Jules Dupire

Pierre-Henri P Puech

Emmanuèle Helfer

Annie Viallat

Mechanical adaptation of monocytes in model lung capillary networks

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

D R A F T

Mechanical adaptation of monocytes in model lung capillary networks.

Jules Dupire a , Pierre-Henri Puech b , Emmanuèle Helfer c , and Annie Viallat c,1 a Aix Marseille University, Marseille, France; b Aix Marseille Univ, CNRS, INSERM, LAI, Marseille, France; c Aix Marseille Univ, CNRS, CINAM, Marseille, France Proper circulation of white blood cells (WBCs) in the pulmonary vascular bed is crucial for an effective immune response. In this branched vascular network, WBCs have to strongly deform to pass through the narrowest capillaries and bifurcations. Although it is known that this process depends on the cell mechanical properties, it is still poorly understood due to the lack of a comprehensive model of cell mechanics and of physiologically relevant experiments. Here, using an in-house microfluidic device mimicking the pulmonary capillary bed we show that the dynamics of THP1 monocytes evolves along successive capillary-like channels, from a non-stationary slow motion with hops to a fast and smooth efficient one. We used actin cytoskeleton drugs to modify the traffic dynamics. This led us to propose a simple mechanical model that shows that a very-finely tuned cortical tension combined with a high cell viscosity govern the fast transit through the network while preserving cell integrity. We finally highlight that the cortical tension controls the steady-state cell velocity via the viscous friction between the cell and the channel walls.

microfluidics | monocyte | cortical tension | pulmonary network T he efficient circulation of white blood cells (WBCs) in the microvasculature is a key factor for proper body protection. The microvasculature is a complex network of interconnected narrow capillaries, that, in many organs, have cross-sections smaller than the diameters of WBCs. A prominent example is the pulmonary bed whose capillary diameters range from 2 to 15 µm (1, 2), while those of neutrophils, the most common type of WBCs (60-70% of all WBCs), range from 6 to 8 µm [START_REF] Doerschuk | Mechanisms of leukocyte sequestration in inflamed lungs[END_REF][START_REF] Doerschuk | Comparison of neutrophil and capillary diameters and their relation to neutrophil sequestration in the lung[END_REF], and those of monocytes (7% of all WBCs), even larger, vary between 15 and 30 µm (4). WBCs must therefore deform when they transit from pulmonary arterioles to pulmonary venules, passing through 8 to 17 alveolar walls, i.e. 40 to 100 alveolar capillaries of a few µm in diameter and length [START_REF] Doerschuk | Mechanisms of leukocyte sequestration in inflamed lungs[END_REF][START_REF] Guntheroth | Pulmonary microcirculation: tubules rather than sheet and post[END_REF]. The deformation process, directly governed by the WBC rheological properties, is slow and results in temporary cell arrests at the entrance of the narrowest capillaries [START_REF] Sa Gebb | Sites of leukocyte sequestration in the pulmonary microcirculation[END_REF]. This characteristic motion by hops [START_REF] Doerschuk | Comparison of neutrophil and capillary diameters and their relation to neutrophil sequestration in the lung[END_REF][START_REF] Sa Gebb | Sites of leukocyte sequestration in the pulmonary microcirculation[END_REF][START_REF] Lien | Physiologic neutrophil sequestration in the canine pulmonary circulation[END_REF] is associated with a prolonged transit time in the lungs compared to plasma or highly deformable red blood cells [START_REF] Hogg | Erythrocyte transit and neutrophil concentration in the dog lung[END_REF][START_REF] Presson | Distribution of pulmonary capillary red blood cell transit times[END_REF]. An increase in the rigidity of WBCs is considered as sufficient for them to be trapped in the pulmonary capillaries, facilitating their subsequent adhesion and degradation on the capillary walls [START_REF] Gs Worthen | Mechanics of stimulated neutrophils: cell stiffening induces retention in capillaries[END_REF][START_REF] Preira | The leukocyte-stiffening property of plasma in early acute respiratory distress syndrome (ards) revealed by a microfluidic single-cell study: the role of cytokines and protection with antibodies[END_REF]. For example, neutrophils stiffened by inflammation molecules released during acute respiratory distress syndrome [START_REF] Preira | The leukocyte-stiffening property of plasma in early acute respiratory distress syndrome (ards) revealed by a microfluidic single-cell study: the role of cytokines and protection with antibodies[END_REF][START_REF] Preira | Microfluidic tools to investigate pathologies in the blood microcirculation[END_REF] are massively trapped in pulmonary capillaries, causing fatal ischemia and lung injury [START_REF] Hogg | Neutrophil kinetics and lung injury[END_REF]. To date, the trapping mechanisms and, more generally, the dynamics of WBCs in pulmonary capillaries remain poorly understood. Indeed, there is no simple unified model describing the dynamical mechanical response of WBCs under various mechanical stresses. Different models have been developed [START_REF] Herant | The mechanics of neutrophils: synthetic modeling of three experiments[END_REF][START_REF] Schmid-Schönbein | Passive mechanical properties of human leukocytes[END_REF][START_REF] Evans | Apparent viscosity and cortical tension of blood granulocytes determined by micropipet aspiration[END_REF][START_REF] Skalak | Passive deformations and active motions of leukocytes[END_REF][START_REF] Tran-Son-Tay | Rheological modelling of leukocytes[END_REF] whose relevance depends on the amplitude and dynamics of the applied stress [START_REF] Shirai | Modeling neutrophil transport in pulmonary capillaries[END_REF]. Another difficulty is the lack of physiologically relevant experiments. In-vivo studies are invasive and do not allow the control of cell mechanics and vascular flow parameters. Most in-vitro approaches have focused on measuring the entry and transit times of individual cells in straight channels [START_REF] Preira | The leukocyte-stiffening property of plasma in early acute respiratory distress syndrome (ards) revealed by a microfluidic single-cell study: the role of cytokines and protection with antibodies[END_REF][START_REF] Gabriele | A simple microfluidic method to select, isolate, and manipulate single-cells in mechanical and biochemical assays[END_REF][START_REF] Preira | Single cell rheometry with a microfluidic constriction: Quantitative control of friction and fluid leaks between cell and channel walls[END_REF][START_REF] Yap | Mechanical deformation of neutrophils into narrow channels induces pseudopod projection and changes in biomechanical properties[END_REF] but have not addressed the dynamics of WBCs in branched capillary networks. The relationships between cell mechanics, motion by hops, retention time in network bifurcations and transport velocity throughout the network remain unknown.

Here, we mimic the capillary bed by a microfluidic network of narrow channels whose size and applied pressure have physiological values. We show that monocytes (from THP1 lineage) have a unique adaptive dynamics in the capillary network with a first non-stationary regime characterized by a slow movement with hops followed by a fast and smooth motion allowing efficient transport. We propose a simple mechanical model that shows how the remarkable combination of high cell viscosity and very finely-tuned cortical tension induces fast and efficient transport of the cell while preserving its integrity. We show that the final cell velocity in the network is controlled by the lubrication layer between the cell membrane and the channel walls, layer itself determined by the cell cortical tension.

Results

The biomimetic channel network. Our microfluidic device consists of a network of microchannels, whose network morphology and channel geometry are similar to those of the lung bed [START_REF] Doerschuk | Mechanisms of leukocyte sequestration in inflamed lungs[END_REF][START_REF] Guntheroth | Pulmonary microcirculation: tubules rather than sheet and post[END_REF]. An interconnected network of horizontal and diagonal channels is created by a series of 50 columns of 7 pillars regularly arranged in a staggered pattern (Figs. 1 andS1A-B). A cell enters the network on the left and exits on the right after passing through 100 channels, which is the typical number

Significance Statement

Proper circulation of white blood cells (WBCs) in the pulmonary blood capillary network is crucial for a healthy immune watch. The remarkable mechanical properties of monocytes, a class of WBCs, allow them to gradually adapt their shape as they pass through the narrowest capillaries, ensuring a transition from a slow motion by hops at the entrance of the capillary network to a smooth and rapid circulation. The key parameters are a low cell cortical tension allowing adaptation of the cell shape, but high enough to prevent cell breakage when passing through the network bifurcations, and a high cell internal viscosity to maintain an elongated shape throughout the network.

D R A F T

of capillaries crossed by a cell in the lung bed. At the exit of each horizontal channel, it has two choices of trajectories, mimicking the multiplicity of possible paths in the alveoli from pulmonary venules to arterioles. The channel's dimensions are in the physiological range with a height of 9.3 µm, a length l of the horizontal channels of 15 µm. The width of horizontal and diagonal channels w is the same and is kept constant in a given device. Various devices were prepared with w varying around physiological values, from 8.6 to 14.6 µm. The network unit mesh is shown in Fig. S1B. The pressure drop ∆Pt applied between the device inlet and outlet was set to 10, 20 or 40 mbar, leading to physiological pressure drops [START_REF] Huang | Computational modeling of rbc and neutrophil transit through the pulmonary capillaries[END_REF] in horizontal and diagonal channels of ∆P h = 1.7∆P mesh and ∆P d = 0.7∆P mesh , respectively, where ∆P mesh = ∆Pt/50 is the pressure drop in a mesh. The streamlines alternate right and left at each bifurcation (Fig. S1C).

A fast periodic monocyte dynamics is achieved in the network. THP1 cells (14.8 ± 1.45 µm in diameter, see Fig. S2) are prestressed in height in the 9.3-µm high inlet channel, before reaching the network. When smaller than the channels' width, they display a smooth periodic motion that follows the streamlines in the network. However, most cells typically have a diameter 25% larger than w and are thus laterally deformed as they pass through the microchannels, leading to a very different behavior (Fig. 1, Movie S1). In the upstream part of the network, cells have a nonperiodic trajectory (Fig. 1A), a significant shape deformation, and a motion by hops with temporary arrests in the bifurcations at the exit of the horizontal channels, which can last up to several seconds. The time to pass through a mesh is very variable (0.1 to a few sec), with the cell sometimes alternating between fast and slow passage through successive meshes (Fig. S3A and Movie S2). Further downstream in the network (typically after 5 to 15 meshes), the cell motion becomes smooth, without arrests, and with a periodic trajectory characterized by alternating right and left turns at the exit of horizontal channels similarly to streamlines (Fig. 1B).

