Pierre Popineau
email: pierre.popineau@inria.fr

Sanket S Kalamkar

Franc ¸ois Baccelli
email: francois.baccelli@inria.fr.

On Velocity-based Association Policies for Multi-tier 5G Wireless Networks

Mobility is a key challenge for beam management in 5G cellular networks due to the overhead incurred at beam switching and base station (BS) handover events. This paper focuses on a network that has a multi-tier structure with two types of BSs operating in the same frequency bands, namely macro BSs that are sparser but with higher transmit power, and micro BSs that are denser and with lower transmit power. We propose a downlink user association policy which is a function of the user mobility. Typically, high mobility users should associate with macro BSs so as to incur less beam switching overhead, whereas low mobility ones should be associated with micro BSs. The main contribution of the paper is a formalization of the optimal threshold association policy, when the optimality is understood with respect to the effective Shannon rate. The analysis is based on stochastic geometry and on an exact representation of the effective Shannon rate of the typical user in this beamforming multi-tier context. Two models are discussed. The simplest one focuses on a single-user optimization problem. We also discuss a more realistic model with bandwidth sharing between all users in the cell. Finally, we identify the mobility and user-density patterns where the velocity-based threshold association policy outperforms the classical best mean power association policy.

I. INTRODUCTION

A key feature of 5G is beam-based communications, which allows one to concentrate the transmit energy in a specific direction and improve the signal power and simultaneously reduce the interference [START_REF] Holma | 5G Technology: 3GPP New Radio[END_REF]. Although more crucial for millimeter wave frequencies, a beam-centric design also helps improve network performance at sub-6 GHz frequencies. However, the key components and procedures of beam management, namely beam selection, beam failure detection, and beam switching, involve different signaling overheads which may degrade the network performance, especially for highly mobile users [START_REF] Giordani | A tutorial on beam management for 3GPP NR at mmWave frequencies[END_REF]. For instance, the signal-to-interference-plus-noise ratio (SINR) could be improved by using narrower beams, but this benefit brings a significant cost in terms of more frequent beam switching and beam misalignment.

The setting of the present paper is that of a network with a multi-tier architecture, namely featuring different types of base stations (BSs) using the same frequency bands, where differences are only in terms of spatial density and transmit powers. The archetype of such an architecture is that with micro cells overlaying macro cells. It is well known that this can quite significantly increase the network performance [START_REF] Dhillon | Modeling and analysis of K-tier downlink heterogeneous cellular networks[END_REF]. When such a multi-tier network is accessed by users with different mobility patterns, e.g., pedestrians, vehicles, and high-speed trains, an important question is that of the association policy to be used. The answer to this question is fundamentally impacted by the beam management used in each tier. For instance, although connecting to micro cells may offer a better coverage and throughput to a mobile user due to some dense deployment of BSs, if this user is moving at a sufficiently high speed, frequent BS handovers and beam switching events combined with a higher chance of beam misalignment may significantly degrade the effective rate obtained by the user.

The present paper addresses this question and describes a mobility-aware association policy within this 5G multi-tier setting. It identifies the optimal association policy within a class of threshold policies which are particularly simple to implement. Optimality is in a sense to be defined precisely in the paper; it is based on a system-level view of different tradeoffs associated with beam management, mobility, load on a cell, spectral efficiency, and the multi-tier parameters. This system-level view allows one to define the notion of the typical user. The optimization coincides with the maximization of the expected effective rate that this typical user gets.

A. Related work

Stochastic geometry is widely used to study several types of wireless networks [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF], [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF], including multi-tier cellular networks [START_REF] Dhillon | Modeling and analysis of K-tier downlink heterogeneous cellular networks[END_REF]. There are several works that study the effect of user mobility on the optimal tier selection in cellular networks [START_REF] Bao | Stochastic geometric analysis of user mobility in heterogeneous wireless networks[END_REF]- [START_REF] Chattopadhyay | Two-tier cellular networks for throughput maximization of static and mobile users[END_REF]. These works have provided important insights into how user mobility affects the performance metrics such as coverage probability, Shannon rate, and BS handover rate. While the works in [START_REF] Bao | Stochastic geometric analysis of user mobility in heterogeneous wireless networks[END_REF], [START_REF] Arshad | Velocity-aware handover management in two-tier cellular networks[END_REF] focus on a single mobile user scenario, [START_REF] Chattopadhyay | Two-tier cellular networks for throughput maximization of static and mobile users[END_REF] considers the case of multiple mobile users who share the network resources. However, to the best of our knowledge, there is no work available on the beam management in a multitier network consisting of mobile users with different speed profiles. There exists one work [START_REF] Kalamkar | Beam management in 5G: A stochastic geometry analysis[END_REF] that has developed a mathematical optimization framework for the beam management in a 5G network and focused on the tradeoffs associated with mobility and beam management. But this last work focuses only on a single-user single-tier cellular network. Overall, the question of optimal user association policy in the presence of intra-cell mobility leading to beam switching, BS handovers, and resource sharing among multiple users is still unanswered.

II. SYSTEM MODEL

A. Network setup

Consider a two-tier downlink cellular network, where we call the first tier the macro tier and the second one the micro tier. We assume that BSs are located according to two independent homogeneous Poisson point processes (PPPs): the macro BSs form a PPP Φ M ⊂ R 2 of intensity λ M and the micro BSs a PPP Φ µ of intensity λ µ , with λ µ > λ M . Mobile users (MUs) are located according to a stationary PPP Φ u of intensity λ u . Each MU moves on a straight line with an orientation chosen at random and independent of other MUs. Also, each MU travels at a constant velocity v, sampled from a given distribution f , independently for each user. Without loss of generality, using the homogeneity and isotropy of the PPP, we can assume that the typical MU is located at the origin and that its motion is along the x-axis.

