Meta Distribution and Secrecy of Partial Non-Orthogonal Multiple Access (NOMA) in Poisson Networks

This work studies the meta distribution in a partial-NOMA network to obtain ﬁne-grained information about the network performance. As the meta distribution is approximated using the beta distribution via moment matching of the ﬁrst two moments, reduced integral expressions are derived for the ﬁrst two moments of the meta distribution. Accurate approximate moments are also proposed to further simplify the calculation. Security is an issue in partial-NOMA because the strong user may decode the weak user’s message in the process of decoding its own message using ﬂexible successive interference cancellation (FSIC). Therefore, a measure of secrecy is deﬁned in this context and the secrecy probability is derived for the case of: 1) a malicious strong user that prioritizes eavesdropping, 2) an innocent strong user that decodes the weak user’s message only when it is required to do so. The obtained results highlight the superiority of partial-NOMA over traditional NOMA in terms of secrecy. They also show that receive ﬁltering and FSIC have a signiﬁcant positive impact on the secrecy of partial-NOMA. Furthermore, partial-NOMA with a small overlap of the resource-block can secure the network from the additional deterioration a malicious eavesdropper may cause.


I. INTRODUCTION
Orthogonal multiple access (OMA) is used to avoid interference between the users' equipment (UEs) being served by a base station (BS) by allocating each UE orthogonal resources. This is performed by allocating each UE a unique time slot and/or frequency channel. The available time and frequency resources are split into a grid of what are referred to as time-frequency resource-blocks [1]. Each UE in OMA can be exclusively allocated one or more resourceblocks. In contrast to OMA, non-orthogonal multiple access (NOMA) allows multiple UEs to share a resource-block by superposing their messages in the power domain, i.e., NOMA UEs transmit their messages using the same resource-block, but at different power levels. Successive interference cancellation (SIC) is used to decode the messages of NOMA UEs. Sharing a resource-block by multiple UEs in NOMA improves the spectral efficiency at the expense of introduction of interference between the UEs being served by the same BS [2], [3]; the interference in this case is referred to as intracell interference. In NOMA, the resource allocation for each UE is based on some measure of channel strength, such that decoding is easier for UEs with weaker channels. Most of the existing work on NOMA orders the UEs channel quality based on the received signal mean power [4]- [9], the quality of the transmission channel such as the fading coefficient [10]- [13], the fading-to-noise ratio [14], the instantaneous received signalto-intercell-interference-and-noise ratio [15], and the instantaneous received signal-to-intercellinterference ratio [16]. Such UE ordering and appropriate resource allocation allow using SIC where a UE decodes messages of UEs weaker than itself and treats the messages of UEs stronger than itself as noise. The achievable sum throughput of NOMA is superior to OMA, which has been extensively shown in the literature [5], [7], [8], [12]- [19].
While NOMA is based on the complete sharing of a resource-block by multiple UEs to improve throughput, the introduced intracell interference deteriorates UE coverage. On the contrary, OMA has no intracell interference, which results in superior coverage, but no spectrum reuse can be applied to UEs served by a BS, which results in lower throughput. In [20], partial-NOMA was introduced as a flexible technique between the two extremes of OMA and NOMA to provide better coverage than NOMA while providing better throughput than OMA. In contrast to NOMA, where UEs share the entire resource-block, partial-NOMA UEs share only a fraction α of the resource-block [20, Fig. 1], thus allowing some spectrum reuse while limiting the intracell interference encountered by UEs. Such a setup is general and allows flexibility to control how to enhance security [30]- [37]. This is typically based on exploiting the random fluctuations in the interference power at the receiver and eavesdropper that give rise to opportunities for secure information transmission. Techniques such as jamming have also been used to improve physical layer security [37]. Partial-NOMA is interesting to study in this context as the nature of the technology can provide additional physical layer security, particularly over traditional NOMA setups. This is because partial-NOMA UEs only share a part of the resource-block making the rest of the information inaccessible, compared to NOMA where information from the entire resource-block is accessible to all UEs. Additionally, receive filtering further suppresses the message not intended for a receiver, making decoding harder for an eavesdropper. While the strong UE always decodes the message of the weaker UE in NOMA, in partial-NOMA with FSIC this is not always the case thereby improving the security of the weaker UE. These factors makes studying the physical layer security achievable in a partial-NOMA setup interesting. It also helps highlight the impact of network parameters on physical layer security. In the second part of this work, we use stochastic geometry to study physical layer security in a large cellular network employing partial-NOMA and compare this with traditional NOMA. Additionally, we study the impact that receive filtering and FSIC have on improving physical layer security.
The main contributions of this paper can be summarized as follows: • We obtain integral expressions for the moments of the meta distribution for a partial-NOMA setup. For the first two moments, which are required to approximate the meta distribution, we are able to reduce the integrations required.
• We propose accurate approximate moments of the meta distribution to further simplify the integral calculation. • We study the physical layer security achievable in a partial-NOMA setup for the weak UE as the strong UE may be able to decode its message. We define 'secrecy' as an event of secure communication for the case of: 1) a malicious strong UE that tries its best to decode the weak UE's message, 2) an innocent strong UE that only decodes the weak UE's message when it is required. We provide the mathematical analysis for the secrecy probability achievable in each case. Such scenarios shed light on the maximum damage that can be done in terms of physical layer security and the ideal situation of innocent users. We find that at small values of the overlap, the malicious eavesdropper is unable to deteriorate secrecy more than its innocent counterpart, highlighting the physical layer security provided by the nature of the partial-NOMA setup to the network.
• We compare the secrecy probability for a partial-NOMA setup employing receive filtering and FSIC with: 1) a setup that always employs traditional SIC, 2) a setup that does not employ receive filtering and 3) traditional NOMA. We find that not employing FSIC or receive filtering has a drastic negative impact on secrecy at lower α values, highlighting their importance on not just coverage but also secure communication. We also observe that partial-NOMA (α ≤ 1) is able to achieve much higher secrecy than traditional NOMA (α = 1).
The rest of the paper is organized as follows. The system model is described in Section II. The signal-to-interference ratio (SIR) analysis for the meta distribution of the coverage probability is studied in Section III. In Section IV, the SIR analysis for physical layer security is studied.
The results are presented in Section V and the paper is concluded in Section VI.
Notation: We denote vectors using bold text, z is used to denote the Euclidean norm of the vector z and b(z, R) denotes a ball centered at z with radius R. The ordinary hypergeometric function is denoted by 2 F 1 . The Laplace transform (LT) of the PDF of the RV X is denoted by is the statistical expectation. The probability is denoted as P, the indicator function, denoted as 1 A , to have value 1 when event A occurs and to be 0 otherwise.

