# Analysis of Traffic Evolution on Road Networks of a Roundabout

Authors: Tolesa Hundesa Muleta; Legesse Lemecha Obsu
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

In this paper, the analyses of traffic evolution on the road network of a roundabout having three entrances and three exiting legs are conducted from macroscopic point of view. The road networks of roundabouts are modeled as a merging and diverging types1×2 and 2×1 junctions. To study traffic evolution at junction, two cases have been considered, namely, demand and supply limited cases. In each case, detailed mathematical analysis and numerical tests have been presented. The analysis in the case of demand limited showed that rarefaction wave fills the portion of the road network in time. In the contrary, in supply limited case, traffic congestion occurs at merging junctions and shock wave propagating back results in reducing the performance of a roundabout to control traffic dynamics. Also, we illustrate density and flux profiles versus space discretization at different time steps via numerical simulation with the help of Godunov scheme.


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## Body

## 1. Introduction

Macroscopic traffic flow model was first introduced in the 1950s due to the seminal work of Lighthill and Whitham [1] and independently by Richards [2]. The model describes the progression of traffic density by the means of macroscopic characteristic governed by scalar conservation law [3–9]. Such modeling approach is also known as LWR model in traffic literature and has been extended to road network with incorporating appropriate boundary condition at junction by several authors, see for example in [10–13] and references cited therein.Worldwide, traffic congestion has been a significant challenging problem related to transportation in most urban areas. This is mainly due to rapid increase of motorizations and limited carrying capacity of urban road networks. Congestion is worsen at road intersections during peak hours. Thus, congestion leads to delays in the movement of both goods and passengers [14, 15]. As a result, it affects the development of a national economy [16]. Real-time and precise prediction models are capable of analyzing traffic flow trends and characteristics on urban road network [17, 18].Roundabouts are special road network with a one-directional circulatory around a central island. It is another means of regulating traffic evolutions at road intersections rendering better safety relative to conventional road intersections [19]. A three entering and three exiting roundabout can be described as a chain of junctions with two incoming and one exiting roads and one incoming and two exiting roads [20]. The dynamics of traffic on the road network of a roundabout is assumed to be governed by nonlinear scalar hyperbolic conservation laws. Usually, vehicles on the main lane of the roundabout have priority over those entering a junction from external entering roads [21].The aim of this paper is to investigate traffic evolution on the road networks of a roundabout having three entering and three exiting roads by extending the work introduced in [22]. Then, we analyze the efficiency of a roundabout in regulating traffic dynamics under different priority parameters. We used Godunov scheme [23] for simulation purposed as detailed in [24].The content of this work is structured as follows. In Section2, we present a mathematical model describing traffic evolution on a road network of a roundabout having three entering and three exiting roads. In Section 3, we discuss Riemann Solver at junctions. In Section 4, we analyze the traffic evolution on the road networks of a roundabout both in demand and supply limited cases, respectively. The obtained analytical results are illustrated through numerical simulations in Section 5. Finally, we give conclusion in Section 6.


## 2. Mathematical Model

In this section, we describe road networks of a roundabout having three entering and three exiting roads as indicated in Figure1. As displayed in Figure 1, the road networks of roundabout can be seen as an oriented graph in which roads are represented by arcs and junctions by vertices.Figure 1
Schematic diagram representing a roundabout with 3 entering and 3 exiting roads.Each road segment forming main lane, entering, and exiting roads of a roundabout can be modeled by intervalsIi=ai,bi⊆ℝ, ai<bi, and Si=ci,di⊆ℝ, ci<di, i=1,⋯,6, where Ii denotes the finite collection of entering and exiting road segments and Si denotes the collection of roads on the main lane of the roundabout as illustrated in Figure 2(a). In this setting, we assume that ai tends to −∞ for all i=2n−1 and bi tends to ∞ for all i=2n, n=1,2,3. For convenience, we also assume that bi=ci, i=2n−1, and ai=di, i=2n,n=1,2,3. The types of merging and diverging junctions are presented in Figure 2(b).Figure 2
Road networks of a roundabout and its corresponding junction.
(a)
Road network under consideration(b)
Types of entering and exiting roads at each junctionThe dynamics of traffic on each road segment is governed by LWR model given as(1)∂ρi∂t+∂fiρi∂x=0,t,x∈ℝ+×Ii,t,x∈ℝ+×Si,∀i=1,2,⋯,6where ρi=ρit,x∈0,ρmax,i is the mean traffic density and ρmax,i is the maximal traffic density allowed on each road. The speed-density relationship is given by
(2)viρ=vmax,i1−ρiρmax,i,where vmax,i is the maximal speed on each road and vi:0,ρmax,i→ℝ+ is a smooth decreasing function denoting the mean traffic speed. The flux density relationship is defined by
(3)fiρi=ρivρi,orequivalently,fiρi=vmax,iρi1−ρiρmax,i.The flux is maximum atρc,i∈0,ρmax,i. In this paper, we use the normalized form of the vehicle density ρt,x to be 0≤ρ≤1, and we assume the following:
(1)
ρmax,i=1(2)
The speedvi depends only on the density ρi(3)
The fluxfi is a strictly concave C2 function(4)
fi0=fi1=0Assumptions (A3) and (A4) imply thatfi has a unique maximum point ρc,i∈0,1 for all i=1,2,3,⋯6..We briefly recall the following definition to be used in later sections.Definition 1 (see [14]).
Letτ:0,1→0,1 be the map such that
(i)
fτρ=fρ for every ρ∈0,1 and τρ≠ρ for every ρ∈0,1\ρc.Definition 2.
Consider a roundabout with 6 road segments on the main lane with 3 entrance and 3 exit links as given in Figure1(b). A collection of functions ρi,i=1,⋯,12∈∏i=112C0ℝ+;L1∩BVIi is an admissible solution to (1) as in [20] if
(1)
ρi is a weak solution on Ii, i.e., ρi:0,+∞×Ii→0,ρmax,i, such that(4)∫ℝ+∫Iiρi∂tφi+fρi∂xφidxdt=0,for every φi∈Cc1ℝ+×Ii,i=1,⋯,12.
(2)
ρi satisfies the Kruzhkov entropy condition [25] on ℝ×Ii, that is(5)∫ℝ+∫Ii∣ρi−k∣∂tφi+sgnρi−k.fρi−fk∂xφidxdt+∫Ii∣ρi,0−k∣φi0,xdx≥0.(3)
At each junction(6)∑i=1nfρit,b−=∑j=n+1n+mfρjt,aj+,where n,m ∈ {1,2} and i and j indicate all the links belonging to a junction. In particular, i indicates the incoming links, and m indicates the exiting ones. Also, bi − represents left side of bi while ai + denotes the right side of ai.


## 3. Riemann Problem at Junction

In this section, we briefly describe construction of Riemann solver at junctions.Definition 3.
A Riemann problem at a junctionJ is a Cauchy problem for constant initial data on each road. Consistency condition:
(7)RSRSρ0=RSρ0.
The detail analysis can be found in [11].Definition 4 (see [20].
A Riemann solver for junctionJ is a map RS:0,12×0,1→0,12×0,1 that associates the Riemann data ρ0 = (ρ1,0,ρ2,0,ρ3,0) at J to a vector ρ^=ρ^1,ρ^2,ρ^3 such that the solution on the incoming road Ii,i=1,2, is given by the wave ρi,ρ^i and on the exiting road Ij,j=3; the solution is given by the wave ρ^j,ρj.
For the entering road, the solutionρi(t,x) over its particular domain x < bi is given by the solution to the Riemann problem
(8)∂tρi+∂xfρi=0t,x∈ℝ+×Iiρi0,x=ρi,0ifx<bi,ρ^iifx≥bi..
Similarly, for the exiting road, the solutionρj(t,x) over its domain x > aj is given by the solution to the Riemann problem
(9)∂tρj+∂xfρj=0t,x∈ℝ+×Ij,ρj0,x=ρj,0ifx>aj,ρ^jifx≤aj.ρ10,.≡ρ1,0,ρ20,.≡ρ2,0,ρ30,.≡ρ3,0.
Moreover, there exists a unique3−tupleρ^1,ρ^2,ρ^3∈0,13 such that
(10)ρ^i∈ρi,0∪τρi,0,1,if0≤ρi,0≤ρc,ρc,1,ifρc≤ρi,0≤1.On the entering road for i =1,2, the solution is given by the wave ρi,0,ρ^i and
(11)ρ^j∈0,ρc,if0≤ρj,0≤ρc,ρj,0∪0,τρi,0,ifρc≤ρi,0≤1.On the exiting road for j =3, the solution is given by the wave ρ^3,ρ3,0.
On the incoming roadi=1,2:
(i)
Ifρi,0<ρc<ρ^i<1,fρi,0>fρ^i and ρc<ρi,0<1, then the solution of the Riemann problem consists of a shock wave with a negative speed(ii)
Ifρi,0<ρc<ρ^i<1,fρi,0=fρ^i, then the solution consists of contact wave(12)ρi,0≤ρc⇒Ωi=0,γi,0,γi,0=fρi,0,ρi,0≥ρc⇒Ωi=0,fρc.
On the exiting roadj=3:
(iii)
Ifρj,0<ρc, then the solution of the Riemann problem consists of a shock wave with a positive speed. If ρ^j<ρc<ρj,0<1, then the solution of the Riemann problem consists of a shock wave with a positive speed and contact wave when fρj,0=fρ^j.(13)ρj,0≤ρc⇒Ωj=0,fρc,γj,0=fρj,0,ρj,0≥ρc⇒Ωj=0,γj,0.


