Jean-Marc Kelif
email: jeanmarc.kelif@orange.com

Jean-Marie Gorce
email: jean-marie.gorce@insa-lyon.fr.

Azeddine Gati
email: azed-dine.gati@orange.com

Performance and Energy in Green Superposition Coding Wireless Networks: An Analytical Model

In this paper, we develop a powerful analytical model of wireless network with Superposition Coding (SC), also referred to as Non Orthogonal Multiple Access (NOMA). This model allows to establish a closed form expression of the minimum power a base station (BS) need to transmit, for a user to achieve a given SINR (signal to interference plus noise ratio) whatever its location in the area covered by the base station. It moreover allows to establish a closed form expression of the minimum total transmit power of a base station. These closed form expressions allow to establish performance of wireless networks, by minimizing the base stations transmit powers. As an application, we show that these closed form expressions allow to quantify the energetic performance, spectral efficiency, total throughput and the coverage of a BS, in a simple and quick way.

I. INTRODUCTION

The design of future 5G wireless networks has to address new challenges such as a latency reduction and capacity increase. In addition, the possibility to connect billions objects receiving data, related to the Internet of Things (IoT), represents also an important issue of the 5G.

In order to meet these requirements, different solutions are envisioned by telecommunication operators allowing to increase networks capacity. Among them, the Superposition Coding (SC) mechanism may represent an interesting approach. Knowing that the achievable throughput of wireless systems is bounded by a fundamental limit [START_REF] Shannon | A mathematical theory of communication[END_REF] [2] [START_REF] Gorce | Spatial continuum model: Toward the fundamental limits of dense wireless networks[END_REF], one interest of SC, also referred to as Non Orthogonal Multiple Access (NOMA) [START_REF] Ding | On the performance of non-orthogonal multiple access in 5g systems with randomly deployed mobiles[END_REF], is that it achieves this fundamental bound, providing more capacity than classical orthogonal multiple access [START_REF] Gorce | Spatial continuum model: Toward the fundamental limits of dense wireless networks[END_REF] [START_REF] Gorce | Energy-capacity trade-off bounds in a downlink typical cell[END_REF].

SC is also promising in the case of IoT, since data need to be transmitted, simultaneously, towards a great number of communicating objects. Several network access technologies have recently been proposed for IoT, as analyzed in [START_REF] Goursaud | Dedicated networks for iot: Phy/mac state of the art and challenges[END_REF] [START_REF] Centenaro | Long-range communications in unlicensed bands: The rising stars in the IoT and smart city scenarios[END_REF], and in uplink scenario [START_REF] Chandrasekhar | Uplink capacity and interference avoidance for two-tier femtocell networks[END_REF] [START_REF] Dhillon | Fundamentals of throughput maximization with random arrivals for M2M communications[END_REF]. An important open question of IoT is to determine the maximum user density that a base station (BS) can serve, taking into account the transmit power, performance and coverage constraints. Considering a spatial density of users, a new approach has been proposed in [START_REF] Gorce | Spatial continuum model: Toward the fundamental limits of dense wireless networks[END_REF] [START_REF] Gorce | Energy-capacity trade-off bounds in a downlink typical cell[END_REF], called spatial continuum model. Based on SC, this approach allowed to analyze the fundamental limits of a dense cell.

In [START_REF] Kim | Superposition Coding is Almost Always Optimal for the Poisson Broadcast Channel[END_REF] SC is shown to be capacity achieving for a continuous-time Poisson broadcast channel with different parameters.

Analyzing a scenario with two users, one close to a BS and the other far from it, the paper [START_REF] Choi | Minimum Power Multicast Beamforming With Superposition Coding for Multiresolution Broadcast and Application to NOMA Systems[END_REF] considers multicast beamforming with superposition coding. An iterative algorithm is developed, based on an optimal power allocation, in order to find beamforming vectors and powers for both users.

In the opposite, assuming Gaussian broadcast channels and taking into account the receiver power constraints, [START_REF] Kim | Energy-Efficient Delay-Tolerant Communication: Revisiting Optimality of Superposition Coding in Broadcast Channels[END_REF] shows that multi-user transmission with SC is not always optimal in terms of spectral efficiency.

In [START_REF] Hwang | Performance Evaluation of Superposition Coding in Downlink LTE[END_REF] SC is used in the aim to increase the system capacity through multiuser diversity exploitation. A joint admission control and SC scheme is proposed which allows to provide a good tradeoff between the QoS level perceived by the user in the system and the utilization of the scarce radio resources.

The paper [START_REF] Abbes | Joint Superposition Coding and Great Fairness Admission Control for QoS provisioning in LTE systems[END_REF] develops an analysis of NOMA scheme in the uplink of wireless powered communication networks. The objective is to maximize the sum rate by jointly designing the time allocation, the downlink energy beamforming and receiver beamforming. It moreover develops an algorithm in the aim to obtain the optimal time allocation. Authors establish results which outperform the fixed time allocation.

