Pierre-Olivier Bauguion
email: pierre-olivier.bauguion@irt-systemx.fr

Claudia D ' Ambrosio
email: dambrosio@lix.polytechnique.fr

Keywords: Transportation optimization, network flows, passengers behavior, supervision

Multimodal transportation plan adjustment with passengers behaviour constraints 1 Introduction

For the last decades, the growth of population in high density areas pressed the need to enhance mobility services. Nowadays, most of the biggest cities have settled public transportation infrastructures to fluidize traffic and ease local mobility. Nevertheless, these infrastructures are likely to meet their saturation limits during rush hours, making the management of such a system a crucial point. This led to consider Transportation Network Optimization Problems (TNOP).

One can divide TNOPs into two main parts, each part corresponding to a different aspect of the problem.

Network flows and user equilibrium

It is usually more convenient to model passengers as flows. The practical interest of flows lies in its ability to aggregate each individual while still capturing the movements on the network. Network flows have been introduced long ago (see, e.g., [START_REF] Ford | Maximal flow through a network[END_REF], enhanced later by [START_REF] Edmonds | Theoretical improvements in algorithmic efficiency for network flow problems[END_REF]). Since then, these models have been generalized for multiple "commodities" [START_REF] Shahrokhi | The maximum concurrent flow problem[END_REF] and grouped under the label of "multi-commodity" flow models. A lot of optimization problems aim at deciding the best routing solution of flows -by minimizing a cost function for instance (see e.g. [START_REF] Ahuja | Network flows: theory, algorithms, and applications[END_REF]), whereas in transportation models these flows are used to describe a user equilibrium, namely a Wardrop's equilibrium. [START_REF] Sheffi | Urban Transportation networks: Equilibrium Analysis with Mathematical Programming Methods[END_REF] sums up equilibrium models by describing its two competitive mechanisms: a "selfish" disutility minimization for each individual on one hand (the objective), and the increase of disutility generated by confronting the strategies of those individuals on the other hand (the constraints). Hence, he shows how to reformulate it with mathematical programming. This will become the base principle from which every other passengers behaviour model will derive.

Network design and Bi-level Programming

Whereas network flows problems consists in using the network, the network design problem consists in deciding how the network should be. The latter can be the case when it comes to choose whether we want to install a network component or not, considering a cost function (see e.g. [START_REF] Gourdin | Telecommunication and location[END_REF]). More generally the impact of such decisions (in terms of cost and/or feasibility) is not always easily computed, which is typically the case in TNOPs. It is typically the case in TNOPs, where we have to solve a network flows problem (as [START_REF] Sheffi | Urban Transportation networks: Equilibrium Analysis with Mathematical Programming Methods[END_REF]) to know what are the consequences in our transportation plan decisions. These two embedded problems (network flows problem and network design problem) can be seen as hierarchical in the sense that the decisions of the transportation plan will set the available itineraries of passengers, but the other way around is not true. From the game theory point of view, this can be seen as Stackelberg competition, as the transport operator decides the transport plan first (and hence is the so-called "leader"), and given that decision, the passengers will decide how to react (they represent the so-called "followers"). However, the leader have insights about the way the follower will react to developp a relevant strategy. To model it as an optimization problem, Bi-level Programming is often used. Bi-level Programming means that two level of decisions exist: one that becomes parameters for the second. [START_REF] Migdalas | Bilevel programming in traffic planning: Models, methods and challenge[END_REF] proposed a large survey of different transportation Bi-level models and algorithm techniques to solve them, completed later by [START_REF] Farahani | A review of urban transportation network design problems[END_REF]. [START_REF] Meng | Transportation network optimization problems with stochastic user equilibrium constraints[END_REF] then [START_REF] Liu | Global optimization method for network design problem with stochastic user equilibrium[END_REF] tackles the stochastic versions of user equilibrium, especially for road networks.

