With the rapid development of the traditional cellular networks and novel Internet-enabled applications, such as multimedia sensors  and electric vehicles [3, 4], the total throughput of mobile networks in 2020 is expected to become 1000-fold larger than that in 2010 . To support the explosive data traffic of future fifth-generation (5G) cellular networks, numerous researches [6, 7, 8, 9] have paid attention to an innovative framework that densifies the traditional networks with massive small base stations (BSs). However, the improvement of these heterogeneous networks (HetNets) is mainly restricted to the capacity of the backhauls. Although the high-speed optical fiber provides a theoretical solution, in practice, connecting the core server to all BSs with fibers is arduous and costly . Moreover, microwave backhauls may pessimistically weaken the throughput gain fetched by the network densification . A recent study  has shown that only 5%-10% of multimedia contents are required by the majority of user equipments (UEs). Additionally, the storage capacity of cache-enabled devices expands promptly at a fairly low cost. Stimulated by such facts, equipping caches at all BSs for storing the most popular contents becomes a promising method to offload the data traffic rather than continuing increasing the networks’ density [13, 14].
Lately, the aforementioned cache-enabled HetNets have been studied in various papers. Authors in  analyzed the energy efficiency and throughput of cellular networks with caches, but they only considered the small cell networks (SCNs) and BSs were modeled following a regular hexagonal grid. Since stochastic geometry is a useful tool to acquire the networks’ randomness , modeling a tier of BSs in SCNs or HetNets with a homogeneous Poisson Point Process (HPPP) is more accurate than the traditional hexagonal scenario [17, 18, 19]. Under this condition, the throughput of multi-tier cache-enabled HetNets was discussed in , where BSs were modeled as mutually independent PPPs. However, the high-capacity backhauls were employed at all nodes including the macro BSs and relays, which is uneconomical in reality. Then the limitation was relaxed by assuming that only macro BSs connected the core networks through backhauls, while BSs in small cells cached the contents via wireless broadcasting . Unfortunately, the further analysis on the impact of backhaul capacity was omitted, which is the key parameter when comparing with the conventional HetNets.
In addition to the network densification, another key capacity-increasing technology for boosting the throughput of future cellular networks is exploiting new spectrum bands, such as millimeter wave (mmWave) [21, 22, 23]. Comparing with the traditional sub-6 GHz networks in 4G, two distinctive characteristics of mmWave are small wavelength and the sensitivity to blockages . Thanks to the short wavelength, steerable antennas with huge scales can be employed at devices to enhance the directional array gain . On the other side, the sensitivity gives rise to severe penetration loss for mmWave signals when passing through building exteriors . Therefore, the path loss law of non-line-of-sight (NLOS) links is substantially different from that of line-of-sight (LOS) links in mmWave communications [25, 27], and it is unrealistic to expect an outdoor-to-indoor coverage from macro mmWave BSs. To compensate the blockage-dependent loss, an ingenious hybrid network is created, where mmWave transmitters contribute to the ultra-fast data rate in short-range small cells, and sub-6 GHz BSs provide the universal coverage .
There exist numerous studies concentrating on the performance of mmWave communications. As discussed in cache-enabled HetNets, stochastic geometry has also been widely utilized in mmWave networks, where the locations of transceivers were modeled following PPPs [24, 29]. With the aid of such structure, the primary article , which employed a simplified flat-top antenna pattern, introduced a stochastic blockage model to represent the actual mmWave communication environment. In fact, this simplified model has limited ability to exactly depict several parameters of a practical antenna, such as beamwidth, front-back ratio and nulls . Therefore, the authors in  proposed an actual antenna pattern for traditional mmWave networks. Considering the hybrid HetNets, a tractable framework with sub-6 GHz macro cells and mmWave small cells was analyzed under two user association strategies in . However, the Rayleigh fading assumption is not accurate for mmWave communications because of the poor scattering feature . Recent works [24, 32] presented a realistic channel model with Nakagami fading to improve the theoretical accuracy.
I-a Motivation and Contribution
Although HetNets with caches have been analyzed under a variety of scenarios with traditional sub-6 GHz networks, there is still lack of articles on a hybrid system with mmWave small cells. Since mmWave has a large range of available bandwidth [33, 34] and it is able to provide fast data rate in short-distance networks , adopting mmWave into a dense pico tier of HetNets is a promising way to increase the throughput of 5G cellular networks. Additionally, utilizing low-cost caches at all macro and pico BSs is capable of offloading the backhaul traffic efficiently and hence providing further improvement regarding the quality of service. The other benefit of such hybrid HetNets is no mutual interferences because each tier uses totally distinctive carrier frequency. These advantages motivate us to create this paper.
