som#1*#3#2[#2]#3#3

# Cache-Aided Communications in MISO Networks with Dynamic User Behavior: A
Universal Solution

Milad Abolpour, MohammadJavad Salehi, and Antti Tölli
This research has been supported by the Academy of Finland, 6G Flagship
program under Grant 346208, 343586 (CAMAIDE), and by the Finnish-American
Research and Innovation Accelerator (FARIA).  Centre for Wireless
Communications, University of Oulu, 90570 Oulu, Finland
E-mail<EMAIL_ADDRESS>
###### Abstract

A practical barrier to the implementation of cache-aided networks is dynamic
and unpredictable user behavior. In dynamic setups, users can freely depart
and enter the network at any moment. The shared caching concept has the
potential to handle this issue by assigning $K$ users to $P$ caching profiles,
where all $\eta_{p}$ users assigned to profile $p$ store the same cache
content defined by that profile. The existing schemes, however, cannot be
applied in general and are not dynamic in the true sense as they put
constraints on the transmitter-side spatial multiplexing gain $\alpha$.
Specifically, they work only if $\alpha\leq\min_{p}\eta_{p}$ or
$\alpha\geq\hat{\eta}$, where in the latter case, $\gamma$ is the normalized
cache size of each user, $\hat{\eta}$ is an arbitrary parameter satisfying
$1\leq\hat{\eta}\leq\max_{p}\eta_{p}$, and the extra condition of $\alpha\geq
K\gamma$ should also be met. In this work, we propose a universal caching
scheme based on the same shared-cache model that can be applied to any dynamic
setup, extending the working region of existing schemes to networks with
$\min_{p}\eta_{p}\leq\alpha\leq\hat{\eta}$ and removing any other constraints
of existing schemes. We also derive the closed-form expressions for the
achievable degrees-of-freedom (DoF) of the proposed scheme and show that it
achieves the optimal DoF for uniform user distributions. Notably, it is the
first scheme to achieve the optimal DoF of $K\gamma+\alpha$ for networks with
uniform user distribution, $\alpha>\hat{\eta}$, and non-integer
$\frac{\alpha}{\hat{\eta}}$, without imposing any other constraints. Finally,
we use numerical simulations to assess how non-uniform user distribution
impacts the DoF performance and illustrate that the proposed scheme provides a
noticeable improvement over unicasting for uneven distributions.

###### Index Terms:

coded caching; shared caching; dynamic networks; multi-antenna communications

##  I Introduction

The increasing volume and diversity of multimedia content require wireless
networks to be enhanced to serve users at higher data rates and with lower
latency [1], while network providers must further develop infrastructure in
anticipation of evolving applications such as wireless immersive viewing [2,
3]. To facilitate the efficient delivery of such multimedia content, coded
caching (CC) has been proposed to increase the data rates by leveraging the
cache memory across the network as a communication resource [4]. Accordingly,
incorporating CC into a single-stream downlink network boosts the achievable
rate by a multiplicative factor proportional to the cumulative cache size in
the entire network via multicasting carefully designed codewords to different
user groups. In light of the significance of multi-antenna connectivity in
deploying next-generation networks [1], various works have studied the
performance of cache-aided multi-input single-output (MISO) configurations [5,
6, 7, 8, 9, 10]. For instance, [8] and [11] discussed the design of optimized
beamformers in finite signal-to-noise-ratio (SNR), and [2] explored the
capability of CC to cope with location-dependent file request applications.

In practice, however, practically achievable CC gains are constrained by the
subpacketization process [12, 13, 14]. That is, in a network with $K$ users,
each file should be split into many smaller parts, the number of which grows
exponentially with $K$. A promising way to overcome this impediment is to use
the shared caching concept, where there exist $P\leq K$ _caching profiles_ ,
and $\eta_{p}$ users are assigned to profile
$p\in\left\\{1,\cdots,P\right\\}$. Even though with this concept, multiple
users with a cache ratio of $\gamma$ could be assigned to the same profile and
cache exactly the same data, in [15], it is shown that in MISO setups with
$\alpha\geq\frac{K}{P}$, the scaling factor in the degrees-of-freedom (DoF)
could be the same as the case of dedicated users’ caches, i.e.,
$K\gamma+\alpha$, where $\alpha$ is the spatial multiplexing gain. However,
for a shared-cache MISO setup with $\alpha\leq\frac{K}{P}$, the optimal DoF is
$\alpha\left(1+P\gamma\right)$ [16].

Interestingly, shared caching can also address another critical issue with
coded caching schemes: handling networks with a dynamic population of users
departing and entering the network at any time. The problem with conventional
CC schemes is that they require the placement phase to be designed based on
the number of users known a priori. By contrast, this problem is alleviated
with the shared-cache model since the cache placement phase is built upon
knowledge of the number of profiles $P$, and not the number of users $K$.
Accordingly, in some cache-aided scenarios, such as extended reality
applications [3], the server is aware of the cache ratio of users rather than
the number of existing users. In this sense, the authors in [17] and [18],
apply the shared caching idea to address the dynamicity issue. In this method,
the server only needs to know the cache ratio of users to determine the number
of caching profiles and design the content placement phase. Although shared
caching is crucial for managing dynamic conditions, the existing models are
not dynamic in the true sense and only support two regions: 1)
$\alpha\leq\min_{p}\eta_{p}$ [16], and 2) $\alpha\geq\hat{\eta}$ with
$\alpha\geq K\gamma$ and arbitrary $\hat{\eta}$ satisfying
$1\leq\hat{\eta}\leq\max_{p}\eta_{p}$ [17, 18]. Therefore, a universal shared-
cache setup that supports any user-to-profile association is not yet
available.

In this work, we design a universal cache-assisted MISO system capable of
handling any instantaneous user distribution among caching profiles. Our
system operation is comprised of two phases: i) Content placement phase, and
ii) content delivery phase. In the placement phase, the server determines the
number of caching profiles according to the cache ratio $\gamma$. Then, upon
connecting to the network, each user is assigned to a single profile and
stores the cache content of that profile. During the content delivery phase,
the server employs a clever combination of multicast and unicast transmissions
to maximize the DoF. In this paper, we obtain closed-form expressions for the
DoF, revealing the DoF loss caused by non-uniformness in users’ distribution.
Particularly, for the uniform user associations, it is shown that our proposed
scheme achieves the optimal DoF not only in the regions covered in the
literature but also in the region $\alpha\geq\hat{\eta}$ with non-integer
$\nicefrac{{\alpha}}{{\hat{\eta}}}$. Notably, our proposed scheme supports any
user distribution, including the regions omitted in the existing literature
[15, 16, 17] such as networks with: i) uneven user association with
$\alpha>\hat{\eta}$ and non-integer $\frac{\alpha}{\hat{\eta}}$ unlike [15],
ii) $\min_{p}\eta_{p}\leq\alpha\leq\hat{\eta}$ unlike [16], and iii)
$\alpha\geq\hat{\eta}$ with $\alpha<K\gamma$ unlike [17].

In this paper, bold lower-case and calligraphic letters show vectors and sets,
respectively. $\left[a:b\right]$ shows the set $\\{a,\cdots,b\\}$,
$[a]=\\{1,\cdots,a\\}$, $\left|{\mathcal{A}}\right|$ is the cardinality of
${\mathcal{A}}$, and for $\Lambda\subseteq{\mathcal{A}}$,
${\mathcal{A}}_{\backslash\Lambda}$ represents ${\mathcal{A}}-\Lambda$.
$\left({\mathcal{A}}\|{\mathcal{A}}\right)_{x}$ denotes $x$ concatenations of
${\mathcal{A}}$ with itself, and ${\mathcal{A}}\|{\mathcal{B}}$ is the
concatenation of ${\mathcal{A}}$ and ${\mathcal{B}}$.

##  II System Model

In this paper, we focus on a dynamic MISO network, where a base station (BS)
equipped with $L$ transmit antennas and the spatial multiplexing gain of
$\alpha\leq L$ serves several cache-enabled single-antenna users. The BS has
access to a library ${\mathcal{F}}$ with $N$ equal-sized files, and each user
is equipped with a large enough memory to store a portion $0<\gamma<1$ of the
entire library. We suppose $\gamma=\frac{{\bar{t}}}{P}$, where ${\bar{t}}$ and
$P$ are natural numbers and $\gcd\left({\bar{t}},P\right)=1$. In this dynamic
setup, users can move, enter and depart the network at any time. Accordingly,
the BS does not have any prior knowledge about the number of available users
during the transmission. When a user $k$ enters the network, it is assigned to
a profile represented by ${\mathsf{p}}[k]\in\left[P\right]$, and the content
of its cache is updated based on a content placement algorithm.

In the placement phase, by following the same way as in [4], each file
$W^{n}\in{\mathcal{F}}$, $n\in[N]$, is split into $\binom{P}{{\bar{t}}}$
equal-sized mini-files $W^{n}_{{\mathcal{P}}}$ such that
$W^{n}\rightarrow\left\\{W^{n}_{{\mathcal{P}}}:{\mathcal{P}}\subseteq[P],\left|{\mathcal{P}}\right|={\bar{t}}\right\\}$.
The cache content associated with profile $p\in[P]$, represented by
${\mathcal{Z}}_{p}$, includes a portion $\gamma$ of each file $W^{n}$ as

${\mathcal{Z}}_{p}=\left\\{W^{n}_{{\mathcal{P}}}:{\mathcal{P}}\ni
p,{\mathcal{P}}\subseteq\left[P\right],\left|{\mathcal{P}}\right|={\bar{t}},\forall
n\in\left[N\right]\right\\}.$

Then, defining ${\mathcal{U}}_{p}$ as the set of users assigned to profile
$p$, i.e.,
${\mathcal{U}}_{p}=\left\\{k:{\mathsf{p}}\left[k\right]=p\right\\}$, each user
$k\in{\mathcal{U}}_{p}$ stores the cache content ${\mathcal{Z}}_{p}$ during
the placement phase.

