# NOMA-Based Integrated Satellite-Terrestrial Networks with Wireless Caching

Authors: Qing Ye; Faxiang Zhao; Weiyang Xu
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

To decrease the transmission delay and alleviate communication congestion, caching has been applied in the integrated satellite-terrestrial networks (ISTNs). In traditional caching schemes, the cache-enabled relays update files during off-peak hours and push them during on-peak hours. However, for the enormous number of end devices in ISTNs, the long-time waiting for off/on-peak hours would decrease the quality of service. To address this issue, this paper proposes a two-tier nonorthogonal multiple access- (NOMA-) based ISTN with wireless caching. Specifically, data transmission of this model consists of two phases. For the first phase, named the file-push-and-placement (FPAP), the satellite employs the NOMA protocol to send information to both the relay and first-tier user. While for the second phase, which is called the file-push-and-delivery (FPAD), second-tier users are served by the relay employing NOMA. Performance analysis of the proposed configuration is carried out, focusing on the exact and asymptotic outage probability (OP), diversity order, and hit probability. Moreover, the influence of key factors on the system performance is also investigated. Compared with the traditional configuration, it is shown analytically and numerically that the proposed scheme achieves lower OP and higher diversity order.


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## Body

## 1. Introduction

Future networks are expected to provide diverse services to cope with the ever-increasing traffic demands of various services. Nevertheless, limited by the network capacity and coverage, only depending on the terrestrial communication systems cannot provide wireless access with high data rate and reliability at every place on the earth, especially in environmentally harsh areas like oceans and mountains [1]. Hence, integrated satellite-terrestrial networks (ISTNs) are deemed to be new network architectures to accommodate diverse services and applications with different quality of service (QoS) requirements [2].Meanwhile, as the latest member of the multiple access family, nonorthogonal multiple access (NOMA) is regarded as a promising technique in the next-generation wireless communication system [3]. In this paper, we focus on the power-domain NOMA, of which the key idea is to superimpose multiple signals in the power domain at the transmitter and employ successive interference cancellation (SIC) at the receiver [4]. With NOMA scheme, more users can be served simultaneously; hence, the spectrum efficiency is improved as a result. Therefore, the application of NOMA in ISTNs has attracted great attention in academia [5–8].In future wireless networks, not only the spectral efficiency but also the transmission delay should be paid special attention to. Stemming from that, the idea of caching is introduced. Instead of retrieving information from a central server, users can ask for the cache-enabled server that has replicated popular information [9]. As a result, the response time required to fetch a content file can be reduced. The authors in [10] analyzed and optimized the outage performance of a multiple cache-enabled amplify-and-forward relay network. Besides, caching can also help to further enhance the spectral efficiency of NOMA [11]. Cache content placement optimization in NOMA networks was investigated in [12, 13] which studied the corresponding coding mode. Furthermore, the authors in [14] proposed a NOMA-based multicast scheme that pushing and multicasting content objects can be accomplished at once, which boosts the spectrum efficiency. On this basis, two NOMA-assisted caching strategies were developed, namely, the push-then-deliver and the push-and-deliver [15]. However, these two strategies were studied individually, and the importance to combine them in practice was not investigated.Recently, the significance of applying caching to the ISTN has become a consensus in the academic community. The authors in [16] compared the outage probability (OP) of a relay-assisted ISTN with two representative cache placement schemes. As for the NOMA-assisted caching schemes applied in ISTNs, [17] investigated the OP of the cache-enabled relays and the hit probability of users in the NOMA-based hybrid satellite-terrestrial content delivery network. Moreover, a satellite-aerial-terrestrial network with cache-enabled aerial relays was introduced in [18], where the NOMA scheme was implemented to deliver content and push other currently most popular content to cache-enabled aerial relays simultaneously.The combination of NOMA and ISTNs can solve the problem of multiple nodes, but in addition to the requirements of massive nodes, future mobile communications also require low latency data transmission; so, cache is introduced to reduce latency. The combination of NOMA technology and cache technology can reduce the delay and increase spectral efficiency. However, the existing literature rarely considers how to make better use of the system spectrum resources and time resources in an environment with complex channel conditions, where the server needs to push files to users and place files to cache nodes at the same time. Therefore, this paper proposes a NOMA-based ISTN with wireless caching. Specifically, the active users request the active contents and are served by the satellite directly. In contrast, proactive users request proactive contents. However, due to heavy shadowing and masking effects, it is highly probable that proactive users lack direct links to the satellite and thus need help from cache-enabled relays. By exploiting the NOMA protocol, the content file placement of relays, and the files pushed to the active users from the satellite can be accomplished simultaneously.The main contributions are summarized as follows:(a)
The overall communication includes the file-push-and-placement (FPAP) and file-push-and-delivery (FPAD) phases. Specifically, in the FPAP phase, active and proactive files are sent to the active users and cache-enabled relays from the satellite by using the NOMA scheme. In the FPAD phase, the satellite pushes new files to active users, while the relays deliver files to the proactive users by using the NOMA technique(b)
The performance of the proposed system is thoroughly analyzed, with emphasis on users’ closed-form OP expressions. Then, the diversity gain is derived from the asymptotic behavior of OP. In addition, the hit probability of the relay node is studied. At last, the influence of key system parameters on outage performance is investigated(c)
A comparison between the proposed scheme and the NOMA-based ISTN without caching is carried out, where the result demonstrates that the system performance is improved by our scheme


## 2. System Model

The proposed two-tier heterogeneous cache-enabled NOMA-based ISTN is depicted in Figure1. Due to the complexity and diversity of channel conditions, users with stable and reliable direct links with satellites are divided into the first tier and defined as active users. Users suffering from severe shadow effect without a direct link are divided into the second tier and defined as proactive users. It is worth noting that the satellite can directly serve the active users, but it needs to use the relay node with good channel conditions to assist the proactive users. Due to the long backhaul link of the satellite communication network, to avoid the high delay of proactive users, the cache-enabled relay nodes are used to help the communication of proactive users. Assuming that there is good channel state quality between the satellite and the cache-enabled relay node, the relay node is divided into the first tier. In addition, content files are divided into active files and proactive files according to different requirements, i.e., the active file is defined as the file currently requested by the user, and the proactive file belongs to the file that the user does not request at present but will request in the future. Therefore, when the satellite directly serves the first-tier users, the pushed files are active; when the satellite places the content file to the cache-enabled relay node, the pushed file is the proactive.Figure 1
System model of the proposed cache-enabled NOMA-based ISTN.Above all, select active users and cache-enabled relays as the first-tier nodes and proactive users as the second-tier nodes. As for the signal transmission, the satellite communicates with the nodes in the first tier directly, while with the nodes in the second tier with the help of relays. To facilitate the analysis, it is supposed that there is one satellite (S), one active user (U1), one cache-enabled relay (R), and two proactive users (U2,U3) in the considered model. All nodes are equipped with a single antenna and operate in half-duplex mode. Moreover, U1 and R in the first tier, as well as U2 and U3 in the second tier, form NOMA clusters. It is worth noting that two-user NOMA-based downlink transmission has been proposed for long-term evolution (LTE) advanced [15].As aforementioned, the overall communication consists of two phases. In the FPAP phase, by using the NOMA scheme,S pushes the active files to U1, and at the same time, pushes the proactive popular contents that will be acquired by U2 to R. In the FPAD phase, U2 and U3 receive their corresponding files from R using the NOMA protocol. Without loss of generality, we assume that a NOMA cluster consists of two users, where the user with good channel condition is denoted by the strong user, while the other one is the weak user. For ease of description, in the following analysis, we denote the FPAP and FPAD as the first and second phases, respectively.


