Sage Journals: Discover world-class research

Abstract

A metering roundabout where traffic is controlled by signals where phase times are influenced by queue detector occupancy may be the solution to reduce queue lengths under unbalanced traffic flows. In the past decades, a number of studies have attempted to evaluate the effectiveness of metering roundabout, especially on the dominant approach. Little studies, however, have been directed on prediction of the queuing lengths, which is essential to determine the detector locations. This paper introduces a queue length estimation model using adaptive neuro fuzzy inference system for unbalanced roundabout traffic flows. The adaptive neuro fuzzy inference system model consists of an input layer representing four parameters as arrival volumes, conflicting volumes, phase green and red time, and output layer with four neuron representing queuing length. MATLAB software and additional statistical tests were used as the tool to develop the models for the data. In order to conduct credible model validations, model output data were compared against the observed data collected using drones. The results from the analysis demonstrated that adaptive neuro fuzzy inference system model is able to estimate the queuing length at metering roundabouts. Thus, it is expected that the adaptive neuro fuzzy inference system model will help practitioners in determining optimal detector locations and will be a foundation research for roundabouts with signals.

Keywords

Metering roundabout queuing length adaptive neuro fuzzy inference system drone data

Introduction

In regard to a metering roundabout, the determination of the detector location is very important, because queuing lengths on each approach can be affected by when the signal should be actuated and the signal’s green/red time, which is determined by when the queue hits the detector on the controlling approach.^1,2

With respect to the metering roundabout, a number of studies have been conducted; however, majority of the studies concentrated on performance evaluations, particularly the analysis of the dominant approach. Carter and Pratt³ introduce a technical review of all applications and aspects of vehicles automatic detection system. According to past research, unbalanced traffic flow conditions can exacerbate the roundabout operational performance. In such cases, the metering roundabout can be a solution leading to a reduction in vehicle delay times. In addition, many studies suggested that vehicle movements on each approach operate individually as a T-junction.⁴ Krogscheepers and Roebuck⁵ present a definition with regard to unbalanced conditions with the precondition that entering vehicles from a specific approach may influence the vehicle movements on other approaches.

There has been relatively few research investigating queuing length estimation, which is an important parameter affecting the optimal detector locations at the roundabout. Under unbalanced traffic flow conditions, roundabouts are likely to experience long queuing problems.^6–8 The other studies on unbalanced traffic conditions were conducted and the unbalanced flow was defined through the origin–destination factor.^9,10 In addition, the origin–destination factor may improve the estimation of roundabout capacity, in particular, with the unbalanced traffic conditions.^9,10

The Flannery et al.¹¹ study performed on queuing estimation at single-lane roundabouts derived a relationship between entering traffic and circulating traffic considering the distribution of time gaps applied into an M/G2/1 queuing model. However, the research did not deal with the queuing length depending on the phase time changes, and the developed model was only for a normal signal roundabout which operational principle is obviously different from a metering roundabout. Fortuijn¹² investigated the detector locations based on a four-leg turbo roundabout with two detectors and one traffic signal based on Pollaczek-Khinchine’s M/M/1 waiting time formula. The author found a linear relationship between entry capacity and the mean queuing length that the precise queuing length estimation is not feasible and an advanced detection system may be required, such as an automatic traffic recorder with three different detectors installed for detecting changeable queues.

Recently, Fuzzy logic and neural networks both play important roles in the modeling of intelligent control in complex systems.^13–15 In fact, fuzzy logic is widely used in combination with neural networks to solve difficult problems in classification, identification and system control.^16–19 Eric and Finst proposed intelligent transportation system technologies for traffic simulation and management including all methods and components. The intelligent agent system has the ability to respond to any incident urban traffic.²⁰ However, there are no clear guidelines for finding the optimal detector locations due to the difficulty of definition between the detector locations and phase time changes, in dynamic situations.

Therefore, this study investigates the operation of metering roundabout and then predicts the queuing length on each approach using adaptive neuro fuzzy inference system (ANFIS). To achieve this purpose, data collected by drones are used in the process of model calibration and validation. Furthermore, root-square (R²) and root mean square error (RMSE) tests will be conducted to assess the fitness of the ANFIS model. Finally, the vehicle queuing lengths were calculated from the stop line.

