Abstract

The aim of this work is to improve urban traffic viability through an appropriate choice of yielding and stop signs or red and green phases for traffic lights in junctions with two entering and one exiting roads (junctions of 2×1 type). We consider a macroscopic fluid-dynamic model able to capture the traffic evolution. We analyze different functionals measuring networks performance in terms of average velocity, average traveling time, total flux, density, stop and go waves, average traveling time, weighted with the number of cars moving on roads, and kinetic energy. Right of way parameters which optimize the latter two functionals are obtained. Simulations of simple junctions of 2×1type have been used to test the correctness of the analytical results. Then, global performance of optimization procedures has been investigated on Re di Roma Square, in Italy. In particular, we discuss cases in which the functionals are optimized locally at each junction for different values of right of way parameters. We show that for the chosen initial data the only algorithm for the maximization of velocity assures globally the best performance for the network, also in terms of average traveling times and kinetic energy.

1. Introduction

The problem of traffic modeling is a very huge task, due to the complexity of the system to analyze. Many methods have been developed based on different approaches ranging from microscopic one, taking into account each single car, to kinetic and macroscopic ones, dealing with averaged quantities. Of course the understanding and control of traffic phenomena by means of simulation and optimization studies can be useful to eventually make decisions which may alleviate congestion, maximize flow traffic, reduce accidents, and other desirable ends.

Here rather than analyzing the movement of individual cars, we describe road networks behaviour using fluid-dynamic models; hence we treat traffic situations resulting from complex interaction of many vehicles. In particular, following [1] we determine the evolution of car traffic on each road with the first-order model of Lighthill, Whitham, and Richards (LWR model), which is on one side simple enough to permit a complete understanding of traffic flows and on the other side rich enough to detect important phenomena as queue formation. A key role in the networks modeling is played by the junctions. In order to capture the dynamics at a node, Cauchy problems with initial data constant on each road, called Riemann Problems at the node, have to be solved. The latter have a unique solution if we introduce some rules:(A)the cars flow from an incoming road to outgoing ones according to the final destination;(B)the number of cars passing the junction is maximized respecting rule (A).

For a junction of 1×2 type, that is, with one incoming road, labeled with 1, and two outgoing roads, labeled with 2,3, the rule (A) is expressed by a coefficient 𝛼 which describes the percentage of cars going from road 1 to road 2. Obviously 1−𝛼 is the percentage of cars moving towards the outgoing road 3.

If the junction has two incoming roads, labeled with 1 and 2, and one outgoing road, labeled with 3, (junction of 2×1 type), in order to solve the dynamics, we have to introduce the right of way parameter 𝑝. Supposing that 𝐶 cars can enter into the junction, then 𝑝𝐶 cars come from road 1 and (1−𝑝)𝐶 from road 2.

Assigning the initial density of all incoming and outgoing roads in a node, we compute the final equilibrium as function of the traffic distribution coefficient or/and of the right of way parameter depending on junction type. Such equilibrium, belonging to the admissible region for the final fluxes, is chosen according to a fixed strategy, and represents the solution of the Riemann Problem.

The aim of this work is the optimization of urban networks performance with junctions of 2×1 type through a suitable choice of the right of way parameter 𝑝. Observe that the choice of 𝑝 corresponds to the use of yielding and stop signs or to the regulation of red and green phases for traffic lights.

Optimization problems for fluid-dynamic models have been already considered for car traffic: [2] is devoted to traffic light regulation, while [3] and [4] are more related to our analysis but focus on the case of smooth solutions (not developing shocks) and boundary control. A specific traffic regulation problem is addressed in [5]. Given a crossing with some expected traffic, is it preferable to construct a traffic circle or a light? The two solutions are studied in terms of flow control and the performances are compared. In [6–8], the traffic behaviour has been analyzed using four cost functionals, 𝐽1, 𝐽2, 𝐽3, and 𝐽5, measuring, respectively, cars average velocity, cars average traveling time, car velocity, weighted with cars quantity, traveling on the roads, that is, fluxes and stop and go waves (see also [9]). The optimization was done over right of way parameters and traffic distribution coefficients with the aim of maximizing 𝐽1 and 𝐽3 and minimizing 𝐽2. In particular, two special cases of junctions have been considered for the optimization: the 2×1 and 1×2 cases.

In this work, we analyze the urban traffic behaviour introducing new functionals. From the solution of the Riemann Problem, we determine the average speeds at which drivers travel and we define additional functionals: 𝐽4 which measures the density, 𝐽6 the average traveling time, seen as the sum of the average traveling times on each road, weighted with the number of cars moving on it, and 𝐽7 which gives information about the kinetic energy. Of course the aim is to optimize the choice of the parameters in order to minimize 𝐽6 and to maximize 𝐽7. Since in case of 1×2 junctions, the functionals 𝐽6 and 𝐽7 are optimized for the same values of the distribution coefficients which maximize and minimize 𝐽1 and 𝐽2, we focused our attention on the optimization of junctions of type 2×1, for which 𝐽1 and 𝐽6 have, in some cases, different optimal values. It is interesting to notice that in many cases (with the extreme case of functionals not depending on 𝑝) there is a set of optimal values of the right of way parameters.

The correctness of analytical optimization algorithms is tested through simulations. First we consider simple junctions of 2×1 type and compare different choices of parameters, statistical parameters, and optimal one. In particular, the initial density values on the roads are chosen in such way that optimization procedures, obtained evaluating the described cost functionals, give different optimal parameters.

