Marine Lallemant
email: marine.lallemant@wanadoo.fr

Tiguida Kadiakhe

Jérôme Chambert

Arnaud Lejeune

Rajeev Ramanah

Nicolas Mottet

Emmanuelle Jacquet

Jeröme Chambert

In-vitro biomechanical properties of porcine perineal tissues to better understand human perineal tears during delivery

Keywords: stress-strain curve, perineum, childbirth, perineal tear, biomechanical properties, rupture, deformation Abbreviations EAS: external anal sphincter, IAS, internal anal sphincter

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

ABSTRACT

Introduction : Data concerning the mechanical properties of the perineum during delivery are very limited. In-vivo experimentations raise ethical issues. The aim of the study was to describe some of the biomechanical properties of each perineal tissue layer collected from sows in order to better understand perineal tears during childbirth.

Material and methods: Samples of each perineal tissue layers were obtained from the skin, the vagina, the external anal sphincter (EAS), the internal anal sphincter (IAS) and anal mucosa of fresh dead sows. They were tested in quasi-static uniaxial tension using the testing machine Mach-1®. The tests were performed at a displacement velocity of 0.1 mm.s -1 . Stress-strain curves of each perineal tissue layers before the first damage for each sow were obtained and modeled by hyperelastic Yeoh model described by three coefficients: C1, C2, and C3. Pearson correlation coefficient was calculated to measure the correlation between C1-coefficient and the duration between the first microfailure and the complete rupture for each perineal tissue layer.

Pearson correlation was computed between C1-coefficient and the number of micro-failures before complete rupture for each tissue.

Results: Ten samples of each perineal tissue layer were analyzed. Mean values of C1coefficient and corresponding standard deviations were 46 ± 15 kPa, 165 ± 60 kPa, 27 ± 10 kPa, 19 ± 13 kPa, 145 ± 28 kPa for the perineal skin, the vagina, the EAS, the IAS and the anal mucosa, respectively. According to this same sample order, the first microfailure in the population of 10 sows appeared at an average of 54%, 27%, 70%, 131% and 22% of strain. A correlation was found between C1 hyperelastic-coefficient and the duration between the first microfailure and the complete rupture (r=0.7, p=0.02) or the number of micro-failures before complete rupture only for the vagina (r=0.7, p=0.02).

Conclusions:

In this population of fresh dead sow's perineum, the vagina and the anal mucosa were the stiffest tissue. The IAS and EAS were the more extensible and the less stiff. A significantly positive correlation was found between C1-coefficient and the duration between the first microfailure and the complete rupture of the vagina, and the duration between the first microfailure and the complete rupture of the vagina.

INTRODUCTION

The perineum is a soft structure made up of skin and muscles closing the pelvic. It is located under the levator ani muscle. Its mechanical properties ensure normal pelvic stability.

During childbirth, the morphological and dynamic adaptation of the perineum to the fetal presentation depends on its resistance to the stresses induced by the presentation. Under the compressive efforts induced by the presentation, the perineum becomes thinner until sometimes it tears. Sometimes, it leads to more or less severe perineal tears. Stages III and IV correspond to obstetrical lesions of the anal sphincter and/or anal mucosa. These perineal lacerations can impact women's quality of life with anal incontinence, perineal pain, dyspareunia, recto-vaginal fistula and depression [START_REF] Viannay | Prise en charge et conséquences des lésions obstétricales du sphincter anal (LOSA) : mise au point[END_REF][START_REF] Attanasio | Recovery from Obstetric Anal Sphincter Injury in a Prospective Cohort of First Births[END_REF]. The rate of obstetrical anal sphincter injuries varies between 0.25 and 6% depending on the study [START_REF] Thubert | Definition, epidemiology and risk factors of obstetric anal sphincter injuries: CNGOF Perineal Prevention and Protection in Obstetrics Guidelines[END_REF].

Data concerning the mechanical properties of the perineum and fetal stresses during delivery are very limited [START_REF] Pu | Effect of different labor forces on fetal skull molding[END_REF][START_REF] Lapeer | Fetal head moulding: finite element analysis of a fetal skull subjected to uterine pressures during the first stage of labour[END_REF][START_REF] Bamberg | Evaluating fetal head dimension changes during labor using open magnetic resonance imaging[END_REF][START_REF] Ami | Three-dimensional magnetic resonance imaging of fetal head molding and brain shape changes during the second stage of labor[END_REF][START_REF] Silva | Study on the influence of the fetus head molding on the biomechanical behavior of the pelvic floor muscles, during vaginal delivery[END_REF][START_REF] Yan | Effects of fetal head shape variation on the second stage of labour[END_REF][START_REF] Zemčík | Stereophotogrammetry of the perineum during vaginal delivery[END_REF]. They result essentially from numerical models of the distension of the levator ani muscles. Only Zemčík et al., managed to quantify perineal strain during vaginal delivery using stereophotogrammetry [START_REF] Zemčík | Stereophotogrammetry of the perineum during vaginal delivery[END_REF]. But the perineum is a structure composed of different tissues whose behavior must be characterized in order to understand the tearing mechanism. Therefore, biomechanical parameters of the perineum are necessary to understand perineal tears. But, in-vivo experimentations raise ethical issues. To begin our work, we decided to study biomechanical properties of porcine perineum. A porcine model was chosen because of its similar morphological and immunohistochemical properties, and the results of microindentation tests as reported by the literature [START_REF] Debeer | Comparative histology and immunohistochemistry of porcine versus human skin[END_REF][START_REF] Ranamukhaarachchi | A micromechanical comparison of human and porcine skin before and after preservation by freezing for medical device development[END_REF]. So the aim of our study was to describe some of the biomechanical properties of each perineal tissue layer collected from sows in order to better understand perineal tears during childbirth.

MATERIAL AND METHODS

We performed an experimental study on porcine perineal tissues. Ten samples of each perineal layer were analyzed. They were dissected from fresh dead sow provided by local slaughterhouse waste. The sow breed was the French pork butcher's pig. Sow were slaughtered 24-48 hours before the experimentation. Sows were not frozen. For each sow, one sample was obtained from the skin perineal layer, the vagina perineal layer, the external anal sphincter (EAS), the internal anal sphincter (IAS) and anal mucosa perineal layer (Figures 1 and2.).

Perineal sows were refrigerated during the time before collection. In order to obtain samples in a reproducible way and preserve fiber integrity, a precise dissection method has been implemented (Figure 2.). The dissection was performed by an urogynecologist expert in anatomy of perineal tears. The used instruments were fine scissors and atraumatic forceps. No traction on the tissues was performed. A careful midline incision next to the vulvar area between the vaginal and anal openings was made. The external anal sphincter was immediately identified and isolated. The perineal skin was dissected from the anus to the ventral extremity into two dorso-ventral samples (right and left). Next, the ventral part of the vagina was incised at 6 o'clock to facilitate vaginal access. A right and left vaginal samples from the perineum were dissected on the rectovaginal septum side. Histologically, the vaginal sample included the vaginal wall composed of mucosa, lamina propria, muscularis and adventitia. Then, the external anal sphincter was carefully dissected and sectioned at 9 o'clock and 3 o'clock.

Finally, the dorsal wall of the rectum and anus was incised at 12 o'clock. Two samples (right then left) of the anal mucosa and the internal anal sphincter were obtained. Only the samples on the right side were analyzed for the perineal skin, the perineal vagina, the perineal anal mucosa and the IAS. The left samples of these sows were analyzed in another study with noncomparable experimental conditions. EAS was taken. The entire EAS was collected between the vagina and the anus. Due to its thinness, it was not possible to divide it into two samples like the other layers. The entire EAS was used for sampling, along the fibers, in this study.

Samples were oriented (dorso-ventral, left-right, cranio-caudal).

To obtain uniform stresses at the center of the sample during tensile testing, standardized samples were cut for each perineal layer with a scalpel and a caliber. Samples were obtained using a rectangular pattern 30 mm long and 10 mm wide in the cranio-caudal direction for all tissues except the EAS, which was in the latero-lateral direction (direction shown in Figure 3.).

Each sample end was covered with instant glue (Loctite 401®) and held in paper.

These ends were inserted into the jaw in such a way as not to over-or under-tighten (Figure 3.).

If the sample slipped, it would not be taken into account in the post-processing analysis.

Uniaxial tensile tests were carried out using Mach-1® mechanical tester (Biomomentum Inc, Canada). Tensile forces, stretching forces acting on the tissue, were measured with a 250 N cell. A preload of 0.3 N was applied during two minutes before each test to compare sample results. Sample sizes were measured after the pre-stress. The initial measurements of thickness (t 0 ), width (w 0 ) and length (l 0 ) measurements were obtain using a digital caliper after preload in the middle of the sample to avoid compression tissue and deformation near the jaws. The moving crosshead displacement (Δl) of the tensile-testing machine was recorded to an accuracy of 0.5 µm in the direction of stress (traction), along with force at a frequency of 100 Hz. The displacement was applied until failure. Tensile tests stopped when the sample broke. The tests were performed at a displacement velocity of 0.1 mm.s -1 and at a constant temperature of 21°C to avoid dehydration.

The uniaxial engineering stress  (kPa) in the loading direction was defined by: where F is the measured force, A 0 is the initial cross-sectional area of the specimen.

The uniaxial engineering strain  (%) in the loading direction is obtained by: Stress-strain curves of each perineal tissue before the first damage for each sow and their mean were obtained. Unlike elastic materials where stress varies linearly with respect to strain, soft tissues are hyperelastic. It means that these tissues have a non-linear elastic behavior [START_REF] Khaniki | Hyperelastic structures: A review on the mechanics and biomechanics[END_REF]. Like elasticity, hyperelasticity models reversible behavior. Non-linearity allows to take account large deformations. Non-linear elastic behavior of these tissues were observed and modeled by Yeoh model [START_REF] Martins | A Comparative Study of Several Material Models for Prediction of Hyperelastic Properties: Application to Silicone-Rubber and Soft Tissues[END_REF]: This hyperelastic law is described by three coefficients: C1, C2, and C3. These coefficients were identified by the method of least mean squares from experimental curves using the Levenberg-Marquardt algorithm before the initial sign of damage appeared in each tissue sample [START_REF] Gill | Algorithms for the Solution of the Nonlinear Least-Squares Problem[END_REF]. The damage, also called microfailure, is due to local and microscale effects.

