I Introduction
With the rapid development of the traditional cellular networks and novel Internetenabled applications, such as multimedia sensors [2] and electric vehicles [3, 4], the total throughput of mobile networks in 2020 is expected to become 1000fold larger than that in 2010 [5]. To support the explosive data traffic of future fifthgeneration (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 highspeed optical fiber provides a theoretical solution, in practice, connecting the core server to all BSs with fibers is arduous and costly [10]. Moreover, microwave backhauls may pessimistically weaken the throughput gain fetched by the network densification [11]. A recent study [12] has shown that only 5%10% of multimedia contents are required by the majority of user equipments (UEs). Additionally, the storage capacity of cacheenabled 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 cacheenabled HetNets have been studied in various papers. Authors in [15] 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 [16], 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 multitier cacheenabled HetNets was discussed in [20], where BSs were modeled as mutually independent PPPs. However, the highcapacity 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 [10]. 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 capacityincreasing 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 sub6 GHz networks in 4G, two distinctive characteristics of mmWave are small wavelength and the sensitivity to blockages [24]. Thanks to the short wavelength, steerable antennas with huge scales can be employed at devices to enhance the directional array gain [25]. On the other side, the sensitivity gives rise to severe penetration loss for mmWave signals when passing through building exteriors [26]. Therefore, the path loss law of nonlineofsight (NLOS) links is substantially different from that of lineofsight (LOS) links in mmWave communications [25, 27], and it is unrealistic to expect an outdoortoindoor coverage from macro mmWave BSs. To compensate the blockagedependent loss, an ingenious hybrid network is created, where mmWave transmitters contribute to the ultrafast data rate in shortrange small cells, and sub6 GHz BSs provide the universal coverage [28].
There exist numerous studies concentrating on the performance of mmWave communications. As discussed in cacheenabled 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 [24], which employed a simplified flattop 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, frontback ratio and nulls [30]. Therefore, the authors in [31] proposed an actual antenna pattern for traditional mmWave networks. Considering the hybrid HetNets, a tractable framework with sub6 GHz macro cells and mmWave small cells was analyzed under two user association strategies in [28]. However, the Rayleigh fading assumption is not accurate for mmWave communications because of the poor scattering feature [30]. Recent works [24, 32] presented a realistic channel model with Nakagami fading to improve the theoretical accuracy.
Ia Motivation and Contribution
Although HetNets with caches have been analyzed under a variety of scenarios with traditional sub6 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 shortdistance networks [35], adopting mmWave into a dense pico tier of HetNets is a promising way to increase the throughput of 5G cellular networks. Additionally, utilizing lowcost 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 [10], we introduce fiberconnections between macro BSs and the multimedia server to evaluate the impact of backhaul capacity in cacheenabled 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 mmWaveenabled 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 sub6 GHz), authors in [38] 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 sub6 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 cacheenabled HetNets are discussed under two user association strategies: 1) Maximum Received Power (MaxRP), where the requesting user chooses the macro BS or pico BS offering the maximum average biased received power^{1}^{1}1The 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 [39]. from all BSs containing the requested file; and 2) Maximum Rate (MaxRate), where the typical UE selects the BS, which provides the highest biased transmitting rate, from all BSs caching the desired content.

We analyze the cacherelated coverage performance of traditional sub6 GHz macro cells and mmWave small cells with the actual antenna pattern. Furthermore, closedform coverage probability equations for the mmWave tier and an interferencelimited case with sub6 GHz are derived. Our analytical expressions can be directly applied into other mmWave or sub6 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 MaxRP has a positive correlation with the serving tier’s biased transmit power. However, the success probability of MaxRate 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 cacheenabled hybrid HetNets outperform the traditional HetNets where macro BSs have no caching capacity, and MaxRate achieves a better performance than MaxRP in terms of the success probability and ASE; 2) the proposed network is an interferencelimited system due to the nature of sub6 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.
