Member, IEEE Sanket S Kalamkar
email: skalamka@nd.edu

Fellow, IEEE Martin Haenggi
email: mhaenggi@nd.edu

Simple Approximations of the SIR Meta Distribution in General Cellular Networks

Keywords: Interference, heterogeneous cellular networks, meta distribution, Poisson point process, signal-to-interference ratio, stochastic geometry

Compared to the standard success (coverage) probability, the meta distribution of the signal-to-interference ratio (SIR) provides much more fine-grained information about the network performance. We consider general heterogeneous cellular networks (HCNs) with base station tiers modeled by arbitrary stationary and ergodic non-Poisson point processes. The exact analysis of non-Poisson network models is notoriously difficult, even in terms of the standard success probability, let alone the meta distribution. Hence we propose a simple approach to approximate the SIR meta distribution for non-Poisson networks based on the ASAPPP ("approximate SIR analysis based on the Poisson point process") method. We prove that the asymptotic horizontal gap G0 between its standard success probability and that for the Poisson point process exactly characterizes the gap between the bth moment of the conditional success probability, as the SIR threshold goes to 0. The gap G0 allows two simple approximations of the meta distribution for general HCNs: 1) the per-tier approximation by applying the shift G0 to each tier and 2) the effective gain approximation by directly shifting the meta distribution for the homogeneous independent Poisson network. Given the generality of the model considered and the finegrained nature of the meta distribution, these approximations work surprisingly well.

I. INTRODUCTION

A. Motivation and Objective

The accurate modeling of base station (BS) locations is important to characterize the performance of cellular networks and obtain useful design insights. Traditionally, in a cellular network, the BS locations were modeled in a deterministic (regular) manner using either triangular or square lattices. The lattice model has been extensively studied using simulations since it is usually analytically intractable. However, to meet an exponential growth in mobile traffic and improve the spatial reuse, the deployment of cellular networks has become more irregular and heterogeneous. For example, in a geographical region, macro, pico, and femto BSs can coexist. The network tiers possess different characteristics such as different BS densities, different path loss exponents, and different deployment structures (e.g., clustered or repulsive deployments).

The Poisson point process (PPP) may be used to model irregular and real-world BS deployments [START_REF] Guo | Spatial stochastic models and metrics for the structure of base stations in cellular networks[END_REF]. The modeling of BS locations by the PPP has become popular due to its analytical tractability, which leads to crisp insights about the network performance. However, in an actual cellular network, the BS deployment is neither completely random (as the PPP) nor completely regular (as the triangular and square lattices)it lies somewhere in between. The BS deployment depends heavily on the topology and the type of the geographical region (urban or rural). As a result, a single point process model may not be applicable in all scenarios. For example, using actual data from the UK, it is shown in [START_REF] Guo | Spatial stochastic models and metrics for the structure of base stations in cellular networks[END_REF] that there exists repulsion among BSs, which can be modeled using hard-core point processes [START_REF] Haenggi | Stochastic Geometry for Wireless Networks[END_REF]Chapter 3]. On the other hand, in [START_REF] Lee | Stochastic geometry based models for modeling cellular networks in urban areas[END_REF], the Poisson cluster process [START_REF] Haenggi | Stochastic Geometry for Wireless Networks[END_REF]Chapter 3] is shown to accurately model the BS deployment in many cities. Especially, at a larger geographical scale, the BSs appear to form a cluster point process due to the high density in urban regions and low density in rural regions. Hence it is important to investigate the performance of non-Poisson cellular networks.

The main impediment to the study of non-Poisson cellular networks is that, compared to the PPP, their analysis is much harder due to the dependence between the BS locations. Thus it would be convenient if the performance of non-Poisson cellular networks could be related (approximately) to that of Poisson cellular networks.

Recently, in [START_REF] Haenggi | The meta distribution of the SIR in Poisson bipolar and cellular networks[END_REF], a new fundamental performance metric called the meta distribution of the signal-to-interference ratio (SIR) is introduced for cellular networks. The meta distribution, defined as the distribution of the conditional success probability given the point process, is an important performance metric as it answers a key question: "How are the individual link success (or coverage) probabilities 1 distributed in a realization of the cellular network?" The answer directly leads to the performance of the "5% user," which corresponds to the performance of the top 95% of users and is an important design criterion for cellular operators. The meta distribution provides much more fine-grained information about the network than the standard success (coverage) probability; the latter provides just the average of individual link success probabilities in each realization of the network and thus yields limited information about the network. In contrast, the meta distribution provides the distribution of the link success probability conditioned on the point process and thus allows the analysis of the network at a finer level.

The goal of this paper is to study the meta distribution in heterogeneous non-Poisson cellular networks, where the cellular networks consist of multiple tiers and the BSs in each tier may form an arbitrary stationary and ergodic point process. To achieve this goal, we have to overcome two main difficulties:

1) The direct calculation of the meta distribution seems infeasible even for the Poisson cellular network [START_REF] Haenggi | The meta distribution of the SIR in Poisson bipolar and cellular networks[END_REF]-one has to calculate the moments of the conditional success probability and then use the Gil-Pelaez theorem [START_REF] Gil-Pelaez | Note on the inversion theorem[END_REF] to calculate the meta distribution.

2) The analysis for non-Poisson cellular networks is significantly more difficult than that for Poisson cellular networks. In fact, obtaining an analytical expression of the (standard) success probability is extremely difficult in non-Poisson networks. Even for the arguably secondsimplest model, the Ginibre point process, it can only be given in the form of an expression in which 3 integrals, an infinite product, and an infinite sum are nested [START_REF] Deng | The Ginibre point process as a model for wireless networks with repulsion[END_REF]Theorem 2]. Consequently, analyzing the meta distribution in non-Poisson networks is very challenging. This problem becomes even worse in the case of heterogeneous cellular networks, where one needs to consider the intra-tier as well as the inter-tier interference. In this paper, we propose two simple and indirect approaches to approximately calculate the meta distribution for general heterogeneous cellular networks (HCNs) by comparing it to that for a homogeneous independent PPP (HIP) model where the BSs in each tier are modeled by an independent and homogeneous PPP.

B. Related Work

The works in [START_REF] Dhillon | Modeling and analysis of K-tier downlink heterogeneous cellular networks[END_REF]- [START_REF] Nigam | Coordinated multipoint joint transmission in heterogeneous networks[END_REF] obtained analytically tractable results for the HIP model. For HCNs with non-Poisson deployments, it is often the case that it is hard to perform an exact mathematical analysis of key performance metrics such as the SIR distribution (sometimes called the coverage probability). Even if an exact expression of the SIR distribution exists, it is available in a complex form that does not help gain insights about the performance of the network for different network parameters [START_REF] Nakata | Spatial stochastic models for analysis of heterogeneous cellular networks repulsively deployed base stations[END_REF]- [START_REF] Saha | 3GPP-inspired HetNet model using Poisson cluster process: Sum-product functionals and downlink coverage[END_REF].

Fortunately, [START_REF] Guo | Spatial stochastic models and metrics for the structure of base stations in cellular networks[END_REF] observed that the SIR distribution for the downlink of cellular networks modeled by different non-Poisson point processes can be closely approximated by simply applying a horizontal shift to the SIR distribution curve for the PPP model. The approximation becomes asymptotically exact as the SIR threshold θ → 0 [START_REF] Guo | Asymptotic deployment gain: A simple approach to characterize the SINR distribution in general cellular networks[END_REF]. The horizontal shift is termed the deployment gain in [START_REF] Guo | Asymptotic deployment gain: A simple approach to characterize the SINR distribution in general cellular networks[END_REF] since the shift is because of the deployment. This method of approximating the SIR distribution for a non-Poisson point process model by that for the PPP model is called "Approximate SIR analysis based on the PPP" (ASAPPP) in [START_REF] Haenggi | ASAPPP: A simple approximate analysis framework for heterogeneous cellular networks[END_REF]. [START_REF] Haenggi | The mean interference-to-signal ratio and its key role in cellular and amorphous networks[END_REF] showed that the deployment gain as θ → 0 can be expressed as the ratio of the mean interference-to-signal ratio (MISR) of the two different point processes under consideration. Further, [START_REF] Ganti | Asymptotics and approximation of the SIR distribution in general cellular networks[END_REF] proved that the deployment gain as θ → ∞ is determined by the expected fading-to-interference ratio (EFIR). A key observation from [START_REF] Ganti | Asymptotics and approximation of the SIR distribution in general cellular networks[END_REF] is that the deployment gain as θ → 0, denoted by G 0 , provides an excellent approximation to the entire SIR distribution. In [START_REF] Takahashi | Simple and fast PPP-based approximation of SIR distributions in downlink cellular networks[END_REF], a formula is derived to approximately calculate G 0 analytically for (stationary) non-Poisson point process models whenever the second moment of the contact distance is available. [START_REF] Wei | Approximate SIR analysis in general heterogeneous cellular networks[END_REF] showed that the ASAPPP approximation works very well for HCNs with non-Poisson deployment of BSs. First, [START_REF] Wei | Approximate SIR analysis in general heterogeneous cellular networks[END_REF] proposed the per-tier ASAPPPbased approximation for general HCNs, where the ASAPPP approximation was used to approximate the SIR distribution corresponding to each non-Poisson tier using the MISR-based deployment gain G 0 . Second, when the path loss of each tier was the same, [START_REF] Wei | Approximate SIR analysis in general heterogeneous cellular networks[END_REF] showed that the SIR distribution for general HCNs could be directly obtained from that for the HIP model by scaling the SIR threshold by the effective gain.