The rest of the paper organized as follows: the “Background” section introduced all related background including the concept of metering roundabout and calibration model using both root-square and RMSE tests. The “Proposed model” section focused on the detail of roundabout model and proposed ANFIS model. The “Data collection using drones” section clearly mentioned the real data collection using drones at Old Belair Road roundabout in Adelaide. Finally, the proposed model structure was optimized and validated using R² and RMSE tests methods.

Background

Concept of metering roundabout

The concept of metering roundabout is illustrated in Figure 1. In this roundabout, two approaches are configured differently from traditional ones: the first is the controlling approach (dominant) where the queue detectors are positioned with a proper setback distance, and the second is the metered approach (sub-dominant) where a traffic signal is installed near the stop line. The mechanism of the metering roundabout is when queue reaches the detector, the traffic signal changes to red on the metered approach. Vehicles on the controlling approach, then, have sufficient time to enter the roundabout. In terms of dispersing the queue length on the controlling approach, there is an inverse relationship between the controlling and metered approaches, as the vehicles from the controlling approach can enter when the signal is red on metered approach.

Figure 1.

Concept of metering roundabout.⁸

When vehicles enter from a dominant approach, they may have fewer opportunities to utilize gaps between circulating vehicles, resulting in longer queues and delays on the busiest approach, and subsequently the entire roundabout carries the expense of the approaches having lower traffic volumes. Thus, a signalized roundabout with traffic signals managing entering vehicle movements from the other approaches is becoming a popular alternative, called metering roundabout. According to previous studies,^21–23 a signalized roundabout can be grouped into three categories: traffic flow control, operation time and number of approaches controlled as described in Table 1.

Table 1.

Categories of signalized roundabout.

Classification	Control type	Description
Traffic flow control	Direct control	Entering traffic flow and circulatory traffic flows are controlled by signals
Traffic flow control	Indirect control	Only entering traffic flows are controlled by signals
Operation time	Full-time control	Signal is operated for 24 h
Operation time	Part-time control	Signal is activated by time of day or by detectors
Number of approaches controlled	Full control	All approaches are controlled by signals
Number of approaches controlled	Partial control	Approaches are controlled by signals while the remaining approaches operate under give way control

Among the categories of signalized roundabouts given in Table 1, the indirect control with part-time operation is the most suitable method for roundabouts with unbalanced flows during peak hour operations.^24,25 In addition, this simple form of metering roundabout has a good cost-benefit ratio due to the low cost of installation.²⁴ It is also the case that the metering signal phase lengths can be controlled by the location of the approach detector.

The numerical model was formulated using the entering volumes and the circulating volumes as the ratio of imbalance (ρ_i). Therefore, the Northern approach standard formula is described as below (vehicles drive on the left side of roads condition)

$ρ_{n} = \frac{Q_{wn}}{Q_{wn} + Q_{sn} + Q_{en}} = \frac{Q_{wn}}{Q_{nc}}$ (1)

where, ρ_n is the proportion of the ratio of imbalance for the Northern approach, Q_wn is traffic from Western to Northern, Q_sn is traffic from Southern to Northern and Q_en is traffic from Eastern to Northern.

From field data validation, three important values of $ρ_{n}$ were found to be 0, 0.5 and 1. In the case that the value of ρ_n is 0.5, the roundabout can be considered to be under a balanced flow. Whereas if ρ_n is 0 or 1, the roundabout is experiencing unbalanced traffic conditions. Furthermore, when ρ_n is close to zero, entering vehicles from the Northern approach do not have enough accepted gaps to enter the roundabout due to high volumes of circulating vehicles from the Southern approach. If ρ_n is close to 1, the circulating vehicles from the Western approach may interrupt the vehicle movements from the Northern approach.

Model calibration and validation with RMSE and root-square tests

This research selected the Old Belair Road metering roundabout as a study site for model calibration and validation based on field data. Two different statistical tests, RMSE and R² were performed to verify the fitness of the ANFIS. RMSE has been widely used in model comparisons due to its precision including its sensitivity, relationship to decisions and validity²⁶

$RMSE = \sqrt{MSE (θ)} = \sqrt{E {(P - O)}^{2}}$ (2)

where, $θ$ is an estimated parameter, P is the model predicted variable and O is the observed variable.