### 2.1. Caching Model

In the light of [14], the content files are divided into two categories: active files which are requested by the active users currently and proactive files not requested by the proactive users now but will be requested soon. Correspondingly, the relay caches the proactive files in the first phase, and then it can provide service to the proactive users in the second phase. Besides, the active files are pushed to active users directly in the first phase.The active files are collected in a finite content catalog denoted byFa=fa1,⋯,faN, and proactive files are collected in another one Fp=fp1,⋯,fpM, where N and M are the numbers of active and proactive files, respectively. Besides, the popularity of the requested files obeys a Zipf distribution, e.g., the popularity of fpm1≤m≤M is given by [17]
(1)Pfpm=m−ξ∑l=1Ml−ξ,where ξ>0 denotes the shape parameter defining the content popularity skewness. The popularity of the file becomes more concentrated when ξ gets larger, while the cache utilization ratio depends on the number of files requested by users [10].


### 2.2. Signal Transmission

#### 2.2.1. The First Phase (FPAP Phase)

During this phase,S transmits the superimposed active signals to U1 and R. After channel propagation, the received signals at U1 or R are
(2)yi=Qihiω1Psx1+ω2Psx2+n1,i∈su1,sr,where hi and Qi denotes the channel coefficient and composite fading distribution describing the links from S to U1 or R, respectively. The satellite transmit power is Ps, and ω1 and ω2 are the power control coefficients with constraints ω1+ω2=1 and ω1>ω2. x1 and x2 are the information-bearing symbols intended for U1 and R, respectively. Without loss of generality, it is assumed that Ex12=Ex22=1. Moreover, n1∼CN0,N0 refers to the additive noise at the receiver, where N0 is the noise variance.According to the principle of NOMA,U1 decodes x1 first by treating x2 as interference; thus, the signal-to-interfere-plus-noise ratio (SINR) of decoding x1 at U1 is given by
(3)λu1,x1=ω1hsu12ρsQsu1ω2hsu12ρsQsu1+1,where the signal-to-noise ratio (SNR) is defined as ρs=Ps/N0.As forR, it first decodes x1 and then removes x1 to decode x2. Hence, the SINR of decoding x1 and x2 at R can be separately expressed by
(4)λr,x1=ω1hsr2ρsQsrω2hsr2ρsQsr+1,λr,x2=ω2hsr2ρsQsr.


#### 2.2.2. The Second Phase (FPAD Phase)

In this phase,U2 and U3 are served by R with the NOMA protocol. In the meanwhile, S continuously sends data symbols to U1. However, the signal reception at U1 is subject to the interference from R. Assume that the file needed by U3 has already been cached in R, this can be achieved if R possesses storage capacity. After the channel propagation, the received signals of U2 and U3 can be
(5)yj=hjτ2Prx2+τ3Prx3+n2,j∈ru2,ru3,where x3 is intended for U3 with Ex32=1, hj indicates the channel coefficient from R to U2 or U3, and we assume that the channel condition of hru2 is better than that of hru3 without the loss of generality. Pr denotes the transmit power of R, and τ2 and τ3 are the power control coefficients, where τ2+τ3=1 and τ3>τ2. At last, n2 follows the same distribution as n1.According to the NOMA principle,U2 first decodes x3 by treating x2 as interference and then removes x3 to decode x2. As for U3, it decodes x3 directly. Hence, the SINR of decoding x3 and x2 at U2 can be expressed as
(6)λu2,x3=τ3hru22ρrτ2hru22ρr+1,λu2,x2=τ2hru22ρr,where ρr=Pr/N0. Besides, the SINR of decoding x3 at U3 is
(7)λu3,x3=τ3hru32ρrτ2hru32ρr+1.As forU1, the signal reception consists of desired signal from satellite S and interference from relay R; thus, its received signal is written as
(8)yu1′=Qsu1hsu1τ1Psx1′+hru1τ2Prx2+τ3Prx3+n3,where hru1 is the terrestrial channel link from R to U1, which suffers more severe fading than hsu1. x1′ is the transmit signal from S in the second phase, with Ex1′2=1. In addition, τ1 is the transmit power coefficient, and n3 follows the same distribution as n1. Hence, U1 decodes x1′ directly, and the corresponding SINR is expressed by
(9)λu1,x1′=τ1hsu12ρsQsu1hru12ρr+1.


### 2.3. Channel Model

For the satellite links, effects such as antenna gain, path loss, and link fading should be taken into account. We consider the GEO satellite; thus, the scaling parameter is given as(10)Qi=cGiφiTi4πfdiKBTBc.where c denotes the speed of light, f is the carrier frequency, di≈35786km is the distance between satellite S and the first-tier user ii∈su1,sr, KB=1.38×10−23J/°K is the Boltzman constant, T is the receiver noise temperature, and Bc is the carrier bandwidth. Besides, Ti denotes the antenna gain at user i, whereas Giφi gives the satellite beam gain based on both the satellite beam pattern and position of user i and is approximated by [19]
(11)Giφi≈GmaxJ1ui2ui+36J3uiui32.With ui=2.07123sinφi/sinφi3dB, Jv· represents the Bessel function of order v, while Gmax denotes the maximal beam gain. φi is the angle between user i and beam center with respect to the satellite.In (2), hi denotes the channel fading vector, which is usually modeled by composite fading distributions to describe more accurately the amplitude fluctuation of the signal envelope. Let the satellite link (hi2,i∈su1,sr) obey Shadowed-Rician fading, and the probability density function (PDF) is given by [20]
(12)fhi2x=αie−βix1F1mi;1;δix,where αi=2bimi/2bimi+Ωimi/2bi, βi=1/2bi, and δi=0.5Ωi/bi/2bimi+Ωi, 2bi is the average power of the multipath, Ωi is the line-of-sight (LoS) component, mimi>0 denotes the Nakagami-m fading parameter, and 1F1a;b;c represents the confluent hypergeometric function [21]. According to [21], the cumulative distribution function (CDF) of the satellite link gain is given by
(13)Fhi2u=αi∑k=0∞mikδikk!2βik+1γk+1,βiu,where ∙k is the Pochhammer symbol defined as xk=xx+1x+k−1.When it comes to the terrestrial link, which follows Nakagami-m fading, the PDF of the channel gain is represented by
(14)fhj2x=mjΩjmjxmj−1Γmje−mj/Ωjx,where mj is the integer fading severity parameter, Ωj is the average power of each link, and Γ∙ is the Gamma function [21]. In addition, the CDF of the terrestrial link gain can be expressed as
(15)Fhj2x=1Γmjγmj,mjΩjx,where γa,x=∫0xe−tta−1dt denotes the lower incomplete Gamma function [21].


### 2.4. Traditional Scheme without Caching

The model highly related to the proposed is the NOMA-based coordinated direct and relay transmission (CDRT) [22], which is depicted in Figure 2. In CDRT, a base station (BS) directly communicates with user equipment 1 (UE1) while communicating with user equipment 2 (UE2) via a relay. The communication consists of two phases, i.e., in the first phase, BS transmits the superposed signal α1x1+α2x2 to UE1 and the relay directly with NOMA protocol. In the second phase, the relay retransmits the decoded signal x2 (required by UE2) to UE2, while the BS transmits a new data symbol to UE1. The main challenge of nonorthogonal CDRT can be solved by using the inherent property of NOMA that allows a receiver to obtain side information such as other UE’s data for interference cancellation. Stemming from that, it is reasonable to apply NOMA-based CDRT to achieve high spectrum resource utilization.Figure 2
Traditional NOMA-based CDRT model.The main difference between proposed model and the traditional CDRT lies in the second phase. Specifically, in the proposed model, relay can serve multiple users simultaneously, while CDRT serves only one. To achieve the same time and spectrum utilization as the proposed scheme, the NOMA-based CDRT without caching, i.e.,S serves U1 and R in the first slot, and R serves two users at the second slot, at the same time, S serves U1 with newly signal, can be described as follows. In the first slot, S transmits the superimposed signal to R and U1. The received signal can be
(16)yi=Qihiς1Psx1+ς2Psx2+ς3Psx3+n,where the power coefficients satisfy ς1+ς2+ς3=1, ς1>ς2>ς3, and ς1>ς2+ς3. Thus, the SINR of U1 to decode x1 is
(17)λu1,x1no=ς1hsu12ρsQsu1ς2+ς3hsu12ρsQsu1+1.Besides, the SINR of decodingx1, x2, and x3 at R is
(18)λr,x1no=ς1hsr2ρsQsrς2+ς3hsrQsr2ρs+1,λr,x2no=ς2hsr2ρsQsrς3hsr2ρsQsr+1,λr,x3no=ς3hsr2ρsQsr.Meanwhile, the second time slot is the same as that of the proposed scheme; thus, the description is omitted for brevity.