The meta distribution of the SIR for cellular networks was proposed in [START_REF] Haenggi | The meta distribution of the SIR in Poisson bipolar and cellular networks[END_REF], where the focus was on the downlink of the Poisson cellular network. Furthermore, the meta distribution of the SIR was calculated for both the downlink and the uplink of the Poisson cellular network with power control in [START_REF] Wang | The meta distribution of the SIR for cellular networks with power control[END_REF], for the downlink Poisson cellular network underlaid with a device-to-device (D2D) network in [START_REF] Salehi | Analysis of D2D underlaid cellular networks: SIR meta distribution and mean local delay[END_REF], for the nonorthogonal multiple access (NOMA) network in [START_REF] Salehi | Meta distribution of the SIR in large-scale uplink and downlink NOMA networks[END_REF], and with base station cooperation in [START_REF] Cui | The SIR meta distribution in Poisson cellular networks with base station cooperation[END_REF]. For general cellular networks with a multi-slope path loss model, [START_REF] Afshang | Equi-coverage contours in cellular networks[END_REF] gave a scaling law involving the parameters of BS and user point processes (e.g., the density of the point process) that keeps the meta distribution of the SIR the same. For the HIP-based K-tier HCN, [START_REF] Wang | SIR meta distribution of K-tier downlink heterogeneous cellular networks with cell range expansion[END_REF] calculated the SIR meta distribution with cell range expansion.

Overall, for general HCNs with non-Poisson deployment of BSs, the focus has been only on the SIR distribution, and the SIR meta distribution is calculated only for Poisson cellular networks. In this paper, we determine the fine-grained network performance of general HCNs through the SIR meta distribution. Since the model for cellular networks considered is quite general, we cannot expect to obtain exact analytical expressions for the SIR meta distribution. Hence we propose simple approximations that enable a quick calculation of the SIR meta distribution for general HCNs. Note that, until now, the ideas of the SIR meta distribution and the ASAPPP have been explored and applied separately. Applying the shift to the SIR distribution (which is just the mean of the SIR meta distribution) does not imply that the shifting approach also works for the entire SIR meta distribution. That said, combining these ideas is of significant importance because "mean to distribution" and "Poisson to non-Poisson" are highly nontrivial extensions.

C. Contributions

This paper makes the following contributions: 1) For cellular networks, we apply the idea of ASAPPP to the meta distribution and propose a simple and novel method, called AMAPPP which stands for "Approximate meta distribution analysis using the PPP," to obtain the meta distribution for an arbitrary stationary and ergodic point process from the meta distribution for the PPP. 2) We prove that, as θ → 0, the bth moment of the conditional success probability for a stationary and ergodic point process model can be obtained exactly by shifting that for the PPP model by the asymptotic deployment gain G 0 . 3) For Rayleigh fading and an unbounded path loss model, we confirm by simulations that applying the horizontal shift by the gain G 0 to the meta distribution for the PPP closely approximates the meta distribution for the stationary triangular lattice, the perturbed triangular lattice, the Gauss-Poisson point process, and the Matérn cluster process (two regular and two cluster point processes). 4) We extend the AMAPPP approach to general HCNs, which approximates the bth moment of the conditional success probability corresponding to each tier using the MISR-based gain G 0 . This per-tier approach is further used to approximately calculate the meta distribution for general HCNs. 5) When the path loss exponents in all tiers are the same, we obtain an effective gain using the MISR-based gain G 0 corresponding to each tier, which can be simply applied to shift the meta distribution for the HIP cellular network to approximately obtain the meta distribution for general HCNs.

D. Organization of the Paper

The rest of the paper is organized as follows. In Sec. II, we provide the network model, describe the point process models considered in this paper, and briefly review the SIR meta distribution and the ASAPPP method. In Sec. III, we propose the AMAPPP approach for single-tier non-Poisson cellular networks, which is extended to general HCNs in Sec. IV. We provide conclusions in Sec. V.

II. SYSTEM MODEL A. Network Model

We consider a general K-tier HCN where the locations of BSs of the kth tier are modeled by arbitrary stationary, ergodic, and independent point processes

Φ k ⊂ R 2 , k = 1, 2, . . . , K.
The density of Φ k is λ k . All BSs are always active. A base station belonging to Φ k transmits at power P k . Due to the stationarity of the point processes, we focus on the cellular user situated at the origin o = (0, 0), henceforth called the typical user. We focus on the downlink with the (on average) strongest-BS association, where the typical user connects to the BS with the strongest received power on average. The other BS transmissions from the same tier as that of the serving BS and those from different tiers cause interference at the typical user. The signal propagation experiences fading as well as path loss. We assume independent and identically distributed (i.i.d.) Rayleigh fading where the channel power gains are exponentially distributed with mean 1. The path loss function corresponding to the tier Φ k is given by (x) = x -α k , where α k > 2 is the path loss exponent.

We focus on an interference-limited network where the received SIR determines the network performance. Let

x 0 arg max{x ∈ Φ k , k ∈ [K] : P k x -α k } be the serving BS of the typical user, where [K] {1, 2, . . . , K}. Also let Φ ! k and [K] ! denote Φ k \ {x 0 } and [K] \ {k}, respectively.
When the typical user connects to a BS of Φ k , the SIR at the typical user is given by

SIR S I Intra + I Inter = P k h x0 x 0 -α k x∈Φ ! k P k h x x -α k + i∈[K] ! y∈Φi P i h y y -αi , (1)
where h x represents the exponential random variable corresponding to the channel power gain between the BS at x and the typical user, S P k h x0 x 0 -α k is the received signal power at the typical user, I Intra

x∈Φ ! k P k h x x -α k and I Inter i∈[K] !
y∈Φi P i h y y -αi denote the interference power received by the typical user from the intra-and inter-tier BS transmissions, respectively.

The standard success probability is given as

p s (θ) P(SIR > θ) = k∈[K] P(SIR > θ, x 0 ∈ Φ k ), (2)
where θ is the SIR threshold.

B. Point Process Models

In this subsection, we describe four stationary and ergodic point processes, two regular (repulsive) and two cluster point processes, that may be used to model BS locations. These four point processes are used as illustrative examples of stationary and ergodic point processes.

1) Stationary Triangular Lattice (TL):

The triangular lattice is the most regular point process. The stationary triangular lattice is obtained by randomly translating the non-stationary triangular lattice L ⊂ R2 given by

L {v ∈ Z 2 : Gv}, (3)
where

G = η 1 1/2 0 √ 3/2
is the generator matrix and η is the distance between any two neighboring points of L. The density of the triangular lattice is 2/( √ 3η 2 ). We translate L by a random vector X distributed uniformly over the Voronoi cell of the origin to obtain the stationary triangular lattice Φ as 2

Φ {v ∈ Z 2 : Gv + X}. (4)
In the rest of the paper, by the "triangular lattice," we mean the "stationary triangular lattice."

2) Perturbed Triangular Lattice (PTL): The second regular point process that we consider is the perturbed triangular lattice (PTL) [START_REF] Haenggi | Stochastic Geometry for Wireless Networks[END_REF]Chapter 2]. The PTL is obtained by perturbing the stationary triangular lattice, i.e.,

Φ {v ∈ Z 2 : Gv + X + Y v }, (5)
where (Y v ) v∈Z 2 is a family of i.i.d. random variables. In this paper, we assume that (Y v ) are uniformly distributed on the disk b(o, R pert ) centered at the origin with radius R pert .

3) Gauss-Poisson Point Process (GaPPP): The GaPPP is a Poisson cluster process where each cluster contains either one or two points with probabilities p and 1 -p, respectively [START_REF] Guo | The Gauss-Poisson process for wireless networks and the benefits of cooperation[END_REF]. For a one-point cluster, the point is at the parent point location, while for a two-point cluster, one of the points is at the parent point location and the other is located at a deterministic distance u from the parent point in a random direction, i.e., the second point lies uniformly at random on the circle of radius u centered at the parent point location.

For a Poisson parent point process Φ p with density λ p , let Φ x with x ∈ Φ p be the clusters of the GaPPP, which are denoted by

Φ x = {x} with probability p {x, x + u x } with probability 1 -p, (6)
where u x = (u sin φ x , u cos φ x ) with φ x uniformly distributed in [0, 2π]. The density of the GaPPP is λ p (2 -p).