We assume that all BSs are always active. Some of our results can be extended to the case where BSs with no users are shut off, but this does not fall in the scope of this work. The formulas of this section are presented for the macro tier, but they hold for the micro tier by swapping the roles of the two tiers.

B. Beamforming

We assume that macro BSs (resp. micro BSs) use directional beamforming to communicate with the typical MU. Each macro BS (resp. micro BS) has n M (resp. n µ ) beams. Each beam has the same angular width ψ M = 2π n M in the macro tier and ψ µ = 2π nµ in the micro tier. The main lobe is restricted to the beamwidth. The antenna gain in the macro tier is assumed to be:

G M (θ) = G M m if |θ| ≤ ψ M /2 G M s else,
where G M m is the main lobe gain and G M s is the side lobe gain. The probability that an MU lies within the main lobe of an interfering macro BS is p M,m = ψ M 2π = 1 n M , and that of an interfering micro BS is p µ,m = 1 nµ . Thus the BS antenna gain g M,x , w.r.t. the typical MU, of an interfering BS located at x in the macro tier is given by

g M,x = G M m w.p. p M,m G M s w.p. 1 -p M,m . (1)

C. Mobility-based beam misalignment

Due to intra-cell mobility, the MU has to reselect the beam when it moves from the coverage of one beam to another. Such a beam reselection in a 5G network occurs during a synchronization signal block (SSB) burst with period τ . If the MU moves out of the main lobe of its original connection beam (also called the reference beam) between two consecutive SSB bursts without selecting a new beam, a beam misalignment occurs, namely, the MU receives from the serving BS via a side lobe. We assume, as in [START_REF] Kalamkar | Beam management in 5G: A stochastic geometry analysis[END_REF], that the probability p M bm that there is a beam misalignment event for an MU moving at velocity v is equal to:

p M bm (v) = 1 -exp (-vν M,b τ ) , (2)
where ν M,b is the time intensity of beam reselections. Hence, taking this beam misalignment into account, the antenna gain g M,0 at the serving BS of the typical MU is given by

g M,0 = G M m w.p. 1 -p M bm (v) G M s w.p. p M bm (v). (3)

D. Access policy and SINR

The micro and macro tiers share the same spectrum, so they interfere with one another. We call the tier with which the typical MU is associated the association tier, and the other tier, the interferer tier. An MU in the network always connects to the closest BS in its association tier. This BS association results in cells of BSs forming a Poisson-Voronoi (PV) tessellation [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF].

Let (x)

Kx -α , with α > 2 and K = c 4πfc 2 , be the path-loss function for the system, where f c is the carrier frequency and c is the speed of light. We assume Rayleigh fading with mean 1 between the BS located at x and the typical MU at the origin, denoted by h x . Let P M and P µ denote the transmit powers for each tier, with P µ < P M , and G M

x and G µ y the antenna gains for macro and micro BSs located at x and y, respectively. We define the interference experienced by the typical MU from macro tier BSs as:

I M (Φ M ) = x∈Φ M h x g M,x P M ( x ).
Let X M,0 be the closest macro BS to the typical MU, and X µ,0 be its micro counterpart. Conditioned on that the closest macro BS to which the typical MU connects is at distance r > 0, the SINR at the typical MU, denoted as SINR M is expressed as

SINR M = h 0 P M g M,0 Kr -α σ 2 + I M (Φ M \{X M,0 }) + I µ (Φ µ ) , (4)
where g M,0 (given by ( 3)) is the gain of the serving BS of the typical MU and σ 2 is the thermal noise density.

E. Parameters

We consider that both tiers use a carrier frequency f c in the 3.5 GHz range with a bandwidth W = 100 MHz. The transmit powers of macro and micro BSs are assumed to be 20 W and 4 W, respectively. The antenna gain profile we consider for numerical results is that the main lobe gain is G M m = n M and the side lobe gain is G M s = 1/n M , but other gain profiles can be used. The thermal noise σ 2 is -174 dBm.Hz -1 . Finally, MU velocities are sampled from an exponential distribution with scale parameter v u . Table I gives the parameters used in the numerical results.

III. COVERAGE PROBABILITY AND SHANNON RATE

A. Coverage probability

The coverage probability is defined as the probability that the SINR received by the typical user is greater than a predefined threshold T . Conditioning on the fact that the

p M (v, T ) = (1 -p M bm (v))q M (G M m , T ) + p M bm (v)q M (G M s , T ), (5)
where

q M (G, T ) = πλ M r≥0 e -πλ M r exp - T σ 2 P M KG r 1/δ exp -πr T P M G δ λ M P δ M ρ M (G, T ) + λ µ P δ µ κ µ dr.
Here δ 2/α, and ρ M and κ µ are given by

ρ M (G, T ) = p M,m (G M m ) δ ∞ T G M m G -δ du 1 + u 1/δ + (1 -p M,m )(G M s ) δ ∞ T G M s G -δ du 1 + u 1/δ , κ µ = p µ,m (G µ m ) δ + (1 -p µ,m ) (G µ s ) δ ∞ 0 du 1 + u 1/δ .
The coverage probability in the micro tier is obtained by swapping the role of both tiers.

Proof: The proof is given in [10]. In an interference-limited network, i.e., when σ 2 = 0, the functions q M and q µ simplify to

q M (G, T ) = 1 1 + T G δ (ρ M (G, T ) + Ωκ µ ) (6)
q µ (G, T ) = 1 1 + T G δ ρ µ (G, T ) + 1 Ω κ M , (7)
where

Ω = λµP δ µ λ M P δ M
. When setting Ω = 0, which corresponds to setting either λ µ = 0 or P µ = 0, i.e., removing the micro tier, we obtain the same formula as the one from [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].