Occurrence of damage results in the weakening of the tissue and makes it more likely to rupture and tear. A microfailure is associated with the inflexion point on the experimental curve and is identified by locating the maximum on the derivative of the curve. The inflexion of the curve shows a reduction in the tissue's ability to withstand mechanical stress, which is interpreted by the appearance of fiber rupture or delaminations in the tissue. This inflection generally precedes macroscopic tissue rupture.

Only C1-coefficients were analyzed because of its meaning: initial slope of stressstrain curve at low strains (less than 5%). A high C1 hyperelastic coefficient means that the tissue is stiff (rigid) with a small deformation in response to an applied force. On the contrary, a low C1 hyperelastic coefficient means that the tissue is easy to deform even in response to a low applied force.

Tangent modulus (kPa) E0 and E1 were calculated during low and high strain respectively. For each tissue, the mean stress and the mean strain were calculated up to the first rupture among all curves for the same tissue. Low strain was defined at the beginning of the master (mean) curve and used to define E0, i.e. at 0.4% deformation on average. E1 was defined as the tangent modulus of the curve at the end of the mean curve, i.e. at 59% deformation on average. Minimal E0 and E1 were calculated from the lowest curve for each layer of perineal tissue. Maximal E0 and E1 were calculated from the highest curve for each layer of perineal tissue.

Coefficients were expressed as mean ± standard deviation. C1-hyperelastic coefficients were compared according to the perineal layer using an ANOVA model. If a statistically difference was found, pairwise comparisons using t-tests and including Bonferroni correction were performed identify which perineal layers were different from the others. In order to determine whether the C1 coefficient was predictive of tissue damage, Pearson correlation coefficient was calculated to measure the correlation between C1-coefficient and the duration between the first microfailure and the complete rupture for each tissue. Pearson correlation was computed between C1-coefficient and the number of micro-failures before complete rupture for each tissue. Statistical analysis was performed using R software (version 4.3.0). For all analyses, we considered a p-value less than 0.05 to be statistically significant.

RESULTS

Ten samples of each layer were analyzed. Samples were described in Table 1. No sample slipped. Mean values of C1-coefficient and corresponding standard deviations were 46 ± 15 kPa, 165 ± 60 kPa, 27 ± 10 kPa, 19 ± 13 kPa, 145 ± 28 kPa for the perineal skin, the vagina, the EAS, the IAS and the anal mucosa, respectively (Figure 4. and Table 2.). C1hyperelastic coefficients were statistically different between perineal layers (p<0.001) (Supporting Information Appendix S1). C1-hyperelastic coefficients of the anal mucosa were statistically different to the C1-hyperelastic coefficients of the perineal skin, the EAS and IAS (p<0.001). C1-hyperelastic coefficients of the perineal vagina were statistically different to the C1-hyperelastic coefficients of the perineal skin, the EAS and IAS (p<0.001). The other perineal layers had statistically comparable C1-hyperelastic coefficients.

The vagina and the anal mucosa had the highest C1 hyperelastic coefficients. The mean ratio of C1 hyperelastic coefficients of the vagina to the anus was 0.3 (standard deviation of 0.1). The anal sphincter (external and internal) had the lowest hyperelastic coefficients.

Tensile tests lasted less than 15 minutes for the IAS and less than 10 minutes for the other perineal layers.

Tangent modulus during low and high strain were presented Table 3. The E1 modulus derived from the mean of the curves tended to triple compared with the E0 modulus for the skin and the vagina. Modulus E1 increased by a factor of 1.3 for SAE and SAI. Anal mucosa increased by a factor of 1.2.

The mean stress-strain curves and their standard deviation for each perineal tissue before the first damage in this population were drawn (Figure 5.). The first microfailure in the population of 10 sows appeared at least at 29%, 13%, 38%, 46% and 14% of strain for the perineal skin, the vagina, the EAS, the IAS and the anal mucosa, respectively (Table 4.). The first microfailure in the population of 10 sows appeared at an average of 54%, 27%, 70%, 131% and 22% of strain for the perineal skin, the vagina, the EAS, the IAS and the anal mucosa, respectively. The external and internal anal sphincters were the most extensible and less stiff tissue. The vagina was the stiffest tissue. The anal mucosa was the less extensible tissue.

The duration between the first microfailure and the complete rupture of each tissue was 79 ± 105 s, 69 ± 86 s, 146 ± 168 s, 151 ± 162 s, 166 ± 108 s for the perineal skin, the vagina, the EAS, the IAS and the anal mucosa, respectively (Table 5.). The number of microfailures before complete rupture of each tissue was 16 ± 14, 6 ± 4, 23 ± 19, 18 ± 17, 38 ± 23 for the perineal skin, the vagina, the EAS, the IAS and the anal mucosa, respectively (Table 5.). The mean ratio of vaginal to anal microfailures before complete rupture was 0.2 (standard deviation 0.2) A significantly positive correlation was found between C1-coefficient and the duration between the first microfailure and the complete rupture of the vagina (r=0.7 p=0.02) (Table 4.). Also, a significantly positive correlation was found between C1-coefficient and the duration between the first microfailure and the complete rupture of the vagina (r=0.7 p=0.02) (Table 5.).

For the others tissue, no correlation was found between C1-coefficient and the duration between the first microfailure and the complete rupture of each tissue (p>0.05) (Table 4.). Also, no correlation between C1-hyperelastic coefficient and the number of micro-failures before complete rupture of each tissue except for vagina (p>0.05) (Table 5).

DISCUSSION

In this study, some biomechanical properties of each perineal layer of the sow has been obtained. C1-coefficients for perineal skin, the vagina, the EAS, the IAS and the anal mucosa of fresh dead sow were measured.

In this population, the vagina and the anal mucosa were the stiffest. In other words, these rigid tissues had smaller deformation in response to an applied force. C1-coefficients of the vagina and the anal mucosa were higher than C1-coefficients of the perineal skin (165 and 145 versus 46 kPa). This difference was also described by Gabriel et al. who compared the hyperelastic coefficient at low strain of the vagina and the abdominal skin of unfrozen cadavers without relevant pelvic organ prolapse (0.35 versus 0.14 MPa, p>0.05) [START_REF] Gabriel | Vagina, abdominal skin, and aponeurosis: do they have similar biomechanical properties[END_REF]. In the literature, no study assessed biomechanical properties of the anal mucosa. Only one rectal tissue evaluation was found [START_REF] Rubod | Biomechanical properties of human pelvic organs[END_REF]. Rubod et al. compared vaginal and rectal tissue on fresh female cadavers without prolapse performing multiaxial tension tests [START_REF] Rubod | Biomechanical properties of human pelvic organs[END_REF]. They demonstrated that the vagina was much more rigid and less extensible than the rectal tissue. In our study, the anal mucosa and the vagina were the less extensible tissues. Their first microfailure appeared at a mean strain of 22% and 27% respectively.

The IAS and EAS were the more extensible and the less stiff. The EAS is a skeletal muscle. Passive extensibility of skeletal muscles is an important component of total muscle function. It is well described in the literature [START_REF] Gajdosik | Passive extensibility of skeletal muscle: review of the literature with clinical implications[END_REF]. Studies with animal muscles have shown that passive extensibility is influenced by the size (mass) and length of muscle fibers, and the amount and arrangement of the connective tissues of the muscle belly. The resistance to passive lengthening is influenced by the readily adaptable amount of muscle tissue, including the contractile proteins and the non-contractile proteins of the sarcomere cytoskeletons. Concerning IAS's biomechanical properties, no study was found in the literature.

But a corroboration with vaginal smooth muscle cells could be done. Smooth muscle cells contribute to soft tissues' quasistatic and viscoelastic mechanical behavior [START_REF] Wagner | Differential Passive and Active Biaxial Mechanical Behaviors of Muscular and Elastic Arteries: Basilar Versus Common Carotid[END_REF][START_REF] Greven | Creep after loading in the relaxed and contracted smooth muscle (taenia coli of the guinea pig) under various osmotic conditions[END_REF][START_REF] Greven | Creep after loading in relaxed and contracted (KC1 or K2SO4 depolarized) smooth muscle (taenia coli of the guinea pig)[END_REF]. According to Clark-Patterson et al., smooth muscle cells provide mobility by allowing the vagina to stretch under sustained pressures. By the same way, IAS could ensure mobility by allowing the anal mucosa to stretch under sustained pressures [START_REF] Clark-Patterson | Smooth muscle contribution to vaginal viscoelastic response[END_REF].

According to the tangent modulus, the skin and vagina seemed more resistant to deformation for high strain (much higher modulus). The stress applied to the tissues increased very quickly in the high strain zone which tends to show that these tissues will tear first during childbirth because the deformations of the perineal tissues are imposed by the passage of the fetus. The great ability of SAE and SAI to deformation (38 and 46%) is highlighted by lower values of E1 modulus for these tissues. The stresses applied was lower for the same level of deformation than those observed for the skin and the vagina. These data are compatible with what is observed in maternity wards. The skin and the vagina tend to tear first before the anal sphincter and the anal mucosa during childbirth.

In our study, the sow's anal sphincter was more extensible and less stiff than the vagina or the perineal skin. It could explain why severe perineal tears are rare in sows. If the same results are confirmed in humans, that could also maybe explain why the anal sphincter is not always torn during childbirth. In the same way, the sow's anal mucosa and its vagina were the less extensible tissues. If these biomechanical properties are also found in humans, it could maybe explain isolated rectal buttonhole perineal tear in obstetrics [START_REF] Roper | Isolated rectal buttonhole tears in obstetrics: case series and review of the literature[END_REF]. It is an isolated tear of the anal epithelium or rectal mucosa and vagina but without involving the anal sphincter. This kind of laceration is rare in human and only described in equine case report [START_REF] Kay | Surgical technique to repair grade IV rectal tears in post-parturient mares[END_REF][START_REF] Watkins | Rectal tears in the Horse: an analysis of 35 cases[END_REF][START_REF] Speirs | The management of rectal tears in horses[END_REF].

