Cyril Nguyen

Van Phu

Nadir Farhi

Estimation of urban traffic state with probe vehicles

Keywords:

We present in this paper a method to estimate urban traffic state with communicating vehicles. Vehicles moving on the links of the urban road network form queues at the traffic lights. We assume that a proportion of vehicles are equipped with localization and communication capabilities, and name them probe vehicles. First, we propose a method for the estimation of the penetration ratio of probe vehicles, as well as the vehicles arrival rate on a link. Moreover, we show that turn ratios at each junction can be estimated. Second, assuming that the turn ratios at each junction are given, we propose an estimation of the queue lengths on a 2-lanes link, by extending a 1-lane existing method. Our extension introduces vehicles assignment onto the lanes. Third, based on this approach, we propose control laws for the traffic light and for the assignment of the arriving vehicles onto the lane queues. Finally, numerical simulations are conducted with Veins framework that bi-directionally couples microscopic road traffic and communication simulators. We illustrate and discuss our propositions with the simulation results.

I. INTRODUCTION A. State of the art

Different techniques are traditionally used to measure road traffic parameters; for example we can cite inductive loops or video cameras. There is nowadays an infrastructure-less technique to estimate traffic flow parameters such as queue lengths : GPS localization system coupled with communicating vehicles, namely probe vehicles. This kind of equipment penetration ratio is increasing and does not need heavy set up.

Probe vehicles were historically studied for measuring travel times [START_REF] Lin | Estimating Nationwide Link Speed Distribution Using Probe Position Data[END_REF]. They also helped to estimate penetration ratio and arrival rate of vehicles (equipped and non equipped) on a link. For example, the author of [START_REF] Comert | Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters[END_REF] derived these estimations from the estimation of queue lengths at junctions, queue lengths being estimated using the information provided by the probe vehicles. Thus, we can see that in order to characterize urban road traffic state and its primary parameters such as arrival rates or penetration ratio, estimating queue lengths at junctions is an important step. Furthermore, Varaiya [START_REF] Varaiya | Max pressure control of a network of signalized intersections[END_REF] has modeled a road network as "a controlled store-and-forward (SF) queuing network" and proposed an algorithm to control this network of queues. Indeed, minimizing delays and waiting times can be done by minimizing queue lengths at junctions controlled with traffic light signals. Hence, queue length estimation is a major measurement input data, used to control traffic light signals, and so transportation road networks.

Concerning queue lengths, in 1963, Miller [START_REF] Miller | Settings for fixed-cycle traffic signals[END_REF] found an approximation of the average queue length at junctions. More recently, the authors of [START_REF] Ramezani | Exploiting probe data to estimate the queue profile in urban networks[END_REF] used shockwave theory to refine queue length estimation. Some works also proposed to use probability distribution of the queues [START_REF] Heidemann | Queue length and delay distributions at traffic signals[END_REF]. Other works used Markov chains to model the dynamics of queue lengths [START_REF] Viti | Modeling queues at signalized intersections[END_REF]. The authors of [START_REF] Comert | Queue length estimation from probe vehicle location and the impacts of sample size[END_REF] and [START_REF] Shahrbabaki | A data fusion approach for real-time traffic state estimation in urban signalized links[END_REF] have addressed the queue length estimation with probe vehicles by proposing a probabilistic analytical model. In [START_REF] Comert | Queue length estimation from probe vehicle location and the impacts of sample size[END_REF], the authors have estimated queue length in under-saturated traffic conditions, with the "a priori knowledge of the marginal distribution of the queue length" and using "the location information of the last probe vehicle in the queue". The authors of [START_REF] Shahrbabaki | A data fusion approach for real-time traffic state estimation in urban signalized links[END_REF] have proposed a method to estimate the queue length, the incoming arrival rate, and the output flow, on a m-lanes link (m ≥ 2). The estimations are given for low or saturated demand with no requirement of information concerning the timings of the traffic light signal. In [START_REF] Shahrbabaki | A data fusion approach for real-time traffic state estimation in urban signalized links[END_REF], all the lanes are assumed to be balanced (i.e. cars share the lanes of the link without any preference). Therefore, all the lanes would have the same length. In [START_REF] Zheng | Estimating traffic volumes for signalized intersections using connected vehicle data[END_REF], the authors estimate arrival rate for low penetration ratio of equipped vehicles. The method proposed in [START_REF] Zheng | Estimating traffic volumes for signalized intersections using connected vehicle data[END_REF] uses as input data "vehicle trajectories approaching to an intersection as well as traffic signal status". The trajectories of equipped vehicles are used to detect if a probe vehicle has stopped at the traffic light and its stopping position. With these information, the arrival rate is estimated and bounds for this arrival rate are given. In [START_REF] Tiaprasert | Queue Length Estimation Using Connected Vehicle Technology for Adaptive Signal Control[END_REF], the authors proposed another method. They have lower-bounded the queue length by "the location of the last stopped connected vehicle" and upper-bounded it, when the bound exists, by the location of the "closest moving connected vehicle". Once bounded, the queue length is estimated using the least-mean-square-error method and the noise is filtered using discrete wavelet transform. In 2015, the authors of [START_REF] Lee | Real-time estimation of lane-based queue lengths at isolated signalized junctions[END_REF] have addressed the two lanes case by combining discriminant models "based on time occupancy rates and impulse memories" from detectors. The proportions of total traffic volume in each lane are estimated with Kalman filter. In 2018, the authors of [START_REF] Rompis | Probe vehicle lane identification for queue length estimation at intersections[END_REF] have also addressed the two lanes case. They have measured " individual probe vehicles' shockwave speed". Then the lane each probe is moving on, is determined by discriminating the two lanes with data clustering methods. They have shown that a bivariate mixture model clustering gives the best results. Shockwave theory and LWR (Lighthill, Whitham and Richards) model [START_REF] Lighthill | On kinematic waves ii. a theory of traffic flow on long crowded roads[END_REF], [START_REF] Richards | Shock waves on the highway[END_REF] refine the queue length estimation.

B. Paper contribution and organization

We present here an extension of an existing method that uses probe vehicles for the estimation of urban traffic state, including penetration ratio of communicating vehicles, vehicular arrival rates, as well as the queue lengths of an urban link. Our extension considers the general case where different destinations can be associated to the lanes, which produces different arrival rates to each lane of the urban link. We propose here to estimate the joint probability distribution of all the queue lengths of the urban link, instead of estimating only one queue length for the link, as done in [START_REF] Shahrbabaki | A data fusion approach for real-time traffic state estimation in urban signalized links[END_REF], [START_REF] Zheng | Estimating traffic volumes for signalized intersections using connected vehicle data[END_REF]. This distinction of the lane queues improves the estimation of the number of cars on the queues. Moreover, it gives us the possibility to control the flows of each queue separately, and then ameliorates the traffic control on the junction. We propose in addition control laws for balancing the queue lengths in a multi-lane link. We present here our method on a link of two lanes. We think that the ideas presented in this particular case could be adapted in order to address the general case (m-lanes, with m > 2). Furthermore, we think the method proposed here could be used and extended in a decentralized manner to the network case because of the low computational effort needed for the one link case. The estimators and the control laws we propose here would permit us to perform multi-level urban traffic control, as initiated in [START_REF] Phu | A vehicle-toinfrastructure communication based algorithm for urban traffic control[END_REF] (local control) and in [START_REF] Farhi | A semi-decentralized control strategy for urban traffic[END_REF] (semi-decentralized control).

In section I we give an introduction with the related works. In section II we describe the problem statement and the notations. In section III we propose an estimation of traffic state parameters : penetration ratio of probe vehicles, vehicles arrival rate (subsection III-A), and queue lengths in the case of two incoming lanes (subsection III-B). Our method estimates queue lengths at junctions with two lanes incoming roads, under the hypothesis of under-saturated traffic (moderate/low demand without overflow queue). We also assume that the GPS localization system is not able to determine which lane a vehicle is moving on because of a typically five meters accuracy [START_REF] Van Diggelen | The world's first gps mooc and worldwide laboratory using smartphones[END_REF]. We extend the analytic model proposed in [START_REF] Comert | Queue length estimation from probe vehicle location and the impacts of sample size[END_REF] to the two-lanes case by introducing a vehicle assignment model onto the lanes. In subsection III-C, we propose a control of the traffic light and an optimal assignment of the vehicles onto the lanes, in order to balance the two lanes queue lengths. In section IV, numerical simulations are conducted with Veins framework [START_REF] Sommer | Bidirectionally Coupled Network and Road Traffic Simulation for Improved IVC Analysis[END_REF] which bi-directionally couples microscopic road traffic and communication simulators. Finally, we conclude in section V.

II. PROBLEM STATEMENT

In this section we describe the main assumptions of our work and the notations used.

A. Assumptions a) Road network topology: We consider a road network composed of junctions controlled by traffic light signals, and links between junctions. We assume that all the incoming and outgoing links to/from a signalized junction have maximum two lanes. We define an entry link of the network as a link which does not have a start node. We assume that the geometry of the road network is known. A typical junction is represented on Fig. 1. We also assume that the timings of the traffic light signal are known, and specially the red times. b) Traffic demand: We assume that the travel demand is exogenous, which means that the demand is located only at the entry links of the network. We assume that the vehicles arrive onto each link l under a Poisson process of rate λ l . We consider in this paper the low/moderate demand case where the Poisson arrival assumption is valid. In [START_REF] Comert | Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters[END_REF] the author discusses the Poisson arrivals assumption and recalls that this assumption is commonly used to describe arrivals at isolated intersections, specially in the case of low/moderate demand with no overflow queue. The vehicles form queues at junctions. Since we assume Poisson arrivals, we consider that the queues are empty at the beginning of each red time (no overflow queue).

