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Neural, Parallel, and Scientific Computations 24 (2016) 317-334

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE

OF A ROUNDABOUT

LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

aDepartment of Mathematics, Adama Science and Technology University,

P. O. Box 1888, Adama, Ethiopia.

bDepartment of Mathematics, TU Kaiserslautern, 67653 Kaiserslautern, Germany.

cDepartment of Mathematics, Addis Ababa University,

P. O. Box 1176, Addis Ababa, Ethiopia.

ABSTRACT. In this paper we investigate the relationship between short time stay of pedestrian

on the crosswalk situated at entrance and exiting roads of the roundabout. In addition, we study its

implication on the performance of the roundabout in regulating traﬃc ﬂow problems. To this aim,

the roundabout is modeled as network of roads with 2×1and 1×2-type junctions and external

incoming and outgoing roads. The evolution of traﬃc ﬂow on the road network at the roundabout

is modeled by Lighthill-Whitham-Richards model with extended ﬂux to capture the presence of

pedestrians on crosswalk. We introduce three cost functionals that measure the total mass of vehicle

, average velocity, and total ﬂux respectively on the network. Then we analyze the performance of

the roundabout with and without pedestrian through numerical simulation using Godunov scheme.

Keywords: Traﬃc ﬂow, roundabout performance, scalar conservation laws, pedestrians.

AMS (MOS) Subject Classiﬁcation. 35L65, 33F05, 90B20.

1. Introduction

A wide variety of mathematical models describing traﬃc ﬂow and separately

pedestrian motion has been intensively investigated in the recent years using micro-

scopic and macroscopic approach. The macroscopic ﬂuid dynamic type description

of traﬃc ﬂow model was introduced in the 1950s by Lighthill and Whitham [21] and

independently by Richards [23], on an inﬁnite single road using a non-linear scalar

hyperbolic conservation laws. This model is commonly referred to as LWR model.

Further improvements are achieved in [1, 2, 15] and successfully extended to networks

in recent years, see for example [8, 10, 12, 13, 17] and references therein. These models

have also been utilized for the optimization of vehicular traﬃc ﬂow on road networks

through various approaches, see for example [6, 7, 16, 22] and references therein.

The ﬁrst macroscopic pedestrian ﬂow model was due to the seminal work of

Hughes [24] where the preferred direction of motion is given by an eikonal equation.

Furthermore, in [9] pedestrian traﬃc modeling using scalar conservation laws based on

Received April 30, 2015 1061-5369 $15.00 c

Dynamic Publishers, Inc.

318 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

the solution of the eikonal equation has been investigated. More recently, the coupling

of traﬃc ﬂow networks with pedestrian motion on the street has been achieved in [3, 4].

Modern roundabouts are now considered as an alternative traﬃc control device

that can improve safety and operational eﬃciency at intersections compared to other

conventional intersection controls [11].

In the present work we are interested to investigate the relation between short

stay of pedestrians on the crosswalk situated on entrance and exiting arms of the

roundabout and its implication on the performance of the roundabout in regulating

traﬃc ﬂow problems. Compared to [6, 22], the model presented in this paper use

extended ﬂux due to pedestrians motion on the crosswalk and focus on investigating

the performance of the roundabout rather than optimizing the traﬃc ﬂow.

Roundabouts can be seen as particular road networks and can be modeled as

alternatively periodic sequences of 2×1and 1×2-type junctions. In this paper, we

consider a roundabout joining m-incoming and m-outgoing roads, m∈N,m≥2,

with pedestrian crosswalk situated both on the entrance and exit roads. The evolution

of the traﬃc ﬂow on the whole network of the roundabout is described by nonlinear

scalar hyperbolic partial diﬀerential equations.

We introduce three cost functionals that measure the total mass of vehicle, aver-

age velocity, and total ﬂux on the network to analyze the performance of the round-

about with and without pedestrian through numerical simulation.

The paper is structured as follows. In Section 2 we describe mathematical model

for the road networks with roundabout, governing equations and the solution of Rie-

mann problem at junctions. Section 3 presents detailed numerical simulations and

comparisons. Finally we conclude in Section 4.

2. The Model

In this work we consider a roundabout joining m-incoming and m-outgoing roads,

m∈N,m≥2, with pedestrian crosswalk situated both on the entrance and exit arms

as illustrated in Figure 1. A roundabout can be seen as a directed graph in which roads

are represented by arcs and junctions by vertexes. The arcs are modeled by intervals

Ii= [ai, bi]⊂R,ai< bi,i= 1,2,...,2m. In the case of incoming and outgoing

roads either bior aican be extended to +∞or −∞. Each junction Jiis described

by its incoming and outgoing roads, where we denote the set of all incoming roads

of junction Jiby Inc(Ji)and the set of all outgoing roads of junction Jiby Out(Ji)

for i= 1,2,...,2m. In this setting, the roundabout illustrated in Figure 1 can be

decomposed into alternatively periodic sequences of 2×1and 1×2-type junctions

(compare Figure 2).