A. Network Model
This work considers a downlink cellular network where BSs are distributed according to a homogeneous Poisson point process (PPP) Φ with intensity λ. As a large network is being studied in this work, we assume an interference-limited regime. A BS serves two UEs in each resource-block via partial-NOMA using a total power budget of P = 1. To the network we add a BS at the origin o, which under expectation over Φ, becomes the typical BS serving UEs in the typical cell. In the remainder of this work, we study the typical cell. Since Φ does not include the BS at o, the set of interfering BSs for the UEs in the typical cell is denoted by Φ. The distance between the typical BS at o and its nearest neighboring BS is denoted by ρ. Since Φ is a PPP, the pdf of ρ is f ρ (x) = 2πλxe −πλx 2 , x ≥ 0. Consider a disk around the BS at o with radius ρ/2, i.e., b(o, ρ/2); this is referred to as the in-disk [9], [15], [20]. The in-disk is the largest disk centered at a BS that fits inside its Voronoi cell. We study a model where the two partial-NOMA UEs are distributed uniformly and independently at random in the in-disk b(o, ρ/2) of the BS at o. The rationale behind using such a model where UEs are not too far from the serving BS in setups where each UE does not have an individual dedicated resource-block was shown in [9].
We assume a Rayleigh fading environment such that the fading coefficients are independent and identically distributed (i.i.d.) with a unit mean exponential distribution. A power-law pathloss model is considered where the signal decays at the rate r −η with distance r, η > 2 denotes the path-loss exponent and δ = 2 η . Fixed rate transmissions are used by the BSs where the transmission rate of each UE can be different. Such transmissions result in effective rates, referred to as the throughput of the UEs, that are lower than the transmission rate because of outage.

