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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 x 1 and 1 x 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.
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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, mN,m2,
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,
mN,m2, 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
J2m2
J4
J5
J2m
J2J3
J2m1
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 0ρρmax = 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=t2t1is 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)xIi,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+;L1BV(Ii)
is an admissible solution to (2.5) if
1. ρisatisfies the Kružhkov entropy condition [20] on (R+×Ii),that is, for every
kRand for all ϕ∈ C1
c(R×Ii), t > 0,
RR+RIi(|ρik|tϕ+ sgn (ρik)( ˜
f(ρi)˜
f(k))xϕ)dxdt
+RIi|ρi,0k|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 iInc(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 xbi.
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 0ρi,0< ρc
[ρc,1],if ρcρi,01
and on the outgoing road
(2.9) ˆρi
[0, ρc]if 0ρi,0ρc
{ρi,0} ∪ [0, τ (ρi,0)) if ρcρi,01.
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 i. For the incoming roads i= 1,2this is
(2.10) ρi,0ρci= [0, γi,0],
ρi,0ρci= [0, f (ρc)].
For the outgoing road i= 3,
(2.11) ρi,0ρci= [0,˜
f(ρc)],
ρi,0ρci= [0, γi,0].
Moreover, we can define cisuch that
i= [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+c2c3:In this case, we have to look for γ1, γ2such that
max γ1+γ2w.r.t.
0γ1c1,0γ2c2, γ1+γ2c3.
The unique solution is found to be γ1=c1, γ2=c2, γ3=c1+c2.
(2) c1+c2c3:In this case, we have to look for γ1, γ2such that
max γ1+γ2w.r.t.
γ1=q
1qγ2
0γ1c1,0γ2c2, 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
1qγ2
Q
(a) Intersection inside i
γ1
γmax
1(ρi,0)
γ2
γmax
2(ρi,0)
γmax
3(ρi,0) = γ1+γ2
γ1=q
1qγ2
Q
S
γ1
γmax
1(ρi,0)
γ2
γmax
2(ρi,0)
γmax
3(ρi,0) = γ1+γ2
γ1=q
1qγ2
Q
S
(b) Intersection outside i
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
2Qi
1=c1
c3(1 c2
c3)>0. Under these
conditions, the unique solution at junction is found to be
(ˆγ1,ˆγ2,ˆγ3) = (c3c2, 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, c3c1, 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 idepending 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) γ11, αj,1γ1jfor j = 2,3;
(2) Maximize γ1w.r.t. (1);
(3) γj=αj,1γ1, j = 2,3.
Using i= [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 (1α)f(ρc)<
c1and ˜
fic2. 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 t1tt2the 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 t1tt2, 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)=(nt, jx)for nNand jZare 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
jZ
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
it
x(FG(ρn
i, ρn
i+1)FG(ρn
i1, ρn
i)), i = 2,3,...,N1n
where the numerical flux FGtakes the following expression:
(3.3) FG(u, v) =
min(f(u), f (v)) if uv,
max(f(u), f (v)) if v < u < ρcρc< 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=t2t1,
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 q0.7the 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.
REFERENCES
[1] A. AW and M. Rascle, Resurrection of second order models of traffic flows, SIAM J. Appl. Math,
60: 916–938, 2000.
[2] F. Berthelin, P. Degond, Delitla and M. Rascle, A model for the formation and evolution of
traffic jams, Arch. Rational Mech. Anal, 187:185-220, 2008.
[3] R. Borsche, A. Klar, S. Kühn and A. Meurer, Coupling traffic flow networks to pedestrian motion,
Mathematical Models and Methods in Applied Sciences, 24(2):359-380, 2014.
334 LEGESSE L. OBSUa, ANNE MEURERb, SEMU M. KASSAc, AND AXEL KLARb
[4] R. Borsche, and A. Meurer, Interaction of road networks and pedestrian motion at crosswalks,
Discrete Contin. Dyn. Syst, 7(3): 363-377, 2014.
[5] , A. Bressan, Hyperbolic systems of conservation laws: The one dimentional Cauchy Problem,
Oxford Lecture Series in Mathematics and Its Application, New York, 2000.
[6] A. Cascone, C. D’Apice, B. Piccoli and L. Raritá, Optimization of traffic on road net-
works,Mathematical Models and Methods in Applied Sciences, 17:1587-1617, 2007.
[7] , Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete and
Continuous Dynamical Systems Series B, 5: 599–630, 2005.
[8] G.M. Coclite, M. Garavello, and B. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal.,
36 (6): 1862-1886, 2005.
[9] R. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Meth. Appl. Sci,
28: 1553-1567, 2005.
[10] R. M. Colombo, P. Goatin, and B. Piccoli, Road network with phase transition, Journal of
Hyperbolic Differential Equations 7:85-106, 2010.
[11] Y. Feng, Y. Liu, P. Deo, and H. J. Ruskin, Heterogeneous Traffic Flow Model For Two Lane
Roundabouts and Controlled Intersection, International Journal of Modern Physics C, 1: 107–
117, 2007.
[12] M. Garavello and B. Piccoli, Traffic Flow on Networks: Conservation Laws Model,American
Institute of Mathematical Sciences Vol. 1 , USA, 2006.
[13] M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn.
Syst., 32: 1915–1938, 2012.
[14] S. Göttlich, A. Klar and P. Schindler, Discontinuous conservation laws for production networks
with finite buffer, SIAM J. Appl. Math, 73: 1117–1138, 2013.
[15] D. Helbing, Traffic and related self-driven many particle systems,Rev. Modern phys., 73:1067–
1141, 2001.
[16] M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM
Journal on Scientific Computing, 25:1066–1087, 2003.
[17] M. Herty, J. P. Lebacque, and S. Moutari, A Novel Model For intersections of Vehicular Traffic
Flow,Networks and Heterogeneous Media, American Institute of Mathematical Sciences,4:813-
826, 2009.
[18] H. Holden and N. H. Nisebro, A mathematical model of traffic flow on a network of unidirectional
roads, SIAM J. Math. Anal., 26:997–1017,1995.
[19] S. K. Godunov,A finite difference method for the numerical computation of discontinuous so-
lutions of the equations of fluid dynamics,Matematicheskii Sbornik, 47: 271–290, 1959.
[20] S. N. Kružhkov, First order quasilinear equations with several independent variables,Mat. Sb.
(N.S.), 81 (123): 228–255, 1970.
[21] 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 of London. Series A. Mathematical and
Physical Sciences, 229:317-345, 1955.
[22] Legesse L. Obsu, M. L. Delle Monache, P. Goatin and Semu M. Kassa, Traffic flow optimization
on roundabouts, Journal of Mathematical Methods and Applied Sciences, 38:3075–3096, 2015.
[23] R. I. Richards, Shock waves on the highway, Operations research, 4:42–51, 1956.
[24] R. L. Hughes, A continuum theory for the flow of pedestrians,Trans. Res. Part B: Methodolog-
ical, 36:507–535, 2002.
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