Considering the downlink of a NOMA wireless network, authors of [START_REF] Fang | Energy-Efficient Resource Allocation for Downlink Non-Orthogonal Multiple Access Network[END_REF] develop resource allocation algorithms Their algorithms are energy-efficient. In their approach, they assign only two users to the same subchannel. Their analysis allows to maximize the system energy efficiency.

None of these papers proposes closed form formulas of the BS transmit power towards a user and total BS transmit power.

Our Contribution: We develop an analytical model allowing to analyze a wireless network with superposition coding, in which successive interference cancellation (SIC) is implemented. Our analysis takes into account the interference induced by all the base stations of the network. We establish a closed form formula of the minimum power a BS needs to transmit, for a user to reach a SINR target, whatever its location in the area covered by the base station. Moreover, we establish a closed form formula of the minimum BS total transmit power. To the best of our knowledge, this is the first time such formulas are established. This allows to analyze in a simple and quick way, the performance of base stations in terms of energy and throughput, and their coverage.

The organization of the paper follows. In Section II, the system model is presented. In Section III, we establish the closed form formulas of the BS transmit power towards a user whatever its location, and the total base station transmit power. Section IV proposes an analysis of these closed form formulas. Section V presents an application of these closed form formulas focused on the analysis of coverage and performance, in terms of energy and throughput. Section VI concludes the paper.

II. SYSTEM MODEL

We consider a wireless network made of M base stations uniformly distributed. All base stations transmit over the same frequency bandwidth. We focus our analysis on the downlink, on the area covered by a BS b, using a superposition coding mechanism. Moreover the successive interference cancellation (SIC) is implemented. Let remind that superposition coding is also referred to as Non Orthogonal Multiple Access (NOMA) [START_REF] Ding | On the performance of non-orthogonal multiple access in 5g systems with randomly deployed mobiles[END_REF], Let us consider:

• M = {1, . . . , M } the set of base stations, uniformly and regularly distributed over the two-dimensional plane. • U = {1, . . . , U } the set of users uniformly distributed over the network. • N = {1, . . . , N } the set of users connected to a BS, uniformly and regularly distributed. • γ * (also denoted γ) the SINR target to be reached in downlink by any receiver (user). • r u the distance between the transmit BS b and the user u, connected to b. • N u the set of users connected to b, located at a distance lower or equal to the distance r u . • N u the set of users connected to b, located at a distance greater than r u . • P u the transmitted power assigned by the base station b towards user u. • g u the propagation gain between the serving BS b and user u. Let consider that a SINR target, denoted γ * , has to be reached at each user, in the aim to reach a given throughput. We aim to establish a mathematical expression of a BS transmit power, in the aim to reach this SINR target. Since users and base stations are uniformly distributed over the network, we can focus the analysis on any given BS.

A. Expression of the SINR with Superposition Coding

In our analysis, by considering SC, we assume that a BS transmits useful power to all the users of the cell it covers, simultaneously, on the same frequency bandwidth.

Considering a given BS denoted b, we can express the total amount of power received by any user (UE) u connected to the BS b, as the sum of: (i) a useful signal P u g u , (ii) the interference power due to the transmit powers of the BS b towards the other users of the cell: v∈N,v =u P v g u , where P u and P v are the transmit powers towards the mobile u and v and g u is the pathloss between the BS b and the mobile u, (iii) the interference power due to the total transmit power of the other BSs j = b of the network which use the same frequency banwidth as b: I N w = j =b P j g j,u where P j is the total transmit power of BS j towards their own users and g j,u is the propagation gain between BS j and the mobile u (connected to b), (iv) and thermal noise power N th . Therefore, we can consider the SINR target γ * (u) defined by:

γ * (u) = P u g u v∈N,v =u P v g u + I N w + N th . (1)
Users are connected to their best server BS, i.e. the BS which provides the best useful signal. Therefore, the shadowing can be neglected as shown in [START_REF] Kelif | Limiting the Power of 4G Dynamic Green Cellular Networks: Impact on Capacity and Quality of Service[END_REF] [START_REF] Baszczyszyn | Using Poisson processes to model lattice cellular networks[END_REF]. Let remind that, in this case, the best server BS is the nearest BS.

Therefore, considering

g v = Kr -η v [10]
, the SINR is expressed as:

γ * (u) = P u Kr -η u v∈N,v =u P v Kr -η u + I N w + N th , ( 2
)
where η is the pathloss parameter and K is a constant.

B. Successive Interference Cancellation

Since the BS transmits useful powers simultaneously to all the users, and on the same frequency, a user u receives a high level of interference due to the powers transmitted towards all the other users of the cell. The SIC mechanism allows to mitigate this interference. Let rewrite the expression (2) by considering the set of users N as the union of the set N u of users v, such that r v ≤ r u and the set N u of users v, such that r v > r u :

