing: A Metric of Computation Freshness in Communication and
Computation Cooperative Networks

Xingran Chen, Member, IEEE, Yusha Liu, Member, IEEE,
Yali Zheng, and Kun Yang, Fellow, IEEE Xingran Chen, Yusha Liu, and Yali Zheng
are with School of Information and Communication Engineering, University of
Electronic Science and Technology of China, Sichuang, 611731, China (E-mail:
{xingranc, yusha.liu, yalizheng}@uestc.edu.cn). Kun Yang is with School of
Computer Science and Electronic Engineering, University of Essex, Essex, CO4
3SQ, U.K (Email: kunyang@essex.ac.uk).

###### Abstract

In communication and computation cooperative networks (3CNs), timely
computation is crucial but not always guaranteed. There is a strong demand for
a computational task to be completed within a given time. The time taken
involves both processing time and communication time. However, a measure of
such timeliness in 3CNs is lacking. In this letter, we introduce the novel
concept, Age of Computing (AoC), to capture computation freshness in 3CNs. We
develop two methods for calculating this metric, applicable to a wide range of
3CNs. These calculations are applied to a queue-theoretic system comprising a
task-initiating source, a task-executing computational node and a
communication link, resulting in the derivation of two expressions for the
AoC. Then a tight upper bound and a tight lower bound are derived.
Subsequently, we investigate the communication-computation tradeoff and the
AoC-delay tradeoff. While our models are simpler than their real-world
counterparts, they provide comprehensive insight in understanding computation
freshness in 3CNs and the relationships between computing and communications.

keywords: Age of Computing, Computation Freshness, Communication and
Comuputing Cooperated Networks

##  I Introduction

In the 6G era, emerging applications such as the Internet of Things (IoT),
smart cities, and cyber-physical systems have significant demands for
communication and computation cooperative networks (3CNs), which provide
faster data processing, efficient resource utilization, and enhanced security
[1]. 3CNs originated from mobile edge computing technology (MEC), which aims
to complete computation-intensive and latency-critical tasks, with the
paradigm deploying distributedly tons of billions of edge devices at the
network edges [2]. Besides MEC, 3CNs include fog computing and computing power
networks. Fog computing can be regarded as a generalization of MEC, where the
definition of edge devices is broader than that in MEC [3]. Computing power
networks refers to a broader concept of distributed computing networks,
including edge, fog, and cloud computing [4].

In all 3CNs, there is no metric for capturing the freshness of computation.
Recently, a noticeable metric, age of information (AoI), is proposed to
describe the information freshness in communication networks [5, 6, 7].
However, applying the AoI in 3CNs is inappropriate because the AoI only
reveals the latency in communication, while it cannot reflect the latency in
computation. In this letter, we propose a novel metric, called age of
computing (AoC), to capture the computation freshness in 3CNs. A primary
requirement in 3CNs is that computational tasks are processed as timely as
possible and within the maximum acceptable threshold. Thus, the core idea of
AoC is to combine the offloading penalty and delay penalty. The former is
incurred by communication and processing of computational tasks, and the
latter stems from delays of tasks exceeding the users’ acceptation threshold,
which is called the maximum acceptable threshold.

To the best of our knowledge, this is the first work to propose a concept to
describe computation freshness in 3CNs. The contributions of this letter are
listed below.

  * •

We propose the concept of the AoC, which is a product of offloading penalty
and delay penalty. The offloading penalty describes the cumulative latency
incurred by communicating and processing of computational tasks, while the
delay penalty introduces punishment for situations where the delay of a task
exceeds the maximum acceptable threshold.

  * •

Based on this concept, we derive two general expressions for time-average AoC
in queue-theoretic first-come-first-served (FCFS) systems, which consist of
task-initiating source, a task-executing computational node and a
communication link. Utilizing the expressions, we establish a tight upper
bound and a tight lower bound. The time-average AoC and bounds are functions
of the maximum acceptable threshold, the delays, the interval between
consecutive offloadings, and the interval between consecutive task
completions.

  * •

Applying the bounds of time-average AoC to M/M/1 systems, we elucidate two
fundamental tradeoffs: the communication-computation tradeoff and the AoC-
delay tradeoff. Regarding the communication-computation tradeoff, when the
communication efficiency is significantly lower than the computation
efficiency, the AoC (computation freshness) decreases with the communication
efficiency because the computing power is being increasingly utilized; when
the communication efficiency exceeds a certain ratio of computation
efficiency, the AoC increases with the communication efficiency because the
computation power is fully loaded. As for the AoC-delay tradeoff, with small
delays, the AoC declines, while delays exceeding a certain threshold cause an
increase in the AoC. The essential reason is that the AoC gauges system-wide
computation freshness, drawing from comprehensive system state, while delay
measures individual task computation freshness, relying on instantaneous
system state.

The remaining parts of this paper are organized as follows. Section II
proposes and discusses the novel concept AoC. Section III derives the time-
average AoC and the upper and lower bounds in queue-theoretic FCFS systems,
and presents the communication-computation and AoC-delay tradeoffs. We
numerically verify our theoretical results in Section IV and conclude this
work in Section V.

##  II Age of Computing

In the section, we introduce the mathematical formulation of the novel
concept, age of computing (AoC), which quantifies computation freshness within
3CNs. Consider a queue-theoretic system with a source and a computational node
(see Fig. 1). First of all, we define offloading as the transfer of
computational tasks from the source to the computational node. The source
offloads tasks to the computational node, which subsequently processes them.
We index computational tasks as $k=1,2,\cdots$. Suppose the computational node
is equipped with caching capabilities, when a task occupies the computational
node, any newly arrived tasks must wait in the queue on the computational
node’s side.

Figure 1: An example of a queue-theoretic system with a source and a
computational node.