Concerning the experimental methodology, No failure or tears near the grip have been noticed. Operators trained before starting the study to avoid the learning phase and reduce these kinds of failure. Perineal stresses during delivery are not unidirectional. However, due to the size of the samples, only unidirectional traction tests at constant speed were performed.

Yeoh model has been chosen because it described accurately the behavior of all the different perineal tissues. A comparison of Yeoh, Mooney-Rivlin and Ogden models on pig skin (belly and back) confirmed that Yeoh model captures accurately the behavior of the pig skin [START_REF] Dwivedi | A hyperelastic model to capture the mechanical behaviour and histological aspects of the soft tissues[END_REF]. It also highlighted the stability issues that arise when using the Ogden model as well as the failure of the Mooney-Rivlin model to properly describe the behavior of the tissues. Moreover, it requires a small number of parameters, its first parameter being interpreted as half the initial shear modulus.

At our knowledge, our study is the first one in the literature to attempt to describe biomechanical properties of each perineal layer in order to better understand perineal tears.

This study focused on deformation. It concerns only the different tissues of the perineum, considered independently. This work is a preliminary step towards understanding the mechanisms involved. The next steps are aimed at identifying predictive parameters for the risk of perineal tears. These results could be used in simulation using Finite Element Model.

Structural tests on the whole perineum are currently underway, and will be used to validate the model of the whole perineum with controlled boundary conditions. This ongoing study aims to identify predictive parameters for the risk of failure associated with extreme loading conditions (>250% deformation).

According to Rosenberg and Trevathan, non-human primates have the most similar pelvic anatomy to humans [START_REF] Rosenberg | Birth, obstetrics and human evolution[END_REF]. But its availability and its cost has led us seek another animal model. Ewes could have been used also but their availability remains difficult [START_REF] Rynkevic | Biomechanical and morphological properties of the multiparous ovine vagina and effect of subsequent pregnancy[END_REF][START_REF] Rynkevic | The effect of consecutive pregnancies on the ovine pelvic soft tissues: Link between biomechanical and histological components[END_REF].

Broodmares with a perineal body and who are also subject to obstetrical anal sphincter injury could also be an animal model [START_REF] Sheard | Comparing two surgical techniques for repairing third-degree perineal lacerations in postpartum broodmares[END_REF]. However, the model is difficult to obtain. The simple, reproducible, low cost and most frequently animal model for pelvic floor research is the sow. It allows comparing measurement of mechanical properties with the literature. Plus, porcine models are usually used to educate physicians in sphincter injury repair [START_REF] Temtanakitpaisan | Obstetrics anal sphincter injury and repair technique: a review[END_REF][START_REF] Andrews | Structured hands-on training in repair of obstetric anal sphincter injuries (OASIS): an audit of clinical practice[END_REF][START_REF] Emmanuelli | Development and assessment of a workshop on repair of third and fourth degree obstetric tears[END_REF]. Similarly to humans, utero-vaginal prolapse are common in sows [START_REF] Supakorn | Prolapse Incidence in Swine Breeding Herds Is a Cause for Concern[END_REF][START_REF] Kiefer | Vaginal microbiota differences associated with pelvic organ prolapse risk during late gestation in commercial sows †[END_REF]. In a recent industry study, 21% of sow deaths in the USA could be attributed to POP [START_REF] Ross | Identification of putative factors contributing to pelvic organ prolapse in sows[END_REF].

The perineal body is a fibromuscular structure located in the midline of the perineum and provides attachments of the perineal muscles in women [START_REF] Siccardi | Anatomy, Abdomen and Pelvis, Perineal Body[END_REF]. Surprisingly, in our experience, sow's perineum seemed to not have a perineal body. In the literature, Bassett et al. study is the only article describing the anatomy of sow's perineum [START_REF] Bassett | The comparative anatomy of the pelvic and perineal regions of the cow, goat and sow[END_REF]. They did not report a perineal body. Plus, in the area between the vagina and the rectum of the sow, the only perineal muscle that was described is the anal sphincter. The retractor clitoridis muscle insertion is located in the ventrolateral rectal wall. But, there is no perineal muscles other than the anal sphincter in this area. During our multiple dissections, the full area between the rectum and the vagina was excised, and the skin and the subcutaneous tissue removed such as described by Kochová et al. [START_REF] Kochová | The histological microstructure and in vitro mechanical properties of pregnant and postmenopausal ewe perineal body[END_REF]. They called the resulting tissue the perineal body. But when we performed the same dissection as described by this research group, it seemed that they took the muscularis of the vagina, the adventitia of the vagina and fascia-like tissue. Kochová studied the perineal body stiffness in pregnant and on pregnant women [START_REF] Rostaminia | Shear Wave Elastography to Assess Perineal Body Stiffness During Labor[END_REF][START_REF] Chen | In Vivo Estimation of Perineal Body Properties Using Ultrasound Quasistatic Elastography in Nulliparous Women[END_REF].

In addition to a relatively small study group, the sow's parity was not known. These sows were nullipara. According to Rynkevic et al. and Drewes et al., biomechanical properties of the vaginal tissue after pregnancy would not recover to those of virgins [START_REF] Rynkevic | Biomechanical and morphological properties of the multiparous ovine vagina and effect of subsequent pregnancy[END_REF][START_REF] Drewes | Pelvic Organ Prolapse in Fibulin-5 Knockout Mice: Pregnancy-Induced Changes in Elastic Fiber Homeostasis in Mouse Vagina[END_REF]. These reasons could explain the interindividual variation among the sows. Furthermore, we did not study interactions between each layer of perineal layer. We investigated sow's cadaveric tissues.

Postmortem changes must be considered. Indeed, stiffening of muscles appears 2 to 6 hours after death because of adenosine triphosphate (ATP) levels, which cause the binding of the muscle filaments of actin and myosin [START_REF] Martins | Necromechanics: Deathinduced changes in the mechanical properties of human tissues[END_REF]. Even if the studied tissues cannot be considered to have intact chemomechanical properties, the results revealed significant differences in biomechanical behavior, providing new insights to better understand the mechanical responses of perineal structures to mechanical stress. In vivo experimentation is non-ethic. In vivo measurements using nondestructive tests should be performed to confirm these results.

CONCLUSION

In these perineal tissue samples of fresh dead sow, the vagina and the anal mucosa were the stiffest tissue. The IAS and EAS were the more extensible and the less stiff. A significantly positive correlation was found between C1-coefficient and the duration between the first microfailure and the complete rupture of the vagina and between C1-coefficient and the duration between the first microfailure and the complete rupture of the vagina. No correlation was found for the other perineal tissues. These results must be confirmed on humans using in vivo nondestructive tests.

Legends

  et al. has written in their work that they did not know the gross anatomy of the structures in the tissue samples. This discrepancy highlights the limits of the sow's model. The perineal body can by studied in humans using non-invasive technologies. Zemčik et al. studied the deformation of the perineum during normal delivery using stereophotogrammetry [10]. Chen et al. and Rostaminia et al.

Table 3 .Table 4 .Table 5 .Figure 1 .

 3451 Figure 1. Anatomy of the sow's perineum. An incision of the perineal skin between the Vagina and the anus was performed to reveal the external anal sphincter (EAS).

Figure 2 .

 2 Figure 2. Obtention of the skin, the vagina, the IAS, the EAS and the anal mucosa samples of the sow. IAS: internal anal sphincter, EAS: external anal sphincter.

Figure 3 .

 3 Figure 3. Test machine Mach-1® (Biomomentum Inc, Canada) and grips for soft tissues.

Figure 4 .

 4 Figure 4. Stress-strain curves of each perineal tissue before first damage for each sow.

Table 1 .

 1 Description of samples and tensile tests for each perineal tissues.

Table 2 .

 2 Hyperelastic coefficients according to the perineal tissue (Yeoh model).

Conflict of interest:

None

Ethics statement

Sow's perineum were dissected from fresh dead sow provided by local slaughterhouse waste.

Animals were not slaughtered for the study. No ethics application was needed since the animals were bred and killed for food production (French decree on the protection of animals used for scientific purposes).

Author contributions:

Marine Lallemant : conceptualization, data curation, formal analysis, methodology, writingoriginal draft; Tiguida Kadiakhe : conceptualization, data curation, formal analysis, methodology, writing-review; Jeröme Chambert : conceptualization, methodology, writingreview ; Arnaud Lejeune : conceptualization, methodology, writing-review; Rajeev Ramanah: writing-review ; Nicolas Mottet : writing-review; Emmanuelle Jacquet: conceptualization, methodology, writing-review.

Supporting Information legend

Appendix S1. P-values of pairwise comparisons using t-tests and including Bonferroni correction to identify which perineal layers were different from the others according to C1hyperelastic coefficient.


Sabri Khamari
email: sabri.khamari@u-bordeaux.fr

Toufik Ahmed

Mohamed Mosbah
email: mohamed.mosbah@u-bordeaux.fr

Green Edge Servers Placement for Intelligent Transport Systems

Keywords: Green computing, Edge computing, Edge server placement, Intelligent Transport Systems. I

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INTRODUCTION

Smart vehicle networks have become essential to deploy innovative transportations services such as dangereous situation warning, special vehicle prioritization, and area reservation. Also known as Cooperative Intelligent Transport Systems (C-ITS), smart vehicle networks improve traffic management and road safety policies, ensure a future with fewer road accidents, and enable optimal driving to reduce fuel consumption.