To recover the behavior of the roundabout we introduce Coclite, Garavello and

Piccoli (CGP) condition at junctions.

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE OF A ROUNDABOUT 319

Crosswalk

J1

J2m−2

J4

J5

J2m

J2J3

J2m−1

Figure 1. Sketch of the roundabout considered in the article.

1

3

2

2

1

3

Figure 2. A junction with two incoming and one outgoing road (left),

and a junction with one incoming and two outgoing roads (right).

For the deﬁnition of road networks and detail description we refer to [12, 18].

Deﬁnition 2.1. A road network is a couple (I,J), where I={Ii= [ai, bi]⊆R,

i= 1,2,...,2m}represents a ﬁnite set of edges (roads), and Jis the collection of

vertexes. Each vertex J is the union of two nonempty subsets Inc(J) and Out(J) of

{1,2,...,2m}representing respectively, the incoming and the outgoing roads.

In the context of the present work, each junction can be identiﬁed as either 2×1

or 1×2type. The evolution of the traﬃc ﬂow on each arc of the roundabout is given

by the scalar hyperbolic conservation law

(2.1) ∂tρi+∂xf(ρi)=0,(t, x)∈R+×Ii, i = 1,2. . . , 2m,

where ρi=ρi(t, x)∈[0, ρmax]is the mean traﬃc density and ρmax the maximal

density on the road. The ﬂux function fi: [0, ρmax]→R+is given by the following

ﬂux-density relation

fi(ρi) = ρiv(ρi)

where v: [0, ρmax]→R+is a smooth decreasing Lipschtiz continuous function denot-

ing the mean traﬃc speed. As given in [18], the basic assumption we will make for

the evolution of traﬃc on the roundabout is that the velocity function is dependent

only on the density ρof the cars. The mean traﬃc speed vattains its maximal value

320 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

vmax for small density and when ρincreases to some maximum capacity ρmax, the

mean traﬃc speed vanishes. For future use we normalize the vehicle density ρ(t, x)

so that 0≤ρ≤ρmax = 1. Furthermore, as presented in [12], we make the following

assumption on the ﬂux function

(A1)fiis smooth strictly concave C2function.

(A2)fi(0) = fi(1) = 0.

Assumption (A1)and (A2)guarantees the existence and uniqueness of critical den-

sity ρc∈(0,1) such that f0

i(ρc) = 0. A typical example of ﬂux function satisfying

assumption (A1)and (A2)is

(2.2) f(ρ) = ρvmax(1 −ρ)

which is commonly referred as fundamental diagram in the transportation literature

(see Figure 3). For the theory of scalar hyperbolic conservation laws we refer to [5].

ρρc1

fmax

f(ρ)

Figure 3. Flux function considered.

More recently, the coupling of traﬃc ﬂow networks with pedestrian motion on

the street has been achieved in [3, 4] using scalar hyperbolic conservation laws in

combination with the solution of the eikonal equation. In the present work we are

interested to investigate the relation between short stay of pedestrian motion on the

crosswalk on entrance and exiting arms of the roundabout and its implication on

the performance of the roundabout in regulating traﬃc ﬂow problems. To achieve

this goal, we assume that the crosswalk is a two-dimensional space orthogonal to the

driving direction and extend the traﬃc ﬂux function at the exit and entrance by

(2.3) ˜

f(ρi) = ρiv(ρi)g(p), i = 1,2, . . . , m

where the function gis deﬁned by

(2.4) g(p, t) =

0if p = 1 , t ∈[t1, t2]

1if p = 0

and p∈ {0,1}. The state p= 1 corresponds to the situation when pedestrians are

on the crosswalk and p= 0 corresponds to no pedestrians are on the crosswalk. The

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE OF A ROUNDABOUT 321

time t1and t2are randomly chosen. The duration t=t2−t1is the time taken

by pedestrians to cross the road. The situation g(p, t) = 0 indicates the occupancy

of road by pedestrians and the interruption of traﬃc ﬂow. The state g(p, t) = 1

corresponds to the absence of pedestrians on the crosswalk at the exit and entrance

of the roundabout and the traﬃc ﬂow behaves normally as given in the LWR model.