Then, we studied the effects of the decentralized approach on the global performance of more complicated networks. According to this approach local optimal parameters at every junction of a complex network have been used. The discussed example regards Re di Roma Square, in Rome. We notice that the global effect of optimization is achieved using the optimal values for the velocity functional. Indeed, it is also shown that, considering the optimal values for the other cost functionals, the network becomes almost completely full, with consequent birth of congestion phenomena and reduction of cars fluxes. Finally we compare optimal algorithms with random ones. In the latter case, the right of way parameters are chosen randomly at every instant of time and for every junction independently. We see that the optimal algorithm based on the maximization of the velocity ensures better performance than random simulation.

The paper is organized as follows. In Section 2, we recall the basic definitions and the construction of solutions to Riemann Problems at junctions. The subsequent section is devoted to the introduction of the cost functionals and to the optimization of 𝐽6 and 𝐽7, compared with 𝐽1 and 𝐽2. Then, Section 4 reports simulation results first for simple junctions and then for Re di Roma Square.

2. Road Network Model

We consider a traffic network that is a finite collection of roads connected together by junctions. Formally, we introduce the following definition.

Definition 2.1. A traffic network is given by a 4-tuple (𝑁,ℐ, ℱ, 𝒥) where
Cardinality
𝑁 is the cardinality of the network, that is, the number of roads in the network;Lines
ℐ is the collection of roads, modeled by intervals 𝐼𝑘=[𝑎𝑘,𝑏𝑘]⊆ℝ,𝑘=1,…,𝑁;Fluxes
ℱ is the collection of flux functions 𝑓𝑘∶[0,𝜌𝑘max]↦ℝ,𝑘=1,…,𝑁, with 𝜌𝑘max the maximal density on road 𝐼𝑘;Nodes
𝒥 is a collection of subsets of {±1,…,±𝑁} representing junctions. If 𝑗∈𝐽∈𝒥, then the road 𝐼|𝑗| is crossing at 𝐽 as incoming road (i.e., at point 𝑏𝑖) if 𝑗>0 and as outgoing road (i.e., at point 𝑎𝑖) if 𝑗<0. For each junction 𝐽∈𝒥, we indicate by Inc(𝐽) the set of incoming roads, that are 𝐼𝑖's such that 𝑖∈𝐽, while by Out(𝐽) the set of outgoing roads, that are 𝐼𝑖’s such that −𝑖∈𝐽. We assume that each road is incoming for (at most) one node and outgoing for (at most) one node.

In what follows, we suppose that 𝑓𝑘=𝑓 for 𝑘=1,…,𝑁, but it is possible to generalize all definitions and results to the case of different fluxes 𝑓𝑘 for each road 𝐼𝑘. In fact, all statements are in terms of fluxes values at junctions, thus it is sufficient that the ranges of fluxes intersect. On each road, we consider the LWR model, described by the equation (see [10, 11]),𝜌𝑡+𝑓(𝜌)𝑥=0,(2.1) where (𝑡,𝑥)∈ℝ+×ℝ,𝜌=𝜌(𝑡,𝑥)∈[0,𝜌max] is the density of cars, 𝜌max is the maximal density, 𝑓(𝜌)=𝜌𝑣 is the flux, and 𝑣=𝑣(𝜌) the average velocity. For roads such that ⋃𝑖∉𝐽∈𝒥Inc(𝐽) and 𝑏𝑖<+∞ or such that ⋃𝑖∉𝐽∈𝒥Out(𝐽) and 𝑎𝑖>−∞, a boundary condition is needed. In throughout the paper, we use the following flux function:[],𝑓(𝜌)=𝜌(1−𝜌),𝜌∈0,1(2.2) with a unique maximum 𝜎=1/2.

For a single conservation law (2.1) on a real line ℝ, a Riemann Problem (RP), the basic ingredient to construct approximate solutions to Cauchy Problems with wave front tracking algorithm, is a Cauchy problem for an initial data piecewise constant with only one discontinuity. In a similar way, an RP at a junction is a Cauchy Problem for an initial data constant on each incoming and outgoing road.

Fix a junction 𝐽 with 𝑛 incoming roads and 𝑚 outgoing roads (junction of 𝑛×𝑚 type), where 𝐼𝑖,𝑖=1,…,𝑛, are the incoming roads and 𝐼𝑗,𝑗=𝑛+1,…,𝑛+𝑚, are the outgoing ones. Let 𝜌=(𝜌1,…,𝜌𝑛+𝑚),𝜌𝑘∈[0,+∞]×𝐼𝑘 be the density vector for 𝐽.