Consider the $k^{\text{th}}$ computational task: we denote the time when the
offloading starts as $t_{k}$, the time when the processing completes as
$t_{k}^{\prime}$, and the delay of the task as $D_{k}=t_{k}^{\prime}-t_{k}$.

######  Definition 1.

(task indices and timestamp). In a system comprising a source and a
computational node, the index of the most recently completed task at time $t$
is

$\displaystyle N(t)=\max\\{k|t_{k}^{\prime}\leq t\\}.$ (1)

and the timestamp of the most recently completed task is

$\displaystyle u(t)=t_{N(t)}.$ (2)

######  Definition 2.

(delay at every moment). In the system defined in Definition 1, define the
delay at time $t$ as

$\displaystyle d(t)=D_{N(t)},$ (3)

where $N(t)$ is defined in (1).

######  Remark 1.

From Definition 2, the delay at time $t$ represents the delay of the most
recently completed task before and including time $t$.

Now, we formally describe the metric AoC. This metric is introduced to
illustrate the concept of fresh informative computational tasks in 3CNs. Here,
“informative” refers to tasks that (i) bring the latest offloading information
and (ii) are completed within the maximum acceptable tolerant delay by users.

######  Definition 3.

(Age of Computing - AoC). In the system defined in Definition 1, the age of
computing is defined as the random process

$\displaystyle\Theta(t)=h(t)\times\psi(t),$ (4)

where $h(t)$ is the offloading penalty function

$\displaystyle h(t)=t-u(t),$ (5)

and $\psi(t)$ is the delay penalty function

$\displaystyle\psi(t)=1_{\\{d(t)\leq\tau\\}}+\theta\cdot 1_{\\{d(t)>\tau\\}}.$
(6)

In (5), $u(t)$ is defined in (2). In (6), $d(t)$ is defined in (3), $\theta>1$
is the penalty coefficient, and $\tau>0$ is the maximum acceptable threshold.

From Definition 3, the AoC $\Theta(t)$ comprises two components: the
offloading penalty $h(t)$ and the delay penalty $\psi(t)$. To enhance
understanding of $\Theta(t)$, we delve into insights behind (4), (5) and (6)
as follows.

The offloading penalty function $h(t)$ (see (5)) reflects the freshness of
offloading. If the $k^{th}$ task has a later offloading start time, i.e.,
$t_{k}$ is not far from $t_{k}^{\prime}$, then $h(t)$ has a substantial
downward jump at time $t_{k}^{\prime}$, otherwise, it exhibits a slight
downward jump. It is worth noting that although $h(t)$ and the age of
information (AoI, refer to [5]) have similar formulas, they are fundamentally
distinct in terms of their physical interpretations. Specifically, $h(t)$
quantifies, at any moment, the time elapsed since the last completed task
began offloading. Thus, $h(t)$ encompasses both the cumulative transmission
latency and the cumulative processing latency, whereas the AoI only captures
the former. In some circumstances, fewer fresh tasks may be more useless. To
address this, we modify the offloading penalty function to be non-linear
rather than linear,

$\displaystyle h(t)=\exp\big{(}a(t-u(t))\big{)},$ (7)

where $a>0$ is a real number.

The delay penalty function $\psi(t)$ (see (6)) comprises two disjoint events.
$1_{\\{d(t)>\tau\\}}$ indicates that the delay of the most recently completed
task exceeds the maximum acceptable threshold, while $1_{\\{d(t)\leq\tau\\}}$
represents the opposite case. Here, $\tau$ is an exogenous coeffecient, which
is determined by users’ requirements. If a task fails to meet the maximum
acceptable threshold, it loses its value to users, thereby incurring greater
penalty ($\theta>1$). In certain circumstances, as the delay exceeds the
maximum acceptable threshold, the penalty increases proportionally. Thus, we
modify the delay penalty function as

$\displaystyle\psi(t)=1_{\\{d(t)\leq\tau\\}}+\theta\cdot(1+d(t)-\tau)\cdot
1_{\\{d(t)>\tau\\}}.$ (8)

The term $1+d(t)-\tau$ suggests that the delay penalty increases as the delay
exceeds the maximum acceptable threshold.

The curve of AoC (see (4)) is depicted in Fig. 2. Consider the $(k-1)^{th}$
and $k^{th}$ tasks. The $(k-1)^{th}$ task is completed within the maximum
acceptable threshold, causing the AoC (see (4)) drops to
$\Theta(t_{k-1}^{\prime})=t_{k-1}^{\prime}-t_{k-1}$ at time
$t_{k-1}^{\prime}$. The delay of the $k^{th}$ task exceeds the maximum
acceptable threshold, so the AoC drops to
$\Theta(t_{k}^{\prime})=\theta(t_{k}^{\prime}-t_{k})$ at time
$t_{k}^{\prime}$.

Figure 2: The curve of AoC.

##  III Theoretical Results for Time-average AoC

To uncover theoretical insights of the AoC, we investigate the AoC in queue-
theoretic systems. The discipline of queuing is characterized by first-come-
first-served (FCFS). In the rest of this section, we analyze the time-average
AoC using scenarios outlined in [8].

###  III-A System Model

On the source side, computational tasks are offloaded via a stochastic process
characterized by an average rate of $\lambda$. The interval between
consecutive offloadings, denoted by $A_{k}=t_{k+1}-t_{k}$, has the expectation
$\mathbb{E}[A_{k}]=\frac{1}{\lambda}$. At the computational node side, the
storage of the tasks at the queue is instantaneous [8]. Upon arrival, each
task is processed at an average rate of $\mu$. We assume that $\lambda$ is the
maximum offloading rate constrained by communication resources (e.g.,
bandwidth) in the network, while $\mu$ is the maximum processing rate
constrained by computation resources. Thus, $\lambda$ can be interpreted as
the communication efficiency, and $\mu$ as the computation efficiency.