The dynamic nature of the network causes a fluctuating user population
throughout time. During regular intervals, the network’s demanding users
reveal their required files from the library ${\mathcal{F}}$ to the BS. Then,
using a content delivery algorithm, the BS constructs and transmits a set of
codewords, enabling users to retrieve their requested files. In this paper, we
focus on the content delivery procedure over a specific time interval, where
it is assumed that the number of present users during the BS’s transmission is
$K$. In line with the general approach in the literature, we utilize the total
DoF as the metric of interest, representing the average number of concurrent
users served in parallel across all transmit intervals. The main contribution
of this paper is to design delivery algorithms that provide a maximum
combination of global caching and spatial multiplexing gains under the
proposed dynamic conditions. As part of the delivery process, we discuss the
transmission strategies to reduce the DoF loss caused by non-uniformness in
user-to-profile association in the following section.

Figure 1: System model for a dynamic coded caching setup, where $P=2$,
$\gamma=\frac{1}{2}$, ${\bar{t}}=1$, $\alpha=4$, $\hat{\eta}=4$, $Q=2$ and
$\beta=3$. During the placement phase, each user assigned to profiles $A$ and
$B$ stores the cache content associated with those profiles. For the delivery
phase, user $5$ is served via unicasting and other users are served via $3$
multicast transmissions.

##  III Resource Allocation and Data Delivery

In this section, we discuss the resource allocation and transmission
strategies during the content delivery phase. This phase commences once the
set of active users reveals their requested files and comprises two
consecutive steps: 1) Coded caching (CC) data delivery; and 2) Unicast (UC)
data delivery.

Let us define the number of users assigned to profile $p$ as the length of
profile $p$ denoted by $\eta_{p}$, where without loss of generality, it is
assumed that $\eta_{1}\geq\eta_{2}\geq\cdots\geq\eta_{P}$. By choosing a
delivery parameter $\hat{\eta}\leq\max_{p}\eta_{p}$111Here, the delivery
parameter plays a similar role as the unifying length parameter in [17]. Both
delivery and unifying length parameters tune the DoF loss caused by the non-
uniformness in the user-to-profile association., the BS builds and transmits a
set of codewords to serve at most $\hat{\eta}$ users assigned to each profile
with a novel CC-based approach. In this regard, for every profile $p$
$\bullet$ if $\hat{\eta}<\eta_{p}$, we exclude $\eta_{p}-\hat{\eta}$ users,
and exempt BS to serve these users during the CC delivery step. Accordingly,
the excluded users are served in the UC delivery step.
$\bullet$ if $\hat{\eta}\geq\eta_{p}$, all $\eta_{p}$ users are served via the
CC delivery step.

Now, let us suppose that the set of users assigned to profile $p\in[P]$ and
served during the CC delivery step is denoted by ${\mathcal{V}}_{p}$ such that
$\left|{\mathcal{V}}_{p}\right|=\delta_{p}$,
${\mathcal{V}}_{p}=\left\\{v_{p,1},v_{p,2}\cdots,v_{p,\delta_{p}}\right\\}$,
and $v_{p,i}\in{\mathcal{U}}_{p}$ for $i\in[\delta_{p}]$. We note that
$\delta_{p}=\min\left(\hat{\eta},\eta_{p}\right)$, and clearly,
$\delta_{1}=\hat{\eta}$ and
$\delta_{1}\geq\delta_{2}\geq\cdots\geq\delta_{P}$.

In order to build the transmission vectors, as depicted in Fig. 1, the BS
selects a parameter $Q$, which represents the number of profiles served in
each transmission, and a parameter $\beta$, which expresses the number of
users chosen from each profile to serve in each transmission. The necessary
conditions for choosing any arbitrary values for $Q$ and $\beta$ are defined
in Remark 1.

###### Remark 1

In order to serve $Q$ profiles each with a maximum of $\beta$ users, the
network parameters should satisfy the constraints ${\bar{t}}+1\leq
Q\leq{\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\beta}}\right\rceil$ and
$\beta\leq\min\left(\alpha,\hat{\eta}\right)$.

###### Proof:

The proof is relegated to Appendix A. ∎

In the following, we present the system operation maximizing the DoF
performance separately for two regimes: i) $\alpha\leq\hat{\eta}$ and ii)
$\alpha>\hat{\eta}$. For the CC delivery step, each of these cases operates
either with Strategy A (cf. Section III-A) or Strategy B (cf. Section III-B).
The chosen transmission strategy only depends on the parameters $\alpha$ and
$\hat{\eta}$, and it is independent of the content placement phase. In other
words, the server performs the placement phase only based on parameter
$\gamma$ without considering which transmission strategy the system will use
for the CC delivery step.

####  III-1 System operation for $\alpha\leq\hat{\eta}$

For the case of $\alpha\leq\hat{\eta}$, we set $\beta=\alpha$, and the only
option for $Q$ to maximize the DoF is $Q={\bar{t}}+1$. Here, Strategy A is
utilized to build the transmission vectors. Replacing the constraint
$\alpha\leq\hat{\eta}$ with $\alpha\leq\min_{p}\delta_{p}$ reduces our system
model to [16], while our proposed scheme also works for the scenarios with
$\min_{p}\delta_{p}<\alpha\leq\hat{\eta}$.

####  III-2 System operation for $\alpha>\hat{\eta}$

For this case, we set $\beta=\hat{\eta}$, and define
$\hat{\alpha}=\frac{\alpha}{\hat{\eta}}$. If $\hat{\alpha}$ is an integer,
then we follow Strategy A to build the transmission vectors. For non-integer
$\frac{\alpha}{\hat{\eta}}$, the server can serve users via Strategy A by
setting ${\bar{t}}+1\leq
Q\leq{\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$,
and via Strategy B by choosing
$Q={\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rceil$. In
this case, if we assume that users are uniformly distributed among caching
profiles, and $\frac{\alpha}{\hat{\eta}}$ is an integer, the system
performance is simplified to [15], while our proposed scheme also covers the
uneven user-to-profile associations with non-integer
$\frac{\alpha}{\hat{\eta}}$.

###  III-A Transmission Strategy A

In this strategy, each mini-file $W_{{\mathcal{P}}}^{n}$ is split into
$\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$ subpackets
$W_{{\mathcal{P}},q}^{n}$, where
$q\in[\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}]$ increases sequentially after
each transmission to ensure none of subpackets is transmitted twice. Next, we
follow the so-called elevation process to serve $Q$ caching profiles each with
at most $\beta$ users.

#### Elevation process

In this process, the aim is to characterize the set of users that are served
in each transmission. Accordingly, we let
$\phi_{p}=\max\left(\beta,\delta_{p}\right)$, and use the so-called
transmission triple $\left(r,c,l\right)$, where $r\in\left[P-Q+1\right]$,
$c\in\left[\phi_{r}\right]$ and $l\in\left[\binom{P-r}{Q-1}\right]$. Also, it
is assumed that $\eta_{1}\geq\eta_{2}\geq\cdots\geq\eta_{P}$, which results in
$\phi_{1}\geq\phi_{2}\geq\cdots\geq\phi_{P}$. This process creates
${\mathcal{T}}_{r,c,l}$, the set of users that are served during the
transmission triple $\left(r,c,l\right)$. In this regard, first, for every
$p\in[P]$, we elevate the set ${\mathcal{V}}_{p}$ to the set
${\mathcal{R}}_{p}$ as follows.

${\mathcal{R}}_{p}={\mathcal{R}}_{p,1}\|\cdots\|{\mathcal{R}}_{p,\phi_{p}},$
(1)

where, for $j\in\left[\phi_{p}\right]$, ${\mathcal{R}}_{p,j}$ is defined as:

${\mathcal{R}}_{p,j}=\begin{cases}{\mathcal{V}}_{p}&\delta_{p}\leq{\beta}\\\
\left\\{v_{p,l}:l=\left(i+j-1\right)\%\delta_{p},1\leq
i\leq\beta\right\\}&\delta_{p}>\beta\end{cases}.$

Here, $\%$ sign demonstrates the mod operator, for which $c\%c=c$ and
$\left(d+c\right)\%c=d\%c$. Furthermore, we use generalized multiset
definition where we allow the same elements to be repeated in the sets, e.g.,
$\left\\{a,a\right\\}$ cannot be reduced to $\left\\{a\right\\}$. Now, for
$p\in[P]$, we define
${\mathcal{S}}_{p}={\mathcal{S}}_{p,1}\|\cdots\|{\mathcal{S}}_{p,\hat{\eta}}$,
while

${\mathcal{S}}_{p,j}=\begin{cases}{\mathcal{R}}_{p,j}&1\leq j\leq\phi_{p}\\\
\varnothing&\phi_{p}+1\leq j\leq\hat{\eta}\end{cases}.$ (2)

Then, for each $r\in[P-Q+1]$, we define the set ${\mathcal{M}}_{r}$ as:

$\displaystyle{\mathcal{M}}_{r}=\left\\{{\mathcal{F}}:{\mathcal{F}}\subseteq\left[r+1:P\right],\,\left|{\mathcal{F}}\right|=Q-1\right\\}.$
(3)