New C-ITS use cases, such as autonomous and remote driving, require intensive AI-based processing and video analytics and necessitate guaranteed Quality of Service (QoS) performance, including ultra-low latency, near-perfect reliability, and extremely high bandwidth [START_REF] Velez | 5g beyond 3gpp release 15 for connected automated mobility in cross-border contexts[END_REF]. Such use cases generate massive amounts of data and require in-network processing capabilities. For example, smart vehicles are equipped with Lidar, radar, and cameras to collect information in the vehicle's vicinity [START_REF] Kocić | Sensors and sensor fusion in autonomous vehicles[END_REF]. However, on-board vehicles' computational resources are insufficient to handle service processing tasks under a high volume of generated data. As a result, some tasks should be offloaded to remote computational nodes [START_REF] Buda | Collaborative vehicular edge computing towards greener ITS[END_REF]. Cloud computing enables the provision of on-demand extensive processing and storage resources capable of managing such enormous amounts of data. Nevertheless, cloud computing capabilities are often offered by data centers located far from the vehicle's position. As a result, end-to-end data transmission often experiences considerable delays [START_REF] Ning | Mobile edge computing-enabled Internet of vehicles: Toward energy-efficient scheduling[END_REF], and networks frequently face bottlenecks. Additionally, transmitting high volumes of data requires high bandwidth usage that consumes a substantial amount of energy [START_REF] Ning | Mobile edge computing-enabled Internet of vehicles: Toward energy-efficient scheduling[END_REF]. MEC (Multi-Access Edge Computing) promises to deliver adequate processing capacity while meeting delay constraints and optimizing energy consumption [START_REF] Ning | Mobile edge computing-enabled Internet of vehicles: Toward energy-efficient scheduling[END_REF]. It entails installing a set of high-performance computing nodes near vehicles on the base stations of cellular operators or the Roadside Units (RSU) of road operators. MEC offers the efficient fulfillment of stringent QoS requirements for a diverse set of services and use cases and the handling of computationally intensive tasks. However, deploying edge servers in urban vehicular networks remains a significant challenge. Indeed, optimal edge server placement implies limiting the number of edge nodes required to meet the service level quality. Such deployment should ensure that appropriate computation and storage resources are always available to meet the C-ITS services' latency, bandwidth, and reliability requirements.

Additionally, optimal placement options are necessary to reduce deployment costs and increase energy efficiency. Inefficient Edge server placement leads to high latency and significantly imbalanced workload distribution across Edge servers. Thus, some Edge servers may be overburdened while others are underutilized. Deploying more edge nodes results in increased deployment costs and reduced energy efficiency.

This study aims to address the issue of efficiently deploying an optimal number of Edge servers while adhering to deployment costs and network performance constraints. This problem is exacerbated in metropolitan environments, where a high volume of data is generated due to user mobility and density.

This paper introduces the Green Optimal Edge Server Placement (GOESP) approach, a mathematical optimization model for a green placement of edge servers in strategic locations. Toward this objective, we formulate the problem as integer linear programming that minimizes the monetary and energy cost associated with the deployment while simultaneously satisfying a specific latency threshold and balancing workload among Edge servers. To provide a realistic Edge server placement solution, we propose a methodology based on open data from real-world automobile traffic in Bordeaux city [START_REF] Opendatabordeaux | [END_REF]. We perform extensive simulations to analyze the suggested approach and demonstrate its effectiveness.

The rest of this paper is organized as follows. Section II introduces some background on Multi-access Edge computing and its application in vehicular networks. Section III presents related work in edge server deployment and energy efficiency solutions. In Section IV, we present our solution and model for the green edge servers' placement. Next, Section V presents the simulation setup and performance evaluation. Finally, we briefly conclude the work and introduce future work in Section VI.

II. BACKGROUND

A. Vehicular applications

In the context of IoT Automotive, V2X (Vehicle to Everything) requires a critical communication infrastructure where reliability and ultra-low latency are determinant factors [START_REF] Zakaria | [END_REF]. V2X covers Vehicle to Vehicle (V2V), Vehicle to Infrastructure (V2I), vehicle to Pedestrian (V2P). Intelligent Transportation System (ITS) applications can be categorized into road traffic safety, road traffic efficiency, and value-added services [8][9]. Use cases in these categories include collision warning, assistance (Intersection Movement, Left Turn, cooperative merging), congestion warning, etc. [START_REF] Singh | A tutorial survey on vehicular communication state of the art, and future research directions[END_REF]. Communication and computation capabilities must be improved to operate an efficient and reliable vehicular application that could include real-time traffic monitoring, continuous sensing, and cooperative and augmented vision [START_REF] Porambage | Survey on multi-access edge computing for internet of things realization[END_REF]. However, the current network and infrastructure based on cloud computing cannot fully satisfy the application's requirements. Recently, MEC (Multi-access Edge Computing) has emerged as an innovative architecture that offers a higher level of flexibility and provides a viable solution that can accelerate the development of ITS applications [START_REF] Datta | Vehicles as connected resources: Opportunities and challenges for the future[END_REF].

B. Multi-access Edge Computing (MEC)

Mobile Edge Computing was introduced by the European Telecommunications Standards Institute (ETSI) Industry Specification Group (ISG) as a means of extending intelligence to the edge of the network along with higher processing and storage capabilities [START_REF] Hu | Mobile edge computing-A key technology towards 5G[END_REF]. From 2017, the ETSI industry group renamed it to Multi-Access Edge Computing (MEC) since the benefits of MEC technology reached beyond mobile towards Wi-Fi and fixed access technologies. The principle of MEC consists of extending cloud computing capabilities to the edge of networks to minimize network congestion and improve resource optimization, user experience, and overall network performance.

C. Deployment of MEC systems

The primary motivation of Multi-access Edge Computing (MEC) is to bring the Cloud computing capability close to the network edges to mitigate cloud computing limits such as high latency and throughput bottleneck. Selecting MEC locations, especially Edge servers, remains the first step towards building up the MEC system and deploying vehicular applications.

However, the site selection for the deployment of Edge servers presents an important challenge. The server's placement strategy differs from the conventional base station site selection's problem. The optimal placement of Edge servers is related to the computational resource provisioning and constrained by the deployment budget [START_REF] Mach | Mobile edge computing: A survey on architecture and computation offloading[END_REF] and vehicles traffic on roads. The MEC providers should consider computational demands of task arrival and deployment costs such as site rentals to make the MEC servers placement strategy costeffective. In general, considering the system deployment budget, more MEC servers should be installed in regions with higher computation demands. Using existing telecom infrastructure, such as base stations, could reduce costs, but it would not solve all the problems. Indeed, due to the continuous increase in computation requirements for vehicles and various smart devices (IoT), application requirements cannot be guaranteed. Therefore, Edge servers must be deployed with a more distributed strategy and closer to the user.

III. RELATED WORK

Existing research works on the usage of edge servers, such as [START_REF] Zhang | An optimal roadside unit placement method for vanet localization[END_REF][15], focus on the placement challenge and load balancing, with little emphasis on meeting QoS requirements and on green issues such as energy consumption and deployment costs. The authors of [START_REF] Premsankar | Efficient placement of edge computing devices for vehicular applications in smart cities[END_REF] tackled the placement problem using a mixed-integer linear programming formulation that minimizes the deployment cost of Edge computing devices by jointly satisfying a target level of network coverage and computational demand. However, this work disregarded the latency requirements of vehicular applications. The work of [START_REF] Cao | Exploring placement of heterogeneous edge servers for response time minimization in mobile edge-cloud computing[END_REF] investigated the problem of deploying heterogeneous edge servers for response time minimization in mobile edge-cloud computing systems. Two stages are proposed: an offline stage using an Integer Linear Programming (ILP) technique to produce an optimal solution of edge server placement, and an online stage that deals with the mobility of users using a game theory-based scheme of base stations. Wang et al. [START_REF] Wang | Edge server placement in mobile edge computing[END_REF] investigated the Edge server placement problem in mobile Edge computing environments for smart cities. They formulated the problem as a multi-objective constraint optimization problem that places Edge servers in strategic locations to balance workload between Edge servers and minimize Edge server access delay. Even though this work did not particularly address the vehicular network, it is the most pertinent to the placement problem discussed in this paper. We consider it for comparison with our proposed solution.

Green edge solutions for vehicular environments focus on tasks and computational resources offloading [20][21]. The authors of [START_REF] Buda | Collaborative vehicular edge computing towards greener ITS[END_REF] proposed a collaborative edge computing scheme that enables vehicle-as-an-edge technology for task processing in the vehicular Internet of Things (IoT). Ning et al. [START_REF] Ning | Mobile edge computing-enabled Internet of vehicles: Toward energy-efficient scheduling[END_REF] presented an incentive-based optimal computation offloading scheme that maximizes the service provider's utilities while reducing smart device energy consumption and task execution time. However, none of these works considers the placement of edge servers as a vital parameter for green edge solutions for automotive networks.

Our study focuses on developing a global solution that considers both the placement strategy required to meet QoS requirements and the energy efficiency limitations associated with reducing the deployment cost.

IV. GREEN OPTIMAL EDGE SERVER PLACEMENT

Figure 1 depicts the proposed reference architecture for vehicular green edge computing. We consider connected vehicles equipped with ITS-G5, cellular communication interfaces, or both. These vehicles collect data from their surrounding environment. Frequently, advanced processing is often required by vehicle services' tasks. To meet the requirements of C-ITS applications, storage and computing capabilities are provided at the network edge close to the vehicles. The edge layer comprises edge servers that can run on renewable energy sources such as solar energy if the deployment area is equipped with appropriate materials. In such a scenario, vehicles offload their processing jobs to the edge servers via cellular connection (4G/5G) using the base station (BS) or via the ITS-G5 communication using roadside equipment (RSU). The edge servers are connected to the cloud and the road operator platform via the core network.

The following subsection describes our solution, namely Green Optimal Edge Server Placement (GOESP), which enables effective green edge server placement while meeting the requirements of automotive applications (latency and load balancing) and taking into account the green environmental elements of communication

A. System model

The system is composed of a set J = {S1, S2, …, Sm} of potential locations for Edge servers and a set I = {R1, R2, …, Rn} of roadside units and/or BS. Some of the proposed sites include renewable green energy sources like solar power. Figure 2 illustrates an example of the vehicular edge computing system where one of the three edge servers (S2) is deployed at a location with a green energy source.

Assume that K Edge servers will be assigned to K different locations, where K is a constant (in Figure 2, K=3). To process the requests sent by the vehicles, we suppose that each Edge server has the same restricted computing resource (storage, processing, and memory). Each RSU/BS is linked to one Edge server to delegate vehicle requests. The objective is to install as few edge servers as possible to minimize energy consumption while serving all vehicle requests without surpassing the servers' capacity. Additionally, the model emphasizes the importance of choosing a location in an area with a green energy supply.

