A mathematical model for headway variance of fixed-route buses

A mathematical model for computing and analysing headway variance of fixed-route buses offering scheduled service is developed. The model assumes a situation where both the passengers' OD pattern and the bus-route structure do not have any particular form but the number of passengers served at any bus station is a stationary Poisson process. Basic probability concepts are mostly used in deriving the model. Innovations in the model are the inclusion of a skip probability at a stop and the correlation factors between successive bus loads at a loading point and between the travel times of successive buses on a link. The model is, therefore, applicable to a wider range of situations than was possible with the earlier models. Results indicate that bus loading conditions and traffic conditions along the route are the major factors responsible for headway variability. The developed model can be used to determine the relative importance of these factors on headway variance at individual stopping station and thereby simplify the task of evolving effective countermeasures in any given situation. A numerical example that illustrates the application of the model is also given.     

Discharge headway at signalized intersections in Hong Kong

This paper reports the analysis and comparisons of discharge headways at 26 sites in Hong Kong. Previous studies here established good understanding of the average discharge headway under various conditions but very few studies dealt with discharge headway of individual vehicles which is a vital component in the traffic simulation at signalized intersections. This study that looked into the discharge headway of individual vehicles found that the discharge headway at different queue position follows the Type I Extreme Value Distribution. A method of estimating site‐specific parameters for this distribution has also been proposed.     

Improved driver responses at intersections with red signal countdown timers

Traffic Signal Countdown Timers (TSCTs) are innovative, practical and cost effective technologies with the potential to improve efficiency at signalized intersections. The purpose of these devices is to assist motorists in decision-making at signalized intersections with real-time signal duration information. This study focused specifically on driver responses in the presence of a Red Signal Countdown Timer (RSCT). A Linear Mixed Effect (LME) model was developed to predict the effect of RSCT on the headway of the first vehicle waiting on a red signal. The model predicted 0.72 s reduction in the headway of the first queued vehicle resulting from the presence of RSCT, while the observed difference in mean headway was 0.82 s. This result is suggestive of a reduction in start-up lost time at signalized intersections, i.e., an improvement in signalized intersection efficiency when an RSCT is present.     

Simple analytical models for estimating the queue lengths from probe vehicles at traffic signals

As mobile traffic sensor technology gets more attention, mathematical models are being developed that utilize this new data type in various intelligent transportation systems applications. This study introduces simple analytical estimation models for queue lengths from tracked or probe vehicles at traffic signals using stochastic modeling approach. Developed models estimate cycle-to-cycle queue lengths by using primary parameters such as arrival rate, probe vehicle proportions, and signal phase durations. Valuable probability distributions and moment generating functions for probe information types are formulated. Fully analytical closed-form expressions are given for the case ignoring the overflow queue and approximation models are presented for the overflow case. Derived models are compared with the results from VISSIM-microscopic simulation. Analytical steady-state and cycle-to-cycle estimation errors are also derived. Numerical examples are shown for the errors of these estimators that change with probe vehicle market penetration levels, arrival rates, and volume-to-capacity ratios.     

Nonlinear gating control for urban road traffic network using the network fundamental diagram

This work proposes a nonlinear model predictive controller for the urban gating problem. The system model is formalized based on a research on existing models of the network fundamental diagram and the perimeter control systems. For the existing models, modifications are suggested: additional state variables are allocated to describe the queue dynamics at the network gates. Using the extended model, a nonlinear model predictive controller is designed offering a ‘non‐greedy’ policy compared with previous, ‘greedy’ gating control designs. The greedy and non‐greedy nonlinear model predictive control (NMPC) controllers are compared with a greedy linear feedback proportional‐integral‐derivative (PID) controller in different traffic situations. The proposed non‐greedy NMPC controller outperforms the other two approaches in terms of travel distance performance and queue lengths. The performance results justify the consideration of queue lengths in dynamic modeling, and the use of NMPC approach for controller design. Copyright © 2014 John Wiley & Sons, Ltd.     