We solve the Cauchy problem

(2.5)

∂tρi+∂x˜

f(ρi)=0,(t, x)∈R+×Ii,

ρi(0, x) = ρi,0(x)x∈Ii,1,2,...,2m.

on each Iiwhere ρi,0(x)is the initial density on the road of the roundabout. We

consider entropic solutions on each of the single roads of the roundabout. A weak

solution of the network and Riemann problem has been given in [12, 18] and refer-

ences therein as a solution in the sense of distributions with test functions, which are

smooth at the junctions. We use entropic solutions on each of the single roads of the

roundabout in the sense of the following deﬁnition.

Deﬁnition 2.2. Consider a roundabout as in Figure 1. Let Abe a distribution

matrix at a ﬁxed junction of the roundabout as given in [12]. A collection of function

ρ= (ρi)i=1,...,2m∈

2m

Y

i=1

C0R+;L1∩BV(Ii)

is an admissible solution to (2.5) if

1. ρisatisﬁes the Kružhkov entropy condition [20] on (R+×Ii),that is, for every

k∈Rand for all ϕ∈ C1

c(R×Ii), t > 0,

RR+RIi(|ρi−k|∂tϕ+ sgn (ρi−k)( ˜

f(ρi)−˜

f(k))∂xϕ)dxdt

+RIi|ρi,0−k|ϕ(0, x)dx ≥0; i= 1,2,...,2m.(2.6)

2. ˜

f(ρi(t, b−

i)) =

n

P

j=1

βj,i ˜

f(ρi(t, a+

i)), at each junction of the roundabout;

3. ˜

f(ρi(t, b−

i)) must be maximized subject to (1) and (2).

Here b−

idenotes left side of biin the interval whereas a+

iindicates the right side

of ai. Condition (1) is concerned with conservation of cars whereas condition (2)and

(3) correspond to the preferences of drivers and the maximization procedure. In the

deﬁnition, n= 1 or 2 depending on the junction type of the roundabout.

For future use we deﬁne the function τas follows, for further properties see [12].

Deﬁnition 2.3. Let τ: [0,1] →[0,1] be the map such that

•˜

f(τ(ρ)) = ˜

f(ρ)for every ρ∈[0,1];

•τ(ρ)6=ρfor every ρ∈[0,1] \ {ρc}.

322 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

Deﬁnition 2.4. A Riemann problem at a junction J is a Cauchy problem for constant

initial data on each road i.

Deﬁnition 2.5. A Riemann solver for junction J is a map RS : [0,1]2×[0,1] →

[0,1]2×[0,1] that associates the Riemann data ρ0= (ρ1,0, ρ2,0, ρ3,0)at J to a vector

ˆρ= (ˆρ1,ˆρ2,ˆρ3)such that the solution on the incoming road Ii, i = 1,2is given by

the wave (ρi,ˆρi)and on the outgoing road Ij, j = 3, the solution is given by the wave

(ˆρj, ρj). Consistency condition: RS(RS(ρ0)) = RS(ρ0).

For a road i∈Inc(J), the solution ρi(t, x)over its spacial domain x<biis given

by the solution to the following Riemann problem

∂tρi+∂x˜

f(ρi) = 0,(t, x)∈R+×Ii,

ρi(0, x) =

ρi,0if x < bi,

ˆρiif x≥bi.

The Riemann problem for outgoing road is deﬁned similarly except for ρi(0, x >

bi) = ρi,0and ρi(0, x ≤bi) = ˆρi. If ρ= (ρ1, . . . , ρn+m),ρi∈[0,+∞]×Iiis a weak

solution as in [12, 18] at the junction such that each x7→ ρi(t, x)has a bounded

variation, then precisely the conservation of cars through the junction J

(2.7)

n

X

i=1

˜

f(ρi(t, bi)) =

n+m

X

j=n+1

˜

f(ρi(t, aj))

holds where for a junction type 1×2,n= 1, m = 2 and for 2×1,n= 2, m = 1 in

the context of the roundabout.

Deﬁnition 2.5 does not guarantee the uniqueness of solution at each junction.

Hence, to ensure uniqueness of the solution we shall introduce a parameter q∈]0,1[

in the system. That is, assume that qis a priority parameter that deﬁnes the amount

of ﬂux that enters the outgoing main lane from each incoming road. In particular,

when the priority parameter is applied, q˜

f(ρ(t, b−)) represents the ﬂux allowed from

the incoming mainline into the outgoing main lane, and (1 −q)˜

f(ρ(t, b−)) represents

the ﬂux from the incoming secondary road of the roundabout.

