D2D-Aware Device Caching in mmWave-Cellular Networks

In this paper, we propose a novel policy for device caching that facilitates popular content exchange through high-rate device-to-device (D2D) millimeter-wave (mmWave) communication. The D2D-aware caching policy splits the cacheable content into two content groups and distributes it randomly to the user equipment devices, with the goal to enable D2D connections. By exploiting the high bandwidth availability and directionality of mmWaves, we ensure high rates for the D2D transmissions, while mitigating the co-channel interference that limits the throughput gains of the D2D communication in the sub-6-GHz bands. Furthermore, based on a stochastic-geometry modeling of the network topology, we analytically derive the offloading gain that is achieved by the proposed policy and the distribution of the content retrieval delay considering both half- and full-duplex modes for the D2D communication. The accuracy of the proposed analytical framework is validated through Monte Carlo simulations. In addition, for a wide range of a content popularity indicator, the results show that the proposed policy achieves higher offloading and lower content-retrieval delays than existing state-of-the-art approaches.


A. Background
O VER the last few years, the proliferation of mobile devices connected to the Internet, such as smartphones and tablets, has led to an unprecedented increase in wireless traffic that is expected to grow with an annual rate of 53% until 2020 [1].To satisfy this growth, a goal has been set for the 5th generation (5G) of mobile networks to improve the capacity of current networks by a factor of 1000 [2].While traditional approaches improve the area spectral efficiency of the network through, e.g., cell densification, transmission in the millimeter-wave (mmWave) band, and massive MIMO [2], studies have highlighted the repetitive pattern of user content requests [3], [4], suggesting more efficient ways to serve them.
With proactive caching, popular content is stored inside the network during off-peak hours (e.g., at night), so that it can be served locally during peak hours [5].Two methods are distinguished in the literature: i) edge caching [6] when the content is stored at helper nodes, such as small-cell base stations (BSs), and ii) device caching [7] when the content is stored at the user equipment devices (UEs).While edge caching alleviates the backhaul constraint of the small-cells by reducing the transmissions from the core network, device caching offloads the BSs by reducing the cellular transmissions, which increases the rates of the active cellular UEs and reduces the dynamic energy consumption of the BSs [8].The UEs also experience lower delays since the cached content is served instantaneously or through D2D communication from the local device caches.
The benefits of device caching in the offloading and the throughput performance have been demonstrated in [7], [9]- [13].In [7], the spectrum efficiency of a network of D2D UEs that cache and exchange content from a content library, is shown to scale linearly with the network size, provided that their content requests are sufficiently redundant.In [9], the previous result is extended to the UE throughput, which, allowing for a small probability of outage, is shown to scale proportionally with the UE cache size, provided that the aggregate memory of the UE caches is larger than the library size.To achieve these scaling laws, the impact of the D2D interference must be addressed by optimally adjusting the D2D transmission range to the UE density.In [10], a cluster-based approach is proposed to address the D2D interference where the D2D links inside a cluster are scheduled with time division multiple access (TDMA).The results corroborate the scaling of the spectrum efficiency that was derived in [7].In [11], a mathematical framework based on stochastic geometry is proposed to analyze the cluster-based TDMA scheme, and the trade-off between the cluster density, the local offloading from inside the cluster, and the global offloading from the whole network is demonstrated through extensive simulations.In [12], the system throughput is maximized by jointly optimizing the D2D link scheduling and the power allocation, while in [13], the offloading is maximized by an interference-aware reactive caching mechanism.
Although the aforementioned works show positive results for device caching, elaborate scheduling and power allocation schemes are required to mitigate the D2D interference, which limit the UE throughput and increase the system complexity.The N. Giatsoglou and E. Kartsakli are with IQUADRAT Informatica S.L., Spain (e-mail: ngiatsoglou@iquadrat.com, ellik@iquadrat.com)K. Ntontin is with the Department of Informatics and Telecommunications, University of Athens (UoA), Greece (e-mail: kntontin@di.uoa.gr). A. Antonopoulos and C. Verikoukis are with the Telecommunications Technological Centre of Catalonia (CTTC/CERCA), Spain (e-mail: aantonopoulos@cttc.es,cveri@cttc.es)high impact of the D2D interference is attributed to the omni-directional transmission patterns that are commonly employed in the sub-6 GHz bands.While directionality could naturally mitigate the D2D interference and alleviate the need for coordination, it requires a large number of antennas, whose size is not practical in the microwave bands.In contrast, the mmWave bands allow the employment of antenna arrays in hand-held UE devices due to their small wavelength.Combined with the availability of bandwidth and their prominence in future cellular communications [2], the mmWave bands are an attractive solution for D2D communication [14], [15].
The performance of the mmWave bands in wireless communication has been investigated in the literature for both outdoor and indoor environments, especially for the frequencies of 28 and 73 GHz that exhibit small atmospheric absorption [16], [17].According to these works, the coverage probability and the average rate can be enhanced with dense mmWave deployments when highly-directional antennas are employed at both the BSs and the UEs.MmWave systems further tend to be noise-limited due to the high bandwidth and the directionality of transmission [18].Recently, several works have conducted system-level analyses of mmWave networks with stochastic geometry [19]- [21], where the positions of the BSs and the UEs are modeled according to homogeneous Poisson point processes (PPPs) [22].This modeling has gained recognition due to its tractability [23].