γ * (u) = P u Kr -η u v∈Nu,v =u P v Kr -η u + v∈N u ,v =u P v Kr -η u + I N w + N th .
(3) For the user u to reach a given target SINR γ * , it is needed to transmit at a given power P u . Due to the pathloss between a transmitter and a receiver, users located at distances r v > r u (from BS b) need a transmit power P v > P u in the aim to reach the same SINR γ * . The successive interference cancellation mechanism allows the user u to decode, using an iterative process, the powers it receives dedicated to users located at distances further than him (from the BS b). Let notice p u u the power dedicated to user u and received by the user u (p u u = P u Kr -η u ). Let notice p v u the power dedicated to user v and received by the user u (p v u = P v Kr -η u ). Let consider a user u needing a useful power p u u in the aim to reach the SINR γ * . Using the SIC mechanism, the user u is able to decode any (interfering) power it receives higher than p u u . And particularly, it is able to decode the power it receives dedicated to the user v. Indeed, since P v ≥ P u , the power p v u has a higher value than p u u . Therefore, p v u can be decoded by the user u. And thus, it does no more represent an interference for u. This is available for any user v for which Figure 1. Cell of radius Rc. SIC: user UE 2, located at r, is able to decode the power dedicated to any user UE 3, located at a distance greater than r (white area). The power dedicated to any user UE 1, located at a distance lower than r (grey area), cannot be decoded by user UE 2. r v > r u . As a consequence, considering all the users of the cell, , the term

v∈N u ,v =u P v Kr -η
u disappears from expression

(3), since it does no more represent an interference (see fig. 1). We thus can write (3) as:

γ * (u) = KP u r -η u v∈Nu,v =u KP v r -η u + I N w + N th . ( 4
)
which can be written:

γ * (u) = P u v,rv≤ru P v -P u + INw+N th K r η u . (5)
From this expression, the total power transmitted to users located at a distance lower or equal to r u can be expressed as:

v,rv≤ru P v = γ * (u) + 1 γ * (u) P u - I N w K r η u - N th K r η u . (6)
Since I N w = j =b P j g j,u = j =b KP j r -η j,u , we can write v,rv≤ru

P v = γ * (u) + 1 γ * (u) P u - j =b P j r -η j,u r η u - N th K r η u . (7)
From this expression, we can express the transmit power towards a user and the total transmit power of a BS.

III. WIRELESS NETWORK ANALYTICAL MODEL

The analytical model proposed in this paper consists in considering the interfering base stations of the network as a continuum of base stations, characterized by a density of base station ρ bs . This approach was validated by comparison with a simulated hexagonal network [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] [20] [START_REF] Kelif | Spatial Outage Probability for Cellular Networks[END_REF]. Considering a uniform distribution of users over the network, we can write P j = P B = constant. Therefore, the interference due to all the base stations of the network j,j =b KP B r -η j,u can be expressed as P B ρ bs Ks -η sdsdθ, where sdsdθ represents the element of integration. The approach developed in [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] [20] [START_REF] Kelif | Spatial Outage Probability for Cellular Networks[END_REF] allows to express this integral as 2π ρ bs PB K(2Rc-ru) 2-η η-2

, where 2R c represents the intersite distance (ISD) between two BS. We refer the reader to [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] [20] [START_REF] Kelif | Spatial Outage Probability for Cellular Networks[END_REF] for the detailed explanation and validation through Monte Carlo simulations. Using this model, it becomes possible to express [START_REF] Centenaro | Long-range communications in unlicensed bands: The rising stars in the IoT and smart city scenarios[END_REF] as follows. Assuming the cell covered by the BS as a disk of radius R c , we consider a user located at a distance r from it (dropping the index u). The discrete sum on users v of ( 7) can be expressed as an integral over the cell. A unique SINR target γ * is considered for all the users of the cell. This corresponds to a service which demands a given throughput for all users. For sake of simplicity, let denote γ = γ * and ζ = γ * +1 γ * . We can write (7) as:

2π 0 r R0 ρ(s)P (s)sdsdθ = ζP (r) - 2πρ bs (2R c -r) 2-η r η η -2 P B - N th K r η . ( 8
)
where P (r) the power transmitted towards the user located at distance r, s is the variable of integration, ρ(s) is the density of users on the area covered by the BS, depending on s and sdsdθ is the element of integration, and R 0 the minimum distance between a user and the base station. Let remind that in expression ( 8) the parameter P B , which represents the total transmit power of a BS, is a constant. Since the users are uniformly distributed over the area of the cell, ρ(s) = ρ does not depend on s, (8) can be written as:

2πρ r R0 P (s)sds = ζP (r) - 2πρ bs (2R c -r) 2-η η -2 r η P B - N th K r η . (9)
Let denote

a(r) = 2π ρ bs (2R c -r) 2-η η -2 r η , (10) b
(r) = N th K r η , (11)
and

F (r) = a(r)P B + b(r). (12)
We thus can write (9) as:

2πρ r R0 P (s)sds = ζP (r) -F (r). (13)
From expression [START_REF] Kim | Superposition Coding is Almost Always Optimal for the Poisson Broadcast Channel[END_REF], two important expressions can be derived: i) the transmit power towards a user located at distance r, ii) the total BS transmit power. The two following theorems can be written:

A. Theorem 1: Transmit Power Towards A User

Considering a density of users ρ, and assuming a unique SINR target γ over the cell, the transmit power towards a user located at a distance r from the BS, denoted P (r), can be expressed as (denote α = γ γ+1 ρπ):

P (r) = γ γ + 1 e αr 2 r R0 F ′ (s)e -αs 2 ds, (14)
where F'(r) is the derivative of F(r) with respect to r:

F ′ (r) = a ′ (r)P B + b ′ (r) (15)
and a ′ (r) = 4π ρ bs (Rcη-r)(2Rc-r) 1-η r η-1 η-2

is the derivative of a(r) with respect to r, and b ′ (r) = η N th K r η-1 is the derivative of b(r) with respect to r. The proof is detailed in Appendix 1.