B. Partial-NOMA Model
A BS serves two UEs in each resource-block via partial-NOMA by multiplexing the signals for each UE with different power levels using the total power budget. While in traditional NOMA, the two UEs have complete access to the full resource-block, i.e., the entire time slot and the whole frequency channel, each UE in partial-NOMA has access only to a part of the resourceblock. The fraction of the resource-block the two UEs share is denoted by α. In particular, in our work, the UEs have full access to the time slot while they share an overlap α of the frequency channel. It should be noted that another way to achieve an overlap α of the resource-block is by having full access to the frequency channel for each UE and only an overlap α of the time slot. Such an overlap scenario is beyond the scope of this work. We refer to the fraction of the resource-block accessible to only UE 1 by β, where 0 ≤ β ≤ 1 − α. Thus, the fraction of the bandwidth available to UE 1 is BW 1 = α + β. The remaining fraction of the bandwidth 1 − α − β is available solely to UE 2 . The total fraction of the bandwidth thus available to UE 2 is BW 2 = 1 − β. With a slight abuse of notation, in the remainder of the manuscript, we will refer to the overlap α in the frequency channel of the resource-block simply as an overlap α of the resource-block. As the entire time slot is available to both UEs, we will disregard this aspect when referring to the partial overlap of a resource-block.
An overlap in the frequency domain allows implementing filtering at the receiver side to further suppress interference. A matched filter that has a Fourier transform equal to the complex conjugate of the Fourier transform of the transmitted signal is used [20], [38], [39]. In this work, we assume that square pulses are used for transmissions of both UEs. Receive filtering results in any message that has an α-overlap with the UE of interest to be scaled by an effective interference factor 0 ≤ I(α, β) ≤ 1. From [20], the effective interference factor as a function of β and the overlap α is calculated as where the center frequency of UE 1 's message is f a = α+β 2 and UE 2 's message is f b = 1+β 2 . The factors E i for i ∈ {1, 2} are used to scale the energy to 1 and are calculated as As 0 < I(α, β) < 1, receive filtering results in suppression of the interference from the other UE partially sharing the resource-block. Since any message that has an α-overlap with the UE of interest is scaled by I(α, β), not only does receive filtering suppress intracell interference, but also reduces intercell interference.
Similar to NOMA, partial-NOMA requires ordering UEs based on some measure of channel strength. This is required for both resource allocation and decoding. In this work, we order the UEs based on the link distance, R, between the typical BS at o and its UEs uniformly distributed in the in-disk with radius ρ; the link distance is thus conditioned on ρ. Ordering UEs based on increasing link distance is equivalent to ordering based on the decreasing received mean signal power, i.e., R −η . From hereon, we refer to the strong (weak) UE, with the shorter (longer) link distance, as UE 1 (UE 2 ). As the order of the UEs is known at the BS, we use ordered statistics for the pdf of R i , the ordered link distance of UE i , where i ∈ {1, 2}. Using the theory of order statistics [40], in the typical cell since the power allocated to a UE in a resource-block is fixed over the resource-block and as the sum power of the two UEs can never exceed the power budget such that P 1 + P 2 = 1. Note that in the non-overlap areas of the resource-block, the power being used will be less than the power budget. The throughput of UE i , i ∈ {1, 2} is defined as where C i is the event that UE i is in coverage. The cell sum throughput of the typical cell is thus As BW i is a function of both α and β, the resources to be allocated in a partial-NOMA setup for a given α are P 1 = (1 − P 2 ), θ 1 , θ 2 and β.

C. SIRs Associated with Partial-NOMA and Coverage Events
Since partial-NOMA uses FSIC decoding, there are multiple SIRs of interest. For the twouser downlink partial-NOMA setup we require SIR i j , the SIR for decoding the j th message at UE i where i ≤ j and the messages of all UEs weaker than UE j have been removed while the messages of all UEs stronger than UE j are treated as noise. In particular, these are The fading coefficient from the serving BS (interfering BS) located at o (x) to UE i is h i (g y i ). For notational convenience, the intercell interference scaled to unit transmission power by each interferer is defined as intercell interference in the partial-NOMA setup is lower than its traditional counterparts. Additionally, since the network model conditions an interferer to exist at a distance ρ from the typical BS at o, we can rewrite I ø i as Note that as there is no interfering BS inside b(o, ρ), the nearest interfering BS from UE i is at least ρ − R i away. As ρ − R i > R i , the in-disk model offers a larger guard zone than the usual guard zone of link distance for UEs in a downlink Poisson network [15].
While SIR 1 1 is the SIR associated with UE 1 decoding its message after the message of UE 2 has been decoded and removed, FSIC also allows UE 1 to decode its own message while treating the message of UE 2 as noise. The SIR associated with UE 1 for decoding its own message when the message of UE 2 has not been removed is As FSIC decoding for UE 2 involves decoding its own message while treating the interference from the message of UE 1 as noise, the event of successful decoding at UE 2 is defined as FSIC decoding for UE 1 , on the other hand, is the joint event as described in Section II-B. The event of successful decoding at UE 1 is thus defined as wherē The event of successful decoding at UE i is thus of the form