Definition 2.2. A Riemann Solver (RS) for the junction 𝐽 is a map RS∶[0,1]𝑛×[0,1]𝑚↦[0,1]𝑛×[0,1]𝑚 that associates to a Riemann datum 𝜌0=(𝜌1,0,…,𝜌𝑛+𝑚,0) at 𝐽 a vector ̂𝜌=(̂𝜌1,…,̂𝜌𝑛+𝑚) so that the solution on 𝐼𝑖,𝑖=1,…,𝑛, is given by the wave (𝜌𝑖,0,̂𝜌𝑖) and on 𝐼𝑗,𝑗=𝑛+1,…,𝑛+𝑚 is given by the wave (̂𝜌𝑗,𝜌𝑗,0). We require the consistency condition: 𝜌(𝐂𝐂)RSRS0𝜌=RS0.(2.3)

If 𝑚≥𝑛, it is possible to introduce an RS, based on the following rules ([1, 12]).(A)At each junction 𝐽, we define a matrix 𝐴=(𝛼𝑗,𝑖), that describes the traffic distribution from incoming to outgoing roads, where, for every 𝑖∈{1,…,𝑛} and 𝑗∈{𝑛+1,…,𝑛+𝑚},  0≤𝛼𝑗,𝑖≤1 and ∑𝑛+𝑚𝑗=𝑛+1𝛼𝑗,𝑖=1. The 𝑖th column of A indicates the percentages of traffic that, from the incoming road 𝐼𝑖, are distributed to outgoing roads.(B)respecting (A), drivers behave so as to maximize the flux through 𝐽.

In the case 𝑚<𝑛, with 𝑛≥2 and 𝑚=1, in order to define an RS, we fix also the right of way parameters vector 𝑝=(𝑝1,…,𝑝𝑛) with ∑𝑛𝑖=1𝑝𝑖=1 and consider the following additional rule.(P)Assume that not all cars can enter the outgoing roads, and let 𝐶 be the amount that can do it. Then 𝑝𝑖𝐶,𝑖=1,…,𝑛 cars come from the road I𝑖 into the node.

For simplicity, we indicate by 𝜌𝑖(𝑡,𝑥), 𝑖=1,…,𝑛, the densities of the cars on the incoming roads and by 𝜌𝑗(𝑡,𝑥), 𝑗=𝑛+1,…,𝑛+𝑚, those on outgoing roads. Let us introduce the notation 𝛾𝑘𝜌=𝑓𝑘,̂𝛾𝑘=𝑓̂𝜌𝑘,𝑘=1,…,𝑛+𝑚.(2.4)

Proposition 2.3. Let (𝜌1,0,…,𝜌𝑛+𝑚,0)∈[0,1] be the initial densities of an RP at 𝐽. The maximum fluxes that can be obtained on the incoming roads and the outgoing ones, respectively, are given by (for a proof see [12]): 𝛾𝑖max=⎧⎪⎨⎪⎩𝑓𝜌𝑖,0,𝑖𝑓𝜌𝑖,0∈10,2𝑓12=14𝑖𝑓𝜌𝑖,0∈12,𝛾,1𝑖=1,…,𝑛,𝑗max=⎧⎪⎨⎪⎩𝑓12=14,𝑖𝑓𝜌𝑗,0∈10,2𝑓𝜌𝑗,0𝑖𝑓𝜌𝑗,0∈12,,1𝑗=𝑛+1,…,𝑛+𝑚.(2.5)

Now we focus on junctions of 2×1 type and indicate with 1 and 2 the entering roads and with 3 the exiting one. In this case, we need only one right of way parameter 𝑝. The solution to the RP with initial data (𝜌1,0,𝜌2,0,𝜌3,0) is constructed in the following way. Since we want to maximize the through traffic (rule (B)), we set̂𝛾3𝛾=min1max+𝛾2max,𝛾3max.(2.6) If ̂𝛾3=𝛾1max+𝛾2max, then the solution of the RP is ̂𝛾=(𝛾1max,𝛾2max,𝛾1max+𝛾2max). Consider now the case ̂𝛾3=𝛾3max and the following conditions:(A1)𝑝𝛾3max<𝛾1max; (A2)(1−𝑝)𝛾3max<𝛾2max.

The solutions of the RP are the following:(i)(𝑝𝛾3max,(1−𝑝)𝛾3max,𝛾3max) if 𝐴1 and 𝐴2 are both satisfied;(ii)(𝛾3max−𝛾2max,𝛾2max,𝛾3max) if 𝐴1 is satisfied and 𝐴2 is not satisfied;(iii)(𝛾1max,𝛾3max−𝛾1max,𝛾3max) if 𝐴2 is satisfied and 𝐴1 is not satisfied.

The case of both 𝐴1 and 𝐴2 false is not possible, since it would be 𝛾3max>𝛾1max+𝛾2max.

From the flux function, we can express ̂𝜌𝑘 in terms of ̂𝛾𝑘. In fact, solving the equation ̂𝜌𝑘(1−̂𝜌𝑘)=̂𝛾𝑘, we get̂𝜌𝑘=1+𝑠𝑘√1−4̂𝛾𝑘2,𝑘=1,2,3,(2.7) with𝑠𝑖=⎧⎪⎪⎪⎨⎪⎪⎪⎩−1𝑖𝑓𝜌𝑖,0<𝜎,𝛾1max+𝛾2max≤𝛾3max,or𝜌𝑖,0<𝜎,𝛾3max<𝛾1max+𝛾2max,𝑝𝑖̂𝛾3≥𝛾𝑖max,+1if𝜌𝑖,0≥𝜎,or𝜌𝑖,0<𝜎,𝛾3max<𝛾1max+𝛾2max,𝑝𝑖̂𝛾3<𝛾𝑖max,𝑠𝑖=1,2,3=⎧⎪⎨⎪⎩−1if𝜌3,0≤𝜎,or𝜌3,0>𝜎,𝛾1max+𝛾2max<𝛾3max,+1if𝜌3,0>𝜎,𝛾1max+𝛾2max≥𝛾3max,(2.8) where 𝑝𝑖=𝑝if𝑖=1,1−𝑝if𝑖=2.(2.9) The velocity, in terms of ̂𝛾𝑘, is given by 𝑣(̂𝜌𝑘)=(1−𝑠𝑘√1−4̂𝛾𝑘)/2,  𝑘=1,2,3.

