1.Introduction CiroD’Apice,RosannaManzo,andLuigiRarit`a SplittingofTrafﬁcFlowstoControlCongestioninSpecialEvents ResearchArticle

(1)

Volume 2011, Article ID 563171,18pages doi:10.1155/2011/563171

Research Article

Splitting of Traffic Flows to Control Congestion in Special Events

Ciro D’Apice, Rosanna Manzo, and Luigi Rarit `a

Dipartimento di Ingegneria Elettronica e Ingegneria Informatica, Universit`a degli Studi di Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Correspondence should be addressed to Rosanna Manzo,[email protected] Received 24 December 2010; Accepted 12 February 2011

Academic Editor: Marianna Shubov

Copyrightq2011 Ciro D’Apice et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We deal with the optimization of traffic flows distribution at road junctions with an incoming road and two outgoing ones, in order to manage special events which determine congestion phenomena. Using a fluid-dynamic model for the description of the car densities evolution, the attention is focused on a decentralized approach. Two cost functionals, measuring the kinetic energy and the average travelling times, weighted with the number of cars moving on roads, are considered. The first one is maximized with respect to the distribution coefficient, and the second is minimized with respect to the same control parameter. The obtained results have been tested by simulations of urban networks. Decongestion effects are also confirmed estimating the time a car needs to cross a fixed route on the network.

1. Introduction

The vehicles congestion is one of the most important problem of modern cities, challenging many researchers to find techniques to control it. A solution to the problem is represented by the use of more lanes and the construction of crossings, but in many areas the solution is not feasible, and moreover the building and expanding of roads to accommodate the increase of vehicles is more expensive. In particular, the presence of unexpected heavy traffic in situations such as accidents leads to delay in the arrival of the emergency services and supplies to where they are needed. In the case of special events, escorts, closures of roads, traffic directions, and control functions can be performed, when necessary, to ensure the safe and efficient movement of vehicles, splitting the traffic flows at intersections in such way to improve the viability. An example is in Figure1, where policemen are involved to manage traffic at junctions.

(2)

A

B

C

Figure 1: Example of car flows redistribution.

In this context, using a fluid dynamic model able to foresee the traﬃc density evolution on road networks see 1–4, we propose a strategy to redistribute in an optimal way flows at junctions. According to the adopted model, the car densities on each road follow a conservation law see 5, while dynamics at junctions is uniquely solved using the following rules:

Athe incoming traﬃc at a node is distributed to outgoing roads according to some distribution coeﬃcients;

Bdrivers behave so as to maximize the flux through the junction.

If a junctionJis of 1×2 typenamely, one incoming road, 1, and two outgoing ones, 2 and 3, ruleAis expressed by a distribution parameterα, indicating the percentage of cars going from road 1 to road 2. Assigning initial densities for incoming and outgoing roads and using ruleB, we finally compute the asymptotic solution as function ofα.

Here, considering the distribution coefficient as control parameter, we aim to redirect traffic at junctions of 1×2 type in order to improve urban traffic and face emergency situations.

In particular, we analyze two optimization problems over a fixed time horizon: minimizing an objective functionW1, estimating the kinetic energy; maximizing a functionalW2, measuring the average travelling time of drivers, weighted with the number of cars moving on roads.

Indeed, we prove that both functionals are optimized for the same value ofα.

Some control strategies for the right of way parameters and distribution coeﬃcients have already been treated in6,7, where three cost functionals, related to average velocity, average travelling time, and flux, have been introduced for 1×2 and 2×1 junctions. Cost functionalsW1 andW2 have been studied in8 for the optimal control of green and red phases of traﬃc lights, while in9parameters of 2×2 junctions have been optimized for the fast transit of emergency vehicles along an assigned path in case of car accidents.

The analysis of the functionalsW1 and W2on a whole network is a very hard task, so we follow a decentralized approach: an exact solution is found for single 1×2 junctions and asymptoticW₁ andW₂. The global suboptimal solution for networks is obtained by localization: the exact optimal solution is applied locally for each time at each junction of 1×2 type.

The analytical optimization results are then tested by simulationsfor numerics, see 10–12, analyzing optimal and random distribution coeﬃcients. The first ones are given by the optimization algorithm; the second ones consider, at the beginning of the simulation process, random values of α, kept constant during the simulation. Then eﬀects of the decentralized approach on the global performance of two networks have been analyzed.

