A Novel Content Caching and Delivery Scheme for Millimeter Wave Device-to-Device Communications

We consider a C-RAN, where the remote radio heads (RRHs) are spatially distributed according to a homogeneous Poisson point process (HPPP) $\Phi_{R}$ with intensity $\lambda_{R}$. The RRHs are wirelessly connected to the edge cloud (fronthaul links) and the edge clouds are connected to the core network (backhaul links). The spatial distribution of the mobile user (MU) devices is modelled using an independent HPPP $\Phi_{u}$ with intensity $\lambda_{u}$. Content to a MU may either be delivered from a RRH through a cellular link that operates in the sub 6 GHz band (transmit power $P_{c}$, wavelength $W_{c}$, bandwidth $B_{c}$) or from another MU through a mmWave D2D link (transmit power $P_{d}$, wavelength $W_{d}$, bandwidth $B_{d}$). Content delivery through a D2D link will only be possible if the requested file is in the cache of another proximity MU. Each MU is capable of caching $M_{d}$ files of equal size.

RRHs are equipped with omni-directional transmitting antennas while the MUs are equipped with omni-directional antennas for cellular communications and directional antennas for mmWave communications. Similar to [5], sectorized antenna pattern is adopted at the transmitters to approximate the antenna pattern for the mmWave links. It is assumed that the antennas of the transmitter and the receiver are perfectly aligned for desired links while the interfering transmitter antenna bore-sight is uniformly distributed over $[0,2\pi]$. This means the probability distribution of the i.i.d. random antenna gain $G^{\prime}$ associated an interfering mmWave link is given by $\mathsf{Pr}\left\{G^{\prime}=G_{m}^{2}\right\}=\left(\frac{\Delta\theta}{2\pi}\right)^{2},$ $\mathsf{Pr}\left\{G^{\prime}=G_{s}^{2}\right\}=\left(\frac{2\pi-\Delta\theta}{2\pi}\right)^{2},$ and $\mathsf{Pr}\left\{G^{\prime}=G_{m}G_{s}\right\}=\frac{2\Delta\theta\left(2\pi-\Delta\theta\right)}{\left(2\pi\right)^{2}},$ where $G_{m}$ and $G_{s}$ denote the main and side lobe gains, respectively, and $\Delta\theta$ denotes the angle of deviation from the antenna bore-sight.

We refer to the MUs that request content as active MUs. At a given time, the probability of an MU being active is $\rho$. The remaining MUs, which we refer to as inactive MUs, can serve as potential D2D transmitters. It is assumed that all RRHs are active at a given time. Without loss of generality, we consider a typical MU located at the origin for our analysis.

For the cellular links that operate below 6 GHz, the simple path loss model with a path loss exponent $\alpha_{c}$ is used to model the location dependent path loss. For mmWave links, the average LoS ball model is used [7, 8]. According to this model, a link is considered as a LoS link if the link is shorter than $D_{L}$. Otherwise, it is considered a non-line of sight (NLoS) link. The average size and the density of the blockages determine $D_{L}$ [9]. For mmWave links, blockage effects induce different path loss exponents $\alpha_{L}$ and $\alpha_{N}$ for LoS and NLoS links, respectively, with $\alpha_{L}<\alpha_{N}$. We assume fast Rayleigh fading where the fading power is exponentially distributed with unit mean [5]. The received signal-to-interference-plus-noise-ratio (${\rm SINR}$) when receiving file $i$ from the D2D transmitter located at $x\in\Phi_{d}$ is given by

where $\Phi_{d}$, $h_{x}$ and $r_{x}$ denote the point process of the active D2D transmitters, fading power and the distance between the MU and the D2D transmitter at $x$, respectively, $\alpha_{d}\in\left\{\alpha_{L},\alpha_{N}\right\}$, $\hat{N}=\frac{16\pi^{2}N_{o}F_{N}B_{d}}{P_{d}W_{d}^{2}}$, $N_{o}$ is the noise power spectral density and $F_{N}$ is the noise figure of the receiver. Similarly, an expression for the received ${\rm SINR}$ when receiving file $i$ through a cellular link from the RRH $x\in\Phi_{R}$, which we denote by ${\rm SINR}_{c,i,x}$, can be obtained by replacing subscript $d$ in (1) with subscript $c$ and by replacing both antenna gains $G_{m}^{2}$ and $G^{\prime}$ by $G_{T}G_{R}$, where $G_{T}$ and $G_{R}$ are the antenna gains of the transmitting RRH and the receiving MU, respectively.