3. Cost Functionals

In this section, we introduce the functionals used to evaluate the network performance and report optimization results for the functionals 𝐽6 and 𝐽7, compared with 𝐽1 and 𝐽2. We focus again on junctions of 2×1 type and we define the following functionals:𝐽1measuring car average velocity: 𝐽1(𝑡)=3𝑘=1𝐼𝑘𝑣𝜌𝑘(𝑡,𝑥)𝑑𝑥,(3.1)𝐽2measuring average traveling time: 𝐽2(𝑡)=3𝑘=1𝐼𝑘1𝑣𝜌𝑘(𝑡,𝑥)𝑑𝑥,(3.2)𝐽3measuring total flux of cars: 𝐽3(𝑡)=3𝑘=1𝐼𝑘𝑓𝜌𝑘(𝑡,𝑥)𝑑𝑥,(3.3)𝐽4measuring car density: 𝐽4(𝑡)=3𝑘=1𝑡0𝐼𝑘𝜌𝑘(𝜏,𝑥)𝑑𝜏𝑑𝑥,(3.4)𝐽5the Stop and Go Waves functional, measuring the velocity variation: 𝐽5(𝑡)=SGW=3𝑘=1𝑡0𝐼𝑘||||𝐷𝑣(𝜌)𝑑𝜏𝑑𝑥,(3.5) where |𝐷𝑣| is the total variation of the distributional derivative 𝐷𝜌, which is a finite Radon measure,𝐽6measuring the kinetic energy: 𝐽6(𝑡)=3𝑘=1𝐼𝑘𝑓𝜌𝑘𝑣𝜌(𝑡,𝑥)𝑘(𝑡,𝑥)𝑑𝑥,(3.6)𝐽7measuring the average traveling time weighted with the number of cars moving on each road 𝐼𝑘: 𝐽7(𝑡)=3𝑘=1𝐼𝑘𝜌𝑘(𝑡,𝑥)𝑣𝜌𝑘(𝑡,𝑥)𝑑𝑥.(3.7) For a fixed time horizon [0,𝑇], our aim is to maximize ∫𝑇0𝐽1∫(𝑡)𝑑𝑡,𝑇0𝐽3∫(𝑡)𝑑𝑡,𝑇0𝐽6(𝑡)𝑑𝑡 and to minimize ∫𝑇0𝐽2∫(𝑡)𝑑𝑡,𝑇0𝐽7(𝑡)𝑑𝑡, choosing the right of way parameter 𝑝𝑘(𝑡). Since the solutions of such optimization control problems are too difficult, we reduce to the following problem.

(𝑃𝑟)Consider a junction 𝐽 of 2×1 type, the functionals 𝐽𝑘,  𝑘=1,2,3,6,7, and the right of way parameter 𝑝𝑘 as controls. We want to minimize 𝐽2(𝑇),𝐽7(𝑇) and to maximize 𝐽1(𝑇),𝐽3(𝑇),𝐽6(𝑇) for 𝑇 sufficiently big.

Given the initial data, solving the RP, we determine the average velocity, the average traveling time, and the flux over the network as function of the right of way parameters 𝑝, then all the functionals depend on 𝑝. As was proved in [6–8], the functional 𝐽3(𝑇) does not depend on the right of way parameter.

3.1. Optimization of J6 and J7

Let us consider the optimization of the functionals, measuring kinetic energy and weighted average traveling time. For 𝑇 sufficiently big the functionals assume the form: 𝐽6(𝑇)=3𝑘=1𝑓̂𝜌𝑘𝑣̂𝜌𝑘=3𝑘=1̂𝛾𝑘1−𝑠𝑘√1−4̂𝛾𝑘2,𝐽7(𝑇)=3𝑘=1̂𝜌𝑘𝑣̂𝜌𝑘=3𝑖=11+𝑠𝑘√1−4̂𝛾𝑘1−𝑠𝑘√1−4̂𝛾𝑘,(3.8) where 𝑠𝑘,𝑘=1,2,3 are defined in (2.8), ̂𝜌=(̂𝜌1,̂𝜌2,̂𝜌3)=RS(𝜌1,0,𝜌2,0,𝜌3,0), and ̂𝛾=𝑓(̂𝜌).