(3)

Simulation results for a symmetric topology show that, assuming random distribution coefficients, the congestion of one road can determine high traffic densities on the whole network, while decongestion phenomena occur when optimalαvalues are used. In the case study of a portion of the Salerno urban network in Italy, characterized by an asymmetric topology, with 1×2 and 2×1 junctions, some interesting aspects arise: random coefficients frequently provoke hard congestions, as expected; optimal distribution coefficients allow a local redistribution of traffic flows. While random simulation curves of the cost functionalW1

are always lower than the optimal one, the optimal curve ofW2is higher than some random ones. This is not surprising because, at 2×1 junctions, traffic densities can remain high. Hence, for such a network,locallyoptimal solutions alleviate critical traffic situations, but the aim of the global optimization ofW1andW2is not achieved. Moreover, using an algorithmsee 13for tracing car trajectories on a network, some simulations are run to test how the total travelling time of a driver is influenced by distribution coefficients. As intuition suggests, the time for covering a path of a single driver decreases when optimalαvalues are used.

The paper is organized as follows. Section2is devoted to the description of the model for road networks and to the construction of solutions to Riemann Problems 1×2 junctions.

In Section3, we define the cost functionalsW1 and W2 and optimize them with respect to the distribution coeﬃcients at a single junction. Simulation results for complex networks are presented in Section4. Section5ends the paper through conclusions.

2. A Riemann Solver for Road Networks

A road network is described by a coupleI,J, whereIrepresents the set of roads, modelled by intervalsai, b_i⊂^Ê,i1, . . . , N, andJis the collection of junctions.

Indicating byρ ρt, x ∈ 0, ρmax the density of cars,ρmax the maximal density, fρ ρvρthe flux withvρthe average velocity, the traﬃc dynamics is described on each road by the conservation lawLighthill-Whitham-Richards model,3,4:

∂_tρ ∂_xf ρ

0. 2.1 We assume that:Ffis a strictly concaveC²function such thatf0 fρmax 0.

Choosingρmax1 andvρ 1−ρ, a flux function ensuringFis f

ρ ρ

1−ρ

, ρ∈0,1, 2.2

which has a unique maximumσ1/2.

In order to capture the dynamics at a junction, we solve Riemann Problems RPs, Cauchy Problems with a constant initial datum for each incoming and outgoing road, the basic ingredient for the solution of Cauchy Problems by Wave-Front-Tracking algorithms.

Consider a junctionJ ofn×mtype, that is, withnincoming roadsI_ϕ,ϕ1, . . . , n, m outgoing roads,Iψ,ψ n 1, . . . , n m, and initial datumρ0 ρ1,0, . . . , ρn,0, ρ_{n 1,0}, . . . , ρ_{n m,0}. Definition 2.1. A Riemann Solver RSfor the junction J is a map RS : 0,1ⁿ×0,1^m → 0,1ⁿ ×0,1^m that associates to Riemann dataρ0 ρ1,0, . . . , ρn,0, ρ_{n 1,0}, . . . , ρ_{n m,0}at J a vectorρ ρ₁, . . . ,ρ_n,0,ρ_{n 1}, . . . ,ρ_{n m}so that the solution on an incoming roadI_ϕ,ϕ1, . . . , n, is the waveρϕ,0,ρϕand on an outgoing oneIψ,ψ n 1, . . . , n mis the waveρψ, ρψ,0.

(4)

We require the following conditions hold true: C1 RSRSρ0 RSρ0; C2 on each incoming roadI_ϕ,ϕ1, . . . , n, the waveρϕ,0,ρ_ϕhas negative speed, while on each outgoing roadI_ψ,ψn 1, . . . , n m, has the waveρ_ψ, ρ_ψ,0has positive speed.

Ifm≥n, a possible RS atJis defined by the following rulessee1:

Atraffic is distributed at J according to some coefficients, collected in a traffic distribution matrix A αj,i, i 1, . . . , n, j n 1, . . . , n m, 0 < αj,i < 1, _{n m}

jn 1α_j,i 1. Theith column ofAindicates the percentages of traﬃc that, from the incoming roadIi, distribute to the outgoing roads;

BfulfillingA, drivers maximize the flux throughJ.

Focus on a 1×2 junctionJ. We indicate the cars density on the incoming road 1 by ρ₁t, x ∈ 0,1,t, x ∈^Ê ×I₁, and on the outgoing roadsψ,ψ 2,3, by ρ_ψt, x ∈ 0,1, t, x∈^Ê ×I_ψ.

Consider the flux function2.2and letρ1,0, ρ_2,0, ρ_3,0be the initial densities atJ. The maximal flux values on roads are defined by

γ₁^max

⎧⎪

⎪⎨

⎪⎪

⎩ f

ρ_1,0

if 0≤ρ_1,0≤ 1 2, f

1 2

if 1

2 ≤ρ1,0≤1,

γ_ψ^max

⎧⎪

⎪⎨

⎪⎪

⎩ f

1 2

if 0≤ρ_ψ,0≤ 1

2, ψ 2,3, f

ρψ,0

if 1

2 ≤ρψ,0≤1, ψ 2,3.