4) Matérn Cluster Process (MCP):

The MCP is a doubly Poisson cluster process, where the parent point process Φ p is a PPP with density λ p and the daughter points are uniformly distributed within a ball of radius r c with each parent point x p ∈ Φ p as its center [START_REF] Haenggi | Stochastic Geometry for Wireless Networks[END_REF]Chapter 3]. The density of the daughter point process of parent x p is given by

λ d (x) = c πr 2 c 1 B(xp,rc) (x), (7)
where

B (xp,rc) (x) {x ∈ R 2 : x -x p ≤ r c }, c
is the average number of daughter points in a cluster, and 1(•) is the indicator function. The density of the MCP is λ = λ p c.

C. The SIR Meta Distribution

For an SIR threshold θ and a reliability threshold x, the meta distribution of the SIR is given by

F (θ, x) = FPs (θ, x) P(P s (θ) > x), θ ∈ R + , x ∈ [0, 1], (8)
where P s (θ) is a random variable that represents the link success probability conditioned on the point process Φ, given by

P s (θ) P(SIR > θ | Φ). (9)
Here the probability is taken with respect to the fading. The SIR is calculated at the receiver of the link under consideration. The meta distribution is the complementary cumulative distribution function (ccdf) of the conditional link success probability P s (θ). Interpreted differently, for a stationary and ergodic point process, the SIR meta distribution yields the fraction of cellular users that achieve an SIR of θ with reliability at least x. Also, note that the meta distribution of the rate R can be obtained from the meta distribution of the SIR using the relation R = W log 2 (1 + SIR) with W denoting the bandwidth [START_REF] Deng | The energy and rate meta distributions in wirelessly powered D2D networks[END_REF].

The standard success (coverage) probability p s (θ) (the SIR distribution) can be obtained from the SIR meta distribution as the mean of the conditional link success probability P s (θ), i.e.,

p s (θ) P(SIR > θ) = E(P s (θ)) = 1 0 F (θ, x)dx. (10)
Clearly, the distribution of P s (θ) provides much more finegrained information than merely its average p s (θ).

1) Exact Calculation of the Meta Distribution: Finding the exact meta distribution directly seems infeasible, but if we can calculate the moments of the conditional link success probability P s (θ)

M b (θ) E(P s (θ) b ), b ∈ C, (11)
as has been done in [START_REF] Haenggi | The meta distribution of the SIR in Poisson bipolar and cellular networks[END_REF] for Poisson cellular networks, we can calculate the exact meta distribution in (8) using the Gil-Pelaez theorem [START_REF] Gil-Pelaez | Note on the inversion theorem[END_REF] as

F (θ, x) = 1 2 + 1 π ∞ 0 (e -jt log x M jt ) t dt, (12)
where (u) is the imaginary part of u ∈ C. Even though the expression of the meta distribution in ( 12) is exact, its complexity makes it hard to gain direct insights, and it is not very convenient to evaluate numerically. 3 With the nearest-BS association and the path loss exponent α, for the downlink of a single-tier Poisson cellular network and K-tier HIP model, the bth moment M PPP b (θ) of the conditional link success probability is simply given by [5]

M PPP b (θ) = 1 2 F 1 (b, -δ; 1 -δ; -θ) , b ∈ C, (13)
where 2 F 1 (•, •; •; •) denotes the Gauss hypergeometric function and δ 2/α. On the other hand, in a non-Poisson cellular network, the calculation of M b (θ)-let alone the meta distribution-is quite difficult. Hence it would be extremely useful to have a simple approximation to obtain M b (θ) and the meta distribution in a non-Poisson cellular network. We propose to do so by combining the meta distribution with the ASAPPP approach, discussed in the next subsection.

2) Approximate Calculation of the Meta Distribution: For different network settings, the meta distribution can be accurately approximated by the beta distribution by matching the first and the second moments of the conditional link success probability [START_REF] Haenggi | The meta distribution of the SIR in Poisson bipolar and cellular networks[END_REF], [START_REF] Wang | The meta distribution of the SIR for cellular networks with power control[END_REF], [START_REF] Salehi | Analysis of D2D underlaid cellular networks: SIR meta distribution and mean local delay[END_REF], [START_REF] Cui | The SIR meta distribution in Poisson cellular networks with base station cooperation[END_REF], [START_REF] Wang | SIR meta distribution of K-tier downlink heterogeneous cellular networks with cell range expansion[END_REF].

D. Approximate SIR Analysis based on the PPP (ASAPPP)

1) Single-tier Network: ASAPPP is the method that provides an approximation of the SIR distributions in non-Poisson networks by that in the Poisson network by applying a horizontal shift to the latter. This method asserts that if the network model under consideration and the Poisson model only differ in the type of the underlying point process, then the SIR ccdf for the network model under consideration can be closely approximated using the SIR ccdf for the PPP by scaling the SIR threshold θ by a certain factor G 0 [START_REF] Ganti | Asymptotics and approximation of the SIR distribution in general cellular networks[END_REF], i.e.,

p s (θ) ≈ p PPP s (θ/G 0 ), (14)
which corresponds to a horizontal shift by G 0 (in dB) if θ is plotted in dB. The subscript in G 0 corresponds to θ → 0, i.e., the shift is calculated for θ → 0. This asymptotic shift G 0 can also be interpreted as an SIR gain, similar to the notion of the coding gain in coding theory [START_REF] Lin | Error Control Coding[END_REF]. As shown in [START_REF] Ganti | Asymptotics and approximation of the SIR distribution in general cellular networks[END_REF], the 3 To numerically evaluate the integral in [START_REF] Deng | Heterogeneous cellular network models with dependence[END_REF], one needs to carefully select the appropriate numerical integration range and the step size [START_REF] Haenggi | Efficient calculation of meta distributions and the performance of user percentiles[END_REF].

gain G 0 provides an excellent approximation to the entire SIR distribution. This approximation becomes exact as θ → 0, i.e.,

p s (θ) ∼ p PPP s (θ/G 0 ), θ → 0. (15)
Moreover, the gain G 0 shows little sensitivity to the path loss exponent or the fading model [START_REF] Ganti | Asymptotics and approximation of the SIR distribution in general cellular networks[END_REF]; it is a robust constant that captures the difference in the network topologies due to the underlying point process models. The gain G 0 can be expressed using the mean interference-to-signal ratios (MISRs) of the point process Φ under consideration and the PPP as [START_REF] Haenggi | The mean interference-to-signal ratio and its key role in cellular and amorphous networks[END_REF] G

0 = MISR PPP MISR , (16)
= 2 α -2 1 MISR , ( 17
)
where α is the path loss exponent for a single-tier network, and

MISR E x∈Φ\{x0} h x x -α x 0 -α = E x∈Φ\{x0} x -α x 0 -α (18)
and MISR PPP = 2/(α -2) are the MISRs of the network model under consideration and the PPP, respectively. The numerical calculation of G 0 is quite easy since it just depends on the network geometry. 4 This motivates us to investigate whether a horizontal shift of G 0 to the meta distribution for the Poisson cellular network approximates the meta distribution for a non-Poisson cellular network. We show that this is indeed the case. We call this approach "Approximate meta distribution analysis using the PPP" (AMAPPP).

2) K-tier General Heterogeneous Network: In a general HCN, as given by (2), the overall standard success probability is the sum of the probabilities of the joint events that SIR > θ and that the typical user is served by the kth tier where

k ∈ [K].
When the typical user connects to a tier modeled by a stationary non-Poisson point process, the tier is treated as a PPP while shifting the SIR threshold θ to θ/G 0 in the SIR distribution. Also, the interference from other tiers modeled by stationary non-Poisson point processes is approximated by that from a PPP. Such an approximation is called the "per-tier ASAPPP" in [START_REF] Wei | Approximate SIR analysis in general heterogeneous cellular networks[END_REF] since each tier is treated "as a PPP" and is shown to be quite accurate for the entire SIR distribution. We extend the per-tier ASAPPP approach to the meta distribution, which we call the "per-tier AMAPPP" approach. It is discussed in Sec. IV-A.

III. AMAPPP APPROACH FOR SINGLE-TIER NETWORKS

In this section, we focus on single-tier cellular networks, where the BS deployment follows a stationary and ergodic non-Poisson point process.

A. Main Result

The goal here is to show that an approximation of the form

F (θ, x) ≈ F PPP (θ/G 0 , x) (19)
is accurate, where F (θ, x) and F PPP (θ/G 0 , x) denote the meta distributions for a stationary and ergodic point process model and a PPP model, respectively. We know that the shift of the SIR threshold θ by G 0 works quite well for the 1st moment of the conditional link success probability, i.e., the standard success probability p s (θ), and the meta distribution can be exactly calculated using the moments M b (θ) as shown in [START_REF] Deng | Heterogeneous cellular network models with dependence[END_REF]. These results give rise to the interesting question how the bth moments M b (θ) for an arbitrary stationary and ergodic point process model and M PPP b (θ) are related to each other, as θ → 0. The following theorem answers it.