If ̂𝛾3=𝛾1max+𝛾2max, the solution of the RP does not depend on the parameter 𝑝 and the same happens for the functionals 𝐽6 and 𝐽7 which do not depend on the parameter 𝑝. Hence we analyze the functionals in the case 𝛾3max<𝛾1max+𝛾2max, that is, ̂𝛾3=𝛾3max. Let us define 𝛽−=𝛾3max−𝛾1max𝛾1max,𝑝+=11+𝛽−=𝛾1max𝛾3max,𝛽+=𝛾2max𝛾3max−𝛾2max,𝑝−=11+𝛽+=𝛾3max−𝛾2max𝛾3max.(3.9) It is easy to check that 𝛽−≤𝛽+. Then, for 𝑝≥𝑝+,𝐴1 is false and 𝐴2 is true, for 𝑝≤𝑝−,𝐴1 is true and 𝐴2 is false and, finally, for 𝑝−<𝑝<𝑝+,𝐴1 and 𝐴2 are both true. From (2.7), neglecting the parts of the costs that do not depend on 𝑝, maximizing 𝐽6 and minimizing 𝐽7 is equivalent to maximize and minimize, respectively, 𝐽6=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝛾3max−𝛾2max1−𝑠1𝛾1−43max−𝛾2max+𝛾2max1−𝑠2√1−4𝛾2max,0≤𝑝≤𝑝−,𝑝𝛾3max1−𝑠1√1−4𝑝𝛾3max+(1−𝑝)𝛾3max1−𝑠2√1−4(1−𝑝)𝛾3max,𝑝−<𝑝<𝑝+,𝛾1max1−𝑠1√1−4𝛾1max+𝛾3max−𝛾1max1−𝑠2𝛾1−43max−𝛾1max,𝑝+𝐽≤𝑝≤1,7=⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩1+𝑠1𝛾1−43max−𝛾2max1−𝑠1𝛾1−43max−𝛾2max+1+𝑠2√1−4𝛾2max1−𝑠2√1−4𝛾2max,0≤𝑝≤𝑝−,1+𝑠1√1−4𝑝𝛾3max1−𝑠1√1−4𝑝𝛾3max+1+𝑠2√1−4(1−𝑝)𝛾3max1−𝑠2√1−4(1−𝑝)𝛾3max,𝑝−<𝑝<𝑝+,1+𝑠1√1−4𝛾1max1−𝑠1√1−4𝛾1max+1+𝑠2𝛾1−43max−𝛾1max1−𝑠2𝛾1−43max−𝛾1max,𝑝+≤𝑝≤1.(3.10) Observe that when the condition 𝐴1 is false and 𝐴2 is true, or vice versa, the cost functionals 𝐽6 and 𝐽7 are constant with respect to 𝑝. To complete the analysis of the costs it is enough to take the derivatives with respect to 𝑝 in the region where both 𝐴1 and 𝐴2 are true. The expressions are a bit long, so we do not report them, but it is straightforward to check that 𝐽6 and 𝐽7 are decreasing for 𝑝<1/2 and increasing for 𝑝>1/2, hence they have a minimum in 𝑝=1/2.

We analyze the functionals 𝐽6 and 𝐽7 in the intervals: [0,𝑝−],]𝑝−,𝑝+[ and [𝑝+,1]. We search for absolute minimum in the case of 𝐽6 and absolute maximum in the analysis of 𝐽7, taking into account the possible values that 𝑠𝑘,𝑘=1,2,3, can assume in the above intervals:(i)𝑠1=𝑠2=1,𝑠3=±1 in [0,𝑝−],𝑠1=𝑠2=1,𝑠3=±1 in ]𝑝−,𝑝+[,𝑠1=𝑠2=1,𝑠3=±1 in [𝑝+,1];(ii)𝑠1=𝑠2=1,𝑠3=±1 in [0,𝑝−],𝑠1=𝑠2=1,𝑠3=±1 in ]𝑝−,𝑝+[,𝑠1=−1,𝑠2=1,𝑠3=±1 in [𝑝+,1];(iii)𝑠1=1,𝑠2=−1,𝑠3=±1 in [0,𝑝−],𝑠1=𝑠2=1,𝑠3=±1 in ]𝑝−,𝑝+[,𝑠1=𝑠2=1,𝑠3=±1 in [𝑝+,1];(iv)𝑠1=1,𝑠2=−1,𝑠3=±1 in [0,𝑝−],𝑠1=𝑠2=1,𝑠3=±1 in ]𝑝−,𝑝+[,𝑠1=−1,𝑠2=1,𝑠3=±1 in [𝑝+,1].

Remark 3.1. Since 𝑠3 assumes the same values, 1 or −1, in all the intervals, (hence in the point of discontinuities) and the terms in which 𝑠3 occurs do not depend on 𝑝, the optimal values are independent from 𝑠3.

The functionals 𝐽2(𝑇) and 𝐽7(𝑇) are maximized for the same values of 𝑝. In fact we get the following theorem.

Theorem 3.2. Consider a junction 𝐽 of 2×1 type. For the flux function (2.2), and 𝑇 sufficiently big, the cost functionals 𝐽2(𝑇) and 𝐽7(𝑇) are optimized for the following values of 𝑝. (1)Case 𝑠1=𝑠2=+1, we have that(a)𝑝=1/2 if 𝛽−≤1≤𝛽+ or 𝛾2max=𝛾3max;(b)𝑝∈[0,𝑝−] if 𝛽−≤𝛽+≤1;(c)𝑝∈[𝑝+,1] if 1≤𝛽−≤𝛽+;(2)Case 𝑠1=−1=−𝑠2, we have that(a)𝑝=1/2 or 𝑝∈[𝑝+,1] if 𝛽−≤1≤𝛽+ or 𝛾2max=𝛾3max;(b)𝑝∈[0,𝑝−] or 𝑝∈[𝑝+,1] if 𝛽−≤𝛽+≤1;(c)𝑝∈[𝑝+,1] if 1≤𝛽−≤𝛽+;(3)Case 𝑠1=+1=−𝑠2, we have that(a)𝑝=1/2 or 𝑝∈[0,𝑝−] if 𝛽−≤1≤𝛽+;(b)𝑝∈[0,𝑝−] if 𝛽−≤𝛽+≤1;(c)𝑝∈[0,𝑝−] or 𝑝∈[𝑝+,1] if 1≤𝛽−≤𝛽+;(d)𝑝=1/2 or 𝑝∈[𝑝+,1] if 𝛾2max=𝛾3max;(4)Case 𝑠1=𝑠2=−1, we have that(a)𝑝=1/2 or 𝑝∈[𝑝+,1] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+>1 or 𝛾2max=𝛾3max;(b)𝑝=1/2 or 𝑝∈[0,𝑝−] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+<1;(c)𝑝=1/2 or 𝑝∈[0,𝑝−]∪[𝑝+,1] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+=1;(d)𝑝∈[0,𝑝−] if 𝛽−≤𝛽+≤1,(e)𝑝∈[𝑝+,1], if 1≤𝛽−≤𝛽+;where 𝛽−=(𝛾3max−𝛾1max)/𝛾1max,𝛽+=𝛾2max/(𝛾3max−𝛾2max),𝑝+=𝛾1max/𝛾3max,𝑝−=(𝛾3max−𝛾2max)/𝛾3max.

In the particular case 𝛾1max=𝛾2max=𝛾3max, the functionals 𝐽2 and 𝐽7 are optimized for 𝑝=1/2.

The maximization of the functionals 𝐽1(𝑇) and 𝐽6(𝑇) is reached, in some cases, for different values of the right of way parameter, as reported in the following theorem, in which the optimization analysis of the new functional 𝐽6(𝑇) is compared with the results obtained in [6] for 𝐽1(𝑇).

Theorem 3.3. Consider a junction 𝐽 of 2×1 type. For the flux function (2.2), and 𝑇 sufficiently big, the cost functionals 𝐽1(𝑇) and 𝐽6(𝑇) are optimized for the following values of 𝑝. (1)Case 𝑠1=𝑠2=+1, we have that(a)𝑝∈[0,𝑝−] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+>1, or 1≤𝛽−≤𝛽+, or 𝛾2max=𝛾3max;(b)𝑝∈[0,𝑝−]∪[𝑝+,1] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+=1;(c)𝑝∈[𝑝+,1] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+<1 or 𝛽−≤𝛽+≤1;(2)Case 𝑠1=−1=−𝑠2, we have thatfor 𝐽1(𝑇),𝑝∈[𝑝+,1];for 𝐽6(𝑇),(a)𝑝∈[𝑝+,1] if 𝛽−≤1≤𝛽+, or 𝛽−≤𝛽+≤1;(b)𝑝∈[0,𝑝−] or 𝑝∈[𝑝+,1] if 1≤𝛽−≤𝛽+, or 𝛾2max=𝛾3max;(3)Case 𝑠1=+1=−𝑠2, we have thatfor 𝐽1(𝑇),𝑝∈[0,𝑝−];for 𝐽6(𝑇),(a)𝑝∈[0,𝑝−] if 𝛽−≤1≤𝛽+, or 1≤𝛽−≤𝛽+, or 𝛾2max=𝛾3max;(b)𝑝∈[0,𝑝−] or 𝑝∈[𝑝+,1] if 𝛽−≤𝛽+≤1;(4)Case 𝑠1=𝑠2=−1, we have thatfor 𝐽1(𝑇),(a)𝑝∈[0,𝑝−] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+<1, or 𝛽−≤𝛽+≤1;(b)𝑝∈[0,𝑝−]∪[𝑝+,1] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+=1;(c)𝑝∈[𝑝+,1] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+>1, or 1≤𝛽−≤𝛽+, or 𝛾2max=𝛾3max;for 𝐽6(𝑇),(a)𝑝∈[0,𝑝−] or 𝑝∈[𝑝+,1] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+>1, or 𝛽−𝛽+<1, or 𝛾2max=𝛾3max;(b)𝑝∈[0,𝑝−]∪[𝑝+,1] if 𝛽−≤1≤𝛽+, with 𝛽−𝛽+=1;(c)𝑝∈[𝑝+,1] if 𝛽−≤𝛽+≤1;(d)𝑝∈[0,𝑝−] if 1≤𝛽−≤𝛽+;where 𝛽−=(𝛾3max−𝛾1max)/𝛾1max,𝛽+=𝛾2max/(𝛾3max−𝛾2max),𝑝+=𝛾1max/𝛾3max,𝑝−=(𝛾3max−𝛾2max)/𝛾3max.

The functionals 𝐽1 and 𝐽6 are maximized for 𝑝=0 or 𝑝=1if 𝛾1max=𝛾2max=𝛾3max.

Remark 3.4. In the cases in which two possible optimal candidates exist, we evaluate the corresponding values of the cost functionals. The optimal 𝑝 is given by the value that minimizes or maximizes, respectively, the functionals 𝐽1(𝑇),𝐽6(𝑇) and 𝐽2(𝑇),𝐽7(𝑇). If we have a set of optimal values [0,𝑝−], we can choose 𝑝=𝑝− or 𝑝=𝑝−−𝜀, while in the case in which the optimal right of way parameter belongs to [𝑝+,1] we can take as optimal value 𝑝=𝑝+ or 𝑝=𝑝++𝜀, with 𝜀 small and positive.

4. Simulations

In this section, we present some simulation results in order to test the optimization algorithms both for single junctions and complex networks. The aim is to verify the correctness of the analytical results and then to analyze the effects of different control procedures, applied locally at each junction, on the global performances of networks. The approximation of the conservation laws, that describe the density evolution for each road of the network (see [13]), is made by the numerical scheme of Godunov ([14]), with space step Δ𝑥=0.01. The time step is determined by the CFL condition ([15]), equal to 0.5.