2.3

Ifα∈0,1and 1−αindicate, respectively, the percentage of cars that, from road 1, goes to the outgoing roads 2 and 3, the fluxes solution to the RP atJare

γ

γ1, αγ1,1−αγ1

, 2.4

where

γ1min

γ₁^max,γ₂^max α ,γ₃^max

1−α

. 2.5

Hence,ρf⁻¹γis found as followssee1,2:

ρ1∈

⎧⎪

⎪⎨

⎪⎪

⎩ ρ_1,0

∪ τ

ρ_1,0 ,1

if 0≤ρ_1,0≤ 1 2, 1

2,1

if 1

2 ≤ρ1,0≤1,

ρ_ψ∈

⎧⎪

⎪⎨

⎪⎪

⎩

0,1 2

if 0≤ρψ,0≤ 1

2, ψ2,3, ρψ,0

∪ 0, τ

ρψ,0

if 1

2 ≤ρψ,0≤1, ψ2,3,

2.6

(5)

whereτ :0,1 → 0,1is the map such thatfτρ fρfor everyρ∈0,1andτρ/ρ for everyρ∈0,1\ {1/2}.

Finally, on the incoming road 1, the solution is given by the waveρ1,0,ρ1, while on the outgoing roadψ,ψ2,3, the solution is represented by the waveρψ, ρψ,0.

3. Distribution Parameters Optimization

Fix a 1×2 junctionJand an initial datumρ1,0, ρ_2,0, ρ_3,0. We define the cost functionalW₁t andW2t, which measure, respectively, the kinetic energy and the average travelling time weighted with the number of cars moving on roads:

W₁t ³

k1

Ik

f

ρ_kt, x v

ρ_kt, x dx,

W2t ³

k1

Ik

ρkt, x v

ρ_kt, xdx.

3.1

For a fixed time horizon 0, T, with T sufficiently big, consider the traffic distribution coefficientαas control. We aim to maximizeW1Tand to minimizeW2Tseparately. The functionals assume the form:

W₁T ³

i1

f ρ_i

v ρ_i

1 2

3 i1

γ_i

1−s_i

1−4γ_i

,

W₂T ³

i1

ρ_i v

ρi

³

i1

1 si

1−4γi

1−si

1−4γi

,

3.2

wheres1andsψ,ψ 2,3, are given by

s1

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1 ifρ1,0≥ 1 2, orρ1,0< 1

2 and γ₁^max>min γ₂^max

α , γ₃^max 1−α

,

−1 ifρ_1,0< 1

2 andγ₁^max≤min γ₂^max

α ,γ₃^max 1−α

,

sψ

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

1 ifρψ,0> 1

2 and γ_ψ^max α_ψ ≤min

γ₁^max,γ_ψ^max

α_ψ

, ψ/ψ,

−1 ifρψ,0≤ 1 2, orρψ,0> 1

2 and γ_ψ^max α_ψ >min

γ₁^max,γ_ψ^max

α_ψ

, ψ/ψ,

3.3

(6)

with

α_ψ

⎧⎨

⎩

α ifψ 2,

1−α ifψ 3. 3.4

According to the solution of the RP atJ, we have W₁T γ₁

2

1−s₁

1−4γ₁ αγ₁

2

1−s₂

1−4αγ₁

1−αγ₁ 2

1−s₃

1−41−αγ₁

,

W₂T 1 s₁ 1−4γ₁ 1−s₁

1−4γ₁

1 s₂

1−4αγ₁ 1−s₂

1−4αγ₁

1 s₃

1−41−αγ₁ 1−s₃

1−41−αγ₁ ,

3.5 whereγ1is given by2.5. The values ofα, which optimizeW1TandW2T, are reported in the following theoremfor the sketch of the proof, see the appendix.

