Hamed Samie
email: hamed.samie@univ-poitiers.fr

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Rodolphe Vauzelle
email: rodolphe.vauzelle@univ-poitiers.fr

Power Control in Wireless Cellular Networks with a Time-Varying Delay

Keywords: Power control, Potential feedback control, Time-varying delay, Wireless Cellular Networks

Uplink transmission power control is an essential task in Wireless Cellular Networks (WCNs) due to the resource limitation of the Mobile Stations (MSs). One remaining problem is the effect of the delay caused by measuring the signal strength and decision making in the Inner-Loop Power Control (ILPC). In this article, we develop the Potential Feedback Controller (PFC) for a linear scalar discrete-time system with disturbance in order to take into account an unknown bounded time-varying input delay for uplink ILPC. The main interest of the PFC is to treat easily a stabilization problem with a constraint on the state space by using a nonlinear feedback control with a short computation time. Simulations illustrate that by applying the PFC, the communication connectivity is ensured by maintaining the signal strength above a required limit.

Introduction

Power control in WCNs is a key degree of freedom in the management of interference, energy and connectivity [START_REF] Chiang | Power control in wireless cellular networks[END_REF]. In uplink transmission, energy efficiency aspect of power control is more dominant due to the limited energy resource of the Mobile Station (MS). Uplink power control in WCNs includes two main objectives [START_REF] Alpcan | CDMA uplink power control as a noncooperative game[END_REF][START_REF] Sesia | The UMTS Long Term Evolution: From Theory to Practice[END_REF][START_REF] Castro | The UMTS network and radio access technology[END_REF]:

• firstly, to ensure the communication connectivity by setting up a power controller to keep the received signal strength above a limit below which the communication is interrupted.

• secondly, to minimize the overall transmitted power in a cell in order to minimize the interferences between users and to maximize the battery life of the MS.

Principally, power control in WCNs consists of an open-loop power control and a closed-loop power con-trol schemes [START_REF] Chiang | Power control in wireless cellular networks[END_REF]. In open-loop power control, the Base Station (BS) selects the transmit power control by exploiting the estimated channel condition like distancedependent attenuation and frequency-independent slow fading at the MS based on the received signal strength of a pilot signal transmitted by the BS [START_REF] Laiho | Radio network planning and optimisation for UMTS[END_REF]. The closed-loop power control itself consists of two different loops: Inner-Loop Power Control (ILPC) and Outer-Loop Power Control (OLPC) [START_REF] Gunnarsson | Dynamical effects of time delays and time delay compensation in power controlled DS-CDMA[END_REF][START_REF] Laiho | Radio network planning and optimisation for UMTS[END_REF]. In ILPC, the received signal strength is compared to a target value at the BS. If the measured signal strength is higher (lower) than the target value, the BS will send a control signal to the MS in order to decrease (increase) the transmission power. The ILPC should be repeated fast enough to cover the fast fading effect. The OLPC (which is slower than the ILPC) provides the target signal strength based on the Quality of Service (QoS) requirements in the higher layers [START_REF] Laiho | Radio network planning and optimisation for UMTS[END_REF]. In this paper, we are interested in the ILPC.

Transmitting and measuring signals and decision making take time which results in time delays in the closedloop power control. The main reasons of these delays come from the power control algorithm itself; time for computing and decision making; time to transmit the power control command to the MS [START_REF] Holma | LTE for UMTS-OFDMA and SC-FDMA based radio access[END_REF][START_REF] Luna-Rivera | Distributed power control with multiuser detection for asynchronous DS-CDMA networks subject to time-delays[END_REF][START_REF] Chiang | Power control in wireless cellular networks[END_REF]. According to the time-varying behavior of the transmission channel quality, load of the MSs and the size of the packets to be sent in a realistic scenario, a time-varying delay in the closed-loop power control must be considered. The delay in uplink ILPC is discussed further in Section 2. Like every feedback control loop, the ILPC will be affected by this time-varying delay. In fact, the power control is more sensitive to the delay than to the signal strength estimator [START_REF] Gunnarsson | Dynamical effects of time delays and time delay compensation in power controlled DS-CDMA[END_REF].

Stability of the OLPC in the presence of communication delays has been studied in [START_REF] Yates | A framework for uplink power control in cellular radio systems[END_REF][START_REF] Sung | A generalized framework for distributed power control in wireless networks[END_REF][START_REF] Feyzmahdavian | Contractive interference functions and rates of convergence of distributed power control laws[END_REF]. In [START_REF] Gunnarsson | Dynamical effects of time delays and time delay compensation in power controlled DS-CDMA[END_REF], the authors proposed a control scheme to highlight the dynamical behavior of the ILPC subject to a constant delay where they proposed a time-delay compensation (TDC) in order to mitigate the oscillations due to the delay. In our paper, we develop the PFC first proposed in [START_REF] Phan | Potential feedback control for the power control in LTE[END_REF] in order to take into account an unknown bounded time-varying input delay for uplink ILPC in WCNs. To the best of our knowledge, this is the first time that a solution for the time-varying delay in the ILPC of WCNs has been proposed.

The rest of the paper is organized as follows. The system model is described in Section 2. In order to take into account a time-varying input delay, the PFC is developed in Section 3. It is applied and simulated for a WCN using the Long-Term Evolution (LTE) communication standard and compared with the existing Transmission Power Control (TPC) algorithm in Section 4. Finally, a conclusion is addressed in Section 5.

Modeling

In our article, we are interested in the uplink ILPC of a cell with a MS denoted by k and a BS denoted by i which is illustrated in Figure 1. The received Signal-to-Fig. 1. Uplink ILPC between a MS and a BS Interference-plus-Noise Ratio (SINR) is commonly used in wireless communication to measure the transmission quality. For simplicity, we neglect the intra-cell interferences (in communication standards like LTE and 802.11n, the use of respectively OFDMA and OFDM avoids the intra-cell interferences [START_REF] Khan | LTE for 4G mobile broadband: air interface technologies and performance[END_REF]). Therefore, after frequency synchronization, only the inter-cell interferences affect the SINR in the model. The received SINR at the BS i of the transmitted signal from the MS k is given by

χ ik (n) = g ik (n)p ik (n) l =k g il (n).p il (n) + σ 2 ik (n) (1)
where p ik is the transmitted power; g ik the channel gain by which the signal between the transmitter k and the receiver i is attenuated and it can be modeled by three components: path-loss, log-normal shadowing and multipath fading [START_REF] Rappaport | Wireless Communications Principles and Practice[END_REF]; σ 2 ik the thermal noise affecting the channel between the MS k and the BS i; n represents the ILPC sample time. The term l =k g il (n).p il (n) corresponds to the inter-cell interferences. As it has been assumed that the intra-cell interferences are neglected, we can rewrite (1) as follows

χ ik (n) = g ik (n).p ik (n) i ik (n) + σ 2 ik (n) (2)
where i ik (n) = l =k g il (n).p il (n) are the inter-cell interferences. Let us define the pathloss pl between the MS k and the BS i by pl ik (n) =

1 g ik (n) and i o t ik (n) = (i ik (n)+σ 2 ik (n)) σ 2 ik (n)
. We can rewrite (2) as follows