B. Shannon rate

We denote the Shannon rate per Hertz in each tier by

R M = E [log(1 + SINR M )] and R µ = E [log(1 + SINR µ )]. Due to
RF imperfections and modulation schemes, we set Q max to be the maximum achievable SINR. Thus, we have

R M = Qmax 0 p M (T ) 1 + T dT (8)
in the macro tier. The same applies in the micro tier by using the appropriate values. Using Theorem 1, the Shannon rate achieved by the typical MU can be expressed as

R M (v) = (1 -p M bm (v))R M,m + p M bm (v)R M,s , (9)
where R M,m is the Shannon rate achieved with no beam misalignment, while R M,s is its equivalent with beam misalignment (as discussed in Section II-C). Notice that, from [START_REF] Chattopadhyay | Two-tier cellular networks for throughput maximization of static and mobile users[END_REF], comparing the Shannon rates in each tier corresponds to comparing the respective coverage probabilities. The term Ω in [START_REF] Arshad | Velocity-aware handover management in two-tier cellular networks[END_REF] plays an important role in determining the values of the Shannon rates in each tier. If the antenna gain profile is the same in both tiers, and if Ω > 1, i.e., λ µ P δ µ > λ M P δ M , the Shannon rate in the micro tier is always larger than that in the micro tier. In the rest of the paper, we assume that we are in this setup.

Equation [START_REF] Chattopadhyay | Two-tier cellular networks for throughput maximization of static and mobile users[END_REF] gives the Shannon rate for the typical MU without considering the overheads associated with beam reselections during intra-cell mobility and BS handovers during inter-cell mobility. To take these overheads into account, we now define the effective Shannon rate.

C. Effective Shannon rate (ESR)

When an MU crosses the boundary of the cell of its serving BS, it performs a handover to the next BS. Beam reselections also happen inside the cell when the MU crosses the boundary between two beams. The time intensity ν M,c of BS handovers and the time intensity ν M,b of beam reselections are given in Theorem 2 of [START_REF] Kalamkar | Beam management in 5G: A stochastic geometry analysis[END_REF]:

ν M,c = 4 √ λ M π v, ν M,b = n M √ λ M π v.
When we take beam misalignment into account, if the time intensity of beam reselections is higher than the frequency 1/τ of SSB bursts, beam reselections happen during each SSB burst. If the time intensity of beam reselections is lower, the user may stay in the same reference beam between two consecutive SSB bursts. Thus the effective time intensity ν M,e of beam reselections in macro tier is

ν M,e = min 1 τ , ν M,b .
Let T M,c and T M,b be the time of a BS handover and a beam reselection, respectively, in the macro tier. Then we can express the total overhead per unit of time T M o as

T M o = ν M,c T M,c + ν M,e T M,b .
Using the previous definitions and results, we define the ESR R M,eff in the macro tier as

R M,eff (v) = R M 1 -T M o (v) + , (10)
where x + max(0, x).

IV. VELOCITY-BASED ASSOCIATION POLICIES IN A SIMPLE CASE

In this section, we introduce the notion of velocity-based association policy in a simple case with exactly one MU per cell. The notions introduced in this section will be extended in the next section to the situation where the density of MUs and the resulting resource sharing are taken into account. The results of this section are nevertheless of independent interest as they allow one to analyze the peak rate that an MU would get in an ideal network with one MU per cell and hence without bandwidth sharing between MUs of the same cell.

Here and in what follows, to decide what tier an MU should associate with, we consider velocity-based association policies, which are defined as follows: let P(v) be a criterion based on velocity (namely a subset of the positive real line). Each MU associates either with the closest macro BS or to the closest micro BS. If the MU meets the criterion P(v), it associates with the micro BS, and if not, to the macro BS.

A. The ESR under a given association policy

The metric we use to compare association policies is the ESR received by the typical MU as given by (10).

Lemma 1: For all velocity-based association policies P, the average ESR received by the typical MU is given by [START_REF] Birkhoff | Proof of the ergodic theorem[END_REF] where 1 (•) is the indicator function and P the negation of P. This result comes from ergodic theory: we compute the spatial average of the ESR over a large ball of radius R > 0 centered at the origin. Using Birkhoff's ergodic theorem [START_REF] Birkhoff | Proof of the ergodic theorem[END_REF], we can prove that this spatial average converges to a constant, R(P), which is also equal to the ESR experienced by the typical MU.

R(P) = ∞ 0 R µ,eff (v)1 P(v) + R M,eff (v)1 P(v) f (v)dv,

B. Threshold association policies

A natural set of policies to explore is that of threshold policies: we select a velocity threshold v T ; all MUs with velocity v < v T associate with micro BSs and those with velocity v ≥ v T with macro BSs. Under such a threshold policy, the average ESR is given by

R(v T ) = v T 0 R µ,eff (v)f (v)dv+ ∞ v T R M,eff (v)f (v)dv. (12)
We have the following Theorem: Theorem 2 (Threshold velocity-based association policy for the one MU per cell case): Under the foregoing assumptions, there exists a unique threshold maximizing the average ESR per user. This optimal threshold v T does not depend on the velocity distribution f of MUs, and is the unique solution to the following equation:

R µ,eff (v T ) = R M,eff (v T ). ( 13
)
The proof for this theorem is available in [10]. Fig. 1 shows the normalized ESR R M,eff / R M,eff ∞ , where • ∞ is the supremum norm for three exponential velocity distributions;

Percentage of maximal ESR

v u =1 v u =10 v u =35
Fig. 1. Normalized ESR in the network as a function of the velocity threshold v T . The optimal threshold does not depend on the scale parameter vu.

we see that the maximum ESR is obtained at the same value for all velocity distributions. This network setup mimics an ideal situation, corresponding to the peak rate that an MU receives in the network. A more realistic setup with multiple MUs is studied in the next section.

V. LOAD-DEPENDENT VELOCITY-BASED ASSOCIATION POLICY

In this section, we address the load sharing question between all MUs of the same cell. We assume a time-division multiple access (TDMA) setup where radio resources are equally shared among all MUs connecting to the same BS.