Theorem 3.1. Fix a 1×2 junctionJ. AssumingTsuﬃciently big, the cost functionalsW1TW2T is maximized (minimized) forα1/2, with the exception of the following cases (for some of them, the optimal control does not exist but it is approximated):

aifγ₃^max≤γ₁^max/2< γ₁^max≤γ₂^max,αα₁ ε;

bifγ₂^max< γ₁^max/2< γ₁^max≤γ₃^max,αα2;

cifγ₂^max< γ₃^max< γ₁^max, we distinguish three subcases:

c1ifγ₁^max−γ₃^max≥γ₂^max,αα3;

c2ifγ₁^max−γ₃^max< γ₂^maxγ₁^max/2,α 1/2−ε;

c3ifγ₁^max−γ₃^max< γ₂^max≤γ₁^max/2,αα₂−ε;

difγ₃^max< γ₂^max< γ₁^max, we distinguish two subcases:

d1ifγ₁^max−γ₃^max≥γ₂^max,αα3 ε;

d2if1/2γ₁^max≤γ₁^max−γ₃^max< γ₂^max,αα1−ε,

whereα1 γ₁^max−γ₃^max/γ₁^max,α2 γ₂^max/γ₁^max,α3 γ₂^max/γ₂^max γ₃^maxand εis small and positive.

Example 1. Discuss the optimal solution for the following initial conditions:

Aρ1,00.35,ρ2,00.2,ρ3,00.9;

Bρ1,00.45,ρ2,00.75,ρ3,00.15;

Cρ1,00.3,ρ2,00.9,ρ3,00.8.

In caseA, we get

γ₁^max0.2275, γ₂^max0.25, γ₃^max0.09, 3.6

(7)

W₁T

α1 1

α

a

W₂T

α1 1

α

b Figure 2: CaseA: behaviour ofW1andW2for suﬃciently big.

so conditionγ₃^max< γ₁^max< γ₂^maxis satisfied. Hence,

γ

γ1, αγ1,1−αγ1

, 3.7

where

γ₁

⎧⎪

⎪⎨

⎪⎪

⎩ γ₃^max

1−α, 0< α≤α1, γ₁^max, α1< α <1,

3.8

withα1 0.6. ForTsuﬃciently big,W1TandW2Thave one discontinuity point atαα1, as shown in Figure2. The optimal control does not exist, but one can chooseαα₁ ε.

In caseB, we have that

γ₁^max0.2475, γ₂^max0.1875, γ₃^max0.25, 3.9 hence conditionγ₂^max< γ₁^max< γ₃^maxholds. Then, the solution to the RP atJis

γ

γ1, αγ1,1−αγ1

, 3.10

where

γ₁

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

γ₁^max, 0< α≤α₂, γ₂^max

α , α2< α <1,

3.11

with α₂ 0.7575. For T suﬃciently big, the cost functionals W₁T and W₂T have one discontinuity point at α α2, as shown in Figure 3. The optimal control exists, and it is α1/2, for which

W1T 0.347816, W2T 1.1565. 3.12

(8)

W1T

0.5 α2 1 α

a

W2T

0.5 α2 1 α

b

Figure 3: CaseB: behaviour ofW1andW2forTsuﬃciently big.

W1T

α1 α2 1

α

a

W2T

α1 α2 1 α

b

Figure 4: CaseC: behaviour ofW1andW2forT suﬃciently big.

In caseC

γ₁^max0.21, γ₂^max0.09, γ₃^max0.16, 3.13 hence conditionγ₂^max< γ₃^max< γ₁^maxis satisfied, and we obtain

γ

γ1, αγ1,1−αγ1

, 3.14

where

γ1

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ γ₃^max

1−α, 0< α≤α₁, γ₁^max, α1< α < α2,

γ₂^max

α , α2< α <1,

3.15

withα₁ 0.238 andα₂ 0.428. The cost functionalW₁TandW₂T, reported in Figure4 forT suﬃciently big, have two discontinuity points atαα1andαα2. Hence, an optimal value forαdoes not exist, but we can chooseαα2−ε.

(9)

a 1

b

c 2

3 g

f e d

Figure 5: Topology of the symmetric network.

4. Road Traffic Simulation

We present some simulation results in order to test the optimization algorithm for the cost functionals. In particular, we analyze the eﬀects of diﬀerent control procedures, applied locally at each junction, on the global performances of networks and compute the travelling time of a car on assigned paths. For simplicity, from now on we drop the dependence onT fromW1andW2.

4.1. A Symmetric Network

In this subsection, we analyze a symmetric network with three simple junctions of 1×2 type, labelled by 1, 2, and 3, see Figure 5. In particular, the network consists of two inner roads,bandc, and five roads, that connect the inner roads to outside:a,d,e,f, andg. The conservation law with flux function2.2is approximated using the Godunov scheme, with space stepΔx0.0125, and time step, determined by the CFL conditionsee10,11, equal to 0.5. We assume initial conditions zero for all roads at the starting instant of simulation t0, a 0.3 Dirichlet boundary datum for roadsa,d,e,f, a 0.9 Dirichlet boundary condition for roadgand a time interval of simulation0, T, whereT 30 min.