Let $\rho=\lambda/\mu$, where $\rho$ signifies the communication-computation
relative efficiency. A small $\rho$ ($\rho\to 0$) indicates limited
communication resources in the network compared to computation resources,
resulting in minimal consumption of computation power. A large $\rho$
($\rho\to 1$) suggests abundant communication resources relative to
computation resources, leading to nearly full utilization of computation
capacity. According to queuing theory [9], the system is stable when
$0\leq\rho<1$; otherwise, the length of the queue explodes with time,
rendering the system unstable.

Consider an observation interval $[0,\mathcal{T}]$. According to Definition 1,
the index of the last task is denoted by $N(\mathcal{T})$, so
$t_{N(\mathcal{T})}^{\prime}\leq\mathcal{T}$. Denote the interval between
consecutive task completions as $C_{k}=t_{k+1}^{\prime}-t_{k}^{\prime}$.

###  III-B Average AoC

In this subsection, we give two general expressions for AoC through graphical
argument, one for determining an upper bound, and the other for determining a
tight lower bound. In the interval $[0,\mathcal{T}]$, the time-average AoC can
be calculated by

$\displaystyle\Theta_{\mathcal{T}}=\frac{1}{\mathcal{T}}\int_{0}^{\mathcal{T}}\Theta(t)dt.$
(9)

Since the integral in (9) equals the area under $\Theta(t)$,
$\Theta_{\mathcal{T}}$ can be rewritten as a sum of disjoint geometric parts.
Thus we use two graphical arguments to compute $\Theta_{\mathcal{T}}$.

The first decomposition is to decompose the whole area into disjoint
$\\{S_{k}\\}_{k}$ (see Fig. 2). From (5) and (6), for $1\leq k\leq
N(\mathcal{T})-1$, we can obtain the expression of $S_{k}$ as

$S_{k}=\frac{(A_{k}+D_{k+1})^{2}}{2}f_{k}(\tau,\theta)-\frac{D_{k+1}^{2}}{2}f_{k+1}(\tau,\theta).$
(10)

where $f_{k}(\tau,\theta)=\theta\cdot
1_{\\{D_{k}>\tau\\}}+1_{\\{D_{k}\leq\tau\\}}$. We adjust the notation a little
by defining $S_{0}$ and $S_{N(\mathcal{T})}$ as the areas beneath the curve
$\Theta(t)$ associated with the interval $[0,t_{1})$ and
$[t_{N(\mathcal{T})},\mathcal{T}]$, respectively.

The second decomposition is to alternatively decompose the whole area into
disjoint right trapezoids $\\{S_{k}^{\prime}\\}_{k}$ (see Fig. 2). For $1\leq
k\leq N(\mathcal{T})-1$, denote

$\displaystyle
S_{k}^{\prime}=\int_{t_{k}^{\prime}}^{t_{k+1}^{\prime}}\Theta(t)dt=(D_{k}C_{k}+C_{k}^{2}/2)f_{k}(\tau,\theta).$
(11)

We adjust the notation a little by defining
$S_{0}^{\prime}=\int_{0}^{t_{1}^{\prime}}\Theta(t)dt$ and
$S_{N(\mathcal{T})}^{\prime}=\int_{t_{N(\mathcal{T})}^{\prime}}^{\mathcal{T}}\Theta(t)dt$.

Substituting (10) and (11) into (9), we have

$\displaystyle\Theta_{\mathcal{T}}=\frac{1}{\mathcal{T}}\sum_{k=0}^{N(\mathcal{T})}S_{k}=\frac{1}{\mathcal{T}}\sum_{k=0}^{N(\mathcal{T})}S_{k}^{\prime}.$
(12)

Let $\Theta_{\text{AoC}}=\lim_{\mathcal{T}\to\infty}\Theta_{\mathcal{T}}$, and
with the proof in Appendix A, (12) is changed to

$\displaystyle\Theta_{\text{AoC}}=\frac{\mathbb{E}[S_{k}]}{\mathbb{E}[A_{k}]}=\frac{\mathbb{E}[S_{k}^{\prime}]}{\mathbb{E}[A_{k}]}.$
(13)

In the following theorem, we give the two general expressions for
$\Theta_{\text{AoC}}$.

######  Theorem 1.

Given any $\tau$ and $\theta$, the general expressions of
$\Theta_{\text{AoC}}$ are given by

$\displaystyle\Theta_{\text{AoC}}=$
$\displaystyle\frac{1}{\mathbb{E}[A_{k}]}\Big{(}\mathbb{E}[\frac{A_{k}^{2}}{2}]\mathbb{E}[f_{k}(\tau,\theta)]+\mathbb{E}[(A_{k}D_{k+1}f_{k}(\tau,\theta)]$
$\displaystyle+$
$\displaystyle\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k}(\tau,\theta)]-\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k+1}(\tau,\theta)]\Big{)},$
(14)

or

$\displaystyle\Theta_{\text{AoC}}=$
$\displaystyle\frac{1}{\mathbb{E}[A_{k}]}\Big{(}\theta\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)1_{\\{D_{k}>\tau\\}}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)1_{\\{D_{k}\leq\tau\\}}.]$
(15)

###### Proof.

The proof is given in Appendix B. ∎

In general, finding the closed form of $\Theta_{\text{AoC}}$ is non-trivial to
find, even in the point-to-point systems. In (1), $D_{k+1}$ and $A_{k}$ are
correlated and $D_{k}$ and $D_{k+1}$ are correlated as well. In (1), $C_{k}$
and $D_{k}$ are correlated. These correlations result in complex calculation
of $\Theta_{\text{AoC}}$. Therefore, we offer a general upper bound and a
general upper bound and a tight lower bound for $\Theta_{\text{AoC}}$ in
Theorem 2.