## 2.1. Caching Model

In the light of [14], the content files are divided into two categories: active files which are requested by the active users currently and proactive files not requested by the proactive users now but will be requested soon. Correspondingly, the relay caches the proactive files in the first phase, and then it can provide service to the proactive users in the second phase. Besides, the active files are pushed to active users directly in the first phase.The active files are collected in a finite content catalog denoted byFa=fa1,⋯,faN, and proactive files are collected in another one Fp=fp1,⋯,fpM, where N and M are the numbers of active and proactive files, respectively. Besides, the popularity of the requested files obeys a Zipf distribution, e.g., the popularity of fpm1≤m≤M is given by [17]
(1)Pfpm=m−ξ∑l=1Ml−ξ,where ξ>0 denotes the shape parameter defining the content popularity skewness. The popularity of the file becomes more concentrated when ξ gets larger, while the cache utilization ratio depends on the number of files requested by users [10].


## 2.2. Signal Transmission

### 2.2.1. The First Phase (FPAP Phase)

During this phase,S transmits the superimposed active signals to U1 and R. After channel propagation, the received signals at U1 or R are
(2)yi=Qihiω1Psx1+ω2Psx2+n1,i∈su1,sr,where hi and Qi denotes the channel coefficient and composite fading distribution describing the links from S to U1 or R, respectively. The satellite transmit power is Ps, and ω1 and ω2 are the power control coefficients with constraints ω1+ω2=1 and ω1>ω2. x1 and x2 are the information-bearing symbols intended for U1 and R, respectively. Without loss of generality, it is assumed that Ex12=Ex22=1. Moreover, n1∼CN0,N0 refers to the additive noise at the receiver, where N0 is the noise variance.According to the principle of NOMA,U1 decodes x1 first by treating x2 as interference; thus, the signal-to-interfere-plus-noise ratio (SINR) of decoding x1 at U1 is given by
(3)λu1,x1=ω1hsu12ρsQsu1ω2hsu12ρsQsu1+1,where the signal-to-noise ratio (SNR) is defined as ρs=Ps/N0.As forR, it first decodes x1 and then removes x1 to decode x2. Hence, the SINR of decoding x1 and x2 at R can be separately expressed by
(4)λr,x1=ω1hsr2ρsQsrω2hsr2ρsQsr+1,λr,x2=ω2hsr2ρsQsr.


### 2.2.2. The Second Phase (FPAD Phase)

In this phase,U2 and U3 are served by R with the NOMA protocol. In the meanwhile, S continuously sends data symbols to U1. However, the signal reception at U1 is subject to the interference from R. Assume that the file needed by U3 has already been cached in R, this can be achieved if R possesses storage capacity. After the channel propagation, the received signals of U2 and U3 can be
(5)yj=hjτ2Prx2+τ3Prx3+n2,j∈ru2,ru3,where x3 is intended for U3 with Ex32=1, hj indicates the channel coefficient from R to U2 or U3, and we assume that the channel condition of hru2 is better than that of hru3 without the loss of generality. Pr denotes the transmit power of R, and τ2 and τ3 are the power control coefficients, where τ2+τ3=1 and τ3>τ2. At last, n2 follows the same distribution as n1.According to the NOMA principle,U2 first decodes x3 by treating x2 as interference and then removes x3 to decode x2. As for U3, it decodes x3 directly. Hence, the SINR of decoding x3 and x2 at U2 can be expressed as
(6)λu2,x3=τ3hru22ρrτ2hru22ρr+1,λu2,x2=τ2hru22ρr,where ρr=Pr/N0. Besides, the SINR of decoding x3 at U3 is
(7)λu3,x3=τ3hru32ρrτ2hru32ρr+1.As forU1, the signal reception consists of desired signal from satellite S and interference from relay R; thus, its received signal is written as
(8)yu1′=Qsu1hsu1τ1Psx1′+hru1τ2Prx2+τ3Prx3+n3,where hru1 is the terrestrial channel link from R to U1, which suffers more severe fading than hsu1. x1′ is the transmit signal from S in the second phase, with Ex1′2=1. In addition, τ1 is the transmit power coefficient, and n3 follows the same distribution as n1. Hence, U1 decodes x1′ directly, and the corresponding SINR is expressed by
(9)λu1,x1′=τ1hsu12ρsQsu1hru12ρr+1.


## 2.2.1. The First Phase (FPAP Phase)

During this phase,S transmits the superimposed active signals to U1 and R. After channel propagation, the received signals at U1 or R are
(2)yi=Qihiω1Psx1+ω2Psx2+n1,i∈su1,sr,where hi and Qi denotes the channel coefficient and composite fading distribution describing the links from S to U1 or R, respectively. The satellite transmit power is Ps, and ω1 and ω2 are the power control coefficients with constraints ω1+ω2=1 and ω1>ω2. x1 and x2 are the information-bearing symbols intended for U1 and R, respectively. Without loss of generality, it is assumed that Ex12=Ex22=1. Moreover, n1∼CN0,N0 refers to the additive noise at the receiver, where N0 is the noise variance.According to the principle of NOMA,U1 decodes x1 first by treating x2 as interference; thus, the signal-to-interfere-plus-noise ratio (SINR) of decoding x1 at U1 is given by
(3)λu1,x1=ω1hsu12ρsQsu1ω2hsu12ρsQsu1+1,where the signal-to-noise ratio (SNR) is defined as ρs=Ps/N0.As forR, it first decodes x1 and then removes x1 to decode x2. Hence, the SINR of decoding x1 and x2 at R can be separately expressed by
(4)λr,x1=ω1hsr2ρsQsrω2hsr2ρsQsr+1,λr,x2=ω2hsr2ρsQsr.


## 2.2.2. The Second Phase (FPAD Phase)

In this phase,U2 and U3 are served by R with the NOMA protocol. In the meanwhile, S continuously sends data symbols to U1. However, the signal reception at U1 is subject to the interference from R. Assume that the file needed by U3 has already been cached in R, this can be achieved if R possesses storage capacity. After the channel propagation, the received signals of U2 and U3 can be
(5)yj=hjτ2Prx2+τ3Prx3+n2,j∈ru2,ru3,where x3 is intended for U3 with Ex32=1, hj indicates the channel coefficient from R to U2 or U3, and we assume that the channel condition of hru2 is better than that of hru3 without the loss of generality. Pr denotes the transmit power of R, and τ2 and τ3 are the power control coefficients, where τ2+τ3=1 and τ3>τ2. At last, n2 follows the same distribution as n1.According to the NOMA principle,U2 first decodes x3 by treating x2 as interference and then removes x3 to decode x2. As for U3, it decodes x3 directly. Hence, the SINR of decoding x3 and x2 at U2 can be expressed as
(6)λu2,x3=τ3hru22ρrτ2hru22ρr+1,λu2,x2=τ2hru22ρr,where ρr=Pr/N0. Besides, the SINR of decoding x3 at U3 is
(7)λu3,x3=τ3hru32ρrτ2hru32ρr+1.As forU1, the signal reception consists of desired signal from satellite S and interference from relay R; thus, its received signal is written as
(8)yu1′=Qsu1hsu1τ1Psx1′+hru1τ2Prx2+τ3Prx3+n3,where hru1 is the terrestrial channel link from R to U1, which suffers more severe fading than hsu1. x1′ is the transmit signal from S in the second phase, with Ex1′2=1. In addition, τ1 is the transmit power coefficient, and n3 follows the same distribution as n1. Hence, U1 decodes x1′ directly, and the corresponding SINR is expressed by
(9)λu1,x1′=τ1hsu12ρsQsu1hru12ρr+1.