In the proceeding, we use ${\mathcal{M}}_{r}\left(l\right)$ to indicate the
$l$-th $\left(Q-1\right)$-tuple of ${\mathcal{M}}_{r}$. Finally, the set
${\mathcal{T}}_{r,c,l}$ for the transmission triple $\left(r,c,l\right)$ is
given by:

$\displaystyle{\mathcal{T}}_{r,c,l}=\left\\{{\mathcal{S}}_{r,c}\|{\mathcal{S}}_{b_{1},c}\|\cdots\|{\mathcal{S}}_{b_{Q-1},c}:b_{i}\in{\mathcal{M}}_{r}\left(l\right),\forall
i\in[Q-1]\right\\}.$ (4)

Generally speaking, for the transmission triple $\left(r,c,l\right)$, if
$\delta_{r}=0$, the BS does not transmit any signal; otherwise, the BS serves
users assigned to
${\mathcal{T}}_{r,c,l}={\mathcal{S}}_{r,c}\|{\mathcal{S}}_{b_{1},c}\|\cdots\|{\mathcal{S}}_{b_{Q-1},c}$,
while $\left\\{b_{1},\cdots,b_{Q-1}\right\\}={\mathcal{M}}_{r}\left(l\right)$.
During the transmission triple $\left(r,c,l\right)$, by defining
${\mathcal{N}}=\left\\{r\right\\}\cup{\mathcal{M}}_{r}\left(l\right)$, the BS
constructs the transmission vector as follows.

$\displaystyle{\mathbf{x}}_{r,c,l}=\sum\limits_{\Lambda\subseteq{\mathcal{N}}:\left|\Lambda\right|={\bar{t}}}\,\,\sum\limits_{k\in{\mathcal{S}}_{p,c}:p\in{\mathcal{N}}_{\backslash\Lambda}}W_{\Lambda,q}^{k}{\mathbf{w}}_{{\mathcal{G}}_{\Lambda}^{k}},$

where ${\mathbf{w}}_{{\mathcal{G}}_{\Lambda}^{k}}\in\mathbb{C}^{L\times 1}$ is
the zero-forcing (ZF) precoder that cancels out the interference of user $k$
at the set
${\mathcal{G}}_{\Lambda}^{k}=\left\\{j\in{\mathcal{S}}_{p,c}:\forall
p\in{\mathcal{N}}_{\backslash\Lambda},j\neq k\right\\}$. In Appendix D, it is
proven that all users served with Strategy A can decode their requested files
at the end of the CC delivery step. Now, assuming
${\mathbf{h}}_{i}\in\mathbb{C}^{L\times 1}$ as the channel gain of user $i$,
at the end of the transmission triple $\left(r,c,l\right)$, the received
signal at user $i$ is given by:

$\displaystyle y_{i}={\mathbf{h}}_{i}^{\mathrm{H}}{\mathbf{x}}_{r,c,l}+n_{i},$

where $n_{i}$ is the zero-mean additive white Gaussian noise of unit variance.
In order to give further insight, an example for data delivery with Strategy A
is provided in Appendix B.

###  III-B Transmission Strategy B

When $\alpha>\hat{\eta}$ and $\frac{\alpha}{\hat{\eta}}$ is not an integer, we
can serve users by setting $\beta=\hat{\eta}$, and
$Q={\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rceil$.222For
$\alpha>\hat{\eta}$ and non-integer $\frac{\alpha}{\hat{\eta}}$, we can still
serve users with Strategy A. However, the achievable DoF is less than the one
with Strategy B. Here, first, we split each mini-file into
$\left(\hat{\eta}{\bar{t}}+\alpha\right)\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\binom{Q-2}{Q-{\bar{t}}-2}$
subpackets. Then, we use the elevation process to serve $Q$ profiles each with
at most $\beta$ users.

#### Elevation process

This process builds the set of users served in each transmission. First, for
$r\in\left[P\right]$, we define ${\mathcal{Y}}_{r}$ as:

$\displaystyle{\mathcal{Y}}_{r}=\begin{cases}{\mathcal{V}}_{r}&\delta_{r}=\hat{\eta}\\\
{\mathcal{V}}_{r}\|\left(f^{*}\|f^{*}\right)_{\hat{\eta}-\delta_{r}}&\mathrm{o.w.}\end{cases},$
(5)

where $f^{*}$ denotes the phantom (non-existent) users. Generally speaking, in
each transmission of Strategy B, we serve the users assigned to $Q$ profiles
such that we select at most $\hat{\eta}$ users from $Q-1$ profiles, and pick
$\theta=\alpha-\hat{\eta}\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$
users from another profile. Next, for $r\in\left[P\right]$ and
$m\in\left[\hat{\eta}\right]$, we consider the set ${\mathcal{E}}_{r}^{m}$ as:

$\displaystyle{\mathcal{E}}_{r}^{m}=\bigcup\nolimits_{i=0}^{\theta-1}{\mathcal{Y}}_{r}\left(\left(i+m\right)\%\hat{\eta}\right),$
(6)

where ${\mathcal{Y}}_{r}(i)$ is the $i$-th element of ${\mathcal{Y}}_{r}$.
Indeed, ${\mathcal{E}}_{r}^{m}$ shifts ${\mathcal{Y}}_{r}$ to the right for
$m$ times, and picks $\theta$ elements from it. Then, for each
$u\in{\mathcal{E}}_{r}^{m}$, we define the set ${\mathcal{K}}_{r}^{m,u}$ as
follows.

$\displaystyle{\mathcal{K}}_{r}^{m,u}=\left(u\|u\right)_{\nu_{1}}\|\left(f^{*}\|f^{*}\right)_{\nu_{2}-\nu_{1}},$
(7)

where $\nu_{1}=\binom{Q-2}{Q-{\bar{t}}-2}$ and
$\nu_{2}=\binom{Q-1}{Q-{\bar{t}}-1}$. Next, by defining
${\mathcal{K}}_{r}^{m,u}(s)$ as the $s$-th element of
${\mathcal{K}}_{r}^{m,u}$, for $s\in\left[\nu_{2}\right]$, it is assumed that
${\mathcal{K}}_{r,s}^{m,u}$ is the $s$ circular shifts of
${\mathcal{K}}_{r}^{m,u}$, such that:

$\displaystyle{\mathcal{K}}_{r,s}^{m,u}=\bigcup\nolimits_{i=0}^{\nu_{2}}{\mathcal{K}}_{r}^{m,u}\left(\left(i+s\right)\%\nu_{2}\right).$
(8)

Moreover, we assume that
$\overline{{\mathcal{P}}}_{r}=\left[P\right]_{\backslash r}$ for
$r\in\left[P\right]$, and $\bar{\delta}_{c}=\overline{{\mathcal{P}}}_{r}(c)$
is the $c$-th element of $\overline{{\mathcal{P}}}_{r}$. The caching profiles
in $\overline{{\mathcal{P}}}_{r}$ are sorted in descending order such that if
$i<j$, then
$\delta_{\overline{{\mathcal{P}}}_{r}(i)}\geq\delta_{\overline{{\mathcal{P}}}_{r}(j)}$.
Next, for a given $r$ and $c\in\left[P-Q+1\right]$, let
${\mathcal{I}}_{c}^{r}$ as:

$\small{\mathcal{I}}_{c}^{r}=\left\\{{\mathcal{F}}:{\mathcal{F}}\subseteq\left\\{\overline{{\mathcal{P}}}_{r}(c+1),\cdots,\overline{{\mathcal{P}}}_{r}(P-1)\right\\},\left\lvert{\mathcal{F}}\right\rvert=Q-2\right\\}.$
(9)

Furthermore, denote the $l$-th $\left(Q-2\right)$-tuple of
${\mathcal{I}}_{c}^{r}$ by ${\mathcal{I}}_{c}^{r}(l)$, where
$l\in\left[\binom{P-c-1}{Q-2}\right]$. For the delivery process with Strategy
B, we use the so-called transmission quintuple $\left(r,c,l,m,s\right)$, where
$r\in\left[P\right]$, $c\in\left[P-Q+1\right]$,
$l\in\left[\binom{P-c-1}{Q-2}\right]$, $m\in\left[\hat{\eta}\right]$ and
$s\in\left[\nu_{2}\right]$. In each transmission quintuple
$\left(r,c,l,m,s\right)$, users assigned to the caching profiles
${\mathcal{B}}=\left\\{\bar{\delta}_{c}\right\\}\cup{\mathcal{I}}_{c}^{r}(l)$,
and users in the set ${\mathcal{E}}_{r}^{m}$ are served. Suppose that
${\mathcal{C}}=\left\\{{\mathcal{F}}:{\mathcal{F}}\subseteq{\mathcal{B}},\left|{\mathcal{F}}\right|=\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor\right\\}$,
and ${\mathcal{C}}(n)$ is the $n$-th
$\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$-tuple of
${\mathcal{C}}$ with $n\in\left[\nu_{2}\right]$.

We define the function
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)$ such that
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)=0$, if
${\mathcal{E}}_{r}^{m}=f^{*}$ and $\bar{\delta}_{c}=0$; otherwise,
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)=1$. If
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)=1$, after
eliminating the impacts of the phantom users $f^{*}$, the BS builds the
transmission vector for the transmission quintuple $\left(r,c,l,m,s\right)$ as
follows.