III. SIR ANALYSIS OF THE META DISTRIBUTION
As mentioned, network operators and vendors are often interested in the percentile performance of UEs, where the fading and activity, if any, change while the network realization is kept constant. In this regard, the CCP was defined as the coverage probability given a fixed network realization [27]. For a fixed, yet arbitrary, realization of the network P C i , the CCP of UE i in a partial-NOMA setup, is defined as, where (a) follows by using the definition of C i in (11) and the cdf of h i ∼ exp(1). Using the MGF of the independent RVs g y i ∼ exp(1), (b) is obtained.
The requirement for more fine-grained information on performance leads to the notion of studying the distribution of the CCP. The meta distribution was thus defined as the ccdf of the CCP [27]. The meta distribution for UE i can be written as The b th moment of the CCP of UE i , by definition, can be calculated using (12) as Note that by definition, the SCP of UE i is the first moment of the CCP of UE i , i.e., M i,1 Calculating the meta distribution is difficult. To combat this, the beta distribution using moment matching was proposed as a very accurate approximation for the meta distribution [27]. This approach only requires the first two moments of the CCP, i.e., M i,1 and M i,2 . In particular, where The variance of the meta distribution of UE i is defined as

A. Exact Moments of the Meta Distribution
This subsection evaluates the moments of the meta distribution of UE i .
Proof: By writing M i,b according to the definition in (13) and separating the intercell interference along the lines of (5), we have From this, we arrive at the first term of (16) by using the the probability generating functional (PGFL) of the PPP; since the in-disk model allows a larger guard zone, the lower limit on the distance from the nearest interferer, in the inner integral A i,b , is ρ − R i . The average distance between a UE in the in-disk and the interfering BS ρ away from o is approximated to be ρ, which was validated as a tight approximation [8], [15]. The approximation in (16) comes from the use of this approximation in the denominator of the second term.
While M i,b for general b requires a triple integration according to (16), for the cases of b = 1 and 2 we are able to reduce the calculation by one integration. In particular, closed-form (16) Corollary 2: The inner integral for calculating the second moment of the CCP of UE i , M i,2 , is

B. Approximate Moments of the Meta Distribution
Another approach to calculating the moments of the CCP is based on the relative distance process (RDP) [41]. Since the partial-NOMA setup deals with ordered link distances, we will use the ordered RDP. The ordered RDP for UE i is defined as Using the ordered RDP for UE i , we can rewrite the moments of the CCP of UE i in (13) as As (13) is in terms of the PPP Φ, evaluating M i,b in (16) required the PGFL of the PPP.
Note that the distribution of the ordered link distance R i and the distribution of the ordered RDP R i are conditioned on ρ. Consequently, the PGFL of R i is also conditioned on ρ.
DRAFT Proof: By definition of the PGFL we have Similar to (5), (a) is obtained by separating the interferers into two types, the interferers that are farther than ρ from o and the interferer conditioned to be a distance ρ away from o. From this, we obtain the first term in (21) using the PGFL of the PPP.
Unfortunately, it is not possible to obtain closed-form expressions for (21). In this subsection, along the lines of [9], we propose the use of two approximations to relax the constraints and simplify the calculation of the PGFL of the ordered RDP for UE i . In particular, the two approximations used are: • A1: The guard zone around the UEs is approximated to be of radius R i although the largest guaranteed guard zone has radius ρ − R i . As ρ − R i > R i this approximation overestimates the intercell interference encountered by the UEs.
• A2: Deconditioning on the BS located a distance ρ from o. This approximation underestimates the intercell interference experienced by the UEs.
Note that the two approximations have opposing effects on how much intercell interference is accounted for. Using these approximations we can calculate the PGFL of R i in closed-form.

Lemma 3:
The PGFLs of the ordered RDP for UE 2 and UE 1 , respectively, using approximations A1 and A2 are Proof: Along the lines of the proof of Lemma 2, the PGFL of the ordered RDP for UE i using A1 and A2 is Here the second term in (21) has been removed due to A2. Additionally, the lower limit of the integral of the first term in (21) is updated to reflect R i , the radius of the guard zone in A1.

Lemma 4:
The b th moments of the CCP for UE 2 and UE 1 , respectively, using approximations A1 and A2 are Proof: Using the definition in (20), we have (23) and (22) for UE 1 and UE 2 , respectively, are then plugged into the equation above. We arrive at (24) and (25) using where (a) is obtained using z → g −1 .