In contrast to , we introduce fiber-connections between macro BSs and the multimedia server to evaluate the impact of backhaul capacity in cache-enabled HetNets. Then, due to the employment of mmWave, the propagation environment and antenna beamforming pattern in the small cells are replaced by Nakagami fading and actual antenna arrays, respectively. Load balancing problems in mmWave-enabled HetNets have been studied in [36, 37]. However, the optimal solutions are based on a simplified framework which ignores the randomness of network nodes. In order to enhance the generality, we use the stochastic geometry to model the locations of transceivers. Regarding the user association scheme in hybrid HetNets (mmWave plus sub-6 GHz), authors in  assumed that the typical user is served by the BS which offers the minimum path loss. In most instances, even mmWave transmissions have more severe path loss than sub-6 GHz scenarios, they are still able to provide faster data rate due to the huge transmit bandwidth. As a result, the maximum data rate is also an important criterion for user association, in addition to the minimum path loss. We consider both criteria in this paper. The main contributions are summarized as below:
The success probability and area spectral efficiency (ASE) of our hybrid cache-enabled HetNets are discussed under two user association strategies: 1) Maximum Received Power (Max-RP), where the requesting user chooses the macro BS or pico BS offering the maximum average biased received power111The motivation for considering the average received power is that the network designer is interested in the average metric at the requested user for the universal coordination . from all BSs containing the requested file; and 2) Maximum Rate (Max-Rate), where the typical UE selects the BS, which provides the highest biased transmitting rate, from all BSs caching the desired content.
We analyze the cache-related coverage performance of traditional sub-6 GHz macro cells and mmWave small cells with the actual antenna pattern. Furthermore, closed-form coverage probability equations for the mmWave tier and an interference-limited case with sub-6 GHz are derived. Our analytical expressions can be directly applied into other mmWave or sub-6 GHz scenarios with negligible changes.
Different association probabilities for two considered schemes are introduced to calculate final algorithms of success probabilities. We theoretically demonstrate that the success probability of Max-RP has a positive correlation with the serving tier’s biased transmit power. However, the success probability of Max-Rate scheme is independent of the two tier’s transmit power and the density of macro BSs. Finally, expressions of ASEs are deduced for analyzing.
We conclude that: 1) our cache-enabled hybrid HetNets outperform the traditional HetNets where macro BSs have no caching capacity, and Max-Rate achieves a better performance than Max-RP in terms of the success probability and ASE; 2) the proposed network is an interference-limited system due to the nature of sub-6 GHz networks and the high density of mmWave small cells; 3) there is an optimum value of rate requirement for obtaining the maximum ASE; and 4) 73 GHz is the best mmWave carrier frequency for two user association strategies because of possessing the largest antenna scale.
We organize the rest of our treatise as follows: In Section II, we present the system model where two-tier BSs and users in the proposed cache-enabled hybrid HetNets are modeled as three independent HPPPs. In Section III, the expressions of signal-to-interference-plus-noise-ratio (SINR) coverage probabilities for two distinctive tiers are derived with the aid of the random content placement scheme. In Section IV, we discuss two different user association strategies, based on which the algorithms of success probabilities and ASEs are deduced. In Section V, the simulation and numerical results are presented for corroborating the analytical conclusions and providing further analysis, respectively. In Section VI, we draw our conclusions.
Ii System Model
Ii-a Network Architecture
In this paper, we present a cache-enabled hybrid HetNet with two-tier BSs as shown in Fig. 1. Macro BSs, pico BSs, and UEs are distributed following three independent HPPPs with density , , and , denoted by , , and , respectively. A randomly selected typical UE
is fixed at the origin such that the probability density function (PDF) of the distance from the typical UE to its nearest BS in the-th tier is given by , where . Apparently, the number of pico BSs in real HetNets is much more than that of macro BSs and thus we consider . In order to compare the performance of the proposed network with traditional HetNets, we provide a server to supply the less-popular contents. Note that deploying wired connections between the core server and all pico BSs is wasted and arduous. We assume the server only connects to each macro BS through a high-capacity wired backhaul.
In order to avoid inter-tier interference, hybrid carrier frequencies are employed in our system. When communicating with UEs, the macro BSs adopt sub-6 GHz, while the pico BSs utilize mmWave. Note that various multiple-access techniques enable the macro BSs to serve multiple users in one time slot. We assume the quantity of UEs is large enough, namely , to ensure all BSs are active when the typical UE is served.
Ii-B Blockage Model
In the first tier, when the communication distance is , the path loss law for sub-6 GHz signals is same as that in traditional cellular networks, which is given by
where is the path loss exponent and is the intercept for the macro tier.