### 3.1. Conditions for Junction Type1×2

In this case, we apply traffic distribution ratioα ∈ (0,1) which express the distribution of traffic among exiting roads based on the preference of drivers at each junction J. The Riemann initial data is also denoted by ρi,0=ρi,0bi for entering roads and ρj,0=ρj,0aj for exiting roads. We represent the unique solution at junction, i.e., at x = bi for entering and x = ai for exiting roads, by ρ^1,ρ^2,andρ^3. Traffic flux on the entering road segment is defined by f(ρi,0) = γi,0 satisfying the following Coclite, Garavello, and Piccoli (CGP) conditions [11, 20] at junction for each distribution ratio α2,1,α3,1∈0,1 with α2,1+α3,1=1 and convex set Ωi:
(1)
γ1∈Ω1,αj,1γ1∈Ωjifj=2,3.(2)
Maximizeγ1 with respect to (1)(3)
γj=αj,1γ1,j=2,3.UsingΩi=0,ci,i=1,2,3, we obtain
(14)γ1=minc1,c2α2,1,c3α3,1.


### 3.2. Conditions for Junction Type2×1

In this case, traffic distribution matrix and conservation law alone cannot give unique solution at junction. For this reason, we introduce the priority parametersp ∈ (0,1) at junction type, 2×1, two entering and one exiting roads. We assume that all cars cannot enter the exiting road at the time. The purpose of priority parameters is to regulate the condition that neither impose insufficient flows nor send excess vehicles than the carrying capacity of the roads. Assume that N is the number vehicles that can enter the junction. Then, pN cars come from the first entering road and (1 − p)N cars from the second entering roads. Then, the solution of the Riemann problem (ρ1,0, ρ2,0, ρ3,0) is formed by a single wave on each road connecting the initial states to (ρˆ1, ρˆ2, ρˆ3) be determined in the following way. The aim is to maximize traffic flux satisfying:
(15)γ^3=minγ1maxρ1,0+γ2maxρ2,0,γ3maxρ3,0.Now letγ1=pN and γ2=1−pN be traffic flux on the exiting road such that
(16)γ1γ2=p1−p.Then, it follows that the intersection point of line (16) and γ1+γ2=γ^3 is occurred at P such that
(17)Ω=γ1,γ2:0≤γi≤γimaxρi,0≤γ1+γ2≤γ^3.In the case of Figure3(a), there is no intersection neither inside nor outside the feasible region. This is occurred in the demand limited case. On the contrary, Figure 3(b) shows intersection occurred inside the feasible region Ω which indicates supply limited. In the case of Figures 3(c) and 3(d), the intersection point is occurred outside Ω which indicates supply limited case. A similar analysis study on Riemann solver also discussed in [20].Figure 3
Cases that illustrates feasible set for solution of the Riemann solver at junctions.
(a)
No intersection(b)
Intersection inside(c)
Intersection outside and lowerside(d)
Intersection outside and uppersideNote that in Figure3, Q is the point of segment Ω ∩ {(γˆ1,γˆ2): γ1 + γ2 = γˆ3} closest to line in equation (16). We compute fluxes at the junction based on the following cases:
(1)
Ifc1+c2≤c3, we have to look for γ1 and γ2 such that(18)maxγ1+γ2,w.r.t:0≤γ1≤c1,0≤γ2≤c2,γ1+γ2≤c3.The unique solution is found to beγ1=c1, γ2=c2, and γ3=c1+c2.
(2)
Ifc1 + c2 ≥ c3, we have to look for γ1 and γ2 such that(19)maxγ1+γ2,w.r.t:γ1=p1−pγ2,0≤γ1≤c1,0≤γ2≤c2,γ1+γ2≤c3.


## 3.1. Conditions for Junction Type1×2

In this case, we apply traffic distribution ratioα ∈ (0,1) which express the distribution of traffic among exiting roads based on the preference of drivers at each junction J. The Riemann initial data is also denoted by ρi,0=ρi,0bi for entering roads and ρj,0=ρj,0aj for exiting roads. We represent the unique solution at junction, i.e., at x = bi for entering and x = ai for exiting roads, by ρ^1,ρ^2,andρ^3. Traffic flux on the entering road segment is defined by f(ρi,0) = γi,0 satisfying the following Coclite, Garavello, and Piccoli (CGP) conditions [11, 20] at junction for each distribution ratio α2,1,α3,1∈0,1 with α2,1+α3,1=1 and convex set Ωi:
(1)
γ1∈Ω1,αj,1γ1∈Ωjifj=2,3.(2)
Maximizeγ1 with respect to (1)(3)
γj=αj,1γ1,j=2,3.UsingΩi=0,ci,i=1,2,3, we obtain
(14)γ1=minc1,c2α2,1,c3α3,1.


## 3.2. Conditions for Junction Type2×1

In this case, traffic distribution matrix and conservation law alone cannot give unique solution at junction. For this reason, we introduce the priority parametersp ∈ (0,1) at junction type, 2×1, two entering and one exiting roads. We assume that all cars cannot enter the exiting road at the time. The purpose of priority parameters is to regulate the condition that neither impose insufficient flows nor send excess vehicles than the carrying capacity of the roads. Assume that N is the number vehicles that can enter the junction. Then, pN cars come from the first entering road and (1 − p)N cars from the second entering roads. Then, the solution of the Riemann problem (ρ1,0, ρ2,0, ρ3,0) is formed by a single wave on each road connecting the initial states to (ρˆ1, ρˆ2, ρˆ3) be determined in the following way. The aim is to maximize traffic flux satisfying:
(15)γ^3=minγ1maxρ1,0+γ2maxρ2,0,γ3maxρ3,0.Now letγ1=pN and γ2=1−pN be traffic flux on the exiting road such that
(16)γ1γ2=p1−p.Then, it follows that the intersection point of line (16) and γ1+γ2=γ^3 is occurred at P such that
(17)Ω=γ1,γ2:0≤γi≤γimaxρi,0≤γ1+γ2≤γ^3.In the case of Figure3(a), there is no intersection neither inside nor outside the feasible region. This is occurred in the demand limited case. On the contrary, Figure 3(b) shows intersection occurred inside the feasible region Ω which indicates supply limited. In the case of Figures 3(c) and 3(d), the intersection point is occurred outside Ω which indicates supply limited case. A similar analysis study on Riemann solver also discussed in [20].Figure 3
Cases that illustrates feasible set for solution of the Riemann solver at junctions.
(a)
No intersection(b)
Intersection inside(c)
Intersection outside and lowerside(d)
Intersection outside and uppersideNote that in Figure3, Q is the point of segment Ω ∩ {(γˆ1,γˆ2): γ1 + γ2 = γˆ3} closest to line in equation (16). We compute fluxes at the junction based on the following cases:
(1)
Ifc1+c2≤c3, we have to look for γ1 and γ2 such that(18)maxγ1+γ2,w.r.t:0≤γ1≤c1,0≤γ2≤c2,γ1+γ2≤c3.The unique solution is found to beγ1=c1, γ2=c2, and γ3=c1+c2.
(2)
Ifc1 + c2 ≥ c3, we have to look for γ1 and γ2 such that(19)maxγ1+γ2,w.r.t:γ1=p1−pγ2,0≤γ1≤c1,0≤γ2≤c2,γ1+γ2≤c3.