Let P be a given velocity-based association policy. Under the association policy P, let Z 0 µ (P) denote the number of users associated with the micro BS corresponding to the 0cell, i.e., the Voronoi cell containing the typical MU located at the origin 0, and Z 0 M ( P) be its macro counterpart. Note that Z 0 µ (P) and Z 0 M ( P) are random variables due to the random network geometry and velocity. For this setup, we can modify [START_REF] Birkhoff | Proof of the ergodic theorem[END_REF] to rewrite the average ESR R load as follows:

R load (P) = E R µ,eff (v) Z 0 µ (P) 1 P(v) + R M,eff (v) Z 0 M ( P) 1 P(v) . (14)
In ( 14), the number of MUs associated to the 0-cell and the ESR both depend on the geometry of each PV tessellation. We use the following heuristic to approximate R load :

Rload (P) = E 1 Z 0 µ (P) E R µ,eff (v)1 P(v) + E 1 Z 0 M ( P) E R M,eff (v)1 P(v) , (15)
where E 1 Z 0 µ (P) and E 1 Z 0 M ( P) are the average inverse load in the 0-cell of micro and macro tiers, respectively.

A. Load-dependent threshold association policies

Like in Section IV-B, threshold policies remain good candidates to maximize the average ESR experienced by the typical MU. Let v T be a velocity threshold. Under the threshold policy with threshold v T , the intensity of the PPP of users associated with the micro tier is equal to λ u F (v T ), with F being the CDF of the velocity distribution, and the intensity of the PPP of users associated with the macro tier is equal to λ u (1-F (v T )).

Although we do not have an analytical formula for the average inverse load in the network, we can use the following heuristic: Lemma 2: Let Φ and Ψ be two PPPs of respective intensities λ and ν. Let V 0 be the 0-cell of the PV tessellation associated with the process Φ and let

Z = 1+|Ψ∩V 0 |. Finally, let L : x → x 1 - 1 1+ 2 7x 7/2
. The moment of order -1 of Z can be approximated by:

E 1 Z ≈ L λ ν .
Proof: The proof is given in [10]. It uses the results from [START_REF] Mecke | On the relationship between the 0-cell and the typical cell of a stationary random tessellation[END_REF] and a heuristic for the distribution of the size of the typical cell of a PV tessellation developed in [START_REF] Ferenc | On the size distribution of Poisson voronoi cells[END_REF]. Using this, we set the following notation:

E 1 Z 0 µ (v T ) = L λ µ λ u F (v T ) ≡ L µ (v T ) E 1 Z 0 M (v T ) = L λ M λ u (1 -F (v T )) ≡ L M (v T ).
Under a velocity-based threshold policy, the average ESR in the TDMA setup becomes

Rload (v T ) = L µ (v T ) v T 0 R µ,eff (v)f (v)dv + L M (v T ) ∞ v T R M,eff (v)f (v)dv. (16)
We give a load-threshold heuristic for the optimal threshold policy in the TDMA setup as follows:

v LT arg max Rload (v T ). (17)
As shown in Fig. 2, we numerically observe that v LT is uniquely defined, meaning that the function Rload (v T ) goes through a unique maximum. Specifically, Fig. 2 compares the simulations under TDMA and the heuristic from (17). The difference is due to the fact that the inverse of the load and the ESR are not independent. When comparing the values of the optimal achieved ESR in simulations with the ESR achieved using the heuristic from (17) (see Table II), we can see that the obtained values do not differ significantly, which justifies the use of the heuristic.

VI. DISCUSSIONS A. Comparison with a classical association policy

We now compare the optimal average ESR using the loadthreshold heuristic based on (17) to a classical association policy based on the maximum average received power (MARP), Effective Shannon rate (bits/s/Hz) where MUs connect to the BS with the highest average received power: the typical MU located at the origin connects to the macro tier BS located at X 0,M if the average power it receives from this BS, RP M = P M G 0,M (|X 0,M |), is higher than the average power it receives from the micro BS located at X 0,µ , RP µ = P µ G 0,µ (|X 0,µ |). Note that the MARP policy is based on the received power, and thus, the network geometry only, whereas the threshold policy is based on velocity only.

v u =1 v u =10 v u =35
As illustrated in Fig. 3, for a given value of the MU density λ u , the average ESR decreases faster with velocity under the MARP policy than under the load-threshold heuristic; in addition, while for low values of v u , the MARP policy outperforms the load-threshold heuristic, the threshold policy provides MUs with a better average ESR for velocities v u higher than 11 m.s -1 , up to a gain of 45%. Table III displays approximate values for the velocity threshold v † above which the load-heuristic threshold policy outperforms the MARP policy. We can see that v † decreases as the value of λ u increases, which supports the claim the threshold policies perform better in densely populated networks with fast-moving MUs. The fact that these results are obtained for parameters representing accurately the sub-6 GHz setting of 5G new radio (NR) supports the importance of the study of velocity-based threshold policies. We now discuss the reasons behind why these policies outperform each other in certain regimes. When the MUs move slowly and the network is sparsely populated, the effects of mobility-induced beam misalignment and overheads due to beam reselections and BS handovers are negligible. Thus, the factor of better average received power outweighs the time overheads. In this case, the MARP policy outperforms the threshold policy. On the other hand, when the MUs have a higher velocity in the network that is more densely populated, the penalty due to the mobility of MUs (BS handovers, beam reselections, and beam misalignment) has more impact on the performance compared to the received power. Thus, it is beneficial to associate users based on their velocities.

B. Effect of the density of MUs in the network

The density λ u of MUs allows us to understand how well the network performs under different load conditions. Specifically, as λ u → 0, the ESR in the network becomes equal to the Shannon rate without TDMA, which is its peak rate. This translates into v LT (given in (17)) becoming equal to v T from Theorem 2. Conversely, as λ u → ∞, the number of MUs associated with the micro tier becomes arbitrarily large. Because its contribution towards the ESR is larger than that from the macro tier, to maximize the average ESR, the optimal velocity threshold approaches zero.