Theorem 1. For any stationary and ergodic point process and b ∈ C,

M b (θ) ∼ M PPP b (θ/G 0 ), θ → 0. ( 20
)
Proof: From [5, [START_REF] Wang | The meta distribution of the SIR for cellular networks with power control[END_REF]], for any stationary and ergodic point process model, we have

M b (θ) = E x∈Φ\{x0} 1 (1 + θ( x 0 / x ) α ) b , b ∈ C (21) (a) ∼ E y∈R (1 -bθy α ), θ → 0 (22) ∼ 1 -bθ E y∈R y α , θ → 0 ( 23
) (b) = 1 -bθ MISR, (24)
where R {x ∈ Φ \ {x 0 } : x 0 / x } is the relative distance process (RDP) [19, Def. 2], (a) follows by letting y = x 0 / x and using Taylor series expansion, and (b) follows from the definition of the MISR for the RDP [START_REF] Ganti | Asymptotics and approximation of the SIR distribution in general cellular networks[END_REF]. Using ( 16) and [START_REF] Salehi | Meta distribution of the SIR in large-scale uplink and downlink NOMA networks[END_REF], we obtain the desired result.

Note that even though all moments are shifted by the same amount G 0 asymptotically, this does not imply that the meta distribution is also shifted by that amount. However, we can expect the shifted meta distribution for the PPP to provide a good approximation. Next, we explore by simulation whether this is the case.

B. Simulation Results

In this subsection, for a single-tier cellular network, using simulations, we verify the accuracy of approximating the meta distribution for stationary and ergodic point processes by shifting the meta distribution for the PPP by G 0 .

Simulation setup: We perform simulations over a square region [-500, 500] 2 . Unless otherwise mentioned, we assume the following simulation parameters pertaining to specific point processes:

• PTL: The perturbation R pert is 0.5η.

• GaPPP: The probability p that a cluster contains one point is 0.5. Hence the probability that a cluster contains two points is 1 -p = 0.5. The density of the parent point process is 1/15. In a two-point cluster, the distance between those two points is u = 1. • MCP: The density of the parent point process λ p is 0.01.

The mean number of points c in a cluster The of the r c around a parent point over which the associated cluster points are distributed is 4. We average over 5 × 10 5 realizations of the point process.

Fig. 1 shows that the bth moment of the conditional link success probability for a stationary and ergodic non-Poisson point process model can be approximately obtained by shifting that for the PPP model by the asymptotic SIR gain G 0 for the path loss exponent α = 4. The approximation becomes extremely accurate as the SIR threshold θ becomes small, as expected from Thm. 1.

Given that the ASAPPP method aims at approximating the first moment M 1 (θ) of the conditional link success probability P s (θ) and the meta distribution can be obtained using the 3. The meta distribution F (θ, x) for the PPP, the TL, the PTL, the GaPPP, and the MCP against the SIR threshold θ for the path loss exponent α = 4 and the reliability threshold x = 0.95. The asymptotic gain G 0 is 3.6099 dB for the TL, 1.8343 dB for the PTL, -1.3768 dB for the GaPPP, and -5.1702 dB for the MCP. For a single-tier cellular network, Fig. 3 plots the meta distribution values for x = 0.95 against different SIR thresholds θ for the PPP, the TL, the PTL, the GaPPP, and the MCP at α = 4. We observe that the meta distribution for the TL, the PTL, the GaPPP, and the MCP can be obtained approximately by simply applying a horizontal shift of G 0 (in dB) to the meta distribution for the PPP, i.e., the AMAPPP can indeed be used to approximately calculate the meta distribution for a non-Poisson network. Especially, for the values of practical interest where a high fraction of users meet the target reliability (e.g., the 5% user performance), the approximation is quite accurate. This is quite remarkable given that we use the asymptotic SIR gain G 0 as θ → 0 corresponding to the mean of the distribution to approximate the distribution itself.

Figs. 4 and 5 confirm the effectiveness of the AMAPPP method for different values of the path loss exponents α and the reliability thresholds x, respectively.

In Fig. 6, to check the accuracy of the AMAPPP method for the entire meta distribution, we take a closer look at the regime where the SIR threshold θ is small. For the GaPPP, the approximation is extremely accurate even for small values of θ, which confirms that the AMAPPP method is accurate for the entire meta distribution for the GaPPP. For the MCP, there is a gap between the simulation and the approximation (the shifted PPP) because the asymptotics kick in slowly compared to that for the GaPPP. This can be confirmed from Fig. 2, which shows that, for the MCP, the gains G 1 (θ), G 2 (θ), and G 3 (θ) corresponding to M 1 (θ), M 2 (θ), and M 3 (θ), respectively, converge to G 0 more slowly than those for the GaPPP as θ becomes small.

For the stationary triangular lattice, Fig. 6 highlights an interesting case; at θ = -16.68 dB, the value of 1 -F (θ, x) drops to zero because all users achieve the target reliability x = 0.95 for θ < -16.68 dB. For all x, such a threshold can be calculated by shifting the lattice such that the typical The asymptotic gain G 0 is 3.6099 dB for the TL, 1.8343 dB for the PTL, -1.3768 dB for the GaPPP, and -5.1702 dB for the MCP. For small θ, the AMAPPP approximation is optimistic for the MCP in that the shifted curve of the PPP is below that of the simulation curve, while the approximation is pessimistic for the regular point processes. user sits at a Voronoi vertex for the triangular lattice, which is the worst-case scenario for the triangular lattice since there are 3 nearest base stations to the user. For all lattices, there exists such a vertical asymptote, and thus the approximation by shifting G 0 breaks down as θ approaches that threshold. However, for values of θ for which the 5% user achieves 95% reliability, the approximation is tight. Such a worst-case scenario could also occur in the case of the perturbed triangular lattice if the perturbation radius R pert is small enough. In this case, the critical θ is smaller compared to that for the stationary triangular lattice. For example, at R pert = 0.5η, the approximation starts to break down at a smaller value of θ compared to that for the stationary triangular lattice. Fig. 7 shows that the critical θ, denoted by θ c (x), depends on the target reliability x-it decreases with an increase in x because, to meet a higher target reliability, the SIR threshold θ at the worst-case user needs to be lowered. Such a dependency of θ c (x) on the target reliability illustrates the rate-reliability trade-off. 5 Fig. 7 also provides insight into the support of the probability density function (pdf) of the conditional link success probability P s (θ). For each x > 0, there is a θ c (x) such that the support is reduced to [x, 1] at θ = θ c (x). For θ < θ c (x), the support gradually reduces to {1}.

IV. HETEROGENEOUS CELLULAR NETWORKS

A. Per-tier AMAPPP Approximation

We consider a K-tier general cellular network where the ith tier is modeled by a stationary and ergodic point process Φ i of density λ i and each tier is independent of the other tiers. For this setup, the following theorem provides an approximation of M b (θ) for general HCNs.

Theorem 2. Let Mb (θ) = k∈[K] ∞ 0 exp -sF (b, δ k , θ/G k ) - i∈[K] ! ρ ik s α k α i F (b, δ i , θ) ds, (25)
where F (b, δ, θ) 2 F 1 (b, -δ; 1-δ; -θ), δ i 2/α i , G k is the asymptotic SIR gain corresponding to the point process that models the BS deployment in the kth tier, and

ρ ik = λiπP δ i ik (λ k π) α k /α i
with P ik P i /P k . For K-tier general HCNs where the typical cellular user is connected to the BS that results in the strongest average received power, the bth moment M b (θ) of the conditional link success probability is approximated as

M b (θ) ≈ Mb (θ). (26)
Proof: See Appendix A. In this approach, the bth moment of the conditional link success probability corresponding to each tier modeled by a non-Poisson point process is approximated by that of the PPP by shifting the SIR threshold by the MISR-based gain G k , which can be further used to calculate the approximate meta distribution using the Gil-Pelaez (GP) theorem or the beta distribution approximation. Hence we call this approach the "per-tier AMAPPP" approach.

In the calculation of the meta distribution using the beta distribution approximation, there are two approximations involved:

1) The approximate calculation of the first and second moments based on the MISR gain as shown in Thm. 2, 2) The inherent approximation resulting from matching only the first and the second moments. 5 The rate is a function of the SIR threshold.

Hence we call the combination of the beta distribution approximation with the per-tier AMAPPP approach the "approximate beta approximation" (ABA). When all tiers have the same path loss exponent, (25) reduces to the expression given in the following corollary.

Corollary 1. If α 1 = α 2 = . . . = α K = α, we have Mb (θ) = k∈[K] 1 F (b, δ, θ/G k ) + i∈[K] ! λi λ k Pi P k δ F (b, δ, θ) . ( 27
)
For a K-tier HIP cellular network, we have G k = 1. Then when all tiers have the same path loss exponent, [START_REF] Wang | SIR meta distribution of K-tier downlink heterogeneous cellular networks with cell range expansion[END_REF] simplifies to

Mb (θ) = 1/F (b, δ, θ), (28)
as stated in [START_REF] Wang | SIR meta distribution of K-tier downlink heterogeneous cellular networks with cell range expansion[END_REF]Cor. 1]. Note that ( 28) is the exact expression of the bth moment of the conditional link success probability for the HIP model, i.e., M HIP b (θ) = Mb (θ).