During the transient regime, cell shape and velocity vary within a mesh but they also evolve from mesh to mesh till steady-state is reached (Fig. 2). The cell deformation is characterized by its reduced volume ν defined as the ratio of the cell volume V to the volume of a sphere with the same surface area A, ν = 3V /4π( A 4π ) 3/2 (ν ≤ 1, ν = 1 for a sphere). Maximum and minimum values of ν are reached in the horizontal and diagonal channels, respectively, and meshto-mesh evolution shows that globally the cell progressively elongates, especially in the diagonal channel. The changes in velocity behavior mainly occur in the horizontal channels: while in the first meshes cells strongly slow down both in bifurcations and horizontal channels, the main slowdown then only takes place in the bifurcations. In the final steady-state regime (mesh 8, brown lines), velocity and deformation become periodic with a period of two unit meshes: the cells move fast and barely slow down at the entrance of the horizontal channels, the variation of ν is the lowest and, on average, the cell is the most stretched. We compared the cell dynamics to that of giant lipid vesicles of internal viscosity close to that of the external fluid (Fig. S4). Vesicles with a diameter larger than w, and with sufficient excess of surface area to ensure the shape deformation required to pass through the channels without breaking, have a random trajectory over the entire network (Movie S3 and see Fig. S4C for a vesicle rupture). Figs. S4A-B show a typical vesicle motion by hops, with a residence time in the bifurcations of about 25% of the total transit time. The D R A F T velocity and deformation do not evolve from mesh to mesh along the network. This highlights the remarkable adaptation of monocyte shapes along the successive meshes of the microchannel network leading to their fast transport during the final steady-state regime.

The mechanical properties of monocytes affect their dynamics. The question therefore arises of whether the adaptive monocyte behavior requires an active response of the cell or whether it can be described only in terms of complex rheological properties. This led us to modulate the cell rheology and, more specifically, to vary the elasticity of the actin cortex by treating cells with Latrunculin A (lat-cells) or Jasplakinolide (jasp-cells) that respectively inhibits and promotes actin polymerization. Thus, Latrunculin A decreases cell stiffness while Jasplakinolide increases it [START_REF] Gabriele | Microfluidic investigation reveals distinct roles for actin cytoskeleton and myosin ii activity in capillary leukocyte trafficking[END_REF]. Strikingly, both treatments significantly affect the cell trajectories and dynamics (Fig. 3). The percentage of cells reaching the steady-state is 95%, 85% and 46 % for lat-, untreated, and jasp-cells, respectively. All lat-cells reach a steady-state periodic regime at very early meshes, with a trajectory alternating right and left turns and a fast traffic without arrests. In contrast, half of jasp-cells do not reach steady-state over the entire network and move slowly with very irregular Untreated cells present an intermediate behavior between lat-and jasp-cells. The specific cell behaviors are illustrated in Fig. 3 (see also Fig. SS5 and Movies S4-S6 for a better understanding of movements). Lat-cells gently adapt their shape as they pass through a mesh. When exiting a horizontal channel, the cells that do not present a strong shape asymmetry crash on the pillar and deform a lot in the bifurcation. The cell part closest to the entrance of a diagonal channel enters it and becomes the new cell front. The cell rear is sheared by the flow in the other diagonal channel and takes an asymmetric shape. Cells that already have an asymmetric shape in the horizontal channel immediately turn in a diagonal channel at the horizontal channel exit. It is interesting to note that in a bifurcation, the ends of the largest lat-cells may be simultaneously engaged in each of the two diagonal channels. The cells are then stretched at both ends and can possibly tear in two parts, which flow downstream (Movie S7).

In contrast, the jasp-cells that do not reach the steadystate regime are more rigid, retain their symmetrical "sausage" shape when they exit the horizontal channel and tumble into the bifurcation in a rigid-like way. There, they often clog the two diagonal channels and hang in the bifurcation for a long and variable time. The cell elongation may increase but the cells never tear. They eventually return to circulation in one of the diagonal channels by a slight displacement of their center of symmetry in a random direction along the pillar. This randomness leads to a slow motion by hops of the cells and a frequent reversal of their direction, which results in an exchange between the cell front and rear.

Most of untreated cells reach the steady-state regime within less than 10 meshes, few requiring more. In the transient regime (see elapsed times in Figs. 3B, andS5) cells that exit the horizontal channel with a symmetrical shape crash on the pillar and may clog the two diagonal channels like a jasp-cell (hopping motion). But when they return to circulation, unlike jasp-cells, they can be asymmetrically deformed by the flow in the unobstructed diagonal channel. Further down the network, in the steady-state regime, the cells slightly turn at the exit of the horizontal channel, thus engaging their front in a diagonal channel. Their rear tumbles over the pillar but does not clog the other diagonal channel, allowing rapid circulation. Overall, untreated cells are slower than lat-cells.

Relevant mechanical model for monocyte dynamics. In order to highlight the mechanical parameters responsible for this very specific monocyte traffic, we analyzed the dynamics of cell entrance in the first channel (Fig. 4).

First, a minimal pressure drop ∆Pc must be applied to drive a cell into the channel, thus revealing a non-zero cell cortical tension. Above ∆Pc, we measured the cell elongation ratio (t) = L(t)-L(0) L(0) during its entrance in the channel, where L(t) is the cell length along the channel (Fig. 4B for a typical untreated cell). The curve initially shows a short non-linear regime at small deformation (for ≤ 0.1), typical of a viscoelastic material, followed by a slow viscous flow (up to ≥ 0.25) at constant velocity. Thus, these cells are considered as viscoelastic drops with a surface tension τ0. The internal composite material is described by the modified Maxwell model which consists in a viscoelastic Kelvin-Voigt body (dashpot of viscosity η1 and spring of elasticity k1) linked to an additional dashpot of viscosity η2 for the long-time viscous dissipation (Fig. 4C) [START_REF] Guevorkian | Aspiration of biological viscoelastic drops[END_REF][START_REF] Wu | A comparison of methods to assess cell mechanical properties[END_REF].

The cell cortical tension τ0 generated by the actin cortex was determined from ∆Pc, the minimal pressure drop allowing a cell to enter in the network first channel (at 37 • C) or in a micropipette (at 23 • C). In the case of a cylindrical pipette of radius Rp, ∆Pc relates to τ0 by the Laplace law [START_REF] Evans | Apparent viscosity and cortical tension of blood granulocytes determined by micropipet aspiration[END_REF][START_REF] Hochmuth | Micropipette aspiration of living cells[END_REF]:

D R A F T

∆Pc = 2τ0( 1 Rp - 1 R cell ) [1]
where R cell is the cell radius. In the case of the network rectangular microchannel, the channel height h, which is smaller than the cell diameter, defines the vertical curvature at the front and rear of the cell (Fig. 4A). ∆Pc thus writes as:

∆Pc = τ0( 4 w - 1 Rrear ) [2]
where Rrear is the horizontal radius at the rear of the cell and w is the microchannel width. The median values of τ0 of untreated cells are 300 and 196 pN/µm for the microfluidic and the pipette methods, respectively, and show no significant effect of temperature (Fig. 4D). They are in the range of those previously reported for THP1 cells [START_REF] Richelme | Aspiration of thp1 into a micropipette. mechanical deformation of monocytic thp-1 cells: occurrence of two sequential phases with differential sensitivity to metabolic inhibitors[END_REF]. In contrast, latcells entered in the microchannels for applied pressure drops as low as 0.3 mbar (i.e. τ0 < 8.5 pN/µm), confirming that Latrunculin A effectively destroyed the actin cortex.

From the modified Maxwell model [START_REF] Guevorkian | Aspiration of biological viscoelastic drops[END_REF], the mechanical constitutive law of the cell is:

(1 + η1 η2 ) dσ dt + k1 η2 σ = k1 d dt + η1 d 2 dt 2 [3]
where the effective applied stress σ for a flowing cell is determined by the pressure balance between ∆Pc and the pressure drop in the horizontal channel ∆P h :

σ = ∆P h -∆Pc = ∆P h -τ0( 4 w - 1 Rrear ) [4]
In specific cases, the second term of the right-hand side of Eq. 4 is negligible, making σ ≈ ∆P h constant. It happens for cells small enough to little deform because R cell ≈ Rrear ≈ w/4, and lat-cells for which τ0 is very small. Integration of Eq. 3 in these two cases yielded the parameters η1, k1, and T1 = η1/k1 by fitting (t) curves in the first horizontal microchannel (see the example of a lat-cell in Fig. S6). In the other cases, only T1 could be estimated from the transition from elastic to viscous regime on the temporal variation of the elongation ratio (see arrow in Fig. 4B). As shown in Fig. 4E, no significant difference was found between untreated, lat-and jasp-cells. The Voigt elasticity k1 and viscosity η1 are of the order of 10 3 Pa and 10 Pa.s, respectively. The raising time for the elastic deformation T1 is very short, of the order of 10 to 20 ms. The viscosity η2 was determined on lat-cells both from the entrance in the microchannel network at 37 • C and from micropipette aspiration, together with untreated cells, at 23 • C (Fig. 4F).

The median viscosity at 23 • C is 203 Pa.s and 156 Pa.s for untreated and lat-cells respectively, in excellent agreement with the values of 185 and 135 Pa.s reported for monocytes [START_REF] Richelme | Aspiration of thp1 into a micropipette. mechanical deformation of monocytic thp-1 cells: occurrence of two sequential phases with differential sensitivity to metabolic inhibitors[END_REF] and granulocytes [START_REF] Evans | Apparent viscosity and cortical tension of blood granulocytes determined by micropipet aspiration[END_REF]. Also in agreement with this latter work, we observed a strong temperature dependence of the viscosity as η2 for lat-cells decreases to 45 Pa.s at 37 • C. This mechanical analysis shows that cells can undergo i) fast small viscoelastic deformations with a characteristic time T1 ≈ 20 ms, and ii) slow viscous deformations with a characteristic flow time Tη 2 ≈ η 2 σ , which is typically of the order of 100 ms for values of = 10%, η2 ≈ 40 Pa.s and σ ≈ 40 Pa. The cell cortical tension, which resists to deformation, is the main parameter affected by actin-drug treatment and is thus considered to be responsible for the different dynamics of lat-, untreated and jasp-cells.

Role of cell cortical tension in the bifurcation.