The behavior of the optimal threshold for intermediate values of λ u depends on the velocity distribution. Fig. 4 gives the optimal velocity threshold for the heuristic given in (17). The MU velocity is a limiting factor in the network: if the velocity of MUs is higher, the average overhead per unit of time in the macro tier increases, leading to a decrease in the contribution of the macro tier to the average ESR. To balance this effect, more MUs are associated with the micro tier, resulting in an increase in the optimal velocity threshold.

VII. CONCLUDING REMARKS

This paper discusses a simple velocity-based association policy for multi-tier 5G networks. The proposed framework captures essential features of 5G networks, namely, beamforming, beam misalignment, delay overheads associated with beam reselections and base station handovers, speed profiles of users, and network geometry. In this association policy, users with a velocity smaller than a threshold connect to the micro tier, otherwise to the macro tier. The threshold is optimized to maximize the effective Shannon rate that the typical user gets. The comparison of this policy with the well-known association policy based on the maximum average received power reveals that, for parameters that are typical to sub-6 GHz 5G networks, when the network is populated with users moving with variable enough speed patterns, the former outperforms the latter in terms of effective Shannon rate. This observation highlights a key tradeoff between the power (affecting the SINR) and the velocity (affecting the delay overheads). The fact that simple velocity-based schemes may significantly outperform the state of the art association policies opens an interesting line of thoughts aiming at combining such velocity-based schemes with those based on maximal received powers to provide a new and better association policy for mobile users in this 5G context.

Fig. 2 . 1 Fig. 3 .

 213 Fig. 2. Comparison of the heuristic in (16) (in plain line) and simulations (circles) for a fixed user density λu = 1.

Fig. 4 .

 4 Fig.[START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF]. Optimal velocity threshold obtained for the heuristic from (17) (in plain line) for three velocity distributions against the density of users. In black, dashed, the equivalent network without load.

Acknowledgements This work was realized at LINCS. It was supported by the ERC NEMO grant, under the European Union's Horizon 2020 research and innovation programme, grant agreement number 788851 to INRIA. The authors thank Luis Uzeda Garcia of Nokia Bell Laboratories for inspiring discussions on this line of thoughts.


Scheme 1 Coordinative Chain Transfer Polymerization (CCTP)

Evidence of Coordinative Chain Transfer Polymerization of Isoprene using Iron Iminopyridine/ZnEt 2 Catalytic Systems

Obaid H. Hashmi, a Marc Visseaux a, * and Yohan Champouret a, * a. Univ. Lille, CNRS, Centrale Lille, Univ. Artois, UMR 8181 -UCCS -Unité de Catalyse et de Chimie du Solide, F-59000 Lille, France. Email: marc.visseaux@univ-lille.fr, yohan.champouret@univ-lille.fr † Electronic Supplementary Informa"on (ESI) available: Experimental part and characterization data. See DOI: 10.1039/d1py00433f

Abstract: Isoprene polymerization using iminopyridine-supported iron complexes combined with Al i Bu3/[B(C6F5)4][CPh3] was carried out in the presence of ZnEt2 as chain transfer agent (CTA). Detailed studies revealed that the Mn of the resulting polyisoprenes and the rate of polymerization decrease steadily as the amount of CTA increases while preserving the catalyst selectivity. These results highlight the first example of a reversible iron-mediated chain transfer polymerization of isoprene.

To date, coordinative chain transfer polymerization (CCTP) processes are predominantly based on olefin monomers and it is only recently that this methodology has been extended to conjugated dienes. 1 For the latter ones, the few examples disclosed in the literature have exclusively focused on rare earth metal-based systems 2 combined with Al or Mg as Chain Transfer Agent (CTA). To our knowledge, the implementation of transition metal-catalyzed CCTP of conjugated dienes remains to be proven, especially for iron complexes. 2b, [3][4][START_REF] See | The activation of the pre-catalyst by MAO can lead, in certain cases, to uncontrolled chain transfer of the poly-1,3-diene chains to Al due to the presence of free AlMe3[END_REF] In this paper, we describe the first successful CCTP of isoprene using iminopyridine iron complexes in the presence of ZnEt2 as CTA, after proper cocatalyst activation.

In the quest of sustainable chemistry, iron catalysis is currently witnessing a major revival given the natural abundance and relative non-toxicity of this metal, which offers a suitable substitute to other metal-based systems involved in the catalytic transformation of organic substrates. 6 Notably, the development of well-defined single-site iron-based complexes has led to an increased attention in coordination-insertion polymerization of polar and nonpolar monomers, 7 with a more focus over the last few decades on the polymerization of ethylene. 8 A significant breakthrough in this field was described by Gibson and coworkers who carried out a polymerization of ethylene using the bis(imino)pyridine iron dichloride/methylaluminoxane catalytic system [(BIP)FeCl2/ MAO, [BIP = bis(imino)pyridine, MAO = methylaluminoxane] in the presence of a large amount of ZnEt2, which allowed accurate control of the macromolecular data (Mn, Đ) of the resulting polymer. 9 In this process, known as CCTP, the growing polymer chain is reversibly transferred from the active Fe center to the dormant CTA Zn center via a putative heterobimetallic Fe/Zn intermediate (Scheme 1). 1 This methodology exhibits living characteristics and enables the control growth of several macromolecular chains per catalyst molecule since i) the chain transfer is fully reversible, ii) its rate is fast compared to the rate of propagation, iii) the formation of the heterobimetallic intermediate has a sufficient lifetime for the CCTP to be effective and iv) the chain transfer occurs in absence of any undesired irreversible transfer/termination pathway. 1 All these requirements are highly dependent on the electronic and steric properties of the catalyst, besides the nature and adequacy of the CTA with the monomer used. As a result, narrow dispersities are obtained with the macromolecular chains terminated by the chain transfer agent, which enables further polymer chain-end functionalization, the preparation of block copolymers and, subsequently, the elaboration by chain shuttling polymerization (CSP) of copolymers featuring multi-block architecture.