Thus, this paper tests another statistical test R², which is also widely applied for goodness-of-fit measurement between two sets of data.^26,27 In addition, the range 0 to 1 represents model fit (i.e. a higher proportion indicates better agreement), as can be seen from Equation (3)

$R^{2} = {\frac{\sum_{i = 1}^{N} (O_{i} - Ó) (P_{i} - P \overset{'}{P})}{{[\sum_{i = 1}^{N} {(O_{i} - Ó)}^{2}]}^{0.5} {[\sum_{i = 1}^{N} {(P_{i} - \overset{'}{P})}^{2}]}^{0.5}}}^{2}$ (3)

where $n$ is the number of periods, $P_{i}$ is the model outputs, $\overset{'}{P}$ is the mean of model outputs, $O_{i}$ is the observation and $Ó$ is the mean of observation.

Proposed model

In order to predict the queuing length with ANFIS, an essential step is to define the effective model input parameters.

Definition of system parameters

The controlling and metered approaches are considered separately as illustrated in Figure 2, the circular movements are in red and entering movements are in green. This is because, as mentioned earlier, the queuing lengths of the two approaches will be in contrast with the signal phase time. Moreover, this study takes into account the four main parameters into consideration, that is, arrival volumes, conflicting volume, phase green and red time as shown in Figure 2.

Figure 2.

Parameters for the queuing length estimation (drive on the left side of roads).

ANFIS queuing length model

Figure 3 shows the proposed model in which the relation can be drawn as a linking diagram. One line in this figure means the association between one element in set X and one element in set Y = (f). If one element x_i is associated with another element y_j, there should be one line in the figure; if one element is not associated with another element, there should now be no line in the figure.

Figure 3.

Proposed ANFIS model on the basis of Takagi–Sugeno.

Different input membership functions for ANFIS in this research are tabulated in Table 2.

Table 2.

Input function membership.

Membership functions	Purpose
trimf	Triangular membership function
trapmf	Trapezoidal membership function
psigmf	Product of two sigmoidal membership functions
pimf	Pi-shaped curve membership function
gbellmf	Generalized bell curve membership function
gaussmf	Gaussian curve membership function
gauss2mf	Two-sided Gaussian curve membership function
dsigmf	Difference of two sigmoid membership functions

Let R be a crisp relation between the two sets X = {arrival volumes, conflicting volumes, Pgreen time and Pred time} and Y = {Queuing length}, such that R: X→Y and R consists of the pairs.

This idea can be extended to define a fuzzy relation R between two sets, X and Y. For each pair (x, y) X Y, a fuzzy relation R can be specified to express a grade of membership (x, y), which is bound to the interval [0, 1]

$μ_{R} : X \times Y \to [0, 1]$ (4)

Thus, a fuzzy relation R can be expressed as a set of pairs, just as the fuzzy set U was earlier

$R = {〈 (x, y), μ_{R} (x, y) 〉 | (x, y) \in X \times Y}$ (5)

If the two sets are finite, R can be specified as a matrix of $μ_{R}$ values as follows

$R = (\begin{matrix} μ_{R} (x_{1}, y_{1}) & μ_{R} (x_{1}, y_{2}) & \dots & μ_{R} (x_{1}, y_{n}) \\ μ_{R} (x_{2}, y_{1}) & μ_{R} (x_{2}, y_{2}) & \dots & μ_{R} (x_{2}, y_{n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ μ_{R} (x_{m}, y_{1}) & μ_{R} (x_{m}, y_{2}) & \dots & μ_{R} (x_{m}, y_{n}) \end{matrix})$ (6)

Data collection using drones

The data collection was done in the Old Belair Road roundabout in the Southern part of Adelaide as shown in Figure 4. This roundabout has four legs and is operated by traffic signals (on the Northern and Eastern approaches) and queue detectors (on the Southern approach) during the morning peak periods (06:30–09:30 AM). Detector distance (on the Southern approach) from the stop line is 115 m.

Figure 4.

The Old Belair Road roundabout in Adelaide.