As known for long [START_REF] Rallison | The deformation of small viscous drops and bubbles in shear flows[END_REF], the dynamics of droplets in linear flows is determined by the nonlinear coupling of their restoring internal tension forces and the deforming hydrodynamic forces. It directly depends on the capillary number Ca = ηout v f luid /τ0, which represents the relative effect of viscous drag forces versus surface tension, where ηout and v f luid are the viscosity and velocity of the external fluid, respectively. The capillary number of THP1 cells increases from jasp-to untreated to lat-cells. At low Ca, for jasp-and untreated cells, τ0 reduces the ability of the viscous force to deform the cells. Part of the work done by viscous force is converted into surface energy, which slows down the cell (Fig. 4G) both in the entry of the network and in bifurcations. Indeed, numerical simulations performed on droplets have clearly reported that the residence time of a droplet in a bifurcation is longer when the value of Ca is low until an instability drives the droplet into a daughter channel [START_REF] Carlson | Droplet dynamics in a bifurcating channel[END_REF][START_REF] Wang | Viscoelastic droplet dynamics in a y-shaped capillary channel[END_REF]. Interestingly, above a critical value of Ca, a droplet is expected to split up in extensional flows [START_REF] Rallison | The deformation of small viscous drops and bubbles in shear flows[END_REF] and in bifurcations [START_REF] Carlson | Droplet dynamics in a bifurcating channel[END_REF][START_REF] Wang | Viscoelastic droplet dynamics in a y-shaped capillary channel[END_REF], the two parts flowing separately in the two daughter channels after the bifurcation, as observed in our case for some very large lat-cells (Movie S7). A low cortical tension corresponding to a high Ca therefore increases the traffic speed of the cells but it must remain sufficiently high to preserve the integrity of the cell as it passes the bifurcations.

Towards the periodic steady-state.

A key to the steady-state regime is that, at the exit of horizontal channels, the cells immediately turn alternatively right and left to engage their front part in the next diagonal channel. This is illustrated in Fig. S5 on an untreated cell in mesh 10 (t = 44 ms) which earlier crashed on the next pillar in the transient regime (mesh 5, t = 51 ms). This is also shown in Fig. 5 where the cell spontaneously turns left at the exit of the network while the flow is symmetrical. This is the result of a higher friction of the cell along one of the two side walls of the horizontal channel, i.e. a thinner lubrication layer along one wall, creating a torque that directs the cell trajectory. It is shown in Fig. S5 on a lat-cell (t = 369 ms) and Fig. 5 on an untreated cell where the relative positions of two intracellular defects alternate in successive horizontal channels. This behavior is associated with an asymmetric deformation of the cell achieved in the non-symmetrical part of the mesh, at the bifurcation exit and in the diagonal channel, hereinafter referred to as the diag position (see circle in Fig. S1B). Since the cell is viscous, the deformation needs time to be reached. Therefore, the residence time of the cell at the diag position T * must be long enough for the deformation to develop and/or for the fluid in the lubrication layer to be partially drained off. Furthermore, the cell transit time through the next symmetric horizontal channel T h must be short enough so that the asymmetry achieved at the diag position does not fully relax until the cell reaches the next bifurcation. As illustrated in Figs. S3C-D, T * and T h are both longer than the Kelvin-Voigt time T1, indicating that the cell has time to undergo a viscoelastic deformation at the diag position but also to relax this deformation in the horizontal channel. On the other hand, T * ≈ Tη 2 > T h , therefore allowing a deformation of about 10% at the diag position, which cannot fully relax in the horizontal channel. Thus, the η2 viscosity allows the cell to maintain an asymmetric deformation at the horizontal channel's exit, making it turn directly towards the downstream channels. The optimal steady-state cell deformation is obtained progressively during the transient regime by alternating rapid and slow cell transit in successive meshes (Fig. S3A-B). Indeed, a cell blocked for a very long time in the diag position (see the untreated cell at t = 117 ms, Fig. 3B) reaches a very high degree of asymmetry that will allow it to engage directly and efficiently at the next diag position. However, the cell will then not have time to develop the degree of mirror asymmetry sufficient to engage effectively in the next diag position and will therefore be slowed down there and may then develop a more pronounced asymmetry again. This alternation of slow and fast passage gradually converges towards an optimal residence time and deformation in the steady-state.

Steady-state. Dynamics of cell transport.

In the steady-state regime, a cell can be considered as progressing from mesh to mesh with an average mesh velocity v cell . This velocity results from the balance between the driving hydrodynamic force due to the pressure difference applied in a mesh, F d ≈ hw∆P mesh and the friction force of the cell membrane on the channel walls. This latter force per unit of membrane area is the shear stress exerted by the lubrication layer between the wall and the cell membrane F f ≈ -S f kvis v cell , where S f is the surface area of the cell close to the walls (estimated by considering that the cell has an almost parallelepipedic shape of length L, height h and width w: S f = 2L(h + w)) and kvis is the viscous coefficient due to the shear force in the lubrication layer between the cell membrane and the channel wall (kvis = ηout/δ where δ is the thickness of the lubrication layer). In the steady-state regime the sum of forces is zero:

hw S f
= kvis v cell ∆P mesh [START_REF] Sa Gebb | Sites of leukocyte sequestration in the pulmonary microcirculation[END_REF] Fig. 6 displays hw/S f versus v cell /∆P mesh for cells, treated or not, transiting through channels of width w < 11 µm. A remarkable linear dependence is observed, validating the approach. The different slopes disclose that the friction increases from lat-to untreated to jasp-cells, resulting in a faster movement of lat-cells. Taking ηout = 10 -3 Pa.s, the values of δ deduced from the experimental determination of kvis are within a reasonable range of a few hundreds of nanometers (see Table 1). Surprisingly, the curves converge to a non-zero value when v cell /∆P mesh tends towards 0, equivalent to a solid friction k sol that remains to be understood. The lubrication theory states that the thickness of the lubrication layer δ is inversely proportional to the tension D R A F T Fig. 6. Experimental determination of kvis and k sol based on Eq. 5. kvis is the curve slope; k sol is the value extrapolated at v cell /∆P mesh = 0. gradient in the membrane [START_REF] Fp Bretherton | The motion of long bubbles in tubes[END_REF]:

τF ≈ ηoutv cell ( δ R cell c0 ) 3 2 [6]
where τF is the tension at the cell front and c0 is a numerical pre-factor that differs only very slightly in the literature (2.123 in (32) and 2.05 in [START_REF] Bruinsma | Rheology and shape transitions of vesicles under capillary flow[END_REF]). Values of τF calculated from Eq. 6 with v cell ≈ 1 mm.s -1 (Table 1) are in excellent agreement with those of τ0 measured in microchannel and micropipette experiments (Fig. 4D). These similar cortical tension values, deduced independently, strongly support our approach and confirms that the friction between the channel walls and the cell governs its speed. Note that the tension estimated from the lubrication layer is that of the cell front, higher than that of the rear. Also note that the tension of jasp-cells is twice that of untreated cells, which explains their difficulty in reaching the steady-state regime.

Discussion and conclusion

The circulation of WBCs in the network of pulmonary capillaries is crucial for immune surveillance. Their slowed circulation leads to an increase in their concentration in the pulmonary bed, which is supposed to allow rapid release and effective response if the immune system is activated. It is known that WBC deformability is an important factor in their proper circulation. For example, neutrophils' multilobed nuclei allow them to better squeeze into the narrowest capillaries. Here, we showed that the very fine combination of the mechanical properties of monocytes allows their rapid transport without blocking in bifurcations. First, their moderate cortical tension allows them to deform in the network. Second, thanks to their fast viscoelastic mechanical response, they adapt their shapes quickly and reversibly to local geometric variations in the capillary network. Third, their slow viscous response allows their shape to evolve from mesh to mesh and finally develop a periodic asymmetry optimized to have a smooth motion without arrests. In comparison, giant vesicles whose inner fluid is 10,000 times less viscous than monocytes immediately adapt their shape to external mechanical stresses. They thus exit the horizontal channel with a symmetrical shape, are blocked against the pillar in the middle of the bifurcation and present a motion by hops. This work also shows that cortical tension is a key parameter. A low cortical tension allows the cell not to hang in the bifurcations and, by reducing friction between the cell membrane and the capillary wall, to increase its speed in the network. Nevertheless, the cortical tension must be high enough to preserve the integrity of the cell by preventing vesiculation and cell rupture in the bifurcations. Finally, it is worth noting that, physiologically, WBCs circulate in a highly concentrated suspension of red blood cells (see Fig. S7). Could the effect of red blood cells on WBC traffic just be equivalent to an increase in the effective viscosity of the suspending fluid, as often considered in hemorheology?

We hope that future theoretical and experimental works will answer these questions, stimulated by the present study.

Materials and Methods

More details are available in SI. The pressure between the inlet and the outlet is controlled by a flow controller (MFCS-8, Fluigent). The cells are observed at 37 • C in brightfield using videomicroscopy. Note that, in comparison, micropipette experiments are performed at room temperature. Video analysis. Video images are analyzed using home-made routines in Matlab (Mathworks) to automatically track the cell contour and derive its projected area, perimeter, length and position.

D R A F T

Fig. 1 .

 1 Fig. 1. Typical behavior of THP1 cells larger than the microchannels: trajectories (top images) and X-position along the network versus time (bottom plots). Red open circles correspond to the cell positions when they are in the center of the horizontal channels. A) A 15-µm cell entering a 10.6-µm channel network moves randomly between the pillars for the first 8 meshes before alternating between two channel rows, ∆P = 40 mbar. The transit time through a mesh is still irregular (with a long stop at the exit of the 13th horizontal channel) before steady-state is reached at mesh 14 (shown by the yellow line). This cell, which stay longer in the transient regime than other cells, was chosen to highlight the initial random motion. B) A 15-µm cell flowing in steady-state through the last 9-µm wide channels of a network, ∆P = 20 mbar.

Fig. 2 .

 2 Fig. 2. Evolution of cell shape and velocity of a 13.2-µm cell through the first 8 meshes of an 8.6-µm channel network, ∆P = 40 mbar. Reduced volume ν (middle) and instantaneous velocity v (bottom) as function of the cell position in the mesh. The lines are derived from sliding averages over 11 points. Before it enters the network, the cell reduced volume is 0.79 (dashed grey line). Top: Images of the cell through the first mesh, at min and max values of ν, and min value of v. Scale bar: 15 µm.

Fig. 3 .

 3 Fig.3. A) THP1 trajectories in the first 10 meshes of the network, each color corresponds to one cell. Most untreated cells display a random trajectory followed by a periodic trajectory after a few meshes, some cells still move randomly after 10 meshes (top); lat-cells immediately display a periodic trajectory (middle); jasp-cells never display periodic trajectory within 10 meshes. B) Selected images from timelapses of lat-, jasp-, and untreated cells (more details in Fig.S5). Scale bar: 15 µm.