In the second tier, the effect of blockages is important due to the employment of mmWave. Therefore, we adopt a LOS ball model222In most urban scenarios, the considered LOS ball model has negligible deviation with the more accurate multi-slope LOS probability scheme, especially when the altitude of pico BSs is less than the average height of obstacles . [24, 41], as shown in Fig. 2(a), to depict the blockage environment. The radius for the LOS ball represents the departure from nearby obstacles. The probability of LOS links is one inside the ball and zero outside that area. A recent study  has advocated that when the density of BSs is large, this blockage pattern has a negligible difference with the commonly used random shape theory model . Note that we consider a dense pico tier. The simplified LOS ball model is capable of providing enough analytical accuracy. Regarding NLOS links, various articles [24, 44] have demonstrated that the impact of NLOS signals can be ignored in mmWave networks due to their severe path loss. As a result, only LOS signals are analyzed in this paper333Regarding the interference via NLOS transmissions, it can still be ignored since the high density of pico BSs enhances not only the interference from NLOS links but also the counterpart from LOS links [24, 32, 44, 45].. Accordingly, the path loss law in the second tier can be expressed as follows
where and are the path loss exponent and the intercept of LOS links, respectively. is the unit step function, which is defined as
Ii-C Cache-enabled Content Access Protocol
In this paper, we assume that a static multimedia content catalog containing files is stored at the server and all files have the same size with bits. Each macro BS and pico BS have restricted storage capacities with and bits, respectively, which means the maximum storage capacity of the proposed HetNet obeys . High-speed fiber backhauls are employed for connecting the core server to macro BSs like traditional HetNets. The backhaul capacity is . When the data traffic load becomes low, the contents are broadcast to all BSs following the random content placement scheme as discussed in  until the storage is fully occupied. On this basis, we introduce the requesting probability, content placement, association strategy, and access protocol in the following part.
Requesting Probability: We apply the Zipf distribution to represent the probability of content being requested [47, 12, 48]. If files are indexed according to the popularity, namely the first and the -th files are the most and the least popular contents, respectively, the requesting probability of the -th file is given by
where is an integer and
is the skew parameter of the popularity distribution.
Content Placement: We assume that the first files are cached in the considered HetNet. The probability that the -th ranked file is cached at the -th tier is denoted by . Based on the optimal solution presented in , such probability obeys:
An example in the macro tier is illustrated in Fig. 3. Note that the pico tier has the same content placement strategy but different storage capacities.
Association Strategy: We consider association strategies depending on both cached files and channel conditions, which is essentially different with traditional HetNets without caches . In the -th tier, the locations of BSs containing the -th file form a set . When the typical UE requests this file, two association strategies are used in the considered HetNet: 1) Max-RP, where the typical UE communicates with the BS at that provides the maximum biased average received power; and 2) Max-Rate, where the typical UE connects to the BS at that provides the maximum biased received data rate.
Access Protocol: If the desired -th file is cached at the two-tier HetNet, which obeys , the typical UE communicates with the macro or pico BSs following the aforementioned association strategies. However, if the demanded content is absent from the BSs due to limited storage capacities, namely , the typical UE turns to the core server via the nearest macro BS.
Ii-D Directional Beamforming
In the -th tier, we employ antenna arrays composed of elements at all cache-enabled BSs and the transmit power is assumed to be a constant . Due to the small wavelength of mmWave signals, the uniform linear array (ULA) pattern can be deployed at all pico BSs . Directional antenna arrays deployed at the pico BSs supply substantial beamforming gains to compensate the path loss. However, we only consider an omnidirectional antenna model at macro BSs and UEs for tractability of the analysis . When the typical UE requests the -th file from the -th tier, the received signal can be expressed as follows
where the typical UE is receiving the message from the corresponding BS at . The locations of all interfering BSs forms a set and each element in such set is donated by the variable
. The channel vector from the BS to the typical UE and the beamforming vector of the BS in the-th tier are denoted by and , respectively. represents the thermal noise.
Combining with the aforementioned assumptions, the product of the fading gain and beamforming gain of the BS located at in the -th tier is shown as below 
where is the small fading term. is the spatial angle of departure (AoD) from the interfering BS to the typical UE, and is the spatial AoD between the BS at location and its corresponding receiver, see Fig. 2(b). is the array gain function. More specifically, the actual array pattern is employed at pico BSs so that , where. and are the antenna spacing and wavelength, respectively . On the other hand, the array gain function for macro BSs is due to the omnidirectional antenna pattern.