The notations utilized in our model are listed in Table 1. cj presents the cost of deploying an Edge server at location j. This cost combines the monetary and energy consumption cost; hence, a lower cost is attributed to a location with a green energy source. The tuple memory, storage, and processing express the Edge server capacity. Similarly, each RSU/BS has its requirements or demands described similarly. Such demand represents the aggregated tasks of vehicles received by this RSU/BS. Accordingly, we formulate an integer linear programming model that captures the model's features under consideration. We begin by defining the decision variables of the model:

1 , if server is deployed at location $ 0 , else (1)
, 1 , if RSU i is linked to server $ 0 , else

* +, ∶ total cost (3) (2)
The objective function is defined as:

./ , .. 1.

./ * +,

s.t

* , 1 , ∀ . 3 4 +, (5)
Beginning with the decision variables in (1) and (2), we describe each expression in our model. denotes a Boolean variable indicating whether a candidate position is selected for Edge server deployment. , represents a Boolean variable that shows which Edge server is connected to each RSU/BS. For instance, :,; = 1 indicates that RSU "5" is connected to the Edge server deployed in position "2". The objective function in (4) aims to minimize the deployment cost by reducing the number of Edge servers to be deployed where cj denotes the deployment cost of Edge server at location j from J.

Following that, we discuss the constraints. Equation ( 5) ensures that each roadside unit is connected to only one server among the deployed servers. Constraint [START_REF] Opendatabordeaux | [END_REF] guarantees that each RSU is connected to an Edge server without exceeding the maximum tolerated latency and is defined by distance. Constraints (7-9) ensure that memory, storage, and processing requirements are met. It guarantees that the total number of requests sent by roadside units and base stations to the Edge server does not exceed its capacity. Finally, constraints [START_REF] Porambage | Survey on multi-access edge computing for internet of things realization[END_REF][START_REF] Datta | Vehicles as connected resources: Opportunities and challenges for the future[END_REF] ensure that the decision variables satisfy the integrality requirements.

B. Use case Scenario

We consider the city of Bordeaux, France, as the target area for Edge servers' placement. A map of Bordeaux city, including information about roads and buildings, is obtained from OpenStreetMap [23]. The map bounds an area of 15.10 km². Figure 3 shows the concerned area. For simplicity but without loss of generality, we assume that 27 roadside units are installed at road intersections at a set spacing of around 800 meters apart. We adopted this strategy because of the lack of data on actual RSU deployment. The following stage consists of identifying potential Edge server sites. We chose those locations using a straightforward technique. Between every three or four RSUs, one site is specified as a possible location for an Edge server. The optimization model chooses the most strategic places for deploying Edge servers based on a set of 19 possible Edge server locations. The candidate locations shown in a green square in Figure 3 are equipped with a renewable energy source. We used the open dataset OpenDataBordeaux [START_REF] Opendatabordeaux | [END_REF] to generate a traffic trace similar to the real one of Bordeaux. Based on this data, we identified three classes of roads: high traffic roads, moderate traffic roads, and low traffic roads. The RSUs send vehicles' tasks to the edge servers according to generated traffic trace. Therefore, we distinguish three classes of RSU: RSU dealing with high demands, RSU with moderate demands, and RSU with low demands, as shown in Table 2.

V. EVALUATION AND RESULTS

A. Methodology and simulation setup

Figure 4 provides a comprehensive overview of the methodology and simulation setup used to evaluate our green placement strategy. To obtain the optimal solution of our model, we use AIMMS (Advanced Interactive Multidimensional Modeling System [START_REF]Download AIMMS developer[END_REF]) with CPLEX12.10 solver. AIMMS is a perspective analytics platform that allows the modeling and development of optimization-based applications. The model requires the following inputs: the geographical coordinates of roadside units, the geographical coordinates of the Edge server's candidate locations, the latency threshold (represented by distance), Edge servers' capacity, and RSUs demands (in terms of processing, memory, and storage). According to the model, the latency, and servers' capacity constraints, each roadside unit is linked to one of the edge servers. As a result, the solver returns the number of Edge servers to deploy as well as the placements of the servers among the possible locations. Therefore, the total cost of deployment is calculated. The machine for this optimization task presents the following configuration: processor intel i7-10610U CPU @ 1.80GHz 2.30 GHz and RAM 16 GO.

Once the optimal placement of edge servers is obtained, we run Omnet++ [START_REF] Varga | Modeling and Tools for Network Simulation[END_REF] and the Artery framework [START_REF] Riebl | Artery: Large scale simulation environment for its applications[END_REF] to simulate vehicle traffic. We use SUMO [START_REF] Lopez | Microscopic traffic simulation using sumo[END_REF] to generate the traffic trace of the simulation. The simulations were conducted on a computer with 16 GO RAM and the Intel i7-9700 CPU @ 3.00GHz x8 processor. Table 3 summarizes the parameters used to find the optimal solution using AIMMS and run simulations. We use two values to represent the cost edge server deployment. It is equal to one (unit) if the candidate location has a green energy source and equal to two (unit) otherwise. We suppose that all Edge servers have the same capacity, and the demands of the RSUs depend on the category to which the RSU belongs.

B. Results and performance evaluation

The optimal solution calculated with the solver CPLEX indicates that four (K=4) Edge servers should be deployed in the concerned area of Bordeaux. The chosen locations are location 3, location 4, location 13, and location 19, as shown in Figure 5. The placement results show that our approach favors the green communication strategy as three (3) among the four (4) locations are powered by renewable green energy. Thus, the total deployment cost is 5.

To better evaluate our solution, we implemented other placement strategies such as Random, Top-K, and K-means as baseline comparisons because they are the most used strategies in such problems. Furthermore, we also implemented Wang et al. approach [START_REF] Wang | Edge server placement in mobile edge computing[END_REF], which is the most relevant state-of-the-art placement strategy compared to our work. The different strategies are described below:

• Random: this method randomly assigns the K Edge servers to candidate locations. We evaluated two different approaches to this problem, based on how the roadside units are connected to the Edge servers. The first is Random-Random, in which RSUs are randomly assigned to servers. The second is Random-Nearest, in which each RSU is connected to the Edge server located closest to it.

• Top-K: The K RSUs with the highest traffic demands are picked. Each RSU is connected to the Edge server located closest to it. Then, the Edge servers are deployed in the places that are closest to each RSU (K locations).

• K-means clustering: This technique is frequently used to automatically cluster a dataset into K groups. We discover K clusters of roadside units and then put K Edge servers in their centers using the K-means clustering technique.

• Wang et al.'s approach [START_REF] Wang | Edge server placement in mobile edge computing[END_REF]: considers the deployment of edge servers in smart cities to balance the workload between Edge servers and minimize the Edge server access delay. They adopted mixed integer programming to find the optimal solution.

Table 4 below summarizes the placement results for the different strategies. We fixed the number of deployed edge servers to K=4 for comparative purposes for the Random, Top-K, and K-means techniques. As a result, our solution has a lower deployment cost than Top-K and K-means. The Wang et al. approach has the lowest deployment cost by deploying just two servers. However, this solution does not ensure the system's performance, and demands may exceed the servers' capacity at peak periods. The link between each roadside unit and the Edge server is shown in Table 5, and the latency between the entities is stated in terms of distance. For instance, Server 1, located at position 3, is used to connect R5 at a distance of 1101 meters, R6 at a distance of 1204 meters, and so on. Additionally, we show the Edge servers' workload, which is the aggregate of the demands of the roadside units connected to this server; it is 120 units for server 1. The obtained results adhere to the server's latency and capacity limits. All RSUs (R1 to R27) are linked to one of the Edge servers without exceeding the model's maximum latency (measured in distance) of 1500m. For example, regarding the capacity constraint, the sum of RSUs linked to server 1 is 120 units, which is less than server 1's capacity (150 units). Our method satisfies the server's capacity constraint while also ensuring optimal load balancing among edge servers.

Edge server

Roadside ID (distance to the edge server)

Server 1 (at location 3) R5 (1101m), R6 (1204m), R7 (582m), R8 (1058m), R11(1117m), R12(1236m) Server 2 (at location 4) R1 (893m), R2(191m), R3 (1030m), R4(1005m), R9(911m), R10(1303m) Server 3 (at location 13) R13 (1337m), R14 (852m), R15 (516m), R16 (1175m), R17 (530m), R18 (721m), R19(496m), R20 (877m) Server 4 (at location 19) R21 (1338m), R22 (534m), R23 (373m), R24 (242m), R25 (1056m), R26 (740m), R27 (793m) Figure 5 summarizes and indicates the chosen locations for deployment of Edge servers for each placement strategy. The Random-Random and Random-Nearest approaches are also not presented here as their results are random. Figure 6 highlights the load balancing results achieved using the different placement strategies by displaying the percentage of tasks handled by each server. Our approach GOESP provides an extremely high level of load balancing where servers process nearly the same number of tasks (maximum difference of 5%). The GOESP approach enhances the fairness among the Edge servers. The other strategies do not ensure good load balancing. For example, server 2 handles 36% of tasks (160 units) in Top-K, surpassing 150 units. We examined the critical nature of load balancing in situations where servers have limited processing capacity. To do this, we fixed the queue size on each server to 150 tasks, and when a server's queue is full, the next arriving task is dropped and must be retransmitted later. The retransmission of tasks from the vehicle to the server consumes additional energy and degrade the latency. To this purpose, minimizing the number of dropped tasks is critical for decreasing and conserving energy consumption on cars and edge servers. The percentage of dropped tasks for each approach is depicted in Figure 7 . The results indicate that our approach can deliver the best performance with the lowest dropped tasks rate. As a result, our method is more energy-efficient than the other placement strategies. Figure 8 shows the additional energy consumption caused by the retransmission of the dropped tasks. Specifically, our method consumes 48% less energy than Random-Random and 69% less energy than Random-Nearest. The Top-K strategy eliminates 5,67% of tasks, resulting in a 95,5% increase in energy consumption when compared to our solution. The Kmeans and Wang et al. approaches consume significantly more energy and increase by up to 394% compared to our solution. Finally, we examine the effect of our solution on the average latency of messages received by the servers. Figure 9 depicts the latency curves generated using the simulation tool within the Artery framework for the approaches mentioned above. Our approach GOESP achieves a latency that varies between 1,51ms in low density and 1,61ms in high density. The other approaches offer a latency in approximately the same interval. When we examine the findings more closely, we see that GOESP outperforms Wang et al. [START_REF] Wang | Edge server placement in mobile edge computing[END_REF] Additionally, our solution presents a higher latency of 0,01ms (maximum) than the Top-K and K-means approaches. This increase could be explained by the fact that those approaches use the nearest server criteria, i.e., each RSU is connected to the nearest server; however, they do not consider other constraints like the server capacity and the workload balancing. For example, in the Top-K approach, server 2, with a capacity of 150 units, cannot handle all the roadside units' tasks (160 units of total demands). Therefore, this minor latency increase is acceptable for our solution to ensure that the server's capacity is not exceeded. On the other hand, the proposed model ensures that the latency threshold ( ) constraint is always respected.