Fig. 4 .

 4 Fig. 4. THP1 mechanical properties. A) Timelapse of a 15.4-µm cell (untreated) entering the first horizontal channel of an 8.6-µm network. The total cell length L is measured from the tracked cell contour (in red). The cell front and rear are approximated with circles (in green and blue, respectively). The front radius R f ront is half the channel width w. Scale bar: 15 µm. B) Elongation ratio of the cell in (A) while it enters the channel. The arrow highlights the transition between the two deformation regimes. C) Rheological model of the cell as a viscoelastic liquid combining a Kelvin-Voigt solid (η1, k1) in series with a dashpot η2, and with a cortical tension τ0. D) τ0 measured from micropipette and microchannels experiments, at 23 • C and 37 • C, respectively. E) Viscoelastic parameters k1 and η1, and raising time T1 = η1/k1 of the elastic deformation. F) Viscosity η2 measured from micropipette and microchannels experiments at 23 • C and 37 • C, respectively. Cells were treated or not with Latrunculin A. **: p < 10 -3 , ns: not significant. G) Entry time into the channel Te as function of normalized to pressure drop, under drug treatment or not.

Fig. 5 .

 5 Fig. 5. Asymmetric motion in the channels. Superimposed images of a cell passing through the network. Red lines drawn between two bright dots located in the cell rear (zoomed in the insets) highlight the shear effect. Scale bar: 15 µm.

  Cells and reagents. THP1 cells (TIB-202, ATCC) are cultured at 37 • C and under 5% CO 2 , in RPMI medium supplemented with 10%vol FBS, 1% GlutaMax, Penicilline/Streptomycine and 25 mM Hepes (RPMI 1640, HEPES, GlutaMAX 100X, Pen/strep, Gibco; FBS, Lonza). During experiments, 1% BSA (Sigma-Aldrich) is added. For actin cortex alteration cells are treated 30 min at 37 • C with 3 µg/mL Latrunculin A (Sigma-Aldrich) or Jasplakinolide (Calbiochem) before injection in the microfluidic network. Microfluidic device and experiments. The microchannel network is made of PDMS (Sylgard, Dow Corning) using standard technique of microfabrication. Microfluidic experiments are performed on an inverted microscope (IX71, Olympus, x20) equipped with Photon-focus and high-speed Photron cameras up to 1000 fps.

Table 1 . Viscous friction and lubrication thickness of flowing cells

 1

	Cells	kvis(mP a.s.m -1 )	δ(nm)	τ F (N/m)
	Lat	2.41 ± 0.09	415 ± 162	1.96 10 -6
	Untreated	4.00 ± 1.18	250 ± 74	4.19 10 -4
	Jasp	7.59 ± 2.29	132 ± 40	1.09 10 -3

To whom correspondence should be addressed. E-mail: annie.viallat@univ-amu.fr 1

ACKNOWLEDGMENTS. This work was supported by the French

National Research Agency (ANR-09-BLAN-ChipCellTrap). We warmly thank F. Beyer, A. Peignier and K. Cosentino for experiments, and P. Tabeling, M. Reyssat, C. Vergne, F. Monti, J. Magnaudet, M. Leroux, for very fruitful discussions.


Vasileios Kotsiou

Member, IEEE, Periklis Chatzimisios, Senior Member, IEEE Georgios Z Papadopoulos
email: georgios.papadopoulos@imt-atlantique.fr

Senior Member, IEEE Fabrice Theoleyre

P Chatzimisios
email: pchatzimisios@ihu.gr

LDSF: Low-latency Distributed Scheduling Function for Industrial Internet of Things

Keywords: Industrial Internet of Things, Distributed Scheduling, End-To-End Delay, High Reliability

The Industrial Internet of Things (IIoT) is expected to be a key enabler for the Industry 4.0. However, networked control automation often requires high reliability and a bounded latency to react properly. Thus, modern wireless protocols for industrial networks, such as IEEE 802.15.4-2015 Time Slotted Channel Hopping (TSCH), rely on a strict schedule of the transmissions to avoid collisions and to make the end-to-end traffic deterministic. Unfortunately, guaranteeing a bounded endto-end latency is particularly challenging since transmissions have to be temporally chained. Even worse, potential degradation of the link quality may result in reconstructing the whole TSCH schedule along the path. In this article, we propose the Low-latency Distributed Scheduling Function (LDSF) that relies on the organization of the slotframe in smaller parts, called blocks. Each transmitter selects the right set of blocks, depending on its hop distance from the border router, so that retransmission opportunities are automatically scheduled. To save energy, a node can still turn off its radio as soon as its packet is correctly acknowledged. Our mathematical analysis as well as our simulation evaluation show the efficiency of the proposed LDSF algorithm compared to three state-of-the-art scheduling functions, the Minimal Scheduling Function (MSF), Low Latency Scheduling Function (LLSF) and Stratum.

I. INTRODUCTION

The Industrial Internet of Things (IIoT) consists of a large collection of wireless sensors and actuators for various industrial applications. Typically, networked control systems comprise of sensors, actuators and a controller [START_REF] Longo | Optimal and robust scheduling for networked control systems[END_REF]. The sensors send their measurements regularly to a controller. According to this feedback, the controller may trigger a reaction by activating some actuators. This control loop has to react in real-time.

Industry 4.0 is expected to rely heavily on the IIoT to make the manufacturing process reconfigurable [START_REF] Wan | Software-Defined Industrial Internet of Things in the Context of Industry 4.0[END_REF]. For instance, industrial robots tend to exploit wireless communications since cables are prone to breakage after a few thousands of flexions [START_REF] Papadopoulos | Raw use cases[END_REF]. The network topology can also be changed easily to reconfigure the production lines [START_REF] Wan | Software-Defined Industrial Internet of Things in the Context of Industry 4.0[END_REF]. Most of the industrial applications require a highly reliable communication network, with a bounded end-to-end latency to react appropriately.

Typically, big data requires to collect a large amount of measurements to create an intelligent manufacturing system [START_REF] Xu | Industrial Big Data Analysis in Smart Factory: Current Status and Research Strategies[END_REF]. For instance, predictive maintenance helps to replace hardware before a failure/breakdown targeting to decrease the shutdown time [START_REF] Yu | A Global Manufacturing Big Data Ecosystem for Fault Detection in Predictive Maintenance[END_REF]. We here focus on the end-devices of the Industrial Internet of Things, that use wireless transmissions to send/receive their packets. To reduce the deployment cost, some of the devices may forward the packets from others in order to reach a gateway with a wired connection. The devices are battery-operated and need to turn off regularly their radio chipset to save energy. Thus, a transmitter must know when the receiver will schedule its next wake-up. In the rest of this article, we will use the term node to designate all these devices, that generate and forward data packets.

To cope with these constraints, deterministic Medium Access Control (MAC) protocols have been proposed in the literature. These proposals rely on a strict schedule of the transmissions: only non-interfering transmissions are triggered at the same time to alleviate collisions. Moreover, slow channel hopping strategies help to increase network capacity and to combat external interference. IEEE 802.15.4-TSCH typically adopts these strategies [START_REF]IEEE Standard for Low-Rate Wireless Networks[END_REF]. A scheduling matrix is composed of cells, defined by a timeslot (when to transmit the packet) and channel offset (which physical channel to use). Thus, the scheduling algorithm has to allocate a certain number of cells to each transmitter. Since a cell is designed to contain at most one packet transmission and its acknowledgment, the number of cells represents directly a number of transmission opportunities. IEEE 802.15.4-TSCH supports both centralized and distributed scheduling algorithms [START_REF] Hermeto | Scheduling for IEEE802.15.4-TSCH and Slow Channel Hopping MAC in Low Power Industrial Wireless Networks[END_REF], denoted as Scheduling Functions.

Since we rely on a wireless infrastructure, the receiver may not always be able to decode a packet due to potential external interference or multi-path fading. Thus, the transmitter has to retransmit the corresponding packet through another cell. Consequently, the scheduling function needs to allocate several cells for each hop of the path, inversely proportional to its link reliability metric [START_REF] Theoleyre | Experimental Validation of a Distributed Self-Configured 6TiSCH with Traffic Isolation in Low Power Lossy Networks[END_REF].

Guaranteeing a maximum end-to-end delay requires to chain the timeslots along the path: a node has to receive the packet before being able to forward it. However, this sequential scheduling has a negative impact on the end-to-end delay: a packet can be forwarded only after all the retransmission cells. In addition to this, a local change of the link quality may imply a whole schedule reconfiguration along the path.

In this article, we propose a Low-latency Distributed Scheduling Function (LDSF) tailored to minimize the latency, while providing high reliability. We address the problem of designing a scheduling algorithm that provides low end-to-end delay even in the presence of retransmissions. LDSF is able to schedule a large number of retransmissions to handle the worst-case situation and to minimize the buffering delay where the over-provisioned cells are chained. The contributions of this article are as follows:

1) We design a novel organization of the slotframes, divided into repetitive short blocks. Chaining the blocks reduces drastically the overall end-to-end delay. Moreover, a transmission opportunity is automatically reserved in consecutive blocks to deal with retransmissions; 2) We define the concepts of primary and ghost cells to save energy. While the primary cell corresponds to the earliest expected reception time, ghost cells are automatically reserved to deal with retransmissions, while limiting the impact on the delay; 3) We provide a mathematical analysis of the average endto-end latency of the state-of-the-art scheduling algorithms defined to minimize latency, including our LDSF algorithm; 4) Simulations help us to investigate more complex scenarios and highlight the practical interest of our proposed scheduling function. The rest of the article is organized as follows. Section II provides the technical background and overviews the related work on slow channel hopping and specifically scheduling. Section III details the impact of the reliability on the delay when scheduling the transmissions. Section IV describes the proposed Low-latency Distributed Scheduling Function to organize properly the transmissions targeting to reduce the end-to-end delay. Section V provides a mathematical analysis of the delay provided by different state-of-the-art solutions, while Section VI presents a performance evaluation based on simulations to assess the performance. Finally, Section VII concludes this article and provides future perspectives.

II. TECHINICAL BACKGROUND AND RELATED WORK

In this section, we will detail the default operations of the IEEE 802.15.4-TSCH and 6TiSCH standards, and present the existing scheduling algorithms that focus on the end-to-end delay minimization.