## 4. Roundabout with Demand and Supply Limited

To study the nature of traffic evolution on the roundabout, we focus separately in the case of light traffic and heavy traffic.Assumption 5.
To avoid unnecessary complexity, we make the following assumptions.(1)
Complete turn or U-turn on the roundabout is not allowed(2)
There is no obstacle at the exit road of the roundabout(3)
Initially, the network of the roundabout is empty
The model proposed in [22] is applied to a traffic circle with two entering and two exiting roads. The authors computed asymptotic fluxes and then compared the result with traffic light. Analyzing traffic dynamics in the case of traffic circle with two entering and two exiting is very simple compared to a roundabout with three entering and three exiting roads. Further, the priority parameter is applied only at two junctions which is easily computed. In the case of a roundabout with three entering and three exiting roads, computing shock wave on the roundabout is very complicated. So, in this section, we will analyze shock wave on the roundabout via demand and supply limited cases.


### 4.1. Demand Limited Case

In this case, we assume that traffic arriving at junctions of a roundabout as illustrated in Figure1 is less, in the sense that the number of cars reaching the roundabout is less than the optimal carrying capacity of a roundabout. Thus, the junction stays demand limited, and we do not need priority rule in this situation. The traffic evolution is only governed by conservation law. However, we assign traffic distribution matrix to describe how traffic coming from the entering roads Si,i=2n−1,n=1,2,3, choose to distribute to their corresponding intermediate roads and external exiting roads. Hence, we have the following assumptions [11].Assumption 6.
We assume three fixed parameterαi ∈ (0,1) for i =1,2,3 such that
(a)
IfN cars reach the roundabout from I1, then α1N drive to road I4 and 1−α1N drive to road I5(b)
IfN cars reach the roundabout from I2, then α1N drive to road I5 and 1−α1N drive to road I6(c)
IfN cars reach the roundabout from I3, then α1N drive to road I6 and 1−α1N drive to road I4
Letρ¯1,ρ¯2, and ρ¯3 be constant densities on the external entering roads Ii,i=1,2,3, respectively. In the demand limited case, this situation is equivalent to equilibrium state. In this setting, we define periodic boundary condition at junction as
(20)ρ¯it,ai≡ρ¯ii=1,2,3.
Further, letf denotes traffic flux on the roads network of a roundabout. Then, the constant fluxes on each road do not exceed the maximum flux fρc, that is, fρ¯1+1−α3fρ¯3<fρc,fρ¯2+1−α1fρ¯1<fρcandfρ¯3+1−α2fρ¯2<fρc. If αifρ¯i of the flux leaves the roundabout at junction Ji, the remaining 1−αifρ¯i will continue on the links of a roundabout until reaching the next exiting road as depicted in Figure 4.Theorem 7.
Consider a roundabout with three entering and three exiting roads as displayed in Figure4. Then, there exists time dependent coefficient αi:0,∞→0,1,i=1,2,3. Provided that the boundary condition hold at t=0 and the solution ρ(t) is constantly equal to that of Figure 4 for t ≥ T >0.Proof.
Since by its nature, the traffic behavior is time dependent, and thus, it is common to assume thatαi varies with time. This means, we move forward in time to construct the solution of Riemann problem at each junction and joined them together. By assumption, the traffic evolution is light on the roundabout. This implies that the cars from roads I1, I2, and I3 reach either of the existing roads I4, I5, and I6 through rarefaction wave. At each time step, rarefaction waves reach either of the junction J1, J3, or J5. We regulate the corresponding traffic distribution coefficient in such a way that cars reach the correct road. Further, the existence of such coefficient is also depends on driver’s preference. Thus, the crossing coefficients at equilibrium situations are
(i)
(S1,I4,S2)(21)αS1,4=α1fρ¯1+1−α3fρ¯3fρ¯1+1−α3fρ¯3,αS1,S2=1−α1fρ¯1fρ¯1+1−α3fρ¯3,(ii)
(S3,I5,S4)(22)αS3,5=α2fρ¯2+1−α1fρ¯1fρ¯2+1−α1fρ¯1,αS3,S4=1−α2fρ¯2fρ¯2+1−α1fρ¯1,(iii)
(S5,I6,S6)(23)αS5,6=α3fρ¯3+1−α2fρ¯2fρ¯3+1−α2fρ¯2,αS5,S6=1−α3fρ¯3fρ¯3+1−α2fρ¯2.
A car started fromI1 reach I4 and I5. Then, we set αS1,4 = α1 and αS1,S2 = (1 − α1). A car started from I2 reach I5 and I6. Then, we set αS3,5 = α2 and αS3,S4 = (1 − α2). A car started from I3 reach I6 and I4. Then, we put αS5,6 = α3 and αS5,S6 = (1 − α3). Under the given assumption, we modify in time the distribution coefficient. With such choice, one can find a time T>0 so that the solution is given by the flux indicated in Figure 4 hold for each t ≥ T. This completes the proof.Figure 4
Free-body diagram sketch for traffic at equilibrium situation.


### 4.2. Supply Limited Case

In this case, we have possible traffic jams if the constant fluxes on each road in Figure4 do exceed the maximum flux f(ρc), such that f(¯ρ1) + (1 − α3)f(¯ρ3) > f(ρc), f(¯ρ2) + (1 − α1)f(¯ρ1) > f(ρc), and f(¯ρ3) + (1 − α2)f(¯ρ2) > f(ρc) conditions hold. Instantaneously, shocks are produced on road networks of the roundabout at junctions J1, J3, and J5.Now suppose that 0< pi< 1,i =1,2,3 be the priority parameter applied at merging junctions J1, J3, and J5. Assume further that
(24)γ1max=f1,γ2max=f2,γ3max=f3,γS1max=fρ¯1+1−α3fρ¯3=fρc,γS3max=f2ρ2+1−α1f1=fρcandγS5max=f2ρ2+1−α1f1=fρc.For simplicity, we takea: = (1 − α1), b: = (1 − α2), c: = (1 − α3), and fi: = f(¯ρi), i =1,2,3 to be used in the following theorem. We summarized junction wave interaction in Table 1.Table 1
Shock wave at merging junctions.
DerivationCase 1Case 2Case 3J1f1+cf3>fρc⇒f1fρc>1−cf3fρc.p1≤1−cf3fρc,γ^1=fρc−cf3,γ^S6=cf3.1−cf3fρc<p1<f1fρc,γ^1=p1fρc,γ^S6=1−p1fρc.p1≥f1fρc,γ^1=f1,γ^S6=fρc−f1.Interpretation→Shock produced onI1 not on S6Shock produced onI1 and S6Shock produced onS6 not on I1J3f2+af1>fρc⇒f2fρc>1−af1fρc.p2≤1−af1fρc,γ^2=fρc−af1,γ^S2=af1.1−af1fρc<p2<f2fρc,γ^2=p2fρc,γ^S2=1−p2fρc.p2≥f2fρc,γ^2=f2,γ^S2=fρc−f2.Interpretation→Shock produced onI2 not on S2Shock produced onI2 and S2Shock produced onS2 not on I2J5f3+bf3>fρc⇒f3fρc>1−bf2fρc.p3≤1−bf2fρc,γ^3=fρc−bf2,γ^S4=bf2.1−bf2fρc<p3<f3fρc,γ^3=p3fρc,γ^S4=1−p3fρc.p3≥f3fρc,γ^3=f3,γ^S4=fρc−f3.Interpretation→Shock produced onI3 not on S4Shock produced onI3 and S4Shock produced onS4 not on I3Assume that we are in case 1 as given in Table1 at junctions J1, J3, and J5. Then, rarefaction waves are produced on roads S2, S4, and S6. These rarefaction waves decrease the traffic density at their respective junctions, and thus, the road network of the roundabout is not congested. Thus, we have proved the follow result.Theorem 8.
Suppose thatpi∈0,1,i=1,2,3 be priority parameters at J1, J3andJ5 of the roundabout, respectively, such thatp1≤1−cf3/fρc,p2≤1−af1/fρcandp3≤1−bf2/fρc at these junctions, then traffic flux on the road networks of a roundabout never jammed.Next, we give detailed analysis at each junction when the situation is fully supply limited.This corresponds to case 2 of Table1 at junctions J1, J3, and J5. Since a shock is produced on all entering roads at each junctions, then rarefaction are produced on roads S2, S4, and S6. We present this situation case by case as follows.