Multimodal transportation plan adjustment with passengers behaviour constraints

With the rise of digital technology and telecommunications, transportation operators can simultaneously be closer to the real time state of transportation infrastructure and have larger vision of the whole (even multimodal) transportation system. This allows transport operators to react at a disturbance dynamically and disseminate quickly a recourse strategy to face it off. This is what we call multimodal supervision. From this point of view, we propose to address a problem of transportation plan readjustment. Readjustment differs from classical optimization by four main points. Firstly, the problem has a variable spatial and temporal perimeter, as we do not necessarily know in advance how long and how far the issue will propagate along the network(s). In other words, the space and time needed to go back to a nominal state is unknown. Secondly, for the previous reason (propagation avoidance), the adjustment model must be as accurate as possible during the horizon it is deployed. It includes passengers strategies and problems caused by their overloads (delays, door blocking...). This can induce interdependency between different transportation systems. Thus, the multimodal network must be considered. Thirdly, the optimization horizon starts with a given infrastructure state. This includes the positions of vehicles and passengers, and whole infrastructure state. Finally, adjustment implies that the solution is a differential plan from the nominal one. In terms of optimizing process, it can induce a set of specific constraints related to the distance with the proposed plan (adjustment) and the nominal one.

Moreover, as the situation is extremely dynamical, to ensure the strategy to be deployed at the time it is provided, the adjustment must be made in a short time-window. This problem can be formalized as a particular TNOP and modelled using (Time-Expanded or Time-Dependent) graphs (see e.g. [START_REF] Liebchen | Some practical aspects of periodic timetabling[END_REF][START_REF] Skutella | An Introduction to Network Flows over Time[END_REF]). We discuss here a first mathematical formulation to tackle this problem in one single Mixed Integer Linear Program, and future methods to achieve these operational objectives.


Ahmad Fadel

Bernard Cousin

Ayman Khalil

User

User Selection in 5G Heterogeneous Networks Based on Millimeter-Wave and Beamforming

  L'archive ouverte pluridisciplinaire

Abstract-Heterogeneous Networks (HetNETs) are considered as an effective solution to improve the coverage and system throughput for future cellular networks. The extremely growing mobile market, together with the arising demand for high data rates, motivate us to open a new spectrum related to millimeter waves (mmwaves) while using beamforming that can serve simultaneously a group of users. In this paper, we formulate an optimization problem for HetNETs multi-user selection in a multi-input-multi-output and orthogonal frequency-division multiple access (MIMO-OFDMA) system, aiming to maximize the total system throughput. We solve the problem by applying a modified version of well-known metaheuristic algorithms. The optimal solution is obtained using an exhaustive search algorithm that provides an ideal solution which is complex to be computed. Greedy zero-forcing dirty-paper gZF-DP and zeroforcing selection ZFS algorithms were selected from literature for the sub-optimal solution. In parallel, a water-filling algorithm has been optimized after adding new power constraint and it has been used for power distribution. Hence, we analyze the throughput performance of our systems using throughput metric. The results show that ZFS outperforms gZF-DP algorithm as it achieves higher total throughput, While gZF-DP outperforms ZFS algorithm in the execution time.

Index Terms-User selection, 5G heterogeneous network, beamforming, milli-meter waves.

I. INTRODUCTION

5G the next generation of the wireless communication system will be able to handle a thousand times more traffic than today's networks, and it will be up to 10 times faster than 4G LTE. There are five new technologies emerging as a foundation of 5G. Millimeter waves (mmwaves), small cells, massive multi-input-multi-output MIMO, beamforming and full duplex [START_REF] Andrews | What will 5g be?[END_REF]. Mmwaves gives us the opportunity to use a new spectrum of band, that fall between 30 and 300 GHz [START_REF] Niu | A survey of millimeter wave (mmwave) communications for 5g: opportunities and challenges[END_REF]. This section of the spectrum has never been used before for mobile devices and opening it up means more bandwidth for everyone, especially after having all frequency spectrum bands under 6 GHz started to get more crowded. But there is a drawback millimeter wave communications suffer from huge propagation loss compared with other communication systems in using lower carrier frequencies. Accordingly, this fact limits their range to approximately 200 m. Therefore to figure out this problem the concept of heterogeneous networks HetNETs has been introduced while densifying the macrocell by small cells. Doubtless, HetNETS is a promising solution for the capacity enhancement in the 5G cellular networks [START_REF] Niu | A survey of millimeter wave (mmwave) communications for 5g: opportunities and challenges[END_REF]. Under those circumstances, small cells are appropriate for the utilization of mmwaves.