B. Motivation and Contribution
Based on the above, it is seen that device caching can significantly enhance the offloading and the delay performance of the cellular network, especially when the UEs exchange cached content through D2D communication.On the other hand, the D2D interference poses a challenge in conventional microwave deployments due to the omni-directional pattern of transmission.While directionality is difficult to achieve in the sub-6 GHz band for hand-held devices, it is practical in the mmWave frequencies due to the small size of the antennas.The high availability of bandwidth and the prominence of the mmWave bands in future cellular networks have further motivated us to consider mmWave D2D communication in a device caching application.To the best of our knowledge, this combination has only been considered in [24], which adopts the cluster-based TDMA approach for the coordination of the D2D links and does not exploit the directionality of mmWaves to further increase the D2D frequency reuse.
In this context, the contributions of our work are summarized as follows: • We propose a novel D2D-aware caching (DAC) policy for device caching that facilitates the content exchange between the paired UEs through mmWave D2D communication and exploits the directionality of the mmWave band to increase the frequency reuse among the D2D links.In addition, we consider a half-duplex (HD) and a full-duplex (FD) version of the DAC policy when simultaneous requests occur inside a D2D pair.• We evaluate the performance of the proposed policy in terms of an offloading metric and the distribution of the content retrieval delay, based on a stochastic geometry framework for the positions of the BSs and the UEs.• We compare our proposal with the state-of-the-art most-popular content (MPC) policy through analysis and simulation, which shows that our policy improves the offloading metric and the 90-th percentile of the delay when the availability of paired UEs is sufficiently high and the content popularity distribution is not excessively peaked.The rest of the paper is organized as follows.In Section II, we present the proposed DAC and the state-of-the-art MPC policy.In Section III, we present the system model.In Section IV and Section V, we characterize the performance of the two policies in terms of the offloading factor and the content retrieval delay respectively.In Section VI, we compare analytically and through simulations the performance of the caching policies.Finally, Section VII concludes the paper.

II. BACKGROUND AND PROPOSED CACHING POLICY
In this section, based on a widely considered model for the UE requests, we present the state-of-the-art MPC policy and the proposed DAC policy.

A. UE Request Model
We assume that the UEs request content from a library of L files of equal size σ f ile [6] and that their requests follow the Zipf distribution.According to this model, after ranking the files with decreasing popularity, the probability q i of a UE requesting the i-th ranked file is given by

B. State-of-the-Art MPC Policy
In device caching, every UE retains a cache of K files, where K << L, so that when a cached content is requested, it is retrieved locally with negligible delay instead of a cellular transmission.This event is called a cache hit and its probability is called the hit probability, denoted by h and given by where C represents the cached contents of a UE, as determined by the caching policy.The MPC policy is a widely considered caching scheme [10], [27], [28] that stores the K most popular contents from the library of L files in every UE, resulting in the maximum hit probability, given by

C. Proposed DAC Policy
Although the MPC policy maximizes the hit probability, it precludes content exchange among the UEs since all of them store the same files.In contrast, a policy that diversifies the content among the UEs enables content exchange through D2D communication, resulting in higher offloading.Furthermore, thanks to the high D2D rate and the enhancement in the cellular rate due to the offloading, the considered policy may also improve the content retrieval delay, despite its lower hit probability compared with the MPC policy.
Based on this intuition, in the proposed DAC policy, the 2K most popular contents of the library of L files are partitioned into two non-overlapping groups of K files, denoted by groups A and B, and are distributed randomly to the UEs, which are characterized as UEs A and B respectively.When a UE A is close to a UE B, the network may pair them to enable content exchange through D2D communication.Denoting by h A and h B the hit probabilities of the two UE types, three possibilities exist when a paired UE A requests content: • the content is retrieved with probability h A through a cache hit from the local cache of UE A.
• the content is retrieved with probability h B through a D2D transmission from the cache of the peer UE B.
• the content is retrieved with probability 1 − h A − h B through a cellular transmission from the associated BS of UE A. The above cases are defined accordingly for a paired UE B. In Proposition 1 that follows, we formally prove that the probability of content exchange for both paired UEs are maximized with the content assignment of the DAC policy.Proof: See Appendix A. When the paired UEs store non-overlapping content, their hit probabilities coincide with their content exchange probabilities, i.e., e A = h B and e B = h A , hence, the DAC policy also maximizes h A and h B over all possible 2K partitions in the sense of Proposition 12 .The 2K most popular contents can be further partitioned in multiple ways, but one that equalizes h A and h B is chosen for fairness considerations.Although exact equalization is not possible due to the discrete nature of the Zipf distribution, the partition that minimizes the difference |h A − h B | can be found.Considering that this difference is expected to be negligible for sufficiently high values of K, h A and h B can be expressed as Finally, since two paired UEs may want to simultaneously exchange content, with probability h 2 dac , we consider two cases for the DAC policy: i) an HD version, denoted by HD-DAC, where the UEs exchange contents with two sequential HD transmissions, and ii) an FD version, denoted by FD-DAC, where the UEs exchange contents simultaneously with one FD transmission.Although the FD-DAC policy increases the frequency reuse of the D2D transmissions compared with the HD-DAC policy, it also introduces self-interference (SI) at the UEs that operate in FD mode [29] and increases the D2D co-channel interference.It therefore raises interesting questions regarding the impact of FD communication on the rate performance, especially in a mmWave system where the co-channel interference is naturally mitigated by the directionality.