## 2.3. Channel Model

For the satellite links, effects such as antenna gain, path loss, and link fading should be taken into account. We consider the GEO satellite; thus, the scaling parameter is given as(10)Qi=cGiφiTi4πfdiKBTBc.where c denotes the speed of light, f is the carrier frequency, di≈35786km is the distance between satellite S and the first-tier user ii∈su1,sr, KB=1.38×10−23J/°K is the Boltzman constant, T is the receiver noise temperature, and Bc is the carrier bandwidth. Besides, Ti denotes the antenna gain at user i, whereas Giφi gives the satellite beam gain based on both the satellite beam pattern and position of user i and is approximated by [19]
(11)Giφi≈GmaxJ1ui2ui+36J3uiui32.With ui=2.07123sinφi/sinφi3dB, Jv· represents the Bessel function of order v, while Gmax denotes the maximal beam gain. φi is the angle between user i and beam center with respect to the satellite.In (2), hi denotes the channel fading vector, which is usually modeled by composite fading distributions to describe more accurately the amplitude fluctuation of the signal envelope. Let the satellite link (hi2,i∈su1,sr) obey Shadowed-Rician fading, and the probability density function (PDF) is given by [20]
(12)fhi2x=αie−βix1F1mi;1;δix,where αi=2bimi/2bimi+Ωimi/2bi, βi=1/2bi, and δi=0.5Ωi/bi/2bimi+Ωi, 2bi is the average power of the multipath, Ωi is the line-of-sight (LoS) component, mimi>0 denotes the Nakagami-m fading parameter, and 1F1a;b;c represents the confluent hypergeometric function [21]. According to [21], the cumulative distribution function (CDF) of the satellite link gain is given by
(13)Fhi2u=αi∑k=0∞mikδikk!2βik+1γk+1,βiu,where ∙k is the Pochhammer symbol defined as xk=xx+1x+k−1.When it comes to the terrestrial link, which follows Nakagami-m fading, the PDF of the channel gain is represented by
(14)fhj2x=mjΩjmjxmj−1Γmje−mj/Ωjx,where mj is the integer fading severity parameter, Ωj is the average power of each link, and Γ∙ is the Gamma function [21]. In addition, the CDF of the terrestrial link gain can be expressed as
(15)Fhj2x=1Γmjγmj,mjΩjx,where γa,x=∫0xe−tta−1dt denotes the lower incomplete Gamma function [21].


## 2.4. Traditional Scheme without Caching

The model highly related to the proposed is the NOMA-based coordinated direct and relay transmission (CDRT) [22], which is depicted in Figure 2. In CDRT, a base station (BS) directly communicates with user equipment 1 (UE1) while communicating with user equipment 2 (UE2) via a relay. The communication consists of two phases, i.e., in the first phase, BS transmits the superposed signal α1x1+α2x2 to UE1 and the relay directly with NOMA protocol. In the second phase, the relay retransmits the decoded signal x2 (required by UE2) to UE2, while the BS transmits a new data symbol to UE1. The main challenge of nonorthogonal CDRT can be solved by using the inherent property of NOMA that allows a receiver to obtain side information such as other UE’s data for interference cancellation. Stemming from that, it is reasonable to apply NOMA-based CDRT to achieve high spectrum resource utilization.Figure 2
Traditional NOMA-based CDRT model.The main difference between proposed model and the traditional CDRT lies in the second phase. Specifically, in the proposed model, relay can serve multiple users simultaneously, while CDRT serves only one. To achieve the same time and spectrum utilization as the proposed scheme, the NOMA-based CDRT without caching, i.e.,S serves U1 and R in the first slot, and R serves two users at the second slot, at the same time, S serves U1 with newly signal, can be described as follows. In the first slot, S transmits the superimposed signal to R and U1. The received signal can be
(16)yi=Qihiς1Psx1+ς2Psx2+ς3Psx3+n,where the power coefficients satisfy ς1+ς2+ς3=1, ς1>ς2>ς3, and ς1>ς2+ς3. Thus, the SINR of U1 to decode x1 is
(17)λu1,x1no=ς1hsu12ρsQsu1ς2+ς3hsu12ρsQsu1+1.Besides, the SINR of decodingx1, x2, and x3 at R is
(18)λr,x1no=ς1hsr2ρsQsrς2+ς3hsrQsr2ρs+1,λr,x2no=ς2hsr2ρsQsrς3hsr2ρsQsr+1,λr,x3no=ς3hsr2ρsQsr.Meanwhile, the second time slot is the same as that of the proposed scheme; thus, the description is omitted for brevity.


## 3. Performance Analysis

### 3.1. Outage Performance

The OP is defined as the probability that the instantaneous SINRλ is less than a predefined threshold λth, i.e.,
(19)Poutλth=Pλ<λth=Fλλth,where λth=22Rth−1 with Rth being the required data rate, and Fλ∙ is the CDF of λ. We emphasize that the analytical derivation is based on the following assumptions. Firstly, the CSI and SIC are crucial to the outage performance, and there have been several studies concerning this issue [23]. The CSI can be obtained with channel estimation methods, which have been adopted in most the existing works [24, 25]. As the focus herein is the caching scheme, we assume that the perfect CSI is available, and the SIC is error-free. Secondly, both the terrestrial and satellite channels are independent and identically distributed (i.i.d.).


#### 3.1.1. OP ofU1

In our model,U1 receives signals within two phases, and then its OP needs to be discussed separately. In the first phase, the outage occurs when the decoding of x1 is failed, i.e.,
(20)Pu1,out1=Pλu1,x1<λth1=Fhsu12ϕ1,where ϕ1=λth1/ω1−ω2λth1ρsQsu1. λth1 is the required SINR of decoding x1, which satisfies λth1<ω1/ω2.In the second phase, the OP is expressed as(21)Pu1,out2=Pλu1,x1′<λth1.According to (21), we can obtain 1−Pu1,out2 as follows:
(22)1−Pu1,out2=Pτ1hsu12ρsQsu1hru12ρr+1>λth1=∫0∞fhru12xdx⏟J1−∫0∞Fhsu12λth1ρrxQsu1+1ρsτ1fhru12xdx⏟J2,where J1 and J2 denote the first and second integrals, respectively. Afterwards, it is easy to find that J1=1, and then J2 is expanded as
(23)J2=∫0∞αsu1∑k=0∞msu1kδsu1kk!2βsu1k+1γk+1,βsu1λth1ρsτ1ρrxQsu1+1mru1Ωru1mru1xmru1−1Γmru1e−mru1/Ωru1xdx.According to [21], one can arrive at
(24)J2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2Γmru1∫0∞e−Ξ2ρrx−mru1/Ωru1x∑m=0kΞ2ρrmm!x+1ρrQsu1mxmru1−1dx.Then, the close-formed expression ofPu1,out2 is shown as
(25)Pu1,out2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2Γmru1∑m=0kΞ2ρrmm!∑n=0mmn1ρrQsu1m−nΞ2ρr+mru1Ωru1−n−mru1Γn+mru1,where
(26)Ξ1=αsu1∑k=0∞msu1kδsu1kk!βsu1k+1,Ξ2=βsu1λth1Qsu1ρsτ1.Hence, the derivation of the OP ofU1 concludes.


#### 3.1.2. OP ofU2

As forU2, outage occurs when the detection of x3 is failed, or the detection of x2 fails provided the successful detection of x3. Therefore, the resultant OP can be derived by
(27)Pu2,out=Pfp2×Pλu2,x3<λth3+Pfp2×Pλu2,x3≥λth3,λu2,x2<λth2=Pfp2×Pλu2,x3<λth3+Pfp2×Pλu2,x3≥λth3×Pλu2,x2<λth2=Pfp2×Fhru22ϕ2+Pfp2×1−Fhru22ϕ2×Fhru22ϕ3,where
(28)ϕ2=λth3τ3−τ2λth3ρr,ϕ3=λth2τ2ρr.Besides,λth2 and λth3 are the corresponding SINRs of decoding x2 and x3, which satisfies λth3<τ3/τ2.