The data collection was carried out on 5 days (Monday to Friday in the morning peak periods) for model calibration validation as follows:

First survey: 7–9 October 2015 between 07:45 and 08:45 (Wednesday to Friday);

Second survey: 16–17 November 2015 between 07:50 and 08:50 (Monday and Tuesday).

The main purpose of the exercise was to measure the queuing length variations on each approach at the same time, 5-min (300 s) intervals were used as the observation unit. Although normally video cameras could be used for the vehicle movement recordings, the limited ability to capture entering vehicle movements for all approaches, especially when the queuing lengths are extended to a large distance, has made this method invalid. Therefore, the maximum queuing length of all approaches in this study were recorded using two DJ Phantom2 drones and they can record a full view of the queues for further process, as shown in Figure 5.

Figure 5.

Drone footage.

Table 3 presents that the maximum queuing length on the Southern (controlling) approach, the queuing length on the Northern approach is longer than the other three approaches.

Table 3.

Sample of the maximum queue length on each approach (7 October).

Time	Q. Northern (m)	Q. Southern (m)	Q. Western (m)	Q. Eastern (m)
07:45–07:50	600	80	60	3
07:50–07:55	800	60	100	14
07:55–08:00	500	150	160	8
08:00–08:05	910	130	160	22
08:05–08:10	620	135	150	8
08:10–08:15	400	155	150	14
08:15–08:20	665	150	180	9
08:20–08:25	490	125	140	7
08:25–08:30	535	135	140	4
08:30–08:35	440	70	100	13
08:35–08:40	420	50	70	0
08:40–08:45	220	50	20	7

Controlling approach: Southern approach; Metered approach: Northern and Eastern approaches.

Table 4 summarizes the maximum queuing length on 17 November. Similar to above, the longer queuing length (up to 1000 m during 08:15–08:20) was observed on the Northern approach.

Table 4.

Sample of the maximum queue length on each approach (17 November).

Time	Q. Northern (m)	Q. Southern (m)	Q. Western (m)	Q. Eastern (m)
07:50–07:55	680	67	58	20
07:55–08:00	768	67	40	15
08:00–08:05	700	53	58	16
08:05–08:10	906	67	58	10
08:10–08:15	790	58	40	11
08:15–08:20	1003	153	115	1
08:20–08:25	835	170	170	25
08:25–08:30	507	128	120	32
08:30–08:35	650	135	180	12
08:35–08:40	60	160	225	16
08:40–08:45	45	130	230	18
08:45–08:50	95	45	85	19

Controlling approach: Southern approach; Metered approach: Northern and Eastern approaches.

In order to perform the model development, Sydney Coordinated Adaptive Traffic System (SCATS) data received from South Australian Department of Planning, Transport and Infrastructure (DPTI), including entering, conflicting volumes and phase durations, were also used as listed in Table 5.

Table 5.

Sample of values of parameters on each approach.

Time	Northern		Southern		Western		Eastern		Phase time (commonness)
Time	V_CN (veh)	V_N (veh)	V_CS (veh)	V_S (veh)	V_CW (veh)	V_W (veh)	V_CE (veh)	V_E (veh)	Pgreen (s)	Pred (s)
7 October
07:45–07:50	5	52	42	69	48	48	88	2	258	42
07:50–07:55	4	50	48	49	79	45	133	5	243	57
07:55–08:00	5	67	44	83	76	43	123	1	170	130
08:00–08:05	8	57	48	80	71	50	120	2	150	150
08:05–08:10	7	60	37	74	70	55	106	3	150	150
08:10–08:15	4	44	42	72	61	50	103	5	182	118
08:15–08:20	7	53	41	63	78	44	135	3	175	125
08:20–08:25	5	65	50	86	64	52	115	1	160	140
08:25–08:30	6	64	44	67	54	44	102	2	160	140
08:30–08:35	5	57	44	57	72	55	119	1	160	140
08:35–08:40	6	57	53	45	56	45	114	2	160	140
08:40–08:45	6	71	39	55	63	55	106	0	160	140
17 November
07:50–07:55	7	64	49	74	73	50	69	4	155	145
07:55–08:00	7	75	59	73	73	47	80	6	165	135
08:00–08:05	8	53	42	90	88	59	59	5	152	148
08:05–08:10	9	52	40	97	95	52	58	4	128	172
08:10–08:15	7	51	40	95	93	49	55	5	110	190
08:15–08:20	8	63	55	73	71	26	66	0	125	175
08:20–08:25	7	70	52	78	77	39	74	2	148	152
08:25–08:30	5	76	55	49	49	43	81	4	170	130
08:30–08:35	7	71	48	64	62	53	77	2	193	107
08:35–08:40	5	63	41	79	77	44	68	3	215	85
08:40–08:45	4	59	36	66	64	58	65	0	237	63
08:45–08:50	6	43	32	66	65	57	49	2	250	50