B. Effective Gain Approximation

In the per-tier AMAPPP method, a non-Poisson tier is approximated by the PPP using the MISR-based gain. In this subsection, we provide an approximation where we directly shift the meta distribution for the K-tier HIP model by the effective gain to approximate the meta distribution for the general K-tier cellular network. This effective gain was introduced in [START_REF] Wei | Approximate SIR analysis in general heterogeneous cellular networks[END_REF] for the standard success probability. The following theorem calculates the effective gain corresponding to M b . where

G eff k∈[K] w k (w k G k + (1 -w k )) = 1 + k∈[K] w 2 k (G k -1), (29)
with w k

λ k P δ k i∈[K] λiP δ i
. For b < 0, we need to replace '≤' by '≥'.

Proof: See Appendix B. Using Thm. 3, we obtain an another approximation of M b (θ) for a stationary and ergodic point process as

M b (θ) ≈ M HIP b θ G eff . ( 30
)
The effective gain G eff corresponds to the overall SIR gain of HCNs which can be obtained from the MISR-based gains of the individual tiers of the HCNs. Hence, similar to approximating the meta distribution for a stationary and ergodic non-Poisson tier by shifting that for the PPP by the MISRbased gain, the meta distribution for general HCNs can be approximated by directly shifting that for the HIP model by the overall SIR gain, i.e.,

F (θ, x) ≈ F HIP (θ/G eff , x). ( 31
)
We call the approximation in [START_REF] Lin | Error Control Coding[END_REF] the "effective gain approximation."

C. Results

The simulation parameters are the same as those for a single-tier cellular network. Unless otherwise mentioned, each tier has a BS density of 0.1. We assume

α 1 = α 2 = . . . = α K = α.
For a 2-tier HCN denoted by first tier/second tier, Fig. 8 shows that the per-tier AMPPPP approximation and the effective gain approximation closely approximate the bth moment M b (θ) of the conditional link success probability. As proved in Thm. 3, the effective gain approximation provides an upper bound on the per-tier approximation. Fig. 8 also shows that the gap between the simulation and the per-tier AMAPPP approximation is larger for the TL compared to that for the PTL since the former is more regular than the latter, and thus the approximation of the interference by that of the PPP is less accurate. Similarly, the gap between the simulation and per-tier AMAPPP approximation is smaller for the GaPPP compared to that for the MCP because the latter is more clustered than the former.

For the path loss exponents of α = 3, 4, Fig. 9 shows the meta distribution for a 2-tier HCN. The BSs of the first tier form a stationary and ergodic non-Poisson point process and those of the second tier a PPP. The effective gain approximation and the per-tier approximation both work quite well over a wide range of θ. Especially, the approximate beta approximation (ABA) method is remarkably accurate given that it involves two approximations. Fig. 10 shows the accuracy of the AMAPPP approximation when each tier of a multi-tier cellular network is modeled by a stationary and ergodic point process of different densities. Fig. 11 verifies the accuracy of both approximations for a 3-tier general HCN.

Figs. 12 and 13 show the contour plots for 2-tier HCNs. These contour plots illustrate the trade-off between the SIR threshold θ and the reliability threshold x. The contours Fig. 9. The effective-gain approximation, the per-tier AMAPPP approximation (GP theorem), the approximate beta approximation (ABA) of the meta distribution F (θ, x) of a "Non-Poisson/PPP" deployment (a two-tier general HCN) against θ.

provide the possible pairs (θ, x) that a fraction v of users achieves. For example, in Fig. 12, the curve corresponding to v = 0.95 shows that 95% users achieve an SIR of -10 dB with probability 0.69 and an SIR of -5 dB with probability 0.34. Furthermore, Figs. 12 and 13 show that the effective gain approximation and the per-tier AMAPPP method (using both the GP theorem and ABA) work quite well for different values of the fraction v of users, the reliability threshold x, and the SIR threshold θ.

V. CONCLUSIONS In this paper, we have proposed AMAPPP, a simple and novel approach to approximately obtain the meta distribution for an arbitrary stationary and ergodic point process as well as general HCNs from that for the PPP and the HIP model, respectively. For the bth moment M b (θ) of the conditional success probability for any stationary and ergodic point process model, we proved that M b (θ) ∼ 1 -bθ MISR, as θ → 0. Through detailed simulations for the triangular lattice, the perturbed triangular lattice, the Gauss-Poisson point process, and the Matérn cluster process, we have shown that the asymptotic deployment gain G 0 of the standard success (coverage) probability can be used to relate the meta distribution to that for the PPP. The approximation of the meta distribution for the triangular lattice by that for the PPP becomes pessimistic in the worst-case scenario, i.e., when the typical cellular user is located such that it has three nearest base stations. For Ktier HCNs, the per-tier approach closely approximates the bth moment of the conditional link success probability, which can be further used to calculate the approximate meta distribution. The other approach directly calculates the approximate meta distribution for general HCNs from that for the HIP model by simply applying a shift by the effective gain. Overall, given the generality of the model and the fine-grained nature of the meta distribution, the AMAPPP approach works surprisingly well.

There are interesting future directions to our work. It is important to obtain PPP-based approximations for the SIR meta distribution for uplink and general fading models. But one has to be careful in the uplink case since there could be an interferer arbitrarily close to a receiver, which makes the uplink problem intricate. Also there is no work on the SIR meta distribution general fading models even for the PPP-based Hence is naturally more appropriate to first analyze the SIR meta distribution for the PPP with fading and then investigate whether the shifting approach for general cellular networks with general fading models. Another interesting line of research is to consider the association with a base station that offers the maximum instantaneous SIR, instead of the nearest-base station association. Averaging over the fading, it follows that

P (k) s (θ) = x∈Φ ! k 1 1 + θ k (x) k (x0) i∈[K] ! y∈Φi 1 1 + θP ik i(y) k (x0) 1 x0∈Φ k , (33)
where P ik = P i /P k . The bth moment of P

where f k (r) = 2πλ k re -λ k πr 2 is the distribution of the distance of the typical user to the nearest BS that belongs to the kth tier, ρ ik λiπP δ i ik (λ k π) α k /α i , and F (b, δ, θ) 2 F 1 (b, -δ; δ; -θ). (a) follows from asymptotically AMAPPP approximation of Φ k where the SIR threshold θ is shifted to k and the point process Φ is replaced by a PPP denoted by Φ PPP

where (a) is due to k∈[K] w k = 1 and 1/F (b, δ, θ/G) being concave in G.

From the definition of the MISR-based gain, (39) can be viewed as

Mb (θ) ≤ M HIP b (θ/G eff ), (40)
where

G eff k∈[K] w k (w k G k + (1 -w k )) (41)
is termed the effective gain for K-tier HCNs. For b < 0, we just need to reverse the inequality, i.e., replace '≤' by '≥.'

Fig. 1 .

 1 Fig. 1. Approximation of M b (θ) for four stationary and ergodic non-Poisson point processes by M PPP b (θ/G 0 ). The path loss exponent α is 4.

Fig. 2 .

 2 Fig.2. The SIR gains G 1 (θ), G 2 (θ), and G 3 (θ) corresponding to the moments M 1 (θ), M 2 (θ), and M 3 (θ), respectively, for the path loss exponent α = 4. The asymptotic gain G 0 is 3.6099 dB for the TL, 1.8343 dB for the PTL, -1.3768 dB for the GaPPP, and -5.1702 dB for the MCP.

  Fig.3. The meta distribution F (θ, x) for the PPP, the TL, the PTL, the GaPPP, and the MCP against the SIR threshold θ for the path loss exponent α = 4 and the reliability threshold x = 0.95. The asymptotic gain G 0 is 3.6099 dB for the TL, 1.8343 dB for the PTL, -1.3768 dB for the GaPPP, and -5.1702 dB for the MCP.

Fig. 4 .

 4 Fig. 4. The meta distribution F (θ, x) for the GaPPP against the SIR threshold θ for different values of the path loss exponent and the reliability threshold x = 0.95. We have G 0 = -1.3768 dB for α = 4, G 0 = -1.3423 dB for α = 3.5, G 0 = -1.2558 dB for α = 3, and G 0 = -0.8661 dB for α = 2.5.

Fig. 5 .

 5 Fig.5. The meta distribution F (θ, x) for the PTL against the SIR threshold θ for different values of the reliability threshold x and the path loss exponent α = 4. The asymptotic gain G 0 for the PTL is 1.8343 dB. The curves are for x = 0.6, 0.7, 0.8, 0.9, 0.95 (from top to bottom).

Fig. 6 .

 6 Fig.6. The meta distribution 1 -F (θ, x) against the SIR threshold θ for the path loss exponent α = 4 and the reliability threshold x = 0.95. The asymptotic gain G 0 is 3.6099 dB for the TL, 1.8343 dB for the PTL, -1.3768 dB for the GaPPP, and -5.1702 dB for the MCP. For small θ, the AMAPPP approximation is optimistic for the MCP in that the shifted curve of the PPP is below that of the simulation curve, while the approximation is pessimistic for the regular point processes.