Effects of countdown timers on queue discharge characteristics of through movement at a signalized intersection

This study investigates how countdown timers installed at a signalized intersection affect the queue discharge characteristics of through movement during the green phase. Since the countdown timers display the time remaining (in seconds) until the onset of the green phase, drivers waiting in the queue at the intersection are aware of the upcoming phase change, and are likely to respond quicker. Thus, the countdown timers could reduce the start-up lost time, decrease the saturation headway, and increase the saturation flow rate. This study observed vehicle flow at an intersection in Bangkok for 24 h when the countdown timers were operating, and for another 24 h when the countdown timers were switched off. The signal plans and timings remained unchanged in both cases. Standard statistical t-tests were used to compare the difference in traffic characteristics between the “with timer” and “without timer” cases. It was found that the countdown timers had a significant impact on the start-up lost time, reducing it by 1.00–1.92 s per cycle, or a 17–32% time saving. However, the effects on saturation headway were found to be trivial, which implies that the countdown timers do not have much impact on the saturation flow rate of signalized intersections, especially during the off-peak day period and the late night period. The savings in the start-up lost time from the countdown timers was estimated to be equivalent to an 8–24 vehicles/h increase for each through movement lane at the intersection being studied.     

Lane-based real-time queue length estimation using license plate recognition data

License plate recognition (LPR) data are emerging data sources that provide rich information in estimating the traffic conditions of urban arterials. While large-scale LPR system is not common in US, last few years have seen rapid developments and implementations in many other parts of world (e.g. China, Thailand and Middle East). Due to privacy issues, LPR data are seldom available to research communities. However, when available, this data source can be valuable in estimating real-time operational metrics in transportation systems. This paper proposes a lane-based real-time queue length estimation model using the license plate recognition (LPR) data. In the model, an interpolation method based on Gaussian process is developed to reconstruct the equivalent cumulative arrival–departure curve for each lane. The missing information for unrecognized or unmatched vehicles is obtained from the reconstructed arrival curve. With the complete arrival and departure information, a car-following based simulation scheme is applied to estimate the real-time queue length for each lane. The proposed model is validated using ground truth information of the maximum queue lengths from the city of Langfang in China. The results show that the model can capture the variations in queue lengths in the ground truth data, and the maximum queue length for each signal cycle can be estimated with a reasonable accuracy. The estimated queue length information using the proposed model can serve as a useful performance metric for various real-time traffic control applications.     

Queue length estimation at metered freeway‐to‐freeway connectors

Freeway‐to‐freeway connector metering is a cost‐effective and proven freeway management strategy for relieving recurrent congestion. However, one of the critical challenges in design and operation of freeway‐to‐freeway connector metering is the lack of up‐to‐date queue storage length design guidance. In this study, it was found that ramp queue is dynamically related to the metering rate, on‐ramp demand, and traffic flow arrival pattern. Hence, simply using an average demand cannot provide accurate queue length estimation and is also not suitable for queue storage design where the maximum or a percentile queue length is generally used. A mesoscopic queue length simulation model was developed based on the input–output method for estimating queue lengths under various demand‐to‐capacity ratio scenarios. Simulation results indicate that for under‐saturated situations, the ramp queue may exist temporally due to the random short‐term surge of traffic arrivals, and the exponential function could best capture the relationship between queue length and demand‐to‐capacity ratio. For over‐saturated situations, the ramp queue tends to prolong linearly with the demand‐to‐capacity ratio. Based on the simulation, it was recommended that queue storage length be designed as 4.3% of on‐ramp demand when demand is lower than 1200 vph or 2.3% when demand is between 1200 and 2400 vph. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic network loading: A stochastic differentiable model that derives link state distributions