c) Probe vehicles: We assume that a ratio p (with 0 ≤ p ≤ 1) of vehicles are equipped with localization and communication systems and we name them probe vehicles. The probe vehicles send their positions and speeds to a road side unit (RSU) coupled with the traffic light signal of the junction. We assume that the transmit power of the communication system embedded in every vehicle is strong enough, and that the sensitivity of the RSU is accurate enough, such that the RSU can detect every vehicle in every incoming or outgoing link of its associated junction. We consider the case where the localization system embedded in the vehicles is not accurate enough to discriminate the lane the vehicle is moving on. d) Turn ratios: Fig. 2 represents the queues we consider, on a link of the road network. Probe vehicles are represented Vehicles going straight can choose both lanes. We will assume that a ratio α (with 0 ≤ α ≤ 1) of such vehicles going straight will choose the queue on lane M . So, on a 2-lanes link, we assume that the main flow λ is composed of three flows : 1) the flow with arrival rate λ n which is necessarily assigned to lane N (vehicles turning right). 2) the flow with arrival rate λ m which is necessarily assigned to lane M (vehicles turning left). 3) the flow with arrival rate λ nm which can be assigned to both lanes N or M (vehicles going straight). We consider that these three flows are independent and identically distributed (iid) stochastic arrivals, each one being a Poisson process. We suppose that a fraction α (with 0 ≤ α ≤ 1) of the flow λ nm is assigned to lane M and the complement (1 -α) of this same flow λ nm is assigned to lane N . As denoted in TABLE I, A t is the random variable representing the assignment onto the two lanes. We assume that A t is following a Bernoulli law such that P (A t = 1) = α and P (A t = 0) = 1 -α. Thus, E(A t ) = α. We define :

M t =8 N t =6 L p t =6
µ N (t) := r N (t)(λ n + (1 -α)λ nm ) (1) µ M (t) := r M (t)(λ m + αλ nm ) (2)
We will show later in Proposition 1 that µ N (t) and µ M (t) represent the average arrival rate multiplied by the red duration on respectively lane N and lane M . Also, we assume that the turn ratios are given. Indeed, it is easy to measure the turn ratios as following : the RSU detects all the probe vehicles in its radio range area. This is because the probe vehicles embed WAVE (Wireless Access in Vehicular Environments) [START_REF]Ieee guide for wireless access in vehicular environments (wave)architecture[END_REF] on OBU (on board unit). In [START_REF]Ieee guide for wireless access in vehicular environments (wave)architecture[END_REF], the basic safety messages (BSM) broadcast periodically the location and speed of probe vehicles. This is a default feature which is also implemented as a basic function in VEINS simulator [START_REF] Sommer | Bidirectionally Coupled Network and Road Traffic Simulation for Improved IVC Analysis[END_REF] that we use in this paper. So, if we know at time t the location of each probe vehicle on a given link and its unique identifier, it is enough to look at a time t + t x (t x being a time shift), where those vehicles are located. With this method, it is possible to estimate the turn ratios l n , l m and l nm which are the proportions of the main flow λ on the incoming link that respectively turn right, left or go straight. We will note : λ n = l n λ, λ m = l m λ, λ nm = l nm λ, with l n + l m + l nm = 1. We assume in this paper that l n , l m , and l nm are given.

B. Notations

We will use the notations of TABLE I.

III. TRAFFIC STATE ESTIMATION

A. Primary parameters estimation

In this section, we give a method for the estimation of the primary traffic parameters p and λ. We assume that every probe vehicle in the RSU radio range area is assigned to an incoming or outgoing link to/from the junction. Thus, the total number of incoming probe vehicles x p (t) in a given link to the junction is known.

TABLE I: Notations

We consider vehicles i moving at speed v i (t) and at a distance ρ i (t) (depending on time t) from the traffic light. Let us consider the following definition. Definition 1. For a given threshold car-speed v * and a given threshold car-distance ρ * to the junction, the vehicles queue

Q = Q(t, v * , ρ * ) is defined by Q = {i, v i < v * and ρ i < ρ * }.
ρ * is useful because if the queue would exceed the bound ρ * , we could know that the assumption of low/moderate demand is not adequate. Furthermore, ρ * is less than the edge length, so the queue keeps bounded. We then denote by Q p the subset of Q that includes only probe vehicles, Q p ⊂ Q. The total number of probes c p in the queue is given by the cardinal (number of elements) of the set Q p . We assume mixed vehicles (equipped and non equipped) with an average vehicle length L V and minimum distance gap G V between vehicles. ρ 0 denotes the offset distance from the RSU to the stop line of the traffic light signal. We propose to compute l p the last probe location in the unit of "number of vehicles" as follows. We have :

max i∈Qp (ρ i ) = ρ 0 + l p L V + (l p -1)G V (3)
Then,

l p = [(max i∈Qp (ρ i ) -ρ 0 + G V )/(L V + G V )] (4)
where [•] denotes the round operator to the nearest integer. Given c p , l p , Comert [START_REF] Comert | Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters[END_REF] has derived many estimators for p, one of them being c p /l p , which is biased for p < 1. We follow here the same idea and propose a variation of the estimator of p. For the one lane case, we propose:

p = (c p -1)/(l p -1), for l p > 1 (5)
We have : N t = l p + 1/p -1, where 1/p -1 represents the average backlog of the queue behind the last probe. Then p = c p /N t = c p /(l p + 1/p -1). Moreover, by following the same arguments of the proof in [START_REF] Comert | Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters[END_REF], it is easy to check that this estimator is unbiased for every p, i.e. E(p) = p, ∀p, 0 ≤ p ≤ 1.

For two lanes, we introduce : κ := min(µ n , µ m )/ max(µ n , µ m ). We can see that κ does depend only on turn ratios but not on the arrival rate for the link, because κ is a ratio.

κ = min(r N (t)(l n + (1 -α)l nm ), r M (t)(l m + αl nm )) max(r N (t)(l n + (1 -α)l nm ), r M (t)(l m + αl nm )) (6)
We consider queue lengths on lanes N and M respectively equal to n and m. We propose :

p = c p n + m (7)
By the way, in our case, the length n of queue N can be estimated with the number of arrivals on lane N during r N (t) which is µ n . As µ n +µ m = max(µ n , µ m )+min(µ n , µ m ) and by estimating max(µ n , µ m ) = l p + 1/p -1, where 1/p -1 represents the backlog of the queue behind the last probe, we can write :

µ n + µ m = max(µ n , µ m ) 1 + min(µ n , µ m ) max(µ n , µ m ) (8)
µ n + µ m = (l p + 1/p -1)(1 + κ) (9)
We introduce c κ = c p /(1 + κ) and replace ( 9) in [START_REF] Viti | Modeling queues at signalized intersections[END_REF]. Finally, we get the following equation :

p = c κ l p + (1 -p)/p (10)
Hence, solving in p, by putting p = p :

p = (c κ -1)/(l p -1), for l p > 1 and c p > 1 (11)
which extends (5) for the case of two lanes. Similarly, we can check that this estimator for two lanes is unbiased, by following again the same arguments as in [START_REF] Comert | Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters[END_REF]. In Appendix A, we propose another method to compute p which is based on the calculus of an unbiased estimator and gives the same result of formula [START_REF] Tiaprasert | Queue Length Estimation Using Connected Vehicle Technology for Adaptive Signal Control[END_REF].

We propose to compute λ with formula (12) by simply accumulating probe vehicles on the entire radio range area of the RSU during red time, and using x = x p /p. λ should be computed when all the lanes of the considered link have a red light at the traffic light. We formulate this as following, where t 0 is the starting time for the red light on both lanes and t 1 is the time after which the set of the two lanes are not at red light.

λ = x p (t 1 ) -x p (t 0 ) p(t 1 -t 0 ) (12)
B. Queue length estimation

Once p and λ are estimated with probe vehicles, we can refine our traffic state estimation (queue lengths). We propose in this section to estimate all the queue lengths associated to all the lanes on a link of the road network. We will propose a model that uses vehicular assignment onto the lanes, for links composed of two incoming lanes. In a first step, an analytical probability distribution formulation of the queue lengths, without using the information from the probe vehicles, will be presented. Then, we will use the information provided by the probe vehicles : while generalizing the work for 1lane road done in [START_REF] Comert | Queue length estimation from probe vehicle location and the impacts of sample size[END_REF] to the 2-lanes case, we will refine our analytical formulation. We recall here that we can not directly detect the lanes on which the probe vehicles are moving because of insufficient accuracy of GPS localization system [START_REF] Van Diggelen | The world's first gps mooc and worldwide laboratory using smartphones[END_REF], which makes the problem not obvious.

a) Distribution probability law of the 2-lanes without having the information provided by the probes: We first propose an estimation of the probability distribution P (N t = n, M t = M ) without having any information from the probe vehicles.

Proposition 1.

P (N t = n, M t = m) = µ N (t) n e -µ N (t) n! µ M (t) m e -µ M (t) m! (13)
Proof. We subdivide the Poisson process of rate λ nm common to the two lanes. The common arrival of rate λ nm is splitted with probability α to lane M and probability (1 -α) to lane N . The two produced flows are independent random flows each one following Poisson process of parameters respectively αλ nm for the flow assigned to lane M and (1 -α)λ nm for the flow assigned to lane N . Furthermore, the splitted Poisson processes are independent; see subdividing Poisson process in reference [START_REF] Gallager | Stochastic Processes: Theory for Applications, ser. Stochastic Processes: Theory for Applications[END_REF]. By combination, arrivals on lane N is the sum of two independent Poisson processes. Using the stationary property of Poisson processes, we can show that the number of arrivals in [0, r N (t)] on lane N is a Poisson process of parameter µ N . Similarly, the number of arrivals on lane M in [0, r M (t)] is a Poisson process of parameter µ M . As these two arrival flows on lanes N and M are independent, then the bivariate distribution probability law of the two queue lengths is the product of two Poisson Law of parameters µ N and µ M .

b) Distribution probability law of the 2-lanes queue lengths with the information provided by the probe vehicles:

We present here the conditional probability law of the two queue lengths, taking into consideration the information provided by the probe vehicles, specially the location of the last probe l p and the total number of probes in the two lanes queues c p . We recall here N t p and M t p are the number of probe vehicles respectively on the lane N and on the lane M , at time t. Proposition 2.