IB Organization
We organize the rest of our treatise as follows: In Section II, we present the system model where twotier BSs and users in the proposed cacheenabled hybrid HetNets are modeled as three independent HPPPs. In Section III, the expressions of signaltointerferenceplusnoiseratio (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
Iia Network Architecture
In this paper, we present a cacheenabled hybrid HetNet with twotier 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 lesspopular 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 highcapacity wired backhaul.In order to avoid intertier interference, hybrid carrier frequencies are employed in our system. When communicating with UEs, the macro BSs adopt sub6 GHz, while the pico BSs utilize mmWave. Note that various multipleaccess 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.
IiB Blockage Model
In the first tier, when the communication distance is , the path loss law for sub6 GHz signals is same as that in traditional cellular networks, which is given by
(1) 
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 model^{2}^{2}2In most urban scenarios, the considered LOS ball model has negligible deviation with the more accurate multislope LOS probability scheme, especially when the altitude of pico BSs is less than the average height of obstacles [40]. [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 [42] 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 [43]. 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 paper^{3}^{3}3Regarding 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
(2) 
where and are the path loss exponent and the intercept of LOS links, respectively. is the unit step function, which is defined as
(3) 
IiC Cacheenabled 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 . Highspeed 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 [46] 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
(4) 
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 [49], such probability obeys:
(5) 
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 [10]. 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) MaxRP, where the typical UE communicates with the BS at that provides the maximum biased average received power; and 2) MaxRate, 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 twotier 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.
IiD Directional Beamforming
In the th tier, we employ antenna arrays composed of elements at all cacheenabled 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 [30]. 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 [28]. When the typical UE requests the th file from the th tier, the received signal can be expressed as follows
(6) 
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 [30]
(7) 
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 [30], where
is a uniformly distributed random variable over
. and are the antenna spacing and wavelength, respectively [50]. On the other hand, the array gain function for macro BSs is due to the omnidirectional antenna pattern.Since sophisticated beam training protocols [33] 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.
IiE Propagation Model
IiE1 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
(8) 
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 [24]. 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 .
IiE2 Association Criteria
When the typical UE requests the th file. For MaxRP, the biased average received power is defined as follows [8]
(9) 
where is a bias factor that aims to balance the load between two tiers under the MaxRP scheme [10]. Then, we consider the biased received data rate for MaxRate, which is given by
(10) 
where is the bandwidth per resource block. is another bias factor to control the data traffic under MaxRate.
When the typical UE requests a file from the core server, the throughput can be limited by both the macrotier conditions and the backhaul capacity [15]. Therefore, the instantaneous downlink data rate from the core server can be expressed as
Iii CacheRelated SINR Coverage Probability
Cacherelated 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 cacherelated 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 nonHPPP 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
(11) 
In the following part, we first analyze the cacherelated SINR coverage performance in the pico tier and then we consider the macro tier.
Iiia SINR Coverage Analysis in The Second Tier
If the typical UE is associated with the pico tier, the corresponding cacherelated coverage probability can be derived with the aid of Laplace Transform of Interference.
IiiA1 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 closedform expressions.
Lemma 1.
When requesting the th content, under the condition , the Laplace transform of interference with the predecided SINR threshold in the pico tier is as follows
(12) 
where
(13) 
are GaussChebyshev nodes over , and is a tradeoff 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
(14) 
where
(15) 
Proof:
See Appendix A. ∎
Remark 1.
The analytical expressions in Lemma 1 show that is independent of the transmit power and the intercept .
IiiA2 CacheRelated Coverage Probability
Considering the biased received power, we define the cacherelated coverage probability , when requesting the th file from the second tier, as follows
(16) 
where represents the probability function. With the aid of Lemma 1, the closedform coverage probability of pico tier is calculated as below.
Theorem 1.
When the typical UE requests the th ranked content from the pico tier, the cacherelated SINR coverage probability in this dense mmWave network can be expressed as follows
(17) 
where
(18) 
and .
Remark 3.
When , in Theorem 1 represents the coverage probability of this dense mmWave network without caching ability.
Remark 4.
Due to the long communicating distance and high path loss, traditional cellular networks with mmWave are noiselimited [28]. However, recent articles [29, 32] have shown that with a high BS density, such systems become interferencelimited. Under this condition, we present the first assumption below and the corroboration is provided in Section V.
Assumption 1.
The dense mmWave network in the pico tier is assumed to be an interferencelimited system, .
Corollary 1.