III. SYSTEM MODEL
In this section, we present the network model, the mmWave channel model, the FD operation of the UEs, and the resource allocation scheme for the cellular and the D2D transmissions.

BS paired UE unpaired UE
Fig. 1: A network snapshot in a rectangle of dimensions 300 m x 300 m consisting of BSs (triangles) and UEs (circles).The paired UEs are shown connected with a solid line.

A. Network Model
We consider a cellular network where a fraction of the UEs are paired, as shown in the snapshot of Fig. 1.We assume that the BSs are distributed on the plane according to a homogeneous PPP Φ bs of intensity λ bs , while the UEs are distributed according to three homogeneous PPPs: the PPP Φ u with intensity λ u representing the unpaired UEs, and the PPPs Φ (1)  p and Φ (2) p with the same intensity λ p representing the paired UEs.We assume that Φ u is independent of Φ and Φ (2)  p are dependent due to the correlation introduced by the D2D pairings.Specifically, for every UE of Φ (1)  p , a D2D peer exists in Φ (2)  p that is uniformly distributed inside a disk of radius r max d2d , or, equivalently, at a distance r d2d and an angle φ d2d that are distributed according to the probability density functions (PDFs) f r d2d (r) and f φ d2d (φ), given by We assume that the D2D pairings arise when content exchange is possible, based on the cached files of the UEs.In the DAC policy, the BSs distribute the content groups A and B independently and with probability 1/2 to their associated UEs, and a fraction δ of them, which are located within distance r max d2d , are paired.Defining the aggregate process of the UEs Φ ue as and its intensity λ ue as3 the ratio δ of the paired UEs is given by Regarding the UE association, we assume that all the UEs are associated with their closest BS 4 , in which case the cells coincide with the Voronoi regions generated by Φ bs .Denoting by A cell the area of a typical Voronoi cell, the equivalent cell radius r cell is defined as and the association distance r of a UE to its closest BS is distributed according to the PDF f r (r), given by [23]

B. Channel Model
Regarding the channel model, we assume that the BSs and the UEs transmit with constant power, which is denoted by P bs and P ue respectively, and consider transmission at the mmWave carrier frequency f c with wavelength λc for both the cellular and the D2D communication through directional antennas employed at both the BSs and the UEs.The antenna gains are modeled according to the sectorized antenna model [30], which assumes constant mainlobe and sidelobe gains, given by where ∆θ is the antenna beamwidth, θ is the angle deviation from the antenna boresight, and i ∈ {bs, ue}.
Because the mmWave frequencies are subject to blockage effects, which become more pronounced as the transmission distance increases [16], the line-of-sight (LOS) state of the mmWave links is explicitly modeled.We consider the exponential model [16], [20], according to which a link of distance r is LOS with probability P los (r) or non-LOS (NLOS) with probability 1 − P los (r), where P los (r) is given by The parameter r los is the average LOS radius, which depends on the size and the density of the blockages [20].We further assume that the pathloss coefficients of a LOS and a NLOS link are a L and a N respectively, the LOS states of different links are independent, and the shadowing is incorporated into the LOS model [31].Finally, we assume Rayleigh fast fading where the channel power gain, denoted by η, is exponentially distributed, i.e., η ∼ E xp (1).

C. FD-Operation Principle
When a UE operates in FD mode, it receives SI by its own transmission.The SI signal comprises a direct LOS component, which can be substantially mitigated with proper SI cancellation techniques, and a reflected component, which is subject to multi-path fading.Due to the lack of measurements regarding the impact of the aforementioned components in FD mmWave transceivers, we model the SI channel as Rayleigh [32], justified by the reduction of the LOS component due to the directionality [33].Denoting by η si the power gain of the SI channel including the SI cancellation scheme, and by κ si its mean value, i.e, κ si = E[η si ], the power of the remaining SI signal, denoted by I si , is given by where η si ∼ Exp 1 κ si .

D. Resource Allocation and Scheduling
We focus on the downlink of the cellular system, which is isolated from the uplink through frequency division depluxing (FDD), since the uplink performance is not relevant for the considered caching scenario.We further consider an inband overlay scheme for D2D communication [34], where a fraction χ d2d of the overall downlink spectrum BW is reserved for the D2D traffic, justified by the availability of spectrum in the mmWave band.Regarding the scheduling scheme, we consider TDMA scheduling for the active cellular UEs, which is suited to mmWave communication [35], and uncoordinated D2D comnunication for the D2D UEs, relying on the directionality of the mmWave transmissions for the interference mitigation.

IV. OFFLOADING ANALYSIS
In this section, the DAC and the MPC policies 5 are compared in terms of their offloading performance, which can be quantified by the offloading factor F, defined as the ratio of the average offloaded requests (i.e., requests that are not served through cellular connections) to the total content requests in the network, i.e.
The offloading factor F is derived for each policy as follows: • In the MPC policy, a content request can be offloaded only through a cache hit, hence • In the DAC policy, in addition to a cache hit, a content request of a paired UE can be offloaded through D2D communication, hence Based on the above, the relative gain of the DAC over the MPC policy in terms of the offloading factor, denoted by F gain , is given by where h r atio represents the ratio of the hit probabilities of the two policies, given by We observe that F gain depends on the fraction of the paired UEs δ, the UE cache size K and the content popularity exponent ξ, but not the library size L. The impact of K and ξ on h r atio and, consequently, F gain is analytically investigated in Proposition 2 that follows.