• If lp ≤ max(n, m) and c p ≤ n + m, then

P (N t = n, M t = m|L t p = l p , N t p + M t p = c p ) = lp-1+min(lp,n,m) cp-1 (1 -p) n+m P (N t = n, M t = m) j,k≥0 subject to max(j,k)≥lp j+k≥cp lp-1+min(lp,j,k) cp-1 (1 -p) j+k P (N t = j, M t = k) . • Otherwise, P (N t = n, M t = m|L t p = l p , N t p + M t p = c p ) = 0.
Proof. By Bayes' rule we have

P (N t = n, M t = m|L t p = l p , N t p + M t p = c p ) = P (N t = n, M t = m, L t p = l p , N t p + M t p = c p ) P (L t p = l p , N t p + M t p = c p ) (14)
Then the numerator in ( 14) is written

P (N t = n, M t = m, L t p = l p , N t p + M t p = c p ) = P (L t p = l p |N t p + M t p = c p , N t = n, M t = m) P (N t p + M t p = c p |N t = n, M t = m) P (N t = n, M t = m) (15)
We have

• P (L t p = l p |N t p + M t p = c p , N t = n, M t = m) = lp-1+min(lp,n,m) cp-1 / n+m cp . • P (N t p + M t p = c p |N t = n, M t = m) = n+m cp p cp (1 -p) n+m-cp . For the calculus of P (L t p = l p |N t p + M t p = c p , N t = n, M t =
m), we followed the same ideas as those of section 3 in [START_REF] Comert | Queue length estimation from probe vehicle location and the impacts of sample size[END_REF]. Indeed, we will use the example of Fig. 2 where l p = 6, M t = 8, N t = 6 and c p = 7. The probability is then computed by selecting the total number of events where L p = l p = 6 divided by the sample space. The sample space, which is composed of all the last probe possible locations is given by n+m cp = 14 7 . For L p = l p = 6 we must have all the probes in the preceding locations. The event space has a number of events corresponding to choosing (c p -1) = 6 probes among l p -1 + min(l p , n, m) = 6 -1 + 6 = 11 positions available. Here, the event space has a total number of elements given by lp-1+min(lp,n,m) cp-1 = 11 6 . This is why :

P (L t p = l p |N t p + M t p = c p , N t = n, M t = m) = l p -1 + min(l p , n, m) c p -1 / n + m c p (16)
For the calculus of P (N t p + M t p = c p |N t = n, M t = m) we have c p probe vehicles among n + m total vehicles. The probability for a vehicle to be a probe vehicle is p and the probability to be unequipped is (1 -p). The configurations considered in this case are c p vehicles equipped and (n + m -c p ) vehicles unequipped. The number of combinations of such configurations is n+m cp . This is why :

P (N t p + M t p = c p |N t = n, M t = m) = n + m c p p cp (1 -p) n+m-cp (17)
So the numerator in ( 14) is given by :

P (N t = n, M t = m, L t p = l p , N t p + M t p = c p ) = l p -1 + min(l p , n, m) c p -1 p cp (1 -p) n+m-cp P (N t = n, M t = m)
The denominator in ( 14) is the marginal distribution probability of P (N t = j, M t = k, L p = l p , N t p + M t p = c p ) on (j, k). Therefore, the ideas to compute this probability are the same as the ideas used to compute the numerator of [START_REF] Lighthill | On kinematic waves ii. a theory of traffic flow on long crowded roads[END_REF]. We notice here that the last probe position (in the unit number of vehicles) is necessarily less than or equal to the maximum of the queue lengths, since the last probe is necessarily in one of the two lanes queues. Similarly, the total number of probes c p is less than or equal to the total number of vehicles in the queues, since the probes are in the queues. Therefore, we can write :

P (L t p = l p , N t p + M t p = c p ) = j,k≥0 max(j,k)≥lp j+k≥cp P (N t = j, M t = k, L p = l p , N t p + M t p = c p ).
c) Estimators: The distribution probability law of the couple (N t , M t ) is known; see Proposition 2 . As, E(N t , M t ) = (E(N t ), E(M t )), one way to estimate the two queue lengths is to derive each queue length separately from the couple, by computing the expectation of N t and M t separately. We propose the following estimator for queue length on lane N :

E(N t |L t p = l p , N t p + M t p = c p ) = n≥0 n k≥0 P (N t = n, M t = k|L t p = l p , N t p + M t p = c p ) (18)
Similarly for the queue length on lane M , we will choose :

E(M t |L t p = l p , N t p + M t p = c p ) = m≥0 m j≥0 P (N t = j, M t = m|L t p = l p , N t p + M t p = c p ) (19)
C. Traffic light control and optimal assignment of vehicles onto the lanes

We are interested here in the equilibration of the two queue lengths with respect to the two parameters α and r := r N /r M . We exclude here the case r N = r M = 0 where both lanes have green light, and where no queue is formed by assumption; and therefore, the assignment onto the queues is meaningless. We use notations E (α,r N ) (N t ) := E(N t ) and E (α,r M ) (M t ) := E(M t ) in order to emphasize the dependence of these two expectations on the parameters α, r N and r M . Let us now define f (α, r) as follows.

f (α, r) := |E (α,r N ) (N t ) -E (α,r M ) (M t )|/E (α,r N ) (N t ) = |µ N -µ M |/µ N = λ|r N (ln+(1-α)lnm)-r M (lm+αlnm)| λr N (ln+(1-α)lnm) = |r(ln+(1-α)lnm)-(lm+αlnm)| r(ln+(1-α)lnm)
.

We are interested here in the minimization of f (α, r) with respect to the two parameters α and r, which permits the equilibration of the two queue lengths. Let us use the notations.

r * (α) := arg min r f (α, r). (20)
α * (r) := arg min

α f (α, r). (21)
Proposition 3 and Proposition 4 below determines r * (α) and α * (r) respectively.

Proposition 3. ∀α ∈ [0, 1], r * (α) = lm+αlnm ln+(1-α)lnm , and f (α, r * (α)) = 0. Proof. ∀α ∈ [0, 1], r = r0 = lm+αlnm ln+(1-α)lnm implies E (α,r N ) (N t ) = r N (λ n + (1 -α)λ nm ) = (r M r)(λ n + (1 -α)λ nm ) = r M (r 0 (λ n + (1 -α)λ nm )) = r M (λ m + αλ nm ) = E (α,r M ) (M t ).
Therefore, f (α, r0 ) = 0. Thus, r * (α) = r0 = lm+αlnm ln+(1-α)lnm .

Proposition 4. ∀r ≥ 0, α * (r) = max 0, min 1, rl n + rl nm -l m l nm (r + 1) .

Moreover, if r ∈ I := [ lm ln+lnm , lm+lnm ln ], then α * (r) = rl n + rl nm -l m l nm (r + 1)
, and f (α * (r), r) = 0.

Proof. For any r ≥ 0, α * (r) is simply the argument of the minimization of f (α, r) with respect to α, projected into the interval [0, 1]. In the case where r ∈ I := [ lm ln+lnm , lm+lnm ln ], we can easily check that the constraint α * (r) ∈ [0, 1] is not activated, and then we do not need to project into the interval [0, 1]. Moreover, in this case, α * (r) cancels f (α, r).

We notice here that the calculus of the optimal assignment proportion α * (r) of the vehicles going straight onto the lanes is done in deterministic at the macroscopic level (proportion of vehicular flow). The realization of the optimal assignment proportion α * (r) is done randomly at the microscopic level, as it has already been mentioned in section II.A.d: every vehicle going straight is randomly assigned to the the lanes M and N with probabilities α and (1-α) respectively, assuming that the vehicles going straight that can choose both lanes will choose the shortest queue. By equilibrating the two queues with r * (α) or α * (r), we can avoid spill-back onto the links of the network and by that reduce the risk of congestion. For example, the optimal r * (α) given in Proposition 3 can be taken into account as an additional constraint in the optimization problem of the traffic light split on every intersection, in such a way that the optimal traffic light setting will automatically balance the queue lengths on the incoming links of the intersections, which should help to avoid spill-back at the network level.

IV. SIMULATION RESULTS, EXAMPLES AND DISCUSSION

We present in this section the results of numerical simulations we conducted with Veins framework [START_REF] Sommer | Bidirectionally Coupled Network and Road Traffic Simulation for Improved IVC Analysis[END_REF] which combines the microscopic road traffic simulator SUMO [START_REF] Krajzewicz | Recent development and applications of SUMO -Simulation of Urban MObility[END_REF] with the communication simulator OMNET++ [START_REF] Varga | The omnet++ discrete event simulation system[END_REF]. The road network is one simple junction with links composed of two incoming lanes described in Fig. 1. The junction is controlled by a traffic light with a cycle duration of 90 s. The traffic demand is coming from West towards East, North and South. We vary the arrival rates and turn ratios depending on the scenarios, as we mentioned in TABLE II. The messages we use to detect the location of the vehicles are the Basic Safety Messages (BSM) that are sent in broadcast by every probe vehicle. Given the road network topology and the data provided by the probe vehicles, we can know on which incoming/outgoing link each probe vehicle is located.

A. Primary parameters estimation

λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.1 λ = 0.
13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.13 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 λ = 0.17 In this part, we illustrate estimation of primary parameters we proposed in section III-A. Fig. 3 represents the estimated penetration ratio p, given by formula [START_REF] Ramezani | Exploiting probe data to estimate the queue profile in urban networks[END_REF], associated in this figure to the real penetration ratio p, in the case of an incoming link of one lane. Fig. 4

λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ = 0.2 λ =
λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0.3 λ = 0
.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 λ = 0.37 0.05 penetration ratio p estimated arrival rate λ (veh/s) λ arrival rate estimator for two incoming lanes, κ =0.5 Fig. 5: n + λ m + λ nm ) in vehicles/second for a two lanes incoming link, depending on penetration ratio for various demand scenarios. Simulated time = 40 min. κ = 0.5. Fig. 5 represents arrival rate estimated as given by formula (12) of section III-A in the case of an incoming link of two lanes for κ = 0.5. We can see here that the estimation is better when p gets higher. As there are more data and as the arrival rate is higher, the estimation of arrival rate is more accurate.

B. Probability distributions

In this part, we illustrate the probability distribution queue lengths as proposed in section III-B. We assume the demand is coming from the West of the junction as described in TABLE II with r N = r M i.e. r = 1.