######  Theorem 2.

Denote $q_{\tau}=\Pr\big{(}D_{k}\leq\tau\big{)}$,
$g_{\tau}=\theta(1-q_{\tau})+q_{\tau}$, and
$h_{\tau}=\max\\{\theta(1-q_{\tau}),q_{\tau}\\}$. The time-average
$\Theta_{\text{AoC}}$ is upper bounded by

$\displaystyle\Theta_{\text{UB}}=$
$\displaystyle\frac{g_{\tau}\mathbb{E}[\frac{A_{k}^{2}}{2}]+\theta\mathbb{E}[A_{k}D_{k+1}]}{\mathbb{E}[A_{k}]},$
(16)

and is lower bounded by

$\displaystyle\Theta_{\text{LB}}=$ $\displaystyle
h_{\tau}\frac{\mathbb{E}[\frac{A_{k}^{2}}{2}]+\mathbb{E}[A_{k}D_{k+1}]}{\mathbb{E}[A_{k}]}.$
(17)

The equality holds in (16) when $\tau=0$, and the equalith holds in (17) when
$\tau\to\infty$ and $\tau=0$,

$\displaystyle\Theta_{\text{AoC}}|_{\\{\tau=0\\}}=\Theta_{\text{UB}}|_{\\{\tau=0\\}}=\Theta_{\text{LB}}|_{\\{\tau=0\\}}$
$\displaystyle=\theta\frac{\mathbb{E}[\frac{A_{k}^{2}}{2}]+\mathbb{E}[A_{k}D_{k+1}].}{\mathbb{E}[A_{k}]}$
(18)
$\displaystyle\lim_{\tau\to\infty}\Theta_{\text{AoC}}=\lim_{\tau\to\infty}\Theta_{\text{LB}}=\frac{\mathbb{E}[\frac{A_{k}^{2}}{2}]+\mathbb{E}[A_{k}D_{k+1}].}{\mathbb{E}[A_{k}]}.$
(19)

###### Proof.

The proof is given in Appendix C. ∎

In scenarios where users are latency-sensitive ($\tau$ is small), both upper
and lower bounds effectively approximate $\Theta_{\text{AoC}}$. However, when
users are more tolerant of delays ($\tau$ is large), the lower bound
$\Theta_{\text{LB}}$ serves as an accurate approximation of
$\Theta_{\text{AoC}}$.

###  III-C Tradeoffs

Now, we aim to reveal two fundamental tradeoffs: the communication-computation
tradeoff and the AoC-delay tradeoff. To ensure analytical tractability, we
focus on M/M/1 systems, although the framework can be extended to analyze
other systems as well. The analysis of the time-average AoC is clearly
difficult due to the correlations in $(C_{k},D_{k})$, $(D_{k},D_{k+1})$, and
$(A_{k},D_{k+1})$ even in M/M/1 systems, here we will proceed by providing
theoretical results in the upper bound in (16) and the lower bound (17).

As discussed in Section III-A, the ratio $\rho$ represents the relative
efficiency of communication to computation. The tradeoff between communication
and computation is analyzed using this ratio.

######  Theorem 3.

(Communication-Computing Tradeoff) In M/M/1 FCFS systems, let $\mu$ be fixed,
denote $g(\rho)=1-e^{-\mu(1-\rho)\tau}+\theta e^{-\mu(1-\rho)\tau}$ and
$h(\rho)=\max\\{\theta e^{-\mu(1-\rho)\tau},1-e^{-\mu(1-\rho)\tau}\\}$.

  * (1)

The upper bound $\Theta_{\text{UB}}$ is expressed by

$\Theta_{\text{UB}}=\frac{1}{\mu}\Big{(}\frac{g(\rho)}{\rho}+\theta+\frac{\theta\rho^{2}}{1-\rho}\Big{)},$
(20)

there is a unique minimum point $\rho_{UB}^{*}$ in $(0,1)$, such that
$\Theta_{\text{UB}}$ decreases with $\rho$ for $\rho\leq\rho_{UB}^{*}$, and
increases with $\rho$ otherwise.

  * (2)

The lower bound $\Theta_{\text{LB}}$ is expressed by

$\Theta_{\text{LB}}=\frac{h(\rho)}{\mu}\big{(}1+\frac{1}{\rho}+\frac{\rho^{2}}{1-\rho}\big{)},$
(21)

there is a unique minimum point $\rho_{LB}^{*}$ in $(0,1)$ when
$\tau\to\infty$, such that $\lim_{\tau\to\infty}\Theta_{\text{LB}}$ decreases
with $\rho$ for $\rho\leq\rho_{LB}^{*}$, and increases with $\rho$ otherwise.

###### Proof.

The proof is given in Appendix D. ∎

Theorem 3 elucidates the communication-computation tradeoff. Note that $\mu$
is fixed, when $\rho\leq\rho_{UB}^{*}$ (respectively,
$\rho\leq\rho_{LB}^{*}$), an increase in $\rho$ implies an increase in
communication efficiency, indicating that more tasks are offloaded to and
processed at the computational node. This leads to improved computation
freshness, resulting in a decreased upper bound $\Theta_{\text{UB}}$
(respectively, limit lower bound $\lim_{\tau\to\infty}\Theta_{\text{LB}}$).
However, when $\rho>\rho_{UB}^{*}$ (respectively, $\rho>\rho_{LB}^{*}$), the
continued rise in communication efficiency causes the computational node to
become fully loaded, leading to a backlog of tasks in the queue. This
congestion adversely affects computation freshness, leading to an increase in
the upper bound $\Theta_{\text{UB}}$ (respectively, the limit lower bound
$\lim_{\tau\to\infty}\Theta_{\text{LB}}$).

Denote $\bar{D}=\mathbb{E}[D_{k}]$, according to [9],
$\bar{D}=\frac{1}{\mu(1-\rho)}$. We investigate the AoC-delay tradeoff in
Theorem 4, with the proof provided in Appendix E.