We present a dynamic network loading model that yields queue length distributions, accounts for spillbacks, and maintains a differentiable mapping from the dynamic demand on the dynamic queue lengths. The model also captures the spatial correlation of all queues adjacent to a node, and derives their joint distribution. The approach builds upon an existing stationary queueing network model that is based on finite capacity queueing theory. The original model is specified in terms of a set of differentiable equations, which in the new model are carried over to a set of equally smooth difference equations. The physical correctness of the new model is experimentally confirmed in several congestion regimes. A comparison with results predicted by the kinematic wave model (KWM) shows that the new model correctly represents the dynamic build-up, spillback and dissipation of queues. It goes beyond the KWM in that it captures queue lengths and spillbacks probabilistically, which allows for a richer analysis than the deterministic predictions of the KWM. The new model also generates a plausible fundamental diagram, which demonstrates that it captures well the stationary flow/density relationships in both congested and uncongested conditions.     

Optimisation of parking guidance and information systems display configurations

Operators of parking guidance and information (PGI) systems often have difficulty in determining the best car park availability information to present to drivers in periods of high demand. This paper describes a behavioural model of parking choice incorporating drivers perceptions of waiting times at car parks based on PGI signs. This model was used to predict the influence of PGI signs on the overall performance of the traffic system.Relationships were developed for estimating the arrival rates at car parks based on trip patterns, driver characteristics, car park attributes as well as the car park availability information displayed on PGI signs. Drivers' perceptions of waiting times at car parks were assumed to be influenced by the PGI signs for observers of the signs and actual car park utilisation levels for non-observers. The model assumes that the choice of car park does not change after entering the city centre, even if conditions observed are different from those initially perceived.A mathematical programme was formulated to determine the optimal display PGI sign configuration to minimise queue lengths and vehicle kilometres of travel (VKT). The model was limited to off-street parking choices and illegal parking was not incorporated. A simple genetic algorithm was used to identify solutions that significantly reduced queue lengths and VKT compared with existing practices.These procedures were applied to an existing PGI system operating in Tama New Town near Tokyo. Significant reductions in queue lengths and VKT were predicted using the optimisation model. This would reduce traffic congestion and lead to various environmental benefits.     

An application of shock wave theory to traffic signal control

A real time control policy minimizing total intersection delays subject to queue length constraints at an isolated signalized intersection is developed in this paper. The policy is derived from a new traffic model which describes the simultaneous evolution of queue lengths of two conflicting traffic streams, controlled by a traffic light, in both time and space. The model is based on the examination of shock waves generated upstream of the stop lines by the intermittent service of traffic at the signal. The proposed policy was tested against the existing pre-timed control policy at a high volume intersection and it was found superior, especially when demands increase well above the saturation level.     

Real time queue length estimation for signalized intersections using travel times from mobile sensors

We study how to estimate real time queue lengths at signalized intersections using intersection travel times collected from mobile traffic sensors. The estimation is based on the observation that critical pattern changes of intersection travel times or delays, such as the discontinuities (i.e., sudden and dramatic increases in travel times) and non-smoothness (i.e., changes of slopes of travel times), indicate signal timing or queue length changes. By detecting these critical points in intersection travel times or delays, the real time queue length can be re-constructed. We first introduce the concept of Queue Rear No-delay Arrival Time which is related to the non-smoothness of queuing delay patterns and queue length changes. We then show how measured intersection travel times from mobile sensors can be processed to generate sample vehicle queuing delays. Under the uniform arrival assumption, the queuing delays reduce linearly within a cycle. The delay pattern can be estimated by a linear fitting method using sample queuing delays. Queue Rear No-delay Arrival Time can then be obtained from the delay pattern, and be used to estimate the maximum and minimum queue lengths of a cycle, based on which the real-time queue length curve can also be constructed. The model and algorithm are tested in a field experiment and in simulation.     