The table shows the parameters as follows:

V_CN: conflicting volumes (from Western to Eastern + from Western to Southern + from Southern to Eastern);

V_N: entering volumes from Northern approach;

V_CS: conflicting volumes (from Eastern to Northern + from Eastern to Western + from Northern to Western);

V_S: entering volumes from Southern approach;

V_CW: conflicting volumes (from Southern to Northern + from Southern to Eastern + from Eastern to Northern);

V_W: entering volumes from Western approach;

V_CE: conflicting volumes (from Northern to Southern + from Northern to Western + from Western to Southern);

V_E: entering volumes from Eastern approach;

Pred (s): total red time (all approaches use common values);

Pgreen (s): total green time (all approaches use common values).

Result and discussion

The data in Tables 3 –5 used for the training and testing were collected by Drone to target for ANFIS model. Sixty samples’ data were used to construct the models of ANFIS. In order to improve the ANFIS model, about 75% of data were used for training and the remainder for testing performance. The optimized structure has been selected as ANFIS, based on the residual error minimum and maximum of 500 epochs. First, the optimized model shown in Figure 6 depicts the convergence characteristics of the training algorithms and training error for constant and linear output membership functions, respectively. Epoch shows the number of iterations.

Figure 6.

Training error with constant output MF type.

Table 6 compares the convergence characteristics and training error of eight different functions. The table obviously presented the information that “trimf” and “gaussmf” membership functions present the best results and they are accurate enough to estimate the queuing length and its seat but the “gaussmf” has smaller RMSE and is chosen as the best algorithm.

Table 6.

Performance and training error of different algorithm.

Function	Iteration output MF type		RMSE output MF type
Function	Constant	Linear	Constant	Linear
psigmf	3350	90	0.08071	0.004006
dsigmf	1100	90	0.152021	0.0050048
pimf	2450	450	0.122082	0.0091116
gaussmf	3900	90	0.011867	0.00017438
gauss2mf	950	950	0.050265	0.0022592
gbellmf	850	625	0.076138	0.0033531
trapmf	1900	420	0.258162	0.0098235
trimf	90	90	0.08071	0.004005

MF: membership function; RMSE: root mean square error.

Table 7 shows the best parameter for the ANFIS model and Figure 7 shows the ANFIS models for training and testing data, where the error (ER) for variable X and the average absolute error (AAE).

Table 7.

ANFIS information of the best models.

Variables	ANFIS model
Variables	trimf	gaussmf
No. of MFs to each input	4	4
Total no. of parameters	45	39
No. of nonlinear parameters	17	13
No. of linear parameters	26	26
No. of nodes	36	36
No. of training data	60	60

ANFIS: adaptive neuro fuzzy inference system; MS: membership function.

Figure 7.

The comparison among the numerical and predicted values of queuing length using ANFIS models: (a) Queuing length on Northern approach, (b) Queuing length on Southern approach, (c) Queuing length on Western approach and (d) Queuing length on Eastern approach.

It can be seen from Table 8 that the ANFIS model predicts queuing length, the maximum error for trained is –9.29% and the range of AAE for all queuing length is 4.61%–5.13%.

Table 8.

Comparison between the drone data and ANFIS model for testing data.