We assume that MUs request content from a finite content library of $N$ files of equal size, and the file requests follow a Zipf distribution of popularity exponent $\epsilon$. Thus, the probability of requesting the $i$-th most popular file is given by

The rate and the delay constraints of the files may vary with the file type and the associated application. The rate constraint for the $i$-th most popular file is denoted by $R_{i}$, and this constraint necessitates an ${\rm SINR}$ greater than $T_{i}=2^{\frac{R_{i}}{B}}-1,$ for a link having a bandwidth of $B\in$ $\left\{B_{d},B_{c}\right\}$.

Due to the limited storage capacity of the MUs, the propagation characteristics of mmWaves, and the rate requirements of different files, caching content at MUs should be done in an efficient manner. In this paper, we focus on designing a content placement scheme that offloads the traffic to the D2D devices without violating the rate (QoS) constraints, which are considered to be crucial in next generation networks. Moreover, considering the effects of blockage, it is preferred to have LoS D2D links. Thus, we focus on the successful LoS reception probability (${\rm SLP}$) for the $i$-th most popular file, which we define as

where $\hat{x}$ denotes the location of the closest D2D transmitter who has the $i$-th file in its cache. The ${\rm SLP}$ depicts the probability of receiving content from the nearest LoS D2D transmitter without violating the rate threshold.

Let the probability of the $i$-th file being stored in the cache of an MU be $q_{i}$. We define $\boldsymbol{q}$, an $N$-dimensional caching probability vector $\boldsymbol{q}=[q_{1},..,q_{N}]$, which denotes the probabilities of an MU caching the $N$ files in the content library. We focus on finding $\boldsymbol{q}$ that maximizes the average successful LoS reception probability (${\rm ASLP}$), defined as

where the ${\rm SLP}_{i}$ values are averaged over the request probabilities, while not violating the MU storage constraints statistically (on average). Note that our objective function captures both popularity and the QoS requirements of different content, and also the effects of wireless propagation. The optimization problem can be formulated as

An expression for ${\rm SLP}_{i}$ can be obtained using fundamentals of stochastic geometry [10]. That is, from (2),

where $\hat{I}=\sum_{y\in\Phi_{d}\setminus\left\{{\hat{x}}\right\}}h_{y}G^{\prime}r_{y}^{-\alpha_{d}}$ and $L_{\hat{I}}\left(S\right)$ is the Laplace transform of the interference from mmWave D2D links. The Laplace transform of the D2D interference is given by

where (a) follows from the probability generating functional (PGFL) of the PPP, $p_{d}$ is the probability of receiving the requested content via a D2D link, which has to be separately calculated, as shown later Section IV, and the expectation in (b) can be evaluated by using the PDF of $G^{\prime}$ given in Section II, which would result in a product of three exponential functions.

It is not hard to see that the resulting expression, which has multiple integrals, makes it prohibitively hard for us to use it in a meaningful manner in the optimization problem. We have an $N$-dimensional non-convex constrained optimization problem, and even obtaining the optimum solution numerically is not trivial. Hence, we make few approximations to obtain a mathematically tractable expression for ${\rm SLP}_{i}$, i.e., we obtain a convex approximation of the objective function such that we can solve the optimization problem in closed form. We note that these approximations are made only to design the content placement policy, and all assumptions are relaxed in the remainder of the paper, which includes the performance evaluation and the numerical evaluations in Sections IV and V, respectively.

Firstly, we neglect small-scale fading with regards to mmWave propagation since it causes only minor changes in received power when the transmitter is within the LoS region [11, 12]. Secondly, we consider the worst case of D2D interference where all the user requests are catered by D2D transmitters, and approximate the random interference using the average worst case interference. To overcome the singularity when computing the D2D interference averaged only over the large-scale fading, we use the bounded path loss model $g(r)=\min\left(1,r^{-\alpha_{d}}\right)$ to model the path loss from the interferers, similar to [13]. With these approximations, and by using Campbell’s theorem, the average interference can be written as

From (2), and by considering the maximum search discovery distance to initiate a D2D communication link to be $D_{R}$, we have

where

On the assumption that the transmitters having the $i$-th content stored in their cache form a PPP of intensity $\lambda_{u}\left(1-\rho\right)q_{i}$, and by using the distribution of the distance to the nearest MU in a PPP, we have

where $D_{i,c}=\min\left\{\hat{D}_{i},D_{L},D_{R}\right\}$.