Neural network modeling of vehicle discharge headway at signalized intersection: model descriptions and results

Vehicle discharge headway at signalized intersections is of great importance in junction analysis. However, it is very difficult to simulate the discharge headway of individual queued vehicle because of the great variations in the driver behaviors, vehicle characteristics and traffic environment. The current study proposes a neural network (NN) approach to simulate the queued vehicle discharge headway. A computer-based three-layered (NN) model was developed for the estimation of discharge headway. The widely used backpropagation algorithm has been utilized in training the NN model. The NN model was trained, validated with field data and then compared with other headway models. It was found that the NN model performed better. Model sensitivity analysis was conducted to further validate the applicability of the model. Results showed that the NN model could produce reasonable discharge headway estimates for individual vehicles.     

Time-dependent queueing at road junctions: Observation and prediction

Measurements of queue lengths and vehicle delays have been used to test the predictions of time-dependent queueing models. Observations were made at eight public road sites for a number of peak periods at each site. In all, sixty-eight peaks were studied. The model predictions of the time-dependent trials-average queues and delays agreed with the observations within the limits set by day-to-day variability for seven of the sites, but for the eighth there was consistent overprediction.     

Modeling and simulation of vehicle projection arrival–discharge process in adaptive traffic signal controls

Real‐time signal control operates as a function of the vehicular arrival and discharge process to satisfy a pre‐specified operational performance. This process is often predicted based on loop detectors placed upstream of the signal. In our newly developed signal control for diamond interchanges, a microscopic model is proposed to estimate traffic flows at the stop‐line. The model considers the traffic dynamics of vehicular detection, arrivals, and departures, by taking into account varying speeds, length of queues, and signal control. As the signal control is optimized over a rolling horizon that is divided into intervals, the vehicular detection for and projection into the corresponding horizon intervals are also modeled. The signal control algorithm is based on dynamic programming and the optimization of signal policy is performed using a certain performance measure involving delays, queue lengths, and queue storage ratios. The arrival–discharge model is embedded in the optimization algorithm and both are programmed into AIMSUN, a microscopic stochastic simulation program. AIMSUN is then used to simulate the traffic flow and implement the optimal signal control by accessing internal data including detected traffic demand and vehicle speeds. Sensitivity analysis is conducted to study the effect of selecting different optimization criteria on the signal control performance. It is concluded that the queue length and queue storage ratio are the most appropriate performance measures in real‐time signal control of interchanges. Copyright © 2010 John Wiley & Sons, Ltd.     

A new methodology for incident detection and characterization on surface streets

In this paper, a new methodology is presented for real-time detection and characterization of incidents on surface streets. The proposed automatic incident detection approach is capable of detecting incidents promptly as well as characterizing incidents in terms of time-varying lane-changing fractions and queue lengths in blocked lanes, lanes blocked due to incidents, and incident duration. The architecture of the proposed incident detection approach consists of three sequential procedures: (1) Symptom Identification for identification of incident symptoms, (2) Signal Processing for real-time prediction of incident-related lane traffic characteristics and (3) Pattern Recognition for incident recognition. Lane traffic counts and occupancy are the only two major types of input data, which can be readily collected from point detectors. The primary techniques utilized in this paper include: (1) a discrete-time, nonlinear, stochastic system with boundary constraints to predict real-time lane-changing fractions and queue lengths and (2) a pattern-recognition-based algorithm employing modified sequential probability ratio tests (MSPRT) to detect incidents. Off-line tests based on simulated as well as video-based real data were conducted to assess the performance of the proposed algorithm. The test results have indicated the feasibility of achieving real-time incident detection using the proposed methodology.     