χ ik (n) = p ik (n) pl ik (n).σ 2 ik (n).i o t ik (n) . (3)
In the following, we rewrite the received SINR equation in the logarithmic domain in dB. Consequently, equation (3) becomes

x ik (n) = P ik (n) -P L ik (n) -Σ 2 ik (n) -I o T ik (n). ( 4
)

Uplink ILPC in WCNs

In uplink ILPC, the SINR is estimated at the MS and it is sent to the BS where it will be compared with a SINR min which is the minimum required value of SINR to maintain the communication connectivity. Based on its difference with the SINR min , a TPC command is sent to the MS to update the transmission power [START_REF] Muhammad | Performance evaluation of uplink closed loop power control for LTE system[END_REF][START_REF] Gunnarsson | Dynamical effects of time delays and time delay compensation in power controlled DS-CDMA[END_REF][START_REF] Sheth | Adaptive power control and selective radio activation for low-power infrastructure-mode 802[END_REF]. So we can describe this procedure as follows:

• Receiver (BS):

err i (n) = x ik (n) -x min u i (n) = f (err i (n)) (5)
where x min is the SINR min ; err i the difference between the available measured SINR at the sample time n and the SINR min ; u i the TPC command computed at the BS to be sent to the MS which is a function of the err i as it is explained above. The SINR min is provided by the OLPC at a lower update rate based on the required value of the Block Error Rate (BLER) in higher layers [START_REF] Chiang | Power control in wireless cellular networks[END_REF]. Hence, this value may be regarded constant for the ILPC. There exist different algorithms based on the different communication standards applied in WCNs [START_REF] Muhammad | Performance evaluation of uplink closed loop power control for LTE system[END_REF][START_REF] Gunnarsson | Dynamical effects of time delays and time delay compensation in power controlled DS-CDMA[END_REF][START_REF] Sheth | Adaptive power control and selective radio activation for low-power infrastructure-mode 802[END_REF]. For example in LTE, the TPC command u i can vary between [-1; 3] dBm (for more information see [START_REF] Muhammad | Performance evaluation of uplink closed loop power control for LTE system[END_REF]). In Section 4, we will compare the PFC with taking into account an unknown bounded time-varying input delay with an existing TPC algorithm in LTE communication standard.

• Transmitter (MS): Evolution of the transmit power at the MS is given by the following equation

P ik (n + 1) = P ik (n) + u i (n -τ (n)) (6)
where P ik is the transmit power adjustment. It should be taken into account that the computed uplink transmit power should not exceed the maximum MS transmit power P max [START_REF] Chiang | Power control in wireless cellular networks[END_REF][START_REF] Castellanos | Performance of uplink fractional power control in UTRAN LTE[END_REF]. For simplicity, we do not take into account the P max in our calculation but we will take it into account in our simulations in Section 4.

Both measuring and signaling in cellular systems take time, which results in delayed signals. Here are the main reasons for the existense of time delay in power control update in WCNs [START_REF] Chiang | Power control in wireless cellular networks[END_REF]:

-the power control algorithm itself results in a delay of one sample time since the power control at sample time n is used to update the power level at sample time n + 1; -it takes certain amount of time to measure the SINR and generate a power control decision at the BS; -the power control update is only allowed to be transmitted at certain time instants; -the time that it takes to transmit the power control command from the BS to the MS.

As it has been explained in Section 1, this delay τ (n) must be considered as a time-varying delay. The capacity of the network and the details of the communication standard are known, therefore the maximum and minimum values of τ (n) are known too:

0 < h 1 ≤ τ (n) ≤ h 2 (7
) where h 1 and h 2 are respectively the minimum and the maximum value of the time-varying delay. In the following, and for simplicity, we will omit the subscripts i and k. By putting (4) into the dynamical equation ( 6), we obtain

x(n + 1) = x(n) + u(n -τ (n)) + P L(n) + Σ 2 (n) + I o T (n) -P L(n + 1) -Σ 2 (n + 1) -I o T (n + 1) (8)
Also, we have

P L(n + 1) = P L(n) + ∆P L(n) Σ 2 (n + 1) = Σ 2 (n) + ∆Σ 2 (n) I o T (n + 1) = I o T (n) + ∆I o T (n) (9)
By putting ( 9) into (8), we obtain

x(n + 1) = x(n) + u(n -τ (n)) + d(n) (10) where d(n) = -∆P L(n) -∆Σ 2 -∆I o T (n).
In the following section, we develop the PFC in order to take into account the time-varying input delay τ (n).

PFC for discrete-time systems with a timevarying input delay

In this section, we develop a new result on stabilization by using the Potential Feedback Controller (PFC) first proposed in [START_REF] Phan | Potential feedback control for the power control in LTE[END_REF], in order to take into account an unknown bounded time-varying input delay for the uplink ILPC of WCNs. Stability of linear discrete-time dynamical systems with an unknown bounded input delay has been discussed in several works. For example [START_REF] González | Robust stabilization of linear discretetime systems with time-varying input delay[END_REF], [START_REF] Chen | Delay-dependent guaranteed cost control for uncertain discrete-time systems with both state and input delays[END_REF] and [START_REF] Yu | Optimal guaranteed cost control of discretetime uncertain systems with both state and input delays[END_REF] propose different solutions by means of Linear Matrix Inequalities (LMIs) and [START_REF] Xia | Sliding mode control of uncertain linear discrete time systems with input delay[END_REF] gives a solution by designing a predictor-based sliding surface. The potential control has originally been applied in robotics to avoid collisions with obstacles [START_REF] Ogren | Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment[END_REF]. The main idea of exploiting the potential control for the ILPC is to maintain the signal strength (SINR) higher than a target value SINR min defined by the OLPC while keeping the SINR as close as possible to the SINR min in order to be more energy efficient. The proof of this result is based on the Artsein's reduction method proposed in [START_REF] González | Robust stabilization of linear discretetime systems with time-varying input delay[END_REF].

Theorem 3.1 Consider the system given by

x(n + 1) = x(n) + u(n -τ (n)) + d 1 (n) (11) where x(n) ∈ R is the state of the system, |u(n)| ≤ M is the bounded input with 0 < h 1 ≤ τ (n) ≤ h 2 ∈ N a bounded time-varying delay and |d 1 (n)| ≤ ε a bounded disturbance. Let c ≤ 0, under the initial condition x(0) > c, system (11) in closed-loop with the feedback control u(n) = k 1 (z(n) -c) + k 2 z(n) -c (12)
where u(n) = 0 for n < 0 and

-1 2 < k 1 < 0.5(-h1-h2+1) h1+h2 k 2 > (k1+1)(ε+2M ) 2 (2k1+1) 2 or 0.5(-h1-h2+1) h1+h2 < k 1 < 0 k 2 > 0.25(ε+2M ) 2 (h1+h2+1) 2 k1+1 and z(n) = x(n)+ 1 2 h1-1 j=0 u(n-h 1 +j)+ 1 2 h2-1 j=0 u(n-h 2 + j)
is asymptotically stable with respect to the attractive and invariant set S = [ 4(k

1 + 1)k 2 -(ε + 2M ) -(h 1 + h 2 )(ε+2M )+c; -(ε+2M )(k1-1)+ √ (ε+2M ) 2 (k1-1) 2 -4k 3 1 k2 2k 2 1 + (ε + 2M ) + (h 1 + h 2 )(ε + 2M ) + c]. Moreover, we have x(n) ≥ 4(k 1 + 1)k 2 -(ε + 2M ) -(h 1 + h 2 )(ε +2M ) + c for all n > h 1 + h 2 .
Proof. We have two possible cases which depend on the sign of the initial condition x(0) of system [START_REF] Khan | LTE for 4G mobile broadband: air interface technologies and performance[END_REF].