#### 4.2.1. Case (i)

The flux on roadS1 composed of p1f(ρc) from road I1 and (1 − p1)f(ρc) from road S6. Since α1 of the flux from road S1 exists through roads I4, the part of the flux on S1 is α1p1f(ρc) + (1 − p1)f(ρc) = (1 − ap1)f(ρc) that exits to road I4. All flux from road S6 leave the system through road I4. The remaining flux ap1f(ρc) will proceed to road S2. In terms of traffic distribution matrix, this is equivalent to saying that αS21 = ap1. Since the shock wave produced on the entering road propagates back with negative speed, then the shock produced at junction J3 reaches J2. Thus, we have a shock on S1. We displayed wave junction interaction in Figure 5.Figure 5
Wave junction intersections on the network of a roundabout.The values of traffic flux on roadS1 just after the shock junction interaction are equal to
(25)αS21x1n=1−p2x2n−1⇒x1n=1−p2ap1x2n−1,where x2n−1 represents the value of the flux on road S2 after the shock junction intersection. Also, after interaction, the traffic flux on I2 is x2n=p2x3n−1. All the flux from road S2 leave the network of the roundabout through I5.


#### 4.2.2. Case (ii)

The flux onS3 is composed of p2f(ρc) from road I2 and (1 − p2) from road S2. Since α2 of the flux exit the system of the roundabout through I5, the flux that leave the network of the roundabout through I5 equal to α2p2f(ρc) + (1 − p2)f(ρc) = (1 − bp2)f(ρc). The remaining flux (1 − α2)p2f(ρc) will go to S4. This is equivalent to saying that αS43 = (1 − α2)p2. As a consequence of backward propagation of shock on the entering road, the shock produced at junction J5 reaches junction J4; then, we have a shock on road S4. On the other hand, the flux on road S5 composed of p3f(ρc) from road I3 and (1 − p3)f(ρc) from road S4. Therefore, the value of traffic flux on road segment S3 just after the shock junction interaction becomes:
(26)αS43x3n=1−p3x4n−1⇒x3n=1−p3bp2x4n−1.Again, just after shock junction interaction, the traffic flux on the entering roadI3 is given by x9n=p3x5n−1. Furthermore, the flux on road S5 contains p3fρc from the entering road I3 and 1−p3fρc from road S4. Thus, the flux on S4 can be expressed as x4n=1−p3x5n−1.


#### 4.2.3. Case (iii)

Similarly, the traffic flux on roadS5 composed of p3f(ρc) from road I3 and (1 − p3)f(ρc) from main road S4. But α3 of the traffic flux from the entering road I3 leaves the roundabout through I6 which amounts to α3p3f(ρc) + (1 − p3)f(ρc) = (1 − cp3)f(ρc). The remaining traffic flux cp3f(ρc) moves to road S6. In terms of traffic distribution matrix, this amounts to αS56 = (1 − α3)p3, with similar analysis as previous cases, the value of the traffic on road S5 just after wave junction interactions
(27)αS65x5n=1−p1x6n−1⇒x5n=1−p1cp3x6n−1.The traffic flux on roadS6 equals to x6n=1−p1x1n−1 after shock junction interaction. Similarly, on the entering road I1, we have x7n=p1x1n−1. Thus, the shocks are produced on the whole roads of the roundabout recursively and compactly in matrix form
(28)A¯=1−p2ap10001−p3bp20001−p1cp3.Since the new shock induced on the roadsS1, S3, and S5 of the roundabout, then we focus only on the matrix formed by x1n, x3n, and x5n denoted by A¯. The eigenvalue of the matrix A¯ are
(29)λ1=1−p2ap1,λ2=1−p3bp2,λ3=1−p1cp3andλ1λ2λ3=1−p11−p21−p3abcp1p2p3.Moreover, at junctionsJ2, J4, and J6, respectively, we can observe that
(30)1−p2fρc<af1,1−p3fρc<bf2and1−p1fρc<cf3.Furthermore, from(31)1−cf3fρc<p1<f1fρc⇒p1fρc<f1,1−af1fρc<p2<f2fρc⇒p2fρc<f2,1−bf2fρc<p3<f3fρc⇒p3fρc<f3.From equations (30) and (31), we obtain
(32)1−p2fρc<af1,p1fρc<f1,1−p3fρc<bf2,p2fρc<f2,1−p1fρc<cf3,p3fρc<f3.Simplifying, we get(33)p11−p2<1a,p21−p3<1b,andp31−p1<1c.Since all are positive quantities, multiplication does not influence the inequalities. That is,(34)p1p2p31−p11−p21−p3<1abc⇒abcp1p2p31−p11−p21−p3<1.Equivalently,(35)1−p11−p21−p3abcp1p2p3>1.From traffic point of view,p1fρc<f1, p2fρc<f2, and p3fρc<f3 suggest that flow is congested but not jammed. That means, γ1max=f1, γ2max=f2, and γ3max=f3, as illustrated in case 2 of J1, J3, and J5. Even though excess traffic flux on the entering roads, the roundabout is not stuck. Thus, we proved the following result.Proposition 9.
Assume that
cf3/fρc<p1<f1/fρc,1−af1/fρc<p2<f2/fρc and 1−bf2/fρc<p3<f3/fρc are satisfied for α1,α2,α3 and for all the right of way parameters p1,p2,p3. If
(36)1−p11−p21−p3abcp1p2p3>1,then the traffic flow on the roundabout does not stuck.Proposition 10.
If(37)f1=p1fρc,f2=p2fρc,f3=p3fρc,and1−p11−p21−p3abcp1p2p3=1,then the traffic flow on the roundabout is not jammed.Proof.
From case (iii) we know thatcf3=fρc−f1, since f1=p1fρc. Then, cf3=fρc−p1fρc=1−p1fρc. On the other hand, traffic flux on S1 equals f1+cf3=fρc⇔p1fρc+1−p1fρc=fρc, similar for other cases. This implies that all the traffic flux are entered the junction smoothly. Thus, it attains maximum flow.
Finally, we consider case 3 given Table1. If the strict inequality hold at all junctions J1I1,S6,S1, J3I2,S2,S3, and J5I3,S4,S5, then rarefactions are produced on the roads S1, S3, and S5 while shocks are produced on roads S2, S4, and S6.Proposition 11.
Ifpi>fi/fρc,i=1,2,3,, then the traffic flux on the roundabout is completely jammed.Proof.
Following the arrangement given under case 2, we get the following inequalities:(38)1−p2fρc<af1,p1fρc>f1,1−p3fρc<bf2,p2fρc>f2,1−p1fρc<cf3,p3fρc>f3.
This is the same as saying(39)1−p2fρc<ap1fρc,1−p3fρc<bp2fρc,and1−p1fρc<cp3fρc.
From this, we conclude that(40)1−p11−p21−p3<abcp1p2p3⇒1−p11−p21−p3abcp1p2p3<1.