Fig. 7 .

 7 Fig. 7. The critical SIR threshold θc against the reliability threshold x for the triangular lattice.

Theorem 3 .

 3 When α 1 = α 2 = . . . = α K = α, for any b > 0, the approximate bth moment Mb (θ) for a general Ktier cellular network is related to the bth moment M HIP b for the HIP model as Mb (θ) ≤ M HIP b θ G eff ,

  TL/PPP, G eff = 1.2190 dB. PTL/PPP, G eff = 0.5361 dB. GaPPP/PPP, G eff = -0.3064 dB. MCP/PPP, G eff = -0.8301 dB.

Fig. 8 .

 8 Fig. 8. The per-tier AMAPPP approximation and the effective-gain approximation of M b (θ) for a "Non-Poisson/PPP" deployment (a two-tier general HCN). The path loss exponent is α = 4.

  TL/PPP, G eff = 1.1951 dB for α = 3, G eff = 1.2190 dB for α = 4. PTL/PPP, G eff = 0.5491 dB for α = 3, G eff = 0.5361 dB for α = 4. GaPPP/PPP, G eff = -0.2819 dB for α = 3, G eff = -0.3064 dB for α = 4. MCP/PPP, G eff = -0.8511 dB for α = 3, G eff = -0.8301 dB for α = 4.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. The meta distribution F (θ, x) for a 2-tier cellular network GaPPP/PPP against the SIR threshold θ for the reliability threshold x = 0.95 and the path loss exponent α = 4. The density of the GaPPP is 0.2, while the density of the PPP is 0.1. For α = 3, G eff = -0.2226 dB and for α = 4, G eff = -0.2287 dB.

Fig. 12 .

 12 Fig.[START_REF] Deng | Heterogeneous cellular network models with dependence[END_REF]. Contour plot of the meta distribution F (θ, x) for the GaPPP/PPP cellular network for the path loss exponent α = 4. The effective gain is G eff = -0.3064 dB. The values at the curves are F (θ, x) = v = 0.95, 0.9, 0.8, 0.7, 0.6, and 0.5 (from bottom to top).

Fig. 13 .

 13 Fig.[START_REF] Suryaprakash | On the modeling and analysis of heterogeneous radio access networks using a Poisson cluster process[END_REF]. Contour plot of the meta distribution F (θ, x) for the PTL/PPP cellular network for the path loss exponent α = 4. The effective gain is G eff = 0.5361 dB. The values at the curves are F (θ, x) = v = 0.95, 0.9, 0.8, 0.7, 0.6, and 0.5 (from bottom to top).

e 1 1 + θP ik r α k t α i b tdt dr = ∞ 0 2πλ k r exp -λ k πr 2 -λ2πλ k r exp -λ k πr 2 F

 1022 -λiπ(P ik ) δ i r α k δ i × exp -2πλ i i π(P ik ) δi r α k δi -(b, δ k , θ/G k ) -i∈[K] ! λ i πP δi ik r α k δi F (b, δ i , θ) dr (e) = ∞ 0 exp -sF (b, δ k , θ/G k ) -i∈[K] ! ρ ik s α k α i F (b, δ i , θ) ds,

k..

  (b) follows from the approximation of Φ i by a PPP Φ PPP i . (c) follows from the probability generating functional (PGFL) of the PPP and averaging over the distance x 0 of the typical user to the nearest BS belonging toΦ PPP k θt -1/δ ) b dt ≡ 2 F 1 (b, -δ; 1 -δ, -θ) -1. (35)(e) uses the substitution s = λ k πr 2 . Finally, by summing over [K], we obtain the result.APPENDIX B PROOF OF THM. 3Letw k λ k P δ k i∈[K] λiP δ i. We can then write Mb (θ) in[START_REF] Wang | SIR meta distribution of K-tier downlink heterogeneous cellular networks with cell range expansion[END_REF] asMb (θ) = k∈[K] w k w k F (b, δ, θ/G k ) + (1 -w k )F (b, δ, θ) . (36) Noticing that for b > 0, F (b, δ, θ/G) is convex in G ∈ (0, ∞), ] w k (w k G k +(1-w k )) ] w k (w k G k + (1 -w k )),

  Approximation of the point process of BSs of kth tier by a PPP of the same density as Φ k

		TABLE I
		NOTATION AND ABBREVIATION
	Notation	Definition/Meaning
	Φ k	Point process of BSs of kth tier
	Φ PPP k
	λ k	Density of BSs of kth tier
	P k	Transmit power of BSs of kth tier
	[K]	{1, 2, . . . , K}
	SIR	Signal-to-interference ratio
	θ	SIR threshold
	α k	Path loss exponent of kth tier
	δ k	2/α k
	G 0	Asymptotic SIR gain as θ → 0
	G k	G 0 for kth tier, k ∈ [K]
	G eff	Effective gain for HCNs
	ps(θ)	Standard success (coverage) probability
	Ps(θ)	Conditional link success probability
	M b (θ) F (θ, x)	bth moment of the conditional link success probability SIR meta distribution for the target reliability of x
	MISR	Mean interference-to-signal ratio
	EFIR	Expected fading-to-interference ratio
	Rpert	Perturbation radius for the perturbed triangular lattice
	p, u	Probability of one-point cluster and the distance between two points in a two-point cluster for
		the Gauss-Poisson point process
	λp, c, rc	Density of parent point process, the mean number of points in a cluster, and the radius of the
		cluster for the Matérn cluster process

The success probability of a link is the probability that the SIR at the receiver of that link is greater than the target SIR threshold θ.

An (easier) alternative to generate the stationary triangular lattice is to generate the stationary square lattice, which has square Voronoi cells, and then multiply its points by the generator matrix G.

For a stationary non-Poisson point process, an analytical expression of the MISR (and hence that of the gain G 0 ) is currently unavailable. An approximate formula for G 0 is obtained in a very recent paper[START_REF] Takahashi | Simple and fast PPP-based approximation of SIR distributions in downlink cellular networks[END_REF].

This work was done when S. S. Kalamkar was with the University of Notre Dame, IN, USA. This work is supported by the US National Science Foundation (grant CCF 1525904). Part of this work was presented at the 2018 IEEE International Conference on Communications (ICC'18) [1].


Angelika Bikner-Ahsbahs

Arthur Bakker
email: bikner@math.uni-bremen.de

Heather Lynn Johnson
email: heather.johnson@ucdenver.edu

Man Ching

Esther Chan

Introduc- Tion

Keywords: Networking of theories, methodology, embodied design, teaching and learning path, epistemology, design genre, generativity, generalizability, generality

la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction to the Thematic Working Group 17 on Theoretical Per-spectives and Approaches in Mathematics Education Research of CERME11

Introduction to the Thematic Working Group 17 on Theoretical Perspectives and Approaches in Mathematics Education Research of CERME11 1 Theory as a fundamental part of mathematical education research

The discussion about what a theory is, why theory is needed, and how theory is used is a necessary and ongoing debate in the community of mathematics education as a scientific discipline. Although there is no clear agreement in the field [START_REF] Assude | The notions and roles of theory in mathematics education research[END_REF][START_REF] Niss | The very multifaceted nature of mathematics education[END_REF], we can gather some of the essential elements of theories from this debate. Theories can be taken as tools or objects [START_REF] Assude | The notions and roles of theory in mathematics education research[END_REF]; as lenses to "see," "observe," and "understand" phenomena of teaching and learning mathematics. Researchers can draw on theory to improve coherence and consistency of scientific argumentation, but researchers also need to guard against becoming blind to aspects that a theory does not capture. While theories provide a language, a more or less structured and coherent system of concepts, theorizing is a way of sense making [START_REF] Mason | The role of theory in mathematics education and research[END_REF]. Background theories encompass a philosophical stance to "understand what are taken to be the things that can be questioned and what counts as an answer to that questioning" [START_REF] Mason | The role of theory in mathematics education and research[END_REF], p. 1056), while foreground theories are about particular research objects addressing a foreground aim such as "what does and can happen within and without educational institutions?" (p. 1056). Some of these foreground theories may developed into an epistemological tool for investigating a specific phenomenon, for example, the epistemic action model of Abstraction in Context (Tabach, Rasmussen, Dreyfus, & Hershkowitz, 2017). Theories are integral to the research process because "[t]heory is the essential product of the research activities, and theorizing, therefore, its essential goal" (Bishop, 1992, p. 711).

2 An overview about the previous work of the CERME theory groups [START_REF] Kidron | Theoretical perspectives and approaches in mathematics education research[END_REF] have offered a comprehensive overview about the theory working groups of CERME over the years, and we would like to highlight some aspects of the groups in view of the Theoretical Perspectives Thematic Working Group (TWG) of CERME11.