For a junction with 1×2type, we consider traﬃc split ratio α∈(0,1) describing

the distribution of traﬃc among outgoing roads depending on the preference of drivers

at each junction J. We denote the Riemann initial data by ρi,0=ρi,0(bi)for incoming

arcs and ρi,0=ρi,0(ai)for outgoing arcs for a single junction. Assuming a unique

solution for the problem at the junction, we denote the solution at the junction, i.e.,

at x=bifor incoming and at x=aifor outgoing roads, by

(ˆρ1,ˆρ2,ˆρ3).

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE OF A ROUNDABOUT 323

Given the constant initial values ρi,0, we need to determine a unique solution ˆρi

satisfying the coupling condition in the context of Coclite, Garavello and Piccoli

(CGP) approach at junction. The possible values of ˆρiare necessarily as follows.

On the incoming road

(2.8) ˆρi∈

{ρi,0} ∪ (τ(ρi,0),1] if 0≤ρi,0< ρc

[ρc,1],if ρc≤ρi,0≤1

and on the outgoing road

(2.9) ˆρi∈

[0, ρc]if 0≤ρi,0≤ρc

{ρi,0} ∪ [0, τ (ρi,0)) if ρc≤ρi,0≤1.

The values of iare easily ﬁxed depending on the junction type in the context of the

present work.

Remark 2.6. On the incoming road:

•if ρi,0< ρc<ˆρi<1,˜

f(ρi,0)>˜

f(ˆρi), and ρc< ρi,0<1, the solution of the

Riemann problem consists of a shock wave with a negative speed. Moreover,

•if ρi,0< ρc<ˆρi<1and ˜

f(ρi,0) = ˜

f(ˆρi), the solution consists of contact wave.

On the outgoing road:

•if ρi,0< ρcthe solution of the Riemann problem consists of a shock wave with a

positive speed.

•if ˆρi< ρc< ρi,0<1, the solution of the Riemann problem consists of a shock

wave with positive speed and contact wave when ˜

f(ρi,0) = ˜

f(ˆρi).

For the following discussion we refer to [14, 16] for detail.

A. Coupling conditions for junction type 2×1:

We consider a junction with two incoming arcs and one outgoing arc. The initial

densities on each roads iare given by ρi,0with i= 1,2,3. The corresponding ﬂuxes

are denoted by γi,0=˜

f(ρi,0). Denote the maximum of the ﬂux by ˜

f(ρc). We denote

the sets of valid resulting ﬂuxes γiby Ωi. For the incoming roads i= 1,2this is

(2.10) ρi,0≤ρc⇒Ωi= [0, γi,0],

ρi,0≥ρc⇒Ωi= [0, f (ρc)].

For the outgoing road i= 3,

(2.11) ρi,0≤ρc⇒Ωi= [0,˜

f(ρc)],

ρi,0≥ρc⇒Ωi= [0, γi,0].

Moreover, we can deﬁne cisuch that

Ωi= [0, ci].

The ﬂuxes at the junction are found in the following way, distinguishing two cases:

324 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

(1) c1+c2≤c3:In this case, we have to look for γ1, γ2such that

max γ1+γ2w.r.t.

0≤γ1≤c1,0≤γ2≤c2, γ1+γ2≤c3.

The unique solution is found to be γ1=c1, γ2=c2, γ3=c1+c2.

(2) c1+c2≥c3:In this case, we have to look for γ1, γ2such that

max γ1+γ2w.r.t.

γ1=q

1−qγ2

0≤γ1≤c1,0≤γ2≤c2, γ1+γ2=c3.

where q∈(0,1) is the priority parameter introduced at the merging junction as given

in [12]. Figure 4 illustrates feasible set for the solution of the Riemann solver. For de-

tail theory regarding Riemann solver at junction we refer the reader to Section(5.2.2)

of [12]. The purpose of the priority parameters is to regulate the condition that nei-

ther impose insuﬃcient ﬂows nor send excess vehicles than the carrying capacity of

the main link of the roundabout.

γ1

γmax

1(ρi,0)

γ2

γmax

2(ρi,0)

γmax

3(ρi,0) = γ1+γ2

γ1=q

1−qγ2

Q

(a) Intersection inside Ωi

γ1

γmax

1(ρi,0)

γ2

γmax

2(ρi,0)

γmax

3(ρi,0) = γ1+γ2

γ1=q

1−qγ2

Q

S

γ1

γmax

1(ρi,0)

γ2

γmax

2(ρi,0)

γmax

3(ρi,0) = γ1+γ2

γ1=q

1−qγ2

Q

S

(b) Intersection outside Ωi

Figure 4. Solutions of the Riemann Solver at the junction.