Q. Northern (m)			Q. Southern (m)			Q. Western (m)			Q. Eastern (m)
Drone	ANFIS	%ER	Drone	ANFIS	%ER	Drone	ANFIS	%ER	Drone	ANFIS	%ER
600	553.8463	–7.69228	80	77.31192	–3.3601	60	62.53001	4.216685	3	2.827591	–5.74695
835	894.7582	7.156676	60	59.02044	–1.6326	100	95.47775	–4.52225	14	13.4128	–4.19426
500	455.5889	–8.88223	150	157.727	5.151364	160	168.1626	5.101646	8	8.190884	2.386051
910	900.3362	–1.06196	130	117.9173	–9.29438	170	172.7447	1.614522	22	23.59585	7.253859
620	558.8809	–1.10912	135	131.565	–2.54443	225	227.8215	1.253996	8	7.430143	–7.12321
400	404.3237	1.080935	155	144.8644	–6.53907	150	156.8023	4.53488	14	14.62363	4.454525
665	638.906	–3.92391	150	154.3353	2.890178	180	184.7163	2.620172	9	9.155853	1.731705
490	472.9737	–3.47475	125	117.0069	–6.39451	140	148.3614	5.972461	7	6.480826	–7.41677
535	498.2204	–6.87469	135	130.5497	–3.29649	230	235.8558	2.546021	4	3.703243	–7.41892
440	421.1408	–4.28617	70	74.44606	6.35151	85	89.82259	5.673637	13	13.23911	1.839291
420	435.3489	3.654501	50	45.81643	–8.36713	70	66.05604	–5.63423	0	0.000234	0
1003	970.3898	–3.25127	67	63.11651	–5.79625	115	117.0301	1.765293	7	7.339042	4.843453
680	635.2597	–6.57945	160	161.7988	1.124225	58	54.09996	–6.72421	15	13.98915	–6.73903
768	795.6632	3.601985	45	43.94053	–2.35439	40	36.83766	–7.90586	16	16.45267	2.829189
906	854.772	–5.65431	53	49.02686	–7.49649	120	130.9415	9.117922	10	10.76125	7.612457
	%AAE	5.135536			4.839541			4.613585			4.772645

ANFIS: adaptive neuro fuzzy inference system; ER: error; AAE: average absolute error.

R² and RMSE statistical tests can be conducted for the whole periods for evaluating the goodness-of-fit between ANFIS output and field data for model training (see Table 9 and Figure 7).

Table 9.

R² and RMSE tests for model training.

Statistic	Northern	Southern	Western	Eastern
R ²	0.9717	0.9849	0.9955	0.9903
RMSE	37.31 m	5.79 m	5.59 m	0.63 m

RMSE: root mean square error.

It is observed that the comparison among the drone and predicted values of queuing length (Q. Northern, Q. Southern, Q. Western and Q. Eastern) using ANFIS model are in good agreement with more than R² value of 98.9%. In addition, values of RMSE are all within 10% (Northern: 37.31 m, Southern: 5.79 m, Western: 5.59 m and Eastern: 0.63 m), which means “acceptable” and “very good.”

Conclusion and future work

This paper attempted to use ANFIS models to predict the queuing length on each approach at metering roundabout based on four parameters, which mainly affect the queuing length on each approach; that is, arrival volumes, conflicting volumes, phase green and red times. The model outputs were compared with the field data from drones with R² and RMSE statistic tests for evaluation of the goodness-of-fit.

The model results showed that the ANFIS can be used effectively to predict the queuing length on each approach at Old Belair Road metering roundabout. The study results can be summarized as follows:

The prediction of the queuing length with the ANFIS models is in good agreement with the drone data and also has a smaller error.

The ANFIS model is best for predicting the queuing length with comprehensive performance.

It can be expected that the proposed ANFIS model would contribute as a platform for determining the optimal detector locations at metering roundabouts.

This study, however, has some limitations that detector locations which can adjust signal phase times is not considered into the ANFIS models. Thus, in order to determine the optimal detector locations at metering roundabouts for capacity enhancement, the relationship between detector locations and four parameters need to be investigated. In addition, more case studies should be performed in the future based on a variety of metering roundabouts with different geometric designs.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship and/or publication of this article.

ORCID iDs

Hong Ki An

Ahmed N Abdalla

References

Yue

Stazic

. Estimation of vehicle queuing lengths at metering roundabouts. J Traffic Transp Eng 2017; 4(6): 545–554.

Emmerson

. A new concept of traffic measurement and control. Meas Control 2008; 41(5): 145–150.

Carter

Pratt

. Vehicle traffic detectors. Meas Control 1993; 26(2): 36–44.