First case: x(0) > 0. We take c = 0. By applying the Artstein transform as explained in [START_REF] González | Robust stabilization of linear discretetime systems with time-varying input delay[END_REF], we can transform system [START_REF] Khan | LTE for 4G mobile broadband: air interface technologies and performance[END_REF] into the following scalar discrete-time system without delay

z(n + 1) = z(n) + u(n) + d 1 (n) + d 2 (n) (13) where z(n) is obtained by the following transformation    z(n) = x(n) + φ u n (h 1 ) + φ u n (h 2 ) φ u n (h) = 1 2 h-1 j=0 u(n -h + j) (14)
and d 2 (n) is an extra disturbance caused by applying the Artstein transform and given by the following equation

d 2 (n) = u(n -τ (n)) - 1 2 (u(n -h 1 ) + u(n -h 2 )) (15) A simple calculation based on bounded input assump- tion |u(n)| ≤ M shows that |d 2 | ≤ 2M .
As it has been explained in [8, Theorem 1], system [START_REF] Holma | LTE for UMTS-OFDMA and SC-FDMA based radio access[END_REF] with the feedback control

u(n) = k 1 z(n) + k 2 z(n) (16)
where -1

2 < k 1 < 0 and k 2 ≥ (ε+2M ) 2 (k1+1) (2k1+1) 2
is asymptotically stable with respect to the attractive set

S 2 = [ 4(k 1 + 1)k 2 -(ε + 2M ); -(ε+2M )(k1-1) 2k 2 1 + √ (ε+2M ) 2 (k1-1) 2 -4k 3 1 k2 2k 2 1 ].
According to [8, Theorem 1], we have min

z(n)∈S2 z(n + 1) ≥ min S 2 . ( 17
)
Therefore, due to the variations of the continuous function z(n + 1) with respect to the variable z(n) which is increasing and the Intermediate Value Theorem, we only have to prove that the following conditions

z(n + 1)| z(n)=min S2 ≤ max S 2 (18) z(n + 1)| z(n)=max S2 ≤ max S 2 (19) where min S 2 = 4(k 1 + 1)k 2 -(ε + 2M ) and max S 2 = -(ε+2M )(k1-1)+ √ (ε+2M ) 2 (k1-1) 2 -4k 3 1 k2 2k 2 1
, are satisfied in order that S 2 be an invariant set.

By applying the control feedback [START_REF] Luna-Rivera | Distributed power control with multiuser detection for asynchronous DS-CDMA networks subject to time-delays[END_REF] to system (13), we obtain

z(n + 1) = (k 1 + 1)z(n) + k 2 z(n) + d 1 (n) + d 2 (n) (20)
Afterwards, by solving the equation ∂z(

= 0, we obtain the critical point z c = k2

k1+1 . An investigation into the behavior of system (13) around z c shows that z(n + 1) is increasing on k2 k1+1 , +∞ and decreasing on 0, k2 k1+1 . Therefore, z c is the minimum value of z(n + 1). We have

z min = z(n + 1)| z(n)=zc = (1 + k 1 ) k 2 k 1 + 1 + k 2 k2 k1+1 -(ε + 2M ) = 4(k 1 + 1)k 2 -(ε + 2M ) = min S 2 (21)
Considering that z(n + 1)| z(n)=zc = min S 2 , and z(n + 1) is increasing on k2 k1+1 , +∞ , we deduce that if min S 2 ≥ k2 k1+1 and the inequality ( 19) is fulfilled then inequality ( 18) is fulfilled too. First, we check that

z min = 4(k 1 + 1)k 2 -(ε + 2M ) ≥ k 2 k 1 + 1 which leads to √ k 2 (2k 1 + 1) √ k 1 + 1 ≥ (ε + 2M ) ( 22
)
where √ k 2 ≥ 0 and √ k 1 + 1 ≥ 0. Thus, we have -1 2 < k 1 < 0. Also from [START_REF] Yates | A framework for uplink power control in cellular radio systems[END_REF], we obtain

k 2 ≥ (ε+2M ) 2 (k1+1) (2k1+1) 2
. Now it is sufficient to show that inequality [START_REF] Yu | Optimal guaranteed cost control of discretetime uncertain systems with both state and input delays[END_REF] is fulfilled. We have

z(n + 1)| z(n)=max S2 = N 1 D where N 1 = (k 1 + 1)[-(ε + 2M )(k 1 -1) + (ε + 2M ) 2 (k 1 -1) 2 -4k 3 1 k 2 ] + 4k 4 1 k 2 + 2(ε + 2M )k 2 1 (-(ε + 2M )(k 1 -1) + (ε + 2M ) 2 (k 1 -1) 2 -4k 3 1 k 2 ) and D = 2k 2 1 (-(ε + 2M )(k 1 -1) + (ε + 2M ) 2 (k 1 -1) 2 -4k 3 1 k 2 ). We can rewrite max S 2 as follows max S 2 = (-(ε+2M )(k1-1)+ √ (ε+2M ) 2 (k1-1) 2 -4k 3 1 k2) 2 2k 2 1 (-(ε+2M )(k1-1)+ √ (ε+2M ) 2 (k1-1) 2 -4k 3 1 k2) = N 2 D .
In order that inequality [START_REF] Yu | Optimal guaranteed cost control of discretetime uncertain systems with both state and input delays[END_REF] be fulfilled, we must have

N 2 > N 1 . Thus, a series of calculations lead to 2k 1 (k 1 -1)(ε + 2M ) 2 - 2(ε + 2M )k 1 (ε + 2M ) 2 (k 1 -1) 2 -4k 3 1 k 2 > 0 (23)
which is satisfied considering that -1 2 < k 1 < 0. Hence, we conclude that

S 2 = [ 4(k 1 + 1)k 2 -(ε + 2M ); -(ε+2M )(k1-1) 2k 2 1 + √ (ε+2M ) 2 (k1-1) 2 -4k 3 1 k2 2k 2 1
] is an attractive and invariant set with -1 2 < k 1 < 0 and

k 2 ≥ (ε+2M ) 2 (k1+1) (2k1+1) 2
. By applying the Artstein transform, we derive that system ( 11) is asymptotically stable with respect to a set S we have to determine. From ( 13) we obtain