Two different choices of the distribution coefficients are considered:locallyoptimal parameters at each junction, given by analytical resultsoptimal case; random parameters random case, that is, the distribution coefficients are taken randomly for each road junction when the simulation starts and then are kept constant.

The evolution of W1 and of W2 are depicted in Figures 6 and 7, reporting with a continuous line the optimal case and with dashed lines various random cases. In some random simulations, the α values are such that a lower traﬃc density goes to road g, with a consequent natural improvement of the network performances. This justifies the fact that some dashed W1 and W2 curves approach the optimal ones. In other cases, W1

rapidly decreases andW₂tends to infinity, indicating that the random choice ofαprovokes congestions on all network roads. However, in any case, the optimal case is better than the others. In fact, it describes the natural situationthat happens on congested real urban networksin which the traﬃc is redirected to less congested roads.

4.2. A Real Urban Network

This subsection is devoted to the simulation on a portion of the urban network of Salerno, Italy. The network topology, depicted in Figure8, is characterized by four principal roads.

Each of them is divided into segments, labelled by letters: Via Torrionesegmentsa,b, andc,

(10)

0.1 0.2 0.3 0.4 0.5

W1t

5 10 15 20 25 30

tmin a

0.42 0.44 0.46 0.5 0.48 0.52 0.54

W1t

5 10 15 20 25 30

tmin b

Figure 6: W1t evaluated for optimal distribution coeﬃcients continuous line and random choices dashed lines.aevolution in0;T.bzoom around the asymptotic optimal values.

1 2 3 4 5

W2t

5 10 15 20 25 30

tmin a

0.9 1 1.1 1.2 1.3

W1t

0 5 10 15 20 25 30

tmin b

Figure 7: W2t evaluated for optimal distribution coeﬃcients continuous line and random choices dashed lines.aevolution in0;T.bzoom around the asymptotic optimal values.

Via Leonino Vinciprovasegmentsdande, Via Settimio Mobiliosegmentsf,g,h, andi, and Via Guerciosegment l. We distinguish inner roads segments, b,e,f,g, andh, and external ones,a,c,d,i,l. Junctionsindicated by numbers1, 3 and 5 are of 1×2 type,while 2 and 4 are of 2×1 types. The evolution of traﬃc flows is simulated by the Godunov method withΔx 0.0125,Δt Δx/2 in a time interval0, T, withT 120 min. Initial conditions and boundary data for densities are in Table1and have been taken in order to simulate a congestion scenario. Notice that, for junctions 2 and 4, right of way parameters are chosen according to measures on the real network.

In Figure 9, we report the behaviour of W1 and W2, where optimal simulations are indicated again by continuous lines, while random cases by dashed ones. Random simulations curves of W1 are always lower than the optimal ones. In fact, when optimal parameters are used, a flows redistribution occurs on roads, with consequent reduction of congestions at junctions of 1×2 type. Focus now on W₂, for which the optimal curve is higher than some random ones. This is not surprising as we deal with the simulation of a high congested asymmetric network. The traﬃc redirection at congested 1×2 junctions is of local type and, as expected, benefits occur only on roads and at junctions where the optimization procedure is applied. This is easy deducible considering Via Torrione, which presents a 2×1 junction, labelled by 2, where traﬃc high densities cannot be redirected.

(11)

a 1 b 2 c

f

g

4 h

5 i

l 3

d

e

Figure 8: Topology of the portion of the real urban network of Salerno.

0.2 0.4 0.6 0.8 1

W1t

20 40 60 80 100 120

tmin a

20 40 60 80 100 120 140

W2t

20 40 60 80 100 120

tmin b

Figure 9: Evolution in0;T ofW1t a andW2t b evaluated for optimal distribution coeﬃcients continuous lineand random choicesdashed lines.

Table 1: Initial conditions, boundary data and right of way parameters for roads of the network.

Road Initial condition Boundary data Right of way parameters

a 0.2 0.3 /

b 0.2 / 0.2

c 0.2 0.9 0.4

d 0.2 0.3 /

e 0.8 / /

f 0.2 / 0.8

g 0.75 / 0.6

h 0.6 / /

i 0.75 0.9 /

l 0.75 0.9 /

Even if the right of way parameters which characterize 2×1 junctions are optimized according to the values in8, traﬃc conditions almost remain the same. Hence, although optimal

(12)

0.2 0.4 0.6 0.8 1

xt

22.5 25 27.5 30 32.5 35 37.5 tmin

a

0.2 0.4 0.6 0.8 1

xt

62.5 65 67.5 70 72.5 75 77.5 tmin

b

Figure 10: Evolution of a car trajectoryxtalong roadgwith initial travel timest0 20aandt0 60 b, evaluated for optimal distribution coeﬃcientscontinuous linesand random choicesdashed lines.

distribution coefficients are used, high traffic densities affect some roads and this justifies that the optimal curve is not the lowest forW₂.