Up to now, coordinative polymerization of conjugated dienes, such as butadiene, isoprene and derivatives, remains a preferred choice to obtain effectively polydiene materials with desired molar masses along with controlled microstructures. 10 In the case of polyisoprene, the resulting polymer can present different arrangements comprising cis-1, 4, trans-1,4, 3,4-and, occasionally, 1,2-vinyl sequences or a combination of these units that dictates the enduse applications of the material in the rubber industry. 11 Within this framework, we [START_REF] Hashmi | [END_REF] and others, 13 have recently described that discrete iron-based complexes bearing an iminopyridine ligand are capable of efficiently catalyze the coordinative polymerization of isoprene after activation by an appropriate cocatalyst, some of them exhibiting high control of stereoselectivity. 7,13a In the present work, we focused our study on the implementation of CCTP of isoprene. For this purpose, we have targeted on the use of ZnEt2 as CTA, since previous studies have shown that this CTA associates successfully with the (BIP)FeCl2/MAO catalyst system for the CCTP of ethylene 9,14 and acetylene. 15 The CCTP of isoprene was initially carried out using the Fe-based complexes 1-4 (Scheme 2) combined with Al

i Bu3/[B(C6F5)4][CPh3] (isoprene/Fe/Al i Bu3/[Ph3C][B(C6F5)4
= 2 000/1/3/1) under optimized conditions, [START_REF] Hashmi | [END_REF] in the absence and the presence of ZnEt2 (Zn/Fe = 10), the results are gathered in Table 1 (see also Figs. S1-S10 for SEC traces, ESI †). As previously reported, [START_REF] Hashmi | [END_REF] activation of complexes 1 to 3 with Al i Bu3/[B(C6F5)4][CPh3], without ZnEt2, produced polyisoprenes with Mn values higher than the theoretical ones, which speaks in favor of a much higher propagation rate with respect to the initiation steps, while the 4/Al i Bu3/[B(C6F5)4][CPh3] catalytic system yielded polymers with Mn experimental close to the theoretical value (vide infra). The expected CCTP effects of the CTA upon the polymerization are i) decreasing Mn values and ii) narrowing the dispersity (Đ value tending to 1). From the comparison of the results of the experiments performed with and without ZnEt2, the first criteria is fulfilled by precatalysts 1 to 4, as for a complete conversion of isoprene, the Mn are reduced by a factor of 2.7 (complex 1, entries 1 vs 2), 1.3 (complex 2, entries 3 vs 4), 3 (complex 3, entries 5 vs 6) and 7.6 (complex 4, entries 7 vs 8), respectively. Although, the dispersity was found to increase slightly in the presence of 10 eq. of ZnEt2 for complexes 1 and 2 (Đ = 1.6 vs 1.7 and 1.3 vs 1.8, respectively) while they remained constant for complexes 3 and 4 (Đ = 1.3).

Furthermore, the selectivity of each catalytic system was notably maintained during CCTP screening (Figs. S11-S20 for NMR spectra, ESI †). For example, the high degree of selectivity of the 4/Al i Bu3/[B(C6F5)4][CPh3] catalytic system produces polyisoprene, which microstructure, predominantly 1,4-, is similar to that obtained in the presence of additional ZnEt2 (Table 1, entries 7 vs 8) and consists of approximately 77 -79 % of trans-1,4 units along with the contribution of 3,4 (ca. 9 -10%) and cis-1,4 fractions (ca. 12 -13%). This observation reveals that the growth of the polymer chain is probably not influenced by the presence of the CTA, suggesting distinct mechanistic elementary steps of insertion (Fe catalyst) and transfer (Fe/Zn heterobimetallic species). Among the four catalysts tested, the highest transfer efficiency was found for 4/Al i Bu3/[B(C6F5)4][CPh3]/ZnEt2 and may be attributed to a good bond strength adequacy, i.e. comparable bond strength value of the metal catalyst-growing polymer chain and of the metal CTA-dormant polymer chain, as previously observed for the catalyzed chain growth of ethylene and acetylene using [(BIP)FeCl2]/MAO catalytic system in the presence of an excess of ZnEt2. 9,15 A similar hypothesis can be drawn in our case from the 4/Al i Bu3/[B(C6F5)4][CPh3]/ZnEt2 catalytic system with the assumption that metal-allyl bonds are involved in the polymerization of isoprene. 16 Further attempts to assess CCTP of isoprene using 4 and 100 eq. of MAO in presence of various amounts of ZnEt2 resulted in a less efficient system (Table S1 and Figs. S21-S24, ESI †). Furthermore, Al and Mg alkyl compounds, uncommonly used as CTAs with Fe precatalysts, were screened in this frame with the most efficient complex 4. The use of an excess of Al i Bu3 with complex 4 (Al/Fe = 10/1) led to practically no change in the resulting polyisoprene Mn (entry 10, Table 1, Mn = 40 000 g/mol) compared to the Mn Scheme 2 Fe-mediated CCTP of isoprene value obtained in the presence of 3 eq. of Al reagent (entry 9, Table 1, Mn = 41 000 g/mol). This result indicates that the transfer is not relevant between Al and Fe under our experimental conditions. Moreover, the broadening of the dispersity was noticeable (entries 9 vs 10, Table 1, Ð = 1.2 vs 1.6); this observation was also apparent when using Al i Bu3 in excess for rare earth catalytic systems. 17 The resulting polyisoprene was acquired in good yields (> 99%), displaying similar microstructure as observed with 3 eq. of Al i Bu3. Furthermore, replacement of Al i Bu3 with a sterically less hindered aluminum CTA, AlEt3, did not significantly improve the transfer efficiency. [START_REF]One of the referees questioned whether the transfer was limited to ZnEt2 as CTA or if a sterically less hindered aluminum reagent, AlEt3, could be used as CTA[END_REF] In contrast, the 4-based system failed to produce polyisoprene (entry 11, Table 1) in presence of 10 eq. of butylethylmagnesium (BuMgEt, BEM), which is similar to that found in the case of ethylene polymerization using (BIP)FeCl2/MAO/Mg n Bu2 (1/100/500). 4b In addition to the question of the similarity of bond strengths between the catalyst-polymer chain and the CTA-polymer chain, it is clear that the monomeric nature of ZnEt2 as compared to aggregated MgR2 in non-polar solution [START_REF] Ducom | [END_REF] could be an explanation for a better CCTP efficiency when associated with the most sterically hindered precatalyst 4. The chain transfer between the catalyst originating from 4 associated with ZnEt2 was also examined through the kinetic profile of isoprene polymerization at 25 °C, without and with 10 eq. of ZnEt2 per Fe compound (Table S2, ESI †). From the kinetic plots (Fig. S25a, ESI †), it is evident that the addi"on of 10 eq. ZnEt2 prolonged the polymerization time substantially, which nearly follows a first order kinetic profile with and without CTA (the process being even better controlled under polymer chain transfer regime, Fig. S25b, ESI †), indica"ng minimal loss of ac"ve species during the polymerization. In addition, the plot of Mn vs conversion (Fig. S25c, Figs S26-S35, ESI †) is consistent with a controlled polymerization process in absence and in the presence of CTA, as assessed by the linear relationship observed. The 4/Al i Bu3/[B(C6F5)4][CPh3] catalytic system was further assessed by conducting a detailed study through the variation in the amount of ZnEt2 during the polymerization process. The quantity of ZnEt2 (0-100 eq. of ZnEt2 vs 4) was monitored and its impact on the molar masses of the resulting polyisoprenes was observed, the results of which are displayed in Table 2 and Fig. 1.