Since sophisticated beam training protocols  can be used at BSs to acquire the location information of the typical UE, we assume the corresponding BS provides the maximum directivity gain by aligning the antenna beam towards the typical UE.
Ii-E Propagation Model
Ii-E1 Channel Model
In the proposed HetNet, since the corresponding BS is interfered by other active BSs located in the same tier, the received SINR at the typical UE for requesting the -th file from the -th tier can be expressed as follows
where and . is the set of locations of BSs that do not store the -th file and it obeys , . follows independent Nakagami fading due to utilizing mmWave and the parameter of Nakagami fading is considered to be a positive integer for simplifying the analysis . Therefore, is a normalized Gamma random variable. On the other side, we assume a Rayleigh fading model for the macro tier so that the fading parameter .
Ii-E2 Association Criteria
When the typical UE requests the -th file. For Max-RP, the biased average received power is defined as follows 
where is a bias factor that aims to balance the load between two tiers under the Max-RP scheme . Then, we consider the biased received data rate for Max-Rate, which is given by
where is the bandwidth per resource block. is another bias factor to control the data traffic under Max-Rate.
When the typical UE requests a file from the core server, the throughput can be limited by both the macro-tier conditions and the backhaul capacity . Therefore, the instantaneous downlink data rate from the core server can be expressed as
Iii Cache-Related SINR Coverage Probability
Cache-related SINR coverage probability is the proportion of the received SINR that surpasses the requested SINR threshold depending on the content distributions. We separately discuss the cache-related SINR coverage probabilities for two tiers in this section, which is the theoretical basis for analyzing the final performance considering the association strategies.
Based on the content placement, can be regarded as an independent non-HPPP with the density . Therefore, the PDF of the distance between the typical UE and its nearest BS that contains the -th file is given by
In the following part, we first analyze the cache-related SINR coverage performance in the pico tier and then we consider the macro tier.
Iii-a SINR Coverage Analysis in The Second Tier
If the typical UE is associated with the pico tier, the corresponding cache-related coverage probability can be derived with the aid of Laplace Transform of Interference.
Iii-A1 Laplace Transform of Interference
Since the path loss exponent of LOS links is no less than 2 for practical scenarios, we divide the analysis on the Laplace transform of interference into two conditions ( and ) in order to achieve closed-form expressions.
When requesting the -th content, under the condition , the Laplace transform of interference with the pre-decided SINR threshold in the pico tier is as follows
are Gauss-Chebyshev nodes over , and is a trade-off parameter between the accuracy and complexity [51, 16]. When , the equality is established. and denotes Gauss hypergeometric function. and is the gamma function.
Numerous actual channel measures [52, 53] have indicated that the LOS path loss exponent is for various carrier frequencies, e.g. GHz, GHz and GHz. Under such condition , the equation (1) is changed to
See Appendix A. ∎
The analytical expressions in Lemma 1 show that is independent of the transmit power and the intercept .
Iii-A2 Cache-Related Coverage Probability
Considering the biased received power, we define the cache-related coverage probability , when requesting the -th file from the second tier, as follows
where represents the probability function. With the aid of Lemma 1, the closed-form coverage probability of pico tier is calculated as below.
When the typical UE requests the -th ranked content from the pico tier, the cache-related SINR coverage probability in this dense mmWave network can be expressed as follows
When , in Theorem 1 represents the coverage probability of this dense mmWave network without caching ability.
Due to the long communicating distance and high path loss, traditional cellular networks with mmWave are noise-limited . However, recent articles [29, 32] have shown that with a high BS density, such systems become interference-limited. Under this condition, we present the first assumption below and the corroboration is provided in Section V.
The dense mmWave network in the pico tier is assumed to be an interference-limited system, .
Under Assumption 1, the corresponding cache-related SINR coverage probability in the dense mmWave network can be simplified as follows
By deleting the part in Theorem 1, which represents the thermal noise effect on the coverage performance, we obtain the equations for this corollary. ∎
Since the association probability of Max-Rate is deduced on the basis of the derivative of the coverage probability , we deduce the derivative of described in Lemma 1, based on which the PDF of the coverage probability can be figured out.
As in is the transform variable of in our calculation, we introduce into the equations to make the notation straightforward. When , the derivative of is given by
and when , such derivative can be expressed as
With the aid of Lemma 2, when the typical UE requires the -th ranked content, the PDF of cache-related SINR coverage probability for the second tier is as follows
See Appendix B. ∎
Iii-B SINR Coverage Analysis in The First Tier
In the macro tier, we utilize the Rayleigh fading channel for sub-6 GHz signals. The exact expression of cache-related SINR coverage probability can be achieved in this part.