Once the approximations are applied, it is straightforward to see that the optimization problem becomes convex. The Lagrangian can be written as

where $\mu$ is the non-negative Lagrangian multiplier. By applying Karush-Kuhn-Tucker (KKT) conditions, we have

and according to the first inequality constraint, the optimum $q_{i}$ should satisfy $q_{i}^{\star}=\min\left\{\max\left\{q_{i}(\mu^{\star}),0\right\},1\right\},$ and according to the second constraint, which is met with equality, gives us $\sum_{i=1}^{N}\min\left\{\max\left\{q_{i}(\mu^{\star}),0\right\},1\right\}=M_{d}.$ This can be used to find $\mu^{\star}$ through a simple root finding algorithm such as bisection search[3, 4].

In a hybrid (multi-tier) wireless network, it is important to associate users with the appropriate tiers to achieve traffic offloading while maintaining QoS constraints. In our system model, an MU may receive content via a LoS D2D link, a NLoS D2D link or a cellular link. Therefore, it is important to recognize the appropriate method of content delivery. From (4), one can see that an MU can successfully receive the $i$-th file from a D2D transmitter in the LoS region if the distance to the D2D transmitter is less than $D_{i,c}$. However, we have assumed the worst case D2D interference when obtaining $D_{i,c}$. This means, it may be possible to increase this threshold further without violating the rate constraints, which will facilitate more offloading. Since $\boldsymbol{q}$ is now defined, the probability of receiving the requested content from a D2D transmitter within the LoS region can be calculated as $\gamma=\sum_{i=1}^{N}\beta_{i}\left(1-q_{i}^{\star}\right)\left(1-\exp{\left(-\pi\lambda_{u}(1-\rho)q_{i}^{\star}D_{L}^{2}\right)}\right),$ where we have used the fact that the D2D mode initiates when the required content is not found in self-cache. We can use $\gamma$ to scale the worst-case average interference to have a tighter approximation of the interference from the active D2D transmitters. Hence, assuming that the interference is dominated by the D2D transmitters in the LoS region, we can obtain an updated distance threshold value that satisfies the QoS requirement as

By replacing subscript $L$ with $N$, we can obtain a similar distance threshold $\hat{D}_{i,N}$ for a transmitter in the NLoS region.

The proposed user association scheme can be summarized as follows. For the $i$-th file, the MU first checks its own cache. If not found, it checks with D2D transmitters who are closer than

for a D2D connection. If both actions fail, the MU connects to the nearest RRH through a cellular link. The process is summarized in Algorithm 1.

The rationale behind the distance thresholds can be explained as follows. When $\hat{D}_{i,L}<D_{L}$, it is straightforward that the threshold is $\hat{D}_{i,L}$, as any transmitter outside this (both LoS and NLoS) will not satisfy the rate constraints. When $\hat{D}_{i,L}>D_{L}$, we particularly focus on transmitters between $D_{L}$ and $\hat{D}_{i,L}$, who are NLoS according to the channel model. Whether these NLoS transmitters can transmit successfully or not will depend on the value of $\hat{D}_{i,N}$. To this end, if $\hat{D}_{i,N}<D_{L}$, none of the NLoS transmitters will be able to transmit successfully. Thus, we set the threshold as $D_{L}$. However, when $D_{L}\leq\hat{D}_{i,N}\leq\hat{D}_{i,L}$, all NLoS transmitters between $D_{L}$ and $\hat{D}_{i,N}$ will satisfy the rate constraints, thus we pick $\hat{D}_{i,N}$ as the threshold. Hence, overall, the distance threshold is given by $\max\left\{\hat{D}_{i,N},D_{L}\right\}$. The user association policy tries to make use of candidate NLoS transmitters as well, to further facilitate offloading. Note that we will have $\hat{D}_{i,L}>\hat{D}_{i,N}$ for all meaningful link lengths as $\alpha_{N}>\alpha_{L}$.