Proportionally, high frequency means small antenna size. This can be seen as a good achievement to have massive MIMO technology, where hundred of antenna elements can be integrated easily. Thus, increasing the throughput of today's networks by the factor N t xN r [START_REF] De Lamare | Massive mimo systems: signal processing challenges and future trends[END_REF], where N t and N r are the number of transmitted and received antenna respectively. Broadcasting information in every direction at once could cause serious interference problem which brings us to the use of beamforming techniques. Beamforming is like a traffic signaling system for cellular signals, so it allows to send a focused stream to a specific direction hence this precision prevent interference [START_REF] Kutty | Beamforming for millimeter wave communications: An inclusive survey[END_REF].

Combining all 5G technologies in one system is required and must be studied. Indeed, orthogonal frequency division multiple access (OFDMA) will be considered as a promising broadband cellular platform to support high-speed wireless communication [START_REF] Andrews | What will 5g be?[END_REF], and it's the unquestionable front-runner for 5G. Moreover, we have a recent interest in transmit beamforming strategies for the cellular downlink that aim to attain the total throughput of the wireless channel.

k-tier heterogeneous downlink Cellular Networks proposed in the literature [START_REF] Park | Feedback design for multiantenna k-tier heterogeneous downlink cellular networks[END_REF] focused on cooperation between different tiers. The authors in [START_REF] Zhu | Joint beamforming, resource allocation, and scheduling for multi-cell multi-user mimo-ofdma systems[END_REF] have formulated an optimization problem for joint beamforming, resource allocation and scheduling for multi-cell multi-user selection, that achieve a total average weighted system throughput maximization. Investigate the design and the associated tradeoffs between energy efficiency and system throughput maximization in a Twotier heterogeneous network. This is obtained by introducing active/sleep (on/off) modes in macrocell base station [START_REF] Quek | Energy efficiency analysis of two-tier heterogeneous networks[END_REF], [START_REF] Soh | Energy efficient heterogeneous cellular networks[END_REF]. A reduced-complexity suboptimal solution to total throughput maximization is suggested in [START_REF] Caire | On the achievable throughput of a multiantenna gaussian broadcast channel[END_REF]. It proposes the use of QR decomposition of the channel combined with dirty-paper DP coding at the transmitter. In that case, this combination of both approaches, nulls interference between data streams, and hence, it is called zero-forcing dirty-paper (ZF-DP) precoding [START_REF] Caire | On the achievable throughput of a multiantenna gaussian broadcast channel[END_REF]. Zero forcing algorithms ZFS was applied in a single cell multi-user selection while using statistical channel model where path loss is normalized and the study is based on the 3rd technology of wireless communication networks [START_REF] Dimic | On downlink beamforming with greedy user selection: performance analysis and a simple new algorithm[END_REF].

Our goal is to investigate low-complexity downlink beamforming solution for the proposed optimization problem to meet the overwhelming demands of network throughput for a practically important case wherein the number of users N u is larger than the number of transmit antennas N t , which entails user selection. Our aim is four-fold: i) Formulate an optimization problem for joint beamforming, power allocation, and user selection, in a heterogeneous cellular network based on two different technologies (4G) and (5G), for the purpose of maximizing the system throughput; ii) compute the optimal solution by applying the ES algorithm that will be considered as the maximum threshold for all other solutions; iii) propose an alternative modified version for both gZF-DP and ZFS, to be adaptable for the suggested channel model; iv) assess the performance of both gZF-DP and ZFS compared to that of ES, while total throughput is considered as the evaluation metric.

The residual of the paper is organized as follows: Section II outlines the system and channel models. In Section III we formulate the joint beamforming, resource allocation, and scheduling optimization problem [START_REF] Zhu | Joint beamforming, resource allocation, and scheduling for multi-cell multi-user mimo-ofdma systems[END_REF]. Section IV describes the optimal and reduced-complexity algorithms. Furthermore, Section V provides a simulation-based comparison of the system throughput performance for ES, gZF-DP and ZFS algorithms. Conclusions and future work are mentioned in Section VI.

II. HETNET MULTI USER MIMO-OFDMA NETWORK MODEL

A. HetNet MULTI USER MIMO-OFDMA System Model

A heterogeneous multi-user MIMO-OFDMA network is considered, consisting of M macrocells densified by P picocells with a K user equipment (UEs) distributed overall the network as depicted in Figure 1. Each base station (BS) is equipped with N t transmit antennas, and each UE with N r receive antennas.