Under Assumption 1, the corresponding cacherelated SINR coverage probability in the dense mmWave network can be simplified as follows
(20) 
where
(21) 
Proof:
By deleting the part in Theorem 1, which represents the thermal noise effect on the coverage performance, we obtain the equations for this corollary. ∎
Remark 5.
Since the association probability of MaxRate is deduced on the basis of the derivative of the coverage probability [28], we deduce the derivative of described in Lemma 1, based on which the PDF of the coverage probability can be figured out.
Lemma 2.
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
(22) 
and when , such derivative can be expressed as
(23) 
where
(24) 
and .
Corollary 2.
With the aid of Lemma 2, when the typical UE requires the th ranked content, the PDF of cacherelated SINR coverage probability for the second tier is as follows
(25) 
where
(26) 
Proof:
See Appendix B. ∎
IiiB SINR Coverage Analysis in The First Tier
In the macro tier, we utilize the Rayleigh fading channel for sub6 GHz signals. The exact expression of cacherelated SINR coverage probability can be achieved in this part.
IiiB1 CacheRelated Coverage Probability
As discussed in the previous part, we keep the SINR threshold in the sub6 GHz tier. With the similar analysis in the pico tier, the cacherelated coverage probability in the second tier for requiring the th ranked content can be expressed as follows
(27) 
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 cacherelated coverage probability in the following paragraph.
Theorem 2.
When the typical UE requests the th ranked file from the macro tier, the exact cacherelated SINR coverage probability is given by
(28) 
Special Case 1: We assume , as it is valid for most sub6 GHz networks [54].
Corollary 3.
Under Special Case 1, the closedform cacherelated coverage probability in the first tier can be expressed as follows
(29) 
where , , and is the complementary error function.
Assumption 2.
Corollary 4.
Under Assumption 2, the cacherelated coverage probability for the first tier in Theorem 2 can be simplified as follows
(30) 
Proof:
By removing the noise part from Theorem 2, the SIR coverage probability can be deduced with the fact that . ∎
Remark 6.
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 closedform equation as a proxy of the exact expression in the remainder of this paper.
With the aid of the closedform expression in Corollary 4, we are able to figure out the closedform derivative of cacherelated 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 cacheenabled HetNet, the data rate at the typical UE exceeding the predecided rate threshold contributes to the success probability [10].
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.
Iva Association Mode
In this mode, since two association strategies (MaxRP and MaxRate) have different judgment standards to decide the corresponding BS, we study them separately.
IvA1 Maximum Received Power Scheme
The MaxRP scheme has been utilized in numerous HetNets proposed in recent articles, for example, the traditional cacheenabled HetNets [10] and the hybrid HetNets with mmWave [28]. Under this scheme, the association procedure is fast and at a low cost due to ignoring the interference effects. We define the MaxRP association probability, when the typical UE connects to the th tier BS for the th file, as follows
(32) 
where and .
Note that the path loss law for the pico tier has a step character. With the aid of similar proof in [17], the PDF of the distance between the typical UE and its corresponding th tier BS with containing th ranked file under MaxRP scheme is given by
(33)  
(34) 
where , , and .
IvA2 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 [28]. Compared with MaxRP, MaxRate is able to provide higher data rate, but the extra knowledge of channel state information is indispensable. For MaxRate, the association probability of the typical UE being associated with the th tier BS for requesting the th file is defined as
(35) 
Instead of analyzing the relationship between the PDF of the corresponding distance as discussed in MaxRP, we are able to directly derive the PDF of considered coverage probability with the SINR threshold .
Lemma 3.
When requesting the th ranked content, the PDF of the cacherelated coverage probability for the th tier under the MaxRate strategy is shown as follows
(36) 
Proof:
See Appendix C. ∎
Remark 7.
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 by^{4}^{4}4This approximation is valid for sub6 GHz scenarios [38] and mmWave scenarios [56]. 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. [28] where , , and .
Remark 8.
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 MaxRP scheme and in the MaxRate scheme, we are able to control the average number of UEs for each tier, thereby balancing the load of the proposed HetNet.
IvB Server Mode
We present the backhaul capacity in order to compare our system with the traditional HetNets [10] 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.
Lemma 4.
The success probability in the server mode is given by
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