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE OF A ROUNDABOUT 325

Since c1+c2> c3,ˆγ3= min(c1+c2, c3) = c3. For c2>(1 −q)c3and c1> qc3,

we set Qi

1= 1 −c2

c3and Qi

2=c1

c3such that Qi

2−Qi

1=c1

c3−(1 −c2

c3)>0. Under these

conditions, the unique solution at junction is found to be

•(ˆγ1,ˆγ2,ˆγ3) = (c3−c2, c2, c3) if q∈(0, Qi

1);

•(ˆγ1,ˆγ2,ˆγ3) = (qc3,(1 −q)c3, c3) if q∈[Qi

1, Qi

2];

•(ˆγ1,ˆγ2,ˆγ3) = (c1, c3−c1, c3) if q∈(Qi

2,1).

B. Coupling conditions for junction type 1×2:

We consider a junction with one incoming and two outgoing arcs. We use the same

notation as before; i.e., we deﬁne γi,0and the sets Ωidepending on whether incoming

or outgoing roads are considered. Using traﬃc distribution rates α2,1, α3,1∈(0,1)

with α2,1+α3,1= 1, then the CGP-conditions are

(1) γ1∈Ω1, αj,1γ1∈Ωjfor j = 2,3;

(2) Maximize γ1w.r.t. (1);

(3) γj=αj,1γ1, j = 2,3.

Using Ωi= [0, ci],i= 1,2,3, we obtain

γ1= min{c1,c2

α2,1

,c3

α3,1

}.

Remark 2.7. Condition (B) is exactly what is known as the FIFO (ﬁrst in, ﬁrst out)

rule of a dispersing junction in the traﬃc literature, see for example [14].

2.1. Analytical Study. In this subsection we give analysis for traﬃc evolution on

the roundabout network with and without the presence of pedestrians on the cross-

walk. The analysis is only limited to a roundabout having four incoming and four

outgoing roads for later numerical study and simpliﬁcation purpose.

2.1.1. In the Absence of Pedestrian. Let ˜

f= ( ˜

f1,˜

f2,˜

f3,˜

f4)be the traﬃc ﬂux on the

incoming roads towards the roundabout and f(ρc)be the maximal traﬃc ﬂux on the

road network. Assume that αf(ρc)of the traﬃc is ﬂowing out of the main roads

of the roundabout through their corresponding exiting arms while the remaining

(1 −α)f(ρc)proceed to ﬂow on the main road of the roundabout towards the next

junction. Suppose that q= (q1, q2, q3, q4)is the applied priority parameters vector at

the respective merging junctions of the roundabout where α= (αji)is the splitting

rate. In the case of supply limited situation,

(2.12) ˜

fi+ (1 −α)f(ρc)> f(ρc), i = 1,2,3,4.

On the other hand from presentation under condition (A) we know that (1−α)f(ρc)<

c1and ˜

fi≤c2. Then we have the following conditions

(a) qi∈(0, Qi

1)

326 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

(b) qi∈[Qi

1, Qi

2]

(c) qi∈(Qi

2,1),i= 1,2. . . , 4.

If qi/∈[Qi

1, Qi

2],i= 1,2. . . , 4, then either qi∈(0, Qi

1)or qi∈(Qi

2,1). If qi∈(0, Qi

1),

all the traﬃc on the incoming external roads of the roundabout enter the junction

while excess vehicles are waiting on the main link of the roundabout. In such a

situation, backward propagating shock waves are produced on the main link of the

roundabout while no wave occur on the incoming secondary road at each junction. On

the contrary, if qi∈(Qi

2,1) all the traﬃc on the main link of the roundabout enter their

corresponding junctions while queues are formed on the external incoming road of the

roundabout. In both situation, the priority rule is violated due to the limited traﬃc

demand entering the junction from either the main road of the roundabout or from the

incoming external road of the roundabout. However, if qi∈[Qi

1, Qi

2]the priority rule

is satisﬁed well due to suﬃcient demand from both main and the incoming secondary

roads of the roundabout at each junction. In this case, backward propagating shock

will be formed on the network and congestion get raised. Furthermore, some of the

junctions could be congested while the others stay demand limited. This fact is due

to the volume of inﬂow traﬃc on the incoming edges.

2.1.2. Pedestrian Involvement. This situation includes all the cases given under Sub-

section 2.1.1 in addition to the presence of pedestrian on the crosswalk. We assume

that the crosswalk is situated both on the incoming and outgoing roads of the round-

about without any traﬃc light.