## 4.1. Demand Limited Case

In this case, we assume that traffic arriving at junctions of a roundabout as illustrated in Figure1 is less, in the sense that the number of cars reaching the roundabout is less than the optimal carrying capacity of a roundabout. Thus, the junction stays demand limited, and we do not need priority rule in this situation. The traffic evolution is only governed by conservation law. However, we assign traffic distribution matrix to describe how traffic coming from the entering roads Si,i=2n−1,n=1,2,3, choose to distribute to their corresponding intermediate roads and external exiting roads. Hence, we have the following assumptions [11].Assumption 6.
We assume three fixed parameterαi ∈ (0,1) for i =1,2,3 such that
(a)
IfN cars reach the roundabout from I1, then α1N drive to road I4 and 1−α1N drive to road I5(b)
IfN cars reach the roundabout from I2, then α1N drive to road I5 and 1−α1N drive to road I6(c)
IfN cars reach the roundabout from I3, then α1N drive to road I6 and 1−α1N drive to road I4
Letρ¯1,ρ¯2, and ρ¯3 be constant densities on the external entering roads Ii,i=1,2,3, respectively. In the demand limited case, this situation is equivalent to equilibrium state. In this setting, we define periodic boundary condition at junction as
(20)ρ¯it,ai≡ρ¯ii=1,2,3.
Further, letf denotes traffic flux on the roads network of a roundabout. Then, the constant fluxes on each road do not exceed the maximum flux fρc, that is, fρ¯1+1−α3fρ¯3<fρc,fρ¯2+1−α1fρ¯1<fρcandfρ¯3+1−α2fρ¯2<fρc. If αifρ¯i of the flux leaves the roundabout at junction Ji, the remaining 1−αifρ¯i will continue on the links of a roundabout until reaching the next exiting road as depicted in Figure 4.Theorem 7.
Consider a roundabout with three entering and three exiting roads as displayed in Figure4. Then, there exists time dependent coefficient αi:0,∞→0,1,i=1,2,3. Provided that the boundary condition hold at t=0 and the solution ρ(t) is constantly equal to that of Figure 4 for t ≥ T >0.Proof.
Since by its nature, the traffic behavior is time dependent, and thus, it is common to assume thatαi varies with time. This means, we move forward in time to construct the solution of Riemann problem at each junction and joined them together. By assumption, the traffic evolution is light on the roundabout. This implies that the cars from roads I1, I2, and I3 reach either of the existing roads I4, I5, and I6 through rarefaction wave. At each time step, rarefaction waves reach either of the junction J1, J3, or J5. We regulate the corresponding traffic distribution coefficient in such a way that cars reach the correct road. Further, the existence of such coefficient is also depends on driver’s preference. Thus, the crossing coefficients at equilibrium situations are
(i)
(S1,I4,S2)(21)αS1,4=α1fρ¯1+1−α3fρ¯3fρ¯1+1−α3fρ¯3,αS1,S2=1−α1fρ¯1fρ¯1+1−α3fρ¯3,(ii)
(S3,I5,S4)(22)αS3,5=α2fρ¯2+1−α1fρ¯1fρ¯2+1−α1fρ¯1,αS3,S4=1−α2fρ¯2fρ¯2+1−α1fρ¯1,(iii)
(S5,I6,S6)(23)αS5,6=α3fρ¯3+1−α2fρ¯2fρ¯3+1−α2fρ¯2,αS5,S6=1−α3fρ¯3fρ¯3+1−α2fρ¯2.
A car started fromI1 reach I4 and I5. Then, we set αS1,4 = α1 and αS1,S2 = (1 − α1). A car started from I2 reach I5 and I6. Then, we set αS3,5 = α2 and αS3,S4 = (1 − α2). A car started from I3 reach I6 and I4. Then, we put αS5,6 = α3 and αS5,S6 = (1 − α3). Under the given assumption, we modify in time the distribution coefficient. With such choice, one can find a time T>0 so that the solution is given by the flux indicated in Figure 4 hold for each t ≥ T. This completes the proof.Figure 4
Free-body diagram sketch for traffic at equilibrium situation.


## 4.2. Supply Limited Case

In this case, we have possible traffic jams if the constant fluxes on each road in Figure4 do exceed the maximum flux f(ρc), such that f(¯ρ1) + (1 − α3)f(¯ρ3) > f(ρc), f(¯ρ2) + (1 − α1)f(¯ρ1) > f(ρc), and f(¯ρ3) + (1 − α2)f(¯ρ2) > f(ρc) conditions hold. Instantaneously, shocks are produced on road networks of the roundabout at junctions J1, J3, and J5.Now suppose that 0< pi< 1,i =1,2,3 be the priority parameter applied at merging junctions J1, J3, and J5. Assume further that
(24)γ1max=f1,γ2max=f2,γ3max=f3,γS1max=fρ¯1+1−α3fρ¯3=fρc,γS3max=f2ρ2+1−α1f1=fρcandγS5max=f2ρ2+1−α1f1=fρc.For simplicity, we takea: = (1 − α1), b: = (1 − α2), c: = (1 − α3), and fi: = f(¯ρi), i =1,2,3 to be used in the following theorem. We summarized junction wave interaction in Table 1.Table 1
Shock wave at merging junctions.
DerivationCase 1Case 2Case 3J1f1+cf3>fρc⇒f1fρc>1−cf3fρc.p1≤1−cf3fρc,γ^1=fρc−cf3,γ^S6=cf3.1−cf3fρc<p1<f1fρc,γ^1=p1fρc,γ^S6=1−p1fρc.p1≥f1fρc,γ^1=f1,γ^S6=fρc−f1.Interpretation→Shock produced onI1 not on S6Shock produced onI1 and S6Shock produced onS6 not on I1J3f2+af1>fρc⇒f2fρc>1−af1fρc.p2≤1−af1fρc,γ^2=fρc−af1,γ^S2=af1.1−af1fρc<p2<f2fρc,γ^2=p2fρc,γ^S2=1−p2fρc.p2≥f2fρc,γ^2=f2,γ^S2=fρc−f2.Interpretation→Shock produced onI2 not on S2Shock produced onI2 and S2Shock produced onS2 not on I2J5f3+bf3>fρc⇒f3fρc>1−bf2fρc.p3≤1−bf2fρc,γ^3=fρc−bf2,γ^S4=bf2.1−bf2fρc<p3<f3fρc,γ^3=p3fρc,γ^S4=1−p3fρc.p3≥f3fρc,γ^3=f3,γ^S4=fρc−f3.Interpretation→Shock produced onI3 not on S4Shock produced onI3 and S4Shock produced onS4 not on I3Assume that we are in case 1 as given in Table1 at junctions J1, J3, and J5. Then, rarefaction waves are produced on roads S2, S4, and S6. These rarefaction waves decrease the traffic density at their respective junctions, and thus, the road network of the roundabout is not congested. Thus, we have proved the follow result.Theorem 8.
Suppose thatpi∈0,1,i=1,2,3 be priority parameters at J1, J3andJ5 of the roundabout, respectively, such thatp1≤1−cf3/fρc,p2≤1−af1/fρcandp3≤1−bf2/fρc at these junctions, then traffic flux on the road networks of a roundabout never jammed.Next, we give detailed analysis at each junction when the situation is fully supply limited.This corresponds to case 2 of Table1 at junctions J1, J3, and J5. Since a shock is produced on all entering roads at each junctions, then rarefaction are produced on roads S2, S4, and S6. We present this situation case by case as follows.


### 4.2.1. Case (i)

The flux on roadS1 composed of p1f(ρc) from road I1 and (1 − p1)f(ρc) from road S6. Since α1 of the flux from road S1 exists through roads I4, the part of the flux on S1 is α1p1f(ρc) + (1 − p1)f(ρc) = (1 − ap1)f(ρc) that exits to road I4. All flux from road S6 leave the system through road I4. The remaining flux ap1f(ρc) will proceed to road S2. In terms of traffic distribution matrix, this is equivalent to saying that αS21 = ap1. Since the shock wave produced on the entering road propagates back with negative speed, then the shock produced at junction J3 reaches J2. Thus, we have a shock on S1. We displayed wave junction interaction in Figure 5.Figure 5
Wave junction intersections on the network of a roundabout.The values of traffic flux on roadS1 just after the shock junction interaction are equal to
(25)αS21x1n=1−p2x2n−1⇒x1n=1−p2ap1x2n−1,where x2n−1 represents the value of the flux on road S2 after the shock junction intersection. Also, after interaction, the traffic flux on I2 is x2n=p2x3n−1. All the flux from road S2 leave the network of the roundabout through I5.