A. Industrial Stack 1) IEEE 802.15.4-TSCH: Relies on channel hopping to combat external interference and signal fading [START_REF]IEEE Standard for Low-Rate Wireless Networks[END_REF]. The protocol relies on a classical Automatic Repeat reQuest (ARQ) approach, where a packet is retransmitted if no acknowledgement is received.

The slotframe contains a fixed number of timeslots, during which a frame and its acknowledgment are transmitted. Each timeslot is labelled with an Absolute Sequence Number (ASN) which serves as a global clock, counting the number of timeslots since the network bootstrapped. Based on a schedule, a node can decide its role at the beginning of each timeslot. Typically, a node can transmit if the cell is in Transmitting mode in the schedule, and stays awake to possibly receive a packet if the cell is in Receiving mode. If the cell is allocated to the node, the physical frequency to use for transmission/reception is derived from the ASN of the timeslot and the channel offset, following a pseudo-random approach.

The standard supports two medium access techniques:

• Shared cells are contention-based cells that can be used by multiple possibly interfering transmitters. A node dequeues a packet and transmits it immediately in the next shared cell. If an acknowledgement was required and was not received, the transmitter assumes a collision occurred and selects a random backoff, skipping the corresponding number of shared cells; • Dedicated cells are contention-free cells and are allocated carefully to avoid collisions. Thus, a transmitter does not implement any random access during these cells. However, a packet can be retransmitted if no acknowledgement is received, due to a bad radio link quality or external interference. Let us consider the topology illustrated in Figure 1. One shared cell is placed at the beginning of the slotframe for best-effort control traffic and broadcast packets. Then, each node has a dedicated cell to send packets to its neighbors.

2) 6TiSCH: The IETF Working Group has defined a stack of protocols to operate IPv6 on top of the IEEE 802.15.4-TSCH standard. In particular, the standard makes a distinction between how to negotiate the cells, with the 6P protocol [START_REF] Wang | 6TiSCH Operation Sublayer (6top) Protocol (6P)[END_REF], and the Scheduling Function (SF) that represents the scheduling algorithm. The Scheduling Function defines a method to compute the number of cells, and which ones to pick in the scheduling matrix for each link. Then, a 6P transaction is engaged: a two-way handshake takes place over the shared cells. More specifically, the sender transmits a list of available cells to the receiver with a 6P request in unicast. Consequently, the receiver verifies if all (or part of) these cells are available in its schedule and sends a 6P reply accordingly.

B. Scheduling Approaches

1) Centralized scheduling algorithms: Determine the cells to be allocated to each link in order to respect end-to-end delay and reliability constraints. TASA [START_REF] Palattella | On Optimal Scheduling in Duty-Cycled Industrial IoT Applications Using IEEE802.15.4e TSCH[END_REF] applies graph theoretical tools to derive a compact schedule, and thus to increase the network capacity. Additional cells can be reserved in the schedule to deal with packet losses due to unreliable links [START_REF] Gaillard | High-reliability scheduling in deterministic wireless multi-hop networks[END_REF]. SchedEx reinforces the schedule by allocating more transmission opportunities to the different flows while still respecting the end-to-end delay constraints [START_REF] Dobslaw | End-to-End Reliability-Aware Scheduling for Wireless Sensor Networks[END_REF]. However, centralized algorithms need a precise view of the network conditions, and generate a large overhead when the schedule has to be updated.

2) Distributed scheduling algorithms: These algorithms are rather traffic adaptive. Typically, a hysteresis function decides how many cells to reserve, (de)allocating some of them to maintain the number of cells between two bounds [START_REF] Chang | CCR: Costaware cell relocation in 6TiSCH networks[END_REF]. Colliding cells are detected autonomously and moved in the slotframe. To deal with unreliable links, additional cells have to be provisioned for retransmissions: the number of cells should be at least equal to the inverse of the packet delivery ratio of the link, multiplied by the number of packets in the queue [START_REF] Theoleyre | Experimental Validation of a Distributed Self-Configured 6TiSCH with Traffic Isolation in Low Power Lossy Networks[END_REF]. PID based scheduling uses the well-known proportional, integral, and derivative algorithm to decide how many cells to reserve when a burst arrives [START_REF] Domingo-Prieto | Distributed PID-Based Scheduling for 6TiSCH Networks[END_REF]. Inversely, cells are maintained in the schedule to possibly retransmit the packets if required in the future. While these solutions decide how many cells to use, the scheduling algorithm has also to select the right ones to reduce the end-to-end delay.

Minimizing the end-to-end delay is also of primary importance since many industrial applications rely on deadline constraints. Stratum scheduling [START_REF] Hosni | Localized scheduling for endto-end delay constrained Low Power Lossy networks with 6TiSCH[END_REF] divides the network into stratums, regrouping the nodes by their hop distance from the border router. By picking the cells in the corresponding stratum, the system guarantees that the packet is delivered before the end of the slotframe, even if the packet is retransmitted. Hosni et al. [START_REF] Hosni | Self-healing distributed scheduling for endto-end delay optimization in multihop wireless networks with 6TiSCh[END_REF] derive the right size of each stratum to minimize the collision probability. However, the packet is guaranteed to be delivered at the end of the slotframe, which may correspond to a very long delay. Furthermore, stratum scheduling does not allow frequency re-use.

ReSF [START_REF] Daneels | ReSF: Recurrent Low-Latency Scheduling in IEEE 802.15.4e TSCH networks[END_REF] targets recurrent traffic with strict latency requirements. Some cells are pre-reserved in the schedule for these sporadic flows, allocating several cells per hop to cope with retransmissions. The cells are chained along the path to minimize the end-to-end delay. The Low Latency Scheduling Function (LLSF) [START_REF] Chang | LLSF: Low Latency Scheduling Function for 6TiSCH Networks[END_REF] attempts to move the transmitting and receiving cells closer in the schedule to reduce the buffering delay. However, a change in the link quality may result in changing the whole schedule in the path.

Anycast helps to allocate the same cell to several receivers: a transmission is successful if any of the receivers is able to decode it. If we select the right number of parents, we can limit the impact on the energy consumption [START_REF] Hosni | Adaptive k-cast Scheduling for High-Reliability and Low-Latency in IEEE802.15.4-TSCH[END_REF]. Sending several copies of the same packet through different paths may help to improve the reliability while reducing the jitter [START_REF] Koutsiamanis | From Best Effort to Deterministic Packet Delivery for Wireless Industrial IoT Networks[END_REF]. These techniques do not focus on the end-to-end delay optimization.

Handling unreliable links is of prime importance for lowpower and lossy networks. We need to propose a scheduling algorithm able to provide a low delay, even if a packet has to be retransmitted along the path to the border router. We will detail hereafter the challenges to tackle to reach this objective.

III. PROBLEM FORMULATION AND OBJECTIVES

Centralized scheduling algorithms require the whole radio and interference topology knowledge, while they are not adaptive to changes. Thus, in this article, we rather propose a Node's lifetime for the node i distributed Scheduling Function to optimize both the following performance metrics: End-to-end delay: The packet has to be delivered to the border router before a certain deadline constraint; Reliability: Even if links are unreliable, the system has to schedule retransmissions to guarantee a minimum endto-end Packet Delivery Ratio (PDR). Guaranteeing end-to-end delay is actually very challenging, since we have to also consider queuing delays. Let us consider Figure 1. Typically the links CD and DE are scheduled in two consecutive cells, and the delay is minimal. On the contrary, the links BE and AB are scheduled inversely and the packet has to be enqueued in B until the next slotframe. The slotframe may be very long since we can handle very infrequent transmissions. The slotframe duration should typically be equal to the inter-packet time. With heterogeneous periods, the slotframe length should be equal to the least common multiplier of all the packet's periods.

A. Providing High-Reliability

Unreliable links require retransmissions. However, most of the algorithms provision cells for the average case. Typically, most scheduling algorithms allocate 2 cells if the PDR is equal to 50%. If a packet needs more retransmissions, they will be scheduled in the next slotframe, which may be very long.

Let us assume that a link provides a PDR of P ll (link), i.e., ratio between the number of packets acknowledged by the receiver with respect to the total amount of sent packets. We assume here that k different cells are allocated, and thus, the transmitter can transmit at most k times this frame. The packet will be delivered from the transmitter A and correctly acknowledged by the receiver B within the current slotframe with the following probability: We assume here that all cells provide the same Packet Error Rate (PER) for a given link. Indeed, the frequency hopping scheme helps to mitigate external interference, so that the PER is the same for any cell [START_REF] Chang | CCR: Costaware cell relocation in 6TiSCH networks[END_REF].

P net (A, B, k) = i∈[1,k] (1 -P ll (A, B)) i-1 * P ll (A, B) (1)
Figure 2 illustrates a numerical analysis of the network reliability (P net ()) achieved, depending on the link quality of the link (P ll (A, B)) and the number of cells provisioned in the schedule for the (re)transmissions. Increasing the number of cells allows the transmitter to retransmit the packet. However, we need a very large number of cells to achieve a very high reliability. For a Packet Delivery Ratio of the link (P ll ()) of 50%, we need 7 cells to achieve a 99% delivery to the next hop, after the retransmissions.

B. Delay Constraint with Dynamic Scheduling

Some scheduling algorithms reserve a range of consecutive cells for retransmissions to optimize the end-to-end delay [START_REF] Gaillard | Kausa: KPI-aware Scheduling Algorithm for Multi-flow in Multi-hop IoT Networks[END_REF]. The number of cells in this range has to be sufficient to handle the worst case situation, with possibly a very large number of cells (Eq. 1). Thus, the delay is proportional to the worst case, since it corresponds to the sum of the ranges along the path. Furthermore, negotiating cells is expensive since 6P control packets use shared cells, which are prone to collisions and, thus, increasing the convergence delay [START_REF] Theoleyre | Experimental Validation of a Distributed Self-Configured 6TiSCH with Traffic Isolation in Low Power Lossy Networks[END_REF].

Even worse, such approach is inaccurate for dynamic network conditions. Indeed, a node may detect a change in the PDR, and has in that case to increase the number of cells with its parent. Unfortunately, the cells are chained along the path.

Thus, inserting a novel cell requires to re-schedule all the cells until the border router. This renegotiation is time-consuming, and implies packet drops before the re-convergence.

Let us now consider the schedule illustrated in Figure 3. The first two hops provide a reliability of 80% and 3 cells have to be provisioned to provide a per link reliability of 99%. However, inserting a novel cell for the link (AB) is expensive: we have to move all the subsequent cells. Otherwise, we have to place the retransmission cell after the 10 th cell, requiring to reserve novel cells for the rest of the path.