######  Theorem 4.

(AoC-Delay Tradeoff) In M/M/1 FCFS systems, with $\mu$ fixed,
$\Theta_{\text{UB}}$ (respectively, $\lim_{\tau\to\infty}\Theta_{\text{LB}}$)
decreases with $\bar{D}$ when $\bar{D}\leq\frac{1}{\mu(1-\rho_{UB}^{*})}$
(respectively, $\bar{D}\leq\frac{1}{\mu(1-\rho_{LB}^{*})}$), and increases
with $\bar{D}$ otherwise.

In Theorem 4, the upper bound $\Theta_{\text{UB}}$ (respectively, the limit
lower bound $\lim_{\tau\to\infty}\Theta_{\text{LB}}$) reveals a tradeoff
between the average AoC and the delay $\bar{D}$. Let $\mu$ be fixed, when
$\rho$ is small, indicating low communication efficiency compared to
computation efficiency, tasks are processed as soon as it arrives at the
computational node since the queue is empty. Consequently, the delay is small,
but the upper bound $\Theta_{\text{UB}}$ (respectively, the limit lower bound
$\Theta_{\text{LB}}$) may be large due to the fact that the arrived task was
offloaded a long time ago. Conversely, when $\rho$ is large, many tasks wait
in the queue, resulting in a large delay and a large $\Theta_{\text{UB}}$
(respectively, $\lim_{\tau\to\infty}\Theta_{\text{LB}}$).

##  IV Simulations

In this section, we verify our findings through simulations. Let $\mu=1$,
$\lambda\in[0.2,0.8]$, $\tau\in\\{8,15,30\\}$, and $\theta=2$.

In Fig. 3, we investigate the communication-computation tradeoff under
different $\tau$ in M/M/1 FCFS systems. All the the upper bounds, lower
bounds, and time-average AoC have the same trends with respect to $\rho$,
which coincides with Theorem 3. As discussed in Section III-C, when $\rho$ is
small, the AoC (and the bounds) decreases with $\rho$ because more tasks are
offloaded to and processed at the computational node. Conversely, when $\rho$
exceeds a certain threshold, the AoC (and the bounds) increases with $\rho$
due to the overloaded computational power and a backlog of tasks in the queue.
Additionally, when $\tau$ is small, both the upper and lower bounds
increasingly approximate the AoC, whereas when $\tau$ is large, the lower
bound becomes a more accurate approximation of the AoC. This coincides with
Theorem 3 as well. Finally, we observe that the AoC decreases with $\tau$.
This is because more tasks are completed within the maximum acceptable
threshold, resulting in a smaller AoC in the system.

Figure 3: The communication-computing tradeoff under different pairs of
$(\tau,\theta)$ in M/M/1 FCFS systems.

In Fig. 4, we explore the AoC-delay tradeoff when $\tau=15$. When the delay is
small, task duration within the system is short. With a fixed computing
efficiency $\mu$, AoC (and the bounds) is large as task offloading from the
source occurs infrequently. On the other hand, when the delay is large, the
queue becomes congested due to numerous tasks, resulting in a large AoC (and
the bounds) as well. The essence of the tradeoff lies in AoC’s ability to
gauge system-wide computation freshness, drawing from comprehensive queuing
state information, while delay measures individual task computation freshness,
relying on instantaneous queuing state information.

Figure 4: The AoC-delay tradeoffs in M/M/1 FCFS systems.

##  V Conclusion

In this letter, we initially introduce the concept of AoC to mathematically
describe computation freshness. Subsequently, we derive two general
expressions of the time-average AoC in queuing-theoretic FCFS networks with
one source and one computational node. By utilizing the expressions of AoC, we
provide a upper bound and a tight lower bound. In M/M/1 systems, we
theoretically illustrate two fundamental tradeoffs in 3CNs: the communication-
computation tradeoff and the AoC-delay tradeoff. The numerical results serve
to validate our theoretical findings. There are two primary future research
directions: 1) deriving and analyzing the time-average AoC in practical
scenarios involving complex graphs, such as sequential dependency graphs,
parallel dependency graphs, and general dependency graphs [1]; 2) identifying
optimal AoC-based resource management policies in 3CNs.

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##  Appendix A Proof of (13)

Recall that $C_{k}$ is the interval between consecutive task completions, we
call it the inter-departure time. Based on Definition 2, the delays of tasks
$\\{D_{k}\\}_{k}$ and the inter-departure time $\\{C_{k}\\}_{k}$ are
identical, respectively. Moreover, from the definition of $A_{k}$, the
sequence $\\{A_{k}\\}_{k}$ is independent and identical (i.i.d.), hence the
sequences $\\{S_{k}\\}_{k}$ and $\\{S_{k}^{\prime}\\}_{k}$ are identical as
well, respectively. Recalling (13), $\Theta_{\text{AoC}}$ can be re-written as

$\displaystyle\Theta_{\text{AoC}}=$
$\displaystyle\lim_{\mathcal{T}\to\infty}\Big{(}\frac{S_{0}+S_{N(\mathcal{T})+1}}{\mathcal{T}}+\frac{1}{\mathcal{T}}\sum_{k=1}^{N(\mathcal{T})}S_{k}\Big{)}$
$\displaystyle=$
$\displaystyle\lim_{\mathcal{T}\to\infty}\frac{N(\mathcal{T})}{\mathcal{T}}\frac{1}{N(\mathcal{T})}\sum_{k=1}^{N(\mathcal{T})}S_{k}$
$\displaystyle\overset{(a)}{=}$
$\displaystyle\lambda\mathbb{E}[S_{k}]\overset{(b)}{=}\frac{\mathbb{E}[S_{k}]}{\mathbb{E}[A_{k}]},$

where (a) holds because
$\lambda=\lim_{\mathcal{T}\to\infty}\frac{N(\mathcal{T})}{\mathcal{T}}$ and
the Law of Large Numbers, and (b) holds because $\mathbb{E}[A_{k}]=1/\lambda$.
The same proof holds for
$\Theta_{\text{AoC}}=\frac{\mathbb{E}[S_{k}^{\prime}]}{\mathbb{E}[A_{k}]}$ by
replacing $S_{k}$ with $S_{k}^{\prime}$ directly.