#### 3.1.3. OP ofU3

The OP ofU3 can be derived straightforward as
(29)Pu3,out=Pfp3×Pλu3,x3<λth3=Pfp3×Fhru32ϕ2.


### 3.2. Diversity Order

Due to their complex forms, it is difficult to have a deep understanding of the closed-form OP expressions derived above. On the other hand, the diversity order, defined asdi=−limρi⟶∞logP~ρi/logρi, is another key parameter to measure the system performance. To facilitate our analysis of diversity order, the asymptotic outage behaviour of OP at high SNR regimes should be investigated. First, the approximation of CDFs of terrestrial and satellite channels is addressed. For terrestrial channels, by expressing the exponential function in (15) in terms of the Maclaurin series, the PDF is approximated as
(30)fhj2x≈mjmjxmj−1ΓmjΩjmj,j∈ru1,ru2,ru3.Thus, the corresponding CDF of (30) is obtained straightforward, shown as
(31)Fhj2x≈1Γmj+1mjxΩjmj.As for satellite links, via using the series representation in [21], one can arrive at
(32)γk+1,βiu=∑k=0∞−1nβiuk+1+nn!k+1+n≈βiuk+1k+1u⟶0,i∈su1,sr.Thus, the approximated CDF expression for the satellite links can be obtained as follows:(33)Fhi2u≈αi∑k=0∞mikδikk!2βik+1×βiuk+1k+1.And the final approximated result is(34)Fhi2u≈αiu,With (31) and (34), diversity orders of users can be derived as follows.


#### 3.2.1. Diversity Order ofU1

According to (20) and (34), the asymptotic OP of U1 in the first phase is
(35)P~u1,out1≈αsu1ϕ1.By assumingρs⟶∞, it is derived that the diversity order of U1 is 1.In the second phase, assumingρs⟶∞ and ρr⟶∞, then λu1,x1′ is approximated as τ1hsu12Qsu1/hru12. Therefore, the asymptotic OP is calculated by
(36)∫0∞fhru12x∫x/τ1∞fhsu12ydydx≈mru1Ωru1mru1αsu1Γmru1∫x/τ1Qsu1∞xmru1−1e−mru1/Ωru1xdx.According to [21], the OP of U1 in the second phase approximates to
(37)P~u1,out2≈αsu1Qsu1Γmru1mru1Ωru1−1mru1.Hence, the diversity order ofU1 in the second phase is 0.


#### 3.2.2. Diversity Order ofU2

According to (27) and (31), the asymptotic OP of U2 is shown as
(38)P~u2,out≈Pfp2Γmru2+1×mru2Ωru2mru2×ϕ2mru2+ϕ3mru2−ϕ2mru2×ϕ3mru2×mru2Ωru2mru2×1Γmru2+1.After straightforward mathematical manipulations, the diversity order is obtained asmru2.


#### 3.2.3. Diversity Order ofU3

First, the asymptotic OP ofU3 is
(39)P~u3,out≈Pfp3×1Γmru3+1×mru3Ωru3×ϕ2mru3.At last, one can obtain that the diversity order ismru3.


### 3.3. Hit Probability

When a user requests a certain file cached inR, a cache hit event happens. Hence, an effective criterion to evaluate the performance of content pushing is the hit probability, which indicates the probability that, during the content delivery phase, a user finds its requested file in the associated cache-enabled R [15]. The hit probability can be expressed as
(40)Phit=∑n=1NPfpn×1−Pr,outfpn,where Pr,outfpn denotes the OP of R to decode file fpn. In our considered system model, the OP of R to decode fp2 is derived as
(41)Pr,outfp2=Fhsr2λth1ω1−ω2λth1ρsQsr+1−Fhsr2λth1ω1−ω2λth1ρsQsr×Fhsr2ϕ4,where ϕ4=λth2/ω2ρsQsr. By substituting (41) into (40), the hit probability of R is acquired.


### 3.4. Comparison to the Scheme without Caching

For comparison, this subsection gives the outage performance of users in the traditional CDRT configuration.


#### 3.4.1. OP ofU1

In the first time slot, the OP of detectingx1 is expressed as
(42)Pu1,outno,1=Fhsu12λth1s1−s2+ς3λth1ρsQsu1.While for the second slot, outage occurs in three cases. First, whenR fails to decode x3 or x2 given the success decoding of x1; thus, the SINR of detecting x1′ at U1 is denoted by λu1,x1′no,1=τ1hsu12ρsQsu1. Second, detection of x1 and x2 is successful at R, but decoding x3 encounters with failure, and then the SINR is λu1,x1′no,2=τ1hsu12ρsQsu1/τ2hru12ρr+1 . At last, all the signals are decoded correctly in the first slot; hence, λu1,x1′no,3 is equal to λu1,x1′. To sum up, the OP can be derived as
(43)Pu1,outno,2=P1+1−P1×P2×Px1′1+1−P1×1−P2×P3×Px1′2+1−P1×1−P2×1−P3×Px1′3,where P1, P2, and P3 denote the OP of decoding x1, x2, and x3 at R, respectively. Besides, Px1′1, Px1′2, and Px1′3 indicate the OP of decoding x1′ in three cases described above, respectively. Those outage probabilities can be calculated by
(44)P1=Fhsr2λth1ς1−ς2+ς3λth1ρsQsr,P2=Fhsr2λth2ς2−ς3λth2ρsQsr,P3=Fhsr2λth3ς3ρsQsr,Px1′1=Fhsu12λth1τ1ρsQsu1,Px1′3=Pu1,out2.Moreover, by using the same approach in (25), Px1′2 is obtained as
(45)Px1′2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2τ2Γmru1∑m=0kΞ2τ2ρrmm!∑n=0mmn1ρrQsu1m−nΞ2τ2ρr+mru1Ωru1−n−mru1Γn+mru1.With (43) and (44), Pu1,outno,2 is finally obtained.


#### 3.4.2. OP ofU2

As forU2, the outage event occurs in situations as follows. First, the decoding of x1 at R fails. Second, the decoding of x2 fails given the success decoding of x1. Third, under the assumption of successfully detecting x1 and x2 at R, the decoding of x3 at R is failed. Finally, the detection at R is successful, but the decoding of x3 or x2 fails at U2. Therefore, the OP can be derived as
(46)Pu2,outno=P1+1−P1×P2+1−P1×1−P2×P3×P4+1−P1×1−P2×1−P3×P5+1−P5×P4,where P4 and P5 denote the OP of decoding x2 and x3 at U2, respectively. The detailed expressions are
(47)P4=Fhru22λth2τ2ρr,P5=Fhru22λth3τ3−τ2λth3ρr.


#### 3.4.3. OP ofU3

As forU3, an outage first occurs if the decoding of x1 at R fails. Then, the decoding of x2 fails given the successful decoding of x1. Moreover, the decoding of x3 is failed when the detection of both x1 and x2 succeeds. At last, the detection at R succeeds, but U3 fails to detect x3. Hence, the overall OP is
(48)Pu3,outno=P1+1−P1×P2+1−P1×1−P2×P3+1−P1×1−P2×1−P3×P6,where P6 denotes the OP of detecting x3 at U3, which is
(49)P6=Fhru32λth3τ3−τ2λth3ρr.


## 3.1. Outage Performance

The OP is defined as the probability that the instantaneous SINRλ is less than a predefined threshold λth, i.e.,
(19)Poutλth=Pλ<λth=Fλλth,where λth=22Rth−1 with Rth being the required data rate, and Fλ∙ is the CDF of λ. We emphasize that the analytical derivation is based on the following assumptions. Firstly, the CSI and SIC are crucial to the outage performance, and there have been several studies concerning this issue [23]. The CSI can be obtained with channel estimation methods, which have been adopted in most the existing works [24, 25]. As the focus herein is the caching scheme, we assume that the perfect CSI is available, and the SIC is error-free. Secondly, both the terrestrial and satellite channels are independent and identically distributed (i.i.d.).