We aim to provide insights as to why continuing work on theory is necessary for an ongoing establishing and re-establishing of mathematics education as a scientific discipline. The founding of this working group can be taken as a reaction to the increasing diversity of theoretical approaches towards the end of the millennium which provided varied perspectives, such as cognitive, sociological, institutional, or activity theory-based views, to name a few. This diversity of theoretical approaches raised the question of how coherence can be maintained in the field since in these approaches, similar words may be used for different things or different words for similar phenomena, thus causing problems when communicating research results [START_REF] Bikner-Ahsbahs | Networking of theories-why and how? Special plenary lecture[END_REF]. From the discussion on the diversity of theories at CERME4, the notion of networking of theories emerged (Artigue, Bartolini Bussi, Dreyfus, Gray, & Prediger, 2006). This notion takes the diversity of theories as a rich feature of the field, thus, respects the identity of different theory cultures, and does not aim at achieving a unifying theoretical understanding in the field [START_REF] Bikner-Ahsbahs | Networking of theories-why and how? Special plenary lecture[END_REF]. Important steps to understand how the networking of theories can maintain coherence in the field of mathematics education research include the use of networking strategies in research involving multiple theories [START_REF] Prediger | Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework[END_REF] and the development of networking methodologies for the investigation of theory use in research. When networking theories, the aim is not to create a uniform theory, but rather to embrace complexities and re-new our theoretical understanding arising from examining networking practices.

Differences in background theoretical views also create differences in the notion of what theories are. For this reason, the networking of theories faces difficulties not only through incommensurable and incompatible assumptions of theories but also because the notions of theory used by researchers may not be the same (cf., [START_REF] Niss | The very multifaceted nature of mathematics education[END_REF]. In many sessions over the years, participants generally agreed upon using the concept of theory developed by [START_REF] Radford | Connecting theories in mathematics education: Challenges and possibilities[END_REF][START_REF] Radford | On the growth and transformation on mathematics education[END_REF], whereby a theory [(P,M,Q)R] is a way of producing understanding based on a set of relatively stable principles (P), methodologies (M) related to the principles, paradigmatic questions (Q), and its use in research produces results (R) that contributes to its further developing. [START_REF] Monaghan | Theoretical genesis of an informal meta-theory to develop a way of talking about mathematics and science education and to connect European and North American literature[END_REF] proposed to consider "theoretical genesis" as similar to "instrumental genesis": A theory is an artifact addressing and developing a specific research practice, which in turn may allow researchers to use the theory more creatively in a new way. Thus, theories or theoretical approaches frame research practices as well as develop further through the act of research.

Main issues raised at the Thematic Working Group 17 at CERME11

The thematic working group on Theoretical Perspectives and Approaches in Mathematics Education Research of CERME11 has taken up the tradition of addressing multi-theoretic approaches but put more emphasis on the interplay between theory and methodology, and how this interplay may advance research practice. Key issues addressed included epistemological and philosophical considerations, ontology of concepts, and contributions on design research involving new directions such as embodied design and the use of technology. To advance our knowledge on the triad of theory-methodology-research, one specific objective was to identify argumentative grammars for design research, and (re-)new theory strands such as embodiment.

Nineteen papers and two posters were presented at the conference as part of the TWG. In the proceedings, these contributions are grouped according to five topics. In what follows, we present vari-ous interconnections between theory and methodology in a diagram and describebased on the diagramkey points raised by the papers to emphasize their contributions to the aims of the TWG.

Essential contributions to the interplay of theory and methodology

Building from [START_REF] Radford | Connecting theories in mathematics education: Challenges and possibilities[END_REF] theory triplets (P, M, Q), we propose a graphic organizer (Fig. 1) illustrating the interplay among researchers' work with theories, their elaboration of methodology, and enactment of research. We place Working with Theories in the top layer, because researchers' methodology-related decisions are driven by their tacit or explicit sensitivities to theoretical assumptions. When working with theories, researchers are expected to make explicit their sensitivities to research epistemologies, grain sizes of theories, kinds of mathematical knowledge, objects of research, ecologies of theories, and research problems, phenomena, or purposes. While we show each of these elements as discrete ovals in the graphic organizer, we view these sensitivities to be overlapping. For example, research-ers may choose to network theories of different grain sizes to investigate phenomena of students' ways of coming to know mathematics as research objects. Yet, networking of theories is not an aim in itself but was developed to solve problems. Further, decisions in networking are never neutral, because scholars developed theories with particular perspectives, for example, based on specific research results and within particular ecological paradigms. Researchers' methodological decisions relate to their research problem, phenomenon, and purpose. We view this relationship as reflexiveresearchers' theorizing informs their research problems, which in turn inform their methodologies, and their methodologies in turn inform their research problems.

There

Reciprocity of theory and methodology: mutual affordances

The first topic, reciprocity of theory and methodology for research and design, sets the scene: The topic emphasizes the epistemological nature of research, highlights and describes the reciprocity of theory and methodology in research, and how such reciprocity may impact on design and research.

Kidron compared the procedures of a priori analyses and their epistemological roles in the application of two theories, suggesting that differences in the a priori analyses reflected different analytical focus priorities based on the theories and, hence, illustrated the reciprocity of theory and methodology. Chan and Clarke conceptualized and explained this reciprocity between theory and methodology in terms of mutual affordance, where affordance refers to "the investigative options made possible (and also constrained) by the choice of theory or methodology". In the case of design research, Hanke and Bikner-Ahsbahs encapsulated the mutual affordances between theory and methodology into a design principle "boundary crossing by design(ing)," linking the theoretical construct of boundary crossing with a course design for pre-service mathematics teacher training.

Philosophical considerations: interplay of epistemology and ontology

The second topic addresses philosophical, epistemological, and ontological considerations. The papers provide examples of the interplay of theory and methodology and this interplay's epistemological and ontological implications on the level of generality. The papers address analytical ways of arguing, thus generality, but also illustrating the generality by concrete examples.

Lensing argued that concepts and discourse form an ontological source of building mathematical objects. Radford illustrated the epistemological power of joint labor on tasks and material as ontological source in the classroom. Kuzniak and Vivier transformed the notion of the work mathematicians do as the epistemological source of concepts to the mathematical work in the classrooms. They conceptualized this work as the dialectic between the cognitive and the epistemological levels when learners build mathematical objects or concepts. Zarianakos developed a phenomenological research methodology to understand epistemic processes and highlighted the usefulness of a phenomenological attitude to investigating such processes. Finally, Flores advocated for the legitimization of a visuality's perspective for mathematical visualization. Using art laboratories as a case example, her poster challenged mathematics education researchers to rethink what mathematics and mathematics education are.

Embodied design: generativity and theory-methodology-design bundle as key ideas

The third topic focuses on embodied design with coordination as a key aspect, yet the nature of coordination presented in the papers varies. The papers reported on theoretical coordination, coordination of scientific criteria of design research, epistemological coordination, and coordination of individual affect and individual movement by a trans-individual activity and of disciplines.

Alberto et al. coordinated embodied design and instrumental genesis into the concept of embodied instrumentation; in this kind of networking, the researchers regarded the body as part of an instrumental genesis with an artifact. Their example illustrated a specific design genre where the movements of both hands must be coordinated to explore a problem of the sine graph. Referring to the same design genre, Bakker, Shvarts and Abrahmason made a plea for attending to generativity in addition to generalizability in education research. Such double attention generates the need for scientific coordination of generalizability and generativity-two criteria resonating with the two parts of design research, the design(ing) and the theorizing based empirical data. Based on a culturalhistorical approach, Shvarts' poster abstract added an example of embodied action-based design. She showed that in her research, three kinds of couplings could be epistemologically coordinated through the whole bundle theory-methodology-design: theory-methodology, theory-design and method-design, where the theory-methodology-design bundle serves as the comprehensive lens for the perception-action system distributed between the tutor and student during the teaching-learning process.

The fourth contribution on embodiment by de Freitas, Ferrara and Ferrari provided a theoretical idea which was based on affectivity as the responsive nature of the human body. This kind of affectivity was substantiated by examining the coordination of students' movements while they collectively performed a task involving body movements to produce and express a circle.

Research on teaching and learning paths: the need to go beyond existing solutions

The fourth topic continues discussions about design research, addressing the need to go beyond a single theoretical framework or existing frameworks. Johnson, McClintock, and Gardner coordinated Variation Theory as a pedagogical theory of learning with the theory of Quantitative Reasoning, as a subject-specific theory for design. This kind of networking was possible because the research linked two different but compatible theory grain sizes. Tasks, shaping a design genre already, were reconsidered for the new design step of transfer. This step required an additional theoretical model that was sensitive to transfer, thus allowing the generativity of the design genre be expanded. Similarly, Fonger, Ellis, and Dogan coordinated radical constructivist theory with Harel's Duality, Necessity, Repeated Reasoning principle for instructional design of a learning trajectory for quadratic growth. While exploring the design of the trajectory, the authors identified shifts of understanding among mathematics students, which demanded the inclusion of an instructional theory into the theorized trajectory, thus expanding the concept of the learning trajectory towards a teaching and learning trajectory.