B. Theorem 2: Total BS Transmit Power

Considering a density of users ρ, and assuming a unique SINR target γ over the cell, the total power transmitted by the BS, simultaneously towards all the users of the cell, denoted P B , can be expressed as:

P B = e αR 2 c Rc R0 b ′ (s)e -αs 2 ds -b(R c ) 1 -e αR 2 c Rc R0 a ′ (s)e -αs 2 ds + a(R c ) , ( 16
)
The proof is detailed in Appendix 2.

Remark

Let notice that if the interference due to the other base stations of the network is neglected, the term a(r) is neglected. The total BS transmit power can thus be expressed as:

P B = e αR 2 c Rc R0 b ′ (s)e -αs 2 ds -b(R c ). (17)
Since b ′ (s) = η N th K s η-1 (from ( 11)), we can write

P B = η N th K e αR 2 c Rc R0 s η-1 e -αs 2 ds - N th K R η c ( 18
)
By considering R 0 << R c and thus R0 Rc ≈ 0, and denoting β = αR 2 c , we can express [START_REF] Fang | Energy-Efficient Resource Allocation for Downlink Non-Orthogonal Multiple Access Network[END_REF] as:

P B = 2β N th K R η c e β Γ( η + 2 2 , β) -Γ( η + 2 2 ) (19)
where

Γ(x) = ∞ 0 t x-1 e -t dt and Γ(u, x) = ∞ x t u-1 e -t dt is the incomplete Γ function.

IV. BASE STATION TRANSMIT POWER ANALYSIS

A. Transmit Power Expressions Interest

The expressions ( 14) and ( 16) highlight the impact of the parameters of the whole network

• on the transmit power towards a user located at a distance r ( 14), • on the total transmit power of the BS [START_REF] Hwang | Performance Evaluation of Superposition Coding in Downlink LTE[END_REF]. The minimum transmitted power needed to reach a given SINR target γ * depends on i) the user location r (transmit power towards a user ( 14)), ii) the size of the cell R c (total transmit power ( 16)), iii) the propagation (pathloss factor) η, iv) the antenna characteristics and frequency (parameter K), v) the thermal noise N th . vi) the density of users and the SINR target value. Considering the network impact, two more parameters play a role: the intersite distance and the density of BS.

B. Network Impact

It can be noticed that the network impact is expressed as the denominator of [START_REF] Hwang | Performance Evaluation of Superposition Coding in Downlink LTE[END_REF]. Since a power is a positive quantity, the denominator of ( 16) needs to be strictly positive:

1 -e αR 2 c Rc R0 a ′ (s)e -αs 2 ds + a(R c ) > 0. (20)
If the denominator is close to 0, the power tends to infinity. From [START_REF]Guidelines for evaluation of radio interface technologies for IMT-Advanced[END_REF], since

ρ bs = 1 πR 2 c
, we can express [START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF] as:

4 η e β 1 R0/Rc (η -u)(2 -u) 1-η u η-1 e -βu 2 du < 1 (21)
This expression shows constraints on the density of users and the SINR target. Moreover, they are strongly correlated (since

β = αR 2 c = γ γ+1 ρπR 2 c
). The SINR target is limited for a given density of users. Therefore the achievable throughput is limited. This means that the behavior observed (fig. 2 and fig. 3, dotted curves) is directly induced by the network impact. Over a given SINR target value, the needed transmit power increases indefinitely. For example, fig. 2 (left side, dotted curve) shows that for a density of 8 users per cell, and a cell radius of 50m, the BS transmit power varies from -15 to 29 dBm for a total cell throughput varying from 1 Mbits/s to a maximum of 30 Mbits/s. Let remind that this value corresponds to a given value of SINR. And then, the transmit power value increases indefinitely, without increasing the throughput.

C. Isolated Cell

Considering a theoretic case where the impact of the network could be neglected, the SINR target and the density of users could reach high level values. Expression [START_REF] Fang | Energy-Efficient Resource Allocation for Downlink Non-Orthogonal Multiple Access Network[END_REF] shows that the density of users and the SINR target are not limited.

The examples of fig. 2 and3 show that the curves without network impact (plain curves) have behaviors completely different than the ones with the impact of network (dotted curves). Without network impact, the reachable SINR target is not limited. It can reach very high values. Therefore, the total throughput can reach, in theory, very high values, by using limited transmit powers. Therefore, the network impact is significant. Let notice that the isolated cell represents a theoretic case, which does not exist (or whose probability is very low) in a real network.

These powerful expressions make it possible to develop network analyzes of great interest in a simple and quick analytical way. Hereafter, we propose an example of application focused on performance and energy. Other applications may be developed such as, for example, dimensioning or outage probability analysis.