Consider vehicular traﬃc and pedestrian ﬂow during peak hours. Assume for

short period of time the crosswalk on the exit arm of the roundabout is occupied by

pedestrians. Consequently, the original traﬃc ﬂux is altered and behaves as given

by equation 2.3. For the random duration t1≤t≤t2the ﬂux is equal to zero on

the crosswalk since the road is occupied by pedestrians. The interruption in ﬂux

function on the outgoing road results in a backward propagating shock wave. Thus,

depending on the amount of traﬃc volume and duration of the pedestrians staying

on the crosswalk the operational performance of the roundabout could be reduced.

On the contrary, when the pedestrians occupy the crosswalk on the entrance arm

of the roundabout in the random time interval t1≤t≤t2, queue would be formed on

the incoming edge behind the crosswalk while the roundabout operate with less traﬃc

compared to its carrying capacity. Further, when the probability of the pedestrians

to be on the crosswalk both at entrance and exit road of the roundabout equal to 1,

some of the circulatory road of the roundabout becomes congested due to backward

propagating shock wave being demand limited on the entrance side. Also, as a result of

priority parameter, drivers wait at the give-way lines for appropriate acceptable gaps

between vehicles already circulating on the roundabout. This could also contribute

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE OF A ROUNDABOUT 327

in producing a backward propagating shock wave. However, under low traﬃc ﬂow

any vehicle can proceed through the roundabout without delay.

From these scenarios, one can infer that traﬃc congestion plays a fundamental

role in the formation of delay because vehicles spend longer periods of time near

the roundabout while queuing, decelerating or accelerating due to the presence of

pedestrian on the crosswalk.

3. Numerical Approximation

In this section we consider the traﬃc regulation problem for a road network given

as in Figure 1. We analyze the impact of pedestrians’ motion on the traﬃc evolution

on the networks of a roundabout. In particular, we want to compare the performance

of a roundabout with and without the involvement of pedestrians during peak hours.

3.1. Network topology. The roundabout will be modeled by

•8 roads from the circle: I5,I6,I7,I8,I9,I10 ,I11,I12 coupled with CGP condition;

•8 roads connecting the roundabout with the rest of the network: 4 incoming

roads and 4 outgoing ones.

3.2. Numerical scheme. From the topology, it can be noted that all the junctions

in the roundabout can be represented as alternatively periodic sequences of 2×1

and 1×2-type for which it might be necessary to deﬁne respectively a right-of-way

parameter qand distribution rate α. The ﬁrst step is then to discretize the junction

model. We deﬁne a numerical grid in (0, T )×Rusing the following notation.

•∆xis the ﬁxed space grid size;

•∆tis the time step given by the CFL condition;

•(tn, xj)=(n∆t, j∆x)for n∈Nand j∈Zare the grid points.

Each road is divided in N + 1 cells numbered from 0 to N. The ﬁrst and last cell of

an edge are always a junction and we assume that these cells are ghost cells.

3.3. Godunov Scheme. The Godunov scheme as introduced in [19] is based on

exact solutions to Riemann problems. The main idea of this method is to approximate

the initial datum by a piecewise constant function, then the corresponding Riemann

problems are solved exactly and a global solution is simply obtained by piecing them

together. Finally, one takes the mean on the cell and proceeds by induction. Under

the CFL condition

(3.1) ∆tmax

j∈Z

λn

j+1

2

≤∆x,

328 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

the waves generated by diﬀerent Riemann problems do not interact. In the above

inequality, λn

j+1

2

is the wave speed of the Riemann problem solution at the interface

xj+1

2at time tn.Under the condition (3.1) the scheme can be written as

(3.2) ρn+1

i=ρn

i−∆t

∆x(FG(ρn

i, ρn

i+1)−FG(ρn

i−1, ρn

i)), i = 2,3,...,N−1∀n

where the numerical ﬂux FGtakes the following expression:

(3.3) FG(u, v) =

min(f(u), f (v)) if u≤v,

max(f(u), f (v)) if v < u < ρc∨ρc< v < u,

f(ρc)if v < ρ < u.

for concave ﬂux f. We introduce the following cost functionals that indicate the

total mass of vehicle on the road networks of the roundabout, average velocity, and

ﬂux respectively to analyze the performance of the roundabout with and without

pedestrian motion on the crosswalk.

(3.4)

J1(t) =

m

P

i=1 RIiρ(t, x)dx,

J2(t) =

m

P

i=1 RIiv(ρi(t, x)) dx,

J3(t) =

m

P

i=1 RIi

˜

f(ρi(t, x)) dx

For a ﬁxed time horizon [0, T ]our aim is to compare RT

0J1(t)dt,RT

0J2(t)dt, and

RT

0J3(t)dt for an appropriate ﬁxed distribution matrix and priority parameters.