Spatial multiplexing is the most attracted characteristic in MIMO systems because each BS can serve up to Nt Nr UEs simultaneously for each resource unit. Frequencies are attributed as follows: we associate for macrocell the 4G frequency spectrum, and for picocells mmwaves that fall in the spectrum range of 5G. Thanks to fiber optic backhaul links, macro and pico BSs are connected to a centralized control unit. We assume that the wireless channel operates in time-divisionmultiplexing (TDD) mode to provide reciprocity, by which the uplink channel is used as an estimate of the downlink channel, and this occurs when receiving a pilot training sequence from terminal devices toward the base station where channel state information (CSI) estimation takes place.

B. Channel Model

An OFDMA system is considered with T resource units with h nk (t) ∈ C Nt×Nr to be the channel vector between the (m, p)th BS and the kth UE of the (m , p )th cell in the tth

y nk(t) = S nk (t)h nk (t)w nk (t)x nk (t) + I nk (t) + n u , ∀ n = (m, p) with m ∈ [1 . . . M ] and p ∈ [0 . . . P ] (1)
where S nk (t) ∈ {0, 1} denotes the resource unit allocation indicator, w nk (t) and x nk (t) are the transmit power-scaled beamforming and the transmit symbol for the link between the (m, p)th BS and the kth UE in the tth resource unit, respectively. Furthermore, h nk (t) = l nk (t)g nk (t) hnk (t), where l nk (t), g nk (t), represent respectively the path loss, shadowing, where they are generated using rapapport model [START_REF] Akdeniz | Millimeter wave channel modeling and cellular capacity evaluation[END_REF], and what is worth to be mentioned is that this model is realistic. Then, hnk (t) small scale fading coefficients are generated using [START_REF] Alkhateeb | Limited feedback hybrid precoding for multi-user millimeter wave systems[END_REF] that provides a statistical channel model. In addition, n u ∼ N (0, σ 2 I nk (t)) is the additive white Gaussian noise (AWGN) corrupting the received signal with zero mean and variance σ 2 and I nk (t) is the interference.

III. PROBLEM FORMULATION

In [START_REF] Zhu | Joint beamforming, resource allocation, and scheduling for multi-cell multi-user mimo-ofdma systems[END_REF], the authors have formulated an optimization problem for a multi-cell network, assuming universal frequency reuse. We reformulate this optimization problem to be compatible with the proposed system model, where we are dealing with HetNets using mmwaves only for picocells.

A. System Performance Evaluation Metric

The system throughput will be used as a metric unit to evaluate the proposed system. The instantaneous channel throughput for the kth user of the nth cell in the tth resource unit is given by

R nk (t) = B w log 2 (1 + ζ nk (t)), ∀n = (m, p) with m ∈ [1 . . . M ] and p ∈ [0 . . . P ] (2)
where B w is the channel bandwidth and ζ denotes the received signal-to-interference-plus-noise-ratio (SINR) given by

ζ nk (t) = |h nk (t)w nk (t)| 2 I nk (t) + σ 2 , ∀n = (m, p) (3)
where I nk (t) represents the received intracell multi-user interference. While picocells are placed far enough from each other to avoid intercell interference.

The performance evaluation metric for the proposed problem formulation is the average system throughput, which is defined by U as the total average number of bit/s/Hz/BS successfully received at the kth UEs, worth mentioning that we are dealing with HetNets on the contrary to [START_REF] Zhu | Joint beamforming, resource allocation, and scheduling for multi-cell multi-user mimo-ofdma systems[END_REF]. Thus respecting the different characteristics of each type of cell (macro, pico) is unquestionable, like the size and different technologies adopted in each cell. The corresponding expression is given by

U(B, P, S) = 1 M * (P + 1) M m=1 P p=0 k∈Un T t=1 R nk (t) ∀n = (m, p) with m ∈ [1 . . . M ] and p ∈ [0 . . . P ] (4)
where B, P and S, refer to the beamforming, power allocation and scheduling policies respectively. U n , ∀n = (m, p), denotes the set of UEs served by each macro and pico BSs. P and M are the total number of picocells and macrocells in the network respectively.