From these results, we could notice a unique trend in the decrease of Mn values, which is consistent with the increasing quantities of CTA. By adding 5 eq. of ZnEt2/4, the molar mass is reduced by a factor of 4.7 (entries 2 vs 1, Table 2), which in turn is lowered by a factor of 7.6 (entries 3 vs 1, Table 2) as the amount of CTA gets doubled. This decreasing trend is regularly maintained throughout the series of experiments as we compare the Mn value of each entry from 5 to 100 eq. of ZnEt2/4. This can also be seen through SEC analysis with the characteristic Gaussian curves shifting to the left (lower masses) as the amount of CTA is increasing (Fig. 1 and also see Figs. S36-S43 for SEC traces, ESI †). Importantly, the selectivity of 4 remained constant in all cases by increasing the quantity of CTA (Figs. S44-S51 for NMR spectra, ESI †). Nevertheless, we can observe that the use of a large amount of ZnEt2 does not decrease the dispersities, which remains constant during the set of experiments (Đ = 1.2-1.5). Furthermore, the situation here is far from ideal since, for the most efficient system with a ratio of 4/ZnEt2 = 1/10, less than 8 chains were formed whereas a number of 21 chains was expected (20 ). This result potentially indicates that at best only one of the two ethyl groups of ZnEt2 is transferred to the iron center to initiate a polymer chain growth, which could preclude the possibility of carrying two polymer chains per Zn, highlighting some of the limitations of this system.

One of the remarkable extensions of CCTP concept is represented by Chain Shuttling Polymerization (CSP), where a dual-catalyst system in presence of CTA is used to generate distinct growing macromolecular chains, either from two comonomers or from a unique monomer that can lead to different chain regularities depending on the regio-, chemo-or stereo-selectivity of each catalyst. Thus, the two distinct growing chains are reversibly transferred from one catalyst to another by means of the CTA (here called the chain shuttling agent), giving rise to multiblock architectures that contain block segments with the microstructural signature of each catalyst. 20 In this context, CSP of conjugated dienes has been independently described by three research groups using either two regio-21 or stereoselective 22,23 rare-earth catalysts in the presence of CTA, producing poly(1,3-dienes) multiblock containing alternated 3,4-/1,4-polyisoprene block or cis/trans-block of either polyisoprene or polybutadiene, respectively. Despite the moderate regio-and stereo-selectivity of our iron-based systems for the CCTP of isoprene, we still attempted to prepare multiblock polyisoprene by CSP. For this purpose, we chose to use complexes 1 and 3 (isoprene/1+3 = 2 000/1) combined with Al i Bu3/[B(C6F5)4][CPh3] and in the presence of 10 eq. of ZnEt2 (Table 3), since the transfer efficiencies are quite similar between these two catalysts (N chains = 2.7 -3 for complex 1 and 3, Table 1, entries 2 and 6) and also result in polymer chains with distinguishable stereo-and regio-regularities (trans-1,4/cis-1,4/3,4 content = 33/46/21 for complex 1 vs 69/22/9 for complex 3). Detailed studies of the CSP tests can be found in Table S3 and Figs. S52-S73 in ESI †.