3.3.1. Comparison of roundabout with and without pedestrian motion. We consider

approximation obtained by Godunov scheme with space step size ∆x= 0.333 and

the time step determined by the CFL condition. The traﬃc and pedestrian ﬂow on

the road network is simulated in a time interval [0, Tmax], where Tmax = 10. For the

initial condition on the roads of the network, we assume that at initial time t= 0

all the roads are empty and inﬂux at boundary of incoming edges is equal to 0.2.

In order to show the diﬀerent state of traﬃc evolution on the network, we assume

Crosswalk

J1

I1

I6I10

I7

I5

I9

I11

I8

I12

I13 I3

I15

J6

J4

J5

J8

I14

I4I16

I2

J2J3

J7

Figure 5. For comparison.

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE OF A ROUNDABOUT 329

that the crosswalk is marked orthogonally at the midpoint of incoming and outgoing

external roads. There is no crosswalk on the main roads forming the roundabout

under consideration. Further, we assume that the crosswalk on the incoming and

outgoing roads are occupied by pedestrians for short period of time step t=t2−t1,

see Figure 5. We now compare the results of the traﬃc ﬂow with and without the

pedestrians. The corresponding pictures on all the external incoming and outgoing

roads, roads forming the roundabout look the same and therefore we just compare

one from each of them. For the ﬁrst few time steps t < 62, the evolution of traﬃc

(a) Roundabout with pedestrians

on the crosswalk.

(b) Roundabout without pedestri-

ans on the crosswalk.

Figure 6. Traﬃc evolution on the incoming roads of the roundabout.

on the incoming roads behave similar in both cases due to the absence of pedestrians

on the crosswalk. As soon as the pedestrians interrupt ﬂow on the incoming roads

at cell position x= 15 at time step t= 62, the situation is immediately changed as

illustrated in Figure 6. Diﬀerent colors in the ﬁgure correspond to diﬀerent states of

traﬃc evolution over simulation period. The blue color corresponds to demand limited

case whereas the red color corresponds to congested state. The shock occurred due

to pedestrians motion on the crosswalk propagating back on the incoming road. The

part of the road between crosswalk and roundabout stay demand limited until the

pedestrians cleared on the road at time step t= 77. Then rarefaction wave ﬁll this

portion of the road. Due to priority at merging junctions of the roundabout, new

shock wave is produced on the incoming road. This shock wave moves back on the

incoming road as depicted in Figure 6a.

Traﬃc congestion can occur at merging junctions in the case of roundabout with-

out pedestrian involvement, see Figure 6b. Similar to the other case shock wave prop-

agating back on the incoming roads. Comparing these two states of the roundabout

one can easily observe the diﬀerence in the magnitude of traﬃc jam. Shock formed

330 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

due to priority at merging junction do not reach the inﬂux boundary in the absence

of pedestrian involvement. In Figure 7a, the blue color at about time step t= 100

(a) Roundabout with pedestrians

on the crosswalk.

(b) Roundabout without pedestri-

ans on the crosswalk.

Figure 7. Traﬃc evolution on the main road between merging and

diverging junctions of the roundabout.

reveals that the interrupted ﬂow reach the main road between merging and diverging

junctions of the roundabout. Rarefaction waves on the main road between merging

and diverging junctions of the roundabout increases the density to its critical density.

The evolution of traﬃc on this portion of the roundabout remains smooth.

(a) Roundabout with pedestrians

on the crosswalk.

(b) Roundabout without pedestri-

ans on the crosswalk.

Figure 8. Traﬃc evolution on the main road between diverging and

merging junctions of the roundabout.

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE OF A ROUNDABOUT 331

Traﬃc congestion appearing on the main road forming the roundabout at merg-

ing junctions. The shocks are moving back as it can be seen from Figure 8. Further,

the impact induced by pedestrians is not clearly reﬂected in this portion of the round-

about. This could be due to αf of the traﬃc exiting the roundabout through outgoing

roads and the inﬂuence of priority parameter value at merging junctions.

(a) Roundabout with pedestrians

on the crosswalk.

(b) Roundabout without pedestri-

ans on the crosswalk.

Figure 9. Traﬃc evolution on the outgoing roads of the roundabout.

Traﬃc jam arises on the outgoing edges when pedestrians are moving on the

crosswalk (compare Figure 9). The inﬂuence due to this jam on the traﬃc circulating

on the inner road of the roundabout is insigniﬁcant because of the short stay of pedes-

trians on the crosswalk. The outgoing secondary roads of the roundabout without

pedestrians interference remain demand limited.