B. Formulation of the Optmization Problem

In this subsection, a new optimization problem (P) is formulated as follows:

(P)                    (B, P, S) * = arg max U(B, P, S) s.t, C1 : S nk (t) ∈ {0, 1}, ∀n = (m, p), k, t C2 : k∈Un S nk (t) ≤ N t N r , ∀n = (m, p), t C3 : k∈Un T t=1 S nk (t) w nk(t) 2 ≤ P n , ∀n = (m, p) C4 : n T t=1 S nk (t) ≤ 1, ∀k, n = (m, p) with m ∈ [1 . . . M ] and p ∈ [0 . . . P ]
where (B, P, S) * refers to the optimal beamforming power allocation, and scheduling policies respectively. C1 indicates whether the resource unit is allocated to the UE or not, in each macro or picocells. C2 guarantees that a resource unit is only assigned to all most N t N r UEs in one resource unit, again in each macro or picocells. C3 guarantees that the individual transmit power constraint for each BS [START_REF] Zhu | Joint beamforming, resource allocation, and scheduling for multi-cell multi-user mimo-ofdma systems[END_REF] will not exceed a certain maximum P n . C4 assures that users could take maximum one and only one resource unit during each selection.

Solving this optimization problem analytically presents a high complexity, thus we propose to use metaheuristic algorithms to find a sub-optimal solution that can give us an overview of the first result in the proposed system. For the rest of this paper, we will consider one resource unit for each BS to be shared between users, hence this means that t = 1 for each BS. The reason from this assumption is to simplify the simulation part.

IV. OPTIMAL AND LOW-COMPLEXITY ALGORITHMS

In this section, we will apply three algorithms ES, gZF-DP and gZF that take into consideration the same constraints and objective of the optimization problem (P). Accordingly, the distributed power between users should be less or equal the maximum power in each BS. The number of users that share the same resource unit should not exceed the number of antenna elements installed on a BS.

A. Exhausitve Search Algorithm

Maximizing the system throughput is the most important target of telecommunication operators in the next generation of cellular networks. Thus, high throughput enables to reply quickly to users, thus it achieves their satisfaction.

The objective of the optimization problem (P), is selecting r ≤ N t users and respecting all the aforementioned constraint in a way to maximize the system throughput, while using beamforming and best power allocation. To achieve the goal, we suggest getting the optimal solution first, through applying the exhaustive search algorithm. This algorithm checks all the possible combinations in a way to get the optimal solution by selecting a set of users that achieve the maximum total throughput.

In Algorithm 1, we propose to select r ≤ N t from N u users, by trying all the possible combinations to achieve the maximum total throughput. Power is uniformly distributed among the selected users. if R(S r ) < R(c) then

7:

Let S r = c 8:

end if 9: end while Consider h nk ∀n = (m, p) where m ∈ [1 . . . M ] and p ∈ [0 . . . P ] the channel vector between the N t transmit antenna elements and the received antenna of k users in each macrocell and picocells for all algorithms.

We denote h k := [ h k,1 h k,2 . . . h k,Nt ] a row vector. Thus, the channel matrix H is

H = [h * 1 h * 2 . . . h * k ] * (5)
Where (.) * is the conjugate-transpose, and the received signal at user k, and W represents a specific type of beamforming which is ZF. Let w k := [ w k,1 w k,2 . . . w k,Nt ] T ((.) T denotes transpose) be the beamforming weight vector for user k. The beamforming weight matrix W is

W = [w 1 w 2 . . . w k ] (6)
The complexity of getting this optimal solution motivate us to find a less complex algorithm. The greedy algorithms get a sub-optimal solution that makes a trade-off for the computing time. This is what will be presented in the upcoming proposed algorithms.

B. Greedy Zero-Forcing Dirty-Paper Algorithm

The authors in [START_REF] Caire | On the achievable throughput of a multiantenna gaussian broadcast channel[END_REF] have proposed a suboptimal solution to maximize (3), based on the QR-type decomposition [START_REF] Horn | Matrix analysis[END_REF] of the channel H = LQ matrix obtained by applying Gram-Schmidt orthogonalization to the rows of H, while computing P ⊥ r the projector matrix. L is a lower triangular matrix, and has orthonormal rows. Setting W = Q * . They have proposed a multiplexing technique based on coding for known interference, known as "Writing on Dirty Paper," Costa precoding [START_REF] Costa | Writing on dirty paper (corresp.)[END_REF], or dirty paper (DP) coding. The combination of both approaches QR-decomposition and DP coding at the transmitter, nulls interference between data stream. Authors of [START_REF] Caire | On the achievable throughput of a multiantenna gaussian broadcast channel[END_REF] have applied the quite same greedy solution, on a single cell for 3G technology, as they considered a statistical channel model with normalized pathloss which is not the case in our proposed system model since we are using the mmwave channel model of Rappaport [START_REF] Akdeniz | Millimeter wave channel modeling and cellular capacity evaluation[END_REF] for picocells and Friis model [START_REF] Lassabe | A friisbased calibrated model for wifi terminals positioning[END_REF] for macrocell.