Proposition 2:
The ratio of the hit probabilities of the two policies, h r atio , decreases monotonically with the popularity exponent ξ and the UE cache size K.In addition, the limit of h r atio with high values of K is equal to Proof: See Appendix B. Proposition 2 implies that h r atio attains its minimum value for ξ → ∞, and its maximum value for ξ = 0, hence This result shows that for δ = 1, representing the case of a fully paired network, the DAC policy always exhibits higher offloading than the MPC policy, while for δ = 0, representing the case of a fully unpaired network, the converse holds.For an intermediate value of δ, the offloading comparison depends on ξ and K and can be determined through (17).Finally, in Fig. 2, the convergence of h r atio to its limit value for high values of K is depicted.This limit is a lower bound to h r atio and serves as a useful approximation, provided that ξ is not close to 1 because, in this case, the convergence is slow.

V. PERFORMANCE ANALYSIS
In this section, the DAC and the MPC policy are characterized in terms of their rate and delay performance.The complementary CDF (CCDF) of the cellular rate is derived in Section V-A, the CCDF of the D2D rate is derived in Section V-B, and the CDF of the content retrieval delay is derived in Section V-C.

A. Cellular Rate Analysis
Justified by the stationarity of the PPP [22], we focus on a target UE inserted at the origin of the network and derive the experienced cellular rate, denoted by R cell , when an uncached content is requested.The rate R cell is determined by the cellular signal-to-interference-plus-noise ratio (SINR), denoted by SI N R cell , and the load of the associated cell, denoted by N cell , through the Shannon capacity formula, modified to include the effect of the TDMA scheduling as [36] Based on (21), the distribution of R cell is derived through the distribution of SI N R cell and N cell as where (i) follows by treating SI N R cell and N cell as independent random variables 6 .The distributions of N cell and SI N R cell are derived in the following sections.

1) Distribution of the cellular load:
The distribution of N cell depends on the cell size A cell and the point process of the active cellular UEs, denoted by Φ cell , as follows: • Regarding A cell , we note that due to the closest BS association scheme, the cells coincide with the Voronoi regions of Φ bs .
Although the area distribution of a typical 2-dimensional Voronoi cell is not known, it can be accurately approximated by [37] The cell of the target UE, however, is stochastically larger than a randomly chosen cell, since the target UE is more probable to associate with a larger cell, and its area distribution can be derived from ( 23) as [38] f • Regarding Φ cell , it results from the independent thinning [22] of Φ ue , considering the probability of a UE being cellular.This probability is denoted by c u and c p for the case of an unpaired and a paired UE respectively, and its values are summarized in Table I for the two considered policies.Although Φ cell is not PPP due to the correlation in the positions of the paired UEs, it can be treated as a PPP with density λ cell , given by This approximation is justified by the small cell radius of the mmWave BSs, which is expected to be comparable to the D2D distance of the paired UEs, so that their positions inside the cell are sufficiently randomized.Based on the above, N cell is approximated with the number of points of one PPP that fall inside the (target) Voronoi cell of another PPP, hence, it is distributed according to the gamma-Poisson mixture distribution [38], given by where µ = λ cell κλ bs + λ cell .

2) Distribution of the cellular SINR:
The cellular SINR is defined as where • S is a random variable representing the received signal power from the associated BS, which is located at a distance r from the target UE.Assuming that the BS and UE antennas are perfectly aligned, S is given by • I is a random variable representing the received interference power from the other-cell BSs of Φ bs .Assuming that the UE density is sufficiently high, all the BSs have a UE scheduled and I is given by where r x and G x are the length and the gain of the interfering link respectively.The latter comprises the antenna gains of the interfering BS and the target UE.
• N is the noise power at the receiver, given by where N 0 is the noise power density, F N is the noise figure of the receiver, and BW cell is the cellular bandwidth.Introducing the normalized quantities and applying ( 28), (29), and ( 30) to ( 27), the expression for SI N R cell is simplified to The CCDF of SI N R cell is subsequently derived as where (i) follows from the CCDF of the exponential random variable, and (ii) from the Laplace transform of Î, denoted by L Î (s).Considering that the impact of the interference is reduced due to the directionality of the mmWave transmissions, and that the impact of noise is increased due to the high bandwidth of the mmWave band, we assume that the system is noiselimited, which means that SI N R cell can be approximated by the cellular signal-to-noise ratio (SNR), denoted by SN R cell , as where f r (r) is given by (10).Although the integral in (34) cannot be solved in closed form, we present a tight approximation in Proposition 3 that follows.Proposition 3: The CCDF of the cellular SINR can be accurately approximated by where with Proof: See Appendix C.