### 4.2.2. Case (ii)

The flux onS3 is composed of p2f(ρc) from road I2 and (1 − p2) from road S2. Since α2 of the flux exit the system of the roundabout through I5, the flux that leave the network of the roundabout through I5 equal to α2p2f(ρc) + (1 − p2)f(ρc) = (1 − bp2)f(ρc). The remaining flux (1 − α2)p2f(ρc) will go to S4. This is equivalent to saying that αS43 = (1 − α2)p2. As a consequence of backward propagation of shock on the entering road, the shock produced at junction J5 reaches junction J4; then, we have a shock on road S4. On the other hand, the flux on road S5 composed of p3f(ρc) from road I3 and (1 − p3)f(ρc) from road S4. Therefore, the value of traffic flux on road segment S3 just after the shock junction interaction becomes:
(26)αS43x3n=1−p3x4n−1⇒x3n=1−p3bp2x4n−1.Again, just after shock junction interaction, the traffic flux on the entering roadI3 is given by x9n=p3x5n−1. Furthermore, the flux on road S5 contains p3fρc from the entering road I3 and 1−p3fρc from road S4. Thus, the flux on S4 can be expressed as x4n=1−p3x5n−1.


### 4.2.3. Case (iii)

Similarly, the traffic flux on roadS5 composed of p3f(ρc) from road I3 and (1 − p3)f(ρc) from main road S4. But α3 of the traffic flux from the entering road I3 leaves the roundabout through I6 which amounts to α3p3f(ρc) + (1 − p3)f(ρc) = (1 − cp3)f(ρc). The remaining traffic flux cp3f(ρc) moves to road S6. In terms of traffic distribution matrix, this amounts to αS56 = (1 − α3)p3, with similar analysis as previous cases, the value of the traffic on road S5 just after wave junction interactions
(27)αS65x5n=1−p1x6n−1⇒x5n=1−p1cp3x6n−1.The traffic flux on roadS6 equals to x6n=1−p1x1n−1 after shock junction interaction. Similarly, on the entering road I1, we have x7n=p1x1n−1. Thus, the shocks are produced on the whole roads of the roundabout recursively and compactly in matrix form
(28)A¯=1−p2ap10001−p3bp20001−p1cp3.Since the new shock induced on the roadsS1, S3, and S5 of the roundabout, then we focus only on the matrix formed by x1n, x3n, and x5n denoted by A¯. The eigenvalue of the matrix A¯ are
(29)λ1=1−p2ap1,λ2=1−p3bp2,λ3=1−p1cp3andλ1λ2λ3=1−p11−p21−p3abcp1p2p3.Moreover, at junctionsJ2, J4, and J6, respectively, we can observe that
(30)1−p2fρc<af1,1−p3fρc<bf2and1−p1fρc<cf3.Furthermore, from(31)1−cf3fρc<p1<f1fρc⇒p1fρc<f1,1−af1fρc<p2<f2fρc⇒p2fρc<f2,1−bf2fρc<p3<f3fρc⇒p3fρc<f3.From equations (30) and (31), we obtain
(32)1−p2fρc<af1,p1fρc<f1,1−p3fρc<bf2,p2fρc<f2,1−p1fρc<cf3,p3fρc<f3.Simplifying, we get(33)p11−p2<1a,p21−p3<1b,andp31−p1<1c.Since all are positive quantities, multiplication does not influence the inequalities. That is,(34)p1p2p31−p11−p21−p3<1abc⇒abcp1p2p31−p11−p21−p3<1.Equivalently,(35)1−p11−p21−p3abcp1p2p3>1.From traffic point of view,p1fρc<f1, p2fρc<f2, and p3fρc<f3 suggest that flow is congested but not jammed. That means, γ1max=f1, γ2max=f2, and γ3max=f3, as illustrated in case 2 of J1, J3, and J5. Even though excess traffic flux on the entering roads, the roundabout is not stuck. Thus, we proved the following result.Proposition 9.
Assume that
cf3/fρc<p1<f1/fρc,1−af1/fρc<p2<f2/fρc and 1−bf2/fρc<p3<f3/fρc are satisfied for α1,α2,α3 and for all the right of way parameters p1,p2,p3. If
(36)1−p11−p21−p3abcp1p2p3>1,then the traffic flow on the roundabout does not stuck.Proposition 10.
If(37)f1=p1fρc,f2=p2fρc,f3=p3fρc,and1−p11−p21−p3abcp1p2p3=1,then the traffic flow on the roundabout is not jammed.Proof.
From case (iii) we know thatcf3=fρc−f1, since f1=p1fρc. Then, cf3=fρc−p1fρc=1−p1fρc. On the other hand, traffic flux on S1 equals f1+cf3=fρc⇔p1fρc+1−p1fρc=fρc, similar for other cases. This implies that all the traffic flux are entered the junction smoothly. Thus, it attains maximum flow.
Finally, we consider case 3 given Table1. If the strict inequality hold at all junctions J1I1,S6,S1, J3I2,S2,S3, and J5I3,S4,S5, then rarefactions are produced on the roads S1, S3, and S5 while shocks are produced on roads S2, S4, and S6.Proposition 11.
Ifpi>fi/fρc,i=1,2,3,, then the traffic flux on the roundabout is completely jammed.Proof.
Following the arrangement given under case 2, we get the following inequalities:(38)1−p2fρc<af1,p1fρc>f1,1−p3fρc<bf2,p2fρc>f2,1−p1fρc<cf3,p3fρc>f3.
This is the same as saying(39)1−p2fρc<ap1fρc,1−p3fρc<bp2fρc,and1−p1fρc<cp3fρc.
From this, we conclude that(40)1−p11−p21−p3<abcp1p2p3⇒1−p11−p21−p3abcp1p2p3<1.


## 4.2.1. Case (i)

The flux on roadS1 composed of p1f(ρc) from road I1 and (1 − p1)f(ρc) from road S6. Since α1 of the flux from road S1 exists through roads I4, the part of the flux on S1 is α1p1f(ρc) + (1 − p1)f(ρc) = (1 − ap1)f(ρc) that exits to road I4. All flux from road S6 leave the system through road I4. The remaining flux ap1f(ρc) will proceed to road S2. In terms of traffic distribution matrix, this is equivalent to saying that αS21 = ap1. Since the shock wave produced on the entering road propagates back with negative speed, then the shock produced at junction J3 reaches J2. Thus, we have a shock on S1. We displayed wave junction interaction in Figure 5.Figure 5
Wave junction intersections on the network of a roundabout.The values of traffic flux on roadS1 just after the shock junction interaction are equal to
(25)αS21x1n=1−p2x2n−1⇒x1n=1−p2ap1x2n−1,where x2n−1 represents the value of the flux on road S2 after the shock junction intersection. Also, after interaction, the traffic flux on I2 is x2n=p2x3n−1. All the flux from road S2 leave the network of the roundabout through I5.


## 4.2.2. Case (ii)

The flux onS3 is composed of p2f(ρc) from road I2 and (1 − p2) from road S2. Since α2 of the flux exit the system of the roundabout through I5, the flux that leave the network of the roundabout through I5 equal to α2p2f(ρc) + (1 − p2)f(ρc) = (1 − bp2)f(ρc). The remaining flux (1 − α2)p2f(ρc) will go to S4. This is equivalent to saying that αS43 = (1 − α2)p2. As a consequence of backward propagation of shock on the entering road, the shock produced at junction J5 reaches junction J4; then, we have a shock on road S4. On the other hand, the flux on road S5 composed of p3f(ρc) from road I3 and (1 − p3)f(ρc) from road S4. Therefore, the value of traffic flux on road segment S3 just after the shock junction interaction becomes:
(26)αS43x3n=1−p3x4n−1⇒x3n=1−p3bp2x4n−1.Again, just after shock junction interaction, the traffic flux on the entering roadI3 is given by x9n=p3x5n−1. Furthermore, the flux on road S5 contains p3fρc from the entering road I3 and 1−p3fρc from road S4. Thus, the flux on S4 can be expressed as x4n=1−p3x5n−1.