$\displaystyle{\mathbf{x}}_{r,c,l}^{m,s}=\sum\nolimits_{n=1}^{\nu_{2}}\,\,\sum\limits_{k\in{\mathcal{K}}_{r,s}^{m,u}(n)\cup{\mathcal{V}}_{p}:u\in{\mathcal{E}}_{r}^{m},p\in{\mathcal{C}}(n)}W_{\Theta_{n},q}^{k}{\mathbf{w}}_{{\mathcal{H}}_{{\mathcal{C}}(n)}^{k}},$

where $\Theta_{n}={\mathcal{B}}_{\backslash{\mathcal{C}}(n)}$ with
$\left|\Theta\right|={\bar{t}}$, and
${\mathbf{w}}_{{\mathcal{H}}_{\mathcal{C}}(n)}\in\mathbb{C}^{L\times 1}$ is
the precoder that suppresses the interference of user $k$ at the set
${\mathcal{H}}_{{\mathcal{C}}(n)}^{k}=\left\\{j\in{\mathcal{E}}_{r}^{m}\cup{\mathcal{V}}_{p}:p\in{\mathcal{C}}(n),j\neq
k,f^{*}\right\\}$. In Appendix D, we prove that all users can decode their
requested files with Strategy B. Accordingly, user $i$ receives the signal

$\displaystyle
y_{i}={\mathbf{h}}_{i}^{\mathrm{H}}{\mathbf{x}}_{r,c,l}^{m,s}+n_{i}.$

In order to give further insight, an example for data delivery with Strategy B
is provided in Appendix C.

###  III-C Unicast (UC) Data Delivery

In the UC delivery step, the BS transmits data to the users excluded from the
CC delivery step. Here, unlike the CC delivery step that benefits from the
global coded caching and spatial multiplexing gains, only local coded caching
and spatial multiplexing gains are available. Suppose that the BS serves
$K_{U}$ users during the UC delivery step such that
$K_{U}=\sum_{p=1}^{P}\max\left(0,\eta_{p}-\hat{\eta}\right)$. Each of the
requested files by these users is split into the same number of subpackets as
in the CC delivery step. Then, in order to transmit these missing subpackets,
we follow a greedy algorithm similar to [17], which comprises 3 processes: 1)
sort users based on the number of their missing subpackets in descending
order; 2) create a transmission vector to deliver one missing subpacket to
each of the first $\min\left(\alpha,K_{U}\right)$ users; 3) repeat processes 1
and 2 until all missing files are transmitted.

##  IV DoF Analysis

In this section, we use DoF as the metric of interest to measure performance.
Here, the DoF is defined as the average number of users served concurrently
during the delivery phase. In CC and UC delivery steps, we denote the total
transmissions by $T_{M}$ and $T_{U}$, respectively, and the number of served
users by $J_{M}$ and $J_{U}$. Therefore, DoF is computed as follows.

$\mathrm{DoF}=\frac{J_{M}+J_{U}}{T_{M}+T_{U}}.$ (10)

Furthermore, we suppose that $K_{M}$ and $K_{U}$ users are served during the
CC and UC delivery steps such that
$K_{M}=\sum_{p}\min\left(\hat{\eta},\eta_{p}\right)$ and
$K_{U}=\sum_{p}\max\left(0,\eta_{p}-\hat{\eta}\right)$. The next theorem
characterizes the DoF for the cache-aided networks operating with strategies A
and B during the CC delivery step.

###### Theorem 1

Consider a dynamic MISO network with the spatial multiplexing gain of
$\alpha$, cache ratio $\gamma$ and the delivery parameter $\hat{\eta}$. If the
system operates with Strategy A in the CC delivery step, the DoF is given by:

$\mathrm{DoF}=\begin{cases}\frac{K\binom{P-1}{Q-1}\beta}{\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}}&K_{U}=0\\\
\frac{K_{M}\binom{P-1}{Q-1}\beta+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}+\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil}&K_{U}\neq
0\end{cases},$ (11)

where $\beta^{\prime}=\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$ and
$D(\delta_{r})=\phi_{r}$ if $\delta_{r}\neq 0$; otherwise,
$D\left(\delta_{r}\right)=0$. If Strategy B is applied during the CC delivery
step, the DoF takes the form as follows.

$\mathrm{DoF}=\begin{cases}\frac{K\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}}{N_{M}}&K_{U}=0\\\
\frac{K_{M}\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}}{N_{M}+N_{U}}&K_{U}\neq
0\end{cases},$ (12)

where
$\alpha^{\prime}=\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{1}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$,
$N_{U}=\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil$
and

$N_{M}=\sum_{r=1}^{P}\sum_{c=1}^{P-Q+1}\sum_{m=1}^{\hat{\eta}}\sum_{s=1}^{\nu_{2}}\binom{P-c-1}{Q-2}I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right).$
(13)

###### Proof:

The proof is relegated to Appendix D. ∎

###### Remark 2

Suppose that $K$ users are uniformly assigned to the caching profiles, i.e.,
$K=P\hat{\eta}$, and all users are served in the CC delivery step, i.e.,
$K_{M}=K$ and $K_{U}=0$. If $\alpha\leq\hat{\eta}$, our schemes achieves the
optimal DoF $\alpha\left(P\gamma+1\right)$ obtained in [16]. If
$\alpha>\hat{\eta}$, the achievable DoF of our scheme is simplified to the
optimal DoF $K\gamma+\alpha$ obtained in [15]. However, unlike the scheme
proposed in [15], our scheme supports the networks with non-integer
$\frac{\alpha}{\hat{\eta}}$ (cf. Appendix E).

Indeed, increasing non-uniformness in user distribution prevents the system
from achieving optimal DoF performance. For instance, for the case
$\alpha>\hat{\eta}$, suppose that all users are served with Strategy A during
the CC delivery step, such that $\beta=\hat{\eta}$ and
$Q\leq{\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$.
In this setup, to boost the DoF performance, we can set
$Q={\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$. By
defining $\eta_{\rm avg}=\frac{K}{P}$ and assuming $\alpha$ is divisible by
$\hat{\eta}$ and $\eta_{\rm avg}$, the DoF loss (compared to uniform user
distribution) is $\alpha\left(1-\nicefrac{{\eta_{\rm
avg}}}{{\hat{\eta}}}\right)$.

Setting
$Q={\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$,
however, requires to implement successive interference cancellation (SIC) at
the receivers. To avoid using SIC, we can set $Q={\bar{t}}+1$, which
simplifies the DoF of the uniform association to $\eta_{\rm
avg}\left({\bar{t}}+1\right)$. Here, we should compare the achievable DoF of
$\eta_{\rm avg}\left({\bar{t}}+1\right)$ with $\alpha$ such that: i) if
$\eta_{\rm avg}\left({\bar{t}}+1\right)\geq\alpha$, we use the proposed CC
scheme to simultaneously benefit from global CC and spatial multiplexing
gains; ii) if $\eta_{\rm avg}\left({\bar{t}}+1\right)<\alpha$, we serve users
via unicasting to not have any loss in DoF.

###### Remark 3

For the non-uniform user-to-profile association with $Q={\bar{t}}+1$ and
$\eta_{1}\leq\alpha$, the best possible DoF is achievable by setting
$\hat{\eta}=\eta_{1}$ (cf. Appendix F).

##  V Numerical Results

In this section, we examine the impacts of non-uniformness in user
distribution on the achievable DoF. In this regard, assume a cache-aided MISO
setup with ${\bar{t}}=1$, $P=6$ and $\gamma=\frac{1}{6}$, in which $K=30$
users are present during the delivery phase. Here, for each association, we
compute the standard deviation $\sigma$ as
$\sigma^{2}=\frac{1}{P}\sum_{p=1}^{P}\left(\eta_{p}-\eta_{\rm
avg}\right)^{2}$, where $\eta_{\rm avg}=5$.

(a) $\sigma=0$

(b) $\sigma=0.3$

(c) $\sigma=3$

Figure 2: The DoF versus $\hat{\eta}$ with different values of $Q$, $K=30$,
$\alpha=7$, $P=6$, $\gamma=\frac{1}{6}$ and ${\bar{t}}=1$ for: (a) $\sigma=0$,
(b) $\sigma=0.3$ and (c) $\sigma=3$.

Fig. 2 illustrates the maximum achievable DoF for different $Q$ and $\sigma$
values with $\alpha=7$. As observed from Fig. 2a, for the uniform user
distribution, i.e., $\sigma=0$, our scheme achieves the optimal DoF
$K\gamma+\alpha=12$ with $\hat{\eta}=\eta_{\rm avg}=5$ and
$Q={\bar{t}}+\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\rceil=3$. Here, we note
that the system performance during the CC delivery step corresponds to
Strategy B for the regime $\alpha>\hat{\eta}$ and non-integer
$\nicefrac{{\alpha}}{{\hat{\eta}}}$, which is missing in the literature. For
small $\sigma$ values (e.g., $\sigma=0.3$), although the achievable DoF for
$\hat{\eta}=6$ and $Q={\bar{t}}+1=2$ is slightly less than $\hat{\eta}=6$ and
$Q=3$, the receiver structure for $Q=2$ is more straightforward, as they do
not need to implement SIC. For large $\sigma$ values (e.g., $\sigma=3$), when
$\max_{p}\eta_{p}>\alpha$, setting $Q={\bar{t}}+1$ and
$\hat{\eta}=\max_{p}\eta_{p}$ maximizes the achievable DoF.

Figure 3: The average of the maximum achievable DoF $\left({\rm
DoF}_{M}\right)$ versus the standard deviation $(\sigma)$ with $K=30$,
$\gamma=\frac{1}{6}$, $P=6$, ${\bar{t}}=1$ and
$\alpha\in\left\\{4,7\right\\}$.

In Fig. 3, for any association, we find $\mathrm{DoF}_{\max}$ which is the
maximum achievable DoF obtained by a line search over $\hat{\eta}$ and $Q$
values. Accordingly, for the associations with the same $\sigma$ value,
$\mathrm{DoF}_{M}$ indicates the average of $\mathrm{DoF}_{\max}$. Here, our
proposed scheme is compared with the optimal case, which corresponds to
uniform user distribution (described as only multicast), and the case that all
users are served via unicasting (described as only unicast). Although the
placement phase was designed solely based on $\gamma$, our scheme boosts the
maximum achievable DoF by $10\%-70\%$ over unicasting for moderate $\sigma$
(e.g., $\sigma=1-4.5$). So, to maximize the achievable DoF, the server should
serve users via the proposed approach for moderate $\sigma$, and serve users
via unicasting for large $\sigma$.

##  VI Conclusion

We proposed a novel coded caching scheme for handling network dynamicity where
the users can freely enter or depart the network at any time. The conventional
schemes in the literature are not truly dynamic as they are only applicable
if: 1) minimum profile length (the number of users assigned to the profile)
exceeds the spatial multiplexing gain $\alpha$, and 2) $\alpha\geq\hat{\eta}$
and $\alpha$ exceeds the global CC gain, where $\hat{\eta}$ can be the length
of any profile. Our proposed scheme addressed this bottleneck by providing a
universal solution applicable to any dynamic network setup, removing all the
constraints imposed by existing solutions. We also analyzed the degrees-of-
freedom (DoF) performance of the proposed scheme, and for the uniform
distribution, we showed that it achieves the optimal DoF not only in the
regions covered in the literature but also in the region
$\alpha\geq\hat{\eta}$ with non-integer $\nicefrac{{\alpha}}{{\hat{\eta}}}$.

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##  Appendix A Proof of Remark 1

As stated earlier, each transmission vector is constructed to serve at most
$Q\beta$ users. Hence, each of these $Q\beta$ users interfered by $Q\beta-1$
interfering users. Since each user has stored a $\gamma$ portion of the entire
library, it can remove at most $\beta P\gamma=\beta{\bar{t}}$ interference
terms using the stored cache content. Therefore, $\alpha$ must be large enough
to suppress the remaining $Q\beta-1-{\bar{t}}\beta$ interference terms, i.e.,

$\alpha-1\geq Q\beta-1-{\bar{t}}\beta,$ (14)

which results in $Q\leq{\bar{t}}+\left\lceil\frac{\alpha}{\beta}\right\rceil$.
Furthermore, $Q\geq{\bar{t}}+1$ should hold to benefit from the maximum coded
caching gain.

Let us assume that profile $p$ is one of the $Q$ selected profiles for the
transmission. Therefore, $\min\left(\beta,\delta_{p}\right)$ users are served
from profile $p$ during this transmission. On the one hand, it is supposed
that the maximum number of users of each profile served in the CC delivery
step is $\hat{\eta}$, which results in having $\beta\leq\hat{\eta}$. On the
other hand, for each profile $p\in\left[P\right]$, we should have
$\min\left(\beta,\delta_{p}\right)\leq\alpha$ to be able to cancel out the
inter-stream interference between users assigned to the same profile. Hence,
according to the fact that $\delta_{1}=\hat{\eta}$, one can say
$\min\left(\beta,\delta_{1}\right)=\min\left(\beta,\hat{\eta}\right)=\beta\leq\alpha$,
which results in having the necessary condition
$\beta\leq\min\left(\alpha,\hat{\eta}\right)$ to select the parameter $\beta$.

##  Appendix B Example for Strategy A

For a cache-aided dynamic setup with $P=3$, $\gamma=\frac{1}{3}$,
${\bar{t}}=1$, $\alpha=6$, $\beta=3$ and
$Q={\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\beta}}\right\rfloor=3$, let us
assume that ${\mathcal{U}}_{1}=\left\\{1,2,3,4,5\right\\}$,
${\mathcal{U}}_{2}=\left\\{6,7,8,9\right\\}$ and
${\mathcal{U}}_{3}=\left\\{10,11,12\right\\}$ are the set of users assigned to
profiles $1,2$ and $3$, respectively. During the placement phase, each file
$W^{n}$ is split into $\binom{P}{{\bar{t}}}=3$ mini-files
$W_{{\mathcal{P}}}^{n}$, where ${\mathcal{P}}\in\left\\{1,2,3\right\\}$. For
$n\in\left[N\right]$, users assigned to profiles $1,2$ and $3$ store the mini-
files $W_{1}^{n}$, $W_{2}^{n}$ and $W_{3}^{n}$, respectively. For the delivery
phase, each mini-file $W_{{\mathcal{P}}}^{n}$ is further split into
$\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}=3$ subpackets
$W_{{\mathcal{P}},q}^{n}$, where $q\in\left[3\right]$.

Assuming, $\hat{\eta}=4$, we have
${\mathcal{V}}_{1}=\left\\{1,2,3,4\right\\}$,
${\mathcal{V}}_{2}=\left\\{6,7,8,9\right\\}$ and
${\mathcal{V}}_{3}=\left\\{10,11,12\right\\}$, while user $5$ will be served
during the UC delivery step333Although it is mentioned in Section III to set
$\beta=\hat{\eta}$ to maximize the achievable DoF for the case of
$\alpha>\hat{\eta}$, here, we set $\beta<\hat{\eta}$ to give further insight
on the system operation with Strategy A.. Then, we obtain
${\mathcal{R}}_{1}-{\mathcal{R}}_{3}$ as follows.

$\displaystyle{\mathcal{R}}_{1}=\left(\left\\{1,2,3\right\\}\|\left\\{2,3,4\right\\}\|\left\\{3,4,1\right\\}\|\left\\{4,1,2\right\\}\right),$
$\displaystyle{\mathcal{R}}_{2}=\left(\left\\{6,7,8\right\\}\|\left\\{7,8,9\right\\}\|\left\\{8,9,6\right\\}\|\left\\{9,6,7\right\\}\right),$
$\displaystyle{\mathcal{R}}_{3}=\left(\left\\{10,11,12\right\\}\|\left\\{10,11,12\right\\}\|\left\\{10,11,12\right\\}\right),$

where ${\mathcal{R}}_{1,1}=\left\\{1,2,3\right\\}$ and so on. Accordingly,
using ${\mathcal{R}}_{1}-{\mathcal{R}}_{3}$,
${\mathcal{S}}_{1}-{\mathcal{S}}_{3}$ are given by:

$\displaystyle{\mathcal{S}}_{1}=\left(\left\\{1,2,3\right\\}\|\left\\{2,3,4\right\\}\|\left\\{3,4,1\right\\}\|\left\\{4,1,2\right\\}\right),$
$\displaystyle{\mathcal{S}}_{2}=\left(\left\\{6,7,8\right\\}\|\left\\{7,8,9\right\\}\|\left\\{8,9,6\right\\}\|\left\\{9,6,7\right\\}\right),$
$\displaystyle{\mathcal{S}}_{3}=\left(\left\\{10,11,12\right\\}\|\left\\{10,11,12\right\\}\|\left\\{10,11,12\right\\}\|\varnothing\right).$

In order to build the transmission vector for the transmission triple
$\left(1,1,1\right)$, first, we obtain
${\mathcal{M}}_{1}={\mathcal{M}}_{1}(1)=\left\\{2,3\right\\}$. Therefore, the
set of users that are served during the transmission triple $(1,1,1)$ is equal
to:

$\displaystyle{\mathcal{T}}_{1,1,1}$
$\displaystyle=\left({\mathcal{S}}_{1,1}\|{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}\right)=\left\\{1,2,3,6,7,8,10,11,12\right\\}.$

Finally, the transmission vector corresponding to the transmission triple
$\left(1,1,1\right)$ is constructed as follows.

$\displaystyle{\mathbf{x}}_{1,1,1}=$
$\displaystyle\sum\limits_{\Lambda\subseteq\left\\{1,2,3\right\\}:\left|\Lambda\right|=1}\,\,\sum\limits_{k\in{\mathcal{S}}_{p,c}:p\in\left\\{1,2,3\right\\}_{\backslash\Lambda}}W_{\Lambda,q}^{k}{\mathbf{w}}_{{\mathcal{G}}_{\Lambda}^{k}}$
$\displaystyle=$
$\displaystyle\sum\limits_{k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}}W_{1,q}^{k}{\mathbf{w}}_{{\mathcal{G}}_{1}^{k}}+\sum\limits_{k\in{\mathcal{S}}_{1,1}\|{\mathcal{S}}_{3,1}}W_{2,q}^{k}{\mathbf{w}}_{{\mathcal{G}}_{2}^{k}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k\in{\mathcal{S}}_{1,1}\|{\mathcal{S}}_{2,1}}W_{3,q}^{k}{\mathbf{w}}_{{\mathcal{G}}_{3}^{k}}.$

Now, in order to show the decoding process, for any
$k,j\in{\mathcal{S}}_{p,c}$ with $p\in{\mathcal{N}}_{\backslash\Lambda}$, if
$j\neq k$, using ZF precoders, we have ${\mathbf{h}}_{k}^{\rm
H}{\mathbf{w}}_{{\mathcal{G}}_{\Lambda}^{j}}=0$; otherwise ($j=k$),
${\mathbf{h}}_{k}^{\rm H}{\mathbf{w}}_{{\mathcal{G}}_{\Lambda}^{k}}=1$. For
example, ${\mathbf{h}}_{1}^{\rm H}{\mathbf{w}}_{{\mathcal{G}}_{3}^{1}}=1$ and
${\mathbf{h}}_{2}^{\rm H}{\mathbf{w}}_{{\mathcal{G}}_{3}^{1}}=0$. Now, the
received signal at user $1$ is given by:

$\displaystyle
y_{1}=\sum\limits_{k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}}{\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{{\mathcal{G}}_{1}^{k}}W_{1,q}^{k}+W_{2,1}^{1}+W_{3,1}^{1}+n_{1}.$

User $1$ have all subpackets $W_{1,q}^{n}$, $\forall n\in[N]$ in its cache
memory, therefore, it can regenerate
$\sum\limits_{k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}}{\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{{\mathcal{G}}_{1}^{k}}W_{1,q}^{k}$ and subtract it from
$y_{1}$, which results in observing the superposition signal
$\tilde{y}_{1}=W_{2,1}^{1}+W_{3,1}^{1}+n_{1}$. Here, we note that user $1$
needs to implement SIC to decode the subpackets [8], [16].