B. D2D Rate Analysis
Similar to the cellular case, we focus on a paired target UE at the origin and derive the experienced D2D rate, denoted by R d2d , when a content is requested from the D2D peer.The following analysis applies only to the DAC policy, which is distinguished for the HD-DAC and the FD-DAC policy in the following sections.
1) Distribution of the D2D rate for the HD-DAC policy: The D2D rate for the HD-DAC policy, denoted by R hd d2d , is determined by the D2D SINR, denoted by SI N R hd d2d , through the Shannon capacity formula as where ψ denotes the HD factor, equal to 1/2 when both paired UEs want to transmit.Subsequently, the CCDF of R hd d2d is determined by the CCDF of SI N R hd d2d as Regarding SI N R hd d2d , it is defined as: • S is a random variable representing the received signal power from the D2D peer, located at a distance r d2d from the target UE.Assuming that the antennas of the two UEs are perfectly aligned, S is given by • I is a random variable representing the received interference power from all transmitting D2D UEs.Denoting by Φ hd d2d the point process of the D2D interferers in the HD-DAC policy, I is given by where r x and G x are the length and the gain of the interfering link respectively.The latter comprises the antenna gains of the interfering UE and the target UE.Since, in the HD-DAC policy, at most one UE from every D2D pair can transmit, the intensity of Φ hd d2d is given by • N is the noise power at the receiver, which depends on the D2D bandwidth BW d2d and is given by Introducing the normalized quantities and applying (41), (42), and ( 44) to (40), the expression for SI N R hd d2d is simplified to The CCDF of SI N R hd d2d is derived similarly to (33) as where L hd Î (s) is the Laplace transform of the interference in the HD-DAC policy, and the expectation over a and r d2d is computed through (12) and (5a) respectively.In contrast to the cellular case, the contribution of the interference in SI N R hd d2d is not negligible, even with directionality, due to the lower bandwidth expected for D2D communication, thus, L hd Î (s) is evaluated according to Proposition 4 that follows.
Proposition 4: The Laplace transform of the D2D interference in the HD-DAC policy, L hd Î (s), is given by where k denotes the order of the approximation, and the averaging is taken over the discrete random variable g with distribution

(50c)
Proof: See Appendix D. As k → ∞, more terms are added in the summation and the approximation becomes exact.Combining (48) with (47) into (39) yields the CCDF of R hd d2d where the final integration over r d2d can be evaluated numerically.2) Distribution of the D2D rate for the FD-DAC policy: As in the case of the HD-DAC policy, the D2D rate for the FD-DAC policy, denoted by R f d d2d , is determined by the D2D SINR, denoted by SI N R f d d2d , through the Shannon capacity formula as Subsequently, the CCDF of R where • S is a random variable representing the received signal power from the D2D peer, given by (41).
• I si is a random variable representing the SI power when the target UE operates in FD mode, given by (13).
• I is a random variable representing the received interference power from all transmitting D2D UEs, given by I = x 1 ∈Φ (1)   p λc 4π 2 where Φ (1)  p and Φ (2)  p are the point processes of the paired UEs, and ψ x denotes the indicator variable for the event that the UE at position x transmits.
• N is the noise power at the receiver, given by (44).Defining g, Ŝ and N as in (45) and introducing Î = x ∈Φ (1)   p ψ x g x η x r −a x x + y ∈Φ (2)   p ψ y g y η y r −a y y , d2d is derived similarly to (47) as where L f d Î (s) and L Î si (s) are the Laplace transforms of the external D2D interference and the SI respectively.Recalling that η si ∼ Exp 1 κ si , L Î si (s) is derived through the Laplace transform of the exponential random variable as where J 3 (s, a) and J 4 (s, a; k) are given by (49).Proof: See Appendix E. Combining (58) and ( 59) with (57) into (56), and applying the result to (52), yields two bounds for the CCDF of R f d d2d .

C. Delay Analysis
In this section, we characterize the delay performance of the MPC and the DAC policies through the content retrieval delay, denoted by D and defined as the delay experienced by a UE when retrieving a requested content from any available source.In the case of a cache hit, D is zero, while in the cellular and the D2D case, it coincides with the transmission delay of the content to the UE 7 .The CDFs of D for the MPC and the DAC policy are derived as follows: • For the MPC policy, the requested content is retrieved from the local cache with probability h mpc , or from the BS with probability 1 − h mpc , hence where the CCDF of R cell is given by ( 22).• For the DAC policy, the case of the paired and the unpaired UE must be distinguished, since the unpaired UE lacks the option for D2D communication.For a paired UE, the requested content is retrieved from the local cache with probability h dac , from the D2D peer with probability h dac , or from the BS with probability 1 − 2h dac , while, for an unpaired UE, the requested content is retrieved from the local cache with probability h dac , or from the BS with probability 1 − h dac , yielding where the CCDF of R cell is given by (22), and the CCDF of R d2d is given by ( 39) for the HD-DAC policy and by (52) for the FD-DAC policy.

VI. RESULTS
In this section, we compare the DAC and the MPC policies in terms of the offloading factor and the 90-th percentile of the content retrieval delay analytically and through Monte-Carlo simulations.Towards this goal, we present the simulation 7 Additional delays caused by the retrieval of the content through the core network are beyond the scope of this work.parameters in Section VI-A, the results for the offloading in Section VI-B, and the results for the content retrieval delay in Section VI-C.

A. Simulation Setup
For the simulation setup of the DAC and the MPC policy, we consider a mmWave system operating at the carrier frequency f c of 28 GHz, which is chosen due to its favorable propagation characteristics [39] and its approval for 5G deployment by the FCC [40].Regarding the network topology, we consider a high BS density λ bs corresponding to an average cell radius r cell of 50 m, which is consistent with the trends in the densification of future cellular networks and the average LOS radius r los of the mmWave frequencies in urban environments [18].The latter is chosen to be 30 m, based on the layout for the Chicago and the Manhattan area [18].Regarding the antenna model of the BSs and the UEs, the gains and the beamwidths are chosen according to typical values of the literature [41], [42], considering lower directionality for the UEs due to the smaller number of antennas that can be installed in the UE devices 8 .Regarding the caching model, we consider a library of 1000 files of size 100 MBs and three cases for the UE cache size: i) K = 50, ii) K = 100, and iii) K = 200, corresponding to the 5%, 10%, and 20% percentages of the library size respectively.The simulation parameters are summarized in Table II.