Optimal control of headway variation on transit routes

Headway control strategies have been proposed as methods for correcting transit service irregularities and thereby reducing passenger wait times at stops. This paper addresses a particular strategy which can be implemented on high frequency routes (headways under 10–12 minutes), in which buses are held at a control stop to a threshold headway. An algorithm is developed which yields the optimal control stop location and optimal threshold headway with respect to a system wait function. The specification of the wait function is based on the development of several empirical models, including a headway variation model and an average delay time model at control stops. A conclusion is reached that the headway variation does not increase linearly along a route, a common assumption made in many previous studies. Furthermore, the location of the optimal control stop and threshold value are sensitive to the passenger boarding profile, as expected. The algorithm itself appears to have practical application to conventional transit operations.     

A block queueing system for slow moving traffic

A common cause of pollution and waste in urban areas is facilities which provide a continuous slow service for motor vehicles. As demand approaches supply, queues can develop and large numbers of engines can often be idling. Typical examples are car park entrances and drive through fast food outlets. This paper proposes a block queueing system to alleviate the problem without using excessive road space and with a minimum of extra infrastructure. The queue is divided into an active section at the front (with engines running) and a passive section at the rear (where drivers switch their engines off). Periodically, as the active queue becomes depleted, the passive queue is momentarily activated and a block of vehicles advances into the active queue. A visual cue can be provided to the drivers using a vehicle actuated traffic signal. It is readily apparent that drivers in the passive queue have to switch their engines on and off at regular intervals. Since this operation has an inherent cost in itself, this argues in favour of a large block size. However, large blocks mean more engines idling in the active queue. A compromise must be therefore reached for the likely range of queue lengths which the system under consideration exhibits. An expression is derived for the optimum block size in steady state conditions. It is shown that the potential benefits of the regime are considerable.     

Queue discharge patterns at signalized intersections with green signal countdown device and long cycle length

Two apparent features that prevail at signalized intersections in China are green signal countdown device and long cycle lengths. The objective of this study is to investigate the impacts of green signal countdown device and long cycle length on queue discharge patterns and to discuss its implications on capacity estimation in the context of China's traffic. At five typical large intersections in Shanghai and Tianjin, 11 through lanes were observed, and 9251 saturation headways were obtained as valid samples. Statistical analyses indicate that the discharge process of queuing vehicles can be divided into three distinct stages according to the discharge flow rate: a start‐up stage, a steady stage, and a rush stage. The average time for queuing vehicles to reach a stationary saturation flow rate, that is, the start‐up stage, was found to be approximately 20–30 seconds; the rush stage usually occurs during the phase transition period. The finding is contrary to the conventional assumption that the discharge rate reaches a maximum value after the fourth vehicle is discharged and then remains constant during the green time until the queue is completely dissolved. The capacity estimation errors that might arise from the conventional methods are discussed through a comparative study and a sensitivity analysis that are based on the identified queue discharge patterns. In addition, a piecewise linear regression method was proposed in order to reduce such errors. The proposed method can be used for capacity estimation at signalized intersections with the identified queue discharge patterns. Copyright © 2017 John Wiley & Sons, Ltd.     

Real-time estimation of lane-based queue lengths at isolated signalized junctions

In this study, we develop a real-time estimation approach for lane-based queue lengths. Our aim is to determine the numbers of queued vehicles in each lane, based on detector information at isolated signalized junctions. The challenges involved in this task are to identify whether there is a residual queue at the start time of each cycle and to determine the proportions of lane-to-lane traffic volumes in each lane. Discriminant models are developed based on time occupancy rates and impulse memories, as calculated by the detector and signal information from a set of upstream and downstream detectors. To determine the proportions of total traffic volume in each lane, the downstream arrivals for each cycle are estimated by using the Kalman filter, which is based on upstream arrivals and downstream discharges collected during the previous cycle. Both the computer simulations and the case study of real-world traffic show that the proposed method is robust and accurate for the estimation of lane-based queue lengths in real time under a wide range of traffic conditions. Calibrated discriminant models play a significant role in determining whether there are residual queued vehicles in each lane at the start time of each cycle. In addition, downstream arrivals estimated by the Kalman filter enhance the accuracy of the estimates by minimizing any error terms caused by lane-changing behavior.