IV. SIR ANALYSIS FOR PHYSICAL LAYER SECURITY
In this section we focus on physical layer security which is based on exploiting the nature of the wireless network to enhance security. As mentioned, this involves exploiting random fluctuations in the interference power at the intended receiver and eavesdropper. In particular, in instances when the eavesdropper receives a deteriorated version of the signal while the legitimate receiver receives a strong signal, the transmitter can send the message of interest at a transmission rate higher than the capacity of the eavesdropper link. This will lead to the event of opportunistic secure spectrum access (OSSA) defined in [37] where the eavesdropper cannot decode the message while the legitimate receiver can. In this work we define the secrecy probability as DRAFT the probability of OSSA and use this as our metric for measuring the physical layer security our setup can achieve.
Partial-NOMA is susceptible to eavesdropping for two main reasons: 1) the overlap α of the resource-block shared by the two UEs, 2) the use of FSIC which at times requires UE 1 to decode the message of UE 2 . While in theory a malicious UE 2 could attempt to decode the message of UE 1 as well, in this work we focus on the security of the message of UE 2 . Thus in the context of our work, we are concerned about the security of the message intended for UE 2 , i.e., UE 2 is the legitimate receiver and UE 1 is the eavesdropper. We study the following two scenarios: • UE 1 is malicious and prioritizes decoding the message of UE 2 .
• UE 1 is innocent and its main goal is not eavesdropping. It therefore only decodes the message of UE 2 when it is required for decoding its own message.
To be explicit, in our setup, we define secrecy as the event that UE 2 is able to decode its own message and that UE 1 is unable to decode the message of UE 2 .
Before delving into the secrecy probabilities, we introduce the LT of the intercell interference encountered by the UEs. The LT of I ø i , the intercell interference at the typical UE i scaled to unit transmission power, conditioned on R i and ρ was approximated in [15,Lemma 1], [20] as Consequently, LĨø i |R i ,ρ (s) = L I ø i |R i ,ρ P i s , where P i = (P i + (1 − P i )I(α, β)). 1) Malicious UE 1 : In the case of a malicious UE 1 , secure communication is achieved when UE 2 is in coverage (i.e., the event C 2 occurs) and UE 1 is in one of the following situations: i) UE 1 can only decode its own message while treating the message of UE 2 as noise and also cannot extract the message of UE 2 .
ii) UE 1 is unable to decode its own message and also cannot extract the message of UE 2 .
Note that as UE 1 is malicious, we consider the possibility of UE 1 decoding UE 2 's message even if it cannot decode its own message after removing the message of UE 2 . Based on these, we can write the secrecy probability of a malicious UE 1 as Theorem 1: The secrecy probability when UE 1 is a malicious eavesdropper is where L I ø i |R i ,ρ (s) for i ∈ {1, 2} is given in (26). Proof: Based on (28), we can rewrite the secrecy probability as Here where (a) is obtained using the cdf of h i ∼ exp(1), i ∈ {1, 2}. Using the LT of I ø i conditioned on R i and ρ, we arrive at (b). Similarly, 2) Innocent UE 1 : In the case of a rather innocent UE 1 , secure communication is achieved when UE 2 is in coverage (i.e., the event C 2 occurs) and UE 1 is in one of the following situations: i) UE 1 can only decode its own message while treating the message of UE 2 as noise.
DRAFT ii) UE 1 can decode both messages and it can also decode its own message while treating the message of UE 2 as noise. However, the latter is chosen as it is easier for UE 1 .
iii) UE 1 is unable to decode its own message, i.e., it is in outage.
Based on these, we can write the secrecy probability of an innocent UE 1 as Theorem 2: The secrecy probability when UE 1 is an innocent eavesdropper is where L I ø i |R i ,ρ (s) for i ∈ {1, 2} is given in (26). Proof: Based on (30), we can rewrite the secrecy probability as Along the lines of the proof of Theorem 1 In addition to studying the secrecy probability for partial-NOMA that employs receive filtering and FSIC, it is also necessary to highlight the impact that receive filtering and FSIC have on secrecy. We thus study the following two cases for both the innocent and malicious UE 1 : (I) When SIC is used by UE 1 instead of FSIC, i.e., UE 1 always decodes and removes the message of UE 2 before decoding its own message.
(II) When receive filtering is not employed prior to the FSIC decoding. In this scenario, I(α, β) takes on the value 1 for all values of α and β.

Corollary 3: The secrecy probability of a malicious UE 1 in (I) is
Proof: With SIC the terms in (29) from A mal become 0. Additionally, as SIC does not allow UE 1 to decode its own message while treating the message of UE 2 as noise,M 1 = M 1 . Corollary 4: The secrecy probability of an innocent UE 1 in (I) is Proof: Following the proof of Coroll. 3, the terms in (31) from A inn become 0 andM 1 = M 1 .

Corollary 6: The secrecy probability of an innocent UE 1 in (II) is
Proof: Following the proof of Coroll. 5, the terms in (31) from A inn become 0 and P i | I(α,β)=1 =

1.
In Section V-B, we also compare the secrecy probability achievable with traditional NOMA (i.e., α = 1) for the both the case of innocent and malicious UE 1 to highlight the benefit that adapting partial-NOMA (i.e., α ≤ 1) can have on physical layer security.