##  Appendix B Proof of Theorem 1

We first prove (1). Taking expectation on both sides of (10), we obtain

$\displaystyle\mathbb{E}[S_{k}]=$
$\displaystyle\mathbb{E}[\frac{(A_{k}+D_{k+1})^{2}}{2}f_{k}(\tau,\theta)]-\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k+1}(\tau,\theta)]$
$\displaystyle=$
$\displaystyle\mathbb{E}[\frac{A_{k}^{2}}{2}f_{k}(\tau,\theta)]+\mathbb{E}[(A_{k}D_{k+1}+\frac{D_{k+1}^{2}}{2})f_{k}(\tau,\theta)]$
$\displaystyle-$
$\displaystyle\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k+1}(\tau,\theta)]$
$\displaystyle\triangleq$ $\displaystyle
E_{1}-\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k+1}(\tau,\theta)].$ (22)

We now focus on $E_{1}$. Recall that $A_{k}$ is the inter-arrval time between
the $k^{th}$ and $(k+1)^{th}$ tasks, and $D_{k}$ is the delay of the $k^{th}$
packet, then $A_{k}$ and $D_{k}$ are independent of each other, so

$\displaystyle
E_{1}=\mathbb{E}[\frac{A_{k}^{2}}{2}]\mathbb{E}[f_{k}(\tau,\theta)]+\mathbb{E}[(A_{k}D_{k+1}+\frac{D_{k+1}^{2}}{2})f_{k}(\tau,\theta)].$
(23)

Substituting (23) into (B), we have

$\displaystyle\mathbb{E}[S_{k}]=$
$\displaystyle\mathbb{E}[\frac{A_{k}^{2}}{2}]\mathbb{E}[f_{k}(\tau,\theta)]+\mathbb{E}[A_{k}D_{k+1}f_{k}(\tau,\theta)]$
$\displaystyle+$
$\displaystyle\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k}(\tau,\theta)]-\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k+1}(\tau,\theta)].$
(24)

Substituting (B) into (13), we obtain (1).

We then prove (1). Taking expectation on both sides of (11), we obtain

$\displaystyle\mathbb{E}[S_{k}^{\prime}]=\mathbb{E}[(C_{k}D_{k}+C_{k+1}^{2}/2)f_{k}(\tau,\theta)].$
(25)

Substituting $f_{k}(\tau,\theta)=\theta\cdot
1_{\\{D_{k}>\tau\\}}+1_{\\{D_{k}\leq\tau\\}}$ into (25), we arrive at (1).

##  Appendix C Proof of Theorem 2

Step 1. We prove the upper bound (16). By modifying (1), we have

$\displaystyle\Theta_{\text{AoC}}=$
$\displaystyle\frac{1}{\mathbb{E}[A_{k}]}\Big{(}\mathbb{E}[\frac{A_{k}^{2}}{2}]\mathbb{E}[f_{k}(\tau,\theta)]+\mathbb{E}[(A_{k}D_{k+1}f_{k}(\tau,\theta)]$
$\displaystyle+$
$\displaystyle\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k}(\tau,\theta)]-\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k+1}(\tau,\theta)]\Big{)}$
$\displaystyle\triangleq$
$\displaystyle\frac{1}{\mathbb{E}[A_{k}]}(X_{1}+X_{2}+X_{3}),$

where

$\displaystyle X_{1}=$
$\displaystyle\mathbb{E}[\frac{A_{k}^{2}}{2}]\mathbb{E}[f_{k}(\tau,\theta)],$
$\displaystyle X_{2}=$
$\displaystyle\mathbb{E}[A_{k}D_{k+1}f_{k}(\tau,\theta)],$ $\displaystyle
X_{3}=$
$\displaystyle\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k}(\tau,\theta)]-\mathbb{E}[\frac{D_{k+1}^{2}}{2}f_{k+1}(\tau,\theta)].$

Taking expectation of $f_{k}(\tau,\theta)$, we get

$\displaystyle\mathbb{E}[f_{k}(\tau,\theta)]=\theta(1-q_{\tau})+q_{\tau}=g_{\tau}.$
(26)

Substituting (26) into $X_{1}$, we have

$\displaystyle X_{1}=g_{\tau}\mathbb{E}[\frac{A_{k}^{2}}{2}].$ (27)

Since $f_{k}(\tau,\theta)\leq\theta 1_{\\{D_{k}>\tau\\}}+\theta
1_{\\{D_{k}\leq\tau\\}}=\theta$, $X_{2}$ is upper bounded by

$\displaystyle X_{2}\leq\theta\mathbb{E}[A_{k}D_{k+1}].$ (28)

Manipulating $f_{k}(\tau,\theta)$, we get

$\displaystyle f_{k}(\tau,\theta)=1+(\theta-1)1_{\\{D_{k}>\tau\\}}.$ (29)

Substituting (29) into $X_{3}$, we have

$\displaystyle
X_{3}=\mathbb{E}[\frac{D_{k+1}^{2}}{2}1_{\\{D_{k}>\tau\\}}]-\mathbb{E}[\frac{D_{k+1}^{2}}{2}1_{\\{D_{k+1}>\tau\\}}].$
(30)

Given that $D_{k}$ is the delay of the $k^{th}$ task and has a finite second
moment in a stationary system, and that $D_{k}$ and $D_{k+1}$ are consecutive
delays, then they are positive correlated, by [10, eqn. (3)], we have
$\mathbb{E}[\frac{D_{k+1}^{2}}{2}1_{\\{D_{k}>\tau\\}}]\leq\mathbb{E}[\frac{D_{k+1}^{2}}{2}1_{\\{D_{k+1}>\tau\\}}]$,
which means that $X_{3}\leq 0$. In summary,

$\displaystyle\Theta_{\text{AoC}}=$ $\displaystyle X_{1}+X_{2}+X_{3}$
$\displaystyle\leq$
$\displaystyle\big{(}\theta(1-q_{\tau})+q_{\tau}\big{)}\mathbb{E}[\frac{A_{k}^{2}}{2}]+\theta\mathbb{E}[A_{k}D_{k+1}]+0,$

which results in (16).