Suppose that a car travels along a path in a network, whose traﬃc evolution is modeled by2.1. The position of the driverxxtis obtained solving the Cauchy problem:

˙ xv

ρt, x ,

xt0 x0, 4.1

wherex0is the initial position at the initial timet0, whilev1−ρ. Using numerical methods, described in13, we aim to estimate the driver travelling time and to prove the goodness of the optimization results. First, we compute the car trajectory along roadg and the time needed for covering it in optimal case and random cases; then, we fix a car path within the Salerno network and study the exit time evolution versus the initial travel timet0the time in which the car enters into the network.

In Figure 10, we assume that the car starts its own travel at the beginning of road g at the initial times t0 20a and t0 60band compute the trajectories xt along roadg, in optimal casecontinuous lineand random casesdashed lines. Although initial timest0are diﬀerent, the evolutionxtin the optimal case has always a higher slope with respect to trajectories in random cases because traﬃc levels are low. When random choices of parametersαare used, higher boundary conditions for roadsc,i, andlcause an increase in density values by shocks propagating backwards. Hence, car velocities are reduced, travel times become longer and so exit times from roadg.

In Table2, we collect the exit timesT_out from roadgthe times needed to go out from roadg, for the optimal choice of distribution coeﬃcientsoptand random choicesri,i 1, . . . ,9, assumingt020.

The exit timeT_outtfrom roadgversus the initial timet₀, assuming that the car starts its path from the beginning of the road is shown in Figure11a. Because of the decongestion eﬀects, the choice of optimal coeﬃcientscontinuous linesallows to obtain an exit time lower than the other casesdashed lines. Notice that the exit time becomes stable after a certain initial time valuet0 18.5 for the optimal distribution choice, unlike the random cases, for whicht0 35.5 andt0 42.

(13)

Table 2: Initial conditions, boundary data and right of way parameters for roads of the network.

Simulations Tout

opt 1.02

r1 9.955

r2 10.89

r3 10.93

r4 11.82

r5 11.895

r6 12.75

r7 13.43

r8 13.585

r9 17.185

2 4 6 8 10 12 14

Texitt

10 20 30 40 50 60 70 80 t0min

a

10 20 30 40

Texitt

10 20 30 40 50 60 70 80 t0min

b

Figure 11: Exit time from road g versus initial travel time t0in 0,80 a and exit time from the patha, g, h, lversus initial travel timest0 in0,80 b, evaluated for optimal distribution coeﬃcients continuous linesand random choicesdashed and dot dashed lines.

Finally, in Figure11b, we consider the exit timeTouttfrom a fixed route,a, g, h, l, versus the initial times t0. Precisely, we study the temporal variation of Tout when a car, starting its trip at the beginning of roadaat timet₀, crosses roadsa,g,h,lin order to exit from the network. Also in this case, different choices of distribution coefficientsthe optimal one is indicated by a continuous line, unlike the othersaffect the time for covering the path.

When optimal parameters are not used,Touttends to infinity at some critical timestctc 13 andtc 27 for the random case represented by dashed line, andtc 23 andtc 36 for the random case with dot-dashed line. This occurs because the car cannot enter roadgor road h, since the traﬃc within them is blocked. For times greater than critical ones, traﬃc densities become stablea light decongestion allows the car to reach the destinationand exit times reach a steady valueTout 31.5 andT_out 30.35at times, respectively,t₀ 42 andt₀ 46.

On the contrary, when optimal parameters are used, the exit time does not tend to infinity as the network is never congestedand reaches the steady valueTout 28.75 at timet0 38 lower, as expected, than steady-state times of simulation with random parameters.

(14)

γ1

α1 1 α

γ₁^max γ₃^max 1−α γ₂^max

α

a

γ1

α2 1 α

γ₁^max γ₃^max 1−α γ₂^max

α

b

Figure 12:acaseγ₃^max< γ₁^max< γ₂^max.bcaseγ₂^max< γ₁^max< γ₃^max.

5. Conclusions

In this paper, an optimization study has been presented to improve the urban traﬃc conditions in case of special events in which splitting of flows is needed.

The optimization has been made over traffic distribution coefficients at junctions, using two cost functionals, that measure, respectively, the kinetic energy and the average travelling times of drivers, weighted with the number of cars moving on roads. An exact solution has been found for simple junctions having one incoming road and two outgoing roads. The obtained analytical results have been tested through simulations, showing that in some cases a total decongestion effect is possible. This is also confirmed by evaluations of cars trajectories on some roads and fixed routes on the network: using optimal distribution coefficients, times needed to cover paths are the lowest.