                                   h1-1 j=0 u(n -h 1 + j) = [z(n) -z(n -h 1 )- h1-1 j=0 d 1 (n -h 1 + j) - h1-1 j=0 d 2 (n -h 1 + j)] = 2φ u n (h 1 ) h2-1 j=0 u(n -h 2 + j) = [z(n) -z(n -h 2 )- h2-1 j=0 d 2 (n -h 2 + j) - h2-1 j=0 d 2 (n -h 2 + j)] = 2φ u n (h 2 ). ( 24
)
From ( 14) and ( 24) we have

x(n) = 1 2 (z(n -h 1 ) + z(n -h 2 )) + γ(n) + λ(n) (25)
where

γ(n) = -1 2 [ h1-1 j=0 d 1 (n-h 1 +j)+ h1-1 j=0 d 2 (n-h 1 +j)] and λ(n) = -1 2 [ h2-1 j=0 d 1 (n -h 2 + j) + h2-1 j=0 d 2 (n -h 2 + j)].
As z(.) is bounded due to [8, Theorem 1] and d 1 and d 2 are bounded by assumption, x(.) is also bounded due to [START_REF] Rappaport | Wireless Communications Principles and Practice[END_REF]. Accordingly, there exists a minimum value x min such that the state x(.) of system ( 13) is always higher than x min . From [START_REF] Rappaport | Wireless Communications Principles and Practice[END_REF] we derive that

x min = z min -(h 1 + h 2 )(ε + 2M ) ( 26
)
where z min = 4(k 1 + 1)k 1 -(ε+2M ) due to [START_REF] Ogren | Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment[END_REF]. This minimum value x min is also the minimum value of the attractive and invariant set S. As x min > 0, a symbolic calculation leads to 4(k

1 + 1)k 1 -(ε + 2M ) -(h 1 + h 2 )(ε + 2M ) > 0 (27)
In order to satisfy [START_REF] Li | A effective SINR link to system mapping method for CQI feedback in TD-LTE system[END_REF], we obtain the following sufficient conditions:

-

1 2 < k 1 < 0.5(-h1-h2+1) h1+h2 k 2 > (k1+1)(ε+2M ) 2 (2k1+1) 2 (28)
or

0.5(-h1-h2+1) h1+h2 < k 1 < 0 k 2 > 0.25(ε+2M ) 2 (h1+h2+1) 2 k1+1 ( 29
)
with ε > 0, 2M > 0, h 2 > 1 and h 1 > 0. Moreover, it is easy to check that the maximum value of S satisfies max S = max S 2 + (h 1 + h 2 )(ε + 2M ).

(30) So we have found the attractive and invariant set S. As we have z(n) > z min for all n > 0, then we have x(n) > x min for all n > h 1 + h 2 .

Second case: x(0) < 0. From ( 11) and ( 12) we have

x(n + 1) = x(n) + k 1 (x(n -τ (n)) -c) + k2 x(n-τ (n))-c + d 1 (n) (31)
By subtracting c in both sides, we obtain

x(n + 1) -c = x(n) -c + k 1 (x(n -τ (n)) -c) + k2 x(n-τ (n))-c + d 1 (n). ( 32
)
Considering the following change of variable

ψ(n) = x(n) -c, it leads to the new system ψ(n + 1) = ψ(n) + k 1 (ψ(n -τ (n))) + k1 ψ(n-τ (n)) + d 1 (n) ( 33
) As x(0) > c, we have ψ(0) > 0, therefore, we can apply the first case of the proof to system (33). Finally we deduce that if x(0) > c, ε > 0, 2M > 0, h 2 > 1, h 2 ≥ τ (n) ≥ h 1 > 0 and k 1 and k 2 are chosen by [START_REF] Xia | Sliding mode control of uncertain linear discrete time systems with input delay[END_REF] or (29) then, system (11) is asymptotically stable with respect to the attractive and invariant set

S = [ 4(k 1 + 1)k 2 -(ε + 2M ) -(h 1 + h 2 )(ε + 2M ) + c; -(ε+2M )(k1-1) 2k 2 1 + √ (ε+2M ) 2 (k1-1) 2 -4k 3 1 k2 2k 2 1 + (ε + 2M ) + (h 1 + h 2 )(ε + 2M ) + c].
This concludes the Proof. 2

Simulation for the LTE standard

The characteristics of the LTE communication standard such as mobility of the MSs which leads to a timevarying transmission channel quality and intense timevarying load of the MSs for each BS imply that the ILPC has a significant time-varying behavior involving a timevarying delay. Consequently, we chose the LTE communication standard as a typical example to illustrate the PFC with taking into account an unknown bounded time-varying input delay.

We suppose that there is an estimation error of 1dB on the d(n) in [START_REF] Gunnarsson | Dynamical effects of time delays and time delay compensation in power controlled DS-CDMA[END_REF]. The assessment of this value depends on the quality of the channel gain estimator (see [START_REF] Phan | Potential feedback control for the power control in LTE[END_REF]). For our simulations it has been assumed that the cell radius is 500m as well as the distance between the MS and the BS which varies between 18.08m and 342m therefore, the P L varies between 63.93dB and 89.18dB.

As it has been described in Section 2, x min which represents the SINR min is provided by the OLPC based on the highest acceptable BLER for communication between the transmitter and the receiver. The maximum BLER given by LTE communication standard is 10 -1 [START_REF] Piro | Simulating LTE cellular systems: an open-source framework[END_REF]. In LTE, Adaptive Modulation and Coding (AMC) module based on the Channel Quality Indicator (CQI) index chooses a specific Modulation and Coding Scheme (MCS) and a Code Rate to transmit the data (for further explanation of the Physical and Medium Access Control (MAC) layers in LTE communication standard, the reader may refer to [START_REF] Cox | An introduction to LTE: LTE, LTE-Advanced[END_REF]). Therefore, based on the relation between the SINR and the BLER for a given MCS and Code Rate, we can define the SINR min [START_REF] Li | A effective SINR link to system mapping method for CQI feedback in TD-LTE system[END_REF]. For our simulations we choose the 64-QAM MCS with Code Rate 2/3. Hence, we find the value of SINR min = 14.37dB (for further details see [START_REF] Phan | Potential feedback control for the power control in LTE[END_REF]).

The coherence time of the channel is given by

T c = 9 16πf 2 max (34)
where f max = v λ is the maximum Doppler Shift with v the velocity of the MS [START_REF] Tse | Fundamentals of wireless communication[END_REF]. The control sample time T s at which we refresh the feedback control cannot exceed the coherence time of the channel. In LTE, the carrier frequency f c is equal to 1.8GHz where f c = c λ . By taking into account the various possible amounts of MS velocity up to 260km.h -1 , we obtain the coherence time T c = 1ms. Hence, we choose the sample time T s = 1ms which is a common value for the sample time in LTE communication standard [START_REF] Khan | LTE for 4G mobile broadband: air interface technologies and performance[END_REF].

In order to take into account the round-trip delay in LTE communication standard, the TPC algorithm takes the maximum round-trip delay as a constant delay. The maximum round-trip delay in LTE is 20ms [START_REF] Holma | LTE for UMTS-OFDMA and SC-FDMA based radio access[END_REF]. Thus, in our simulations we compare the performances of a TPC algorithm considering a maximum round-trip delay of 20ms [START_REF] Muhammad | Performance evaluation of uplink closed loop power control for LTE system[END_REF][START_REF] Song | Evolved cellular network planning and optimization for UMTS and LTE[END_REF], the PFC considering a maximum round-trip input delay of 20ms (PFC c ) [START_REF] Phan | Potential feedback control for the power control in LTE[END_REF] and the PFC with taking into account an unknown time-varying input delay up to a maximum round-trip delay of 20ms (PFC v ) developed in Section 3.