VI. CONCLUSION

Edge computing remains a new emerging technology that can boost the performance of various applications and help reduce power consumption by shifting the processing workload from the cloud to the edge. The energy-efficient Edge server placement problem has received little attention to the best of our knowledge. This paper investigated this topic in urban vehicle networks to optimize deployment costs, energy consumption, and balance workload between Edge servers while minimizing latency. We formulated the Edge server placement problem as an integer linear programming model. We used a methodology based on real-world Bordeaux traffic statistics to give a realistic Edge server placement. Our model considers Edge servers' capacities to balance servers' workloads efficiently. Using the AIMMS analytics platform, we proposed an effective placement solution. The obtained results indicate that our solution outperforms existing solutions in terms of energy efficiency. Workload balancing between servers prevents excessive energy consumption. Thus, our proposed placement technique balances deployment cost, energy consumption, workload balancing, and latency. As future work, we intend to investigate the service migration between Edge servers to ensure service continuity, decrease application delay, and further optimize energy consumption.

Figure 1 .

 1 Figure 1. Vehicular green Edge computing reference architecture

Figure 2 .

 2 Figure 2. Example of the vehicular green edge computing system

  set of RSU/BS J: a set of the potential location of Edge servers. G = (V,E) E: a set of links between roadside units and Edge servers n Number of roadside units m Number of potential locations for Edge servers K The number of deployed Edge servers Cost of deployment of Edge server at location j Maximum latency (proportional to the distance) , The latency between RSUi/BSi and Edge server location j Processing demands of RSUi/BSi Memory demands of RSUi/BSi Storage demands of RSUi/BSi Processing capacity of server j Memory capacity of server j Storage capacity of server j Binary variable for Edge server placed at location j , Binary variable for RSUi/BSi linked to Edge server at location j The total cost of Edge server's deployment

Figure 3 .

 3 Figure 3. Use case scenario: Edge servers' placement on Bordeaux

Figure 4 :

 4 Figure 4: Overview of the used methodology to evaluate "GOESP".

Figure 5 :

 5 Figure 5: Edge server's placement locations for different approaches: GOESP, Top-K, K-Means and Wang et al.[START_REF] Wang | Edge server placement in mobile edge computing[END_REF]

Figure 6 .

 6 Figure 6. Workload results of GOESP, Top-K, K-means, and Wang et al. Approaches

Figure 7 .Figure 8 .

 78 Figure 7. Dropped tasks results

Figure 9 :

 9 Figure 9: Latency simulation results: GOESP, Top-K, K-Means and Wang et al. [18].

TABLE 2 .

 2 RSU CLASSIFICATION

	Category	RSU
	High demand	R9, R10, R12, R23
	Moderate demand	R3, R4, R5, R6, R7, R11, R15, R16, R17, R22
	Low demand	R1, R2, R8, R13, R14, R18, R19, R20, R21, R24,
		R25, R26, R27

TABLE 3 .

 3 MODEL AND SIMULATION PARAMETERS

	AIMMS parameters
		Location with	1 unit
		a green energy
	Edge server deployment cost at	source
	location i	Location	2 unit
		without a green
		energy source
		High Demand	30 unit
	(processing/Memory/Storage) RSU demands	Demand	20 unit
		Low Demand	10 unit
	Edge server capacity (Processing/Memory/Storage)	150 unit
	Latency threshold (in distance)	1500 m
	Omnet/Artery parameters
	RSU transmission power	47,9 mW
	RSU antenna high	5m
	Vehicle transmission	200 mW
	power
	Frequency band	5.9 GHz
	Propagation model	ConstantSpeedPropagation (speed
		of light)
	Pathloss model	GEMv2 [28]
	Obstacle loss model	DielectricObstacleLoss
	Background noise Model	IsotropicScalarBackgroundNoise
		(-110 dBm)
	Message size (payload)	1000 Bytes
	Message frequency	1 Hz
	Vehicles max density	729 Vehicles

TABLE 4 .

 4 EDGE SERVERS' PLACEMENT RESULTS

	Placement strategy	Edge servers' locations	Deployment
			cost
	GOESP	Loc 3, Loc 4, Loc 13, Loc 19	5
	Top-K	Loc 7, Loc 8, Loc 10, Loc 17	7
	K-means	Loc 6, Loc 8, Loc 13, Loc 17	7
	Wang et al. [18]	Loc 3, Loc 10	3

TABLE 5 .

 5 EDGE SERVERS' PLACEMENT AND RSU CONNECTIONS RESULTS


Sabri Khamari
email: sabri.khamari@u-bordeaux.fr

Toufik Ahmed

Mohamed Mosbah
email: mohamed.mosbah@u-bordeaux.fr

Efficient Edge Server Placement under Latency and Load Balancing Constraints for Vehicular Networks

Keywords: Edge servers' placement, Vehicular networks, Edge Computing, Optimization. I

Vehicular applications in smart cities, such as assisted and autonomous driving, require sophisticated data processing, low latency, and high throughput data transmission. Edge Computing is a leading approach designed to meet those application requirements. By deploying Edge servers at the network's edge, close to the vehicles, such applications can be successfully delivered while adhering to low-latency and highthroughput requirements. However, optimal placement of Edge servers is challenging since it necessitates a trade-off between quality of service and deployment cost. Latency can be reduced by placing as many Edge servers as feasible close to the vehicles, however, this results in significant deployment costs. This work addresses the problem of optimal Edge server placement. It solves this problem using integer linear programming, considering the relation between delay and cost, as well as the capacity of Edge servers in realistic road traffic scenarios. The proposed generic methodology is designed to reduce the cost of deploying Edge servers by combining the achievement of the desired latency threshold with workload balancing between Edge servers. We evaluate the efficiency of the proposed solution mathematically and through simulations based on open data from real vehicles traffic on roadways of Bordeaux, France. The obtained results demonstrate that our solution outperforms existing Edge server placement approaches, especially on workload balancing.

INTRODUCTION

Vehicular networks are considered a significant component of future Intelligent Transportation Systems (ITS). Vehicular applications provide safer and more comfortable driving, better traffic efficiency, and a variety of value-added applications. Although vehicular applications are emerging and becoming computation-intense, they require complex processing of data and substantial storage space [START_REF] Liu | Vehicular edge computing and networking: A survey[END_REF]. Usually, vehicles generate and transmit data to faraway cloud computing infrastructures with abundant resources for processing. Cloud computing enables the provision of resources on-demand, cost savings, scalability, and rapid application development [START_REF] Apostu | Study on advantages and disadvantages of Cloud Computing-the advantages of Telemetry Applications in the Cloud[END_REF]. However, the data transmission delays between users and the Cloud can be long and unpredictable, rendering them unsuitable for time-sensitive vehicular applications [START_REF] Javadzadeh | Fog computing applications in smart cities: A systematic survey[END_REF].

To overcome these limitations, the edge computing concept has been developed. Indeed, placing storage and compute resources at the network's boundary, as close as possible to endusers, results in lower latency and more responsive real-time operation [START_REF] Cao | An overview on edge computing research[END_REF]. The fundamental concept of Edge computing is to deploy a collection of Edge nodes or servers across a geographical region. As a result, processing and storage resources are made accessible to end-users, typically within a single hop.

Edge server deployment in urban vehicular networks is a major challenge. Increasing the number of Edge servers helps to reduce latency for the envisioned vehicle applications but increases cost. Edge server physical locations are crucial for ensuring high performance. Inefficient Edge server placement results in increased latency and drastically uneven workloads among Edge servers, i.e., some Edge servers are overwhelmed while others are underutilized [START_REF] Li | Load balancing edge server placement method with QoS requirements in wireless metropolitan area networks[END_REF]. This raises concerns regarding the optimal placement of Edge servers in metropolitan environments, where a massive volume of data is generated due to the high density of users, while considering deployment cost and network performance constraints. This paper aims to establish an optimization model and a scientific methodology for efficiently placing Edge servers. More particularly, it proposes an integer linear programming approach for minimizing the cost of deploying Edge servers while simultaneously fulfilling the desired latency threshold and balancing workloads between Edge servers. We elaborate a methodology based on open data from real-world vehicles traffic from the city of Bordeaux to provide a realistic solution for Edge server placement [6]. This methodology can be extended to other sources of data traffic. The proposed solution is evaluated through extensive simulations and the obtained results show that our solution can effectively solve the Edge server placement problem in a real-world vehicular environment. It outperforms existing Edge server placement strategies, including Random, Top-K, K-means, and Wang et al. [START_REF] Wang | Edge server placement in mobile edge computing[END_REF]. Our solution is based on multi criteria optimization, we satisfy the latency, the workload balancing and the cost.

The remainder of this paper is organized as follows. Section II presents related works in Edge servers' deployment. Then, we introduce the proposed model for the efficient placement of Edge servers in Section III. Section IV presents our methodology and simulation setup. Finally, we conduct performance analysis in Section V, and we provide some concluding remarks in Section VI.