##  Appendix C Example for Strategy B

With the given ${\mathcal{U}}_{1}-{\mathcal{U}}_{3}$ and
${\mathcal{V}}_{1}-{\mathcal{V}}_{3}$ in Appendix B, suppose that $P=3$,
$\gamma=\frac{1}{3}$, ${\bar{t}}=1$, $\alpha=6$, $\beta=\hat{\eta}=4$,
$\theta=\alpha-\hat{\eta}\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor=2$
and $Q={\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rceil=3$.
For the placement phase, similarly to Appendix B, each file $W^{n}$ is split
into 3 mini-files, and users assigned to profiles $1,2$ and $3$, store the
mini-files $W_{1}^{n}$, $W_{2}^{n}$ and $W_{3}^{n}$, respectively. Then,
during the content delivery phase, each mini-file is split into
$\left(\hat{\eta}{\bar{t}}+\alpha\right)\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\binom{Q-2}{Q-{\bar{t}}-2}=10$
subpackets. Now, for the transmission quintuple $(1,1,1,1,1)$, we have:

$\displaystyle{\mathcal{Y}}_{1}=\left\\{1,2,3,4\right\\},\,\,{\mathcal{Y}}_{2}=\left\\{6,7,8,9\right\\},\,\,{\mathcal{Y}}_{3}=\left\\{10,11,12,f^{*}\right\\}.$

At the next step, for $r=c=1$, it is found that
$\overline{{\mathcal{P}}}_{1}=\left\\{2,3\right\\}$,
$\overline{{\mathcal{P}}}_{1}(1)=2$, and
${\mathcal{I}}_{1}^{1}={\mathcal{I}}_{1}^{1}(1)=\left\\{3\right\\}$. During
this transmission, the users of the set
${\mathcal{E}}_{1}^{1}=\left\\{1,2\right\\}$, and the users assigned to the
profiles
${\mathcal{B}}=\overline{{\mathcal{P}}}_{1}(1)\cup{\mathcal{I}}_{1}^{1}(1)=\left\\{2,3\right\\}$
are served. Furthermore, ${\mathcal{K}}_{1,1}^{1,1}=\left\\{1,f^{*}\right\\}$
and ${\mathcal{K}}_{1,1}^{1,2}=\left\\{2,f^{*}\right\\}$. In addition, it is
found that
${\mathcal{C}}=\left\\{\left\\{2\right\\},\left\\{3\right\\}\right\\}$ such
that ${\mathcal{C}}(1)=\left\\{2\right\\}$ and
${\mathcal{C}}(2)=\left\\{3\right\\}$. Accordingly, we have $\Theta_{1}=3$ and
$\Theta_{2}=2$.

Hence, after removing the phantom users $f^{*}$, the transmission vector for
the transmission quintuple $(1,1,1,1,1)$ takes the form of:

$\displaystyle{\mathbf{x}}_{1,1,1}^{1,1}$
$\displaystyle=\sum\limits_{n=1}^{2}\,\sum\limits_{k\in{\mathcal{K}}_{r,s}^{m,u}(n)\cup{\mathcal{V}}_{p}:u\in\left\\{1,2\right\\},p\in{\mathcal{C}}(n)}W_{\Theta_{n},q}^{k}{\mathbf{w}}_{{\mathcal{H}}_{{\mathcal{C}}(n)}^{k}}$
$\displaystyle=\sum\limits_{k\in\left\\{1,2,6,7,8,9\right\\}}W_{3,q}^{k}{\mathbf{w}}_{{\mathcal{H}}_{2}^{k}}+\sum\limits_{k\in\left\\{10,11,12\right\\}}W_{2,q}^{k}{\mathbf{w}}_{{\mathcal{H}}_{3}^{k}},$

where ${\mathcal{H}}_{2}^{6}=\left\\{1,2,7,8,9\right\\}$,
${\mathcal{H}}_{3}^{10}=\left\\{1,2,11,12\right\\}$ and so on. Similarly to
Appendix B, all users that are served via Strategy B are able to decode their
requested files.

##  Appendix D Proof of Theorem 1

Here, first, we present a lemma to simplify the calculations, then, prove the
theorem for strategies A and B in Sections D-A and D-B, respectively.

###### Lemma 1

Given $P$ and $Q$, we have
$\sum\limits_{r=1}^{P-Q+1}\binom{P-r}{Q-1}=\binom{P}{Q}$.

###### Proof:

Assume a scenario at which we have a set
${\mathcal{X}}=\left\\{b_{1},\cdots,b_{P}\right\\}$ with
$\left|{\mathcal{X}}\right|=P$, and we want to select the subsets
$\bar{{\mathcal{X}}}\subseteq{\mathcal{X}}$ such that
$\left|\bar{{\mathcal{X}}}\right|=Q$. Here, our aim is to count the
possibilities for the subsets $\bar{{\mathcal{X}}}$. One way to count these
possibilities is to simply pick the subsets $\bar{{\mathcal{X}}}$ from
${\mathcal{X}}$, and then count them. Here, clearly, the number of
possibilities is equal to $\binom{P}{Q}$. Another way to choose
$\bar{{\mathcal{X}}}$ is to define $\left(P-Q+1\right)$ subsets
${\mathcal{L}}_{r}$ first, such that $r\in\left[P-Q+1\right]$,
${\mathcal{L}}_{r}=\left\\{b_{i}:r+1\leq i\leq P\right\\}$ and
$\left|{\mathcal{L}}_{r}\right|=P-r$. Then, the selection of
$\bar{{\mathcal{X}}}$ consists of $\left(P-Q+1\right)$ rounds. In round
$r\in\left[P-Q+1\right]$, we pick $b_{r}$ and choose $\left(Q-1\right)$
remaining elements from ${\mathcal{L}}_{r}$. In this regard, the number of
possibilities to choose $\bar{{\mathcal{X}}}$ in round $r$ is equal to
$\binom{P-r}{Q-1}$. By following this approach, we have
$\sum\limits_{r=1}^{P-Q+1}\binom{P-r}{Q-1}$ possibilities to choose
$\bar{{\mathcal{X}}}$. Hence, it is sure that
$\sum\limits_{r=1}^{P-Q+1}\binom{P-r}{Q-1}=\binom{P}{Q}$. ∎

###  D-A Proof of Theorem 1 for Strategy A

In order to prove the theorem, first, we compute the number of times that each
user is served with Strategy A during the CC delivery step. Suppose that user
$k$ with $p\left[k\right]=p$ is present in the CC delivery step. As per (3)
and (4), user $k$ is served in the transmission triple $\left(r,c,l\right)$
such that $r\in\left[p\right]$. On the other hand, according to (1) and (2),
for each $r\in[p]$, there are $\beta$ values for $c$ that user $k$ is served
in the transmission triple $\left(r,c,l\right)$. Now, for any $r\in[p]$, let
${\mathcal{A}}_{k}^{(r)}$ be the set of values for $c$ at which user $k$ is
served in the transmission triple $\left(r,c,l\right)$ such that
$\left|{\mathcal{A}}_{k}^{(r)}\right|=\beta$. As mentioned in Section III-A,
for $r=p$ and $c\in{\mathcal{A}}_{k}^{(p)}$, there are $\binom{P-p}{Q-1}$
values for $l$ at which user $k$ is served during the transmission triple
$\left(r,c,l\right)$. Furthermore, for $r\in\left[p-1\right]$ and
$c\in{\mathcal{A}}_{k}^{(r)}$, there exist $\binom{P-r-1}{Q-2}$ transmission
triples in which user $k$ receives the subpackets. Therefore, the number of
times that user $k$ is served with Strategy A during the CC delivery step is
given by:

$\displaystyle\beta\left(\binom{P-p}{Q-1}+\sum\limits_{r=1}^{p-1}\binom{P-r-1}{Q-2}\right).$
(15)

In the proceeding, we simplify (15). We know that
$\binom{P-p}{Q-1}+\binom{P-p}{Q-2}=\binom{P-p+1}{Q-1}$, which results in
having
$\binom{P-p}{Q-1}+\binom{P-p}{Q-2}+\binom{P-p+1}{Q-2}=\binom{P-p+1}{Q-1}+\binom{P-p+1}{Q-2}=\binom{P-p+2}{Q-1}$.
Therefore, by following a similar procedure, we can say:

$\displaystyle\binom{P-p}{Q-1}+\sum\limits_{r=1}^{p-1}\binom{P-r-1}{Q-2}=\binom{P-1}{Q-1}.$
(16)

By substituting (16) into (15), it is observed that $\beta\binom{P-1}{Q-1}$
transmissions serve user $k$. Now, since there are $K_{M}$ users in the CC
delivery step, the total number of times that all users served with Strategy A
is equal to:

$J_{M}=K_{M}\beta\binom{P-1}{Q-1}.$ (17)

In the proceeding, we show that all users can decode their requested files
with Strategy A. As per (1)-(4), user $k$ served with Strategy A receives
$\binom{Q-1}{Q-{\bar{t}}-1}$ subpackets in each transmission. Hence, the total
number of received subpackets at user $k$ is given by:

$\displaystyle\beta\binom{P-1}{Q-1}\binom{Q-1}{Q-{\bar{t}}-1}=\left(1-\gamma\right)\beta\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}.$
(18)

On the other hand, each file is split into
$\beta\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$ subpackets,
while a $\gamma$ portion of these subpackets is stored in cache memory of user
$k$ during the placement phase, and
$\left(1-\gamma\right)\beta\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$
missing subpackets must be delivered to user $k$ with Strategy A. As observed
from (18), user $k$ receives all
$\left(1-\gamma\right)\beta\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$
missing subpackets with Strategy A, hence, one can conclude that all users
attending the CC delivery step with Strategy A are able to decode their
requested files.