Step 2. We prove the lower bound (17). By modifying (1), if we want to prove

$\displaystyle Y_{0}\triangleq$
$\displaystyle\theta\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)1_{\\{D_{k}>\tau\\}}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)1_{\\{D_{k}\leq\tau\\}}]$
$\displaystyle\geq$
$\displaystyle\max\\{\theta(1-q_{\tau}),q_{\tau}\\}\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)],$
(31)

it suffices to prove two separate inequalities:
$Y_{0}\geq\theta(1-q_{\tau})\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)]$ and
$Y_{0}\geq q_{\tau}\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)]$. For the former
ineuqality,

$\displaystyle Y_{0}\geq$
$\displaystyle\theta\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)1_{\\{D_{k}>\tau\\}}]$
$\displaystyle=$
$\displaystyle\theta\Pr(D_{k}>\tau)\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)|D_{k}>\tau]$
$\displaystyle\geq$
$\displaystyle\theta\Pr(D_{k}>\tau)\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)]$
$\displaystyle=$
$\displaystyle\theta(1-q_{\tau})\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)].$ (32)

For the latter inquality, after some algebra, $Y_{0}$ is re-written as

$\displaystyle Y_{0}=$
$\displaystyle(\theta-1)\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)1_{\\{D_{k}>\tau\\}}]+\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)]$
$\displaystyle\geq$
$\displaystyle(\theta-1)\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)1_{\\{D_{k}>\tau\\}}]$
$\displaystyle+$ $\displaystyle q_{\tau}\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)]$
$\displaystyle\geq$ $\displaystyle
q_{\tau}\mathbb{E}[(D_{k}C_{k}+C_{k}^{2}/2)]$ (33)

due to $\theta>1$. Based on (C) and (C), we complete the proof of (C).

Next, we need to prove
$\mathbb{E}[C_{k}^{2}/2]+\mathbb{E}[D_{k}C_{k}]=\mathbb{E}[A_{k}^{2}/2]+\mathbb{E}[A_{k}D_{k+1}]$.
In fact, let us consider an extreme case where $\theta=1$. In this case, we
denote the counterparts of $A_{k}$, $D_{k}$, $C_{k}$, $S_{k}$, and
$\Theta_{\text{AoC}}$ as $A_{k}^{0}$, $D_{k}^{0}$, $C_{k}^{0}$, $S_{k}^{0}$,
and $\Theta_{\text{AoC}}^{0}$, respectively. From the definitions of $A_{k}$,
$D_{k}$, and $C_{k}$, it is obvious that $A_{k}^{0}$, $D_{k}^{0}$, and
$C_{k}^{0}$ have the same distributions as $A_{k}$, $D_{k}$, and $C_{k}$,
respectively. Note that $\psi(t)$ in (6) is reduced to $\psi^{0}(t)=1$ for all
$t$, $\Theta(t)$ in (4) is reduced to $\Theta^{0}(t)=t-u(t)$. Therefore,
$S_{k}$ in (11) is reduced to
$S_{k}^{0}=\frac{1}{2}(D_{k}^{0}+D_{k}^{0}+C_{k}^{0})\cdot C_{k}^{0}$, and via
(13),

$\displaystyle\Theta_{\text{AoC}}^{0}=\frac{\big{(}\mathbb{E}[D_{k}^{0}C_{k}^{0}]+\frac{\mathbb{E}[(C_{k}^{0})^{2}]}{2}\big{)}}{\mathbb{E}[A_{k}^{0}]}.$
(34)

Note that $\Theta^{0}(t)=t-u(t)$ has the same formula as the AoI in [8], then
$\Theta_{\text{AoC}}^{0}$ in (34) has another closed form

$\displaystyle\Theta_{\text{AoC}}^{0}=\frac{\mathbb{E}[A_{k}^{0}D_{k+1}^{0}]+\frac{\mathbb{E}[(A_{k}^{0})^{2}]}{2}}{\mathbb{E}[A_{k}^{0}]}.$
(35)

Note that $A_{k}^{0}$ is the inter-arrival process, which follows an
exponential distribution with parameter $\lambda$. Comparing with (34) and
(35), we have
$\mathbb{E}[C_{k}^{2}/2]+\mathbb{E}[D_{k}C_{k}]=\mathbb{E}[A_{k}^{2}/2]+\mathbb{E}[A_{k}D_{k+1}]$.

Step 3. We prove the limit (LABEL:eq:_AoClimit). Let $\tau\to\infty$, for any
$k$ and $\theta$, $\lim_{\tau\to\infty}f_{k}(\tau,\theta)=1$,
$\lim_{\tau\to\infty}q_{\tau}=1$, and $\lim_{\tau\to\infty}h_{\tau}=1$. Note
that the sequences $\\{D_{k}\\}$ are identical, so substituting
$\lim_{\tau\to\infty}f_{k}(\tau,\theta)=1$ into (1), and
$\lim_{\tau\to\infty}h_{\tau}=1$ into (17), respectively, we obtain
(LABEL:eq:_AoClimit).