II. RELATED WORK

Most existing studies on vehicular networks have addressed the cost-effective placement of roadside units principally from a communication perspective [START_REF] Zhang | An optimal roadside unit placement method for vanet localization[END_REF]. A large share of works targeting Edge computing architecture focused on resource allocation and scheduling [START_REF] Sorkhoh | Workload scheduling in vehicular networks with edge cloud capabilities[END_REF]. Even though there are a few solutions explicitly targeting Edge servers' placements in vehicular networks. The paper presented in [START_REF] Premsankar | Efficient placement of edge computing devices for vehicular applications in smart cities[END_REF] developed an optimization problem and a methodology that the infrastructure providers can use to deploy Edge computing devices in a smart city. The authors proposed a mixed-integer linear programming formulation that minimizes the deployment cost of Edge computing devices by jointly satisfying a target level of network coverage and computational demand. However, this work considers the deployment of Edge servers alongside roadside units, leading to the deployment challenges of the RSUs to ensure a good coverage ratio of the network. Even if this work aims to ensure a certain level of computational demand, it did not consider the latency requirements of vehicular applications. Wang et al. [START_REF] Wang | Edge server placement in mobile edge computing[END_REF] studied the Edge server placement problem in mobile Edge computing environments for smart cities. They formulated the problem as a multi-objective constraint optimization problem that places Edge servers in some strategic locations to balance the workload between Edge servers and minimize the Edge server access delay. Then, they adopted mixed integer programming to find the optimal solution. Even if this work did not target the vehicular network specifically, it is the most related to the problem considered in this paper. We will consider it for comparison with our proposed solution. Laha et al. [START_REF] Laha | Edge nodes placement in 5g enabled urban vehicular networks: A centrality-based approach[END_REF] considered the Edge server placement in 5G-enabled urban vehicular networks. They addressed the efficient deployment of a limited number of Edge servers in an urban scenario under a restricted budget. They considered the structural properties of the road network using complex-network-based centrality metrics and the vehicular traffic distribution to rank the candidate sites for the Edge servers. The problem is mapped to the 0-1 Knapsack problem to maximize the total profit based on the previously cited metrics. However, this work focuses on the coverage ratio, the ratio of the number of vehicles covered by the deployed Edge servers to the total number of vehicles in the network and did not consider the latency and workload balancing constraints.

Our work aims to provide an efficient Edge servers placement approach to minimize the deployment cost of Edge servers by jointly satisfying a target latency threshold and ensuring workload balancing between Edge servers.

III. EFFICIENT PLACEMENT OF EDGE SERVERS

Before proceeding further, let us introduce the reference architecture illustrated in Figure 1. We consider vehicles moving in an urban area. These vehicles run AI-based applications and rely on the Edge servers for advanced learning capabilities depending on the collected data from the surrounding environment. Thus, vehicles communicate with roadside units (RSUs) wirelessly using ITS-G5/IEEE802.11p, while edge servers communicate with RSUs via IP broadband connection, typically via optical fiber. Multiple RSUs can be connected to a single Edge server. Edge servers process the data offloaded by vehicles. Additionally, these Edge servers are connected to the Internet, which provides extra storage and processing resources to enable the Cloud-Edge continuum. Edge servers are also connected to the road operator's platform that provides V2X services and security mechanisms (e.g. AAA and PKI).

Figure 1. Vehicular Edge computing reference architecture

In the next section, we propose "OptPlacement", our solution for an efficient Edge server placement under real traffic and server's capacity constraints, followed by the scientific methodology used for evaluation.

A. System model

The system is composed of a set J = {S1, S2, …, Sm} of potential locations for Edge servers and a set I ={R1, R2, …, Rn} of roadside units. Vehicles transmit data requests or messages to a roadside unit, which offloads processing to the Edge server. The total number of requests received by an RSU will be utilized to calculate its demands. RSU demand represents the amount of processing, memory, and storage resources that should be available on the edge server to handle all requests received by this RSU. We suppose that vehicles send their requests with the same frequency.

Given the notation in Table 1, our objective is to minimize the cost of deploying Edge servers while ensuring a target latency requirement and a load balancing to prevent Edge servers from being overloaded. K Edge servers are to be placed among m potential locations. The following constraints are considered in our formulation:

• Each roadside unit is connected to only one Edge server.

• Each Edge server handles vehicles' requests from the roadside units.

• Each RSU has a demand which represents the sum of requests of vehicles communicating with this RSU. Accordingly, we formulate an integer linear programming model that captures the features of the model under consideration. We begin by defining the decision variables of the model:

1 , 0 , (1) 1
, " # $ 0 , (2) ∑ &' : total cost (3)
The objective function is defined as:

, . . ) &' (4)
s.t

) 1 , ∀ + , &' (5)
) - , &' ∀ + , (6) ) &.
-, ∀ + / (7)

) &. - ∀ + / (8)
)

&.

-∀ + / (9)

+ 00,11 ∀ + / (10) + 00,11 ∀ + / , ∀ + , (11) We explain each expression in our model, starting from the decision variables in (1) and ( 2), denotes a boolean variable indicating whether a candidate position is selected to deploy the Edge server. is a boolean variable indicating which Edge server each roadside unit is connected to. For example, 23 = 1 indicates that RSU "3" is connected to the Edge server deployed in position "4". The total cost function which depends on the number of edge servers and their deployment costs is presented in equation ( 3). The objective function in (4) aims to minimize the deployment cost by decreasing the number of Edge servers to be deployed where denotes the cost of deploying an Edge server at location j from J. Next, we describe the constraints. Equation ( 5) ensures that each roadside unit is connected to one of the deployed servers. Therefore, no RSU is left unconnected or connected to more than one server. Constraint (6) guarantees that each RSU is connected to an Edge server without exceeding the maximum tolerated latency which is defined by distance. Therefore, this constraint ensures that each RSU is within the deployed Edge servers' coverage range.

Constraints [START_REF] Wang | Edge server placement in mobile edge computing[END_REF][START_REF] Zhang | An optimal roadside unit placement method for vanet localization[END_REF][START_REF] Sorkhoh | Workload scheduling in vehicular networks with edge cloud capabilities[END_REF] ensure that memory, storage, and processor requirements are met. They ensure that the total number of requests submitted to the Edge server by roadside units does not exceed its capabilities. Finally, constraints [START_REF] Premsankar | Efficient placement of edge computing devices for vehicular applications in smart cities[END_REF][START_REF] Laha | Edge nodes placement in 5g enabled urban vehicular networks: A centrality-based approach[END_REF] ensure the integrality requirements of the decision variables.

IV. METHODOLOGY AND SIMULATION SETUP

This section describes the preliminary processing necessary to generate the input data for our model. Then, the evaluation methodology and simulation setup are described.

A. Preliminaries

We consider the city of Bordeaux, France, as the target area to place Edge servers. A map of the Bordeaux center which bound an area of 15.10 km² is acquired from OpenStreetMap [12] (see Figure 2). To begin, we examine the placement of roadside units in this map using a uniform strategy. We used this approach because there is no available data on actual RSUs deployment. 27 roadside units were installed at road crossroads spaced by around 800 meters to achieve optimum network coverage. Next, we identify potential location for Edge servers using a straightforward technique. Between every three or four RSU, one location is assigned. This ensures that each RSU has a possible Edge server associated with it without exceeding the maximum acceptable latency (which is defined by distance). A set of m=19 Edge server candidate locations is fed into an optimization model that determines the most strategic locations for Edge server deployment.

Figure 2 . Use case scenario: RSU locations and Edge server candidate locations on Bordeaux map

To perform simulations that are the most accurate representations of the real environment, we need to generate traffic traces that are identical to the actual traffic on Bordeaux's roads. We used the open dataset OpenDataBordeaux [6], which provides vehicles' numbers and speed at a geographical point. Based on this data, we identified three classes of roads: high traffic roads, moderate traffic roads, and low traffic roads. After that, we generate the road traffic trace using the RandomTrips tool provided by Simulation of Urban Mobility (SUMO) [START_REF] Lopez | Microscopic traffic simulation using sumo[END_REF] along with the road classification as an input (by integrating the option Customized Weights of RandomTrips).

Once the traffic trace for our simulation is created, we determine the demands for each roadside unit. The RSU demands are expressed by the number of messages or tasks transmitted to the RSU by the vehicles. Assuming that all vehicles transmit messages at the same rate, the RSU demands are calculated based on the number of vehicles in each RSU's covered region. Next, we perform a simulation to record the number of messages received by each RSU. Based on the results presented in Figure 3, the roadside units are grouped into three categories: high demand, moderate demand, and low demand. Table 2 presents the obtained results (Ri represents roadside unit at location i).

B. Methodology and simulation setup

Figure 4 provides a comprehensive overview of the methodology and simulation setup used to evaluate our placement strategy. The solution comprises two major parts. The first part is dedicated to solve the optimization problems utilizing AIMMS (Advanced Interactive Multidimensional Modeling System) [START_REF]Download AIMMS developer[END_REF] in conjunction with CPLEX12.10 solver. AIMMS is a perspective analytics platform that allows the modeling and development of optimization-based applications. An optimal solution is determined using the following inputs: the geographical coordinates of roadside units, the geographical coordinates of Edge server's candidate locations, the latency threshold (measured in distance), Edge servers' capacity, and RSUs demands (in terms of processing, memory, and storage). As a result, the solver provides the number of Edge servers to be deployed and servers' locations among the candidate locations. Additionally, the total cost of deployment is determined. Finally, one of the Edge servers is assigned to each roadside unit. The connection between the RSU and the server adheres to the latency constraint, and the aggregate of RSU demands linked to a single server must not exceed the capacity of the server. The computer used for this optimization problem has the following configuration: processor intel i7-10610U CPU @ 1.80GHz 2.30 GHz and RAM 16 GO.