B. Offloading Comparison
As shown analytically in Section IV, the offloading gain of the DAC policy over the MPC policy F gain increases monotonically with the UE pairing probability δ, and decreases monotonically with the UE cache size K and the content popularity ξ, while it is not affected by the library size L. In this section, we validate the impact of δ, K, and ξ on F gain by means of simulations.
In Fig. 3, we plot F gain in terms of ξ for K = 100 and for δ = 0.5, 0.75, 1, corresponding to three different percentages of paired UEs inside the network.We observe that the simulation results validate the monotonic increase and decrease of F gain with δ and ξ respectively.The former is attributed to the higher availability of D2D pairs, which improves the opportunities for offloading in the DAC policy and does not affect the MPC policy, while the latter is attributed to the increasing gap in the hit probabilities of the two policies, as illustrated with the decrease of h r atio with ξ in Fig. 2  the above, we observe that the maximum offloading gain of the DAC over the MPC policy is equal to 2 and it is achieved when δ = 1 and ξ = 0, which corresponds to the case of a fully paired network and uniform content popularity respectively.For δ = 1, we further observe that the DAC policy outperforms the MPC policy regardless of the value of ξ, while for lower values of δ, the DAC policy is superior only when ξ < 0.63 for δ = 0.75, and when ξ < 0.97 for δ = 0.5.Based on these observations, we can generalize that for a network with δ < 1 the DAC policy offers higher offloading than the MPC policy for ξ up to a threshold value, which decreases with δ.
In Fig. 4, we plot the minimum δ that is required for the DAC policy to outperform the MPC policy, in terms of ξ and for K = 50, 100, 200.We can observe that the requirements for δ become more stringent with increasing ξ and K, which widen the gap between the hit probability of the two policies, but the impact of K is weaker than the impact of ξ, which is attributed to the low sensitivity of h r atio with K.This behavior can be explained with the bound of h r atio in (19), which represents the limit of h r atio when K → ∞.The minimum δ for K → ∞ is also depicted in Fig. 4, as well as the convergence of the other curves to it.When ξ < 0.5 or ξ > 1.5, the gap between the curves for finite K are close to the bound, because h r atio converges quickly to its limit value.In contrast, when 0.5 < ξ < 1.5, the gap between the curves and the bound is wider, because h r atio converges slowly to its limit value.Due to the slow convergence, for practical values of K, similar to ones considered in this work, h r atio is insensitive to K.
In Fig. 5, we plot F gain in terms of K for δ = 0.75 and ξ = 0.3, 0.6, 0.9.We observe that, as K increases, F gain decreases fast at low values of K and, afterwards, tends slowly to its limit value, calculated by applying (19) to (17).For ξ = 0.9, the gap between the curve and the limit is high because of the slow convergence of (19), validating that F gain is insensitive to K, provided that K is sufficiently high.In contrast, lower values of K favor the DAC over the MPC policy.