In order to minimize the extra disturbance d 2 (n) in [START_REF] Holma | LTE for UMTS-OFDMA and SC-FDMA based radio access[END_REF], the input of the system which represents the Closed Loop Transmit Power (CLTP) should be limited to the minimum possible variation. Therefore, in our simulations we bound the CLTP by [-1; 1] dBm [START_REF] Cox | An introduction to LTE: LTE, LTE-Advanced[END_REF].

Due to Theorem 3.1, S is an invariant set if -1 2 < k 1 < 0, and k 2 satisfies the following inequality

k 2 ≥ (ε + 2M ) 2 (k 1 + 1) (2k 1 + 1) 2 . ( 35
)
However, equation ( 35) is only a sufficient condition.

In order to consume less transmit power and have less interferences, we choose k 1 so that we have less possible received SINR. A numerical computation shows that the closer that k 1 is to -0.5, the lower the level of the received SINR will be (see Figure 2). Accordingly, we choose k 1 = -0.49 for our simulations. As already mentioned, equation (35) gives us a sufficient condition for choosing k 2 , which for when k 1 = -0.49, k 2 = 636. But by performing an offline adjustment we can find an optimal value for k 2 = 300. The TPC keeps the average SINR lower than the other two schemes but it can not maintain the connectivity for the whole communication duration. The PFC v passes the SINR min after a short period of time and stays above it to maintain the communication connectivity (see Figure 3(b)). The PFC c has the same behavior than the PFC v with less oscillation compared to the PFC v due to the extra disturbance d 2 (n). However, the PFC v has a better performance in start-up compared to the PFC c which due to the transmit power saturation can not maintain the SINR above the SINR min till 0.17s. As it has been explained in Section 2, the computed uplink transmit power P should not exceed the maximum MS transmit power P max which in LTE communication standard is 23dBm [START_REF]Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio transmission and reception[END_REF]. As the PFC v limits the input of the system, it does not exceed P max . The SINR min is provided by the OLPC which also could be time-varying but with a lower rate than the ILPC (see Section 2), therefore this start-up lag can be an issue in this case. Finally, Table 1 compares performances of all three ILPC schemes for 10s simulation. The PFC (PFC v and PFC c ) saves 48 % of the computation time compared to the TPC due to the fact that the TPC algorithm performs a comparison for each ILPC sample time which is costly in computation time. However, the average received SINR of the PFC v and the PFC c are higher than the TPC which is the cost of ensuring the communication connectivity. Furthermore, the average received SINR for the PFC v is slightly higher than the PFC c while it passes the SINR min and stays above it significantly faster than the PFC c (see Figure 3(b) and Table 1). As it has been mentioned, the SINR min in LTE communication standard is provided based on the chosen MCS by the AMC module in each OLPC sample time. In more realistic cases, considering the time-varying behavior of the transmission channel quality, the AMC module chooses different MCSs which leads to a time-varying SINR min . In this case, the PFC v is more reactive in the start-up compared to the other two ILPC schemes.

Conclusion

In this paper, we have developed the PFC based on the discrete-time Artstein transform for scalar discrete-time systems having a constraint on the state space variable in order to take into account an unknown bounded timevarying input delay. It is well suited for the ILPC of WCNs due to the constraint on the received SINR imposed by the communication standard and the delay caused by decision making and propagation time which is time-varying. Our simulations in LTE show that the PFC with taking into account an unknown bounded time-varying input delay (PFC v ) decreases the computation time and could ensure a better QoS by maintain-ing the communication connectivity compared to the TPC algorithm. Furthermore, it is significantly faster in maintaining the connectivity at start-up compared to the PFC c which takes into account the maximum round trip delay as a constant input delay. The average received SINR of the PFC v is higher than the other two methods. However we can not be sure that the PFC v consumes more energy because the re-transmission procedure also must be taken into account for measuring the global energy consumption.

Fig. 2 .

 2 Fig. 2. Average of received SINR for different values of k1 for 5s simulation.

Figure 3 (Fig. 3 .

 33 Figure3(a) illustrates the evolution of the received SINR at the BS by using the TPC, the PFC c and the PFC v as the transmission power control scheme.

Table 1

 1

	Performance comparison
		TPC PFCc PFCv
	Execution Time (s)	0.98	0.51	0.51
	Average Received SINR (dB)	16.7	24.7	24.9
	Time to pass the SINRmin	-	0.17	0.01
	and stay above it (s)


Yoann Couble
email: yoann.couble@enseeiht.fr

Emmanuel Chaput

Thibault Deleu
email: thibault.deleu@thalesaleniaspace.com

Cédric Baudoin
email: cedric.baudoin@thalesaleniaspace.com

Jean-Baptiste Dupe
email: jean-baptiste.dupe@cnes.fr

Caroline Bès
email: caroline.bes@cnes.fr

André-Luc Beylot
email: beylot@enseeiht.fr

High throughput satellites have proven to be an excellent solution to provide Internet services to white spots or to complement other existing infrastructures. With the use of multi-beam antennas, the same frequency may be reused dozens of times across a single satellite coverage area, increasing the system capacity and profitability. Legacy frequency reuse patterns such as uncoordinated 4-color scheme ensure interference isolation at the expense of important capacity reduction, using only one fourth of the operator's share in the scarce Ka band. Thus, to increase the per-beam available bandwidth, it is necessary to look for more aggressive and efficient Frequency Reuse schemes, generating higher Co-Channel Interference which has to be analyzed and handled. In this paper, we focus on the DVB-RCS2 return link of a multi-beam satellite and investigate the possibility of reaching the upper bound of a coordinated 2-color system capacity through the use of Interference-aware User Scheduling techniques. We first formalize the problem as an Integer Linear Program and study the impact of greedy simplifications on the optimality and processing times of our optimization models.

I. INTRODUCTION

With the development of High Throughput Satellite and Multiple Beam Antennas, the potential amount of clients accessing simultaneously the satellite has increased drastically. However, there are many challenges to overpass before reaching this new potential, especially at the MAC layer to avoid or to deal with new risks of collisions. In ETSI's DVB-RCS2 [1], Multiple Frequency Time Division Multiple Access (MF-TDMA) combined with Demand Assigned Multiple Access (DAMA) is used as the access scheme for reliable data transmissions. Using more numerous narrow beams means more aggregated bandwidth for the same coverage, but also a more dense usage of each uplink frequency. Reusing a subcarrier in two beams close to each other generates higher Co-Channel Interference (CCI) which is not entirely mitigated by the antenna directivity.

The first and straightforward solution, used for dimensioning current systems, is to use static frequency reuse patterns, dedicating a fraction of the available bandwidth to a subset of the beams, in a geometrically regular way. Frequency Reuse X (FRX) schemes (reusing a frequency band in one out of X beams, with commonly X ∈ 3, 4, 7) have proven to be easy to deploy but at the cost of a low per-beam bandwidth, since only 1/X-th of the bandwidth is available in each beam. Then, Full or Fractional frequency reuse schemes define a central area where sub-carriers are reused in every beam, i.e. FR1, plus an FRX scheme for the edge areas. These schemes have been extensively studied in the terrestrial networks, thoroughly presented in [2], and sometimes adapted to satellite systems.