V. RESULTS
In this section, we consider BS intensity λ = 10 and η = 4. Simulations are repeated 10 5 times. As the power budget is P = 1, P 2 = 1 − P 1 . Fixed resource allocation is used in some of the figures while the other figures use the optimum resource allocation associated with a problem that aims to maximize cell sum throughput while constrained to a threshold minimum throughput (TMT) according to [20,Algorithm 1]. Note that solving such a problem results in resource allocation such that the minimum required resources are spent on UE 2 to attain throughput equal to the TMT and the remaining resources are given to UE 1 to maximize its throughput with. In Section V-A, the exact moments of the meta distribution are used unless specified otherwise.
A. Meta Distribution of Partial-NOMA shows the impact of increasing P 1 on the meta distribution. In particular, when P 1 = 0.1, 81.8% of UE 1 and 79.1% of UE 2 achieve a coverage probability of at least 0.8. Increasing P 1 to 1/3 improves the percentage to 93% for UE 1 and degrades it to 78.2% for UE 2 . As anticipated, increasing P 1 (decreasing P 2 ) increases the percentage of UE 1 and decreases the percentage of UE 2 that can achieve a certain coverage probability. We observe that the impact on UE 1 is higher than that on UE 2 in this scenario. This can be attributed to the fact that the transmission rate for UE 2 is lower in this scenario. Additionally, UE 1 generally has to make more of an effort in decoding its message as it has to decode a message with lower power and at times has to decode two messages, thus the impact of changing P 1 is more drastic for UE 1 .    We observe that the meta distribution of the UEs is not plotted for α = 0.6. This is because for  in general, when resource allocation is fixed, increasing α deteriorates the performance of UE 2 as the percentage of UE 2 that can achieve atleast a certain coverage probability µ decreases.
We also observe that the amount of deterioration increases at first (from α = 0 to 0.8) and then decreases (from α = 0.8 to 1). The deterioration of UE 2 's performance with α can be attributed to the increasing interference associated with increasing I(α, β). The trend in amount of deterioration corresponds to the rate at which I(α, β) grows with α. The performance of UE 1 , on the other hand, deteriorates as α increases from 0 to 0.5 and then improves from α = 0.5 to α = 1. Note that α = 0.6 occurs just in between the switch from performance deteriorating with α to improving with α. The performance trend of UE 1 is more complex as at lower α values the message of UE 2 is treated as noise; increasing α thus increases I(α, β) and therefore the interference encountered from the message of UE 2 . At α = 0.6, I(α, β) is too high for the message of UE 2 to be treated as noise but too low to decode the message of UE 2 before decoding the message of UE 1 ; thus outage is experienced. After α = 0.6 the message of UE 2 is decoded by UE 1 before decoding its own message; increasing α and therefore I(α, β) in this case increases the power of the message of UE 2 making it easier to decode and improving the performance of UE 1 . Fig. 4 is a plot of the meta distribution for different α values using the optimum resource allocation associated with a TMT value of 0.25. Unlike Fig. 3, the resource allocation here is not the same for all α values. This is also why unlike Fig. 3, α = 0.6 is not in outage. The performance of UE 2 first decreases from α = 0 to 0.1 and then increases from α = 0.1 to 0.5.
The initial decrease in performance is attributed to I(α, β) increasing from 0. The performance increase after this is associated with the fact that increasing α increases the bandwidth of UE 2 , resulting in the requirement of a lower θ 2 to achieve a TMT of 0.25. From α = 0.5 to 0.9 we observe a decrease in performance; this is due to the dominance of the growing impact of I(α, β) increasing interference. From α = 0.9 to 1 there is a slight improvement in performance of UE 2 ; although I(α, β) = 1 (the maximum), the small improvement comes from the bigger gain in bandwidth which decreases the θ 2 required thereby improving performance.
While the goal of UE 2 is to use the minimum resources to achieve TMT, the goal of UE 1 is to achieve the largest possible throughput. The performance in terms of the meta distribution for UE 1 in Fig. 4 initially increases from α = 0 to 0.3, as the increasing bandwidth reduces the θ 1 required to achieve maximum throughput, resulting in an improvement of performance in terms of the meta distribution. An abrupt decrease is seen in the performance of UE 1 from α = 0.3 to 0.4, this corresponds to a switch in decoding so that at α = 0.4, UE 1 decodes the message of UE 2 before decoding its own message. As I(α, β) is still not very high, P 2 increases a lot to decode the message of UE 2 ; this in turn leaves behind less P 1 for the message of UE 1 , thereby deteriorating the performance of UE 1 . An increase in performance is seen from α = 0.4 to 0.5.
This coincides with the increase in P 1 as less P 2 is required by UE 1 to decode the message of UE 2 as I(α, β) grows. The performance again decreases from α = 0.5 to 0.6; this is because θ 1 increases. Note that while performance here decreases in terms of the percentage of UE 1 that can achieve at least a certain minimum coverage µ, θ 1 increases because a larger throughput can be supported because of the increasing bandwidth. The performance in general increases from 0.6 to 1 with minor fluctuations as the increasing bandwidth for both UEs allows more power to be left for UE 1 , resulting in improved performance. We observe that for both UEs, at lower values of θ the approximate moments underestimate the SCP while at higher θ they overestimate the SCP. The fact that at some values the approximate moments underestimate the exact and overestimate at others comes from the opposing nature of approximations A1 and A2 and the impact they have. It should be noted that overall in terms of SCP, the approximate is close to the exact. We observe that the approximate moments always seem to overestimate the variance of the meta distribution. The difference between the exact and approximate variance grows with θ and then drops when θ is too high and all the UEs are   in outage. This increase in difference with θ can be attributed to the fact that when θ is low more UEs can guarantee similar performance thus the variance is lower, and the error in the approximate variance grows with θ. increases with α while the variance of UE 1 first increases and then decreases with α. In Fig. 6b, we observe that using the approximate moments we get a very close match to the exact SCP.
The variance of the meta distribution obtained using the approximate moments is less tight but follows the same trends as the exact moments, overestimating them for all α.  fixed, the variance of UE 1 increases and then decreases. We observe that the maximum variance does not correspond with the optimum θ 1 which maximizes the throughput of UE 1 for either α value. We also observe that while in Fig. 7a, the SCP of UE 1 decreases continuously with θ, this is not the case in Fig. 7b, where from θ 1 = 8 dB to 16 dB, the SCP is constant. The variance of UE 1 in Fig. 7b for α = 0.4 is also constant in this range of θ 1 unlike its α = 0.2 counterpart. This happens because for α = 0.4 at θ 1 = 8 dB, the decoding technique switches from UE 1 treating the message of UE 2 as noise to UE 1 decoding and removing UE 2 's message before decoding its own. As P 1 (= 1 − P 2 ) and θ 2 are fixed, it appears that until θ = 16 dB, the resources are more than sufficient for UE 1 such that increasing θ 1 does not reduce SCP. Note that α = 0.2 does not have a switch in decoding technique as the lower value of α makes it difficult for UE 1 to decode the message of UE 2 and thus UE 1 always treats the message of UE 2 as noise.  B. Physical Layer Security in Partial-NOMA Fig. 9 is a plot of the secrecy probability vs. α for the case of an innocent and a malicious UE 1 using different values of θ 1 and θ 2 . The figure validates our analysis in Section IV as the simulations are a tight match. For both the case of the innocent and the malicious UE 1 , the secrecy probability is high at low α. This happens due to the physical layer security provided by the nature of partial-NOMA which makes it difficult to decode the message of the other UE at low α due to low I(α, β). As α increases, a steep drop in secrecy probability is observed for both cases. The drop in secrecy probability occurs at lower α for the malicious UE 1 than its innocent counterpart, highlighting the increased susceptibility to eavesdropping in the case of a malicious UE 1 . Additionally, at higher α, the secrecy probability in the case of the malicious UE 1 is lower. As the secrecy probability of the innocent and malicious UE 1 is identical and large at low α, it should be highlighted that partial-NOMA using low α can be used to reduce security threats from a malicious eavesdropper to the levels of an innocent eavesdropper; this is in contrast to what can be done in the case of traditional NOMA (α = 1).
In Fig. 9 we also observe that decreasing θ 2 increases secrecy probability for the innocent UE 1 case as it becomes easier for UE 2 to decode its message. The secrecy for the malicious UE 1 case also increases at low α due to this. At higher α, however, the secrecy probability decreases as it also becomes easier for the malicious UE 1 to decode UE 2 's message. Decreasing θ 1 decreases the secrecy probability for the innocent UE 1 case at high α as decoding becomes easier for UE 1 and in the process it decodes UE 2 's message, decreasing secrecy. curves represent the setup that employs both receive filtering and FSIC (i.e., the partial-NOMA decoding approach). For this approach and (I), the secrecy probability of the innocent and malicious UE 1 at low α is identical, while at high α the malicious UE 1 has lower secrecy probability. For (II) and NOMA, the malicious UE 1 always has lower secrecy probability than the innocent UE 1 . At low α, we observe a drastic difference between the secrecy probabilities for a partial-NOMA setup that employs receive filtering and FSIC compared to the setups in (I) and (II) that do not employ both of these, highlighting the importance of each on the physical layer security in a network. While the performance of (II) is unaffected by α since I(α, β) is set to 1 and fixed resource allocation is used, we observe that (I) has a local maxima at α = 0.5.