4.1. Single Junctions

We consider single junctions of 2×1 type, namely, junctions consisting of two incoming roads, 1 and 2, and one outgoing road, 3, in order to verify the goodness of optimization procedures. Then we compare cost functionals behaviour using right of way parameters, that optimize the functionals (optimal case) and fixed right of way parameters ( fixed case), according to which 𝑝 is chosen by the user. The evolution of the traffic is simulated in a time interval [0,𝑇], where 𝑇=20min for the flux function (2.2). As for the initial conditions on the roads, we assume that, at the starting instant of simulation (𝑡=0), all roads are empty. Moreover, for roads 1, 2, and 3, also boundary data, 𝜌𝑖,𝑏,𝑖∈{1,2,3}, have to be considered. Precisely, we choose 𝜌𝑖,𝑏=0.3,  𝜌2,𝑏=0.4,  and  𝜌3,𝑏=0.1.

The motivation for the choice of such boundary data is that it allows to capture the case in which different optimization procedures, one for 𝐽1, indicated as "opt𝐽1", one for 𝐽6, "opt𝐽6", and one for both 𝐽2 and 𝐽7, "opt𝐽2𝐽7", give origin to different optimal values.

As you can see from Figures 1 and 2, the optimal algorithms for 𝐽1, 𝐽2, 𝐽6, and 𝐽7 improve traffic conditions with respect to the fixed cases. Notice that, when we are not in the steady state, it could occur that some fixed simulations can behave better than the optimal ones. This is due to the fact that the analytical results are always obtained for big times. In Figure 3, we present the differences between the optimization algorithms for cost functionals 𝐽1 and 𝐽6. As expected, 𝐽1 and 𝐽6 are higher, respectively, applying opt𝐽1 and opt𝐽6. Moreover, simulating the junction with the parameter 𝑝 obtained using the control procedures opt𝐽1, opt𝐽6, and opt𝐽2𝐽7, the behaviours of 𝐽1 and 𝐽6 are different, confirming the theoretical results that, depending on the initial data, the functionals can be optimized for different values of the right of way parameter.

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Figure 1 
Comparison among 𝐽 1 computed by opt 𝐽 1 (a), 𝐽 6 computed by opt 𝐽 6 (b), and fixed cases for 𝑝 = 0 . 4 and 𝑝 = 0 . 8 .
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Figure 2 
Comparison among 𝐽 2 (a) and 𝐽 7 (b) computed by opt 𝐽 2 𝐽 7 , and fixed cases for 𝑝 = 0 . 4 and 𝑝 = 0 . 8 .
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Figure 3 
Comparison among 𝐽 1 (a) and 𝐽 6 evaluated by opt 𝐽 1 , opt 𝐽 2 𝐽 7 and opt 𝐽 6 (b).

As for a single junction of 2×1 type, various optimization algorithms can be applied to set the priority parameters, we tried to understand which is the best policy to adopt for the improvements of traffic conditions. In order to discriminate among the different optimization approaches, we analyze the behaviour of the Stop and Go Waves functional (SGW), which measures the velocity variations on roads. We simulate the traffic evolution for a junction in four different situations, denoted by 𝐴, 𝐵, 𝐶, and 𝐷, in a time interval [0,𝑇], where 𝑇 is 30 min for cases 𝐴 and 𝐵 and 50 min for cases 𝐶 and 𝐷. We assume that at the beginning of the simulation all the roads are empty. Boundary data are reported in Table 1, and chosen in such way to test various cases of Theorems 3.2 and 3.3.

Table 1 
Boundary data for the four simulation cases.

For such case studies, optimal values for the right of way parameter 𝑝 are as shown in Table 2.

Table 2 
Optimal right of way parameters for simulation cases 𝐴, 𝐵, 𝐶, and 𝐷.

In Figures 4 and 5, the behaviour of SGW, applying opt𝐽1, opt𝐽2𝐽7, and opt𝐽6, in the different simulation cases, is depicted. In the steady state, SGW is lower when the algorithm opt𝐽1 is used, and this implies a lower probability of car accidents. We can conclude that it is suitable to choose the parameter 𝑝 which maximizes the average velocities. Notice that in cases 𝐵 and 𝐶, the optimal algorithms opt𝐽1 and opt𝐽2𝐽7 give the same set of 𝑝, hence curves overlap.

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Figure 4 
Comparison among S G W , evaluated by opt 𝐽 1 , opt 𝐽 2 𝐽 7 , and opt 𝐽 6 , in case A (a) and case B (b).
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Figure 5 
Comparison among S G W , evaluated by opt 𝐽 1 , opt 𝐽 2 𝐽 7 , and opt 𝐽 6 , in case 𝐶 (a) and case 𝐷 (b).
4.2. Simulation of Traffic for Re Di Roma Square

This subsection is devoted to the description of simulative results for a real urban network, Re di Roma Square, a big traffic circle inside the urban network of Rome, in Italy. The choice of this case study is justified by the presence of congestion phenomena, which could be avoided through an opportune choice of network parameters.