V. APPLICATION: ENERGY, PERFORMANCE AND COVERAGE

In this section, the energetic performance, throughput, and coverage of a BS, are analyzed and quantified in an analytical way (considering values given by Table I). The minimum transmit power needed for a BS to cover an area of radius R c is given by expression [START_REF] Hwang | Performance Evaluation of Superposition Coding in Downlink LTE[END_REF]. In other words the coverage can directly be calculated with accuracy by using ( 16), for a given transmit power. And moreover, this allows to minimize the energy needed to cover a given area. This expression also gives the reachable level of SINR in the cell, and consequently the maximum throughput. Let remind that, for a given SINR γ, the spectral efficiency SE is given by SE = log 2 (1 + γ) and the throughput D is given by D = W log 2 (1 + γ), where W is the frequency bandwidth.

Fig. 2 and 3 compare the reachable performances, in terms of energy and total throughput per cell, by considering or not, the interference induced by the network.

1) Considering the network impact : the maximum total throughput of the cell reaches 25 Mbits/s (Fig. 2 and3, dotted curves), for 8 and 15 users/cell area, for R c = 50m (Fig. 2) and R c 500m (Fig. 3). The BS transmit power needed to reach the maximum throughput of 25 Mbits/s increases with R c . It reaches about 10 dBm for R c = 50m, and 46 dBm for R c = 500m. Therefore, the expression ( 16) also allows to show that the transmit power is an increasing function of the total throughput. Moreover, it increases with R c , too.

2) Without network impact : the total throughput of the cell can theoretically increase indefinitely (Fig. 2 and3, plain curves). In the aim to quantify the network impact, we consider the same transmitting values as before, 10 dBm and 46 dBm. For R c = 50m (Fig. 2), and a BS transmit power of 10 dBm, a total throughput of 100 Mbits/s is reached for a density of 8 users per cell area (resp. 80 Mbits/s for 15 users per cell area). For R c = 500m (Fig. 3), and a BS transmit power of 46 dBm, a total throughput of 100 Mbits/s is reached for a density of 8 users per cell area (resp. 80 Mbits/s for 15 users per cell area). Therefore, due to network impact, a decrease of up to 75% of the total throughput is observed in Figures 2 and3. This confirms the significant impact of the network.

VI. CONCLUSION

We establish closed form expressions of the minimum total transmit power of a BS and of the transmit power needed to achieve a given SINR, whatever the user location, in a wireless network, by considering the Superposition Coding and SIC mechanism. These expressions allow to highlight the impact of the parameters of the system such as the cell radius, the propagation parameters, and the intersite distance. In particular they allow taking into account the impact of the whole network. These expressions are powerful since they make it possible to carry out analyzes in a simple and quick way. As an application, we quantified with accuracy the energetic performance, throughput and coverage of a base station. In a future work, the impact of a dynamic traffic scenario will be analyzed. APPENDIX 1 First, the method to determine the transmitting power consists in derivating expression [START_REF] Kim | Superposition Coding is Almost Always Optimal for the Poisson Broadcast Channel[END_REF] with respect to r. It thus can be written:

2πρP (r)r = ζP ′ (r) -F ′ (r), (22)
where P ′ (resp. F ′ ) means the derivative of P (resp. F ) with respect to r. This differential equation with non-constant coefficients allows to express the transmit power towards a user located at a distance r from its serving BS as:

P (r) = C(r)e

Figure 2 .

 2 Figure 2. Base station Transmit Power vs Total throughput of the cell.Comparison for a density of 8 users per cell area (left) and 15 users per cell area (right), and for a radius of the zone covered of Rc= 50m, considering the impact of the network (dotted curve) and neglecting the impact of the network (plain curve).

Figure 3 .

 3 Figure 3. Base station Transmit Power vs Total throughput of the cell. Comparison for a density of 8 users per cell area (left) and 15 users per cell area (right), and for a radius of the zone covered of Rc= 500m, considering the impact of the network (dotted curve) and neglecting the impact of the network (plain curve).

γ γ+1 ρπr 2 ,

 2

APPENDIX 2

The total transmit power of a BS is expressed as:

In expression (25), the third equality denoted (3) comes from [START_REF] Kim | Superposition Coding is Almost Always Optimal for the Poisson Broadcast Channel[END_REF], where r = Rc. We denote α = γ γ+1 ρπ. Let express ζP (Rc) from (23):

where (from 24), we can write (remind that ζ = γ+1 γ ):

From ( 12), we can express:

where

r η and b(r) = N th K r η . Therefore F ′ (r) can be expressed as:

And thus, we can express:

where

and

From expressions (25) to (32), we can express the total BS transmit power as:

And thus:


Jean-Marie Gorce
email: jean-marie.gorce@insa-lyon.fr

Philippe Mary

Jean-Marc Kélif

TOWARDS FUNDAMENTAL LIMITS OF BURSTY MULTI-USER COMMUNICATIONS IN WIRELESS NETWORK

Considering an isolated wireless cell containing a high density of nodes, the fundamental limit can be defined as the maximal number of nodes the associate base station can serve under some system level constraints including maximal rate, reliability, latency and transmission power. This limit can be investigated in the downlink, modeled as a spatial continuum broadast channel (SCBC) as well as in the uplink modeled as a spatial continuum multiple access channel (SCMAC).