While the first two contributions within the fourth topic addressed designing and investigating the (designed) learning path empirically, Gosztonyi's contribution turned this focus around. She described reverse engineering as a reverse design process of already designed series of mathematical problems: The aim is to reconstruct the rationale behind these series so that teachers can design new series of (selected) problems. Gosztonyi's research is an example of coordination between historical and empirical research methods.

Finally, Bampatsikou et al. took a unique approach to analyze students' learning paths applying Peirce's complete classification of sign relations. According to Bampatsikou et al., the take-up of these classes of signs may "mark those characteristics of the tasks that turn students' conceptualizations to higher level signs" in their semiosis, thus, resonating with the results of Fonger et al.

Theorizing the new: advancing research in terms of theory and methodology

The final topic aimed at showing how new requirements in educations (e.g., new mathematical problems, programming, social media), coming from outside or inside a theory culture, have become driving forces for new ways of theorizing and, hence, advancing research. Makar and Fielding-Well observed that socio-mathematical norms have not been investigated in recent developments of inquiry-based learning even though such norms can strongly impact on teaching. Similarly, Itsios and Barzel observed that the fundamental concept of Grundvorstellungen

has not yet been applied to students' difficulties in understanding exponential expressions. The two papers provide examples of ways to strengthen theoretical approaches that had emerged in different ecologies by expanding their scope and carefully consider the ontological status of the concepts involved. The lack of a theory's applicability can also lead to a theoretical expansion. Exactly this was presented by Otaki, Asami-Johansson and Bahn showing an extension of the Anthropological Theory of the Didactics to embrace and investigate the para-didactic system of the Japanese lesson study.

As in previous ERME conferences, this group agreed upon the dynamic, evolving nature of theories in research. From this perspective, scholars should neither demand that theories be used with absolute rigor nor allow arbitrarily applications of theory. To form coherent research frameworks, scholars engage in reconsidering, reinterpreting and reusing theories to investigate new phenomena, solve new problems and serve new purposes. Thus, theories develop and evolve through research.

Working on coherence and consistency is an ongoing research task, particularly necessary for the Networking of Theories, in which reconsidering the compatibility of the theories or theoretical approaches is an additional epistemological necessity. Achieving generativity, generalizability, and generality affords the potential of the research results to be useful for answering new questions.

Bridging theory and methods to advance research is methodological work. To do so in a coherent and consistent manner, researchers need sensitivity, which is both, brought in by scholars in the field and required from the field. Our graphic organizer (Fig. 1) shows the different sensitivities that were considered during our group's discussions. The presentations on embodied design challenged the group to discuss what counts as mathematics (4.3). Theory ecology was an implicit topic in several contributions (4.5, 4.2). In the presentations focused on philosophical contributions, the group more explicitly addressed sensitivities to the ontology of concepts (4.2). In the presentations on teaching and learning paths, the group examined links between the grain size of the research focus and what researchers took up as research objects, whether the focus was on a micro-step while solving a task, a whole learning path, series of problems or on transfer of knowledge (4.4).

The different sensitivities intertwine, following the epistemology of research in mathematics education-what counts as new knowledge, how that knowledge is achieved and substantiated in the field. New knowledge was gained in terms of epistemological criteria (affordance, generativity, generalizability, generality, phenomenological methodology), extending the scope of theoretical constructs (Grundvorstellungen, socio-mathematical norms, para-didactic system, and interdisciplinary research), the nature of design research (on transfer, on teaching learning trajectories, on embodied design), strengthening theoretical constructions analytically (mathematical work, Theory of Objectification, formation of objects), or in terms of advancing the networking of theories strand by adding networking cases.

With our graphic organizer (Fig. 1), we offer a space for the interplay among researchers' theoretical working, elaborating of methodology, and enactment of research. Notably, we are not trying to find a homogenous, one size fits all epistemological path. Rather, we are working to navigate in the field in order to advance our research in different directions, for example, in the directions of methodology, methods, theory, scope, design, epistemology and ontology. Looking ahead, as topics on epistemology and ontology were addressed and the problem of discerning different argumentation grammars is not yet solved, both should be included in future calls of the TWG on Theoretical Perspectives and Approaches in Mathematics Education Research of ERME.

Artigue, M., Bartolini Bussi, M., Dreyfus, T., Gray, E., & Prediger, S. (2006). Different theoretical perspectives and approaches in research in mathematics education: Report from Working Group

Figure 1 :

 1 Figure 1: A graphic organizer outlining the interplay among researchers' theoretical working, elaborating methodology, and enactment of research.

  are three overarching kinds of research which inform researchers' methodological decisions: design research, empirical research, and theoretical research. For these three kinds of research we distinguish three quality criteria: generativity, generalizability, and generality. To illustrate what we mean by generativity, generalizability, and generality, we draw on research presented in the Theoretical Perspectives TWG of CERME11. The design research carried out by two different groups-Bakker, Shvarts, and Abrahamson, as well as Alberto, Bakker, Walker-van Aalst, Boon, and Drijvers-resulted in new tasks generating a design genre. The theoretical research conducted by de Freitas, Ferrara, and Ferrari addressed affectivity as a general concept of the body's responsivity, illustrating it through an empirical example. Liljekvist, van Bommel, Randahl, and Olin-Scheilier addressed a particular research problem, engaging in empirical research to develop generalizable knowledge. The base layer of the diagram focuses on the enactment of research. Researchers' theoretical frameworks afford and constrain their methodology-driven and methods-driven decisions, when collecting and analyzing research data, and vice versa.

  Connecting mathematics and programming, Lagrange and Laval explained how a framework of connected working spaces may account for students' work across domains; hence, contributing to the generativity of designs for cross-domain research. Liljekvist et al. illustrated how they combined theoretical frameworks for investigating professional development through the social media as a new way of interacting. Lagrange and Laval as well as Liljekvist et al. provided new dimensions to the work of research, suggesting new complexities that may emerge.


scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1 Department of Geography, Université Côte d'Azur, CNRS, ESPACE, Nice, France. mchapuis@unice.fr 2 CEA Cadarache, Saint Paul lez Durance, France. kelian.lagreve@gmail.com ; lylia.kateb@cea.fr 3 Department of Geography, Université Rennes 2, Rennes, France. simon.dufour@univ-rennes2.fr

Context

The Durance River is a dynamic, wandering gravel-bed river which sediment dynamics has been heavily modified in the last century because of dam infrastructures and gravel mining.

Results

5a. Active width and lateral mobility -Fig.3

Before dam (1958)

• Lateral mobility controled by geologic structure (confluence, cluse)

After dam (after 1959)

• 1958-1993: reduction of active width (64%), also linked to low hydrological regime • 1994-2016: increase in active width (11%), linked to:

-intense hydrological regime (9 floods > 900 m 3 /s) -new dam management procedures (discharge and sediment continuity)

Methods

Aim

To study the morphodynamics of a wandering reach located downstream a dam that lets sediment go through, we:

• assessed the bed morphodynamics in relation to the modified hydrology downstream the dam (study period: 1949-2016, quantification of the active width) • quantified the mid-term (7 years: 2010-2017) sediment fluxes downstream the dam.

Field

  -based gravel fluxes measurements in a wandering river to assess sediment mobility downstream a dam (Durance River, Southern French Alps) K. Lagreve, M. Chapuis, L. Kateb, S. Dufour, B. Couvert, C. Doddoli, M. Provansal To cite this version: K. Lagreve, M. Chapuis, L. Kateb, S. Dufour, B. Couvert, et al.. Field-based gravel fluxes measurements in a wandering river to assess sediment mobility downstream a dam (Durance River, Southern French Alps). RCEM 2017-10th Symposium on River, Coastal and Estuarine Morphodynamics, Sep 2017, Trente, Italy. RCEM 2017-10th Symposium on River, Coastal and Estuarine Morphodynamics, 2017

  Fig.3

•

  5b. Bed topographic evolution and RFID tracking -Fig.4 2009-2017 period • Bed reworking (erosion vs. deposition phases) linked to hydrology • Local general trend: deposition • Outer banks translations by bank erosion  ≡ sediment input downstream 5c. Longitudinal profile -Fig.5 1909-2008 period • Direct consequence of the dam: incision up to 2 m (Chapuis et al., 2015) • Decrease in bed slope: adjustment to sediment discontinuity due to the dam (downstream erosion) 2008-2016 period • Increase in bed slope to adjust from dam new management procedures: validation of the « equilibrium slope » to match transport capacity and upstream bedload supply (see hypothesis in Lefort, 2007, p.13) Large (mean width: 240 m) and steep (mean slope: 0.23%) wandering river • 10 km-long reach located in the lower part of the Durance River (Southern French Alps), downstream Cadarache Dam • Intense planning within the floodplain since 1960 (dam, highway, diversion canal) • Peculiar hydrological regime: both Mediterranean and alpine influences  intense and short floods • Bed material: gravel and fine matrix (coarse fraction: D 50 = 40 mm ; sand and silt fraction: D 50 = 33 µm) • Since 2008, new management procedures at the dam: sediment transport is allowed when flooding above 500 m 3 /s  sediment continuity

Field

  Fig.2