The CSP attempt of isoprene using complexes 1 and 3 in a ratio of 1/1 leads to a polyisoprene with Mn = 180 000 g/mol (Đ = 1.5) and a microstructure in which the content of trans-1,4/cis-1,4/3,4 = 40/45/15 reflects the average 1,4/3,4 selectivities of both catalysts (85/15, Table 3, entry 1, between 79/21 and 91/9, entries 2 and 6, Table 1, respectively). Additionally, the determination of the Tg shows a value of -56.8 °C, which is an intermediate value of the polymers produced independently by complexes 1 and 3 under CCTP conditions (between -51.6 °C and -61.7 °C, entries 1 and 5, Table S3, ESI †, respectively). The same reaction was also carried out without ZnEt2 and resulted in a polyisoprene presenting a stereo-, regio-regularity along with a Tg value quasi similar to that obtained in the presence of CTA (Table 3, entry 2). The unique glass transition found at -55.5 °C was expected given that the physical mixture of the two homopolymers, produced independently by CCTP, displays a single Tg of -57.1 °C (Fig. S70) due to the miscibility of the blend. However, the resulting Mn of the polymer produced under CSP was found to be 3 times lower than that achieved without ZnEt2 (Mn = 180 000 vs 380 000 g/mol, respectively), which is similar to the number of chains generated for each independent complex under CCTP (entries 5-6, Table 1). Moreover, the dispersity was 1.5 vs. 1.6 in absence of Zn-CTA. Thus, these results partially suggest the occurrence of a CSP process, however, the moderate selectivity of both iron complexes prevents us from asserting unequivocally that it has indeed taken place. Alternatively, we cannot totally exclude that the growing chains on each iron catalyst are not transferred from one to the other through the Zn CTA, thus producing a mixture of two homopolymers that distinctively incorporate the signature of each catalyst rather than a single polyisoprene chain displaying a multiblock structure. In this study, we have successfully demonstrated the first CCTP of isoprene with an iron-based catalytic system. Various metal alkyls were screened as potential CTAs during the CCTP of isoprene, among which it was observed that the transfer is only efficient with ZnEt2, owing to an appropriate combination between the catalyst and the chain transfer agent. The ketiminopyridine iron complex 4 bearing sterically hindered N-aryl substituents was the most efficient under CCTP conditions when compared to the less sterically crowded complex 3 and the aldiminopyridine iron complexes 1 and 2, although the transfer efficiency between 4 and ZnEt2 was not total (36%, considering alkyls-CTA). Attempts of performing a polymerization of isoprene using a mixture of complexes 1 and 3 combined with 10 eq. ZnEt2, after cocatalyst activation, might suggest the occurrence of a CSP process. Further development of iron-catalyzed reversible chain transfer coordinative polymerization, such as the preparation of block copolymers or the functionalization of polymer chain-end, is currently in progress and will be reported in due course.

aFig. 1

 1 Fig. 1 Effect of quantity of ZnEt2 (0 -100 eq./Fe) on the Mn using 4/Al i Bu3/[B(C6F5)4][CPh3] catalytic system (according to data inTable 2)

a

  Polymerization conditions: 10 µmol of complex 1+3, isoprene/1+3/ Al i Bu3/[B(C6F5)4][CPh3] = 2 000/1/3/1, toluene = 20 mL, [C]isoprene = 1 mol/L, temperature = 25 °C, time = 6 h. b Determined by SEC analysis in THF using polystyrene standards. c Determined by 1 H NMR and 13 C NMR. d Determined by DSC.

Table 1

 1 Screenings 1-4/Al i Bu3/[B(C6F5)4][CPh3]/ZnEt2 catalytic systems for the CCTP of isoprene a

	Entry a	Complex	[Zn]/[Fe]	Time (min)	Conv. (%)	Mn(exp) b (g/mol)	Đ b	N chains b	Microstructure c (%) 1,4 (trans/cis)	3,4
	1 2	1	0 10	360 360	> 99 > 99	487 500 180 000	1.6 1.7	2.7	79 (38/41) 79 (33/46)	21 21
	3 4	2	0 10	360 360	> 99 > 99	250 000 195 000	1.3 1.8	1.3	75 (35/40) 75 (32/43)	25 25
	5 6	3	0 10	360 360	> 99 > 99	261 000 86 000	1.3 1.3	3.0	90 (69/21) 91 (69/22)	10 9
	7 8	4	0 10	360 360	> 99 > 99	155 000 20 500	1.3 1.3	7.6	90 (77/13) 91 (79/12)	10 9
	9 d		0	360	> 99	41 000	1.2		92 (76/16)	8
	10 d	4	7 Al i Bu3	960	> 99	40 000	1.6	1.0	92 (78/14)	8
	11 d		10 BuMgEt 1080	0	-	-	-	-	-

a Polymerization conditions: 10 µmol of Fe(II) complex, isoprene/Fe/Al i Bu3/[Ph3C][B(C6F5)4] = 2 000/1/3/1, toluene = 20 mL, [C]isoprene = 1 mol/L, temperature = 25 °C. b Determined by size exclusion chromatography (SEC) analysis in THF using polystyrene standards, N chains = Mn(exp) without CTA/Mn(exp) with CTA. c Determined by 1 H NMR and 13 C NMR. d Isoprene/Fe/Al i Bu3/[B(C6F5)4][CPh3] = 500/1/3/1.

Table 2

 2 CCTP of isoprene using 4/Al i Bu3/[B(C6F5)4][CPh3] catalytic system with different quantities of ZnEt2 a

	Entry	[ZnEt2]/[4]	Conv. (%)	Mn(exp) b (g/mol)	Đ b	N chains c
	1	0	>99	155 000	1.3	-
	2	5	>99	33 000	1.5	4.7
	3	10	>99	20 500	1.3	7.6
	4	20	>99	15 000	1.2	10.3
	5	30	>99	12 000	1.3	13
	6	40	>99	10 000	1.3	15.5
	7	50	>99	8 000	1.3	19.4
	8	100	64	6 600	1.2	15.0 d

Table 3

 3 Attempt CSP of isoprene using 1+3/Al i Bu3/[B(C6F5)4][CPh3] catalytic system with and without ZnEt2 a

Acknowledgements

We warmly thank the University of Lille for the financial support of O. H. Hashmi and the CNRS for partially funding this work. Aurélie Malfait and Dr. Jonathan Potier are acknowledged for SEC analyses. Dr. Eric Buisine and Patrick Daubias are gratefully thanked for their assistance with NMR spectroscopy.