## 4.2.3. Case (iii)

Similarly, the traffic flux on roadS5 composed of p3f(ρc) from road I3 and (1 − p3)f(ρc) from main road S4. But α3 of the traffic flux from the entering road I3 leaves the roundabout through I6 which amounts to α3p3f(ρc) + (1 − p3)f(ρc) = (1 − cp3)f(ρc). The remaining traffic flux cp3f(ρc) moves to road S6. In terms of traffic distribution matrix, this amounts to αS56 = (1 − α3)p3, with similar analysis as previous cases, the value of the traffic on road S5 just after wave junction interactions
(27)αS65x5n=1−p1x6n−1⇒x5n=1−p1cp3x6n−1.The traffic flux on roadS6 equals to x6n=1−p1x1n−1 after shock junction interaction. Similarly, on the entering road I1, we have x7n=p1x1n−1. Thus, the shocks are produced on the whole roads of the roundabout recursively and compactly in matrix form
(28)A¯=1−p2ap10001−p3bp20001−p1cp3.Since the new shock induced on the roadsS1, S3, and S5 of the roundabout, then we focus only on the matrix formed by x1n, x3n, and x5n denoted by A¯. The eigenvalue of the matrix A¯ are
(29)λ1=1−p2ap1,λ2=1−p3bp2,λ3=1−p1cp3andλ1λ2λ3=1−p11−p21−p3abcp1p2p3.Moreover, at junctionsJ2, J4, and J6, respectively, we can observe that
(30)1−p2fρc<af1,1−p3fρc<bf2and1−p1fρc<cf3.Furthermore, from(31)1−cf3fρc<p1<f1fρc⇒p1fρc<f1,1−af1fρc<p2<f2fρc⇒p2fρc<f2,1−bf2fρc<p3<f3fρc⇒p3fρc<f3.From equations (30) and (31), we obtain
(32)1−p2fρc<af1,p1fρc<f1,1−p3fρc<bf2,p2fρc<f2,1−p1fρc<cf3,p3fρc<f3.Simplifying, we get(33)p11−p2<1a,p21−p3<1b,andp31−p1<1c.Since all are positive quantities, multiplication does not influence the inequalities. That is,(34)p1p2p31−p11−p21−p3<1abc⇒abcp1p2p31−p11−p21−p3<1.Equivalently,(35)1−p11−p21−p3abcp1p2p3>1.From traffic point of view,p1fρc<f1, p2fρc<f2, and p3fρc<f3 suggest that flow is congested but not jammed. That means, γ1max=f1, γ2max=f2, and γ3max=f3, as illustrated in case 2 of J1, J3, and J5. Even though excess traffic flux on the entering roads, the roundabout is not stuck. Thus, we proved the following result.Proposition 9.
Assume that
cf3/fρc<p1<f1/fρc,1−af1/fρc<p2<f2/fρc and 1−bf2/fρc<p3<f3/fρc are satisfied for α1,α2,α3 and for all the right of way parameters p1,p2,p3. If
(36)1−p11−p21−p3abcp1p2p3>1,then the traffic flow on the roundabout does not stuck.Proposition 10.
If(37)f1=p1fρc,f2=p2fρc,f3=p3fρc,and1−p11−p21−p3abcp1p2p3=1,then the traffic flow on the roundabout is not jammed.Proof.
From case (iii) we know thatcf3=fρc−f1, since f1=p1fρc. Then, cf3=fρc−p1fρc=1−p1fρc. On the other hand, traffic flux on S1 equals f1+cf3=fρc⇔p1fρc+1−p1fρc=fρc, similar for other cases. This implies that all the traffic flux are entered the junction smoothly. Thus, it attains maximum flow.
Finally, we consider case 3 given Table1. If the strict inequality hold at all junctions J1I1,S6,S1, J3I2,S2,S3, and J5I3,S4,S5, then rarefactions are produced on the roads S1, S3, and S5 while shocks are produced on roads S2, S4, and S6.Proposition 11.
Ifpi>fi/fρc,i=1,2,3,, then the traffic flux on the roundabout is completely jammed.Proof.
Following the arrangement given under case 2, we get the following inequalities:(38)1−p2fρc<af1,p1fρc>f1,1−p3fρc<bf2,p2fρc>f2,1−p1fρc<cf3,p3fρc>f3.
This is the same as saying(39)1−p2fρc<ap1fρc,1−p3fρc<bp2fρc,and1−p1fρc<cp3fρc.
From this, we conclude that(40)1−p11−p21−p3<abcp1p2p3⇒1−p11−p21−p3abcp1p2p3<1.


## 5. Numerical Simulations

In this section, we present numerical scheme used to solve problem given in (1). We define a numerical grid in 0,T×ℝ using the following notation: Δx is the fixed grid space, Δt is the time step given by the CFL condition and tn,xj=nΔt,jΔx for n∈ℕ, and j∈ℤ are the grid points. Each road is divided into N+1 cells numbered from 0 to N. The first and last cells of an edge are always a junction, and we assume that these cells are ghost cells. The scheme used for solving equation (1) is the Godunov scheme as introduced in [20, 23] and it is based on exact solutions to the Riemann problem. 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 obtained by piecing them together. Under the CFL (Courant-Friedrichs-Lewy) condition, it holds:
(41)Δtmaxj∈ℤ∣λj+12n∣≤Δx,where λj+1/2n is the speed of the wave of the Riemann problem solution at the interface xj+1/2 at the time tn, the numerical scheme can be written as
(42)ρjn+1=ρjn−ΔtΔxFρjn,ρj+1n−Fρj−1n,ρjn,where the numerical flux F for a concave flux function is given by
(43)Fu,v=minfu,vifu≤v,fuifv<u<ρc,fmaxifv<ρc<u,fvifρc<v<u.


### 5.1. Boundary Conditions

Each road is divided inN+1 cells numbered from 0 to N. For the entering roads, we proceed by setting:
(44)ρ0n+1=ρ0n−ΔtΔxFρ0n,ρ0n−Fρ0n,ρ1n,while for the exiting ones, we set
(45)ρNn+1=ρNn−ΔtΔxFρN−1n,ρNn−FρNn,ρNn.


### 5.2. Conditions at the Junctions

From the entering roads which are connected at the junction at the right endpoint, we set(46)ρNn+1=ρNn−ΔtΔxγ^in−FρN−1n,ρNn,while for the exiting ones, connected at the junctions at the left point, we set
(47)ρ0n+1=ρ0n−ΔtΔxFρ0n,ρ1n−γ^out.


### 5.3. Simulation Results

For simulation purpose, we chose a concave fundamental diagram as introduced in equation (3) with the following values for parameters: vmax=1;ρmax=1;L=50;ρc=0.5;T=20, Δx=0.0196, Δt=0.0196. We assume that at initial time t=0 all the roads are empty and influx at boundary of entering roads are equal to 0.9. In the case of demand limited, the traffic evolution is governed by conservation law and the splitting rate to describe how traffic coming from the entering roads choose to distributed to their corresponding intermediate roads and the external exiting roads. Thus, in this case, rarefaction wave fill the portion of the roads of the roundabout. But, in the supply limited case, the traffic congestion can occur at merging junctions and shock wave propagating back, which is the one we simulated in the results.The results obtained are the evolution of density and flux on entrance roads, on exit roads, roads between merging and diverging junctions, and roads between diverging and merging junctions versus space discretization at different time steps as shown in Figures6–8. Figure 9 displays the evolution of traffic density profiles versus priority parameters. In these figures, different colors show different traffic scenarios over simulation time. The blue color corresponds to demand limited case while the red color denotes congested state with value indicated on the color bars. In Figure 6, we illustrate the simulation result corresponds to case 1 at merging junctions J1,J3, and J5.Figure 6
Density and flux profiles on the entrance and main lane of a roundabout in case 1. In case (a), shock waves are produced on roadsI1,I2,andI3, and rarefaction waves are created on roads S2, S4, and S6 as shown in (c) with corresponding fluxes displayed in (b) and (d), respectively.
(a)
Density on the entrances road(b)
Flux on the entrances road(c)
Density between the diverging and merging main lane of the roundabout(d)
Flux between the diverging and merging main lane of the roundaboutFigure 7
Density and flux profiles on the entrance and main lane of a roundabout in case 2. Due to traffic congestion, shock waves are produced on roads (I1,I2,I3) in the case of (a) and on (S2,S4,S6) in the case of (c) with corresponding fluxes as shown in (b) and (d), respectively.
(a)
Density on the entrances road in case 2(b)
Flux on the entrances road in case 2(c)
Density between the diverging and merging main lane of the roundabout in case 2(d)
Flux between the diverging and merging main lane of the roundabout in case 2Figure 8
Density and flux profiles between the merging and diverging main lane of the roundabout under consideration and on the existing roads. In the case (a), shocks are produced on roadsS1,S3,andS5. Further, rarefaction waves are produced on roads I4,I5,andI6 as shown in (c). Corresponding traffic fluxes are shown in (b) and (d), respectively.
(a)
Density between the merging and diverging main lane of the roundabout(b)
Flux between the merging and diverging main lane of the roundabout(c)
Density on the exits road(d)
Flux on the exits roadFigure 9
Evolution of density profiles on the entrance road and main lane of a roundabout versus priority parameters for time simulationT between 2≤T≤6.
(a)
Density profile on the entrance road of a roundabout when priority is given to entering traffic(b)
Density profile on the entrance road of a roundabout when priority is given to traffic on the main lane of a roundabout(c)
Density profile on the main lane of a roundabout when priority is given to entering traffic(d)
Density profile on the main lane of a roundabout when priority is given to traffic on the main lane itselfSimilar to case 1, we performed numerical simulation for case 3. In this case, shock waves are produced on the roadsS2,S4,andS6 propagating back while rarefaction waves on I1,I2,andI3 propagating forward in time. In the next, we describe the simulation results for case 2. In this case, the roundabout under consideration is fully supply limited case as shown in Figures 7 and 8. The shock wave occurred on the entrance roads (I1,I2,I3), and between diverging and merging junctions of roads (S2,S4,S6) is depicted in Figure 7. As a result, backward propagating shock wave produced between merging and diverging junctions, that is, on the roads (S1,S3,S5) as displayed in Figure 8(a).Figure9 shows the evolution of traffic profiles on the entrance road and main roads of a roundabout versus priority parameters for simulation time T between 2≤T≤6. The impact of priority parameters on the external entrance roads is shown in Figure 9(a), and on the main lanes of the roundabout are also displayed in Figure 9(b). Figures 9(c) and 9(d) present density profile on the main lane of a roundabout when priority is given to entering traffic and when priority is also given to traffic on the main lane itself, respectively.