These different scenarios include different possibilities concerning the demand such as : symmetric (Scenario S3) or asymmetric arrivals (scenarios S1, S2, S4, S5). We vary the arrival demand λ n , λ m and λ nm and derive the optimal α * (1) by Proposition 4. We obtain the values α * (1) = [0.1, 0.25, 0.5, 0.75, 0.9]. We do not consider the cases where r = 1. Fig. 6: On the top : probability law of Proposition 1. On the bottom : probability law of Proposition 2. Scenario with λ n = 1/6, λ m = 1/12, λ nm = 1/24 (veh/s)(thus α * (1) = 1), r(t) = 41 s, p = 0.55, c p = 8, l p = 9, at time t = 760s. Red dot is the expectation of the probability distribution. Simulation time=1200s.

a) Example: probability distribution of the two queues lanes without and with the information provided by the probe vehicles: We draw on top of Fig. 6 the probability distribution law P (N t , M t ) (Proposition 1), and on bottom of Fig. 6 the conditional probability distribution law P (N t , M t |L p , N p + M p ) (Proposition 2 ) for scenario of Fig. 6. On top of Fig. 6, we can see that the total number of vehicles in the queue is estimated to (N t = 6, M t = 4), for a total of 10 vehicles in the queue. There is an asymmetry in the distribution probability law because of the asymmetric demand and because the common flow is not strong enough to equilibrate the two queue lanes. We can see on bottom of Fig. 6 that the conditional distribution clearly discriminates the two queue lengths and keeps track of the asymmetry. In this case, the parameters are p = 0.55, c p = 8, l p = 9. We compute κ = 0.75. Following the same ideas as above, we have p = (8/1.75 -1)/(9 -1) = 0.45. It is probable that there are not many cars behind l p , maybe 1 vehicle. Therefore, as l p = 9, the biggest lane should contain around 10 vehicles. Given the asymmetry of the distribution law P (N t , M t ), the conditional probability P (N t , M t |L p , N p + M p ) will favour the lane with the highest arrival rate (lane N ). Then the queue on lane M should contain very few vehicles and will be around the same lane length estimation as in the top figure . In this example, the conditional distribution probability calculus emphasizes the asymmetry of the two lanes. b) Results for the scenarios of TABLE II: For each scenario we measure the maximum and average queue lengths as estimated by SUMO microscopic road traffic simulator. We have used different seeds for the random number generator of the simulators, taking the average value for TABLE III and IV. We notice that SUMO queue length is measured in such a way that any vehicle with a speed greater than 0.1 m/s is not considered in the queue. We vary p ∈ [0, 1] for each scenario and compute :

• MAE(P2):=the mean absolute error between the estimated queue lengths as given by the estimator based on Proposition 2 and SUMO queue lengths on a subset of the data. • MAE(P1):=the mean absolute error between the estimated queue lengths as given by N = µ n , M = µ m (Proposition 1) and SUMO queue lengths on a subset of the data. • MAE(l p ):=the mean absolute error between the estimated queue lengths as given by max( N , M ) = l p and min( N , M ) = κl p , for the queue lengths N and M, and SUMO queue lengths on a subset of the data.

We give here the results for the scenarios of TABLE II in TABLE III and TABLE IV. We comment and emphasize some tendencies on the results. We notice on these tables that the error is decreasing as p tends to 1 : Proposition 2 is getting more accurate as more data is given in input. Even if the estimator based on l p (MAE(l p )) is getting more accurate as p increases, we notice that the performances of our estimator based on Proposition 2 are generally better. Furthermore, if we compare MAE(P1) and MAE(P2) we notice that the performances of Proposition 2 based estimator are in general more accurate than the ones of the other estimators, especially when p increases. We notice finally that MAE(l p ) gives in general less accurate results than the estimators based on Propositions 1 and 2. 1 . We think that the main source of difference between our estimations and SUMO queue lengths is the assignment model. Indeed, drivers do not always choose the shortest queue for their assignment in SUMO. In fact, in SUMO, there is a kind of thresholds on the difference between the queue lengths, beyond which drivers choose the shortest queue. Moreover, the drivers do not have the same behavior, i.e. the choice is stochastic, i.e. the probability of taking the shortest queue increases with the threshold on the difference on the queue lengths. Another source of uncertainty and error is the variance of the arrival flows. Indeed, estimation of the queue length takes into account the average arrival flow λ, but in the simulation there is a variance of the time arrivals, which is then retrieved as an error of measurement.

We also give for information the order of magnitude of the following indicator in TABLE III and TABLE IV: MAPE(P2,R):=the mean percentage absolute error between the estimated queue lengths as given by the estimator based on Proposition 2 and SUMO queue lengths on a subset of the data, at the end of the red time. Indeed, taking into account the MAPE at the beginning of the red time is meaningless (since the queues are not formed yet). This indicator rather emphasizes the difference between our assignment model α * and SUMO assignment model. This indicators varies between around 10% to 30% depending on the penetration ratio and the simulation scenarios.

In Fig. 7 and Fig. 8 we give the two lanes queue lengths in the scenario S4 where r N = r M . We estimate queue lengths for r N > 0 and r M > 0. We notice that the estimation is more accurate as p gets higher.

C. Traffic light control and vehicles assignment onto the lanes

In this part, we assume symmetric demand, λ n = λ m = λ nm = 0.17 vehicles/second. We assume the traffic light cycle includes a phase of 8 seconds, where green light is given to lane M , while red light is given to lane N . Hence, the red duration on lanes N and M are different : r N = r M , r = 1. We also assume α = α * (r) given by Proposition 4.

On Fig. 9 we draw r in blue and α * (r) in green depending on time. At the beginning of the cycle, the red is only for the lane N . The lane M is at this time with green light. Then, the two red lights are simply increasing as a line of slope 1. Starting from 20 seconds, r ∈ I and α * (r) = 1. This is because all the vehicles are assigned to lane M which is the shortest queue (we recall it was at green light until then). Then, α * (r) decreases slowly to reach approximately 0.6 which means that the two queue lengths are more equilibrated as the red durations on lanes N and M are getting less different. We notice that α * (r) would tend to 1/2 if the red time goes to infinity. Concerning r = r n /r m , it is representing how the 1 A significant source of error for this model is the fact that the estimation is done in real numbers, while the measured number of vehicles is done in integer ones. In the general case, the error due to discretization is about 0.5 vehicles. This is very big in the cases where the queue length is small, in particular at the beginning of the red time. If for example, the queue length is 0.5 vehicles in average, then we have 100% error, due only to the discretization. difference in the red lights durations is decreasing as time is going on. r decrease is due to a constant offset (corresponding to the duration where lane M is at green light while lane N is at red light) which becomes less significant as the red light durations are increasing. We notice that r would tend to 1 if the red time goes to infinity. The interval I = [ λm λn+λnm , λm+λnm λn ] as a function of λ nm is represented on Fig. 10. We notice on Fig. 10 that as the common flow λ nm gets higher, the interval I gets larger. Therefore, as the common flow λ nm gets higher, there is more freedom to assign the vehicles onto a lane or another. the queue length estimation application as it could happen in scenarios where more vehicles would communicate, and cause significant delays such as described in [START_REF] Phu | A vehicle-toinfrastructure communication based algorithm for urban traffic control[END_REF].

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Real • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Estimated Mean Absolute
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Real • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Estimated Mean Absolute Error=0.

V. CONCLUSION AND PERSPECTIVES

In this paper, we have proposed a method for the estimation of urban traffic state. We give estimations for the penetration ratio of probe vehicles and for the vehicles arrival rate, on any link of the road network. Knowing the arrival rate of the incoming flow and its composition, we have computed the joint probability distribution of the queue lengths in the case of two lanes link. For this purpose, we have proposed a simple assignment model of vehicles onto the lanes. In addition, we have refined the probability distribution of the queue lengths with the information provided by the probe vehicles. A control of the traffic light has been proposed in order to balance the queues of the two lanes. Moreover, we have proposed a formula for computing the optimal assignment of the vehicles onto the lanes. Numerical simulations have been conducted with Veins framework, and the work presented here has been evaluated. Road traffic control could benefit from the queue length estimations we presented in the present paper, in order to improve travel conditions. We think the ideas we have given in this paper could be extended to a link of any number of lanes.

Fig. 1 :

 1 Fig. 1: A signalized junction of the road network.

  the left lane and lane N is the right lane

Fig. 2 :

 2 Fig. 2: Queues in 2-lanes incoming link. Vehicles that can choose both lanes are assigned onto lane M with probability α and onto lane N with probability (1 -α).

0. 2 λFig. 3 :

 23 Fig.3: Estimated penetration ratio p for a one lane incoming link, depending on penetration ratio for various demand scenarios. The arrival demand levels (λ) are given in (vehicles/s). Simulated time = 40 min.

5 Fig. 4 :

 54 Fig.4: Estimated penetration ratio p for a two lanes incoming link, depending on penetration ratio for various demand scenarios. The arrival demand levels (λ) are given in (vehicles/s). Simulated time = 40 min. κ = 0.5.

  of vehicles for 1200 sTABLE II: Demand for different scenarios (simulated time=1200 s) and for r N = r M

7 Fig. 7 :

 77 Fig. 7: Queue lengths estimator as given by Proposition 2, for varying p = 0.2, p = 0.7 and lane N, r = 1

Fig. 9 :Fig. 10 :

 910 Fig. 9: α * and r depending on cycle time of 90s for λ n = λ m = λ nm = 200 1200 = 0.17 vehicles/second.

  the number of probe vehicles on all the lanes of the considered link at time t. p, 0 ≤ p ≤ 1 the penetration ratio of probe vehicles. N t the total number of vehicles in the queue at time t and lane N. In this paper, N t is assumed to be a random variable. M

	Name	Definition
	L V	the average vehicle length
	G V	the minimum distance gap between vehicles
	R	the total red time in one cycle
	r N (t)	the time since the beginning of the red phase for lane
		N (it is 0 if we are not in red phase at time t), 0 ≤
		r N (t) ≤ R.
	r M (t)	the time since the beginning of the red phase for lane
		M (it is 0 if we are not in red phase at time t), 0 ≤
		r M (t) ≤ R.
	λn, λm,λnm	the average arrival rate in vehicles/second for vehicles
		that respectively turn right, left or go straight.
	λ = λn + λm + λnm	the total arrival rate for the incoming link in vehi-
		cles/second.
	ln, lm, lnm	the proportions (turning ratios) of the main flow λ on
		the incoming link that respectively turn right, left or go
		straight, with λn = lnλ, λm = lmλ, λnm = lnmλ
	x(t)	the total number of vehicles on all the lanes of the
		considered link at time t.
	xp(t)

t the total number of vehicles in the queue at time t and lane M. In this paper, M t is assumed to be a random variable. A t the assignment of a vehicle entering the edge at time t. A t = 1 if the vehicle is assigned on lane M and A t = 0 if it is assigned on lane N. A t is assumed to be a random variable. L t p the location (in number of vehicles) of the last probe in the queue, namely the last connected vehicle, at time t. L t p is assumed to be a random variable, taking value lp. N t p the total number of probe vehicles in the queue at time t and lane N. M t p the total number of probe vehicles in the queue at time t and lane M. cp the total number of probe vehicles in all the lanes and all the queues at time t.

  represents the estimated penetration

	p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p p ^= p

TABLE III :

 III Results in the unit "number of vehicles" for queue on lane N, estimated vs SUMO queue length.