DRAFT
This occurs because at α < 0.5, the low I(α, β) results in a negativeP 1 2 (i.e., M 1 = 0) and thus transmission does not occur in (I); we therefore have zero secrecy in this range for both the case of the malicious and the innocent UE 1 . When α ≥ 0.5, M 1 > 0 and we have positive secrecy probability. As α increases, the malicious UE 1 's secrecy probability decreases because decoding the message of UE 2 becomes easier for UE 1 with increasing I(α, β). Additionally, the increasing I(α, β) also increases interference for UE 2 making it harder for UE 2 to decode its own message. In the case of the innocent UE 1 , although the increasing I(α, β) makes decoding harder for UE 2 , the impact of the increasing I(α, β) increasing intercell interference and making it harder for UE 1 to decode the message of UE 2 is stronger. This results in a slow increase in the secrecy probability in the innocent UE 1 case as α increases from 0.5. Note that after α = 0.5 (α = 0.6) the malicious (innocent) UE 1 's secrecy probability in (I) coincides with its counterpart that uses both FSIC and receive filtering. This occurs due to the switch in decoding with FSIC at high α which obligates UE 1 to decode and remove the message of UE 2 , like in the case of (I). We also observe that the secrecy probability of partial-NOMA at lower α significantly outperforms that of NOMA for both the case of the malicious and innocent UE 1 , highlighting the significance of a partial overlap. Fig. 11 is a plot of the secrecy probabilities vs. α using the optimum resource allocation associated with each α for a TMT of 0.25. Partial-NOMA outperforms traditional NOMA (i.e., α = 1) in terms of secrecy probability for all α values in case of both the innocent and malicious UE 1 . For each scenario the secrecy probability for the malicious UE 1 case never exceeds the innocent UE 1 case. Similar to Fig. 10, the black and the blue curves for both the cases of innocent and malicious UE 1 coincide at higher α values as in this regime UE 1 employing FSIC also always decodes and removes the message of UE 2 , as in the case of SIC, i.e., (I). Before this range of α, the secrecy probability using (I) is zero for both the case of the innocent and malicious UE 1 asP 1 2 < 0 ,i.e., M 1 = 0, and thus transmission does not occur. In the case of (II) where there is no receive filtering ,i.e., I(α, β) = 1, at low α the bottleneck is UE 2 decoding its own message due to the high interference it encounters from UE 1 's message. This results in outage for UE 2 until α = 0.3 and therefore the secrecy probability is zero. Due to I(α, β) = 1, M 1 is never M 0 , i.e., UE 1 can never decode its own message without decoding UE 2 's message.
For the malicious UE 1 , secrecy therefore comes from UE 1 being in outage and unable to decode UE 2 's message, while in the case of the innocent UE 1 , by UE 1 simply being in outage. The high I(α, β) = 1 makes the bottleneck of UE 1 's outage decoding its own message as decoding the message of UE 2 is easy due to the high I(α, β). Thus, the secrecy probability for the case of the malicious UE 1 in (II) is much lower than the innocent as decoding the message of UE 2 is easy for the malicious UE 1 even when it cannot decode its own message.