## 5.1. Boundary Conditions

Each road is divided inN+1 cells numbered from 0 to N. For the entering roads, we proceed by setting:
(44)ρ0n+1=ρ0n−ΔtΔxFρ0n,ρ0n−Fρ0n,ρ1n,while for the exiting ones, we set
(45)ρNn+1=ρNn−ΔtΔxFρN−1n,ρNn−FρNn,ρNn.


## 5.2. Conditions at the Junctions

From the entering roads which are connected at the junction at the right endpoint, we set(46)ρNn+1=ρNn−ΔtΔxγ^in−FρN−1n,ρNn,while for the exiting ones, connected at the junctions at the left point, we set
(47)ρ0n+1=ρ0n−ΔtΔxFρ0n,ρ1n−γ^out.


## 5.3. Simulation Results

For simulation purpose, we chose a concave fundamental diagram as introduced in equation (3) with the following values for parameters: vmax=1;ρmax=1;L=50;ρc=0.5;T=20, Δx=0.0196, Δt=0.0196. We assume that at initial time t=0 all the roads are empty and influx at boundary of entering roads are equal to 0.9. In the case of demand limited, the traffic evolution is governed by conservation law and the splitting rate to describe how traffic coming from the entering roads choose to distributed to their corresponding intermediate roads and the external exiting roads. Thus, in this case, rarefaction wave fill the portion of the roads of the roundabout. But, in the supply limited case, the traffic congestion can occur at merging junctions and shock wave propagating back, which is the one we simulated in the results.The results obtained are the evolution of density and flux on entrance roads, on exit roads, roads between merging and diverging junctions, and roads between diverging and merging junctions versus space discretization at different time steps as shown in Figures6–8. Figure 9 displays the evolution of traffic density profiles versus priority parameters. In these figures, different colors show different traffic scenarios over simulation time. The blue color corresponds to demand limited case while the red color denotes congested state with value indicated on the color bars. In Figure 6, we illustrate the simulation result corresponds to case 1 at merging junctions J1,J3, and J5.Figure 6
Density and flux profiles on the entrance and main lane of a roundabout in case 1. In case (a), shock waves are produced on roadsI1,I2,andI3, and rarefaction waves are created on roads S2, S4, and S6 as shown in (c) with corresponding fluxes displayed in (b) and (d), respectively.
(a)
Density on the entrances road(b)
Flux on the entrances road(c)
Density between the diverging and merging main lane of the roundabout(d)
Flux between the diverging and merging main lane of the roundaboutFigure 7
Density and flux profiles on the entrance and main lane of a roundabout in case 2. Due to traffic congestion, shock waves are produced on roads (I1,I2,I3) in the case of (a) and on (S2,S4,S6) in the case of (c) with corresponding fluxes as shown in (b) and (d), respectively.
(a)
Density on the entrances road in case 2(b)
Flux on the entrances road in case 2(c)
Density between the diverging and merging main lane of the roundabout in case 2(d)
Flux between the diverging and merging main lane of the roundabout in case 2Figure 8
Density and flux profiles between the merging and diverging main lane of the roundabout under consideration and on the existing roads. In the case (a), shocks are produced on roadsS1,S3,andS5. Further, rarefaction waves are produced on roads I4,I5,andI6 as shown in (c). Corresponding traffic fluxes are shown in (b) and (d), respectively.
(a)
Density between the merging and diverging main lane of the roundabout(b)
Flux between the merging and diverging main lane of the roundabout(c)
Density on the exits road(d)
Flux on the exits roadFigure 9
Evolution of density profiles on the entrance road and main lane of a roundabout versus priority parameters for time simulationT between 2≤T≤6.
(a)
Density profile on the entrance road of a roundabout when priority is given to entering traffic(b)
Density profile on the entrance road of a roundabout when priority is given to traffic on the main lane of a roundabout(c)
Density profile on the main lane of a roundabout when priority is given to entering traffic(d)
Density profile on the main lane of a roundabout when priority is given to traffic on the main lane itselfSimilar to case 1, we performed numerical simulation for case 3. In this case, shock waves are produced on the roadsS2,S4,andS6 propagating back while rarefaction waves on I1,I2,andI3 propagating forward in time. In the next, we describe the simulation results for case 2. In this case, the roundabout under consideration is fully supply limited case as shown in Figures 7 and 8. The shock wave occurred on the entrance roads (I1,I2,I3), and between diverging and merging junctions of roads (S2,S4,S6) is depicted in Figure 7. As a result, backward propagating shock wave produced between merging and diverging junctions, that is, on the roads (S1,S3,S5) as displayed in Figure 8(a).Figure9 shows the evolution of traffic profiles on the entrance road and main roads of a roundabout versus priority parameters for simulation time T between 2≤T≤6. The impact of priority parameters on the external entrance roads is shown in Figure 9(a), and on the main lanes of the roundabout are also displayed in Figure 9(b). Figures 9(c) and 9(d) present density profile on the main lane of a roundabout when priority is given to entering traffic and when priority is also given to traffic on the main lane itself, respectively.


## 6. Conclusion

In this paper, we presented the analysis of traffic evolution on road network of roundabout with three entering and three exiting roads. The flow of traffic on the whole road network of the roundabout is governed by nonlinear scalar hyperbolic conservation laws or the LWR model. We used traffic distribution matrix at diverging and priority parameters at merging junctions. Also, we investigated the performance of a roundabout in regulating traffic evolution both in the case of demand and supply limited cases with details mathematical analysis. In demand limited case, the number of cars reaching the roundabout is less than the optimal capacity of the roundabout. Thus, the junction stays demand limited, and we did not need priority rule in that situation. The traffic evolution is only governed by hyperbolic conservation laws. However, we assigned traffic distribution ratio to express how traffic coming from the entering roads choose to distributed to their corresponding intermediate roads and the external exiting roads. Since the number of cars reaching the roundabout was greater than the optimal capacity of the roundabout in supply limited case, we have been used priority rule at merging junctions. We solved the problem using the Godunov scheme. Finally, we presented results of simulations of traffic evolution on road network in terms of density and flux versus space discretization at different time steps. The simulation result indicated that rarefaction waves fill the portion of the roads of the roundabout in demand limited case, and the traffic congestion occurs at merging junctions and shock wave propagating back in supply limited case (congested state) which reduced the efficiency of a roundabout in controlling traffic flow problem. The experiment with field study and validation of the model with real data will be considered in the future work.



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*Source: 3959613-2020-10-05.xml*