Hagring

. Effects of OD flows on roundabout entry capacity. In: Proceedings of the 4th international symposium on highway capacity, Maui, HI, 27 June–1 July 2000.

Krogscheepers

Roebuck

. Unbalanced traffic volumes at roundabouts. In: Proceedings of the fourth international symposium on highway capacity (Transportation Research Circular E-C018), Maui, HI, 27 June–1 July 2000, pp. 446–458. Washington, DC: Transportation Research Board.

Stevens

. Signals and meters at roundabouts. In: Proceedings of the 2005 mid-continent transportation research symposium, Ames, IA, 18–19 August 2005.

Akçelik

. Roundabout metering signals: capacity, performance and timing. Procd Soc Behv 2011; 16: 686–696.

Hummer

Milazzo

Joseph

. The potential for metering to help roundabouts manage peak period demands in the US. In: Transportation research board 93rd annual meeting, Washington, DC, 12–16 January 2014.

Akçelik

. Roundabouts with unbalanced flow patterns. In: ITE 2004 annual meeting, Lake Buena Vista, FL, 1–4 August 2004.

10.

Akçelik

. Capacity and performance analysis of roundabout metering signals. In: TRB national roundabout conference, Vail, CO, 22–25 May 2005, pp. 22–25. Washington, DC: Transportation Research Board.

11.

Flannery

Kharoufeh

Gautam

et al . Queuing delay models for single-lane roundabouts. Civ Eng Environ Syst 2005; 22(3): 133–150.

12.

Fortuijn

. Robustness of roundabout metering systems (RMS). In: 4th international roundabout conference, Seattle, WA, 16–18 April 2014.

13.

Zhu

Huang

Yang

et al . Context-based prediction for road traffic state using trajectory pattern mining and recurrent convolutional neural networks. Inform Sciences 2019; 473: 190–201.

14.

Chatterjee

Chan

S. Real time traffic delay optimization using shadowed type-2 fuzzy rule base. Appl Soft Comput 2019; 74: 226–241.

15.

et al . Daily long-term traffic flow forecasting based on a deep neural network. Expert Syst Appl 2019; 121: 304–312.

16.

Tomar

Singh

Sharma

et al . Traffic management using logistic regression with fuzzy logic. Procedia Comput Sci 2018; 132: 451–460.

17.

Shaojiang

Shuyuan

Jing

. Exploring spatial–temporal relations via deep convolutional neural networks for traffic flow prediction with incomplete data. Appl Soft Comput. Epub ahead of print 29 October 2018. DOI: 10.1016/j.asoc.2018.09.040.

18.

Weihong

Jiyao

Renfa

. A novel fuzzy deep-learning approach to traffic flow prediction with uncertain spatial–temporal data features. Future Gener Comp Sy 2018; 89: 78–88.

19.

Yuankai

Huachun

Lingqiao

. A hybrid deep learning based traffic flow prediction method and its understanding. Transport Res C: Emer 2018; 90: 166–180.

20.

Olivier

Shoaib

Pawel

et al . Urban traffic simulators for intelligent transportation systems. Meas Control 2013; 46(10): 309–314.

21.

Akçelik

. SIDRA intersection user guide: part 4—output guide. Greythorn, VIC, Australia: Akcelik & Associates Pty Ltd., 2010.

22.

Azhar

Svante

. Signal control of roundabouts. Procd Soc Behv 2011; 16: 729–738.

23.

Mosslemi

. Using metering signals at roundabouts with unbalanced flow patterns to improve the traffic condition. Master’s Thesis, Linköping University, Linköping, 2008.

24.

Akçelik

. Analysis of roundabout metering signals. In: The 25th AITPM national conference, Melbourne, VIC, Australia, 3–4 August 2006.

25.

Armstrong

Collopy

. Error measures for generalizing about forecasting methods: empirical comparisons. Int J Forecasting 1992; 8: 69–80.

26.

Cameron

Windmeijer

. An R-squared measure of goodness of fit for some common nonlinear regression models. J Econometrics 1997; 77(2): 329–342.

27.

Legates

McCabe

. Evaluating the use of “goodness-of-fit” measures in hydrologic and hydroclimatic model validation. Water Resour Res 1999; 35(1): 233–241.