### 3.1.1. OP ofU1

In our model,U1 receives signals within two phases, and then its OP needs to be discussed separately. In the first phase, the outage occurs when the decoding of x1 is failed, i.e.,
(20)Pu1,out1=Pλu1,x1<λth1=Fhsu12ϕ1,where ϕ1=λth1/ω1−ω2λth1ρsQsu1. λth1 is the required SINR of decoding x1, which satisfies λth1<ω1/ω2.In the second phase, the OP is expressed as(21)Pu1,out2=Pλu1,x1′<λth1.According to (21), we can obtain 1−Pu1,out2 as follows:
(22)1−Pu1,out2=Pτ1hsu12ρsQsu1hru12ρr+1>λth1=∫0∞fhru12xdx⏟J1−∫0∞Fhsu12λth1ρrxQsu1+1ρsτ1fhru12xdx⏟J2,where J1 and J2 denote the first and second integrals, respectively. Afterwards, it is easy to find that J1=1, and then J2 is expanded as
(23)J2=∫0∞αsu1∑k=0∞msu1kδsu1kk!2βsu1k+1γk+1,βsu1λth1ρsτ1ρrxQsu1+1mru1Ωru1mru1xmru1−1Γmru1e−mru1/Ωru1xdx.According to [21], one can arrive at
(24)J2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2Γmru1∫0∞e−Ξ2ρrx−mru1/Ωru1x∑m=0kΞ2ρrmm!x+1ρrQsu1mxmru1−1dx.Then, the close-formed expression ofPu1,out2 is shown as
(25)Pu1,out2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2Γmru1∑m=0kΞ2ρrmm!∑n=0mmn1ρrQsu1m−nΞ2ρr+mru1Ωru1−n−mru1Γn+mru1,where
(26)Ξ1=αsu1∑k=0∞msu1kδsu1kk!βsu1k+1,Ξ2=βsu1λth1Qsu1ρsτ1.Hence, the derivation of the OP ofU1 concludes.


### 3.1.2. OP ofU2

As forU2, outage occurs when the detection of x3 is failed, or the detection of x2 fails provided the successful detection of x3. Therefore, the resultant OP can be derived by
(27)Pu2,out=Pfp2×Pλu2,x3<λth3+Pfp2×Pλu2,x3≥λth3,λu2,x2<λth2=Pfp2×Pλu2,x3<λth3+Pfp2×Pλu2,x3≥λth3×Pλu2,x2<λth2=Pfp2×Fhru22ϕ2+Pfp2×1−Fhru22ϕ2×Fhru22ϕ3,where
(28)ϕ2=λth3τ3−τ2λth3ρr,ϕ3=λth2τ2ρr.Besides,λth2 and λth3 are the corresponding SINRs of decoding x2 and x3, which satisfies λth3<τ3/τ2.


### 3.1.3. OP ofU3

The OP ofU3 can be derived straightforward as
(29)Pu3,out=Pfp3×Pλu3,x3<λth3=Pfp3×Fhru32ϕ2.


## 3.1.1. OP ofU1

In our model,U1 receives signals within two phases, and then its OP needs to be discussed separately. In the first phase, the outage occurs when the decoding of x1 is failed, i.e.,
(20)Pu1,out1=Pλu1,x1<λth1=Fhsu12ϕ1,where ϕ1=λth1/ω1−ω2λth1ρsQsu1. λth1 is the required SINR of decoding x1, which satisfies λth1<ω1/ω2.In the second phase, the OP is expressed as(21)Pu1,out2=Pλu1,x1′<λth1.According to (21), we can obtain 1−Pu1,out2 as follows:
(22)1−Pu1,out2=Pτ1hsu12ρsQsu1hru12ρr+1>λth1=∫0∞fhru12xdx⏟J1−∫0∞Fhsu12λth1ρrxQsu1+1ρsτ1fhru12xdx⏟J2,where J1 and J2 denote the first and second integrals, respectively. Afterwards, it is easy to find that J1=1, and then J2 is expanded as
(23)J2=∫0∞αsu1∑k=0∞msu1kδsu1kk!2βsu1k+1γk+1,βsu1λth1ρsτ1ρrxQsu1+1mru1Ωru1mru1xmru1−1Γmru1e−mru1/Ωru1xdx.According to [21], one can arrive at
(24)J2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2Γmru1∫0∞e−Ξ2ρrx−mru1/Ωru1x∑m=0kΞ2ρrmm!x+1ρrQsu1mxmru1−1dx.Then, the close-formed expression ofPu1,out2 is shown as
(25)Pu1,out2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2Γmru1∑m=0kΞ2ρrmm!∑n=0mmn1ρrQsu1m−nΞ2ρr+mru1Ωru1−n−mru1Γn+mru1,where
(26)Ξ1=αsu1∑k=0∞msu1kδsu1kk!βsu1k+1,Ξ2=βsu1λth1Qsu1ρsτ1.Hence, the derivation of the OP ofU1 concludes.


## 3.1.2. OP ofU2

As forU2, outage occurs when the detection of x3 is failed, or the detection of x2 fails provided the successful detection of x3. Therefore, the resultant OP can be derived by
(27)Pu2,out=Pfp2×Pλu2,x3<λth3+Pfp2×Pλu2,x3≥λth3,λu2,x2<λth2=Pfp2×Pλu2,x3<λth3+Pfp2×Pλu2,x3≥λth3×Pλu2,x2<λth2=Pfp2×Fhru22ϕ2+Pfp2×1−Fhru22ϕ2×Fhru22ϕ3,where
(28)ϕ2=λth3τ3−τ2λth3ρr,ϕ3=λth2τ2ρr.Besides,λth2 and λth3 are the corresponding SINRs of decoding x2 and x3, which satisfies λth3<τ3/τ2.


## 3.1.3. OP ofU3

The OP ofU3 can be derived straightforward as
(29)Pu3,out=Pfp3×Pλu3,x3<λth3=Pfp3×Fhru32ϕ2.


## 3.2. Diversity Order

Due to their complex forms, it is difficult to have a deep understanding of the closed-form OP expressions derived above. On the other hand, the diversity order, defined asdi=−limρi⟶∞logP~ρi/logρi, is another key parameter to measure the system performance. To facilitate our analysis of diversity order, the asymptotic outage behaviour of OP at high SNR regimes should be investigated. First, the approximation of CDFs of terrestrial and satellite channels is addressed. For terrestrial channels, by expressing the exponential function in (15) in terms of the Maclaurin series, the PDF is approximated as
(30)fhj2x≈mjmjxmj−1ΓmjΩjmj,j∈ru1,ru2,ru3.Thus, the corresponding CDF of (30) is obtained straightforward, shown as
(31)Fhj2x≈1Γmj+1mjxΩjmj.As for satellite links, via using the series representation in [21], one can arrive at
(32)γk+1,βiu=∑k=0∞−1nβiuk+1+nn!k+1+n≈βiuk+1k+1u⟶0,i∈su1,sr.Thus, the approximated CDF expression for the satellite links can be obtained as follows:(33)Fhi2u≈αi∑k=0∞mikδikk!2βik+1×βiuk+1k+1.And the final approximated result is(34)Fhi2u≈αiu,With (31) and (34), diversity orders of users can be derived as follows.


### 3.2.1. Diversity Order ofU1

According to (20) and (34), the asymptotic OP of U1 in the first phase is
(35)P~u1,out1≈αsu1ϕ1.By assumingρs⟶∞, it is derived that the diversity order of U1 is 1.In the second phase, assumingρs⟶∞ and ρr⟶∞, then λu1,x1′ is approximated as τ1hsu12Qsu1/hru12. Therefore, the asymptotic OP is calculated by
(36)∫0∞fhru12x∫x/τ1∞fhsu12ydydx≈mru1Ωru1mru1αsu1Γmru1∫x/τ1Qsu1∞xmru1−1e−mru1/Ωru1xdx.According to [21], the OP of U1 in the second phase approximates to
(37)P~u1,out2≈αsu1Qsu1Γmru1mru1Ωru1−1mru1.Hence, the diversity order ofU1 in the second phase is 0.