Yet, static frequency reuse schemes are based on the estimation of the worst-case scenario, which limits their flexibility in several ways:

• Between colors, static bandwidth allocation means less flexibility in case of demand fluctuation. • Inside a color, the worst case interference estimation will cause a terminal to use a less efficient Modulation and Coding scheme (ModCod) when in reality, it would have been possible to use a higher order ModCod. In other words, there are plenty of situations where resources could be used more greedily. However, introducing flexibility through the use of Dynamic Inter Beam Interference Coordination (D-IBIC) requires a high degree of coordination between users and gateways, as well as solving the User Scheduling Problem which is well known to be NP-hard.

This paper intends to model and analyze the achievable improvements possible thanks to D-IBICs on the Return Link (RL). In section II, we will briefly review and comment existing coordinated interference management schemes. Then in section III we model and formulate the problem of globally allocating MF-TDMA bursts to users. The problem complexity and scalability are also discussed in section IV. In section V we present our numerical results and analyze the impact of several chosen approximations on the performance.

II. DYNAMIC INTERFERENCE COORDINATION

Using D-IBIC in the Forward Link (FL) (resp. RL) requires a Channel State Information (CSI) exchange mechanism to inform the gateway of the reception (resp. transmission) condition and interference levels experienced by users. Obtaining CSI is then at the heart of D-IBIC in the FL (resp. RL), requiring thorough studies to find the best tradeoff between CSI consistency and generated signaling traffic. Then, the next issue is to use this CSI to perform an optimal scheduling of the S2/S2X BBFRAMES (resp. MF-TDMA Frame). In the FL, [START_REF] Lei | Joint Power and Carrier Allocation for the Multibeam Satellite Downlink with Individual SINR Constraints[END_REF], [START_REF] Dimitrov | Radio resource management techniques for high throughput satellite communication systems[END_REF] and [START_REF] Cocco | Radio resource management strategies for dvb-s2 systems operated with flexible satellite payloads[END_REF] among others tackle this issue with different approaches, the former by successively allocating sub-carriers and then optimizing the power for each while the latter uses a heuristic benefiting from the beam-hopping flexibility. What differs between the FL and RL in solving this problem is the nature of interference. Indeed, as shown in Fig 1, interference power in the FL only depends on the user's position in each beam antenna pattern. Whereas in the RL, the interference experienced by a user is given by the interfering users position in the interfered user beam antenna pattern. In other words, the interference generated in the FL doesn't rely on which user is scheduled, when in the RL it is at the heart of the problem. This schedule/interference enslavement brings a great diversity which may be used to improve further the system throughput, along with a huge combinatorial complexity.

There is a lot of literature around this kind of issues in terrestrial networks [START_REF] Giese | ARTIST4g: Innovative scheduling and crosslayer design techniques for interference avoidance[END_REF], whereas there are only a few studies on the matter for the RL of a DVB-S2/RCS2 system. First, RCS/RCS2 frame optimization has been studied in, among others, [START_REF] Pietrabissa | Dynamic uplink frame optimization with adaptive coding and modulation in DVB-RCS2 satellite networks[END_REF] and [START_REF] Dupe | Ordonnancement et gestion des ressources pour un système de télécommunications haut débit : Optimisation de la bande passante satellite[END_REF], but without considering interference management. The authors in [2] propose a simple scheduling heuristic aiming to improve the minimum Carrier power to Interference power ratio (C/I). The authors focus on frame reorganization and assume that users have already been assigned a number of Bandwidth Time Unit (BTU) and that all the Super Frame BTUs will be allocated. This approach has numerous practical advantages such as being able to implement it without modifying the user-BTU allocation algorithm. On the other hand, they don't take ModCod into account, nor do they deal with the user-BTU allocation. Additionally, in many cases, it may be more interesting to leave a few BTU unused (especially when FR1 or FR2 schemes are used) so that other users experience a better C/I and may use more efficient ModCods. In [START_REF] Boussemart | User scheduling for large multi-beam satellite MIMO systems[END_REF] the Multi-user MIMO (Mu-MIMO) paradigm is used to tackle the uplink user scheduling problem. Beams are clustered by groups of 7, and Successive Interference Cancellation (SIC) is used to combine the seven simultaneous user signals. User scheduling is then performed through the use of multi- [START_REF] Dimitrov | Radio resource management techniques for high throughput satellite communication systems[END_REF] pushes further these principles and puts forward multiples algorithms to perform interference aware user scheduling based on an FR1 system. While these algorithms offer large gains (from +50% without SIC to +99% with SIC throughput vs. FR4-random scheduling), their bandwidth efficiency is not so high since they need +300% bandwidth for that. Moreover, [START_REF] Boussemart | User scheduling for large multi-beam satellite MIMO systems[END_REF] and [START_REF] Dimitrov | Radio resource management techniques for high throughput satellite communication systems[END_REF] do not allow more than one empty slot which, as we will show in this paper, may not benefit the system.

III. SYSTEM MODEL AND FORMALIZATION A. System Model

We consider a system composed of a Gateway, a transparent multi-beam satellite and RCS Terminals (RCST). The satellite has N beams directive antennas, defining N beams beams. The beams are organized in a hexagonal grid and split over the two polarizations. To simplify the notations and modelisation, we will limit our study to one of the polarization. The models and optimization that we propound thereafter can be applied to either polarization in the same way. Each beam k has a set of RCST {1, . . . , i k , . . . , N users (k)}, able to transmit using a set of ModCods defined in Table II. We consider a given DVB-RCS2 MF-TDMA frame, composed of N tti timeslots and N carriers equal bandwidth subcarriers. We call BTU (timeslots, sub-carrier) couples. In this paper we concentrate on CCI, and will treat adjacent channel interference (ACI) and cross-polarization interference (CPI) as constant noises. As shown in Fig. 2, DVB-RCS2 allows the use of different ModCod independently on every BTU, which is an important degree of freedom to cope with different transmission conditions.

B. Problem Formulation

First we introduce the binary variables x t,c i k ,m to indicate whether RCST i k is transmitting on BTU (t, c) with ModCod m or not.

1) User Link Budget: For each RCST i k , transmitting with ModCod m on a BTU (t, c), we use the following Signal to Noise plus Interference Ratio (SNIR) expression:

SN IR(i k , m) = P i k G k (i k ) N + k ′ =k j k ′ ,m ′ x t,c j k ′ ,m ′ P j k ′ G k (j k ′ ) (1)

TABLE I NOTATIONS

Notation Description 1 ≤ k ≤ N beams Beam identifier 1 ≤ m ≤ N modcods ModCod identifier 1 ≤ t ≤ N tti Time transmit interval (TTI) 1 ≤ c ≤ N carriers Sub-carrier identifier I(k) Set of users in beam k i k ∈ I(k) User identifier in beam k x t,c i k ,m or x c i k ,m
Binary variable In the coming sections, the link budget expression is reworked in order to formulate an Integer Linear Program, taking advantage of these SNIR margins to maximize the resource utilization and ultimately the system throughput.