Now, we compute the number of transmitted subpackets during the UC delivery
step. By following a similar way as in [17], each user served with UC delivery
step must receive
$\left(1-\gamma\right)\beta\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$
subpackets. Since there are $K_{U}$ users in the UC delivery step, the number
of transmitted subpackets during the UC delivery step is

$J_{U}=K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}.$ (19)

Using (17) and (19), the total transmitted subpackets in the data delivery
phase is given by:

$\displaystyle
J_{M}+J_{U}=K_{M}\binom{P-1}{Q-1}\beta+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}.$
(20)

Next, we calculate the total number of transmissions in the CC delivery step.
As per (4), for fixed $r=1$ and $c=c_{0}\in\left[\phi_{1}\right]$, users
assigned to profiles ${\mathcal{S}}_{1,c_{0}}$,
${\mathcal{S}}_{b_{1},c_{0}},\cdots,{\mathcal{S}}_{b_{Q-1},c_{0}}$ are served
during the transmission triple $\left(1,c_{0},l\right)$, where
$\left\\{b_{1},\cdots,b_{Q-1}\right\\}={\mathcal{M}}_{1}\left(l\right)$ and
$l\in\left[\binom{P-1}{Q-1}\right]$. Hence, the number of transmissions for
$r=1$ is equal to $\phi_{1}\binom{P-1}{Q-1}$. Assuming $\delta_{2}>0$, for
$r=2$ and $c=c_{0}\in\left[\phi_{2}\right]$, the users assigned to
${\mathcal{S}}_{1,c_{0}}$ are not served in the transmission triple
$\left(2,c_{0},l\right)$, and the BS serves users assigned to
${\mathcal{S}}_{2,c_{0}},{\mathcal{S}}_{b_{1},c_{0}}\cdots,{\mathcal{S}}_{b_{Q-1},c_{0}}$,
such that
$\left\\{b_{1},b_{2},\cdots,b_{Q-1}\right\\}={\mathcal{M}}_{2}\left(l\right)$,
and $l\in\left[\binom{P-2}{Q-1}\right]$. Therefore, the number of
transmissions for $r=2$ is $\phi_{2}\binom{P-2}{Q-1}$. Similarly, assuming
$\delta_{r}>0$, for any $r\in\left[P-Q+1\right]$ and
$c_{0}\in\left[\phi_{r}\right]$, users assigned to the set
${\mathcal{T}}_{r,c_{0},l}={\mathcal{S}}_{r,c_{0}}\|{\mathcal{S}}_{b_{1},c_{0}}\|\cdots\|{\mathcal{S}}_{b_{Q-1},c_{0}}$
are served, where
$\left\\{b_{1},\cdots,b_{Q-1}\right\\}={\mathcal{M}}_{r}\left(l\right)$ and
$l\in\left[\binom{P-r}{Q-1}\right]$. So, supposing $\delta_{r}>0$, the total
number of transmissions for any $r\in\left[P-Q+1\right]$ is equal to
$\phi_{r}\binom{P-r}{Q-1}$. As a result, the total number of transmissions in
the CC delivery step with Strategy A is given by:

$\displaystyle
T_{M}=\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}.$ (21)

Then, we obtain the total number of transmissions during the UC delivery step.
As stated earlier, during this step, we have to deliver
$K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}$ subpackets via
$T_{U}$ transmissions, such that in each transmission,
$\min\left(K_{U},\alpha\right)$ subpackets are transmitted. So, given
$K_{U}\neq 0$, the total number of transmissions to deliver these subpackets
is equal to:

$\displaystyle
T_{U}=\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil.$
(22)

Now, for $K_{U}\neq 0$, by substituting (20)-(22) into (10), the DoF is
obtained as follows.

$\displaystyle\mathrm{DoF}=\frac{K_{M}\binom{P-1}{Q-1}\beta+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}+\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil}.$
(23)

Moreover, for $K_{U}=0$, we have $K_{M}=K$, and by applying (17) and (21) into
(10), the DoF takes the form as:

$\displaystyle\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\beta}{\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}}.$
(24)

###  D-B Proof of Theorem 1 for Strategy B

In order to obtain the DoF for the dynamic MISO network operating with
Strategy B, first, we count the number of times that users are served during
the CC delivery step. In this regard, assume that user $k$ with
$p\left[k\right]=p$ is present in the CC delivery step, and receives its
subpackets with Strategy B. According to (6), (8) and (9), when
$c\in\left[P-Q+1\right]$, $l\in\left[\binom{P-c-1}{Q-2}\right]$,
$m\in\left[\theta\right]$ and $s\in\left[\nu_{2}\right]$ , user $k$ is served
during the transmission quintuples $\left(p,c,l,m,s\right)$. Hence, for $r=p$,
the number of transmission quintuples $\left(r,c,l,m,s\right)$ serving user
$k$ is given by:

$\displaystyle\sum\limits_{c=1}^{P-Q+1}\binom{P-c-1}{Q-2}\theta\nu_{2},$ (25)

which by using Lemma 1 is simplified to:

$\displaystyle\sum\limits_{c=1}^{P-Q+1}\binom{P-c-1}{Q-2}\theta\nu_{2}=\theta\nu_{2}\binom{P-1}{Q-1}.$
(26)

Furthermore, as per (6), (8) and (9), when $r\neq p$,
$c\in\left[P-Q+1\right]$, $l\in\left[\binom{P-c-2}{Q-3}\right]$,
$m\in\left[\hat{\eta}\right]$ and $s\in\left[\nu_{2}\right]$, the transmission
quintuples $\left(r,c,l,m,s\right)$ serve user $k$. Therefore, the number of
transmission quintuples $\left(r,c,l,m,s\right)$ serving user $k$ with $r\neq
p$ is obtained as:

$\displaystyle\sum\limits_{r=1}^{P-1}\sum\limits_{c=1}^{P-Q+1}\binom{P-c-2}{Q-3}\hat{\eta}\nu_{2},$
(27)

which by using Lemma 1 takes the form as follows.

$\displaystyle\sum\limits_{r=1}^{P-1}\sum\limits_{c=1}^{P-Q+1}\binom{P-c-2}{Q-3}\hat{\eta}\nu_{2}=\hat{\eta}\nu_{2}\left(P-1\right)\binom{P-2}{Q-2}.$
(28)

As a result, by using (26), (28) and according to that
$\theta=\alpha-\hat{\eta}\left\lfloor\frac{\alpha}{\hat{\eta}}\right\rfloor$,
the number of times that user $k$ is served with Strategy B is equal to:

$\displaystyle\theta\nu_{2}\binom{P-1}{Q-1}+\hat{\eta}\nu_{2}\left(P-1\right)\binom{P-2}{Q-2}=$
(29)
$\displaystyle\binom{P-1}{Q-1}\nu_{2}\left(\hat{\eta}{\bar{t}}+\alpha\right).$

Since $K_{M}$ users are present in the CC delivery step, the total number of
times that all $K_{M}$ users are served with Strategy B is given by:

$\displaystyle
J_{M}=K_{M}\binom{P-1}{Q-1}\nu_{2}\left(\hat{\eta}{\bar{t}}+\alpha\right).$
(30)

In the proceeding, we count the total number of transmissions with Strategy B.
As stated in Section III-B, if
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)=1$, the server
delivers the subpackets to the users during the transmission quintuple
$\left(r,c,l,m,s\right)$; otherwise, it does not transmit any signal. Now, by
following a similar approach to (21), since $r\in[P]$,
$c\in\left[P-Q+1\right]$, $l\in\left[\binom{P-c-1}{Q-2}\right]$,
$m\in\left[\hat{\eta}\right]$ and $s\in\left[\nu_{2}\right]$, the total number
of transmissions in Strategy B is given by:

$\displaystyle
T_{M}=\sum_{r=1}^{P}\sum_{c=1}^{P-Q+1}\sum_{m=1}^{\hat{\eta}}\sum_{s=1}^{\nu_{2}}\binom{P-c-1}{Q-2}I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right).$
(31)

At the next step, we show that user $k$ receives all its subpackets during the
CC delivery step with Strategy B. As observed from (29), each user $k$ is
served for $\binom{P-1}{Q-1}\nu_{2}\left(\hat{\eta}{\bar{t}}+\alpha\right)$
times during the CC delivery step. As per (6)-(9), in each transmission, user
$k$ receives $\nu_{1}$ subpackets. Therefore, the total number of subpackets
that user $k$ receives with Strategy B is
$\binom{P-1}{Q-1}\nu_{1}\nu_{2}\left(\hat{\eta}{\bar{t}}+\alpha\right)$, which
by following the same way as in (18), can be reformulated as follows.