C. Delay Comparison
In this section, we validate the analytical expressions of Section V and compare the two caching policies in terms of the 90-th percentile of the content retrieval delay.
1) Performance of the HD-DAC policy: In Fig. 6a, we illustrate for the HD-DAC policy the CCDFs of the cellular rate R cell and the D2D rate R hd d2d , derived through analysis and simulations, for δ = 1, K = 200, and ξ = 0.4.A second order approximation (k = 2) was sufficient for L hd Î (s) in (49).We observe that R hd d2d is stochastically larger than R cell for rates below 5 Gbps, yielding an improvement of 1.52 Gbps in the 50-th percentile, which means that the D2D UEs experience a rate that is higher than the cellular rate by at least 1.5 Gbps for the 50% of the time.This improvement creates strong incentives for the UEs to cooperate and is attributed to the small D2D distance between the D2D UEs and the reduction of R cell due to the TDMA scheduling.In contrast, the cellular UEs are more probable to experience rates above 5 Gbps, owing to the high difference between the cellular and the D2D bandwidth.Specifically, it is possible for a cellular UE to associate with a BS with low or even zero load and fully exploit the high cellular bandwidth, while a D2D UE is always limited by the 20% fraction of bandwidth that is reserved for D2D communication.
In Fig. 6b, we illustrate for the HD-DAC policy the CDFs of the cellular delay D cell , the D2D delay D hd d2d , and the total delay D that is experienced by a UE without conditioning on its content request.We observe that D hd d2d is significantly lower than D cell , which is consistent with Fig. 6a, while the curve of D is initiated at the value 0.286 due to the zero delay of cache-hits.We further observe that the simulations for D cell do not match the theoretical curve as tightly as in the case of R cell , which is attributed to the reciprocal relation between the rate and the delay that magnifies the approximation error for the delay.Nevertheless, the match is improved in the case of the total delay due to the contribution of the D2D delay, which is approximated more accurately.
2) Performance of the FD-DAC policy: In Fig. 7, we illustrate for the FD-DAC policy the rate and the delay distribution for δ = 1, K = 200, ξ = 0.4, and a second order approximation for L f d Î (s).As seen in Fig. 7a, both bounds for the CCDF of R f d d2d are very close to the simulation curve, hence, only the upper bound is considered for D f d d2d in Fig. 7b.Compared with the HD-DAC policy, the FD-DAC policy yields a minor improvement in the 50-th percentile of R f d d2d , which is higher than the percentile of R cell by 1.62 Gbps, that is attributed to the absence of the HD factor that decreases R hd d2d by half.Nevertheless, the probability of bidirectional content exchange, equal to 0.08 for the considered parameters, is small to significantly influence the results.The same observation holds for the CDFs of the content retrieval delay.
Motivated by the previous observation, in Fig. 8, we illustrate for the FD-DAC policy the rate and the content retrieval delay for ξ = 1.0, in which case h dac = 0.44, resulting in a non-negligible probability for bidirectional content exchange.As seen in Fig. 8a, R f d d2d is reduced due to the higher D2D interference, while R cell is significantly improved due to the higher offloading.Consequently, R f d d2d is higher than R cell , and the total delay is determined by the cache hits and the curve of the cellular delay, as seen in Fig. 8b.Since the FD-DAC and the HD-DAC policy are distinguished when h dac is high, in which case the performance is not influenced by the D2D communication, only the HD-DAC policy is considered in the delay comparison with the MPC policy.
3) Delay Comparison between the MPC and the HD-DAC policy: The MPC policy maximizes the probability of zero delay through cache hits, but the HD-DAC policy may still offer lower delays due to the improvement in the transmission rates.Based on this observation, the two policies are compared in terms of the 90-th percentile of the content retrieval delay, which is an important QoS metric, representing the maximum delay that is experienced by the target UE for 90% of the time.
In Fig. 9, we plot the delay percentiles for the HD-DAC and the MPC policy as a function of the popularity exponent ξ for the cases: a) K = 50, b) K = 100, and c) K = 200.As a general observation, the 90-th percentile of delay for both policies  decreases with higher values of K, since both the hit probability and, in the case of the HD-DAC policy, the probability of D2D content exchange, are higher.The delay percentile of the HD-DAC policy also decreases with δ, since the opportunities for D2D communication are improved with a larger number of D2D pairs, while the MPC policy is not affected.In Fig. 9a, the performance is comparable between the HD-DAC policy with δ = 1.0, and the MPC policy, for ξ < 1.0.In Fig. 9b, the performance is comparable between the HD-DAC policy with δ = 0.75, and the MPC policy, for ξ < 0.8.In Fig. 9c, the performance is comparable between the HD-DAC policy with δ = 0.5, and the MPC policy, for ξ < 0.4.Based on these these observations, we conclude that, for low values of ξ, the HD-DAC policy is favored by larger UE caches and requires fewer D2D pairings to outperform the MPC policy, while for high values of ξ, the MPC policy is favored by larger UE caches due to the wide gap in the hit probabilities of the two policies, which justifies the superior performance of the MPC policy in these cases.

VII. CONCLUSION
In this work, we have proposed a novel policy for device caching that combines the emerging technologies of D2D and mmWave communication to enhance the offloading and the delay performance of the cellular network.Based on a stochasticgeometry modeling, we have derived the offloading gain and the distribution of the content retrieval delay for the proposed DAC policy and the state-of-the-art MPC policy, which does not exploit content exchange among the UEs.By comparing analytically and through Monte-Carlo simulations the two policies, we have shown that the proposed policy exhibits superior offloading and delay performance when the availability of pairs in the system is sufficiently high and the popularity distribution of the requested content is not excessively skewed.In addition, motivated by the prospect of bidirectional content exchange, we presented an FD version of the proposed policy, which exhibits a small improvement over the HD version in terms of the delay performance, due to the low probability of bidirectional content exchange.According to the simulation results, increasing this probability does not yield a proportional improvement in performance due to the resulting prevalence of the cellular rate over the D2D rate, attributed to offloading.As future work, we plan to generalize the proposed caching scheme to a policy that divides the cacheable content to an arbitrary number of groups and study the impact on performance.

ACKNOWLEDGEMENTS
The authors would like to cordially thank the editor and the anonymous reviewers for their constructive suggestions that helped to improve the quality of this work.

APPENDIX A PROOF OF PROPOSITION 1
Denoting by A and B the users of a D2D pair and by C A and C B their caches, the hit probabilities h A and h B of the two users can be expressed as and the exchange probabilities e A and e B as where • signifies the complement in terms of the set of the library contents.
To prove that e A and e B are maximized when C A and C B form a partition of the 2K most popular contents, we need to show that i) the optimal C A and C B do not overlap, i.e., C A ∩ C B = ∅, and ii) the optimal C A and C B cover the 2K most popular contents, i.e., C A ∪ C B = {i ∈ N : 1 ≤ i ≤ 2K }.We prove both i) and ii) by contradiction.Regarding i), if the optimal C A and C B contained a common content, say c ∈ C A ∩ C B , we could simultaneously increase e A and e B by replacing c in C A with a content from C A ∩ C B .Therefore, C A and C B must not overlap.Regarding ii), if C A contained a content c that did not belong in the 2K most popular, then we could replace c with an uncached content from {i ∈ N : 1 ≤ i ≤ 2K }, which would increase e B and h A , while leaving e A and h B unaffected.Therefore, if C A and C B form a partition of {i ∈ N : 1 ≤ i ≤ 2K }, neither e A , e B nor h A , h B can be increased simultaneously with a different partition.