Power is allocated using the water-filling algorithm, whereas it allocates the power among users taking into account the channel radio condition, where it distributes high power for users that have good radio condition and low power if not, see Appendix A. Indeed, using this algorithm is more effective than distributing the power equally between users, since attributing a high power for bad radio condition is useless. We have formulated a water-filling optimization problem, adding a new constraint that guarantees a minimum power α i for all selected users. λ is the solution of the water-filling equation and g i is the radio condition.

K i=1 [λ - 1 g i ] = P max (7)
p i = [λ - 1 g i ] (8)
if p i < α i then p i = α i (9)
The algorithm 2 is called gZF-DP because it may produce a local maximum solution and not the optimal one. Let U n = {1, 2, . . . , K} be the set of indices of all UEs, and Sr = {s 1 , . . . , s r } ⊂ U r .

Algorithm 2 Greedy zero-forcing dirty paper [START_REF] Dimic | On downlink beamforming with greedy user selection: performance analysis and a simple new algorithm[END_REF] Input: N u , N t , P max Output: The set S r 1: Initialize r = 1 2: Select user s 1 such that s 1 = arg max u∈Ur HH * 3: S 1 = {s 1 } 4: while r < N u do 5:

r = r + 1 6: P ⊥ r = I N -H(S * r-1 )(H(S r-1 )H(S r-1 ) * ) -1 H(S r-1 )
7:

H(S r-1 ) = [h * 1 h * 2 . . . h * sr-1 ] * 8: Compute Z r,u = H u P ⊥ n H * u 9:
Find s r = arg max u∈U \Sr-1 Z r,u 10:

Set S r = S r-1 {s r } 11: end while 12: Let W = Q * (beamforming), where H(S r ) = LQ, by applying QR-type decomposition 13: Allocate power: P (S r ) using water-filling algorithm (8)

C. Greedy Zero-Forcing Algorithm

ZF beamforming is a method of spatial signal processing by which the multiple antenna transmitters can null multiuser interference signals in wireless communications. This occurs by inverting the channel matrix at the transmitter so that orthogonal channels between transmitter and receivers are created. This problem is solved by ES, which maximizes R(S r ) the total throughput of the set S r . However such algorithm has prohibitive complexity. Again power is distributed among users using water-filling [START_REF] Soh | Energy efficient heterogeneous cellular networks[END_REF]. We have modified the algorithm in [START_REF] Dimic | On downlink beamforming with greedy user selection: performance analysis and a simple new algorithm[END_REF], at the beginning SNR is calculated for the first selected user since it does not experience any kind of interference. Next users are selected after calculating their SINR, in a way to achieve a maximum throughput. Algorithm 3 does not use dirty paper DP coding, as orthogonality is not the only criterion for selecting users, hence it selects the user that provides higher throughput.

V. EVALUATION AND PERFORMANCE ANALYSIS

In this section, we evaluate the performance of gZF-DP and that of ZFS. Then we compare both algorithms with the optimal one, the exhaustive search, via Monte-Carlos simulations. Hence we average the total throughput over 1,000 random channel realizations. All BSs are assumed to transmit with identical power when using the exhaustive algorithm. Water-filling is adopted for power allocation in both gZF-DP if i = S r then 12:

Find s r = arg max u∈Ur\Sr-1 R(S r-1 {u})

13:

Set S r = S r-1 {s r } R(S r ) max [START_REF] Niu | A survey of millimeter wave (mmwave) communications for 5g: opportunities and challenges[END_REF] 16:

if R(S r ) max ≤ R(S r-1 ) max then 17:

Set r = r -1 and that of ZFS. The simulation parameters are given in Table II.

A. System Throughput versus number of Users in Macro Cells

Figure 2 depicts the total system throughput for both ZFS and gZF-DP to be compared with the optimal solution ES for the macrocells. As shown the three curves are dramatically increasing to reach a maximum threshold. We can detect that ZFS outperforms the gZF-DP, as it is close to the optimal solution. In Table I

B. Average System Throughput versus number of Users in Pico cells

In Figure 3 we show the average system throughput for both ZFS and gZF-DP to be compared with the optimal solution ES for the picocells. As shown the three curves are dramatically increasing to reach a saturated maximum threshold. We can clearly observe that ZFS outperforms the gZF-DP, as it is close to the optimal solution. In Table I we have shown the relative error ZFS with ES. Referring to the calculated values, 0.596 is an acceptable error compared to ES, especially if we take into account that we would avoid the complexity of the exhaustive search.