Stratum [START_REF] Hosni | Localized scheduling for endto-end delay constrained Low Power Lossy networks with 6TiSCH[END_REF] tackles this reconfiguration problem by dividing the whole slotframe into stratums (one stratum per hop). However, this organization reduces the network capacity, and the end-to-end delay can only be the slotframe length.

C. Objectives of LDSF

Our proposed Low Latency Distributed Scheduling Function (LDSF) relies on the following features:

• Slotframe organization (in blocks): We reduce the endto-end delay by chaining appropriately the blocks. All nodes that are at an even number of hops (respectively odd) from the border router are scheduled during the even blocks (respectively odd). This way, the packet progresses on average by one hop at the end of each block. • Cell allocation in sequence: A node identifies automatically the expected earliest arrival of a packet to schedule cells to forward the packet. Then, the node reserves a cell in the next block, to minimize the buffering delay. Additionally, it also reserves the same cell every two blocks to cope with retransmissions. This way, a node can proceed to a large number of retransmissions, to handle the worst case. • Energy Saving: The receiver wakes-up only when a packet may be received (earliest time of arrival). We also keep a deterministic behavior, without any false negative.

IV. LOW-LATENCY DISTRIBUTED SCHEDULING FUNCTION

We here consider very long slotframes, where packets are generated infrequently but in a periodic manner. We propose to organize the long slotframe to reduce the end-to-end delay, by using smaller parts, that we call blocks (Fig. 4).

A. Slotframe organization

We here consider sporadic traffic, where each sensor reports periodically its measurements to a border router. Thus, the slotframe length has to be equal to the least common multiplier of all the traffic periods. Consequently, in each long slotframe, shared cells are reserved for control traffic, such as 6P packets. One dedicated cell is also reserved for each node to send its unicast control packets corresponding to routing or synchronization. All data packets are transmitted through dedicated cells, that each pair of nodes has to reserve.

We propose to organize the long slotframe into small blocks that repeat over time. To reduce the end-to-end delay, we have to limit the buffering delay when a packet is retransmitted. By reserving consecutive blocks for retransmissions, the buffering delay is proportional to the block size. Thus, we divide here the slotframe into small blocks with a few timeslots.

A packet is typically received during a block, and forwarded during the next one. Thus, a transmitter selects its block according to its hop count from the border router. More precisely, each block has a block id, that counts the number of blocks since the beginning of the slotframe. We have consequently even blocks (with an even block id), and odd blocks (with an odd block id). A transmitter has to select a block, so that the remainder of the Euclidean divisions of the hop count and of the block id by 2 are equal. More formally, a transmitter can select any cell in the blocks which respect the following property:

HC (mod 2) = B id (mod 2) (2)
where HC denotes the hop count from the transmitter to the border router, and B id the block id.

Let us consider the topology and the LDSF schedule illustrated in Figure 4. The node A is two hops away from the border router, and the packet is assumed to be generated in the timeslot 0. It must select a block with an even block id (2 (mod 2) = 0). In our example, it selects the timeslot 2. The node C is 1 hop away from the border router, and considers only the blocks with an odd block id. It selects the block 1 (the consecutive block), and reserves one cell (here, the timeslot 5) to forward the packets from A. The blocks are daisy-chained, the packet received during the block i being forwarded in the block i + 1.

We make a distinction between the following types of cells:

• Shared cell for control packets in broadcast (Enhanced Beacons, routing advertisements), and control packets in unicast when no dedicated cell has been reserved (i.e., 6P requests/replies); • Primary (dedicated) cell corresponds to the earliest expected reception time of the data packet from the previous hop (or from the application layer); • Ghost (dedicated) cells correspond to the retransmission opportunities, that are automatically used by the transmitter if it did not receive an acknowledgment for its previous transmission. The ghost cells are scheduled in the same timeslot and channel offset as for the corresponding primary cell, but in the subsequent blocks. When a node reserves a primary cell in a block, a fixed number of ghost cells is automatically reserved every two blocks. Thus, we can daisy chain the transmission opportunities along the path: a node is able to receive a packet during a block, and forward it during the subsequent blocks. This way, we maintain a low end-to-end delay.

A link quality degradation means more retransmissions: in classical scheduling algorithms, we would need to reserve additional cells. Here, we pre-reserve a large number of cells, at the very beginning, to cope with the worst link qualities. Thus, the number of ghost cells (for retransmissions) is fixed, whatever the link quality.

Besides, the impact of the retransmissions on the end-to-end delay is limited since the blocks comprise a small number of timeslots. We use the Automatic Repeat reQuest (ARQ) feature of IEEE 802.15.4-TSCH: the transmitter schedules a retransmission in the next ghost cell only if it does not receive an acknowledgement for its previous transmission.

B. Number of Ghost Cells

We have now to compute the number of ghost cells to provision for the retransmissions.

1) Standard case: The delay induced by the retransmissions is cumulative along the path. Thus, we have to cope with the worst case: a packet may be retransmitted at most M axRetries times by each transmitter in the path. The latest time of arrival corresponds to the last RX ghost cell (e.g., C receives the packet from A at the latest during the timeslot 8 in Fig. 4).

We make here a distinction between the source of the flow that generates a data packet, and a transmitter that forwards this packet. The first transmitter in the path corresponds trivially to the source. We can note that the number of ghost cells is proportional to the hop distance from the source. More precisely, a transmitter has to provision (M axRetries * (Hops + 1)) ghost cells for the retransmissions, where Hops denotes the hop distance from the source to the transmitter.

In Figure 4, A is the source (Hops = 0) and provisions one primary cell (timeslot 2) and one ghost cell (timeslot 8). For the node C, it is one hop away from the source (Hops = 1). Thus, C allocates for the flow F 1 one primary cell (timeslot 5) and 2 ghost cells (M axRetries * (Hops + 1)). We can note that the node C can receive the packet through the primary or the ghost cells. Thus, even if it receives a packet during the last ghost cell (timeslot 8), it has still two transmission opportunities (timeslots 11 and 17) for one transmission, and one retransmission.

2) Overlapping case: Some flows may overlap, i.e., one relaying node uses the same ghost cells for two different flows. For instance, flows F 1 and F 2 are both forwarded by the node C, where some ghost cells are in common for both flows. A node can easily detect an overlap when receiving a 6P request: the primary cell corresponds to a ghost cell already reserved for another flow.

Even with this overlap, we must be sure to have enough ghost cells to handle the worst case. Let us consider the two following cases:

Case 1) the node receives a packet from the novel flow (F 2 ) while a packet from the previous flow (F 1 ) is already in the queue. By construction, the first flow F 1 has still enough ghost cells to handle M axRetries retransmissions. At the latest, the packet for the flow F 2 is received while only M axRetries + 1 ghost cells remain in its schedule (primary transmission + retransmissions). Thus, we need to provision M axRetries+1 ghost cells for the novel flow F 2 , after the ghost cells that would have been allocated to the flow F 2 .

Case 2) the node receives a packet from the novel flow (F 2 ) while the packet from the other flow (F 1 ) was not yet received. For the same reason, the node has enough ghost cells for F 2 for M axRetries retransmissions. Thus, we have also to insert in that case M axRetries + 1 additional ghost cells at the end of the range, but they will be used to forward the packet for the flow F 1 .

In conclusion, it is sufficient to provision M axRetries + 1 additional ghost cells when an overlap is detected, whatever the hop distance from the source.

C. Scheduling process

We now detail how the cells are reserved by each pair of nodes. Shared cells are only used for signaling, i.e., sending/receiving the 6P packets to negotiate which dedicated cells to use. A 6P request typically piggybacks a list of possible (dedicated) primary cells. The (dedicated) ghost cells are automatically derived from a primary cell. The receiver sends a 6P reply to the transmitter to validate the reservation. Since no dedicated cell is present in the schedule, the 6P packets use the shared cell.

A novel allocation is triggered when a node receives either a 6P request from the previous hop or directly the packet from the application layer. Thus, we propose the following procedure (see Algorithm. 1):

1) First of all, we need to allocate the receiving cells if we receive a 6P request. We reserve in Receiving mode all the timeslots every 2 blocks (lines 1-7), located after the timeslot specified in the 6P request.

We allocate one primary cell, and (M axRetries * Hops) ghost cells for the retransmission. comes from the fact that the receiver for the 6P request is one hop farther from the source than the transmitter. Please note that the pseudo-random function pseudoRandom() is executed with the same argument by the receiver and the transmitter to derive the same channel offset (lines 3, and 16).

2) We identify the block which is directly located after the block of the first receiving timeslot (line 8). We will schedule the TX cells after this block; 3) We have to allocate the primary cell, and (M axRetries * Hops) ghost cells (line 10); 4) We have then to make a distinction between the two possible cases:

• Overlapping flows: A cell in this block is already assigned (line 11). Thus, the allocation will re-use the same timeslot and channel offset (line 12). We have also to allocate (M axRetries + 1) additional ghost cells to handle the worst queuing delay with the overlapping flow (line 13).

• No-overlap: We select randomly the timeslot and channel offset (line 15-16). 5) When we have determined the number of ghost cells, and the first timeslot to allocate, we can proceed to the allocation (lines 18-21). It is worth noting that with overlapping flows, some TX cells may be reserved by several flows (e.g., timeslot 11 in Fig. 4). Let us illustrate this scheduling algorithm with Figure 4. For the sake of better representation, we assume that the maximum number of retransmissions is equal to one (M axRetries = 1):

• As explained previously, A selects a cell in the block 0. It also reserves automatically one associated ghost cell (M axRetries * 1 hop) in block 2. After the 6P reservation, the primary and ghost cells are reserved for both the transmitter and the receiver. • The earliest time of arrival for the node C corresponds to the timeslot 2. Thus, it reserves a cell in the next block [START_REF] Longo | Optimal and robust scheduling for networked control systems[END_REF]. It also reserves two ghost cells (M axRetries * 2hops), in the blocks 3 and 5; • We can note that C is also forwarding the flow F 2 (from B). Its primary cell for the flow F 2 is already reserved as ghost cell for the flow F 1 : C detects an overlap. Thus, it first reserves one primary cell (timeslot 11) and 2 ghost cells (timeslots 17 and 23) for the flow F 2 .