P i k Transmit power for user i k G k ′ (i k ) Reception gain of beam k ′ for user i k N Sub-
2) SNIR constraint formulation: As used in [START_REF] Lopez-Perez | Optimization Method for the Joint Allocation of Modulation Schemes, Coding Rates, Resource Blocks and Power in Self-Organizing LTE Networks[END_REF], we rearrange the terms in equation ( 1) combined with the SNIR threshold inequality to obtain the following condition to the successful transmission for any user i k using ModCod m:

Pi k G k (i k ) Γ thresh m ≥ N + k ′ =k j k ′ ,m ′ x t,c j k ′ ,m ′ P j k ′ G k (j k ′ ) (2)
This constraint must be verified when BTU (t, c) of beam k is assigned to RCST i k for a transmission with ModCod m. However, when this BTU is not assigned to user i k , the inequality constraint (2) has no meaning whatsoever and must be ignored. Hence, we introduce a sufficiently large number B to ensure that the inequality constraint ( 2) is always true when x t,c i k ,m = 0. So, ∀k, i k , t, c, m:

x t,c i k ,m • Pi k G k (i k ) Γ thresh m ≥ (x t,c i k ,m -1) • B + N + k ′ =k j k ′ ,m ′ x t,c j k ′ ,m ′ P j k ′ G k (j k ′ ) (3d) When x t,c i k ,m = 1
, the constraint is equivalent to (2). And when x t,c i k ,m = 0, the (x t,c i k ,m -1) • B term guarantees a negative right term to satisfy the constraint.

3) Global Frame Optimization Problem: From constraint (3d) and the system instant throughput maximization objective, we formulate the following Integer Program:

Problem GFO: Maximize X F (X) (3a) ∀k, t, c i k ,m x t,c i k ,m ≤ 1 (3b) ∀k, i k , t c,m x t,c i k ,m ≤ 1 (3c) ∀k, i k , m, t, c x t,c i k ,m • Pi k G k (i k ) Γ thresh m ≥ (x t,c i k ,m -1) • B + N + k ′ =k j k ′ ,m ′ x t,c j k ′ ,m ′ P j k ′ G k (j k ′ ) (3d) ∀k, i k , m, t, c x t,c i k ,m ∈ {0, 1} (3e)
where X is the vector of all x t,c i k ,m and F (X) is an objective function to maximize the system throughput while guaranteeing fairness among users. The first constraint (3b) comes from the MF-TDMA frame structure: there cannot be more than one RCST transmitting on a BTU, whichever the ModCod. We consider here relatively simple terminals, with only one transmission chain, as expressed in constraint (3c)

IV. DECOMPOSITION AND DERIVED MODELS

A. Time-wise decomposition

The GFO problem may be non-linear, has a large amount of variables and its solution space has a lot of symmetries. In the following, we propose to decompose it in N tti identical subproblems, to be treated on a timeslot to timeslot basis rather than at the frame level, with a causality relationship from one instance to the other.

While it may also remove a degree of freedom in the allocation, we expect that the loss of optimality is negligible and the gain of simplicity is important. Also, treating the whole frame at once would either produce N tti times the same timeslotoptimized set of user or require a fairness constraint/utility function to limit the amount of timeslots received by a user. In the latter case, the fairness constraint/utility function would then interfere with a SuperFrame to SuperFrame long-term fairness mechanism implemented in the Gateway. Moreover, the timeslot-local throughput maximization may not even be unfair as it tends to fulfill the demands more quickly and hence save resource to serve less "profitable" users. Anyhow, in this paper we focus on maximizing the Fig. 3. Interference power generated by users located in beams of the same polarization as the reference beam k system capacity by "filling" sub-carriers as much as possible, and to put the fairness aside for further works. ]] \ {k} -inf dB † Beam letters refer to Fig. 3 In Fig. 3 we can see that most of the interference experienced by a beam comes from RCST located in a very limited set of beams, and that after a certain distance, the interference generated by RCSTs can reasonably be neglected or at least estimated and considered as constant noise. Beams a and b are the most impacting beams and should thus always be taken into account in the link budget. Second order beams (c, d ande, f, g, h, i, j) are less influential, but their cumulative interference power should not be neglected.

B. Neighbor orders

Let us call Ω s (k) ⊂ [[1, N beams ]] \ {k} the restricted set of beams which will be accounted in further alternate versions of the problem, where s is the restricted scheme of accounted neighbors. For these restricted models, we ignore users outside of the considered beams, meaning that errors in the allocation may happen. It is thus necessary to estimate the interference I est generated by the ignored beams, and to include it in the link budget constraint (3d). In this section, we will consider the interfering beam sets and corresponding (empirically determined) estimated interference defined in Table III.

As a consequence, we define a collection of TimeSlot Optimisation (TSO(s)) Integer Linear Programs, one for each restriction scheme s that we will solve and compare in the next section.

Problem TSO(s) :

Maximize X k,i k c,m x c i k ,m r m (4a) s.t.: ∀k, c i k ,m x c i k ,m ≤ 1 (4b) ∀k, i k c,m x c i k ,m ≤ 1 (4c) ∀k, i k , m, c x c i k ,m Pi k G k (i k ) Γ thresh m ≥ (x c i k ,m -1) • B + N +I est + k ′ ∈Ωs(k) j k ′ ,m ′ x c j k ′ ,m ′ P j k ′ G k (j k ′ ) (4d) ∀k, i k , m, c x c i k ,m ∈ {0, 1} (4e)
where r m is the instant throughput of MocCod m.

C. Sub-carrier-wise decomposition

Following the same approach as previously, we now consider the possibility of dividing the problem into N carriers sub-problems. Yet, contrarily to the time-wise decomposition, the N carriers sub-problems are not independent. To cope for the absence of constraint (4c), solving these sub-problems must be done sequentially, and users having received an allocation in the c-th resolution must be removed from the user pool that (c + 1)-th instance of seqC-TSO(s) takes as input.

Hence, we define a second collection of sequential sub-Carrier, TimeSlot Optimization (seqC-TSO(s)) Problems, defined by a sequence of Integer Linear sub-Programs, as shown in Eq. ( 5) and Algorithm 1:

Sub-problem sub-seqC-TSO(s) : Maximize X k,i k m x i k ,m r m (5a) s.t.
:

∀k i k ,m x i k ,m ≤ 1 (5b) ∀k, i k , m x i k ,m Pi k G k (i k ) Γ thresh m ≥ (x i k ,m -1) • B + N +I est + k ′ ∈Ωs(k) j k ′ ,m ′ x j k ′ ,m ′ P j k ′ G k (j k ′ ) (5c) ∀k, i k , m x i k ,m ∈ {0, 1} (5d)

V. NUMERICAL RESULTS

A. Test case

We consider a hexagonal grid with 2 × 15 beams equally distributed between the two polarizations. In each beam, II on any of the N carriers sub-carriers available for each beam in its beam polarization.

B. Test methodology

For a given scenario, TSO(s) and seqC-TSO(s) problems formulation are performed and numerically solved using Gurobi ILP optimizer [START_REF]Gurobi optimizer reference manual[END_REF] Gurobi ILP optimizer is configured to use a branch and bound algorithm. As a comparison reference, we also performed a 4 color scheme optimization with the same objective, where the interference is estimated in the worst case scenario. This reference is what is classically done in current systems. In each test, the same process is repeated 50 to 200 times (depending on the test case) with reinitialized users location to narrow the 95% confidence intervals. Tests data is described in Table IV. • ρ = 0.5 Cases where there are only a few active users compared to the amount of available sub-carriers. This may correspond to a system with few professional users with high bandwidth demand or to a low charge period. • ρ = 1 Cases where there are as many active user as subcarriers. • ρ = 2 Cases where there are a lot of active users compared to the available sub-carriers, which could correspond to a system with a huge of users, or a high charge period. The results of this test case are shown in Fig. 4 and Fig. 5. Performance is measured in terms of total system throughput relatively to the FR4 scheme with interference estimation. The first figure points out that neighbor restricted schemes (green, yellow and blue) perform fairly well compared to the global scheme (red), for both TSO (plain) and seqC-TSO models (hatched), with only a 2% system throughput difference between TSO(global) and seqC-TSO(ord 1) at ρ = 2. A second observation is that, as could be expected, the seqC-TSO problem family performs poorly for low values of ρ. Indeed when there are less active RCSTs than available subcarriers, seqC-TSO will not be able to spread RCST across the sub-carriers. Yet, as soon as there are more active RCST, the difference between TSO and seqC-TSO is negligible. Fig. 4 also points out that the more active users, the higher the system throughput: spatial diversity increases with the amount of active users and more interesting RCST sets can be found. This also means that on average, we can expect that users will be able to use more efficient ModCods on their BTU and in the long term, this will increase the mean throughput per user. On Fig. 5 on the other hand, we represented the sub-carriers utilization. The FR4 reference scheme (black dashed line) is obviously limited to 50% of the polarization sub-carriers, while other schemes manage to use more. This confirms that in an FR4 scheme resources are under-utilized and that it is possible to use them more greedily. Also note that sub-carriers utilization is higher for seqC-TSO schemes, meaning that more users are served simultaneously, but with lower order ModCods. 2) Test 2 -Scalability: Secondly, to emphasize the scalability of each formulation, we increased the problem size (i.e. sub-carrier count) progressively while maintaining the user to sub-carrier ratio (ρ = 2). Fig. 6 shows that for TSO(s) models it is quickly (N carriers ≈ 10) impossible to even evaluate the upper bound of the system throughput, while for seqC-TSO(s) models it is possible to reach much higher sizes of scenarios. In particular, seqC-TSO(ord ≤ 1.5) and seqC-TSO(ord 1) show promising resolution times, even though there is still a 1000-fold gap before being ready to use in practice.

3) Test 3 -Larger scenario: Additional simulations have been made for larger systems and solved for seqC-TSO(ord 1), which offers a +47% throughput increase compared to the FR4 scheme on systems with 90 beams, 30 active users per beam and 25 sub-carriers with a mean resolution time of 40s. Results are given in Fig. 7, showing consistent results, but with a slight loss of throughput partly due to the lack of precision in the interference estimation (especially for center beams). Last but not least, it is interesting to note that overall, using an FR2 scheme, even coordinated generates higher interference levels, meaning that the bandwidth efficiency is reduced: additional throughput comes at the expense of even larger bandwidth to carry on the feeder link.

VI. CONCLUSION AND FUTURE WORKS

In this paper, we modeled the user scheduling problem through the use of an integer program, with a ModCodaware interpretation of the SNIR constraint. After discussing the relevance of decomposing the problem time-wise and carrier-wise, we studied the importance and contribution of different beam neighbors and defined several integer linear programs. Finally, we presented our numerical results, showing promising performances regarding system throughput and sub-carriers utilization.

The proposed simplifications presented very consistent results, with a low loss of optimality compared to the computational gains. Moreover, combined with a time and carrier-wise decomposition, the problem becomes solvable for real-scale systems.

Even though we managed to reduce drastically the computational complexity for solving the user scheduling problem, The complexity of the proposed methods should still be reduced before considering a real-time (slot-by-slot) implementation. However, our observations and conclusions remain valuable to formulate new heuristics and scheduling algorithms, as well as to combine our methods with advanced signal processing techniques such as SIC. In further works, it would be crucial to study these models behaviors in the long term and to implement different fairness mechanisms to evaluate their performance.

InterferenceFig. 1 .

 1 Fig. 1. Interference in the Forward Link and Return Link

FFig. 2 .

 2 Fig. 2. DVB-RCS2 MF-TDMA Structure, with individual ModCod setting depending on the reception power

U

  init (1) = k {i k ∈ I(k)} ; // all users for c = 1 to N carriers do From U init (c), define sub-seqC-TSO(s); U sol (c) = solve sub-seqC-TSO(s) ; U init (c + 1) = U init (c) \ U sol (c) ; end Algorithm 1: seqC-TSO(s) definition N users active users are placed uniformly across the beam coverage. Each active user is assumed to transmit with the same transmission power P T x . Users are able to use any ModCod in Table

Fig. 4 .Fig. 5 .

 45 Fig. 4. Performance of the different models solution according to the user to sub-carrier ratio. Sub-carrier count is kept constant (N carriers = 8) so that physical capacity is kept constant, while the user diversity varies with user count (Nusers ∈ {4, 8, 16}). (Ntests = 200)

Fig. 6 .Fig. 7 .

 67 Fig. 6. Models computational complexity related to user and sub-carrier count. In this figure the user to sub-carrier ratio is kept constant to avoid effects highlited in Fig. 4. (Ntests = 50)

  where P i k is the transmission power for user i k of beam k, G k ′ (i k ) is the reception gain of beam k ′ for user i k . G k ′ (i k ) takes into account the antenna gain, its directivity but also every other signal attenuation phenomenon like antenna pointing error, atmospheric losses, path loss, etc. Here N incorporates the thermal noise, and the ACI and CPI noises.A transmission is only useful if it achieves the SNIR threshold (Γ thresh m , see TableII) of the ModCod m it is transmitting at, i.e. if SN IR(i k , m) ≥ Γ thresh m . Any additional SNIR margin will not increase the throughput, except if it overpasses the next ModCod threshold.

		carrier noise power (incl. ACI/CPI)
	Γ thresh m	Threshold SNIR for modcod m
	rm	Instant throughput for modcod m

  First, we compared the performance of the different model families in three situations (ρ = N users /N carriers ):

		TABLE IV
		TEST DATA
	Parameter	Value	Parameter	Value
	Beam aperture θ 3dB	0.27 deg	Pointing error	0.035 deg
	RCST max gain	50.5 dBi	RCST power	1 W
	Satellite max gain	20.9 dBi	Frequency band	Ka (29.5 GHz)
	Sub-carrier Bw.	21 MHz	BTU size	1616 symbols
	N beams	15 for each polarization
		Optimization parameters
	Large number B	10 5	Gurobi Tolerance	0.05
	Hardware	Intel Core i7-4790 (3.60 GHz), 16 GB RAM
	C. Test description and comments
	1) Test 1 -Performance gains: