Predicting queue variability to enable analysis of overload risk

ABSTRACT

Predicting the risk of traffic demands and delays exceeding critical limits at road junctions, airports, hospitals, etc., requires knowing how both mean and variance of queue size vary over time. Microscopic simulation can explore variability but is computationally demanding and gives only sample results. A computationally efficient approximation to the mean is used in many modelling tools, but only empirical extensions for variance in particular situations have been available. The paper derives theoretical formulae for time-dependent and equilibrium variance, believed to be novel and to apply generally to queues covered by the Pollaczek–Khinchin mean formula, and offering possible structural insights. These are applied in an extended approximation giving mutually consistent mean and variance estimates with improved accuracy. Tests on oversaturated peak demand cases are compared with Markov probabilistic simulation, demonstrating accuracy (R^2?>?0.99) for typical random, priority-like (M/M/1) and traffic-signal-like (M/D/1) queues. Implications for risk analysis, planning and policy are considered.     

Overflow delay in signalized networks

Traffic arrivals tend to be random at signals near to the perimeter of a network (or near to traffic generators in a network). Within the signal network, however, surges in traffic demand are reduced due to limitations on the amount of traffic passing through intersections imposed by signals, resulting in more uniform arrivals from cycle to cycle. Such uniformity is a desirable property at signals as underutilization of green periods may be reduced and levels of service improved. This may have serious implications within networks where it may be possible to improve the capacity of critical intersections by the strategic placing and timing of signals at less critical locations. The analysis of such options is, however, restricted by most, if not all, of the currently available evaluation methods. Relatively simple modifications of delay formulae are proposed to overcome these restrictions.     

The impact of signal control on land-side traffic patterns at the Dallas/Fort worth airport

The Parkway Corridor is the primary ground facility at the Dallas/Fort Worth (D/FW) International Airport. It consists of an expressway for passengers to-and-from terminals and a service road mainly for employees of airline-associated agencies to access the service areas. As the Parkway Corridor is a natural short cut in the region, it has attracted more and more through commuters who usually choose to use the service road so as to circumvent the toll plazas on the expressway. The dramatic increase in both the volume and speed of through commuting traffic in recent years not only impedes service activities, but also raises safety concerns. The D/FW Airport Board is considering the installation of a series of traffic signals to regulate the cruise speed and thereby discourage use of the service road by commuters. This research is conducted to evaluate its effectiveness and potential impact on traffic patterns along the Parkway Corridor. A systematic approach, integrating queueing theories (G/M/m), discrete models, and traffic assignment, is proposed in this research, which allows the user to project the traffic distribution on each route with only the information obtained by traffic counts. Numerical results of alternate control strategies and a sensitivity analysis with respect to key parameters, such as the toll and cycle length of traffic signals are also reported.     

Developing passenger car equivalents for heavy vehicles on freeways during queue discharge flow

This paper investigates the hypothesis that the effect of heavy vehicles on traffic is greater during congestion than during undersaturated conditions. A new approach was developed to quantify this effect by deriving passenger car equivalents (PCEs) using queue discharge flow (QDF) capacity as the equivalency criterion. This approach is based on the premise that QDF capacity observations can be expected to show minimal variation if traffic stream was uniform and consisted of passenger cars only. Two sites in Ontario, Canada were used for this research. The first is located at an entrance ramp merge area and the second at a long-term freeway reconstruction zone. Nonlinear programming was utilized to perform optimizations on a number of data sets at each site. Results strongly suggest that the research hypothesis is true and that the approach developed by this research is both plausible and feasible. The mean PCE factor at the first site was 2.36 versus 1.5 in the Highway Capacity Manual (HCM) 2000. At the second site, the mean PCE factors in the two directions of travel were 3.21 and 2.7 versus 2.0 in the HCM 2000. Results also showed that the PCE factor developed from the optimization runs behaves as a random variable that generally follows the normal distribution. Furthermore, the PCE factor was neither a function of weather conditions nor of roadside maintenance work.     

Surrounding traffic complexity analysis for efficient and stable conflict resolution

The constant increase in air traffic demand increases a probability of the separation minima infringements in certain areas as a consequence of increased traffic density. The Annual Safety Report 2016 reports that in recent years the number of infringements, measured per million flight hours, had been increased at a lower rate (Eurocontrol, 2018). However, this level of infringements still generates a continuous pressure on the air traffic control (ATC) system and seeks for more control resources ready to tactically solve potential conflicts, while increasing at the same time the operational costs. Considering present air traffic management (ATM) trade-off criteria: increased airspace capacity and traffic efficiency but reducing the cost while preserving safety, new services must be designed to distribute the separation management ATC task loads among other actors. Based on the Single European Sky Air Traffic Management Research and Next Generation Air Transportation System initiatives, this paper proposes an innovative separation management service to shift the completely centralized tactical ATC interventions to more efficient decentralized tactical operations relying on an advanced surrounding traffic analysis tool, to preserve the safety indicators while considering the operational efficiency. A developed methodology for the proposed service is an application-oriented, trying to respond to characteristics and requirements of the current operational environment. The paper further analysis the traffic complexity taking into consideration the so-called domino effect, i.e. a number of the surrounding aircraft causally involved in the separation management service by the means of identification of the spatiotemporal interdependencies between them and the conflicting aircraft. This complexity is driven by the interdependencies structure and expressed as a time-criticality in quantifying the total number of the system solutions, that varies over time as the aircraft are approaching to each other. The results from two randomly selected ecosystem scenarios, extracted from a simulated traffic, illustrate different avoidance capacities for a given look-ahead time and the system solutions counts, that in discrete moments reach zero value.     

Investigating empirical implications of hysteresis in day-to-day travel time variability

Day-to-day travel time variability plays a significant role in travel time reliability. Nowadays, travelers not only seek to minimize their travel time on average, but also value its variation. The variation in the mean and the variance of travel time (across days, for the same departure time) has not been thoroughly investigated. A temporary decrease in capacity (e.g. congestion caused by an active bottleneck) leads to a quite significant difference in the variance of travel time for congestion onset and offset periods. This phenomenon results in hysteresis loops where the departure time periods in congestion offset exhibit a higher travel time variance than the ones in congestion onset with the same mean travel time. The aim of this paper is to identify empirical implications that yield to the hysteresis phenomenon in day-to-day travel times. First, empirical hysteresis loop observations are provided from two different freeway sites. Second, we investigate the potential link with the hysteresis observed in traffic networks on macroscopic fundamental diagram (MFD). Third, we build a piecewise linear function that models the evolution of travel time within the day. This allows us to decompose the problem into its components, e.g. start time of congestion, peak travel time, etc. These components, along with their probability distribution functions, are employed in a Monte Carlo simulation model to investigate their partial effects on the existence of hysteresis. Correlation among critical variables is the most influential factor in this phenomenon, which should be further investigated regarding traffic flow and traffic equilibrium principles.     

Traffic flow on signalized streets

This paper considers a signalized street of uniform width and blocks of various lengths. Its signals are pretimed in an arbitrary pattern, and traffic on it behaves as per the kinematic-wave/variational theory with a triangular fundamental diagram. It is shown that the long run average flow on the street when the number of cars on the street (i.e. the street’s density) is held constant is given by the solution of a linear program (LP) with a finite number of variables and constraints. This defines a point on the street’s macroscopic fundamental diagram. For the homogeneous special case where the block lengths and signal timings are identical, all the LP constraints but one are redundant and the result has a closed form. In this case, the LP recipe matches and simplifies the so-called “method of cuts”. This establishes that the method of cuts is exact for homogeneous problems. However, in the more realistic inhomogeneous case the difference between the two methods can be arbitrarily large.The paper uses the LP method to obtain the macroscopic fundamental diagrams arising under four different traffic coordination schemes for streets with four different block length configurations. It is found that the best scheme depends on the prevailing density. Curiously, the popular scheme in which all the traffic green phases are started synchronously wins only in rare circumstances. Its performance is particularly underwhelming when the street’s blocks are long. The paper also presents density-aware numerical methods to optimize the signal offsets for 1-way and 2-way streets. For 1-way streets operated with a common cycle the method reduces to a simple graphical construction . In this case the resulting flow matches the flow that would arise if all of the street’s intersections except one with the shortest green phase had been eliminated.     

Traffic signal timing problems with environmental and equity considerations

Traffic signal timings in a road network can not only affect total user travel time and total amount of traffic emissions in the network but also create an inequity problem in terms of the change in travel costs of users traveling between different locations. This paper proposes a multi‐objective bi‐level programming model for design of sustainable and equitable traffic signal timings for a congested signal‐controlled road network. The upper level of the proposed model is a multi‐objective programming problem with an equity constraint that maximizes the reserve capacity of the network and minimizes the total amount of traffic emissions. The lower level is a deterministic network user equilibrium problem that considers the vehicle delays at signalized intersections of the network. To solve the proposed model, an approach for normalizing incommensurable objective functions is presented, and a heuristic solution algorithm that combines a penalty function approach and a simulated annealing method is developed. Two numerical examples are presented to show the effects of reserve capacity improvement and green time proportion on network flow distribution and transportation system performance and the importance of incorporating environmental and equity objectives in the traffic signal timing problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Metamodels for estimating waterway delays through series of queues

A numerical method has been developed for estimating delays on congested waterways. Analytic and numerical results are presented for series of G/G/1 queues, i.e. with generally distributed arrivals and service times and single chambers at each lock. One or two-way traffic operations are modelled. A metamodelling approach which develops simple formulas to approximate the results of simulation models is presented. The structure of the metamodels is developed from queueing theory while their coefficients are statistically estimated from simulation results. The numerical method consists of three modules: (1) delays, (2) arrivals and (3) departures. The first estimates the average waiting time for each lock when the arrival and service time distributions at this lock are known. The second identifies the relations between the arrival distributions at one lock and the departure distributions from the upstream and downstream locks. The third estimates the mean and variance of inter-departure times when the inter-arrival and service time distributions are known. The method can be applied to systems with two-way traffic through common bi-directional servers as well as to one-way traffic systems. Algorithms for both cases are presented. This numerical method is shown to produce results that are close to the simulation results. The metamodels developed for estimating delays and variances of inter-departure times may be applied to waterways and other series of G/G/1 queues. These metamodels for G/G/1 queues may provide key components of algorithms for analyzing networks of queues.     

Single track line capacity model

The objective of the research described in this paper was to develop a model for computation of an ultimate capacity of a single track line and to provide a sensitivity analysis of this capacity to the parameters which influence it. The model is based in a concept of mathematical expectation of capacity and can be applied under saturation conditions i.e. a constant demand for service. It can serve for planning purposes, computation of single track line capacity on the base of which estimations are possible concerning a single track line performance under given conditions, as well as commercial time‐tables planning, decisions about a partial or complete construction of the second parallel track along the line in service, intermediate stations locations planning and the necessary facilities along the line under construction.

In the sensitivity analysis, the model allows a change of parameters upon which the capacity depends. These are: the length of the line segment which is considered to be bottleneck for calculation of capacity, traffic distributions per directions, train mix, train velocities and train spacing rules applied by the dispatching service when regulating the traffic on a line.     

The prigogine-herman kinetic model predicts widely scattered traffic flow data at high concentrations

The classical derivation of a traffic stream model (e.g. speed/concentration relation) from the equilibrium solutions of the Prigogine–Herman kinetic equation invokes the nontrivial assumption that the underlying distribution of desired speeds is nonzero for vanishingly small speeds. In this paper we investigate the situation when this assumption does not hold. It is found that the Prigogine–Herman kinetic equation has a one-parameter family of equilibrium solutions, and hence an associated traffic stream model, only for traffic concentrations below some critical value; at higher concentrations there is a two-parameter family of solutions, and hence a continuum of mean velocities for each concentration. This result holds for both constant values of the passing probability and the relaxation time, and for values that depend on concentration in the manner assumed by Prigogine and Herman. It is hypothesized that this result reflects the well-known tendency toward substantial scatter in observational data of traffic flow at high concentrations.     

Recursive estimation of average vehicle time headway using single inductive loop detector data

Vehicle time headway is an important traffic parameter. It affects roadway safety, capacity, and level of service. Single inductive loop detectors are widely deployed in road networks, supplying a wealth of information on the current status of traffic flow. In this paper, we perform Bayesian analysis to online estimate average vehicle time headway using the data collected from a single inductive loop detector. We consider three different scenarios, i.e. light, congested, and disturbed traffic conditions, and have developed a set of unified recursive estimation equations that can be applied to all three scenarios. The computational overhead of updating the estimate is kept to a minimum. The developed recursive method provides an efficient way for the online monitoring of roadway safety and level of service. The method is illustrated using a simulation study and real traffic data.     