C. System Execution Time

Execution time is an important factor to be taken into consideration. Regarding Figure 4, execution time gZF-DP does not exceed 1 ms when N u = 40 users, while ZFS need 40.8421 ms for the same number of users. But regarding the ES algorithm, its curve increases exponentially and tends toward infinity as the number of users increases. To interpret this result, we can see that gZF-DP outperforms ZFS and ES using the metric of time, but ZFS gives an acceptable throughput compared to optimal thus it could be considered as a trade-off algorithm between gZF-DP and ES.

relative error = ES -X ES ∀X ∈ {ZF S, gZF -DP } (10)

VI. CONCLUSION

In this paper we have formulate an optimization problem aiming to total throughput maximization for downlink OFDMA-MIMO system, using beamforming techniques and 5G HetNETs. We have presented our system model where we used a realistic channel model for mmwaves 5G. We have selected two different metaheuristic algorithms from literature and we have modified them to be more efficient taking into account beamforming and mmwaves, as we have applied the exhaustive search algorithm to get the optimal solution. Numerical results revealed that ZFS algorithm outperforms gZF-DP as it provides higher system throughput and close to the optimal solution. But gZF-DP outperforms ZFS where it needs less execution time.

VII. APPENDIX A A. Water filling

Formulation of water filling algorithm that guarantees for users that suffer from a bad radio condition a minimum power α i to send their data streams. This could be considered as ensuring fairness between the users while replying on their requests. The optimization problem is formulated as follows:

(P)

        
max n i log 2 (1 + g i p i ) s.t, n i=1 p i = P max where α i = 1/g i (2 Cmin -1) p i ≥ α i where g i and p i are the radio condition and power respectively for user i. Whereas C min is a minimum capacity that user must achieve in the worst case.

∂ϕ ∂P = g i ln2(1 + p i g i ) -λ = 0 (11)
Knowing that λ refers to the water level, similarly, it is the solution of the water-filling.

λln2 = g i 1 + g i p i ⇔ g i λln2 -1 = p i g i (12)
Thus,

1 λln2 - 1 g i = p i (13) then,
p i = P max + n i=1 1 gi n - 1 g i (17)
Particular condition:

if p i < α i then p i = α i ( 18
)
This algorithm is necessary especially while dealing with mmwaves technologies. Because channel model of mmwaves experience high pathloss which affect the radio condition where the user with bad channel does not get power where p i = 0, thus to avoid this case and respecting the constraint of the optimization problem, in worst case p i = α i .

Figure 1 .

 1 Figure 1. Heterogeneous network with MIMO base stations.

Algorithm 1 2 : 3 :

 123 Exhaustive searchInput: N u , N t , P max Output: The set S r 1: while c ∈ C Nt Nu do Compute ∀x ∈ c, p x = Pmax Nu ∀x ∈ c, W (x) = H * (HH * ) -1

Algorithm 3 1 : 2 : 4 : 5 : 6 : 7 : 9 :r = r + 1 10 :

 3124567910 Greedy zero-forcing Input: N u , N t , P max Output: The set S r Initialize r = 1, i = 0 Select user s 1 such that s 1 = arg max u∈Ur HH * 3: S r = {s 1 } Compute W = H * (HH * ) -1 ZF-beamforming Allocate power using water-filling algorithm (8) Compute SN R(S r ) Compute R(S 1 ) max 8: while r < N u do for i ∈ N u do 11:

Figure 2 .

 2 Figure 2. Total throughput for macrocells versus Nu users.

Figure 4 .

 4 Figure 4. Execution time versus Nu users

  we have shown the relative error ZFS, gZF-DP and ES respectively. And after analyzing the result and referring to the calculated values, 0.118 is an acceptable Figure 3. Average total throughput for picocells versus Nu users.

		2.2	10	9
				ES-P
	Average system total throughput (bit/s)	0.6 0.8 1 1.2 1.4 1.6 1.8 2		gZFDP-P ZFS-P
		0.4
		0.2
		10		15	20	25	30	35	40	45	50
							Nu