3.3.2. Comparison. In this subsection we compute changes in total density, average

velocity and total ﬂux of the cost functional introduced in equation (3.4); that is, in

the case of traﬃc evolution on the roundabout without pedestrian and with pedestrian

involvement. More precisely, we consider ﬁxed distribution rate in both cases and

diﬀerent simulations cases which vary according to the values of the priority parameter

q∈ {0.2,0.3,0.4,0.5,0.6,0.7,0.8}. Then we compute separately the values of the cost

functional and take their respective diﬀerences for comparison.

From Table 1 one can easily infer that due to presence of pedestrian on the

crosswalk there are more vehicles waiting on the network. In the case of roundabout

without pedestrian involvement the role played by priority parameters are insigniﬁ-

cant in altering total density. This is due to the fact that congestion which propagate

backwards on the incoming roads do not reach the other end over the given time

interval. In the contrary, in the case of roundabout with pedestrian involvement the

priority parameter plays remarkable role in reﬂecting changes in the cost functionals.

332 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb

(a) In the absence

of pedestrians

q

T

P

t=0

J1

T

P

t=0

J2

T

P

t=0

J3

0.2 36.1013 119.0809 19.1204

0.3 36.1013 119.0809 19.5410

0.4 36.1013 119.0809 19.6240

0.5 36.1013 119.0809 19.4765

0.6 36.1013 119.0809 19.2031

0.7 36.0878 119.0944 18.988

0.8 36.0877 119.0945 18.9765

(b) In the presence

of pedestrians

q

T

P

t=0

J1P

T

P

t=0

J2P

T

P

t=0

J3P

0.2 43.6625 111.5197 19.0936

0.3 43.6625 111.5197 19.1127

0.4 43.3753 111.8065 18.9086

0.5 42.6935 112.4887 18.6102

0.6 42.3726 112.8096 18.3428

0.7 42.2879 112.8943 18.1639

0.8 42.2879 112.8943 18.1574

Table 1. Values of cost functional over time horizon. J1,J2and J3

respectively denotes cost functionals that measure total density, average

velocity and total ﬂux on the network.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

6.5

7

7.5

Priority parameter q

Change in the values of cost functional

(a) Change in total density

0.2 0.3 0.4 0.5 0.6 0.7 0.8

−7.5

−7

−6.5

Priority parameter q

Change in the values of cost functional

(b) Change in average velocity

0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.8

−0.6

−0.4

−0.2

0

Priority parameter q

Change in the values of cost functional

(c) Change in total Flux

Figure 10. Diﬀerence in the values of cost functional measuring total

density of vehicles, average velocity and total ﬂux on the network.

MODELLING PEDESTRIANS’ IMPACT ON THE PERFORMANCE OF A ROUNDABOUT 333

To minimize traﬃc congestion, we give more priority for cars circulating on main

road of the roundabout in both cases. Consequently, the change in the total density of

vehicles initially constant and then it starts decreasing, see Figure 10a. Furthermore,

for q≥0.7the change in the total mass of vehicles on the network becomes constant.

Similarly, the change in the average velocity of vehicles is initially constant and then

it starts increasing as depicted in Figure 10b. Figure 10c describes the change in the

total ﬂux due to priority parameters on the whole network.

Comparing these tables we can deduce that the interruption by pedestrian de-

creases the average velocity of vehicles on the network. Similarly, it reduces the

traﬃc ﬂux on the whole network. Thus, the simulation result indicates that the pres-

ence of pedestrian on the crosswalk inﬂuence the performance of the roundabout in

controlling traﬃc ﬂow problem.

4. Conclusions

In this article we studied the performance of the roundabout in regulating traﬃc

ﬂow problems in the presence and absence of pedestrians on the crosswalk located

at entrances and existing roads. The evolution of traﬃc ﬂow on the whole road net-

work of the roundabout is described by nonlinear scalar hyperbolic partial diﬀerential

equations. After we descritized the equations via the Godunov scheme we computed

the values of cost functionals which measure the total mass of vehicles on the road

networks of the roundabout, average velocity and total ﬂux for both cases. Then

we compared the values of cost functionals. The simulation result indicated that the

presence of pedestrian on the crosswalk reduce the performance of roundabout in

controlling traﬃc ﬂow problem. Optimization and validation of the model with real

data will be considered in the future work.

ACKNOWLEDGEMENTS

We kindly acknowledge Dr. Raul Borsche for his constructive comments dur-

ing this work. This research was supported by “German Academic Exchange Service

(DAAD)”. The ﬁrst author thanks Technical University of Kaisersalutern for its hos-

pitality.

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