	Results	Scenarios	S1	S2	S3	S4	S5
	Average SUMO	4.79	3.39	4.20	2.89	3.84
	Queue Length
	Max SUMO	15.22	9.71	14.11	8.61	12.28
	Queue Length
	p = 0.05 MAE(P2)	1.98	1.24	1.39	1.26	1.36
	MAE(P1)	2.02	1.25	1.35	1.22	1.30
	MAE(lp)	5.71	3.94	4.27	3.23	4.22
	MAPE(P2,R)(%)	21.08	28.26	22.48	32.15	29.79
	p = 0.10 MAE(P2)	1.89	1.17	1.30	1.25	1.37
	MAE(P1)	2.02	1.25	1.37	1.22	1.30
	MAE(lp)	2.81	2.53	1.93	1.88	1.93
	MAPE(P2,R)(%)	17.38	24.20	19.45	30.06	27.66
	p = 0.15 MAE(P2)	1.68	1.09	1.22	1.20	1.29
	MAE(P1)	2.02	1.25	1.37	1.22	1.30
	MAE(lp)	1.94	1.54	1.45	1.44	1.68
	MAPE(P2,R) (%)	15.85	19.08	18.25	27.27	26.71
	p = 0.20 MAE(P2)	1.55	1.05	1.19	1.12	1.28
	MAE(P1)	2.02	1.25	1.37	1.22	1.30
	MAE(lp)	1.54	1.25	1.33	1.20	1.58
	MAPE(P2,R)(%)	17.17	21.62	17.82	24.47	31.12
	p = 0.50 MAE(P2)	1.14	0.90	1.03	0.89	1.23
	MAE(P1)	2.02	1.25	1.37	1.22	1.30
	MAE(lp)	0.89	1.15	1.41	1.30	2.00
	MAPE(P2,R)(%)	11.89	16.53	16.11	18.88	27.11
	p = 0.70 MAE(P2)	1.23	0.82	0.96	0.77	1.14
	MAE(P1)	2.02	1.25	1.37	1.22	1.30
	MAE(lp)	0.96	1.13	1.44	1.40	2.16
	MAPE(P2,R)(%)	13.90	15.71	14.25	13.47	25.14
	p = 0.90 MAE(P2)	1.28	0.80	0.91	0.72	1.00
	MAE(P1)	2.02	1.25	1.37	1.22	1.30
	MAE(lp)	0.96	1.19	1.56	1.50	2.30
	MAPE(P2,R)(%)	16.89	15.19	13.81	12.96	21.44

TABLE IV :

 IV Results in the unit "number of vehicles" for queue on lane M, estimated vs SUMO queue length.

The end-to-end delay is a communication indicator of performance that measures the delay from the time a message is sent from a communicating vehicle until the time it is received by the receiver (in our case the receiver is the RSU).

D. Communication network performances

We give here some information on the communication network performances that we have measured in simulation. We have checked that the communication performances are not disturbing the estimation of the queue lengths. As the order of magnitude of the end-to-end-delay 2 is very low (around 0.2 ms), we don't expect significant consequence on

APPENDIX A CALCULUS OF THE BIAS OF p ESTIMATOR

We consider queue lengths on lanes N and M respectively equal to n and m. We propose :

By the way, in our case, the length n of queue N can be estimated with the number of arrivals on lane N during r N (t) which is µ n . As µ n +µ m = max(µ n , µ m )+min(µ n , µ m ) and by estimating max(µ n , µ m ) = l p +i c /p, where 0 ≤ i c ≤ 1 and i c /p represents the backlog of the queue (unequipped vehicles following l p ). We want to determine 0 ≤ i c ≤ 1 and compute p such that the estimation of p is without bias. We can write :

We introduce c κ = c p /(1+κ) and replace (24) in [START_REF] Krajzewicz | Recent development and applications of SUMO -Simulation of Urban MObility[END_REF]. Finally, we get the following equation :

We know from [START_REF] Comert | Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters[END_REF] how to compute the expectation of p and we follow the same ideas below :

To derive the next equation we use two arguments :

• the expectation of a binomial probability distribution law E(B(n x , p)) = n x p with n x = l p -1 + min(l p , n, m) in our case.

E(

We replace min(l p , n, m) = κ max(l p , n, m) = κ(l p + i c /p) :

(p(l p -1+κ(l p +i c /p))+

To get an estimator without bias, we write :

Solving this equation gives :

Finally, we replace i c in (26) :


Christine Reynier

Emma Ea741

Joseph Wiesenfarth Ford

Madox Ford

Women Violet Hunt

Jean Rhys

Stella Bowen

Janice Biala

Texte Intégral

Afficher l'image Michael Robertson's book is definitely a good read. In The Last Utopians, Robertson focuses on four writers who between 1825 and 1915 developed an interest in utopian schemes and social experimentation in Great Britain and the United States. Edward Bellamy (1850-1898), William Morris (1834-1896), Edward Carpenter (1844-1929) and Charlotte Perkins Gilman were all democratic socialists, believing in an egalitarian alternative to industrial capitalism and in women's independence; impressed by the rapid and profound alterations industrial development and new technologies were introducing in their social and ecological environment, they responded by devising alternative ways of life in utopian writings, most of the time implementing them in their own lives. This accounts for Robertson's method which is both biographical and analytical, his concern being to read the lives as well as the texts of these visionaries. At a time when utopian studies are flourishing, the choice Robertson makes is original since he brings together two renowned British figures and two American ones, delineating the specificity of their utopian writings, their distinctive roots in Carlyle's and Ruskin's or in Emerson's and Whitman's ideas, their gendered perspectives, but also the cross-cultural influences they were submitted to, Charles Fourier's and Robert Owen's theories having travelled over the years across the Channel and the Atlantic. In the first chapter, 'Locating Nowhere,' Robertson traces the genealogy of utopian works from Plato's Republic to Thomas More's Utopia and Tommaso Campanella's The City of the Sun, and finally, to Henri de Saint-Simon's, Charles Fourier's and Robert Owen's works. He locates the 'utopian turn' in fiction in the late 1880s when Bellamy's Looking Backward (1888) and many other works of speculative fiction were published that offered visions of egalitarian worlds contrasting with the economic depression both the United States and Great Britain were suffering from at the time.

Looking Backward was a bestseller-'the Uncle Tom's Cabin of the industrial era' (38)-which met with huge success in the United States and Europe and inspired a political movement, Nationalism, that played a significant role. However, today, Looking Backward is quite unpalatable, as Robertson shows in chapter 2; it is a somewhat contradictory work that may owe much to Bellamy's complex personality. Shocked by social inequality, Bellamy first turned to socialism in his youth. Brought up in a Baptist family, he rejected his father's religion in favour of an Emersonian mystical approach of the natural world, but unlike Emerson, he rejected individualism. He studied law but then earned his living as a journalist for the Daily Union before becoming a writer of fiction. The success of Looking Backward lured the reticent Bellamy out of his secluded existence and he then attempted to realize his vision by championing Nationalism and populist ideas in the 1892 election. Published ten years after his first novel, Looking Backward is a utopian romance set in the year 2000 where economic equality and the absence of social classes are the rule; however, it is organized around an industrial army of conscripted workers that ensures the disappearance of individualism in favour of spiritual unity-Bellamy's 'religion of solidarity'. Robertson argues that what read like a utopia in the late 19th century reads today like a dystopia. What appealed mostly to workers and women, such as Gilman, for whom it promised a reassuring and slow transformation towards universal equality, now reads as a frightening regimented state where there is no room for innovation, freedom, difference or dissent. But in spite of its limitations and contradictions, and mainly through its denunciation of inequality, Looking Backward had a lasting influence, on very diverse movements or personalities, from the Fabian movement to John Dewey or Martin Luther King.

News from Nowhere, published in the socialist magazine Commonweal in 1890, was William Morris's famous response to Bellamy's work, which, according to him, was a mere 'Cockney paradise' and a eulogy of modern civilization. It exposes Bellamy's utopian dream of machine production and specialized labour, as Robertson explains in chapter 3. Morris's utopia together with its criticism of modern civilization, technology, mass production and social inequalities, is well known. What Robertson insists on is that it was the outcome of a long commitment to a form of life and standards at odds with the Victorian ones. He thus evokes at length Morris's childhood, his infatuation with Medieval times and lore, his avid reading of Ruskin when he was at Oxford, his friendship with Burne-Jones and D. G. Rossetti. He expatiates on the reforming zeal of Morris that led him to build Red House and then create the Firm, his famous design company that dissolved 'the boundary between the domestic and the natural' (93) in its wallpaper design and challenged industrial methods. Morris, the artist and poet, became a political writer and activist who defended a return to the worker as a craftsman. An active socialist propagandist, he believed that socialism 'was the path to a utopian future' (108). News from Nowhere presents such a utopian world, set in a pastoral future, rid of all the industrial trappings of Victorian times.

In the next chapter, the author turns to Edward Carpenter. After a vocational crisis that made him resign his ordination as an Anglican clergyman in Cambridge, Carpenter visited Walt Whitman whose poems on same-sex love and democracy had come as a revelation to him. This visit to Whitman was crucial and shaped Carpenter's work. Indeed, he then wrote 'Towards Democracy' (1883), the foundation of his utopian project. This long poem in free verse criticizes Victorian respectability and social inequalities and calls for a bright future when the body will be on a par with the mindthe key, according to Carpenter, to freedom and democracy. He then bought a plot of land in Millthorpe, near Sheffield, and became a market gardener. His home became a meeting-place for people from all walks of life who believed, like him, in simple life and a classless society. Carpenter was indeed interested in socialism, more exactly, the 'ethical socialism' that had developed in northern industrial towns and that was bent on awakening people to the ethics of fellowship rather than the class struggle or the struggle against capitalism. A most popular political lecturer, he gave, for example, a lecture on 'Civilisation' in 1889, where he advocated a return to a more natural way of life, that was published and much read until 1938. In 1896, he published Love's Coming of Age, a utopian tract that adopts the genre of the Victorian marriage manual and depicts a future of free sexuality. A few years later came The Intermediate Sex (1908), Carpenter's reflections on same-sex love, which he connected with democracy but which have since been shown to be steeped in Victorian prejudice about gender. As Robertson shows very well, Carpenter's utopianism combined theory and praxis, in Millthorpe.

Chapter 5 unveils the life and work of Charlotte Perkins Gilman, which goes far beyond 'The Yellow Wallpaper' (1892), the short story she is usually associated with. Gilman was the unofficial poet laureate of Nationalism, the movement created by the supporters of Bellamy's utopia. However, unlike Bellamy, she thought that women's independence was a precondition for socialism and the 'mother instinct' could lead to utopia. Her own troubled first marriage, the difficulties she faced in combining domestic duties and her work for the 'betterment of humanity' are inextricably connected with the utopian feminism she defended. In Women and Economics (1898), she showed how women's independence rests on their participation in the economic sphere: once they were involved, they would become 'World-Mothers'. For that purpose, she advocated kitchenless homes so as to liberate women from domestic work, and imagined cookshops, laundries and children's facilities. In 1903, she published The Home and then turned to fiction, such as Herland (1915), an all-female eugenic paradise that came in the wake of Moving the Mountain (1911), a chilling utopia where https://journals.openedition.org/cve/5668 4/5 the genetically defective are sterilized and the 'degenerates, insane, idiots, and real perverts' killed.

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The last chapter gives a panoramic view of visionary writings since the 1920s. Once the First World War had shattered all hope in a better world, anti-utopias were written, such as Aldous Huxley's Brave New World and George Orwell's 1984. In the 1960s, utopianism came back with a vengeance and utopian literature developed again until it was superseded by a turn to dystopia in the 1980s and then, after 9/11. On the whole, the book is well-documented, skilfully written and pleasant to read. Robertson, a former freelance journalist who 'has written for the New York Times or the Village Voice' as the book cover tells us, has certainly retained his mastery on language. He also knows how to grab his readers' attention by inserting anecdotes at the beginning of his chapters (E. M. Forster's decisive visit to Carpenter is a particularly good one). The biographical and analytical method he adopts is certainly justified by the close connections there were between the utopians' ideas and ways of life but the biographical details he gives are sometimes lengthy and tedious, in Morris's case especially. They may in fact appeal more to students discovering the utopians' lives than to the Victorian scholar who will be already familiar with them. The choice of bringing together British and American utopians is original and the analysis sometimes bold. Indeed, unlike most critics, Robertson addresses the thorny problem of Gilman as a supporter both of feminism and eugenics; he also points out that her feminism is combined with racial and ethnic prejudice, anti-Semitism and a desire to control immigrants from southern and eastern Europe. However, the differences between the texts broached, the way in which utopia flirts with anti-utopia and dystopia could have been addressed more clearly at the outset rather than in the last chapter, and backed more strongly by theoretical or philosophical writings. What is most striking though, is Robertson's commitment to his subject: not content with his academic knowledge of utopia, he has acquired an insider's knowledge of it, by staying in intentional communities in Britain and the United States and approaching such movements as Occupy Wall Street or supporters of alternative food or educational experiments: he can thus emphasize what they owe to their 19th-century predecessors and the crucial role they play in our society.

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Jessica Feldman, Victorian Modernism. Pragmatism and the Varieties of Aesthetic Experience

  Introduction [Texte intégral] The Transformative Power of the Arts in Victorian and Edwardian Culture and Society Paru dans Cahiers victoriens et édouardiens, 89 Spring | 2019

Référence électronique

Christine Reynier, « Michael Robertson. The Last Utopians. Four Late Nineteenth-Century Visionaries and their Legacy », Cahiers victoriens et édouardiens [En ligne], 89 Spring | 2019, mis en ligne le 01 juin 2019, consulté le 09 février 2024. URL : http://journals.openedition.org/cve/5668 ; DOI : https://doi.org/10.4000/cve.5668 [Texte intégral] Paru dans Cahiers victoriens et édouardiens, 74 Automne | 2011 Martin Hipsky (éd.), Modernism and the Women's Popular Romance in Britain, 1885-1925 [Texte intégral]


Outhmane Rassili
email: outhmanerassili@gmail.com

Mikhail Ordin
email: m.ordin@bcbl.eu

The effect of regular rhythm on the perception of linguistic and nonlinguistic auditory input

Keywords: rhythm, rhythmicity, isochrony, attentional cycles, rhythmic cognition

Regular distribution of auditory stimuli over time can facilitate perception and attention.

However, such effects have to date only been observed in separate studies using either linguistic or non-linguistic materials. This has made it difficult to compare the effects of rhythmic regularity on attention across domains. The current study was designed to provide an explicit within-subject comparison of reaction times and accuracy in an auditory target-detection task using sequences of regularly and irregularly distributed syllables (linguistic material) and environmental sounds (non-linguistic material). We explored how reaction times and accuracy were modulated by regular and irregular rhythms in a sound-(non-linguistic) and syllable-monitoring (linguistic) task performed by native Spanish speakers (N=25). Surprisingly, we did not observe that regular rhythm exerted a facilitatory effect on reaction times or accuracy. Further exploratory analysis showed that targets that appear later in sequences of syllables and sounds are identified more quickly. In late targets, reaction times in stimuli with a regular rhythm were lower than in stimuli with irregular rhythm for linguistic material, but not for nonlinguistic material. The difference in reaction times on stimuli with regular and irregular rhythm for late targets was also larger for linguistic than for non-linguistic material. This suggests a modulatory effect of rhythm on linguistic stimuli only once the percept of temporal isochrony has been established. We suggest that temporal isochrony modulates attention to linguistic more than to non-linguistic stimuli because the human auditory system is tuned to process speech. The results, however, need to be further tested in confirmatory studies.

Introduction

The perceptual system does not process continuous sensory input equally at all times: some elements of the input are more attended than others [START_REF] Landau | Attention Samples Stimuli Rhythmically[END_REF].

Attention samples the continuously changing environment in discrete chunks, which correspond to the periods of neural oscillations in the 4-8Hz frequency band [START_REF] Vanrullen | Perceptual Cycles[END_REF]. Perception thus operates on these chunks of sensory information, while the phase of neural oscillations modulates attentional intensity, and thus the probability of perceiving a certain element in the environment.

The central [START_REF] Doelling | Acoustic landmarks drive delta-theta oscillations to enable speech comprehension by facilitating perceptual parsing[END_REF][START_REF] Ghitza | The theta-syllable: a unit of speech information defined by cortical function[END_REF] and peripheral [START_REF] Greenberg | Speech Processing in the Auditory System: An Overview[END_REF] auditory neural systems are also sensitive to rhythmic patterns in the environment. This sensitivity plays an important role in processing auditory information, including both linguistic and non-linguistic auditory input. Some segments in the input are better attended because their occurrence is predicted by a repetitive rhythmic pattern. Several theories (e.g. Dynamic Attentional Theory, [START_REF] Jones | Time, our lost dimension: Toward a new theory of perception, attention, and memory[END_REF] The Attentional Bounce Hypothesis, Shields, McHugh, & Martin, 1974, discussed below) have been put forward to explain how these rhythmic patterns might modulate attention allocated for processing sensory input. Attentional rhythms (most likely based on neural oscillations; [START_REF] Fiebelkorn | A Rhythmic Theory of Attention[END_REF]Haegens & Golumbic, 2019;[START_REF] Hickok | The Rhythm of Perception[END_REF][START_REF] Vanrullen | Perceptual Cycles[END_REF] and environmental rhythms can become synchronized, such that more attentional resources are allocated to the expectancy cues provided by regularly occurring events in the environmental input [START_REF] Obleser | What do we talk about when we talk about rhythm?[END_REF]. Regular metrical patterns in environmental input facilitate establishing and maintaining synchronization between environmental and attentional oscillations, which may lead to entraining even spontaneously occurring neural oscillations to environmental rhythms.

Neural rhythms sample the environment rhythmically, and the effect of environmental rhythms, which entrain neural rhythms and thus lead to resampling the world based on what the environment is like in particular circumstances, and drawing attention to more relevant aspects of the world at a particular moment. This, in turn, allows for faster and more accurate processing of continuous sensory inputs [START_REF] Jones | Time, our lost dimension: Toward a new theory of perception, attention, and memory[END_REF].

The coupling between neural and environmental rhythms can also enhance processing of auditory linguistic information which contains more regularly distributed salient acoustic events (stressed syllables, vowel onsets). Attention is drawn to stressed syllables more than their unstressed counterparts [START_REF] Cutler | On the Role of Sentence Stress in Sentence Processing[END_REF]Cutler & Foss, 1976), and mispronounced phonemes are more likely to be perceived as deviant in stressed positions [START_REF] Bond | Misperceptions of fluent speech[END_REF][START_REF] Cole | How are syllables used to recognize words?[END_REF]. [START_REF] Shields | Reaction time to phoneme targets as a function of rhythmic cues in continuous speech[END_REF] asked people to monitor a particular phoneme in connected speech. Target phonemes were detected faster and more reliably in stressed syllables than in unstressed syllables.

Moreover, the facilitatory effect of stress was not observed when the same words containing the target phoneme were embedded in nonsense sentences. The authors suggested that stressed syllables in meaningful sentences are temporarily predictable and thus attract more attentional resources and facilitate target detection. In meaningless sentences, expectancy cues do not exist (participants cannot predict upcoming words and stressed syllables). Thus, it is the expectancy cues rather than acoustic correlates or the perceptual salience of stressed syllables that enhance phoneme detection in speech. Their results led to the Attentional Bounce Hypothesis: attention locks onto the quasi-isochronous distribution of stressed syllables and moves from one stressed syllable to another [START_REF] Shields | Reaction time to phoneme targets as a function of rhythmic cues in continuous speech[END_REF]. The longer the preceding rhythmic pattern leading to the target, the better the percept of temporal isochrony is established, and the stronger expectancy and its facilitatory effect [START_REF] Pitt | The use of rhythm in attending to speech[END_REF]. In line with this, [START_REF] Ordin | The role of native language and the fundamental design of the auditory system in detecting rhythm changes[END_REF] showed that in AX discrimination experiments, regular rhythm in the A stimulus (first stimulus in a stimulus pair) led to faster and more accurate responses, regardless of whether the X stimulus (second stimulus in a stimulus pair) was rhythmically similar to or different from the A stimulus.

They suggested that rhythmic regularity in the first stimulus enhances attention and thus results in better performance.

A succession of stressed syllables creates a metrical grid that is used to facilitate speech processing. The perceptual system can rely on this grid to predict the occurrence of the next stressed syllable, allocating more resources to these syllables so as to process input more efficiently. In addition to metrical expectancy, which relies on the metrical grid created by the occurrence and predictability of stressed syllables, [START_REF] Quené | Effects of Timing Regularity and Metrical Expectancy on Spoken-Word Perception[END_REF] explored the effect of timing expectancy on the processing of linguistic input. Participants performed a phoneme monitoring task while listening to a sequence of isolated words. Timing expectancy was manipulated by variation in the inter-word duration: regular inter-word intervals provided stronger expectancy cues for word onsets than variable inter-word intervals. Metrical expectancy was achieved by modifying the stress patterns in the preceding word sequence. A similar stress pattern (iambic or trochaic) across all the words in a sequence allowed for expectancy cues to emerge, whereas varying stress patterns within a sequence disrupted metrical expectations. For timing expectancy, the resulting reaction times were shorter in the regular than in the irregular condition, showing a clear timing expectancy effect when inter-word durations were constant. On the other hand, metrical expectancy did not exert an effect on reaction times. These results suggest that speech perception may be more affected by timing than by metrical patterns.

Although the facilitatory effect of regularity and expectancy on auditory perception has been observed on linguistic [START_REF] Pitt | The use of rhythm in attending to speech[END_REF][START_REF] Quené | Effects of Timing Regularity and Metrical Expectancy on Spoken-Word Perception[END_REF][START_REF] Shields | Reaction time to phoneme targets as a function of rhythmic cues in continuous speech[END_REF] and non-linguistic [START_REF] Jones | Time, our lost dimension: Toward a new theory of perception, attention, and memory[END_REF]) materials, there are important differences in how linguistic and non-linguistic sounds are processed [START_REF] Warren | Auditory Sequence: Confusion of Patterns Other Than Speech or Music[END_REF]. Speech is segmented into phonemes which belong to classes defined by finegrained boundaries, while non-linguistic sounds are not [START_REF] Hendrickson | The organization of words and environmental sounds in memory[END_REF]. Humans engage in attentive processing of linguistic sounds in contrast to non-linguistic sounds [START_REF] Warren | Auditory Sequence: Confusion of Patterns Other Than Speech or Music[END_REF]. Even if the latter are heard on a daily basis, they are mostly processed unconsciously by the auditory system; humans tend to filter out and ignore passive sounds they are accustomed to hearing constantly, but which hold no significance for them (e.g., birds chirping outside the house). The human auditory system may be better honed for processing linguistic than nonlinguistic acoustic material. Thus, the magnitude of the expectancy effect on attentional rhythms might vary for linguistic and non-linguistic material. The current study provides an explicit within-subject comparison of reaction times and accuracy in an auditory target-detection task in a sequence of regularly and irregularly distributed syllables (linguistic material) and environmental sounds (non-linguistic material). We hypothesized that rhythmic expectancy modulates attention to linguistic more than to non-linguistic stimuli. To test this hypothesis, we set up a syllable-monitoring (linguistic sounds) and sound-monitoring (non-linguistic sounds) task for 25 native Spanish participants. They were instructed to listen for auditory targets within a series of auditory sequences, characterized by either regular or irregular inter-stimulus intervals.

Methodology Participants

Twenty-five native Spanish speakers (mean age = 22.72 years, median age = 22 years; 17 women) were recruited. None of them reported any speech or hearing problems. The experiment was approved by the BCBL ethical review board. All subjects signed written consent in accordance with the Declaration of Helsinki and received €8 as financial compensation.

Material

The linguistic material included 25 distinct consonant-vowel syllables recorded by a native Spanish speaker. The syllables were composed of 5 consonants (l, m, n, s, f) and 5 vowels (a, o, i, e, u), all of which exist in Spanish. We avoided plosive consonants because their durations cannot be manipulated without compromising naturalness.

Each syllable was recorded in a separate audio file. The durations of consonants and vowels were manipulated such that all consonants lasted 100ms and all vowels lasted 200ms (300ms per syllable). All of the audio files were equalized in terms of their average level of intensity (with the upper threshold set to 80dB), to ensure that none of the syllables in relatively louder (i.e., has relatively higher intensity level) than other syllables.

The non-linguistic stimuli comprised 25 distinct transient sounds (e.g. a drop of water).

These sounds were carefully picked to differ in terms of spectral characteristics and to be clearly and easily discriminated by the participants. These sounds were equalized in terms of duration (300ms) and average intensity (with an upper threshold of 80dB).

We created sequences of 8 syllables or sounds to be used as experimental trials. Each sequence started and finished with a 140ms period of silence. We counterbalanced sounds/syllables across sequences. The inter-syllable/sound intervals were manipulated to generate rhythmic differences: constant inter-syllable/sound durations for regular rhythm and jittered inter-syllable/sound durations for irregular rhythm stimuli.

In the regular rhythm condition, all 7 inter-syllable/sound intervals were constant at 140 ms. For the irregular condition, we used seven interval values: 50ms, 80ms, 110ms, 140ms, 170ms, 200ms, 230ms. Within each stimulus, these values occurred in randomized order, with each value used only once, such that the total duration of each trial sequence was 3660ms. In total, 300 stimuli were created, 75 per each of the 4 conditions defined by rhythm type (regular and irregular) and stimulus type (linguistic and non-linguistic).

Procedure

The experiment consisted of a syllable-(for linguistic stimuli) and sound-(for nonlinguistic stimuli) monitoring task. The participants were seated in front of a screen in a soundproof room. The stimuli were presented via headphones in PsychoPy.

On each trial, participants heard the sentence "Now listen for X" (where X stands for the specific target sound or syllable used); after a 1-second pause a sequence of syllables or sounds was played. On each trial, a target was embedded in the presented sequence. The target positions varied between the third, fourth, fifth, sixth, and seventh positions within the sequence. (Figure 1). Syllables/sounds were counterbalanced to appear as targets the same number of times in each position. The targets and their positions within the sequences were counterbalanced across trials, with every single syllable/sound and position being selected three and fifteen times respectively in each condition.

Participants were instructed to press the button on a response box as soon as they heard the specified target sound/syllable. The trial was interrupted when the button was pressed. The participants manually initiated every trial at their own discretion. Nothing was presented on the screen when the sounds were being played. The order of trials with linguistic and non-linguistic material and with regular and irregular rhythms, was randomized for each participant. Prior to the experiment, 4 practice trials were initiated as a training session, in order to familiarize participants with the task and to allow them to adjust the volume to a comfortable level. Volume adjustment changed the loudness of the stream overall, but due to the intensity normalization procedure the relative loudness of the sounds/syllables in sequences was kept constant.

Results

Reaction time was calculated as the delay between the onset of the target syllable/sound and the time when the participant pressed the button to signal that the target had been detected. Listeners failed to detect the targets 72 times (0.96% of all trials lacked responses; a response was considered to be missing if it was not given within 1000ms after the end of a sequence). On 212 trials (2.82% of all trials), a response was given before the target was presented. Missing responses and premature responses amounted to a total of 284 errors (3.7%) trials in the entire experiment. The number of errors across rhythm types was the same. In the linguistic stimuli, there were exactly 43 errors for both the regular and irregular rhythms, while in the non-linguistic stimuli, there were exactly 99 errors for both the regular and irregular rhythms (Table 1). =.13. The interaction between rhythm and stimulus type was also not significant, F(1,24)=.386, p=.368, η p 2 =.015. Thus, the data does not confirm the original hypothesis:

we did not find evidence that regular rhythm modulates performance more on linguistic than non-linguistic stimuli. Moreover, to our surprise, we did not observe any effect of rhythm type: the data did not provide evidence that metrical regularity facilitated performance on monitoring tasks. Therefore, we decided to explore the data further in an exploratory study.

We expected to find a significant effect of rhythm based on earlier studies and theoretical assumptions. The absence of this effect might indicate that it takes time to establish the percept of isochrony between syllable or sound onsets. This would result in either a weak or non-existent effect on targets closer to the beginning of the trial sequences. Therefore, we decided to explore the effect of rhythm in early and late positions. The third and fourth positions were considered the early target positions, while the fifth, sixth, and seventh positions were considered late target positions. We calculated the average reaction time for each participant separately for early and late positions per stimulus type and rhythm (Figure 2), and introduced position (early vs. late) as a factor in the model (in addition to the effects of stimulus type and rhythm). The analysis revealed a significant and substantial effect of position, F(1,24)=166.24, p<.0005, η p 2 =.874. Targets in late positions were detected with shorter reaction times than targets in early positions (Table 2). However, the effect of stimulus type, F(1,24)=2.994, p=.096, η p 2 =.111, and the effect of rhythm, F(1,24)=.919, p=.347, η p 2 =.037 were not significant. None of the interactions were significant. Following the pattern, we compared the effect of rhythm on reaction times for linguistic and non-linguistic stimuli, separately, using paired t-tests (two-tailed, assumption of normality verified by the Kolmogorov-Smirnov tests, which showed no significant deviations in the data distribution from the normal distribution in any sample, all p values >.9). On linguistic material, targets in late positions were identified faster in the regular than the irregular condition, t(24)=2.49, p=.02 (corrected by the Bonferroni method), mean difference M=.009(SD=0.19), SE=.004, 95%CI of the difference [.002: .017], d=.28. On non-linguistic stimuli, however, the effect of rhythm in late positions was not significant, t(24)=.688, p=.498, M=.005(0.34), SE=.007, 95%CI of the difference [-.019:

.001], d=.1. To directly test the hypothesis that the difference in RTs in late position between regular and irregular conditions was larger for linguistic than non-linguistic material, we ran a paired t-test, t(24)=1.958, p=.031, d=.506. This pattern suggests that the effect can only be observed on linguistic stimuli, and at least four inter-syllable intervals are required for the effect of isochrony to emergepossibly because it takes this long for the percept of regularity to become established.

To further understand this relationship, we ran Spearman's correlations between the positional order when the target was detected (all positions were considered, without splitting them into bins) and reaction times, separately for linguistic regular (ρ=-.635), linguistic irregular ρ=-.535), non-linguistic regular (ρ=-.489) and non-linguistic irregular (ρ=-.459) stimuli. The fact that all these correlations were negative shows that reaction time decreases as ordinal position increases (people are faster towards the end of a sequence). However, the only significant difference between the correlation coefficients was between correlations for regular linguistic and regular non-linguistic stimuli, z=1.7, p=.045 (2-tailed). This shows that regularity is associated with a sharper decline in reaction times in linguistic stimuli than non-linguistic stimuli as a sequence progresses (and the ordinal number of the target position increases). This is in line with the earlier conclusion that regularity has a stronger effect on linguistic than on non-linguistic materials. We did not observe differences in correlation strengths in other tests (Table 3).

Discussion

Surprisingly, we did not observe any effect of temporal regularity on performance (accuracy and reaction times) in the sound-monitoring task, which we used as a proxy for online attention. Our exploratory analysis showed that targets towards the end of the sequences were detected faster. The way we randomized inter-sound/syllable intervals might explain why we observed better performance both on regular and irregular stimuli in late positions. To create irregularity, we used a set of 7 possible values, ranging from 50ms to 230ms (with 30-ms steps) as inter-syllable and inter-sound intervals, with each value used only once per sequence. Consequently, participants might have figured out that the inter-sound intervals were not completely random, since with each passing sound, the degrees of freedom for the remaining values reduced, increasing the predictability of the duration of the next inter-sound interval. That is, at the beginning of any sequence, there are seven possible interval durations, but for late positions, only three possible durations remain; if participants kept track of durations that had already occurred as the sequence progressed, they could better predict the onset of the following sound/syllable, and prepare to make a behavioral response if a target was detected. This predictive effect might mask any effect of rhythmic regularity, if the latter only emerges in late positions. In early positions, a regularity effect has not yet been established in the regular condition, nor has a predictability effect been established in the irregular condition, thus explaining why no significant differences between conditions or material types were observed. A similar phenomenon has also been reported in the visual perceptual modality, when targets were embedded in a sequence of isochronously and non-isochronously presented sequences of images [START_REF] Coull | Neural Substrates of Mounting Temporal Expectation[END_REF].

Reaction times decreased over time because the conditional probability that a target would appear given that it has not yet appeared increases over time, thus alerting participants' attention towards the end of the sequence. In future experiments, the position of the target could be kept constant, while manipulating the presence vs.

absence of the target. This would control for differences in attention at the beginning and the end of the sequences due to the increase of the conditional probability that the target will occur, given that it has not yet occurred.

Further tests showed that in late positions, targets were detected faster when syllables in linguistic stimuliwere distributed with isochronous inter-syllable intervals.

Importantly, significant differences in reaction times for detecting target soundsin nonlinguistic materialwere not observed. The difference in reaction times between regular and irregular conditions for linguistic stimuli was significantly bigger than the difference in reaction times between regular and irregular conditions for non-linguistic stimuli. This suggests that a modulatory effect of regularity can only be observed on linguistic material later in sequences (which is in line with [START_REF] Pitt | The use of rhythm in attending to speech[END_REF], who observed a stronger facilitatory effect of regular rhythm in longer than in shorter sequences).

However, due to the exploratory nature of this analysis in our study, a confirmatory study using a more targeted design should be conducted.

It is not clear why we found no effect of regularity on non-linguistic material even in late positions. We can suggest several factors that could interfere with the modulatory effect of rhythm on the sound monitoring task. Individual differences in familiarity with sounds could be an interfering factor; some participants might have experienced more exposure to certain sounds than others. Another factor might be the problem of classifying nonlinguistic sounds. Such sounds might not be concretely classified in the mental lexicon, and thus make it more difficult to exactly determine their identity, resulting in semantic misinterpretations [START_REF] Hendrickson | The organization of words and environmental sounds in memory[END_REF]. Also, humans interact with linguistic and non-linguistic sounds differently. They allocate more attention to sounds which are deemed more important; linguistic sounds are likely to be accorded more importance as they are a means for communication.

Spontaneous neural oscillations lead to rhythmic sampling of the world [START_REF] Vanrullen | Perceptual Cycles[END_REF][START_REF] Fiebelkorn | A Rhythmic Theory of Attention[END_REF]. However, acoustic rhythms in environmental input can modulate excitability patterns in the auditory system, entraining neural rhythms to environmental rhythms [START_REF] Hickok | The Rhythm of Perception[END_REF][START_REF] Lakatos | A New Unifying Account of the Roles of Neuronal Entrainment[END_REF][START_REF] Obleser | What do we talk about when we talk about rhythm?[END_REF].

This entrainment could change how the world is sampled, and what aspects of the environmental stimuli are zoomed in on and selected for attention. Because of the functional importance of speech, it is also possible that speech-like stimuli lead to stronger modulatory effects on neural oscillations, producing better entrainment of neural oscillations than non-linguistic stimuli. If the initial non-linguistic sound in a regular condition in our experiment was aligned with a phase of the perceptual cycle where the probability for perception was low, the following sounds in the sequence, which are distributed at equal temporal intervals, might also have been aligned with that same phase, where they were unlikely to be perceived [START_REF] Vanrullen | Perceptual Cycles[END_REF]. By contrast, syllables, which are quickly recognized as functionally important linguistic inputs, could lead to a more rapid re-adjustment of the phase of neural oscillators. Even in cases where initial syllables in a sequence were aligned with a phase in the perceptual cycle that afforded only a low probability for perception, syllables later in the sequence could become aligned with a phase with a higher probability of perception. Slower phase resets for non-linguistic stimuli and faster phase resets for syllables might explain why we found better task performance for linguistic than non-linguistic material towards the end of the sequence. This interpretation, however, calls for additional empirical testing.

Whether the effect of linguistic stimuli would be observed in the visual modality presents an interesting question for further investigation. As an evolutionarily ancient phenomenon, spoken language might involve finely tuned perceptual and cognitive mechanisms specifically adapted for speech processing and comprehension. Regular rhythm (in the auditory modality) not only affects auditory processes related to speech processing but also higher-level mechanisms involved in language comprehension, including lexico-semantic integration [START_REF] Rothermich | Rhythm's gonna get you: Regular meter facilitates semantic sentence processing[END_REF].

Writing is a relatively recent cultural innovation and, in the visual unlike the auditory modality, linguistic stimuli might not exert a stronger effect on rhythmic attention than non-linguistic stimuli. This hypothesis, if verified, might throw light on how the speech and language faculty influence general cognitive processes in humans.

Figure 1 :

 1 Figure 1: Schematic overview of stimuli in the regular and irregular rhythm stimuli. Each box represents a syllable/sound. Possible positions of target syllables/sounds are marked by a darker colour.

Figure 2 .

 2 Figure 2. Average reaction times categorized by stimulus type (linguistic or non-linguistic), rhythm (regular or irregular), and position (early or late). Each circle stands for the data from an individual participant, red dots represent the means and horizontal lines represent the medians.Boxes contain 50% of the datapoints, downward and upward whiskers each span 25% of datapoints.

Table 1 .

 1 Errors categorized by type, rhythm, and stimuli

		Errors in linguistic stimuli		Errors in non-linguistic stimuli
	Regular	Irregular	Regular	Irregular
	43		43		99		99
	Early	No	Early	No	Early	No	Early	No
	Response	Response	Response	Response	Response	Response	Response	Response
	31	12	36	7	75	24	70	29
	For premature responses, the effect of stimulus type (linguistic vs. non-linguistic) on the
	number of errors was significant and strong, F(1.24)=8.977, p=.006, η p 2 =.272, while the
	effect of rhythm (regular vs. irregular), F(1,24)<.0005, p=1.0, η p 2 <.0005, and the
	interaction between rhythm and stimulus type, F(1,24)=.623, p=.438, η p 2 =.025 were not
	significant. For missing responses, the effect of stimulus type was significant and
	strong, F(1.24)=11.352, p=.003, η p 2 =.321, while the effect of rhythm, F(1,24)<.0005,
	p=1.0, η p 2 <.0005, and the interaction between rhythm and stimulus type, F(1,24)=1.263,
	p=.272, η p 2 =.05 were not significant. This reveals a significantly and substantially larger
	number of errors on non-linguistic than on linguistic material, and this pattern is not
	modulated by the rhythm implemented in the stimuli.
	Data screening was performed on remaining trials to detect the outlying RT values
	defined as the values exceeding 2SD from the mean in each rhythm*stimuli combination
	(22 trials or 0.3% of all trials). Including trials discarded due to errors, this brings the
	total percentage of discarded trials to 4% (306 out of 7.500 trials).

Table 2 .

 2 Averaged reaction time per stimuli type, rhythm, and position

	Condition	Mean (in	SD	95% confidence
		ms)		interval around M
	Linguistic Irregular Early Position	.4928	.044	.48:.51
	Linguistic Irregular Late Position	.4446	.033	.43:.46
	Linguistic Regular Early Position	.5054	.044	.49:.52
	Linguistic Regular Late Position	.4353	.035	.42:.45
	Non-Linguistic Irregular Early Position	.508	.065	.48:.54
	Non-Linguistic Irregular Late Position	.4483	.052	.43:.47
	Non-Linguistic Regular Early Position	.5122	.071	.48:.54
	Non-Linguistic Regular Late Position	.4529	.039	.44:.47

Table 3 .

 3 Comparing strengths of non-parametric correlations between the ordinal position of the target and reaction times in stimuli with isochronous (regular) and non-isochronous (irregular) distribution of speech-like syllables (linguistic stimuli) and non-linguistic sounds (non-linguistic stimuli).

	Condition	z	p	Interpretation
	Linguistic Regular vs. Linguistic Irregular	1.192 .117	Tests linguistic materials to
				ascertain if regular rhythm is
				associated with a sharper
				decline in reaction times than
				irregular rhythm
	Non-Linguistic Regular vs. Non-Linguistic Irregular .282	.389	Tests non-linguistic materials
				to ascertain if regular rhythm is
				associated with a sharper
				decline in reaction times than
				irregular rhythm
	Linguistic Regular vs. Non-Linguistic Regular	1.7	.045	Tests whether regular rhythm
				is associated with a sharper
				decline in reaction times for
				linguistic compared to non-
				linguistic stimuli
	Linguistic Irregular vs. Non-Linguistic Irregular	.79	.215	Tests if irregular rhythm is
				associated with a sharper
				decline in reaction times for
				linguistic compared to non-
				linguistic stimuli

Acknowledgements

This research was supported by the Spanish State Research Agency through BCBL's Severo Ochoa excellence accreditation SEV-2015-0490. The authors are thankful to Prof. Arthur Samuel (Stony Brook University, USA) and to Dr. Efthymia Kapnoula, Dr.

Data accessibility

Data will be made available on the figshare repository.

Competing Interests

The authors have no competing interests to declare.

Authors contributions

OR and MO conceived the study, OR prepared and ran the experiment and analyzed the data, OR and MO interpreted the results and wrote the manuscript.