The topology of the square, represented in Figure 6, is described by 12 roads, which form the circle: 1𝑅, 2𝑅, 3𝑅, 4𝑅, 5𝑅, 6𝑅, 7𝑅, 8𝑅, 9𝑅, 10𝑅, 11𝑅, and 12𝑅; 12 roads, connecting the inner roads with outside: aosta_ent, aosta_exi, vercelli, pinerolo, appia_sud_ent, appia_sud_exi, albalonga_ent, albalonga_exi, cerveteri_ent, cerveteri_exi, appia_nord_ent, appia_nord_exi.


Figure 6 
Topology of Re di Roma Square.

As it is shown, Re di Roma square is formed by junctions of 2×1 type (1, 3, 5, 7, 9, 11), in white, and junctions of 1×2 type (2, 4, 6, 8, 10, 12), in black. The traffic distribution coefficients at 1×2 junctions are completely determined by road capacities (and the characteristics of the nearby portion of the Rome urban network); hence only right of way parameters for 2×1 junctions can be chosen as control parameters. Table 3 reports the distribution coefficients used for simulations.

Table 3 
Traffic distribution parameters for junctions of 1×2 type.

The evolution of the traffic flows is simulated in a time interval [0,𝑇], where 𝑇=30min. We assume that, at the starting instant of simulation, all roads are empty. We use boundary conditions for roads with not infinite endpoints. In particular, roads cerveteri_exi, albalonga_exi, appia_sud_exi, vercelli, aosta_exi, and appia_nord_exi have a boundary data equal to 0.4, while the other ones equal to 0.35.

We analyze two simulation cases: right of way parameters, that optimize the cost functionals (optimal case); dynamic random parameters (dynamic random case), which means that right of way parameters change randomly at every step of the simulation process.

The aim is to investigate the effects of opt𝐽1, opt𝐽6, and opt𝐽2𝐽7 on the global performances of the network in order to understand which is the best optimization algorithm (the algorithm realizing the better viability conditions on the whole Re di Roma square). In the following pictures, we report the time behaviour evolutions of the cost functionals 𝐽1, 𝐽2, 𝐽6, and 𝐽7, fixing at every junction the values of the right of way parameters obtained by the optimization procedures.

The optimizations of local type, like the ones that we are considering here, could not necessarily imply global optimization for big networks, as we can see in Figures 7, 8, 10, 11. This is due to various factors, mainly depending on the network topology and on traffic loads.

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Figure 7 
Behaviour of the optimal cost functionals 𝐽 1 (a) and 𝐽 2 (b).
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Figure 8 
Behaviour of the optimal cost functionals 𝐽 6 (a) and 𝐽 7 (b).

First of all, notice that in the case study the algorithm opt𝐽1 allows a global optimization for the whole network. It is evident in Figure 7, where 𝐽1 and 𝐽2 are, respectively, the highest and the lowest. Hence, the use of opt𝐽2𝐽7 and opt𝐽6, as control procedures, does not guarantee better performance of traffic flows. The goodness of opt𝐽1 for global performances is confirmed by the behaviour of 𝐽2. In fact, opt𝐽2𝐽7 and opt𝐽6 can let 𝐽2 explode, that is, the traffic circle is stuck and the time to run inside goes to infinity. This situation is more evident in Figure 8, where we can capture another important aspect: the total kinetic energy 𝐽6, on the whole network, tends to zero when opt𝐽1 is not used. This means that the cars flux is going to zero, as evident from Figure 9, hence roads inside the circle are becoming full. A consequence of this phenomenon is also visible in 𝐽7 evolution, that tends to infinity.


Figure 9 
Behaviour of the cost functional 𝐽 3 .
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Figure 10 
Behaviour of the optimal cost functionals 𝐽 4 (a) and S G W (b).
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Figure 11 
Behaviour of optimal 𝐽 2 (a) and 𝐽 7 (b) versus dynamic random simulations.

In Figure 10, 𝐽4 and 𝐽5 (Stop and Go Waves functional, SGW) behaviour are depicted. There are no optimization algorithms for these functionals, and they are computed directly using opt𝐽1, opt𝐽2𝐽7, and opt𝐽6. It is evident that the amount of traffic load, visible in 𝐽4, tends to decrease using opt𝐽1. Moreover, the behaviour of 𝐽5, that measures the velocity variation, indicates that the use of opt𝐽1 leads to more regular densities on roads, giving advantages in terms of security.

To complete the discussion, we focus the attention on the characteristics of dynamic random simulations (Figure 11). As proved in [8], the dynamic random simulation is similar to a fixed simulation with all right of way parameters equal to 0.5. In fact, dynamic random choices fit well the optimizations obtained with 𝐽2 and 𝐽7, since in the considered case the optimal value is 0.5 for each junction of 2×1 type. Unlike the case presented in [6, 8], opt𝐽1 does not guarantee an average optimal right of way parameter equal to 0.5, and this justifies the dissimilarities among the optimal performances obtained using opt𝐽1, opt𝐽2𝐽7 and the global effects due to dynamic random simulations.

From all the previous observations, it is clear that for the case study and with the chosen initial data, the best performances on the whole network are given by the optimal algorithm for 𝐽1. Such algorithm is preferable for maximizing traffic flows, since it allows not only the locally optimization of each node of 2×1 type, but also the global improvement of traffic conditions over the whole network. The other algorithms are able only to improve locally traffic viability.

Acknowledgment

This work is partially supported by MIUR-FIRB Integrated System for Emergency (InSyEme) project under the Grant RBIP063BPH.