Appendix

We report the proof of Theorem3.1. Consider new functionals, W1 and W2, in which the terms not depending on αare neglected. Since the solution to the RP atJ depends on the value of the parameterα, we distinguish various cases. Here, for sake of brevity, we analyze some of them.

Assumesee Figure12athat

γ₃^max< γ₁^max< γ₂^max, A.1

then

γ1

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ γ₃^max

1−α, 0< α≤α1, γ₁^max, α1< α <1,

A.2

(15)

whereα1 γ₁^max−γ₃^max/γ₁^max. As for 0 < α ≤ α1,s1 1, s2 −1, and forα1 < α < 1, s2s3−1,W1andW2assume the form:

W1

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1 1−α

⎛

⎝1−

1−4γ₃^max 1−α

⎞

⎠ α 1−α

1

1−4 α

1−αγ₃^max

, 0< α≤α1,

α

1

1−4αγ₁^max 1−α

1

1−41−αγ₁^max , α₁ < α <1,

W2

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1

1−4

γ₃^max/1−α 1−

1−4

γ₃^max/1−α

1−

1−4α/1−αγ₃^max

1

1−4α/1−αγ₃^max, 0< α≤α₁, 1−

1−4αγ₁^max

1

1−4αγ₁^max

1−

1−41−αγ₁^max 1

1−41−αγ₁^max, α₁< α <1.

A.3

Our aim is to maximizeW1and to minimizeW2separately with respect to the parameterα.

If 0< α≤α₁, then

∂W1

∂α 2 γ₃^max 1−α³

1

μ1−α− α

μ1−α/α

1 1−α²

2−μ1−α μ 1−α

α

,

∂W2

∂α 4γ₃^max

μ1−α/α

1 μ1−α/α₂

1−α² − 4γ₃^max μ1−α

μ1−α−1₂

1−α²,

A.4

whereμα

1−4γ₃^max/1−α, and hence the functionalsW₁andW₂are, respectively, an increasing and a decreasing function.

Ifα1< α <1, we get

∂W1

∂α 2γ₁^max

η1−α−ηα

ζα−ζ1−α,

∂W₂

∂α 2γ₁^max

χα−χ1−α θα−θ1−α ,

A.5

where

ζα

1−4αγ₁^max, ηα α ζα,

χα 1−ζα

1 ζα²ζα, θα 1 1 ζαζα.

A.6

(16)

Since

∂W1

∂α ≥0⇐⇒α∈

0,1 2

, ∂W2

∂α ≥0⇐⇒α∈ 1

2,1

, A.7

we conclude thatW1andW2are maximized and minimized, respectively, forα1/2.

Finally, we obtain that

iifα1<1/2, for allα∈0, α1∪α1,1/2,

∂W1

∂α ≥0, ∂W2

∂α ≤0; A.8

iiifα1≥1/2, for allα∈0, α1,

∂W₁

∂α >0, ∂W₂

∂α <0. A.9

Moreover,

αlim→α₁

!

W1α−W1α1"

>0, lim

α→α₁

!

W2α1−W2α"

>0. A.10

Hence, we conclude that

iifα1<1/2,W1andW2are optimized forα1/2;

iiifα1 ≥ 1/2, the optimal value for bothW1andW2does not exist. One can choose αα₁ ε, withεsmall and positive constant.

In the particular caseγ₃^max < γ₁^max γ₂^max, the analysis is unchanged. Ifγ₁^max γ₃^max <

γ₂^maxbothW1andW2are optimized forα1/2.

Assume thatFigure12b

γ₂^max< γ₁^max< γ₃^max. A.11

We have

γ1

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

γ₁^max, 0< α≤α2, γ₂^max

α , α₂< α <1,

A.12

(17)

whereα2 γ₂^max/γ₁^max. Then, as for 0 < α ≤ α2,s2 s3 −1, and for α2 < α < 1,s1 1, s₃−1, we have to maximize

W1

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ α

1−4αγ₁^max 1−α

1−41−αγ₁^max, 0< α≤α2,

1 α

⎛

⎝1−

1−4γ₂^max α

⎞

⎠ 1−α α

# 1

1−41−α α γ₂^max

$

, α₂< α <1,

A.13

and to minimize

W2

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ 1−

1−4αγ₁^max

1

1−4αγ₁^max

1−

1−41−αγ₁^max

1

1−41−αγ₁^max, 0< α≤α2, 1

1−4

γ₂^max/α 1−

1−4

γ₂^max/α

1−

1−41−α/αγ₂^max

1

1−41−α/αγ₂^max, α2< α <1,

A.14

separately with respect toα.