##  Appendix D Proof of Theorem 3

Step 1. We provide the expression of $\Theta_{\text{UB}}$, and prove the
uniqueness and existence of the minimum point $\rho_{UB}^{*}$. According to
[8], we have

$\displaystyle\mathbb{E}[A_{k}]=1/\lambda,\,\,\mathbb{E}[A_{k}^{2}/2]=1/\lambda^{2},$
(36)
$\displaystyle\mathbb{E}[D_{k+1}A_{k}]=\frac{1}{\mu^{2}\rho}+\frac{\rho}{\mu^{2}(1-\rho)}.$
(37)

Substituting (36) and (37) into (16), we obtain (20).

To demonstrate the uniqueness and existence of the minimum point
$\rho_{UB}^{*}$, we need show the function $\Theta_{\text{UB}}$ is convex (in
terms of $\rho$) in the interval $(0,1)$. Equivalently, we need to show the
second derivative of $\Theta_{\text{UB}}$ is positive. It is easy to verify

$\displaystyle\frac{\partial^{2}\Theta_{\text{UB}}}{\partial\rho^{2}}=$
$\displaystyle\frac{1}{\mu}\cdot\frac{g^{\prime\prime}(\rho)\rho^{3}-2g^{\prime}(\rho)\rho^{2}+2g(\rho)\rho}{\rho^{4}}$
$\displaystyle+$
$\displaystyle\frac{\theta}{\mu}\cdot\frac{2(1-\rho)^{3}+2(1-\rho)(2\rho-\rho^{2})}{(1-\rho)^{4}}.$
(38)

Since $\rho\in[0,1]$, the second term in (D) is non-negative, thus we only
need to focus on the first term. By calculation, we have

$\displaystyle g^{\prime}(\rho)=\mu\tau\big{(}g(\rho)-1\big{)}$ (39)
$\displaystyle g^{\prime\prime}(\rho)=(\mu\tau)^{2}\big{(}g(\rho)-1\big{)}$
(40)

From (39) and (40) into the first term, we have

$\displaystyle\frac{1}{\mu}\cdot\frac{g^{\prime\prime}(\rho)\rho^{3}-2g^{\prime}(\rho)\rho^{2}+2g(\rho)\rho}{\rho^{4}}$
$\displaystyle=\frac{1}{\mu\rho^{3}}\Big{(}(\mu\tau)^{2}\rho^{2}\big{(}g(\rho)-1\big{)}-2\mu\tau\big{(}g(\rho)-1\big{)}\rho+2g(\rho)\Big{)}.$
(41)

Since $g(\rho)>1$, then (D) is modified to

$\displaystyle\frac{1}{\mu}\cdot\frac{g^{\prime\prime}(\rho)\rho^{3}-2g^{\prime}(\rho)\rho^{2}+2g(\rho)\rho}{\rho^{4}}$
$\displaystyle\geq\frac{g(\rho)-1}{\mu\rho^{3}}\big{(}(\mu\tau)^{2}\rho^{2}-2\mu\tau\rho+2\big{)}$
$\displaystyle=\frac{g(\rho)-1}{\mu\rho^{3}}\big{(}(\mu\tau\rho-1)^{2}+1\big{)}\geq
0.$ (42)

Subsituting (D) into (D), the second derivative of $\Theta_{\text{UB}}$ is
non-negative in $[0,1]$, so it is convex in terms of $\rho$, thus there exists
an unique minimum point.

Step 2. We provide the expression of $\Theta_{\text{LB}}$, and prove the
uniqueness and existence of the nimimum point $\rho_{LB}^{*}$. Substituting
(36) and (37) into (17), we obtain (21). Let $\tau\to\infty$,
$\lim_{\tau\to\infty}h(\rho)=1$, then

$\displaystyle\lim_{\tau\to\infty}\Theta_{\text{LB}}=\frac{1}{\mu}\big{(}1+\frac{1}{\rho}+\frac{\rho^{2}}{1-\rho}\big{)}.$
(43)

By calculation, we have

$\displaystyle\frac{\partial^{2}\lim_{\tau\to\infty}\Theta_{\text{LB}}}{\partial\rho^{2}}=\frac{2}{\rho^{3}}+\frac{2}{1-\rho}+\frac{2\rho(2-\rho)}{(1-\rho)^{3}}>0.$
(44)

It is evident that $\lim_{\tau\to\infty}\Theta_{\text{LB}}$ is convex in
$\rho\in(0,1)$, thus there exists an unique minimum point.

##  Appendix E Proof of Theorem 4

Since $\bar{D}=\frac{1}{\mu(1-\rho)}$, $\bar{D}$ increases with $\rho$ when
$0<\rho<1$. By algebraic manipulation, we find $\rho=1-\frac{1}{\mu\bar{D}}$,
indicating that $\rho$ increases with $\bar{D}$. Let
$D^{*}=\frac{1}{\mu(1-\rho_{UB}^{*})}$. Therefore, when $\bar{D}\leq D^{*}$
(or equivalently $\rho\leq\rho_{UB}^{*}$), $\rho$ increases with $\bar{D}$ but
$\Theta_{\text{UB}}$ decreases with $\rho$, resulting in $\Theta_{\text{UB}}$
decreases with $\bar{D}$; when $\bar{D}>D^{*}$ (or equivalently
$\rho>\rho_{UB}^{*}$), $\rho$ increases with $\bar{D}$ and
$\Theta_{\text{UB}}$ increases with $\rho$, leading to $\Theta_{\text{UB}}$
increases with $\bar{D}$. The proof holds for $\Theta_{\text{LB}}$ as well, we
only need to replace $(\Theta_{\text{LB}},\rho^{*}_{UB})$ with
$(\Theta_{\text{UB}},\rho^{*}_{LB})$ directly. This completes the proof.