In this short paper, we summarize the different steps towards the characterization of this fundamental limit, considering four figures of merit: energy efficiency, spectral efficiency, latency, reliabilty.

INTRODUCTION

The recent evolution of the cellular market towards internet of things (IoT) connecting objects instead of humans induces significant changes in the mathematical modeling of wireless networks. Indead, the IoT paradigm relies on bursty but massive distributed transmissions, where billions of communicating objects will be spread over large radio cells, transmitting only few packets per day, per month or even per year. In such a scenario, the classical fundamental limit of communication systems derived since the seminal work of Claude E. Shannon [START_REF] Ce Shannon | A mathematical theory of communication[END_REF] needs to be revised. As we will see, the capacity, or the capacity region in case of multi-user communications, becomes less important in regard to other metrics.

It is worth that Shannon's second theorem on capacity can be expressed as a fundamental tradeoff between energy efficiency (EE) and spectral efficiency (SE). For a real Gaussian channel the received power needs to verify P ≥ 2 η S /2 -1 N 0 , where η s , the spectral efficiency, is the number of bits-per-channel-use and N 0 the receiver noise. The normalized energy efficiency is given by η E = η S /(P/N 0 ) . But this fundamental limit is achievable in the asymptotic regime, i.e. when the encoding time spread over an infinite number of channel uses. This is nothing but a steady-state analysis and relies on a set of assumptions, not further valid in the context of this study:

1. A unique source and a unique destination are considered.

2. The traffic is characterized by a fixed and continuous data rate.

3. Synchronization and transmitter identification costs are dropped off since the communication is assumed to be established.

4.

The encoding time is infinite and so is the latency.

Information theory has been extended since its infancy to multi-user scenarios including the multiple access channel (MAC) or the broadcast channel (BC) well appropriate to study the uplink and the downlink, respectively. Most noticeable results with these schemes have been provided in the asymptotic regime, providing the expression of achievable rates region [START_REF] Thomas | Elements of information theory[END_REF] characterizing achievable joint rates. For Gaussian channels, these capacity regions are known even for a large number of nodes and strategies based on superposition coding and successive decoding are capacity achieving [START_REF] Thomas | Elements of information theory[END_REF]. But in the context of IoT cells with bursty traffic, these joint rates are not representative of the real scenarios that have to be supported. It comes out to be necessary to study these fundamental limits in the finite block length (FBL) regime where the information transmitted in the network is divided in finite size quantities that have to be transmitted under finite time transmission constraints. Yuri Polyanskiy et al [START_REF] Polyanskiy | Channel coding rate in the finite blocklength regime[END_REF] recently studied the fundamental limits of the point to point Gaussian channel in the FBL regime, paving the way to the study of latency and reliability constraints from a fundamental point of view. At the best of our knowledge the unique extension of this work for Gaussian channels has been done by Molavian-Jazi and Laneman [START_REF] Molavianjazi | A Second-Order Achievable Rate Region for Gaussian Multi-Access Channels via a Central Limit Theorem for Functions[END_REF] for the Gaussian MAC. We analysed the Gaussian BC in [START_REF] Unsal | The Dispersion of Superposition Coding for Gaussian Broadcast Channels[END_REF].

In addition to the FBL regime it is also necessary to study the behavior of BC/MAC models when the nodes are modeled through a spatial distribution as we proposed in [START_REF] Gorce | Spatial continuum model: Toward the fundamental limits of dense wireless networks[END_REF][START_REF] Gorce | Fundamental limits of a dense iot cell in the uplink[END_REF] to establish the fundamental EE-SE limit under equal rates condition in the asymptotic regime. We called these models SCMAC or SCBC, where SC stands for spatial continuum. This fundamental limit can be interpreted as the equivalent of the asymptotic Shannon capacity for a wireless cell.

If these results provided interesting insights, latency and reliability where kept off the study. Indeed, the asymptotic regime relies on error free but infinite latency transmission. So, the next question relates on the price of introducing a latency constraint. Such a constraint comes essentially with a reliability penalty, but in addition, the capacity region also shrinks.

We therefore anticipate three issues to be addressed in the framework of dense IoT networks. Firstly, the cost of the transient regime necessary to synchronize and detect which nodes are transmitting cannot be neglected. This corresponds to an additional information to be transmitted that should be considered in large scale low rate networks. Secondly, transmitting small packets over finite time slots leads necessarily to transmission errors. The minimal error can be modeled in information theory by working in the finite block length (FBL) regime, introducing an additional latency-reliability tradeoff.

SYSTEM MODEL

Let consider a unique BS, serving a cell area denoted by Ω ⊂ R 2 . We denote by (Ω, A, m) the corresponding measurable space with A the Lebesgue σalgebra and m the Lebesgue measure. Let x be a point in Ω.