The second part of this solution uses the obtained results from the first part as an input to conduct simulations using OMNeT++ [START_REF] Varga | OMNeT++ Discrete Event Simulation System[END_REF] and the Artery framework [START_REF] Riebl | Artery: Large Scale Simulation Environment for ITS Applications[END_REF]. In addition, the traffic trace generated by SUMO [START_REF] Lopez | Microscopic traffic simulation using sumo[END_REF] is also required as input to the simulation. Finally, the simulation also integrates the radio model, which comprises the following characteristics: propagation model, obstacle loss model, path loss model, and the background noise model as presented in Table 3. The simulations were conducted on a computer with 16 GO RAM and the Intel i7-9700 CPU @ 3.00GHz x8 processor. Table 3 presents the different parameters used to find the optimal solution and to conduct simulations. The RSU's processing, memory, and storage demands depend on the roadside unit. RSU demands values are given by unit, which represents the number of received messages by second. The server capacity is expressed by unit, representing the capacity needed in terms of processing, memory, and storage to handle the received messages by second. Our methodology and solution are generic and may be used with any map or source of data traffic. According to the different inputs, the model provides the optimal solution for the efficient placement of edge servers.

V. PERFORMANCE EVALUATION

We compared the performance of our proposed approach OptPlacement with other placement strategies in terms of both latency and workload balancing. We select the following strategies: Random, top-K, and K-means strategies, as baseline comparisons because they are the most used strategies in such problems. Furthermore, Wang et al. approach [START_REF] Wang | Edge server placement in mobile edge computing[END_REF] is also used for comparison since it is the most relevant work identified in the literature. We evaluated two variants of the random approach according to how the roadside units are connected to the Edge servers. The first one is Random-Random, where the RSUs are linked to the servers randomly. The second is Random-Nearest, where each RSU is linked to the nearest Edge server.

We measured for each strategy two critical metrics: the latency and the workload balancing. The latency metric represents the time needed to send a message from the vehicle to the Edge server. The workload balance is represented by the percentage of dropped tasks for each approach. For that, we assume that each Edge server has limited computing capacity and if the server is overloaded the arriving tasks are dropped. The workload balancing will depend principally on the number of roadside units connected to each server and their associated demands.

A. Placement results

According to our scenario, the results obtained by the AIMMS solver indicate that four (K=4) Edge servers should be placed in locations 1, 6, 12, and 19 (see Figure 2). This means that the total deployment cost of Edge servers is equal to 4 units (the optimal solution). The link between each roadside unit and the Edge server is shown in Table 4, as is the latency between the entities stated in terms of distance. For instance, Server 1, located at position 1, is used to connect R1 at a distance of 591 meters. Additionally, we show the Edge servers' workload, which is the aggregate of the demands of the roadside units connected to this server. It is 110 units for server 1.

Table 4. Optimal placement results

Edge server

Roadside ID (distance to the edge server) Workload (units) Server 1 (at location 1) R1 (591m), R2 (1088m), R3 (458m), R5 (736m), R7 (1210m), R9 (1465m) 110 Server 2 (at location 6) R4 (193m), R6 (1272m), R8 (975m), R11 (841m), R13 (1148m), R15 (1332m), R16 (1384m) 120 Server 3 (at location 12) R10 (1441m), R12 (983m), R14 (293m), R17 (967m), R18 (439m), R19 (1036m), R20 (1120m) 120 Server 4 (at location 19) R21 (1462m), R22 (1090m), R23 (1152m), R24 (563m), R25 (367m), R26 (550m), R27 (1275m) 100 This optimal solution is obtained in 42,11 seconds and with 129,1 Mb of used memory. The obtained results satisfy the latency and server's capacity constraints. All RSUs (R1 to R27) are connected to one of the Edge servers without exceeding the maximum latency (expressed in the distance) of 1500m which is used as an input to the model. Regarding the capacity constraint, for example, the sum of RSUs linked to server 1 is 110 units, which is less than server 1's capacity (150 units).

Table 5 below summarizes the obtained results using other placement approaches. For comparison purposes, in the Random, Top-K, and K-means approaches, we fixed K = 4, as the number of Edge servers to be deployed. This allows these approaches to be aligned with the solution obtained by our approach. Therefore, we consider that they have the exact deployment cost (K=4). In Wang et al. approach [START_REF] Wang | Edge server placement in mobile edge computing[END_REF], two servers are deployed according to the optimal solution of the proposed model. In this approach, the maximum latency threshold is not considered, and therefore, two servers are deployed to reduce the cost.

B. Performance results

As previously stated, the considered metrics are the latency and the percentage of the dropped tasks to evaluate workload balance between servers. Figure 6 and Figure 7 depict the latency results obtained by the simulations for the aforementioned approaches. The findings show the average latency by second of messages received by the servers. Our approach OptPlacement achieves a latency that varies between 1,51ms in low density and 1,61ms in high density. The other approaches offer a latency in approximately the same interval. When we examine the findings more closely, we see that OptPlacement outperforms Random-Random and Wang et al. [START_REF] Wang | Edge server placement in mobile edge computing[END_REF]. Additionally, Random-Nearest and OptPlacment have superposed curves and provide similar results. On the other hand, our solution presents a higher latency of 0,01ms (maximum) compared to Top-K and K-means approaches. This could be explained by the fact that those approaches use the nearest server criteria, i.e., each RSU is connected to the nearest server without considering other constraints like the server capacity and the workload balancing. For example, as depicted in Table 5, in the Top-K approach, server 2 with the capacity of 150 units, cannot handle all the tasks of the roadside units connected to it (160 units of total demands). Therefore, this minor increase of latency is acceptable for our solution to ensure that the server's capacity is not exceeded. On the other hand, the proposed model ensures that the latency threshold ( ) constraint is always respected. To conduct a more thorough examination of the placement methods' performance, we investigated workload balancing amongst the Edge servers. If a server is overloaded the next arriving task is dropped. Figure 8 presents the percentage of dropped tasks for each approach. The results show that our approach can deliver the best performance as 2,90% of tasks are dropped. The K-Means and Top-K approaches have shown slightly better results in terms of latency. However, their performance in terms of workload balancing is not promising. The Top-K approach has 5,67% of dropped tasks, which is 95,5% higher than our approach, and the K-means strategy has 13,06% of dropped tasks. The percentage of dropped tasks in the Wang et al. approach [START_REF] Wang | Edge server placement in mobile edge computing[END_REF] is increased by 394% compared to our approach. Those results validate that our proposed model can provide a better workload balancing compared to the other Edge server placement strategies while ensuring the latency constraint.

VI. CONCLUSION

Edge computing is an important emerging technology that can be used to improve the performance of a wide variety of applications, particularly vehicular applications. This paper provides an effective strategy for Edge servers' placement. To give a realistic Edge server location, we used a methodology based on the real-world traffic data set for Bordeaux. Our model is generic and applicable to other maps and different data set of traffic. Using AIMMS analytical platform, we obtained the optimal solution to the placement problem. Then, we conducted simulations to evaluate the solution's performance. The obtained results showed that our proposed solution is effective and outperforms existing approaches by ensuring load balancing among Edge servers and minimizing deployment costs. Therefore, our solution decreases the number of dropped tasks by an average of 56% compared to the other placement strategies. Furthermore, our solution satisfies the constraint of the maximum latency that could not be exceeded. Accordingly, our proposed strategy for Edge server placement achieves an optimal trade-off between deployment cost, workload balancing, and latency. As future work, we intend to investigate the service migration between Edge servers to ensure service continuity and decrease application delay.

Figure 3 .

 3 Figure 3. Roadside unit demands on Bordeaux's roads.

Figure 4 .

 4 Figure 4. Overview of the used methodology to evaluate "OptPlacement".

Figure 5

 5 Figure 5 presents the vehicle density variation over simulation time according to the traffic trace generated from the OpenDataBordeaux dataset.

Figure 5 .

 5 Figure 5. Vehicles' density over the simulation time.

  R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25 R26

Figure 6 .

 6 Figure 6. Latency simulation results: OptPlacement, Random-Random and Random-Nearest.

Figure 7 .

 7 Figure 7. Latency simulation results: OptPlacement, Top-K, K-Means and Wang et al. [7].

Figure 8 .

 8 Figure 8. Dropped tasks results.

  • Each Edge server has limited computing and storage capacity (considered equal in this work). Binary variable for Edge server placed at location j Binary variable for RSUi linked to Edge server at location j The total cost of Edge server's deployment

	Memory capacity of server j
	Storage capacity of server j
		Table 1. Notations
	Symbol	Description
	I	Set of Roadside units (RSU)
	J	Set of the potential location of Edge servers
	n	Number of roadside units
	m	Number of potential locations for Edge servers
	K	The number of deployed Edge servers
		Cost of deployment of Edge server at location j
		Maximum latency (proportional to the distance)
		The latency between RSUi and Edge server location j
		Processing demands of RSU i
		Memory demands of RSU i
		Storage demands of RSU i
		Processing capacity of server j

Table 2 .

 2 RSU classification

	Category	RSU
	High demand (>=1500)	R9, R10, R12, R23
	Moderate demand (400=<x<1500)	R3, R4, R5, R6, R7, R11, R15, R16,
		R17, R22
	Low demand (<400)	R1, R2, R8, R13, R14, R18, R19,
		R20, R21, R24, R25, R26, R27

Table 3 .

 3 Simulation parameters

	Aimms parameters
	Edge server deployment cost at location i	1 unit
	RSU demands (processing/Memory/Storage)	High Demand Moderate Demand Low Demand	30 unit 20 unit 10 unit
	Edge server capacity (Processing/Memory/Storage)	150 unit
	Latency threshold (in distance)	1500 m
	OMNeT/Artery parameters
	Pathloss model	GEMv2 [17]
	Propagation model	Constant Speed Propagation (speed of light)
	Obstacle loss model	Dielectric Obstacle Loss
	Background noise Model	IsotropicScalarBackgroundNoise (-110 dBm)
	Message size (payload)	1000 Bytes
	Message frequency	1 Hz

Table 5 .

 5 Placement results of the other approaches

	Approach	Edge server (placement location)	Workload
		Server 1 (at location 7)	110
	Top-K	Server 2 (at location 8) Server 3 (at location 10)	160 80
		Server 4 (at location 17)	100
		Server 1 (at location 6)	150
	K-means	Server 2 (at location 8) Server 3 (at location 13)	80 90
		Server 4 (at location 17)	90
	Wang et al. [7]	Server 1 (collocated with RSU 3)	250
	approach	Server 2 (collocated with RSU 15)	200