Observe that W₁ and W₂ have both a jump atα α₂. If 0 < α ≤ α₂,∂W₁/∂αand

∂W2/∂αhave the same expressions already examined in the previous case. Ifα2< α <1,

∂W₁

∂α −2γ₂^max α³

1 ωα

1

ωα/1−α

1 α²

ωα−ω α

1−α

−2

,

∂W2

∂α 4γ₂^max

α²ωα−1²ωα− 4γ₂^max

α²1 ωα/1−α²ωα/1−α,

A.15

whereωα

1−4γ₂^max/α, and it follows thatW₁andW₂are, respectively, a decreasing and an increasing function. We get

∂W1

∂α ≥0, ∀α∈0,α,% ∂W2

∂α ≤0, ∀α∈0,α,% A.16

where

% α

⎧⎪

⎪⎨

⎪⎪

⎩

α2, α2< 1 2, 1

2, α2≥ 1 2,

α→limα₂

!

W₁α2−W₁α"

>0, lim

α→α₂

!

W₂α2−W₂α"

<0.

A.17

(18)

We conclude that

iifα2<1/2, the optimal value forW1andW2isαα2; iiifα₂≥1/2,W₁andW₂are optimized forα1/2.

The obtained optimization results also hold if γ₂^max < γ₁^max γ₃^max; on the contrary, assumingγ₁^maxγ₂^max< γ₃^max, the optimal value for both functionals isα1/2.

Acknowledgment

This work is partially supported by MIUR-FIRB Integrated System for EmergencyInSyEme project under Grant RBIP063BPH.

References

1 G. M. Coclite, M. Garavello, and B. Piccoli, “Traﬃc flow on a road network,” SIAM Journal on Mathematical Analysis, vol. 36, no. 6, pp. 1862–1886, 2005.

2 M. Garavello and B. Piccoli, Traﬃc Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical SciencesAIMS, Springfield, Mo, USA, 2006.

3 M. J. Lighthill and G. B. Whitham, “On kinematic waves. II. A theory of traﬃc flow on long crowded roads,” Proceedings of the Royal Society A, vol. 229, pp. 317–345, 1955.

4 P. I. Richards, “Shock waves on the highway,” Operations Research, vol. 4, pp. 42–51, 1956.

5 A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, vol. 20 of Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford, UK, 2000.

6 A. Cascone, C. D’Apice, B. Piccoli, and L. Rarit`a, “Optimization of traﬃc on road networks,”

Mathematical Models & Methods in Applied Sciences, vol. 17, no. 10, pp. 1587–1617, 2007.

7 A. Cascone, C. D’Apice, B. Piccoli, and L. Rarit`a, “Circulation of car traﬃc in congested urban areas,”

Communications in Mathematical Sciences, vol. 6, no. 3, pp. 765–784, 2008.

8 A. Cutolo, C. D’Apice, and R. Manzo, “Traﬃc optimization at junctions to improve vehicular flows,”

Preprint DIIMA, no. 57, 2009.

9 R. Manzo, B. Piccoli, and L. Rarit`a, “Optimal distribution of traﬃc flows at junctions in emergency cases,” submitted to European Journal on Applied Mathematics.

10 E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, vol. 118 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1996.

11 S. K. Godunov, “A diﬀerence method for numerical calculation of discontinuous solutions of the equations of hydrodynamics,” Matematicheskii Sbornik, vol. 4789, no. 3, pp. 271–306, 1959Russian.

12 J. P. Lebacque, “The Godunov scheme and what it means for first order traﬃc flow models,” in Proceedings of the 13th ISTTT, J. B. Lesort, Ed., pp. 647–677, Pergamon, Oxford, UK, 1996.

13 G. Bretti and B. Piccoli, “A tracking algorithm for car paths on road networks,” SIAM Journal on Applied Dynamical Systems, vol. 7, no. 2, pp. 510–531, 2008.

(19)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}

1.Introduction CiroD’Apice,RosannaManzo,andLuigiRarit`a SplittingofTrafﬁcFlowstoControlCongestioninSpecialEvents ResearchArticle

Research Article

Splitting of Traffic Flows to Control Congestion in Special Events

Ciro D’Apice, Rosanna Manzo, and Luigi Rarit `a

1. Introduction

2. A Riemann Solver for Road Networks

3. Distribution Parameters Optimization

4. Road Traffic Simulation

5. Conclusions

Appendix

Acknowledgment

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis