Proposal of a risk model for vehicular traffic: A Boltzmann-type kinetic approach
PPROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC: ABOLTZMANN-TYPE KINETIC APPROACH
PAOLO FREGUGLIA AND ANDREA TOSIN
Abstract.
This paper deals with a Boltzmann-type kinetic model describingthe interplay between vehicle dynamics and safety aspects in vehicular traffic.Sticking to the idea that the macroscopic characteristics of traffic flow, includ-ing the distribution of the driving risk along a road, are ultimately generatedby one-to-one interactions among drivers, the model links the personal (i.e.,individual) risk to the changes of speeds of single vehicles and implements aprobabilistic description of such microscopic interactions in a Boltzmann-typecollisional operator. By means of suitable statistical moments of the kineticdistribution function, it is finally possible to recover macroscopic relationshipsbetween the average risk and the road congestion, which show an interestingand reasonable correlation with the well-known free and congested phases ofthe flow of vehicles. Introduction
Road safety is a major issue in modern societies, especially in view of the con-stantly increasing motorisation levels across several EU and non-EU countries. Al-though recent studies suggest that this fact is actually correlated with a generaldecreasing trend of fatality rates, see e.g., [1, 32], the problem of assessing quant-itatively the risk in vehicular traffic, and of envisaging suitable countermeasures,remains of paramount importance.So far, road safety has been studied mainly by means of statistical models aimedat fitting the probability distribution of the fatality rates over time [24] or at fore-casting road accidents using time series [3, 21]. Efforts have also been made to-wards the construction of safety indicators, which should allow one to classify thesafety performances of different roads and to compare, on such a basis, differentcountries [15, 16]. However, there is in general no agreement on which procedure,among several possible ones, is the most suited to construct a reliable indicatorand, as a matter of fact, the position of a given country in the ranking turns outto be very sensitive to the indicator used. Despite this, a synthetic analysis is ulti-mately necessary: a mere comparison of the crash data of different countries maybe misleading, therefore a more abstract and comprehensive concept of risk has tobe formulated [28].A recent report on road safety in New Zealand [2] introduces the following defin-itions of two types of risk:
Collective risk: is a measure of the total number of fatal and serious injurycrashes per kilometre over a section of road, cf. [2, p. 13];
Personal risk: is a measure of the danger to each individual using the statehighway being assessed, cf. [2, p. 14].While substantially qualitative and empirical, these definitions raise nevertheless animportant conceptual point, namely the fact that the risk is intrinsically multiscale . Mathematics Subject Classification.
Primary: 90B20; Secondary: 35Q20, 35Q70.
Key words and phrases.
Fundamental and risk diagrams of traffic, safety and risk regimes,kinetic equations, stochastic microscopic interactions, Wasserstein spaces. a r X i v : . [ phy s i c s . s o c - ph ] M a y PAOLO FREGUGLIA AND ANDREA TOSIN
Each driver ( microscopic scale) bears a certain personal level of danger, namely ofpotential risk, which, combined with the levels of danger of all other drivers, formsan emergent risk for the indistinct ensemble of road users ( macroscopic scale).Hence the large-scale tangible manifestations of the road risk originate from small-scale, often unobservable, causes. Such an argument is further supported by somepsychological theories of risk perception, among which probably the most popularone in the context of vehicular traffic is the so-called risk homeostasis theory . Ac-cording to this theory, each driver possesses a certain target level of personal risk,which s/he feels comfortable with; then, at every time s/he compares their per-ceived risk with such a target level, adjusting their behaviour so as to reduce thegap between the two [31]. Actually, the risk homeostasis theory is not widely accep-ted, some studies rejecting it on the basis of experimental evidences, see e.g., [11].The main criticism is, in essence, that the aforesaid risk regulatory mechanism ofthe drivers (acting similarly to the thermal homeostatic system in warm-bloodedanimals, whence the name of the theory) is too elementary compared to the muchricher variety of possible responses, to such an extent that some paradoxical con-sequences are produced. For instance, the number of traffic accidents per unittime would tend to be constant independently of possible safety countermeasures,because so tends to be the personal risk per unit time. Whether one accepts ornot this theory, there is a common agreement on the fact that the background ofall observable manifestations of the road risk is the individual behaviour of thedrivers. In this respect, conceiving a mathematical model able to explore the linkbetween small and large scale effects acquires both a theoretical and a practicalinterest. In fact, if on the one hand data collection is a useful practice in orderto grasp the essential trends of the considered phenomenon, on the other hand theinterpretation of the data themselves, with possibly the goal of making simulationsand predictions, cannot rely simply on empirical observation.The mathematical literature offers nowadays a large variety of traffic models atall observation and representation scales, from the microscopic and kinetic to themacroscopic one, see e.g., [25] and references therein for a critical survey. Neverthe-less, there is a substantial lack of models dedicated to the joint simulation of trafficflow and safety issues. In [23] the authors propose a model, which is investigatedanalytically in [18] and then further improved in [22], for the simulation of car ac-cidents. The model is a macroscopic one based on the coupling of two second ordertraffic models, which are instantaneously defined on two disjoint adjacent portionsof the road and which feature different traffic pressure laws accounting for moreand less careful drivers. Car collisions are understood as the intersection of thetrajectories of two vehicles driven by either type of driver. In particular, analyticalconditions are provided, under which a collision occurs depending on the initialspace and speed distributions of the vehicles.In this paper, instead of modelling physical collisions among cars, we are moreinterested in recovering the point of view based on the concept of risk discussed atthe beginning. Sticking to the idea that observable traffic trends are ultimately de-termined by the individual behaviour of drivers, we adopt a Boltzmann-type kineticapproach focusing on binary interactions among drivers, which are responsible forboth speed changes (through instantaneous acceleration, braking, overtaking) and,consequently, also for changes in the individual levels of risk. In practice, as usualin the kinetic theory approach, we consider the time-evolution of the statisticaldistribution of the microscopic states of the vehicles, taking into account that suchstates include also the personal risk of the drivers. Then, by extracting suitablemean quantities at equilibrium from the kinetic distribution function, we obtainsome information about the macroscopic traffic trends, including the average risk
ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 3 and the probability of accident as functions of the road congestion. To some extent,these can be regarded as measures of a potential collective risk useful to both roadusers and traffic governance authorities.In more detail, the paper is organised as follows. In Section 2 we present theBoltzmann-type kinetic model and specialise it to the case of a quantised space ofmicroscopic states, given that the vehicle speed and the personal risk can be con-veniently understood as discrete variables organised in levels. In Section 3 we detailthe modelling of the microscopic interactions among the vehicles, regarding themas stochastic jump processes on the discrete state space. Indeed, the aforemen-tioned variety of human responses suggests that a probabilistic approach is moreappropriate at this stage. In Section 4 we perform a computational analysis of themodel, which leads us to define the risk diagram of traffic parallelly to the more cel-ebrated fundamental and speed diagrams. By means of such a diagram we displaythe average risk as a function of the vehicle density and we take inspiration for pro-posing the definition of a safety criterion which discriminates between safety and risk regimes depending on the local traffic congestion. Interestingly enough, suchregimes turn out to be correlated with the well-known phase transition between freeand congested flow regimes also reproduced by our model. In Section 5 we drawsome conclusions and briefly sketch research perspectives regarding the applicationof ideas similar to those developed in this paper to other systems of interactingparticles prone to safety issues, for instance crowds. Finally, in Appendix A wedevelop a basic well-posedness and asymptotic theory of our kinetic model in meas-ure spaces (Wasserstein spaces), so as to ground the contents of the paper on solidmathematical bases.2.
Boltzmann-type kinetic model with stochastic interactions
In this section we introduce a model based on Boltzmann-type kinetic equa-tions, in which short-range interactions among drivers are modelled as stochastictransitions of microscopic state. This allows us to introduce the randomness of thehuman behaviour in the microscopic dynamics ruling the individual response tolocal traffic and safety conditions.In particular, we consider the following (dimensionless) microscopic states ofthe drivers: the speed v ∈ [0 , ⊂ R and the personal risk u ∈ [0 , ⊂ R , with u = 0, u = 1 standing for the lowest and highest risk, respectively. The kinetic(statistical) representation of the system is given by the (one particle) distributionfunction over the microscopic states, say f = f ( t, v, u ), t ≥ f ( t, v, u ) dv du is the fraction of vehicles which at time t travelat a speed comprised between v and v + dv with a personal risk between u and u + du . Alternatively, if the distribution function f is thought of as normalisedwith respect to the total number of vehicles, f ( t, v, u ) dv du can be understood asthe probability that a representative vehicle of the system possesses a microscopicstate in [ v, v + dv ] × [ u, u + du ] at time t . Remark . The numerical values of u introduced above are purely conventional:they are used to mathematise the concept of personal risk but neither refer nor implyactual physical ranges. Hence they serve mostly interpretative than strictly quant-itative purposes: for instance, the definition and identification of the macroscopicrisk and safety regimes of traffic, cf. Section 4.2. The quantitative information onthese regimes will be linked to a more standard and well-defined physical quantity,such as the vehicle density. Remark . The above statistical representation does not include the space co-ordinate among the variables which define the microscopic state of the vehicles.
PAOLO FREGUGLIA AND ANDREA TOSIN
This is because we are considering a simplified setting, in which the distributionof the vehicles along the road is supposed to be homogeneous . While being arough physical approximation, this assumption nonetheless allows us to focus onthe interaction dynamics among vehicles, which feature mainly speed variations,here further linked to variations of the personal risk. Hence the two microscopicvariables introduced above are actually the most relevant ones for constructing aminimal mathematical model which describes traffic dynamics as a result of con-current mechanical and behavioural effects.2.1.
Use of the distribution function for computing observable quantities.
If the distribution function f is known, several statistics over the microscopic statesof the vehicles can be computed. Such statistics provide macroscopic observablequantities related to both the traffic conditions and the risk along the road.We recall in particular some of them, which will be useful in the sequel: • The density of vehicles at time t , denoted ρ = ρ ( t ), which is defined as thezeroth order moment of f with respect to both v and u : ρ ( t ) := (cid:90) (cid:90) f ( t, v, u ) dv du. Throughout the paper we will assume 0 ≤ ρ ≤
1, where ρ = 1 repres-ents the maximum (dimensionless) density of vehicles that can be locallyaccommodated on the road. • The average flux of vehicles at time t , denoted q = q ( t ), which is definedas the first order moment of the speed distribution, the latter being themarginal of f with respect to u . Hence:(1) q ( t ) = (cid:90) v (cid:18)(cid:90) f ( t, v, u ) du (cid:19) dv. • The mean speed of vehicles at time t , denoted V = V ( t ), which is definedfrom the usual relationship between ρ and q , i.e., q = ρV , whence(2) V ( t ) = q ( t ) ρ ( t ) . • The statistical distribution of the risk , say ϕ = ϕ ( t, u ), namely the marginalof f with respect to v : ϕ ( t, u ) := (cid:90) f ( t, v, u ) dv, (3) which is such that ϕ ( t, u ) du is the number of vehicles which, at time t ,bear a personal risk comprised between u and u + du (regardless of theirspeed). Using ϕ we obtain the average risk , denoted U ( t ), along the roadat time t as: U ( t ) := 1 ρ ( t ) (cid:90) uϕ ( t, u ) du = 1 ρ ( t ) (cid:90) u (cid:18)(cid:90) f ( t, v, u ) dv (cid:19) du. (4) Notice that (cid:82) ϕ ( t, u ) du = ρ ( t ), which explains the coefficient ρ ( t ) in (4).2.2. Evolution equation for the distribution function.
A mathematical mo-del consists in an evolution equation for the distribution function f , derived con-sistently with the principles of the kinetic theory of vehicular traffic.In our spatially homogeneous setting, the time variation of the number of vehicleswith speed v and personal risk u is only due to short-range interactions, whichcause acceleration and braking. Since the latter depend ultimately on people’sdriving style, they cannot be modelled by appealing straightforwardly to standardmechanical principles. Hence, in the following, speed and risk transitions will be ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 5 regarded as correlated stochastic processes. This way both the subjectivity of thehuman behaviour and the interplay between mechanical and behavioural effects willbe taken into account.In formulas we write ∂ t f = Q + ( f, f ) − f Q − ( f ) , where: • Q + ( f, f ) is a bilinear gain operator , which counts the average number ofinteractions per unit time originating new vehicles with post-interactionstate ( v, u ); • Q − ( f ) is a linear loss operator , which counts the average number of inter-actions per unit time causing vehicles with pre-interaction state ( v, u ) tochange either the speed or the personal risk.Let us introduce the following compact notations: w := ( v, u ), I = [0 , ⊂ R .Then the expression of the gain operator is as follows (see e.g., [29]):(5) Q + ( f, f )( t, w ) = (cid:90) (cid:90) I η ( w ∗ , w ∗ ) P ( w ∗ → w | w ∗ , ρ ) f ( t, w ∗ ) f ( t, w ∗ ) d w ∗ d w ∗ where w ∗ , w ∗ are the pre-interaction states of the two vehicles which interact and: • η ( w ∗ , w ∗ ) > interaction rate , i.e., the frequency of interactionbetween a vehicle with microscopic state w ∗ and one with microscopic state w ∗ ; • P ( w ∗ → w | w ∗ , ρ ) ≥ transition probability distribution . Moreprecisely, P ( w ∗ → w | w ∗ , ρ ) d w is the probability that the vehicle withmicroscopic state w ∗ switches to a microscopic state contained in the ele-mentary volume d w of I centred in w because of an interaction with thevehicle with microscopic state w ∗ . Conditioning by the density ρ indicatesthat, as we will see later (cf. Section 3), binary interactions are influencedby the local macroscopic state of the traffic.For fixed pre-interaction states the following property holds:(6) (cid:90) I P ( w ∗ → w | w ∗ , ρ ) d w = 1 ∀ w ∗ , w ∗ ∈ I , ∀ ρ ∈ [0 , . Likewise, the expression of the loss operator is as follows (see again [29]): Q − ( f )( t, w ) = (cid:90) I η ( w , w ∗ ) f ( t, w ∗ ) d w ∗ . On the whole, the loss term f Q − ( f ) can be derived from the gain term (5) byassuming that the first vehicle already holds the state w and counting on averageall interactions which, in the unit time, can make it switch to whatever else state.One has: f ( t, w ) Q − ( f )( t, w ) = (cid:90) (cid:90) I η ( w , w ∗ ) P ( w → w (cid:48) | w ∗ , ρ ) f ( t, w ) f ( t, w ∗ ) d w (cid:48) d w ∗ , then property (6) gives the above expression for Q − ( f ).Putting together the terms introduced so far, and assuming, for the sake ofsimplicity, that η ( w ∗ , w ∗ ) = 1 for all w ∗ , w ∗ ∈ I , we finally obtain the followingintegro-differential equation for f :(7) ∂ t f = (cid:90) (cid:90) I P ( w ∗ → w | w ∗ , ρ ) f ( t, w ∗ ) f ( t, w ∗ ) d w ∗ d w ∗ − ρf. PAOLO FREGUGLIA AND ANDREA TOSIN
Notice that, with constant interaction rates, the loss term is directly proportionalto the vehicle density. Moreover, owing to property (6), it results (cid:90) I (cid:16) Q + ( f, f )( t, w ) − ρf ( t, w ) (cid:17) d w = 0 , therefore integrating (7) with respect to w gives that such a density is actuallyconstant in time (conservation of mass).For the study of basic qualitative properties of (7) the reader may refer to Ap-pendix A, where we tackle the well-posedness of the Cauchy problem associatedwith (7) in the frame of measure-valued differential equations in Wasserstein spaces.Such a theoretical setting, which is more abstract than the one usually consideredin the literature for similar equations, see e.g., [6, Appendix A] and [26], is heremotivated by the specialisation of the model that we are going to discuss in thenext section.2.3. Discrete microscopic states.
For practical reasons, it may be convenientto think of the microscopic states v, u as quantised (i.e., distributed over a setof discrete , rather than continuous, values). This is particularly meaningful forthe personal risk u , which is a non-mechanical quantity naturally meant in levels,but may be reasonable also for the speed v , see e.g., [10, 12], considering that thecruise speed of a vehicle tends to be mostly piecewise constant in time, with rapidtransitions from one speed level to another.In the state space I = [0 , × [0 , ⊂ R we consider therefore a lattice ofmicroscopic states { w ij } , with w ij = ( v i , u j ) and, say, i = 1 , . . . , n , j = 1 , . . . , m .For instance, if the lattice is uniformly spaced we have v i = i − n − , u j = j − m − , with in particular v = u = 0, v n = u m = 1, and v i < v i +1 for all i = 1 , . . . , n − u j < u j +1 for all j = 1 , . . . , m − f ( t, w ) = n (cid:88) i =1 m (cid:88) j =1 f ij ( t ) δ w ij ( w ) , where δ w ij ( w ) = δ v i ( v ) ⊗ δ u j ( u ) is the two-dimensional Dirac delta function, while f ij ( t ) is the fraction of vehicles which, at time t , travel at speed v i with personalrisk u j (or, depending on the interpretation given to f , it is the probability that arepresentative vehicle of the system possesses the microscopic state w ij = ( v i , u j )at time t ). In order to specialise (7) to the kinetic distribution function (8), werewrite it in weak form by multiplying by a test function φ ∈ C ( I ) and integratingover I : ddt (cid:90) I φ ( w ) f ( t, w ) d w = (cid:90) (cid:90) I (cid:18)(cid:90) I φ ( w ) P ( w ∗ → w | w ∗ , ρ ) d w (cid:19) f ( t, w ∗ ) f ( t, w ∗ ) d w ∗ d w ∗ − ρ (cid:90) I φ ( w ) f ( t, w ) d w ;next we read f ( t, w ) d w as an integration measure, not necessarily regular withrespect to Lebesgue, and from (8) we get: n (cid:88) i =1 m (cid:88) j =1 f (cid:48) ij ( t ) φ ( w ij ) ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 7 = n (cid:88) i ∗ ,i ∗ =1 m (cid:88) j ∗ ,j ∗ =1 (cid:18)(cid:90) I φ ( w ) P ( w i ∗ j ∗ → w | w i ∗ j ∗ , ρ ) d w (cid:19) f i ∗ j ∗ ( t ) f i ∗ j ∗ ( t ) − n (cid:88) i =1 m (cid:88) j =1 ρf ij ( t ) φ ( w ij ) . In view of the quantisation of the state space, the transition probability distributionmust have a structure comparable to (8), i.e., it must be a discrete probabilitydistribution over the post-interaction state w . Hence we postulate: P ( w i ∗ j ∗ → w | w i ∗ j ∗ , ρ ) = n (cid:88) i =1 m (cid:88) j =1 P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) δ w ij ( w ) , where P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) ∈ [0 ,
1] is the probability that the vehicle with microscopic state w i ∗ j ∗ jumps to the microscopic state w ij because of an interaction with the vehiclewith microscopic state w i ∗ j ∗ , given the local traffic congestion ρ . Plugging this intothe equation above yields n (cid:88) i =1 m (cid:88) j =1 f (cid:48) ij ( t ) φ ( w ij )= n (cid:88) i =1 m (cid:88) j =1 n (cid:88) i ∗ ,i ∗ =1 m (cid:88) j ∗ ,j ∗ =1 P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) f i ∗ j ∗ ( t ) f i ∗ j ∗ ( t ) − ρf ij ( t ) φ ( w ij ) , whence finally, owing to the arbitrariness of φ , we obtain(9) f (cid:48) ij = n (cid:88) i ∗ ,i ∗ =1 m (cid:88) j ∗ ,j ∗ =1 P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) f i ∗ j ∗ f i ∗ j ∗ − ρf ij . Remark . This discrete-state kinetic-type equation has been studied in the liter-ature, see e.g., [8, 10]. In particular, it has been proved to admit smooth solutions t (cid:55)→ f ij ( t ), which are unique and nonnegative for prescribed nonnegative initialdata f ij (0) and, in addition, preserve the total mass (cid:80) ni =1 (cid:80) mj =1 f ij ( t ) in time.The arguments above can be made rigorous by appealing to the theory for (7)developed in Appendix A. In particular, we can state the following result: Theorem 2.4.
Let the transition probability distribution have the form P ( w ∗ → w | w ∗ , ρ ) = n (cid:88) i =1 m (cid:88) j =1 P ij ( w ∗ , w ∗ , ρ ) δ w ij ( w ) , where the mapping ( w ∗ , w ∗ , ρ ) (cid:55)→ P ij ( w ∗ , w ∗ , ρ ) is Lipschitz continuous for all i, j , i.e., there exists a constant Lip( P ij ) > such that (cid:12)(cid:12) P ij ( w ∗ , w ∗ , ρ ) − P ij ( w ∗ , w ∗ , ρ ) (cid:12)(cid:12) ≤ Lip( P ij ) (cid:16) | w ∗ − w ∗ | + | w ∗ − w ∗ | + | ρ − ρ | (cid:17) for all w ∗ , w ∗ , w ∗ , w ∗ ∈ I , ρ ∈ [0 , .Let moreover f ( w ) = n (cid:88) i =1 m (cid:88) j =1 f ij δ w ij ( w ) PAOLO FREGUGLIA AND ANDREA TOSIN be a prescribed kinetic distribution function at time t = 0 over the lattice of micro-scopic states { w ij } ⊂ I , such that f ij ≥ ∀ i, j, n (cid:88) i =1 m (cid:88) j =1 f ij = ρ ∈ [0 , . Set P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) := P ij ( w i ∗ j ∗ , w i ∗ j ∗ , ρ ) . Then the corresponding unique solutionto (7) is (8) with coefficients f ij ( t ) given by (9) along with the initial conditions f ij (0) = f ij . In addition, it depends continuously on the initial datum as stated byTheorem A.5.Proof. The given transition probability distribution satisfies Assumption A.3, infact W ( P ( w ∗ → · | w ∗ , ρ ) , P ( w ∗ → · | w ∗ , ρ ))= sup ϕ ∈ C b, ( I ) ∩ Lip ( I ) n (cid:88) i =1 m (cid:88) j =1 ϕ ( w ij ) (cid:0) P ij ( w ∗ , w ∗ , ρ ) − P ij ( w ∗ , w ∗ , ρ ) (cid:1) ≤ n (cid:88) i =1 m (cid:88) j =1 Lip( P ij ) (cid:16) | w ∗ − w ∗ | + | w ∗ − w ∗ | + | ρ − ρ | (cid:17) . Furthermore, f ∈ M ρ + ( I ). Then, owing to Theorem A.4, we can assert that theCauchy problem associated with (7) admits a unique mild solution .The calculations preceding this theorem show that if the f ij ( t )’s satisfy (9)then (8) is indeed such a solution, considering also that it is nonnegative (cf. Re-mark 2.3) and matches the initial condition f . (cid:3) Theorem 2.4 requires the mapping ( w ∗ , w ∗ , ρ ) (cid:55)→ P ij ( w ∗ , w ∗ , ρ ) to be Lipschitzcontinuous in I × I × [0 ,
1] but the solution (8) depends ultimately only on the val-ues P ij ( w i ∗ j ∗ , w i ∗ j ∗ , ρ ), cf. (9). Therefore, when constructing specific models, wecan confine ourselves to specifying the values P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ), taking for granted thatthey can be variously extended to points ( w ∗ , w ∗ ) (cid:54) = ( w i ∗ j ∗ , w i ∗ j ∗ ) in a Lipschitzcontinuous way. 3. Modelling microscopic interactions
From now on, we will systematically refer to the discrete-state setting ruledby (9). In order to describe the interactions among the vehicles, it is necessary tomodel the transition probabilities P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) associated with the jump processesover the lattice of discrete microscopic states.As a first step, we propose the following factorisation: P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) = Prob( u j ∗ → u j | v i ∗ , v i ∗ , ρ ) · Prob( v i ∗ → v i | v i ∗ , ρ )=: ( P (cid:48) ) ji ∗ j ∗ ,i ∗ ( ρ ) · ( P (cid:48)(cid:48) ) ii ∗ i ∗ ( ρ ) , which implies that changes in the personal risk (first term at the right-hand side)depend on the current speeds of the interacting pairs while the driving style (i.e.,the way in which the speed changes, second term at the right-hand side) is notdirectly influenced by the current personal risk. In a sense, we are interpreting thechange of personal risk as a function of the driving conditions, however linked tothe subjectivity of the drivers and hence described in probability. By subjectivitywe mean the fact that different drivers may not respond in the same way to thesame conditions. More advanced models may account for a joint influence of speed For the definition of mild solution to (7) we refer the reader to (15) in Appendix A.
ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 9 and risk levels on binary interactions, but for the purposes of the present paper theapproximation above appears to be satisfactory.As a second step, we detail the transition probabilities ( P (cid:48) ) ji ∗ j ∗ ,i ∗ ( ρ ), ( P (cid:48)(cid:48) ) ii ∗ i ∗ ( ρ )just introduced. It is worth stressing that they will be mainly inspired by a proto-typical analysis of the driving styles. In particular, they will be parameterised bythe vehicle density ρ ∈ [0 ,
1] so as to feed back the global traffic conditions to thelocal interaction rules. Other external objective factors which may affect the flow ofvehicles and the personal risk, such as e.g., weather or road conditions (number oflanes, number of directions of travel, type of wearing course), will be summarisedby a parameter α ∈ [0 , Remark . The interpretation of α is conceptually analogous to that of u dis-cussed in Remark 2.1: its numerical values do not refer to actual physical (meas-ured) ranges but serve to convey, in mathematical terms, the influence of externalconditions on binary interactions.3.1. Risk transitions.
In modelling the risk transition probability( P (cid:48) ) ji ∗ j ∗ ,i ∗ ( ρ ) = Prob( u j ∗ → u j | v i ∗ , v i ∗ , ρ )we consider two cases, depending on whether the vehicle with state ( v i ∗ , u j ∗ ) in-teracts with a faster or a slower leading vehicle with speed v i ∗ . • If v i ∗ ≤ v i ∗ we set( P (cid:48) ) ji ∗ j ∗ ,i ∗ ( ρ ) = αρδ j, max { , j ∗ − } + (1 − αρ ) δ j,j ∗ , the symbol δ denoting here the Kronecher’s delta. In practice, we assumethat the interaction with a faster leading vehicle can reduce the personalrisk with probability αρ , which raises in high traffic congestion and goodenvironmental conditions. The rationale is that the headway from a fasterleading vehicle increases, which reduces the risk of collision especially whenvehicles are packed (high ρ ) or when speeds are presumably high (goodenvironmental conditions, i.e., high α ). Alternatively, after the interactionthe personal risk remains the same with the complementary probability. • If v i ∗ > v i ∗ we set( P (cid:48) ) ji ∗ j ∗ ,i ∗ ( ρ ) = δ j, min { j ∗ +1 , m } , i.e., we assume that the interaction with a slower leading vehicle can onlyincrease the personal risk because the headway is reduced or overtaking isinduced (see below).3.2. Speed transitions.
In modelling the speed transition probability( P (cid:48)(cid:48) ) ii ∗ i ∗ ( ρ ) = Prob( v i ∗ → v i | v i ∗ , ρ )we refer to [27], where the following three cases are considered: • If v i ∗ < v i ∗ then( P (cid:48)(cid:48) ) ii ∗ i ∗ ( ρ ) = α (1 − ρ ) δ i,i ∗ +1 + (1 − α (1 − ρ )) δ i,i ∗ , i.e., the vehicle with speed v i ∗ emulates the leading one with speed v i ∗ byaccelerating to the next speed with probability α (1 − ρ ). This probabilityincreases if environmental conditions are good and traffic is not too muchcongested. Otherwise, the speed remains unchanged with complementaryprobability. • If v i ∗ > v i ∗ then( P (cid:48)(cid:48) ) ii ∗ i ∗ ( ρ ) = α (1 − ρ ) δ i,i ∗ + (1 − α (1 − ρ )) δ i,i ∗ , i.e., the vehicle with speed v i ∗ maintains its speed with probability α (1 − ρ ).The rationale is that if environmental conditions are good enough or trafficis sufficiently uncongested then it can overtake the slower leading vehiclewith speed v i ∗ . Otherwise, it is forced to slow down to the speed v i ∗ andto queue up, which happens with the complementary probability. • If v i ∗ = v i ∗ then( P (cid:48)(cid:48) ) ii ∗ i ∗ ( ρ ) = (1 − α ) ρδ i, max { , i ∗ − } + α (1 − ρ ) δ i, min { i ∗ +1 , n } + (1 − α − (1 − α ) ρ ) δ i,i ∗ . In this case there are three possible outcomes of the interaction: if environ-mental conditions are poor and traffic is congested the vehicle with speed v i ∗ slows down with probability (1 − α ) ρ ; if, instead, environmental con-ditions are good and traffic is light then it accelerates to the next speed(because e.g., it overtakes the leading vehicle) with probability α (1 − ρ );finally, it can also remain with its current speed with a probability whichcomplements the sum of the previous two.4. Case studies
Fundamental diagrams of traffic.
Model (9) can be used to investigate thelong-term macroscopic dynamics resulting from the small-scale interactions amongvehicles discussed in the previous section. Such dynamics are summarised by thewell-known fundamental and speed diagrams of traffic, see e.g., [20], which expressthe average flux and mean speed of the vehicles at equilibrium, respectively, asfunctions of the vehicle density along the road. This information, typically obtainedfrom experimental measurements [7, 19], is here studied at a theoretical level inorder to discuss qualitatively the impact of the driving style on the macroscopicallyobservable traffic trends.In Appendix A we give sufficient conditions for the existence, uniqueness, andglobal attractiveness of equilibria f ∞ ∈ M ρ + ( I ) of (7), cf. Theorems A.7, A.8.Here we claim, in particular, that if the transition probability distribution P hasthe special form discussed in Theorem 2.4 then f ∞ is actually a discrete-statedistribution function. Theorem 4.1.
Fix ρ ∈ [0 , and let P ( w ∗ → w | w ∗ , ρ ) be like in Theorem 2.4.Assume moreover that Lip( P ) < (cf. Assumption A.3). Then the unique equilib-rium distribution function f ∞ ∈ M ρ + ( I ) , which is also globally attractive, has theform f ∞ ( w ) = (cid:80) ni =1 (cid:80) mj =1 f ∞ ij δ w ij ( w ) with the coefficients f ∞ ij satisfying (10) f ∞ ij = 1 ρ n (cid:88) i ∗ ,i ∗ =1 m (cid:88) j ∗ ,j ∗ =1 P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) f ∞ i ∗ j ∗ f ∞ i ∗ j ∗ , i = 1 , . . . , nj = 1 , . . . , m. Proof.
We consider directly the case ρ >
0, for ρ = 0 implies uniquely f ∞ ≡ P ) < , we know from Theorems A.7, A.8 that (7) admits a uniqueand globally attractive equilibrium distribution f ∞ ∈ M ρ + ( I ), which is found asthe fixed point of the mapping f (cid:55)→ ρ Q + ( f, f ). In particular, defining the subset D := f ∈ M ρ + ( I ) : f ( w ) = n (cid:88) i =1 m (cid:88) j =1 f ij δ w ij ( w ) , f ij ≥ , n (cid:88) i =1 m (cid:88) j =1 f ij = ρ , ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 11 it is easy to see that if P has the form indicated in Theorem 2.4 then the operator ρ Q + maps D into itself. In fact, for f ∈ D we get1 ρ Q + ( f, f ) = n (cid:88) i =1 m (cid:88) j =1 ρ n (cid:88) i ∗ ,i ∗ =1 m (cid:88) j ∗ ,j ∗ =1 P iji ∗ j ∗ ,i ∗ j ∗ ( ρ ) f i ∗ j ∗ f i ∗ j ∗ δ w ij ( w ) . From here we also deduce formally (10). Therefore, in order to get the thesis, it issufficient to prove that D is closed in M ρ + ( I ). In fact this will imply that ( D, W )is a complete metric space, and Banach contraction principle will then locate thefixed point of ρ Q + in D .Let ( f k ) ⊆ D be a convergent sequence in M ρ + ( I ) with respect to the W metric. It is then Cauchy, hence given (cid:15) > N (cid:15) ∈ N such that if h, k > N (cid:15) then W (cid:0) f h , f k (cid:1) < (cid:15) . This condition means (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) I ϕ ( w ) (cid:0) f k ( w ) − f h ( w ) (cid:1) d w (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) i =1 m (cid:88) j =1 ϕ ( w ij ) (cid:0) f kij − f hij (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) for every ϕ ∈ C b, ( I ) ∩ Lip ( I ). In particular, taking a function ϕ which vanishesat every w ij but one, say w ¯ ı ¯ , we discover | ϕ ( w ¯ ı ¯ ) | (cid:12)(cid:12) f h ¯ ı ¯ − f k ¯ ı ¯ (cid:12)(cid:12) < (cid:15) for all h, k > N (cid:15) .Thus we deduce that ( f kij ) k is a Cauchy sequence in R , hence for all i, j there exists f ij ∈ R such that f kij → f ij ( k → ∞ ). Clearly f ij ≥ f kij ’s are all non-negative by assumption; moreover, (cid:80) ni =1 (cid:80) mj =1 f ij = lim k →∞ (cid:80) ni =1 (cid:80) mj =1 f kij = ρ .Therefore f := (cid:80) ni =1 (cid:80) mj =1 f ij δ w ij ∈ D .We now claim that f k → f in the W metric: W (cid:0) f k , f (cid:1) = sup ϕ ∈ C b, ( I ) ∩ Lip ( I ) (cid:90) I ϕ ( w ) (cid:0) f ( w ) − f k ( w ) (cid:1) d w = sup ϕ ∈ C b, ( I ) ∩ Lip ( I ) n (cid:88) i =1 m (cid:88) j =1 ϕ ( w ij ) (cid:0) f ij − f kij (cid:1) ≤ n (cid:88) i =1 m (cid:88) j =1 (cid:12)(cid:12) f ij − f kij (cid:12)(cid:12) k →∞ −−−−→ . This implies that D is closed and the proof is completed. (cid:3) Under the assumptions of Theorem 4.1, (10) defines a mapping [0 , (cid:51) ρ (cid:55)→{ f ∞ ij ( ρ ) } , i.e., for every ρ ∈ [0 ,
1] there exist unique coefficients f ∞ ij solving (10)such that { f ∞ ij } is the equilibrium of system (9) with moreover (cid:80) ni =1 (cid:80) mj =1 f ∞ ij = ρ .Owing to the argument above, for each ρ it is possible to compute the corres-ponding average flux and mean speed at equilibrium by means of formulas (1), (2)with the kinetic distribution f ∞ . This generates the two mappings(11) ρ (cid:55)→ q ( ρ ) := n (cid:88) i =1 v i m (cid:88) j =1 f ∞ ij ( ρ ) , ρ (cid:55)→ V ( ρ ) := q ( ρ ) ρ , which are the theoretical definitions of the fundamental and speed diagrams, re-spectively, of traffic. Furthermore, it is possible to estimate the dispersion of the mi-croscopic speeds at equilibrium by computing the standard deviation of the speed: σ V ( ρ ) := (cid:118)(cid:117)(cid:117)(cid:116) ρ n (cid:88) i =1 ( v i − V ( ρ )) m (cid:88) j =1 f ∞ ij ( ρ ) , that of the flux being ρσ V ( ρ ), which gives a measure of the homogeneity of thedriving styles of the drivers. Figure 1 shows the diagrams (11), with the corresponding standard deviations,for different values of the constant α parameterising the transition probabilities, cf.Sections 3.1, 3.2. Each pair ( ρ, q ( ρ )), ( ρ, V ( ρ )) has been computed by integratingnumerically (9) up to a sufficiently large final time, such that the equilibrium wasreached.For α = 1 (best environmental conditions) the diagrams are the same as thosestudied analytically in [13]. In particular, they show a clear separation between theso-called free and congested phases of traffic: the former, taking place at low density( ρ < . ρ > . ρ = 0 . α <
1, analogously detailed analytical results are not yet available and, toour knowledge, Theorems A.7, A.8 in Appendix A are the first results giving atleast sufficient conditions for the qualitative characterisation of equilibrium solu-tions to (7) in the general case. According to the graphs in Figure 1, the modelpredicts lower and lower critical values for the density threshold triggering the phasetransition. In addition to that, coherently with the experimental observations, cf.e.g., [19], some scattering of the diagrams appears also for low density, along witha capacity drop visible in the average flux (i.e., the fact that the maximum fluxin the congested phase is lower than the maximum one in the free phase), whichseparates the free and congested phases as described in [33].Compared to typical experimental data, the most realistic diagrams seem to bethose obtained for α = 0 .
8, which denotes suboptimal though not excessively poorenvironmental conditions. It is worth stressing that such a realism of the theoreticaldiagrams is not only relevant for supporting the derivation of macroscopic kinematicfeatures of the flow of vehicles at equilibrium out of microscopic interaction rulesfar from equilibrium. It constitutes also a reliable basis for interpreting, in a similarmultiscale perspective, the link with risk and safety issues, for which synthetic andinformative empirical data similar to the fundamental and speed diagrams are, tothe authors knowledge, not currently available for direct comparison.4.2.
Risk diagrams of traffic.
Starting from the statistical distribution of therisk given in (3), we propose the following definition for the probability of accidentalong the road:
Definition 4.2 (Probability of accident) . Let ¯ u ∈ (0 ,
1) be a risk threshold abovewhich the personal risk for a representative vehicle is considered too high. Wedefine the instantaneous probability of accident P = P ( t ) (associated with ¯ u ) asthe normalised number of vehicles whose personal risk is, at time t , greater than orequal to ¯ u : P ( t ) := 1 ρ (cid:90) u ϕ ( t, u ) du = 1 ρ (cid:90) u (cid:90) f ( t, v, u ) dv du. Asymptotically, using the discrete-state equilibrium distribution function f ∞ = f ∞ ( ρ ) found in Theorem 4.1, we obtain the mapping(12) ρ (cid:55)→ P ( ρ ) := 1 ρ (cid:88) j : u j ≥ ¯ u n (cid:88) i =1 f ∞ ij ( ρ ) , which shows that the probability of accident is, in the long run, a function of thetraffic density. We call (12) the accident probability diagram . ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 13 -0.100.10.20.30.40.50.6 q -0.200.20.40.60.811.2 V -0.0500.050.10.150.20.250.30.350.4 q -0.200.20.40.60.811.2 V -0.04-0.0200.020.040.060.080.10.120.14 q -0.200.20.40.60.811.2 V -0.04-0.0200.020.040.060.080.1 0 0.2 0.4 0.6 0.8 1 q ρ -0.100.10.20.30.40.50.60.70.80.9 0 0.2 0.4 0.6 0.8 1 V ρ (cid:1) = (cid:1) = . (cid:1) = . (cid:1) = . Figure 1.
Average flux (left column) and mean speed (rightcolumn) vs. traffic density for different levels of the quality ofthe environment. Red dashed lines are the respective standarddeviations. n = 6 uniformly spaced speed classes have been used.Definition 4.2 and, in particular, (12) depend on the threshold ¯ u , which needs tobe estimated in order for the model to serve quantitative purposes. If, for a givenroad, the empirical probability of accident is known (for instance, from time series on the frequency of accidents, see e.g., [3, 21, 24]) then it is possible to find ¯ u bysolving an inverse problem which leads the theoretical probability (12) to matchthe experimental one. This way, the road under consideration can be assigned the risk threshold ¯ u .The question then arises how to use the information provided by the risk threshold¯ u for the assessment of safety standards. In fact, the personal risk u , albeit a prim-itive variable of the model, is not a quantity which can be really measured for eachvehicle: a macroscopic synthesis is necessary. Quoting from [2]:Personal risk is most of interest to the public, as it shows the riskto road users, as individuals.To this purpose, we need to further post-process the statistical information broughtby the kinetic model. Taking inspiration from the fundamental and speed diagramsof traffic discussed in Section 4.1, a conceivable approach is to link the personalrisk, conveniently understood in an average sense, to the macroscopic traffic densityalong the road. For this we define: Definition 4.3 (Risk diagram) . The risk diagram of traffic is the mapping (cf. (4))(13) ρ (cid:55)→ U ( ρ ) := 1 ρ m (cid:88) j =1 u j n (cid:88) i =1 f ∞ ij ( ρ ) ,f ∞ = f ∞ ( ρ ) being the equilibrium kinetic distribution function of Theorem 4.1.The related standard deviation is σ U ( ρ ) := (cid:118)(cid:117)(cid:117)(cid:116) ρ m (cid:88) j =1 ( u j − U ( ρ )) n (cid:88) i =1 f ∞ ij ( ρ ) . Using the tools provided by Definition 4.3, we can finally fix a safety criterion which discriminates between safety and risk regimes of traffic depending on thetraffic loads:
Definition 4.4 (Safety criterion) . Let ¯ u ∈ (0 ,
1) be the risk threshold fixed byDefinition 4.2. The safety regime of traffic along a given road corresponds to thetraffic loads ρ ∈ [0 ,
1] such that U ( ρ ) + σ U ( ρ ) < ¯ u. The complementary regime, i.e., the one for which U ( ρ ) + σ U ( ρ ) ≥ ¯ u , is the riskregime . Remark . An alternative, less precautionary, criterion might identify the safetyregime with the traffic loads such that U ( ρ ) < ¯ u and the risk regime with thosesuch that U ( ρ ) ≥ ¯ u .Figure 2 shows the risk diagram (13) and the probability of accident (12) atequilibrium obtained numerically for various environmental conditions as describedin Section 4.1, using the heuristic risk threshold ¯ u = 0 . α = 1 the average risk and the probability of accident are deterministicallyzero for all values of the traffic density in the free phase ( ρ < . ρ > . ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 15 (cid:1) = (cid:1) = . (cid:1) = . (cid:1) = . -0.100.10.20.30.40.50.60.70.80.9 U P -0.0500.050.10.150.20.250.30.350.4 U -0.100.10.20.30.40.50.60.70.80.91 00.10.20.30.40.50.60.7 P -0.200.20.40.60.811.2 U P -0.200.20.40.60.811.21.4 0 0.2 0.4 0.6 0.8 1 U ρ P ρ Figure 2.
Average risk (left column), with related standard devi-ation (red dashed lines), and probability of accident (right column)vs. traffic density for different levels of the quality of the environ-ment. n = 6 and m = 3 uniformly spaced speed and risk classes,respectively, have been used ρ approaches 1, for in a full traffic jam vehicles do not move. Notice that themaximum of the average risk and of the probability of accident is in correspondenceof the critical density value ρ = 0 . Figure 3. a. Comparison of the curves (11), (12), (13) in the case α = 0 . b. Determination of safety and risk regimes of traffic for α = 0 .
8, with corresponding probabilities of accident.For α <
1, the main features of the diagrams ρ (cid:55)→ U ( ρ ) and ρ (cid:55)→ P ( ρ ) describedin the ideal prototypical case above remain unchanged. In particular, a comparisonwith Figure 1 shows that the maximum of both diagrams is still reached in corres-pondence of the density value triggering the transition from free to congested traffic,see also Figure 3a. This is clearly in good agreement with the intuition, indeed itidentifies the phase transition as the most risky situation for drivers. It is worthremarking that such a macroscopically observable fact has not been postulated inthe construction of the model but has emerged as a result of more elementary mi-croscopic interaction rules. For α = 0 . , .
8, the average risk and the probabilityof accident take, in the free phase of traffic, a realistic nonzero value, which firstincreases before the phase transition and then decreases to zero in the congestedphase. Notice that the standard deviation of the risk is higher in the free than in thecongested phase, which again meets the intuition considering that at low densitythe movement of single vehicles is less constrained by the global flow. For α = 0 . U ( ρ ) , P ( ρ ) → ρ → + , namelyin correspondence of the maximum mean speed (cf. the corresponding panels ofFigure 1).According to [2]:Personal risk is typically higher in more difficult terrains, wheretraffic volumes and road standards are often lower.By looking at the graphs in the first column of Figure 2, we see that the resultsof the model match qualitatively well this experimental observation: for low ρ anddecreasing α the average personal risk tends indeed to increase.Again, the most realistic (namely suboptimal, though not excessively poor) scen-ario appears to be the one described by α = 0 .
8. In this case, cf. Figure 3b, thesafety criterion of Definition 4.4, i.e., U ( ρ ) + σ U ( ρ ) < . , individuates two safety regimes for traffic loads ρ ∈ [0 , ρ ) and ρ ∈ ( ρ , ρ ≈ .
275 and ρ ≈ .
51, respectively. The first one corresponds to a probabilityof accident P ( ρ ) (cid:46) P ( ρ ) (cid:46) ρ < ρ ) is lower than the ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 17 one in congested flow ( ρ > ρ ), meaning that the safety criterion of Definition 4.4turns out to be more restrictive in the first than in the second case. This canbe understood thinking of the fact that in free flow speeds are higher and themovement of vehicles is less constrained by the global flow, which imposes tightersafety standards. 5. Conclusions and perspectives
In this paper we have proposed a Boltzmann-type kinetic model which describesthe influence of the driving style on the personal driving risk in terms of microscopicbinary interactions among the vehicles. In particular, speed transitions due to en-counters with other vehicles, and the related changes of personal risk, are describedin probability, thereby accounting for the interpersonal variability of the humanbehaviour, hence ultimately for the subjective component of the risk. Moreover,they are parameterised by the local density of vehicles along the road and by theenvironmental conditions (for instance, type of road, weather conditions), so as toinclude in the mathematical description also the objective component of the risk.By studying the equilibrium solutions of the model, we have defined two macro-scopic quantities of interest for the global assessment of the risk conditions, namelythe risk diagram and the accident probability diagram . The former gives the averagerisk along the road and the latter the probability of accident both as functions ofthe density of vehicles, namely of the level of traffic congestion. These diagramscompare well with the celebrated fundamental and speed diagrams of traffic, alsoobtainable from the equilibrium solutions of our kinetic model, in that they predictthe maximum risk across the phase transition from free to congested flow, when sev-eral perturbative phenomena are known to occur in the macroscopic hydrodynamicbehaviour of traffic (such as e.g., capacity drop [33], scattering of speed and fluxand appearance of a third phase of “synchronised flow” [19]). Moreover, withinthe free and congested regimes they are in good agreement with the experimentalfindings of accident data collection campaigns: for instance, they predict that thepersonal risk rises in light traffic and poor environmental conditions, coherentlywith what is stated e.g., in [2].By using the aforesaid diagrams we have proposed the definition of a safetycriterion , which, upon assigning to a given road a risk threshold based on theknowledge of real data on accidents, individuates safety and risk regimes dependingon the volume of traffic. Once again, it turns out that the risk regime consists ofa range of vehicle densities encompassing the critical one at which phase transitionoccurs. This type of information is perhaps more directly useful to the publicthan to traffic controlling authorities, because it shows the average risk that arepresentative road user is subject to. Nevertheless, by identifying traffic loadswhich may pose safety threats, it also indicates which densities should be preferablyavoided along the road and when risk reducing measures should be activated.This work should be considered as a very first attempt to formalise, by a math-ematical model, the risk dynamics in vehicular traffic from the point of view of simulation and prediction rather than simply of statistical description. Severalimprovements and developments are of course possible, which can take advantageof some existing literature about kinetic models of vehicular traffic: for instance,one may address the spatially inhomogeneous problem [10, 12] to track “risk waves”along the road; or the problem on networks [14] to study the propagation of the riskon a set of interconnected roads; or even the impact of different types of vehicles,which form a “traffic mixture” [27], on the risk distribution. On the other hand,the ideas presented in this paper may constitute the basis for modelling risk andsafety aspects also of other systems of interacting agents particularly interested by such issues. It is the case of e.g., human crowds, for which a quite wide, thoughrelatively recent, literature already exists (see [9, Chapter 4] for a survey), that insome cases [4] uses a kinetic formalism close to the one which inspired the presentwork.
Appendix A. Basic theory of the kinetic model in Wasserstein spaces
Equation (7), complemented with a suitable initial condition, produces the fol-lowing Cauchy problem:(14) (cid:40) ∂ t f = Q + ( f, f ) − ρf, t > , w ∈ I f (0 , w ) = f ( w ) , w ∈ I with the compatibility condition (cid:82) I f ( w ) d w = ρ . Recall that I = [0 , ⊂ R is the space of the microscopic states. The problem can be rewritten in mild formby multiplying both sides of the equation by e ρt and integrating in time: f ( t, w ) = e − ρt f ( w )+ (cid:90) t e ρ ( s − t ) (cid:90) (cid:90) I P ( w ∗ → w | w ∗ , ρ ) f ( s, w ∗ ) f ( s, w ∗ ) d w ∗ d w ∗ ds, (15)where we have used that, in view of (6), ρ = (cid:82) I f ( t, w ) d w is constant in t .In order to allow for measure-valued kinetic distribution functions, as it happensin the model discussed from Section 2.3 onwards, an appropriate space in which tostudy (15) is X := C ([0 , T ]; M ρ + ( I )), where T > M ρ + ( I ) isthe space of positive measures on I having mass ρ ≥
0. An element f ∈ X is thena continuous mapping t (cid:55)→ f ( t ), where, for all t ∈ [0 , T ], f ( t ) is a positive measurewith (cid:82) I f ( t, w ) d w = ρ . X is a complete metric space with the distance sup t ∈ [0 , T ] W ( f ( t ) , g ( t )), where(16) W ( f ( t ) , g ( t )) = sup ϕ ∈ Lip ( I ) (cid:90) I ϕ ( w )( g ( t, w ) − f ( t, w )) d w . is the between f ( t ) , g ( t ) ∈ M ρ + ( I ). In particular,Lip ( I ) = { ϕ ∈ C ( I ) : Lip( ϕ ) ≤ } , Lip( ϕ ) denoting the Lipschitz constant of ϕ . Remark
A.1 . The definition (16) of W follows from the Kantorovich-Rubinsteinduality formula, see e.g., [5, Chapter 7]. However, since in M ρ + ( I ) all measurescarry the same mass and, furthermore, the domain I is bounded with diam( I ) ≤ C b, ( I ) ∩ Lip ( I ), where C b, ( I ) = { ϕ ∈ C ( I ) : (cid:107) ϕ (cid:107) ∞ ≤ } , see [30, Chapter 1]. Hence we also have:(17) W ( f ( t ) , g ( t )) = sup ϕ ∈ C b, ( I ) ∩ Lip ( I ) (cid:90) I ϕ ( w )( g ( t, w ) − f ( t, w )) d w . In the following we will use both (16) and (17) interchangeably.
Remark
A.2 . Let λ > ϕ ∈ C ( I ) with Lip( ϕ ) ≤ λ . Then (cid:90) I ϕ ( w )( g ( t, w ) − f ( t, w )) d w = λ (cid:90) I ϕ ( w ) λ ( g ( t, w ) − f ( t, w )) d w ≤ λW ( f ( t ) , g ( t )) , ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 19 considering that ϕλ ∈ Lip ( I ).We will occasionally use this property in the proofs of the forthcoming theorems.Notice that, if λ <
1, this estimate is stricter than that obtained by using directlythe fact that ϕ ∈ Lip ( I ) (i.e., the one without λ at the right-hand side).To establish the next results, we will always assume that the transition probab-ility distribution P satisfies the following Lipschitz continuity property: Assumption A.3.
Let P ( w ∗ → · | w ∗ , ρ ) ∈ P ( I ) for all w ∗ , w ∗ ∈ I , ρ ∈ [0 , P ( I ) is the space of probability measures on I . We assume that there existsLip( P ) > which may depend on ρ (although we do not write such a dependenceexplicitly), such that W ( P ( w ∗ → · | w ∗ , ρ ) , P ( w ∗ → · | w ∗ , ρ )) ≤ Lip( P ) ( | w ∗ − w ∗ | + | w ∗ − w ∗ | )for all w ∗ , w ∗ , w ∗ , w ∗ ∈ I and all ρ ∈ [0 , Existence and uniqueness of the solution.
Taking advantage of the mildformulation (15) of the problem, we apply Banach fixed-point theorem in X toprove: Theorem A.4.
Fix ρ ∈ [0 , and let f ∈ M ρ + ( I ) . There exists a unique f ∈ C ([0 , + ∞ ); M ρ + ( I )) which solves (15) .Proof. We assume ρ > ρ = 0, the unique solution to (15) is clearly f ≡ T > S defined on X as S ( f )( t, w ) := e − ρt f ( w )+ (cid:90) t e ρ ( s − t ) (cid:90) (cid:90) I P ( w ∗ → w | w ∗ , ρ ) f ( s, w ∗ ) f ( s, w ∗ ) d w ∗ d w ∗ ds. Then we restate (15) as f = S ( f ), meaning that solutions to (15) are fixed pointsof S on X . Now we claim that: • S ( X ) ⊆ X .Let f ∈ X . The non-negativity of f (by assumption) and that of P (byconstruction) give immediately that S ( f )( t ) is a positive measure for all t ∈ [0 , T ]. Moreover, a simple calculation using property (6) shows thatthe mass of S ( f )( t ) is (cid:90) I S ( f )( t, w ) d w = ρe − ρt + ρ (cid:90) t e ρ ( s − t ) ds = ρ. Therefore we conclude that S ( f )( t ) ∈ M ρ + ( I ) for all t ∈ [0 , T ].To check the continuity of the mapping t (cid:55)→ S ( f )( t ) we define I ( ϕ )( s ) := (cid:90) (cid:90) I (cid:18)(cid:90) I ϕ ( w ) P ( w ∗ → w | w ∗ , ρ ) d w (cid:19) f ( s, w ∗ ) f ( s, w ∗ ) d w ∗ d w ∗ , for ϕ ∈ C b, ( I ) ∩ Lip ( I ), then we take t , t ∈ [0 , T ] with, say, t ≤ t and we compute: W ( S ( f )( t ) , S ( f )( t )) = sup ϕ ∈ C b, ( I ) ∩ Lip ( I ) (cid:34)(cid:0) e − ρt − e − ρt (cid:1) (cid:90) I ϕ ( w ) f ( w ) d w + (cid:90) t e ρ ( s − t ) I [ ϕ ]( s ) ds − (cid:90) t e ρ ( s − t ) I [ ϕ ]( s ) ds (cid:35) ≤ ρ (cid:12)(cid:12) e − ρt − e − ρt (cid:12)(cid:12) (cid:18) ρ (cid:90) t e ρs ds (cid:19) + ρ e − ρt (cid:90) t t e ρs ds ≤ ρ | t − t | , where we have used the Lipschitz continuity of the exponential functionand the fact that | I ( ϕ )( s ) | ≤ ρ . Finally, this says that S ( f ) ∈ X . • If T > S is a contraction on X .Let f, g ∈ X , ϕ ∈ Lip ( I ), and define(18) ψ ( w ∗ , w ∗ ) := (cid:90) I ϕ ( w ) P ( w ∗ → w | w ∗ , ρ ) d w . Notice preliminarily that, owing to Assumption A.3, | ψ ( w ∗ , w ∗ ) − ψ ( w ∗ , w ∗ ) | ≤ Lip( P ) ( | w ∗ − w ∗ | + | w ∗ − w ∗ | ) . Then: (cid:90) I ϕ ( w )( S ( g )( t, w ) − S ( f )( t, w )) d w = (cid:90) t e ρ ( s − t ) (cid:90) (cid:90) I ψ ( w ∗ , w ∗ )( g ∗ g ∗ − f ∗ f ∗ ) d w ∗ d w ∗ ds ( f ∗ , f ∗ being shorthand for f ( s, w ∗ ), f ( s, w ∗ ) and analogously g ∗ , g ∗ )= (cid:90) t e ρ ( s − t ) (cid:90) I (cid:18)(cid:90) I ψ ( w ∗ , w ∗ ) g ∗ d w ∗ (cid:19) ( g ∗ − f ∗ ) d w ∗ ds + (cid:90) t e ρ ( s − t ) (cid:90) I (cid:18)(cid:90) I ψ ( w ∗ , w ∗ ) f ∗ d w ∗ (cid:19) ( g ∗ − f ∗ ) d w ∗ ds. Notice that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) I (cid:16) ψ ( w ∗ , w ∗ ) − ψ ( w ∗ , w ∗ ) (cid:17) g ∗ d w ∗ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ Lip( P ) | w ∗ − w ∗ | and that the same holds also for the mapping w ∗ (cid:55)→ (cid:82) I ψ ( w ∗ , w ∗ ) f ∗ d w ∗ .Hence we continue the previous calculation by appealing to Remark A.2(at the right-hand side) and to the arbitrariness of ϕ to discover: W ( S ( f )( t ) , S ( g )( t )) ≤ ρ Lip ( P ) (cid:90) t e ρ ( s − t ) W ( f ( s ) , g ( s )) ds ≤ P ) (cid:0) − e − ρt (cid:1) sup t ∈ [0 , T ] W ( f ( t ) , g ( t )) , whence finallysup t ∈ [0 , T ] W ( S ( f )( t ) , S ( g )( t )) ≤ P ) (cid:0) − e − ρT (cid:1) sup t ∈ [0 , T ] W ( f ( t ) , g ( t )) . From this inequality we see that: − if Lip ( P ) > then it suffices to take T < ρ log P )2 Lip ( P ) − to obtain that S is a contraction on X ; − if Lip ( P ) ≤ then S is a contraction on X for every T > f ∈ X of S which solves (15). If Lip ( P ) ≤ thissolution is global in time, whereas if Lip ( P ) > it is only local. However, a simplecontinuation argument, based on taking f ( T ) ∈ M ρ + ( I ) as new initial conditionfor t = T and repeating the procedure above, shows that we can extend it uniquelyon the interval [ T, T ]. Proceeding in this way, we do the same on all subsequentintervals of the form [ kT, ( k + 1) T ], k = 2 , , . . . , and we obtain also in this case aglobal-in-time solution. (cid:3) ROPOSAL OF A RISK MODEL FOR VEHICULAR TRAFFIC 21
A.2.
Continuous dependence.
By comparing two solutions to (15) carrying thesame mass ρ we can establish: Theorem A.5.
Fix ρ ∈ [0 , and two initial data f , f ∈ M ρ + ( I ) . Let f , f ∈ C ([0 , + ∞ ); M ρ + ( I )) be the corresponding solution to (15) . Then: W ( f ( t ) , f ( t )) ≤ e − ρ (1 − P )) t W ( f , f ) ∀ t ≥ . Proof. Let ϕ ∈ Lip ( I ). Using (15) we compute: (cid:90) I ϕ ( w )( f ( t, w ) − f ( t, w )) d w ≤ e − ρt W ( f , f )+ (cid:90) t e ρ ( s − t ) (cid:90) (cid:90) I ψ ( w ∗ , w ∗ ) ( f ∗ f ∗ − f ∗ f ∗ ) d w ∗ d w ∗ , where ψ ( w ∗ , w ∗ ) is the function (18) defined in the proof of Theorem A.4. Bymeans of analogous calculations we discover W ( f ( t ) , f ( t )) ≤ e − ρt W ( f , f )+ 2 ρ Lip( P ) (cid:90) t e ρ ( s − t ) W ( f ( s ) , f ( s )) ds, whence the thesis follows by applying Gronwall’s inequality. (cid:3) Remark
A.6 . From Theorem A.5 we infer that if Lip( P ) < thenlim t → + ∞ W ( f ( t ) , f ( t )) = 0 . That is, all solutions to (15) approach asymptotically one another. This fact pre-ludes to the result about the equilibria of (15) proved in Theorem A.8 below.A.3.
Asymptotic analysis.
In this section we study the asymptotic trends of (7),in particular we give sufficient conditions for the existence, uniqueness, and attract-iveness of equilibria. It is worth stressing that equilibria of the kinetic model are atthe basis of the computation of fundamental and risk diagrams of traffic discussedin Section 4.Besides the methods presented here, we refer the reader to [17] and referencestherein for other ways to study the trend towards equilibrium of space homogeneouskinetic traffic models and for the identification of exact or approximated steadystates.A.3.1.
Existence and uniqueness of equilibria.
Equilibria of (7) are time-independentdistribution functions f ∞ = f ∞ ( w ) ∈ M ρ + ( I ) such that Q + ( f ∞ , f ∞ ) − ρf ∞ = 0 . If ρ = 0 then it is clear that the unique equilibrium is the trivial distributionfunction f ∞ ≡
0. Assuming instead ρ >
0, from the previous equation we see thatequilibria satisfy(19) f ∞ = 1 ρ Q + ( f ∞ , f ∞ ) , i.e., they are fixed points of the mapping f (cid:55)→ ρ Q + ( f, f ). The next theorem givesa sufficient condition for their existence and uniqueness, relying on the Banachcontraction principle in M ρ + ( I ). Theorem A.7.
Let
Lip( P ) < . For all ρ ∈ [0 , , (7) admits a unique equilibriumdistribution function f ∞ ∈ M ρ + ( I ) . Throughout the proof, we will adopt the shorthand notations f ∗ = f ( t, w ∗ ), f ∗ = f ( t, w ∗ ). Proof.
Throughout the proof we will assume ρ > ρ Q + maps M ρ + ( I ) into itself, in fact, given f ∈ M ρ + ( I ), it isclear that ρ Q + ( f, f ) is a positive measure and moreover (cid:90) I ρ Q + ( f, f )( w ) d w = 1 ρ (cid:90) (cid:90) I f ( w ∗ ) f ( w ∗ ) d w ∗ d w ∗ = ρ. Moreover we claim that, under the assumptions of the theorem, it is a contractionon M ρ + ( I ). Indeed, let f, g ∈ M ρ + ( I ) and ϕ ∈ Lip ( I ), then:1 ρ (cid:90) I ϕ ( w ) (cid:0) Q + ( g, g )( w ) − Q + ( f, f )( w ) (cid:1) d w = 1 ρ (cid:90) I (cid:18)(cid:90) I ψ ( w ∗ , w ∗ ) g ( w ∗ ) d w ∗ (cid:19) ( g ( w ∗ ) − f ( w ∗ )) d w ∗ + 1 ρ (cid:90) I (cid:18)(cid:90) I ψ ( w ∗ , w ∗ ) f ( w ∗ ) d w ∗ (cid:19) ( g ( w ∗ ) − f ( w ∗ )) d w ∗ , where ψ is the function (18). From the proof of Theorem A.4, we know that bothmappings w ∗ (cid:55)→ (cid:82) I ψ ( w ∗ , w ∗ ) g ( w ∗ ) d w ∗ and w ∗ (cid:55)→ (cid:82) I ψ ( w ∗ , w ∗ ) f ( w ∗ ) d w ∗ areLipschitz continuous with Lipschitz constant bounded by ρ Lip( P ), hence from theprevious expression we deduce1 ρ (cid:90) I ϕ ( w ) (cid:0) Q + ( g, g )( w ) − Q + ( f, f )( w ) (cid:1) d w ≤ P ) W ( f, g ) . Taking the supremum over ϕ at the left-hand side yields finally W (cid:18) ρ Q + ( f, f ) , ρ Q + ( g, g ) (cid:19) ≤ P ) W ( f, g ) , which, in view of the hypothesis Lip( P ) < , implies that ρ Q + is a contraction.Banach fixed-point theorem gives then the thesis. (cid:3) A.3.2.
Attractiveness of equilibria.
Under the same assumption of Theorem A.7,the equilibrium distribution function f ∞ is globally attractive. This means that allsolutions to (14) converge to f ∞ asymptotically in time. The precise statement ofthe result is as follows: Theorem A.8.
Let
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