Iii-B1 Cache-Related Coverage Probability
As discussed in the previous part, we keep the SINR threshold in the sub-6 GHz tier. With the similar analysis in the pico tier, the cache-related coverage probability in the second tier for requiring the -th ranked content can be expressed as follows
Due to the Rayleigh fading channel, it is effortless to derive the Laplace transform of interference for the first tier. Therefore, we directly provide the cache-related coverage probability in the following paragraph.
When the typical UE requests the -th ranked file from the macro tier, the exact cache-related SINR coverage probability is given by
Special Case 1: We assume , as it is valid for most sub-6 GHz networks .
Under Special Case 1, the closed-form cache-related coverage probability in the first tier can be expressed as follows
where , , and is the complementary error function.
By removing the noise part from Theorem 2, the SIR coverage probability can be deduced with the fact that . ∎
is a monotonic increasing function with . Moreover, is independent of , , , and . Due to the negligible difference with the simulation shown in Section V, we use this closed-form equation as a proxy of the exact expression in the remainder of this paper.
With the aid of the closed-form expression in Corollary 4, we are able to figure out the closed-form derivative of cache-related coverage probability for the first tier effortlessly.
Iv Success Probability and Area Spectral Efficiency Analysis
From the perspective of customers, the success probability is an important parameter to appraise the quality of service. In our cache-enabled HetNet, the data rate at the typical UE exceeding the pre-decided rate threshold contributes to the success probability .
As discussed in the previous sections, we conclude that the considered system has two different processes in sending multimedia contents: 1) Association Mode, when the requested -th file obeys , the typical UE chooses the suitable BS as the corresponding BS depending on two association strategies; and 2) Server Mode, when the demanded -th content only exists in the server due to limited storage capacity at BSs, which means , the typical UE requests such content from the server via the nearest macro BS. We detailedly discuss these two modes below.
Iv-a Association Mode
In this mode, since two association strategies (Max-RP and Max-Rate) have different judgment standards to decide the corresponding BS, we study them separately.
Iv-A1 Maximum Received Power Scheme
The Max-RP scheme has been utilized in numerous HetNets proposed in recent articles, for example, the traditional cache-enabled HetNets  and the hybrid HetNets with mmWave . Under this scheme, the association procedure is fast and at a low cost due to ignoring the interference effects. We define the Max-RP association probability, when the typical UE connects to the -th tier BS for the -th file, as follows
where and .
Note that the path loss law for the pico tier has a step character. With the aid of similar proof in , the PDF of the distance between the typical UE and its corresponding -th tier BS with containing -th ranked file under Max-RP scheme is given by
where , , and .
Iv-A2 Maximum Rate Scheme
Since the path loss laws and bandwidth for two tiers are dissimilar, the received data rates are totally different even they have the same average received power . Compared with Max-RP, Max-Rate is able to provide higher data rate, but the extra knowledge of channel state information is indispensable. For Max-Rate, the association probability of the typical UE being associated with the -th tier BS for requesting the -th file is defined as
Instead of analyzing the relationship between the PDF of the corresponding distance as discussed in Max-RP, we are able to directly derive the PDF of considered coverage probability with the SINR threshold .
When requesting the -th ranked content, the PDF of the cache-related coverage probability for the -th tier under the Max-Rate strategy is shown as follows
See Appendix C. ∎
Since , if , the PDF of this coverage probability is same as , which means the typical UE is associated with the -th tier BS invariably. As a consequence, if the bandwidth of one tier is far more than the other, the typical UE always connects to the tier with large bandwidth.
Average Load Approximation: When all UEs are associated with the HetNet, the average number of UEs served by the -th tier BSs can be approximated by444This approximation is valid for sub-6 GHz scenarios  and mmWave scenarios . Due to the low probability of requesting the content from the core server, we ignore the corresponding load in the server mode for simplifying the analysis.  where , , and .
Since is a monotonic increasing function with the corresponding bias factor or , the small value of such bias factor is able to offload the data traffic in the -th tier. As a result, by adjusting the bias factors in the Max-RP scheme and in the Max-Rate scheme, we are able to control the average number of UEs for each tier, thereby balancing the load of the proposed HetNet.
Iv-B Server Mode
We present the backhaul capacity in order to compare our system with the traditional HetNets  in which the macro BSs have no caching ability. The comparison is illustrated in Section V. In the server mode, the backhaul capacity restricts the performance of our system. More specifically, if the required rate exceeds the backhaul capacity , no content can be sent successfully due to low system rate. On the other hand, if is larger than , the success probability under this case is limited by the received data rate from the relay macro BS. As a result, we provide the success probability in this mode as follows.
The success probability in the server mode is given by