Because, of the overlap, it has also to allocate additional ghost cells to consider the latest time of arrival (cf. section IV-B2). Thus, it allocates two additional ghost cells (one primary cell + M axRetries) after its last ghost cells. In conclusion, it selects the timeslots 29 and 35 as ghost cells. Since we rely on a distributed random scheduling algorithm, two interfering links may select the same cell. A collision may occur if both transmitters select the same primary cells, or if the ghost and primary cells overlap. For instance, the links AB and CD in Figure 1 may reserve the same timeslot and channel offset. Then, if both nodes A and C send the packet over the same cell, there will be a collision. Since, in our scenarios, the packet generation period is sporadic, and we because consider long slotframes, the probability two or more transmitters to transmit over the same cell is very low. In case of collision, a relocation mechanism [START_REF] Chang | CCR: Costaware cell relocation in 6TiSCH networks[END_REF] will be applied.

D. Energy Savings using Ghost Cells

Reserving ghost cells allows the network to improve the reliability: LDSF can efficiently handle the fast link quality changes since ghost cells are a priori over-provisioned. Concerning the energy efficiency, the transmitter can safely sleep when it has no data packet to transmit. For the receiver side, we have to limit idle listening [START_REF] Vilajosana | A Realistic Energy Consumption Model for TSCH Networks[END_REF], forcing the node to wakeup at the beginning of the timeslot because it is not aware that the transmitter has nothing to transmit.

Under the LDSF algorithm, we configured a fixed number of ghost cells, based on the hop distance from the source, and a constant, whatever the link quality. A receiving node must wake-up at the beginning of each primary cell, to possibly receive a packet. Then, it must also wake-up for all the subsequent ghost cells until a packet has been received and correctly acknowledged. Once, a packet has been received or the last ghost cell is encountered, the receiver can safely save energy until the next primary cell. The receiver has then to forward the packet, and becomes a transmitter. It selects the corresponding cell in the next blocks, and starts to transmit the packet to the next hop.

Let us consider the scenario illustrated in Fig. 4. Let us assume that the node C has been able to decode the packet from the node A in the timeslot 2. It can stop listening to the ghost cell in timeslot 8. However, it will wake-up during the next block (timeslot 5) to forward the packet to the node D.

The primary cell corresponds to the earliest time arrival to optimize the end-to-end delay. Thus, we do not have any false negatives: the receiver is always awake when the transmission takes place, we thus keep the deterministic behavior of IEEE 802.15.4-TSCH.

V. MATHEMATICAL ANALYSIS

Let us analyze next the end-to-end delay, i.e., average time required to deliver a data packet from a node A to the border router S via a path p = {N i } i∈ [1,||p||] , where ||p|| denotes the number of nodes in the path p.

A. Model

To obtain a fair evaluation, we use the same mathematical model to compare each scheduling approach (cf. notation in Table I). We have made the following considerations:

• We consider the worst case delay, when the data packet is enqueued at the beginning of the slotframe; • We assume that the inter packet time is sufficiently large, and the queue for the nodes is empty at the beginning of the slotframe. We apply here the same methodology as in [START_REF] Hosni | Localized scheduling for endto-end delay constrained Low Power Lossy networks with 6TiSCH[END_REF] to compute the average time before a packet is delivered to the border router (i.e., the end-to-end delay).

1) MSF: Since cells are randomly scheduled, we consider a uniform distribution of the cells in the slotframe.

To be generic, we compute the normalized delay in number of timeslots. We have to multiply the normalized delay by the timeslot duration T slot to compute the actual delay. Compared with [START_REF] Hosni | Localized scheduling for endto-end delay constrained Low Power Lossy networks with 6TiSCH[END_REF], we have the following normalized delay:

hop delay (N i , N i+1 ) = SFlength 2 * Cells(N i , N i+1 ) (3)
where SFlength denotes the slotframe length, i.e., number of timeslots.

If the radio link between N i and N i+1 is unreliable, the transmitter needs 1 P ll (Ni,Ni+1) transmissions before the packet is successfully decoded and acknowledged. So using eq. 3:

hop delay (N i , N i+1 ) = SFlength * 1 P ll (Ni,Ni+1) 2 * Cells(N i , N i+1 ) (4)
Finally, the average end-to-end delay to deliver a packet from a source node to the border router along the routing path p = {N i } i∈ [1,||p||] is:

E2E delay (p) = SFlength * ||p||-1 i=1 1/P ll (N i , N i+1 ) 2 * Cells(N i , N i+1 ) (5)
2) Stratum: If we consider the length of each stratum is sufficient to handle retransmissions, the maximum end-to-end delay corresponds to the slotframe length (SFlength). In this case, the packet is delivered successfully before the end of the stratum:

E2E delay (p) = SFlength (6)
If the number of cells is insufficient, the retransmissions may be scheduled during the subsequent slotframe, increasing the end-to-end delay by one slotframe length. We may use the distribution of the packet delivery success according to eq 1 to derive the distribution of this additional delay. It depends also on the length of each stratum, since it upper bounds the number of cells we can allocate. While eq. 6 corresponds to an optimistic case, we consider that the stratum length is correctly sized to handle the worst case situation.

3) LDSF: The cells are pseudo-randomly allocated in a block. Besides, the node N i+1 will forward a packet it received from N i in the just immediately consecutive block (by construction). Thus, the average buffering time is equal to the block length:

hop delay (N i , N i+1 ) = BLength (7)
If the radio link between N i and N i+1 is unreliable, they require 1/P ll (N i , N i+1 )-1 retransmissions. Since consecutive primary/ghost cells are interspaced by 2 * BLength timeslots, we have:

hop delay (N i , N i+1 ) = BLength + BLength 2 P ll (N i , N i+1 ) -1 (8)
Finally, the average end-to-end delay is:

E2E delay (p) = BLength ||p||-1 i=1 2 P ll ((N i , N i+1 ) -1 (9)
4) LLSF: the approach daisy-chains the cells along the path, relying on relocation when they are not contiguous because of retransmission cells. Thus, we will here focus on the steady-state, after the convergence, i.e. we neglect the effect of suboptimal relocations.

The first hop delay is obtained similarly to MSF. Since the nodes select the transmission cells randomly on the first hop, the delay of the first hop is:

hop delay (N 1 , N 2 ) = SFlength * 1 P ll (N1,N2) 2 * Cells(N 1 , N 2 ) (10)
Then, the end-to-end delay is similar to LDSF. Indeed, two cells for the retransmissions are now consecutive instead of being separated by 2 * BLength timeslots. The average endto-end delay is thus:

E2E delay (p) = hop delay (N 1 , N 2 )+ ||p||-2 i=2 2 P ll (N i , N i+1 ) -1 (11)

B. Numerical results

We consider here a scenario, with unreliable radio links (i.e., P ll = 66%). Thus, two cells are allocated for each link (Cells = 2) for possible retransmissions, and the block length is fixed to 5 timeslots (BLength = 5).

Figure 5a illustrates the impact of the slotframe length. The end-to-end delay of both MSF and Stratum is linear with the slotframe length. Indeed, Stratum can only guarantee a delay equal to the slotframe length. As can be observed, the performance of LLSF is affected by the slotframe length due to the selection of the transmission cells randomly at the first hop. On the contrary, the delay of LDSF is independent of the slotframe length: the cells are chained along the path.

Similarly, we can see that the delay increases with the number of hops for MSF (Figure 5b): the TX and RX cells are uniformly distributed, and a node has to buffer the packet for a long time before being able to forward it. On the contrary, Stratum provides a delay independent of the hop length. However, this hop distance cannot exceeds the number of stratums, which is 10 in this case. LDSF achieves to provide a much smaller delay, increasing only slightly for long routes. More precisely, the delay is increased by one block length per hop, i.e., 5 timeslots. Finally, the LLSF achieves low delay too especially when the number of hops increases.

VI. PERFORMANCE EVALUATION

We now assess the performance of our distributed scheduling function in a more realistic environment, where packets can be dropped because of collisions and low link qualities.

A. Simulation Setup

We use here the 6TiSCH Simulator [START_REF] Municio | Simulating 6TiSCH networks[END_REF], a discrete-event simulator written in Python. We consider the following scenarios:

1) Constant Bit Rate (CBR) flows 2) Topologies with a variable number of nodes (by default 39 nodes and one border router) to assess the scalability of our Scheduling Function We generate random topologies, where each node is randomly located in an area of 2000X2000 m 2 . Thus, each node has at least 3 neighbors. The propagation model of 6TiSCH Simulator is based on the Pister-Hack model [START_REF] Le | Energy-aware routing in wireless sensor networks with adaptive energy-slope control[END_REF]. Table II regroups all the values of the different parameters.

We extended the 6TiSCH Simulator with three additional scheduling functions (LDSF, LLSF [START_REF] Chang | LLSF: Low Latency Scheduling Function for 6TiSCH Networks[END_REF] and Stratum [START_REF] Hosni | Localized scheduling for endto-end delay constrained Low Power Lossy networks with 6TiSCH[END_REF]). For the LDSF and Stratum algorithms, each node has to know its hop distance from the border router. We implement in Stratum the opportunistic de-allocation of cells described in [START_REF] Theoleyre | Experimental Validation of a Distributed Self-Configured 6TiSCH with Traffic Isolation in Low Power Lossy Networks[END_REF], where an unused cell is removed after a long timeout to avoid schedule's inconsistencies.

The node's lifetime is Lf i = Battery Capacity ×SF duration Q slotf rame * 3600 * 24 * 365 . Where SF duration is the duration of the slotframe (in seconds), Battery Capacity is the node's battery capacity (in µC), Q slotf rame is the average energy drawn during a slotframe (in µC), and (3600 * 24 * 365) allows to convert seconds into years. To compute the lifetime of each node, we rely on the energy model described in [START_REF] Vilajosana | A Realistic Energy Consumption Model for TSCH Networks[END_REF] that relies on real measurements. The model provides the power of the node in each state: T xDataRxAck: the transmitter sends a packet and waits for an acknowledgement; T xData: the transmitter sends a packet without waiting for an ACK (e.g., a broadcast packet); RxDataT xAck: the receiver decodes a packet and sends an ACK; RxData: the receiver decodes the packet, and switches directly to sleeping mode, without sending an ACK; Idle: the receivers listens to the medium but does not sense anything; Sleep: the nodes turns off its radio chipset. Thus, the simulator [START_REF] Municio | Simulating 6TiSCH networks[END_REF] has just to count the time spent in each state, combined with the values measured in [START_REF] Vilajosana | A Realistic Energy Consumption Model for TSCH Networks[END_REF] and reported also in Table II to compute the lifetime metric. We measure here the network lifetime as the time until the first node dies because it runs out of energy (n-of-n lifetime in [START_REF] Dietrich | On the Lifetime of Wireless Sensor Networks[END_REF]). We assume that the system reaches a steady state, and we just have to identify the node with the largest energy consumption. Finally, the network's lifetime is:

N L = min{Lf i } i∈[1,||N odes||-1] (12)

B. Scheduling Algorithms

We compare the following approaches: MSF: [START_REF] Chang | 6TiSCH Minimal Scheduling Function (MSF)[END_REF] is the default scheduling function of 6TiSCH, where autonomous (pseudo-random) cells are used for control traffic and dedicated cells are used for data packets; Stratum: [START_REF] Hosni | Localized scheduling for endto-end delay constrained Low Power Lossy networks with 6TiSCH[END_REF] divides the slotframe in blocks (i.e., stratums).