### 3.2.2. Diversity Order ofU2

According to (27) and (31), the asymptotic OP of U2 is shown as
(38)P~u2,out≈Pfp2Γmru2+1×mru2Ωru2mru2×ϕ2mru2+ϕ3mru2−ϕ2mru2×ϕ3mru2×mru2Ωru2mru2×1Γmru2+1.After straightforward mathematical manipulations, the diversity order is obtained asmru2.


### 3.2.3. Diversity Order ofU3

First, the asymptotic OP ofU3 is
(39)P~u3,out≈Pfp3×1Γmru3+1×mru3Ωru3×ϕ2mru3.At last, one can obtain that the diversity order ismru3.


## 3.2.1. Diversity Order ofU1

According to (20) and (34), the asymptotic OP of U1 in the first phase is
(35)P~u1,out1≈αsu1ϕ1.By assumingρs⟶∞, it is derived that the diversity order of U1 is 1.In the second phase, assumingρs⟶∞ and ρr⟶∞, then λu1,x1′ is approximated as τ1hsu12Qsu1/hru12. Therefore, the asymptotic OP is calculated by
(36)∫0∞fhru12x∫x/τ1∞fhsu12ydydx≈mru1Ωru1mru1αsu1Γmru1∫x/τ1Qsu1∞xmru1−1e−mru1/Ωru1xdx.According to [21], the OP of U1 in the second phase approximates to
(37)P~u1,out2≈αsu1Qsu1Γmru1mru1Ωru1−1mru1.Hence, the diversity order ofU1 in the second phase is 0.


## 3.2.2. Diversity Order ofU2

According to (27) and (31), the asymptotic OP of U2 is shown as
(38)P~u2,out≈Pfp2Γmru2+1×mru2Ωru2mru2×ϕ2mru2+ϕ3mru2−ϕ2mru2×ϕ3mru2×mru2Ωru2mru2×1Γmru2+1.After straightforward mathematical manipulations, the diversity order is obtained asmru2.


## 3.2.3. Diversity Order ofU3

First, the asymptotic OP ofU3 is
(39)P~u3,out≈Pfp3×1Γmru3+1×mru3Ωru3×ϕ2mru3.At last, one can obtain that the diversity order ismru3.


## 3.3. Hit Probability

When a user requests a certain file cached inR, a cache hit event happens. Hence, an effective criterion to evaluate the performance of content pushing is the hit probability, which indicates the probability that, during the content delivery phase, a user finds its requested file in the associated cache-enabled R [15]. The hit probability can be expressed as
(40)Phit=∑n=1NPfpn×1−Pr,outfpn,where Pr,outfpn denotes the OP of R to decode file fpn. In our considered system model, the OP of R to decode fp2 is derived as
(41)Pr,outfp2=Fhsr2λth1ω1−ω2λth1ρsQsr+1−Fhsr2λth1ω1−ω2λth1ρsQsr×Fhsr2ϕ4,where ϕ4=λth2/ω2ρsQsr. By substituting (41) into (40), the hit probability of R is acquired.


## 3.4. Comparison to the Scheme without Caching

For comparison, this subsection gives the outage performance of users in the traditional CDRT configuration.


### 3.4.1. OP ofU1

In the first time slot, the OP of detectingx1 is expressed as
(42)Pu1,outno,1=Fhsu12λth1s1−s2+ς3λth1ρsQsu1.While for the second slot, outage occurs in three cases. First, whenR fails to decode x3 or x2 given the success decoding of x1; thus, the SINR of detecting x1′ at U1 is denoted by λu1,x1′no,1=τ1hsu12ρsQsu1. Second, detection of x1 and x2 is successful at R, but decoding x3 encounters with failure, and then the SINR is λu1,x1′no,2=τ1hsu12ρsQsu1/τ2hru12ρr+1 . At last, all the signals are decoded correctly in the first slot; hence, λu1,x1′no,3 is equal to λu1,x1′. To sum up, the OP can be derived as
(43)Pu1,outno,2=P1+1−P1×P2×Px1′1+1−P1×1−P2×P3×Px1′2+1−P1×1−P2×1−P3×Px1′3,where P1, P2, and P3 denote the OP of decoding x1, x2, and x3 at R, respectively. Besides, Px1′1, Px1′2, and Px1′3 indicate the OP of decoding x1′ in three cases described above, respectively. Those outage probabilities can be calculated by
(44)P1=Fhsr2λth1ς1−ς2+ς3λth1ρsQsr,P2=Fhsr2λth2ς2−ς3λth2ρsQsr,P3=Fhsr2λth3ς3ρsQsr,Px1′1=Fhsu12λth1τ1ρsQsu1,Px1′3=Pu1,out2.Moreover, by using the same approach in (25), Px1′2 is obtained as
(45)Px1′2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2τ2Γmru1∑m=0kΞ2τ2ρrmm!∑n=0mmn1ρrQsu1m−nΞ2τ2ρr+mru1Ωru1−n−mru1Γn+mru1.With (43) and (44), Pu1,outno,2 is finally obtained.


### 3.4.2. OP ofU2

As forU2, the outage event occurs in situations as follows. First, the decoding of x1 at R fails. Second, the decoding of x2 fails given the success decoding of x1. Third, under the assumption of successfully detecting x1 and x2 at R, the decoding of x3 at R is failed. Finally, the detection at R is successful, but the decoding of x3 or x2 fails at U2. Therefore, the OP can be derived as
(46)Pu2,outno=P1+1−P1×P2+1−P1×1−P2×P3×P4+1−P1×1−P2×1−P3×P5+1−P5×P4,where P4 and P5 denote the OP of decoding x2 and x3 at U2, respectively. The detailed expressions are
(47)P4=Fhru22λth2τ2ρr,P5=Fhru22λth3τ3−τ2λth3ρr.


### 3.4.3. OP ofU3

As forU3, an outage first occurs if the decoding of x1 at R fails. Then, the decoding of x2 fails given the successful decoding of x1. Moreover, the decoding of x3 is failed when the detection of both x1 and x2 succeeds. At last, the detection at R succeeds, but U3 fails to detect x3. Hence, the overall OP is
(48)Pu3,outno=P1+1−P1×P2+1−P1×1−P2×P3+1−P1×1−P2×1−P3×P6,where P6 denotes the OP of detecting x3 at U3, which is
(49)P6=Fhru32λth3τ3−τ2λth3ρr.


## 3.4.1. OP ofU1

In the first time slot, the OP of detectingx1 is expressed as
(42)Pu1,outno,1=Fhsu12λth1s1−s2+ς3λth1ρsQsu1.While for the second slot, outage occurs in three cases. First, whenR fails to decode x3 or x2 given the success decoding of x1; thus, the SINR of detecting x1′ at U1 is denoted by λu1,x1′no,1=τ1hsu12ρsQsu1. Second, detection of x1 and x2 is successful at R, but decoding x3 encounters with failure, and then the SINR is λu1,x1′no,2=τ1hsu12ρsQsu1/τ2hru12ρr+1 . At last, all the signals are decoded correctly in the first slot; hence, λu1,x1′no,3 is equal to λu1,x1′. To sum up, the OP can be derived as
(43)Pu1,outno,2=P1+1−P1×P2×Px1′1+1−P1×1−P2×P3×Px1′2+1−P1×1−P2×1−P3×Px1′3,where P1, P2, and P3 denote the OP of decoding x1, x2, and x3 at R, respectively. Besides, Px1′1, Px1′2, and Px1′3 indicate the OP of decoding x1′ in three cases described above, respectively. Those outage probabilities can be calculated by
(44)P1=Fhsr2λth1ς1−ς2+ς3λth1ρsQsr,P2=Fhsr2λth2ς2−ς3λth2ρsQsr,P3=Fhsr2λth3ς3ρsQsr,Px1′1=Fhsu12λth1τ1ρsQsu1,Px1′3=Pu1,out2.Moreover, by using the same approach in (25), Px1′2 is obtained as
(45)Px1′2=Ξ1−Ξ1mru1Ωru1mru1exp−Ξ2τ2Γmru1∑m=0kΞ2τ2ρrmm!∑n=0mmn1ρrQsu1m−nΞ2τ2ρr+mru1Ωru1−n−mru1Γn+mru1.With (43) and (44), Pu1,outno,2 is finally obtained.