Without lack of genericity, the BS is assumed to be located at point (0, 0). Nodes appear randomly in time and space on Ω. As such, they are not described by a discrete set but through a spatial density u(x) (the density of users per m 2 per time unit). Denoting by ũ(x, t) a realization of the ergodic random process with probability density function (pdf) u(x), then, for any subset B ∈ A, the average number of users per time unit is given by

U (B) = lim T →∞ 1 T t∈T x∈B ũ(x, t) • dx • dt. (1)
The global average number of nodes associated with the whole cell service area Ω is denoted by U T .

Definition 2.1 (Requested rate density)

The requested rate density ρ(x) : Ω → R is a measurable function that represents the information rate spatial density requested at point x.

• Note 1: ρ(x) is normalized 1 by the system bandwidth and is expressed in bits per channel use (bpcu) per m 2 .

• Note 2: The requested rates are either in downlink (SCBC) or uplink (SCMAC) modes.

The cell sum-rate per channel use is denoted the spectral efficiency (SE) of the cell:

η s = Ω ρ(x) • m(dx). (2)
1 ρ(x) and related quantities (R (.)...) are given in bit-per-channeluse (bpcu) throughout this paper. For practical applications, the results can be converted in bits or in bps, with appropriate system parameters.

The uniform rate condition corresponds to the case where each node requests the same quantity of information denoted by I 0 , with

ρ(x) = I 0 • u(x) N cu , (3)
where N cu is the number of channel uses per time unit, allocated to the system. In this special case, one have η s = I0•U T Ncu .

UNIFORM CAPACITY IN THE ASYMPTOTIC REGIME

The access capacity region is defined in [START_REF] Gorce | Spatial continuum model: Toward the fundamental limits of dense wireless networks[END_REF] for the uplink and the downlink, as the set of rate spatial distributions for which a joint encoding-decoding scheme exists such that the transmission error tends to 0 when the encoding time tends to infinity (characterized by the number of channel uses n). Compared to the classical Shannon's approach, the asymptotic regime herein integrates another dimension: when n → ∞, the number of nodes tends to infinity but individual rates tend to 0 since each packet is spread over an infinite time. Note that the cell's sumrate tends to its average spectral efficiency. In [START_REF] Gorce | Spatial continuum model: Toward the fundamental limits of dense wireless networks[END_REF], based on an iterative splitting process, the sum-rate that the cell can achieve when a continuum of users is considered has been obtained. The corresponding fundamental limit is expressed through the total transmisison power required to serve a given distribution and was found identical for the GSCBC and the GSCMAC with transferable powers [START_REF] Gorce | Fundamental limits of a dense iot cell in the uplink[END_REF]:

Theorem 3.1 (GSCBC and GSCMAC fundamental limit)

The achievable EE-SE tradeoff for a given rate spatial density ρ(x) is given by

η e ≤ a ν M νm t • f ν (t) • e a•ηs•Gν (t) • dt -1 , (4)
where

• a = 2 log(2).
• f ν (t) and G ν (t) are respectively the probability density function (pdf) and the complementary cumulated density function (ccdf) of the rate as a function of the equivalent noise (an exemple is represented in Fig. 1).

EXTENSIONS IN THE NON ASYMPTOTIC REGIME

The former result was derived in the asymptotic regime under no latency constraint. Modeling IoT networks requires additional constraints that will reduce the fundamentl limit.

Finite Time Transmission

The asymptotic regime used in Th. 3.1 means that all packets are transmitted simultaneously in the network to or from all nodes and spread over an infinite time. Let now be imposed a finite time constraint formalized as follows:

Figure 1. The rate request is distributed according to the density f ν (.), where ν(x) for a node located in x is the equivalent noise power given by the ratio between the receiver noise and the channel gain ν(x) = N 0 /g(x). Then, G ν (t) is the fraction of the cell with an equivalent noise power greater or equal to t. This distribution for a regular circular cell is represented where the x axis is the equivalent noise and the y axis represents the rate density.

Property 4.1 A multi-user network with packets of constant information quantities I 0 , is said FTT constrained if each transmission is constrained to be done in at most n * channel uses.

The average spectral efficiency (i.e. the sum-rate) of the cell is noted ηs and is nothing but:

ηs = I 0 • U (Ω) N cu . ( 5
)
In the downlink, when BS transmits packets of I 0 bits in n * channel uses, individual rates are given by η u = I0 n * . The BS can achieve the target spectral efficiency by using superpostion coding (SC) to simultaneously transmit several packets. Therefore, optimizing the transmission strategy relies on optimizing the scheduling of the different packets in order to minimize the total power.

Let us define a frame of length N cu , made of L slots, each slot s l ; ∀l ∈ {1, 2, . . . , L} being made of n * channel uses. For consistency, all time units are counted in channel uses. In the downlink, the BS has in its queue a random number of packets to be transmitted to a set of nodes N U,L , with lim L→∞

|N U,L | L•n * = U (Ω).
In each slot s l , a subset of nodes N u (l) ⊂ N U,L is selected and the corresponding packets are simultaneously transmitted. Let N u,l = |N u,l | be the number of nodes allocated to the slot s l , the spectral efficiency is :

η s (l) = N u,l • η u . (6)
To achieve the desired rate η u = I 0 /n * , the SINR at each decoding step has to verify γ * ≥ e aηu -1 according to the Shannon channel capacity. To be optimal, the power allocation is computed such that each node gets a SINR exactly equal to γ * . Using a superposition coding strategy, each node decodes its signal after having decoded all greater signals.