APPENDIX B PROOF OF PROPOSITION 2
The ratio of the hit probabilities of the two policies, h r atio , is given by ( 18), which we repeat here for easier reference: To prove that h r atio decreases monotonically with ξ, we differentiate h r atio in terms of ξ as where (a) follows because the first sum is eliminated due to symmetry, and (b) follows because ln(i) > ln( j) for the remaining indexes.Since the derivative of h r atio in terms of ξ is negative, h r atio decreases monotonically with ξ.
To prove that h r atio decreases monotonically with K, we need to show that h r atio (K + 1) < h r atio (K).Introducing the notation S K K i=1 i −ξ for clarity, the aforementioned inequality is transformed as Manipulating the inequality yields Splitting the odd and even indexes in S 2K as the inequality is further simplified to Comparing the sums term-by-term, the inequality holds provided that Since the final inequality is true and all the steps of the derivation were reversible, the initial inequality is also proven.
To calculate the limit of h r atio for high values of K, we distinguish the cases ξ > 1 and ξ ≤ 1.
• For ξ ≤ 1, the sums in (64) diverge as K → ∞, nevertheless, the limit can be calculated through an asymptotic expression of the sums, based on the Euler-McLaurin summation formula [44].According to this formula, the discrete sum can be approximated with a continuous integral as where ǫ(ξ) represents the asymptotic error of the approximation, also known as the generalized Euler constant 9 .Applying (72) to (64) yields Finally, the limits in ( 71) and ( 73) can be combined in one compact expression as the CCDF of the cellular SINR is expressed as Since the integrals in (75) and (76) cannot be evaluated in closed form, they are approximated as follows.
• Regarding J 1 , the exponential term e yielding where r 1 is chosen so that the approximated value of J 1 is exact for T = 0, i.e., where r 2 is chosen so that the approximated value of J 2 is exact for T = 0, i.e., where (i) follows from the probability generating functional (PGFL) of the PPP [22], and (ii) from the Laplace transform of the exponential random variable.Defining where r 4 is chosen so that the approximated value of J 4 is exact for s = 0, i.e., d2d , (99) cannot be evaluated in closed form, nevertheless, it can be approximated with the following bounds [45].

Proposition 1 :
Denoting by C A and C B the caches of UE A and B inside a D2D pair, and by e A and e B their probabilities of content exchange, e A and e B are maximized when C A and C B form a non-overlapping partition of the 2K most popular contents, i.e., C A ∪ C B = {i ∈ N : 1 ≤ i ≤ 2K } and C A ∩ C B = ∅, in the sense that no other content assignment to C A and C B can simultaneously increase e A and e B .

Fig. 2 :
Fig.2: The hit probability ratio h r atio in terms of the UE cache size K and the popularity exponent ξ.

L f d Î
(s) is derived in Proposition 5 that follows.Proposition 5: The Laplace transform of the D2D interference in the FD-DAC policy, L f d Î (s), can be bounded as

Fig. 3 :
Fig.3: The offloading gain of the DAC policy over the MPC policy, F gain , in terms of the content popularity exponent ξ.

Fig. 4 :Fig. 5 :
Fig.4: The minimum fraction of pairs (δ) required for the DAC policy to achieve higher offloading than the MPC policy in terms of the content popularity exponent ξ.

Fig. 7 :
Fig. 7: Rate and delay performance of the FD-DAC policy for δ = 1, K = 200, and ξ = 0.4 (Ana.stands for Analysis, Sim. for Simulation, UB for Upper Bound, and LB for Lower Bound).

Fig. 8 :
Fig. 8: Rate and delay performance of the FD-DAC policy for δ = 1, K = 200, and ξ = 1.0 (Ana.stands for Analysis, Sim. for Simulation, UB for Upper Bound, and LB for Lower Bound).

Fig. 9 :
Fig. 9: The 90-th percentile of the content retrieval delay D in terms of ξ for a) K = 50, b) K = 100, and c) K = 200.
74) APPENDIX C DERIVATION OF THE CCDF OF THE CELLULAR SINR Defining J 1 (T, a) e − NT r a dr,

J 4 ( 4 krdr ⇒ r 4 == h dac λ p = δ 2 h
(k + 1)(k + 2)r los .(94)Applying (94) to (92), we also observe that the approximation becomes exact as k grows, since lim (91) and (93) into (90) and the result to (87) yields the final result.Please note that the remaining expectation over g is trivial, since g is a discrete random variable with the distribution 2∆θ ue (2π−∆θ ue ) OF THE LAPLACE TRANSFORM OF THE FD INTERFERENCE The D2D interference in the FD-DAC policy is given byÎ = x ∈Φ f d(1) d2d g x η x r −a x x dac λ ue ,(98)but dependent to each other due to the D2D pairings.Based on (97), the Laplace tranform of the D2D interference in the FD-DAC policy, denoted by Lf d Î (s), is expressed as L f d Î (s) = E e − Î s = E Π x ∈Φ f d(1) d2d e −g x η x r −ax x s • Π y ∈Φ f d(2) d2d e −g y η y r −ay y

TABLE II :
SIMULATION PARAMETERS