## 3.4.2. OP ofU2

As forU2, the outage event occurs in situations as follows. First, the decoding of x1 at R fails. Second, the decoding of x2 fails given the success decoding of x1. Third, under the assumption of successfully detecting x1 and x2 at R, the decoding of x3 at R is failed. Finally, the detection at R is successful, but the decoding of x3 or x2 fails at U2. Therefore, the OP can be derived as
(46)Pu2,outno=P1+1−P1×P2+1−P1×1−P2×P3×P4+1−P1×1−P2×1−P3×P5+1−P5×P4,where P4 and P5 denote the OP of decoding x2 and x3 at U2, respectively. The detailed expressions are
(47)P4=Fhru22λth2τ2ρr,P5=Fhru22λth3τ3−τ2λth3ρr.


## 3.4.3. OP ofU3

As forU3, an outage first occurs if the decoding of x1 at R fails. Then, the decoding of x2 fails given the successful decoding of x1. Moreover, the decoding of x3 is failed when the detection of both x1 and x2 succeeds. At last, the detection at R succeeds, but U3 fails to detect x3. Hence, the overall OP is
(48)Pu3,outno=P1+1−P1×P2+1−P1×1−P2×P3+1−P1×1−P2×1−P3×P6,where P6 denotes the OP of detecting x3 at U3, which is
(49)P6=Fhru32λth3τ3−τ2λth3ρr.


## 4. Numerical Results

In this part, numerical simulations are performed to showcase the performance of the proposed cache-enabled NOMA-based ISTNs. First, the satellite link parameters (Qi) are shown in Table 1. In addition, we consider different Shadowed-Rician fading conditions for satellite links (hi), i.e., frequent heavy shadowing (FHS), average shadowing (AS), and infrequent light shadowing (ILS) [17]. The sets of channel coefficients bi,mi,Ωi are 0.063,0.739,8.97×10−4, 0.126,10.1,0.835, and 0.158,19.4,1.29 for FHS, AS, and ILS, respectively. Moreover, hsu1 and hsr are both modeled as ILS. Besides, mru1=2,Ωru1=1,mru2=2,Ωru2=1,mru2=1,Ωru2=1. Furthermore, the transmit power of S is twice that of R, namely, Ps=2Pr. A fixed power allocation scheme is applied that the power coefficients are set by ω1=0.6,ω2=0.4,τ1=1,τ2=0.3,τ3=0.7, and ς1=0.6,ς2=0.3,ς3=0.1. Moreover, the content popularity parameter is ξ=0.7, and the number of files is M=10. At last, the transmission rate requirement for x1, x2, and x3 is 0.5 (bps/Hz), 0.8 (bps/Hz), and 0.8 (bps/Hz), respectively.Table 1
The satellite link parameters (Qi).
ParametersValueOrbitdsjGEO (36000 km)Carrier frequencyf2 GHzCarrier bandwidthBc15 MHz3 dB angle ofφi0.3°Angle betweenU1 and satellite beam center φsu10.3°Angle betweenR and satellite beam center φsr0.8°Antenna gain of user and relayGj4 dBMaximal beam gainGmax52.1 dBReceiver noise temperatureT300°CFigures3 and 4 illustrate the outage performance of users for different schemes. First of all, the good match between the analytical and numerical results validates our derivations. As for the outage performance of U1, due to the same power control and signal transmission, the OP is the same in the first phase for both the proposed and traditional schemes. While in the second phase, it is shown that the OP for the scheme without caching has a turning point at SNR=10dB. The reason behind this is that the decoding at R fails more frequently with small SNRs (<10dB), causing less interference to U1; thus, the OP will drop when SNR increases. On the other hand, when the SNR becomes large, the decoding at R encounters much less failure, and then the OP will increase because the interference at U1 is more severe. For comparison, it is clear that the OP of proactive users (U2 and U3) in our scheme is lower than the scheme without caching. This is because the proactive users are affected by two factors, namely, the channel links and files’ popularity, in the proposed scheme, while they are influenced by the channel links of hru1, hru2, and hru3 in the scheme without caching.Figure 3
The OP ofU1 for the proposed scheme and the scheme without caching.Figure 4
The OP ofU2 and U3 for the proposed scheme and the scheme without caching.To evaluate the diversity order, we concentrate on OP values at SNR=25dB and SNR=30dB. Specifically, the comparison of the diversity order between the two schemes is shown in Table 2. Note that the diversity order of U1 shows no difference in two schemes, which is not the case when considering U2 and U3. In particular, if mru2<1 and mru3<1, then the diversity orders achieved are the same for two schemes. While for larger mru2 and mru3 values, the proposed scheme is superior to the counterpart. This observation lies in that the signal transmission of proactive users is subject to the channel quality of hru2 or hru3 for the proposed scheme, while it is impacted by all the channel links for the scheme without caching.Table 2
Diversity order comparison.
UserProposed schemeWithout cachingAnalysisSimulationAnalysisSimulationU1, first phase10.999911.0264U1, second phase0000U2mru22.0406min1,mru21.1606U3mru31.0116min1,mru30.9935Figure5 illustrates the influence of channel parameters on the OP of U1. In phase one, the OP is affected by hsu1, which is set as ILS or AS. It is clearly observed that the OP reduces when hsu1 changes from AS to ILS, since the channel condition becomes better. In contrast, the OP of U1 in the second phase is decided by both hsu1 and hru1. The channel quality concerning hru1 is tuned by modifying parameter mru1. It is demonstrated that the OP almost keeps unchanged when mru1 varies from 1 to 2, indicating that hru1 has little impact on the OP of U1. Furthermore, the impacts of hru2 on U2 and that of hru3 on U3 are depicted in Figure 6, where phase two is considered. As expected, the outage event occurs more frequently if the channel condition gets worse.Figure 5
The impact of channel parameters on the OP ofU1 in two phases.Figure 6
The impact of channel parameters on the OP ofU2 and U3 in phase two.Figure7 depicts the hit probability of R, where different content popularity parameters are included. The number of content files M is set as 10, and ξ varies from 5 to 1, and then to 0.1. A large ξ indicates the request for files with high popularity, whereas a small one corresponds to the requests with heavy-tailed popularity. Hence, a higher hit probability is obtained with a larger ξ, which is consistent with the numerical results shown in Figure 7. It is also worth noting that enhancing the SNR value is beneficial to level up the hit probability.Figure 7
Hit probability of the proposed scheme.In the simulations above, we assume that the perfect CSI is available to all nodes. However, in practice, the channel estimation can never be error-free; thus, it is necessary to investigate the impact of uncertainty in CSI on the outage performance. To this end, Figure8 draws the outage performance of U1 in phase one, where the CSI uncertainty is modeled as a Gaussian noise independent of the actual channel coefficients. Note that the independence between the actual CSI and estimation error holds if the MMSE channel estimation method is employed. In this figure, the case of zero variance indicates the perfect CSI acquisition. From this figure, the CSI uncertainty harms the outage probability, and the larger the uncertainty is, the higher the outage probability will be.Figure 8
OP ofU1 in phase one subject to the channel estimation error.


## 5. Conclusion

In this paper, a NOMA-based ISTN with wireless caching is proposed to reduce the transmission delay and improve spectrum efficiency. Specifically, the overall signal transmission includes two phases. In the first phase, the active user and cache-enabled relay are served by the satellite using NOMA protocol. In the second phase, the proactive users are served by the relay, and at the same time, the active user is served by the satellite. The exact-form and asymptotic OP, diversity order, and the hit probability are derived. In addition, the impact of key parameters on the system performance is discussed. Both simulation and analytical results are provided to validate the superior performance of the proposed scheme over the conventional one.



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*Source: 6788449-2022-09-13.xml*