Each node selects a block according to its hop distance, so that a packet is delivered in the current slotframe; LLSF: [START_REF] Chang | LLSF: Low Latency Scheduling Function for 6TiSCH Networks[END_REF] aims to reduce the end-to-end latency by allocating receiving and transmitting cells as close as possible in the schedule. LDSF: Our scheduling function described in Section IV.

Stratum uses a slotframe length of 101 timeslots, to be able to provide an end-to-end delay equal to 1010 ms (=101 * 10ms). MSF and LLSF also uses the default slotframe length (101). LDSF uses rather a slotframe length proportional to the maximum flow rate, since it was designed for this purpose. The same cell is used every two blocks for transmissions and retransmissions. Since each block comprises 5 timeslots in LDSF, the transmitter has to wait on average 10ms * 5 timeslots * 2blocks = 100ms.

Our implementation (simulation code, scripts, and raw data) is freely available (https://github.com/vkotsiou/Scheduling for the implementation, and https://doi.org/10.5281/zenodo.3748712 for our dataset).

C. Traffic Rate

We first measure the average end-to-end delay (Fig. 6a). LDSF is very robust to large traffic rates: it keeps on providing a very low delay. Stratum presents a very stable end-to-end delay: packets are delivered at the end of the slotframe exactly (101 timeslots * 10ms). MSF presents the highest end-to-end delay because it does not have any cell allocation strategy to minimize the delay: it picks randomly the cells. LLSF provides also a very stable delay. While the cells are reserved consecutively along the path, the first cell is picked randomly and generates a large buffering delay (half of the slotframe = 505ms).

We can make the same remarks concerning the jitter (Fig. 6b). LDSF provides the lowest jitter performance which is less than 150 ms, even for very high traffic rates. Collisions are accurately handled, and the packets are retransmitted efficiently in the subsequent blocks to minimize the buffering delay. LLSF achieves a larger jitter: the schedule is modified as soon as some retransmission cells have to be inserted. However, this optimization has a cost since the whole schedule has to be modified along the path Stratum provides jitter performance similar to LLSF, corresponding to the length of the last stratum (i.e., block). Indeed, a packet is randomly scheduled in the last stratumto be received by the border router. Since this stratum is typically much larger than the LDSF's block, the jitter is mechanically increased. Finally, MSF provides the worst jitter since retransmission cells can create a cumulative effect along the path, since they can be allocated after the cells of the next hop.

Figure 6c focuses on the reliability. The three schemes are able to guarantee end-to-end reliability above 96% in most cases. Stratum achieves the highest reliability for low traffic rates since the blocks are large to avoid collisions. However, the number of collisions starts to increase for high traffic rates (inter packet time < 10s). LDSF is able to provide an end-toend packet delivery ratio higher than 98%. LLSF provides also a good reliability, except for high-traffic rates: many collisions arise and are particularly challenging to resolve since the cells are contiguous. Moreover, the scheduling process needs to solve the collisions for each cell, while LDSF is more robust since the same cell is pre-reserved also for the retransmissions.

Fig. 6d illustrates the network lifetime. We extrapolate the average energy consumption for the most loaded node to derive the network lifetime. MSF generates a large number of control packets with many (de)allocations, which impact negatively the lifetime. Stratum increases slightly the lifetime, by reducing the renegotiation of cells. LLSF achieves the same characteristics since it uses a short slotframe (101 timeslots) and that shared cells consume energy. Finally, LDSF is very efficient to handle unreliability: ghost cells are automatically reserved after a 6P transactions, minimizing the amount of control traffic. Thus, the lifetime increases for very low traffic conditions.

D. Scalability

We then measure the scalability of these distributed scheduling functions by increasing the number of nodes up to 100. LDSF is very scalable; even a large number of nodes does not increase significantly the collisions. The delay remains below 200 ms. The delay of MSF and Stratum is much higher, while the delay of LLSF scales smoothly with respect to the number of nodes. MSF provides the lowest end-to-end reliability (Fig. 7b): some packets are dropped because of an excessive number of retransmissions, or because of a buffer overflow. Stratum, LLSF and LDSF achieve to still provide a very high reliability even with 100 nodes generating one packet every 20 seconds. More than 98,5% of the packets are delivered to the border router.

LDSF achieves the higher network lifetime whatever the number of nodes (Figure 7c). The ghost cells are pre-reserved, but the transmitter and the receiver do not have to wake-up in every ghost cells. More precisely, they will wake-up only if no acknowledgement was received correctly. This way, LDSF provides the larger lifetime compared against MSF, LLSF and Stratum approaches.

E. Slotframe occupation

Finally, Figure 8 illustrates the percentage of the slotframe occupied, i.e., the cells are allocated to at least one transmitter. We can note that MSF and LLSF use short slotframes, thus the ratio of shared cells is higher compared to long slotframes. Thus, idle listening will consume a large quantity of energy. Stratum may generate more collisions since interfering links have often to select their cell in the same stratum. Thus, the number of allocated cells is reduced, but keeps larger than LDSF. LDSF organizes the cells appropriately and can exploit long slotframes (as long as the CBR period). While LDSF reserves ghost cells for the retransmissions, they are used only if the transmission has failed. If the packet was acknowledged, neither the transmitter nor the receiver will wake-up during the next ghost cells (cf. amount of unused cells in Figure 8).

VII. CONCLUSION AND FUTURE WORK

Minimizing the end-to-end delay requires to chain the cells along a path, so that the buffering delay is minimized. However, achieving high reliability in the context of IEEE 802.15.4-TSCH is particularly challenging since increasing the number of retransmissions may imply to reschedule all the cells in the path toward the destination. We propose here LDSF which divides the long slotframe into small blocks that repeat over time. Each node selects the right block corresponding to its hop distance from the border router to minimize the delay. In order to still provide high reliability, ghost cells are automatically reserved in the slotframe and the delay is still minimized even if the packet has to be retransmitted. Our simulation results demonstrate that LDSF achieves low latency and jitter with high reliability, even for multihop topologies.

In this article, we have assumed that a cell can be used to send a packet from any flow. Considering flow isolation, each flow can use only a specific set of cells. In other words, ghost cells have to be reserved per flow, impacting negatively the network capacity. Inversely, using the same ghost cells for all flows may impact both the end-to-end delay and the reliability.

There are several techniques in the literature that improve the reliability, such as multipath routing [START_REF] Ahrar | Multipath aware scheduling for high reliability and fault tolerance in low power industrial networks[END_REF], whitelisting [START_REF] Kotsiou | Whitelisting without Collisions for Centralized Scheduling in Wireless Industrial Networks[END_REF], or even at the PHY layer [START_REF] Saxena | Efficient IoT Gateway over 5G Wireless: A New Design with Prototype and Implementation Results[END_REF]. For future work we aim at investigating how these techniques can be combined with LDSF in order to cope with time-variable delivery rates.

Fig. 1 :

 1 Fig. 1: TSCH schedule for a 5 nodes topology.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Probability to receive correctly the packet (P net ()) depending on the link reliability of the link (P ll ()) and the number of cells allocated in the schedule.

  Flow through a route of 5 hops.

Fig. 5 :

 5 Fig.5: Impact of the slotframe length and number of hops on the end to end delay with a PDR per link of 66%.

  End-to-end delay performance.

  End-to-end packet delivery ratio.

Fig. 6 :

 6 Fig.6: Impact of the traffic rate (i.e., inter packet time).

Fig. 7 :Fig. 8 :

 78 Fig. 7: Impact of the number of nodes.

TABLE I :

 I Notation.

	P ll (A, B)	Probability that A receives the acknowledgement
		from B (packet delivery ratio at the link layer)
	Pnet(A, B, k)	Probability that A receives at least one ack from B
		after k retransmissions (packet delivery ratio at the
		network layer)
	M axRetries	Maximum number of retransmissions
	nbretx	Current number of retransmissions for a given packet
	Cells(N i , N i+1 ) number of cells from N i to N i+1
	SF length	Slotframe length (in timeslots)
	BLength	Block length (in timeslots)
	p = {A..B}	path from A to B
	hop delay	hop delay, considering the link-layer retransmissions
	E2E delay	end-to-end delay
	Hops	the hop distance between the forwarding node and
		the source of the packet
	B id	block id (number of blocks from the beginning of
		the slotframe)
	RxSlotId	receiving timeslot (in the 6P request if received by
		the radio chipset, or given by the application layer
		if the packet is generated locally)
	N bGhostCells	the number of ghost cells to add in the schedule for
		this flow
	N L	Network lifetime
	Lf i

TABLE II :

 II Simulation setup.

	Topology	Parameter	Value
		# of nodes	39 + 1 border router
		# of Experiments	20 per algorithm
	Simulation
		Duration	60 min
		Payload size	90 bytes
	Protocol
	CoAP	CBR (Unicast)	1 pkt/[5, 10, 20..120]sec
	RPL	DAO period	60 s
		DIO period	8.5 s
	TSCH	NShared cells	1
		Timeslot duration	10 ms
		Maximum retries	5
	MSF,	Slotframe length	101 timeslots
	Stratum &
	LLSF
	LDSF	Slotframe length	CBR * 101 timeslots
		Block length	5 timeslots
	Queues	Timeout	10 s
		Queue size	10 packets
	Energy
		Energy Model	[22]
		Battery Capacity 10157.4 × 10 6 µC
		Idle	6.4 µC
		Sleep	0 µC
		T xDataRxAck	54.5 µC
		T xData	49.5 µC
		RxDataT xAck	32.6 µC
		RxData	22.6 µC

This work was supported by the French National Research Agency (ANR) project Nano-Net under contract ANR-18-CE25-0003.

V. Kotsiou and F. Theoleyre are with the ICube Laboratory,