VI. CONCLUSION
In the first part of this paper, the meta distribution of a partial-NOMA network is studied to obtain fine-grained information on the network performance. Integral expressions are obtained for the moments of the meta distribution. We are able to reduce the integrals for the first two moments; these are the two moments required for approximating the meta distribution using the beta distribution via moment matching. By proposing the use of two approximations, accurate approximate moments of the meta distribution are derived that further simplify the integral calculation. The impact of different parameters on the percentile performance of each UE in a partial-NOMA network with different overlap values is shown. The second part of this paper focused on physical layer security achievable in a partial-NOMA network. As UE 2 is susceptible to eavesdropping because of the nature of FSIC, we defined the event of secure communication or secrecy as UE 2 being able to decode its message while UE 1 is unable to decode UE 2 's message.
Two scenarios were studied: 1) where a malicious UE 1 prioritizes decoding UE 2 's message, 2) where an innocent UE 1 only decodes UE 2 's message when it is required for decoding its own message. Our results show that as anticipated, at high α secrecy in the case of the innocent UE 1 is higher than that of the malicious UE 1 ; at lower α, however, the secrecy probability is equal. Additionally, we find that for both the innocent and malicious UE 1 , secrecy at lower α values is significantly higher than at higher α values including α = 1 (i.e., traditional NOMA) highlighting the superiority of a smaller overlap to a full overlap in NOMA. These results shed light on the fact that partial-NOMA with low α can not only be used to achieve higher secrecy but also be used to offer physical layer security to the network from a malicious eavesdropper.
We also study the secrecy achieved when traditional SIC is used instead of FSIC and when there is no receive filtering. A significant deterioration in secrecy is observed in both scenarios particularly at lower α. This highlights the importance of receive filtering and FSIC, not just for coverage, but also for secure communication.