Under these conditions of equal rate, the minimal power associated to each slot is obtained as: Theorem 4.1 (Sum-power in the downlink) For a given set of users indexed by {1, . . . , N u,l }, the minimal sumpower necessary to transmit reliably to each node independent informations of size I 0 in n * channel uses, is given by:

P m (l) = γ * N u,l k=1 νk ,l • (1 + γ * ) k-1 (7)
with γ * = e aI0/n * -1, and k = N u,l -k + 1.

Therefore, comparing numerically Th. 4.1 to Th. 3.1 allows to evaluate the performance loss due to the FTT constraint.

Finite Block Length Coding

In the former section, we assumed n * to be large enough to have error-free transmission. This is not the case in a band limited system.

Strassen [START_REF] Strassen | Asymptotische Abschaetzungen in Shannon's Informationstheorie[END_REF] studied the finite blocklength regime and has shown for a Discrete Memoryless Channel (DMC) that:

log M * (n, ) = nC -Q -1 ( ) √ nV + O(log n) (8)
where n, and M are the blocklength, the error probability and the codebook size, respectively. The second term in the right hand side of ( 8) is the rate penalty, namely the back-off from the channel capacity. This expression gives the fundamental trade-off between latency and reliability for short packet transmissions. Unfortunately, Strassen's result was not compliant to be generalized, particularly to the AWGN channel, generally to any type of channel with input constraints. Almost half a century later, Polyanskiy et al. [START_REF] Polyanskiy | Channel coding rate in the finite blocklength regime[END_REF] has generalized Strassen's approach, i.e. treating the mutual information density as a random variable and using second order statistics to approximate the channel coding rates in the FBL regime, for various channel models such as the AWGN channel, for which the following holds:

Theorem 4.2 [START_REF] Polyanskiy | Channel coding rate in the finite blocklength regime[END_REF] For the AWGN channel with signal-to-noise ratio (SNR) γ, for ∈ (0, 1) and for equal, maximal and average-power constraints, the following holds,

log M * (n, , γ) = nC(γ) -Q -1 ( ) nV (γ) + O(log n) (9
) where C(γ) is the Shannon capacity defined as

C(γ) = 1 2 log(1 + γ) (10)
and

V (γ) = γ 2 γ + 2 (γ + 1) 2 log 2 e. (11)
However, the extension of this result to the case of multiuser channels is not straightforward. The exact capacity regions of Gaussian MAC or Gaussian BC are not known but achievable regions for the 2-user BC or 2-user MAC have been characterized recently in [START_REF] Molavianjazi | A Second-Order Achievable Rate Region for Gaussian Multi-Access Channels via a Central Limit Theorem for Functions[END_REF][START_REF] Unsal | The Dispersion of Superposition Coding for Gaussian Broadcast Channels[END_REF]. As a first approximation, the use of a superposition coding strategy [START_REF] Unsal | The Dispersion of Superposition Coding for Gaussian Broadcast Channels[END_REF] allows to extrapolate the FBL expression in Th. 4.2 to compute a modified SNR threshold γ * as a function of some latency and reliability constraints. This γ * can be used in Th. 4.1 to modify the fundamental limit.

Additional real time constraints

In the former paragraph, we detailed how to combine the dense nodes distribution assumption and a finite time transmission constraint. To model real-time transmissions, where the nodes have to transmit or receive instantaneously, additional issues have to be considered. In the downlink, the main challenge is to indicate to the nodes if they have something to decode and how it is encoded. Clearly, signaling cannot be ignored and is even a part of the information to be transmitted. In the uplink, the randomness of the active nodes distribution and their identification is necessary. Stochastic geometry coupled with FBL information theory can provide insights about the impact of randomness [START_REF] Harpreet S Dhillon | Fundamentals of throughput maximization with random arrivals for m2m communications[END_REF]. From a protocol point of view, integrating data transmission and signaling as a whole, approaches based on compressive sensing [START_REF] Fletcher | Necessary and Sufficient Conditions for Sparsity Pattern Recovery[END_REF][START_REF] Xie | Many Access for Small Packets Based on Precoding and Sparsity-Aware Recovery[END_REF] are appealing although their optimality is not granted.

CONCLUSION

In this paper, we paved the way to derive fundamental limits of IoT cells where dense distributions of users are spread over large cells and where a sporadic traffic has to be supported. Starting from the EE-SE fundamental limit in the asymptotic regime, we defined the different steps towards the derivation of the EE-SE loss in the FBL regime and also how to balance it with respect to the reliabilitylatency trade-off.

ACKNOWLEDGMENTS

This work has been partly supported by Orange Labs under CRE nF 05151, and by the French National Agency for Research (ANR) under project Arburst.