$\displaystyle\binom{P-1}{Q-1}\binom{Q-2}{Q-{\bar{t}}-2}\binom{Q-1}{Q-{\bar{t}}-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)=$
(32)
$\displaystyle\left(1-\gamma\right)\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\binom{Q-2}{Q-{\bar{t}}-2}\left(\hat{\eta}{\bar{t}}+\alpha\right),$

which shows that user $k$ receives all subpackets of its requested file during
the CC delivery step with Strategy B.

Next, in order to obtain $J_{U}$ and $T_{U}$, we follow the same way as in
(19) and (22). Accordingly, as stated in Section III-B, each file is split
into
$\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\binom{Q-2}{Q-{\bar{t}}-2}\left(\hat{\eta}{\bar{t}}+\alpha\right)$
subpackets. Therefore, according to the fact that $K_{U}$ users are present in
the UC delivery step and the cache ratio at each user is $\gamma$, the total
number of times that users served during the UC delivery step is given by:

$\displaystyle J_{U}$
$\displaystyle=K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\binom{Q-2}{Q-{\bar{t}}-2}\left(\hat{\eta}{\bar{t}}+\alpha\right)$
(33)
$\displaystyle=K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}.$

Furthermore, as mentioned in Section III-C, assuming $K_{U}\neq 0$, each
transmission of the UC delivery step serves $\min\left(K_{U},\alpha\right)$
users. Hence, the total number of transmissions in the UC delivery step is
obtained as follows.

$\displaystyle
T_{U}=\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil.$
(34)

As a result, for $K_{U}\neq 0$, by applying (30), (31), (33) and (34) into
(10), the DoF is obtained as follows.

$\displaystyle\mathrm{DoF}=\frac{K_{M}\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}}{N_{M}+N_{U}}.$
(35)

In addition, when $K_{U}=0$ and $K_{M}=K$, by substituting (30) and (31) into
(10), the DoF is simplified to:

$\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}}{N_{M}}.$
(36)

##  Appendix E Proof of Remark 2

When $K$ users are uniformly assigned to $P$ caching profiles, i.e.,
$K=P\hat{\eta}$, and all users are served via CC delivery step, i.e.,
$K_{M}=K$ and $K_{U}=0$, we prove Remark 2 for three regimes: 1)
$\alpha\leq\hat{\eta}$, 2) $\alpha>\hat{\eta}$ with integer
$\frac{\alpha}{\hat{\eta}}$ and 3) $\alpha>\hat{\eta}$ with non-integer
$\frac{\alpha}{\hat{\eta}}$.

1) $\alpha\leq\hat{\eta}$: In this regime, the system operates with Strategy A
during the CC delivery step, where $\beta=\alpha$ and $Q={\bar{t}}+1$. By
considering $\hat{\eta}=\eta_{r}$ for $\forall r\in[P]$, we know that
$D\left(\delta_{r}\right)=\hat{\eta}$. Next. we reformulate (11) as follows.

$\displaystyle\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\alpha}{\hat{\eta}\sum\limits_{r=1}^{P-Q+1}\binom{P-r}{Q-1}}.$
(37)

Now, using Lemma 1 and setting $Q={\bar{t}}+1$, the DoF in (37) is simplified
to:

$\displaystyle\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\alpha}{\hat{\eta}\binom{P}{Q}}=\frac{KQ\alpha}{P\hat{\eta}}=\alpha\left(P\gamma+1\right).$
(38)

2) $\alpha>\hat{\eta}$ and integer $\frac{\alpha}{\hat{\eta}}$: In this
regime, we know that the system uses Strategy A to deliver data during the CC
delivery step, where $\beta=\hat{\eta}$ and ${\bar{t}}+1\leq
Q\leq{\bar{t}}+\frac{\alpha}{\hat{\eta}}$. Here, in order to maximize the
achievable DoF, we set $Q={\bar{t}}+\frac{\alpha}{\hat{\eta}}$. Then,
similarly to the previous case, we consider $\hat{\eta}=\eta_{r}$ and
$D\left(\delta_{r}\right)=\hat{\eta}$ for all $r\in[P]$. Accordingly, the DoF
in (11) is given by:

$\displaystyle\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\hat{\eta}}{\hat{\eta}\sum\limits_{r=1}^{P-Q+1}\binom{P-r}{Q-1}}.$
(39)

Finally, by using Lemma 1 and (39), the maximum achievable DoF corresponding
to this regime is as follows.

$\displaystyle\mathrm{DoF}=\nicefrac{{K\binom{P-1}{Q-1}}}{{\binom{P}{Q}}}=\nicefrac{{KQ}}{{P}}=\hat{\eta}{\bar{t}}+\alpha=K\gamma+\alpha.$
(40)

3) $\alpha>\hat{\eta}$ with non-integer $\frac{\alpha}{\hat{\eta}}$: In this
regime, in order to maximize the achievable DoF, the server can transmit data
via Strategy B during the CC delivery step, such that $\hat{\eta}=\eta_{r}$
for all $r\in[P]$, $\beta=\hat{\eta}$ and
$Q={\bar{t}}+\left\lceil\frac{\alpha}{\hat{\eta}}\right\rceil$. Accordingly,
we have $I^{+}\left(\bar{\delta_{c}},{\mathcal{E}}_{r}^{m}\right)=1$ for all
$r\in[P]$, $c\in\left[P-Q+1\right]$, $l\in\left[\binom{P-c-1}{Q-2}\right]$,
$m\in\left[\hat{\eta}\right]$ and $s\in\left[\nu_{2}\right]$. Hence, by using
Lemma 1 and (13), $N_{M}$ is simplified to:

$N_{M}=P\hat{\eta}\binom{P-1}{Q-1}\nu_{2}.$ (41)

Finally, by substituting (41) into (36) and using ${\bar{t}}=P\gamma$, the DoF
of this regime for the uniform user-to-profile association takes the form of:

$\displaystyle\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}}{P\hat{\eta}\binom{P-1}{Q-1}\nu_{2}}=\hat{\eta}{\bar{t}}+\alpha=K\gamma+\alpha.$
(42)

##  Appendix F Proof of Remark 3

For $Q={\bar{t}}+1$ and $\eta_{1}\leq\alpha$, the system operates with
Strategy A during the CC delivery step, and $\beta^{\prime}=\beta$, where
$\beta=\alpha$, if $\alpha\leq\hat{\eta}$; otherwise, $\beta=\hat{\eta}$.
Assuming $\hat{\eta}<\eta_{1}$ (i.e. $K_{U}>0$) and using (22), clearly, it is
sure that $\binom{P}{Q}+\frac{1}{\beta}T_{U}>\binom{P}{Q}$. So, by using Lemma
1, we have:

$\displaystyle\frac{1}{\beta}\sum\limits_{r=1}^{P-Q+1}\beta\binom{P-r}{Q-1}+\frac{1}{\beta}T_{U}>\frac{1}{\eta_{1}}\sum\limits_{r=1}^{P-Q+1}\eta_{1}\binom{P-r}{Q-1}.$
(43)

Now, according to the fact that for any arbitrary parameters $w$, $x$, $y$ and
$z$, if $w+x\geq y$, then $\max\left(w,z\right)+x\geq y$, we rearrange (43) as
follows.

$\displaystyle\frac{1}{\beta}\sum\limits_{r=1}^{P-Q+1}\max\left(\beta,\delta_{r}\right)\binom{P-r}{Q-1}+\frac{T_{U}}{\beta}>\frac{1}{\eta_{1}}\sum\limits_{r=1}^{P-Q+1}\eta_{1}\binom{P-r}{Q-1}.$
(44)

We know that $D\left(\delta_{r}\right)=D\left(\eta_{r}\right)=0$ if
$\eta_{r}=0$; otherwise,
$D\left(\delta_{r}\right)=\max\left(\beta,\delta_{r}\right)$ and
$D\left(\eta_{r}\right)=\max\left(\eta_{r},\eta_{1}\right)=\eta_{1}$. Hence,
we reformulate (44) as follows.

$\displaystyle\frac{K\binom{P-1}{Q-1}\beta}{\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}+T_{U}}<\frac{K\binom{P-1}{Q-1}\eta_{1}}{\sum\limits_{r=1}^{P-Q+1}D\left(\eta_{r}\right)\binom{P-r}{Q-1}}.$
(45)

Furthermore, using $\beta^{\prime}=\beta$, $Q={\bar{t}}+1$ and (20), we have:

$\displaystyle J_{M}+J_{U}=K\binom{P-1}{Q-1}\beta.$ (46)

According to (46) and (11), the left side of (45) demonstrates the DoF of a
setup with $Q={\bar{t}}+1$ and $\hat{\eta}<\eta_{1}$ (i.e., $K_{U}\neq 0$),
while the right side corresponds to the DoF with $Q={\bar{t}}+1$ and
$\hat{\eta}=\eta_{1}$ (i.e., $K_{U}=0$). As a result, for $Q={\bar{t}}+1$ and
$\eta_{1}\leq\alpha$, we should set $\hat{\eta}=\eta_{1}$ to maximize the
achievable DoF.