Maximum car ownership under constraints of road capacity and parking space

This paper presents a model for determining the maximum number of cars by zones in view of the capacity of the road network and the number of parking spaces available. In other words, the proposed model is to examine whether existing road network and parking supply is capable of accommodating future zonal car ownership growth (or the reserve capacity in each zone); i.e. the potential maximum zonal car ownership growth that generates the road traffic within the network capacity and parking space constraints. In the proposed model, the vehicular trip production and attraction are dependent on the car ownership, available parking spaces and the accessibility measures by traffic zones. The model is formulated as a bi-level programming problem. The lower-level problem is an equilibrium trip distribution/assignment problem, while the upper-level problem is to maximize the sum of zonal car ownership by considering travellers’ route and destination choice behaviour and satisfying the network capacity and parking space constraints. A sensitivity analysis based heuristic algorithm is developed to solve the proposed bi-level car ownership problem and is illustrated with a numerical example.     

Direct formulation and algorithms for the probit-based stochastic user equilibrium traffic assignment problem

This paper proposes simple and direct formulation and algorithms for the probit-based stochastic user equilibrium traffic assignment problem. It is only necessary to account for random variables independent of link flows by performing a simple transformation of the perceived link travel time with a normal distribution. At every iteration of a Monte-Carlo simulation procedure, the values of the random variables are sampled based on their probability distributions, and then a regular deterministic user equilibrium assignment is carried out to produce link flows. The link flows produced at each iteration of the Monte-Carlo simulation are averaged to yield the final flow pattern. Two test networks demonstrate that the proposed algorithms and the traditional algorithm (the Method of Successive Averages) produce similar results and that the proposed algorithms can be extended to the computation of the case in which the random error term depends on measured travel time.     

Multi-class percentile user equilibrium with flow-dependent stochasticity

Travelers often reserve a buffer time for trips sensitive to arrival time in order to hedge against the uncertainties in a transportation system. To model the effects of such behavior, travelers are assumed to choose routes to minimize the percentile travel time, i.e. the travel time budget that ensures their preferred probability of on-time arrival; in doing so, they drive the system to a percentile user equilibrium (UE), which can be viewed as an extension of the classic Wardrop equilibrium. The stochasticity in the supply of transportation are incorporated by modeling the service flow rate of each road segment as a random variable. Such stochasticity is flow-dependent in the sense that the probability density functions of these random variables, from which the distribution of link travel time are constructed, are specified endogenously with flow-dependent parameters. The percentile route travel time, obtained by directly convolving the link travel time distributions in this paper, is not available in closed form in general and has to be numerically evaluated. To reveal their structural properties, percentile UE solutions are examined in special cases and verified with numerical results. For the general multi-class percentile UE traffic assignment problem, a variational inequality formulation is given and solved using a route-based algorithm. The algorithm makes use of the diagonal elements in the Jacobian of percentile route travel time, which is approximated through recursive convolution. Preliminary numerical experiments indicate that the algorithm is able to achieve highly precise equilibrium solutions.     

A systematic analysis of multimodal transport systems with road space distribution and responsive bus service

A smart design of transport systems involves efficient use and allocation of the limited urban road capacity in the multimodal environment. This paper intends to understand the system-wide effect of dividing the road space to the private and public transport modes and how the public transport service provider responds to the space changes. To this end, the bimodal dynamic user equilibrium is formulated for separated road space. The Macroscopic Fundamental Diagram (MFD) model is employed to depict the dynamics of the automobile traffic for its state-dependent feature, its inclusion of hypercongestion, and its advantage of capturing network topology. The delay of a bus trip depends on the running speed which is in turn affected by bus lane capacity and ridership. Within the proposed bimodal framework, the steady-state equilibrium traffic characteristics and the optimal bus fare and service frequency are analytically derived. The counter-intuitive properties of traffic condition, modal split, and behavior of bus operator in the hypercongestion are identified. To understand the interaction between the transport authority (for system benefit maximization) and the bus operator (for its own benefit maximization), we examine how the bus operator responds to space changes and how the system benefit is influenced with the road space allocation. With responsive bus service, the condition, under which expanding bus lane capacity is beneficial to the system as a whole, has been analytically established. Then the model is applied to the dynamic framework where the space allocation changes with varying demand and demand-responsive bus service. We compare the optimal bus services under different economic objectives, evaluate the system performance of the bimodal network, and explore the dynamic space allocation strategy for the sake of social welfare maximization.     

Dynamic user equilibrium with side constraints for a traffic network: Theoretical development and numerical solution algorithm

This paper investigates a traffic volume control scheme for a dynamic traffic network model which aims to ensure that traffic volumes on specified links do not exceed preferred levels. The problem is formulated as a dynamic user equilibrium problem with side constraints (DUE-SC) in which the side constraints represent the restrictions on the traffic volumes. Travelers choose their departure times and routes to minimize their generalized travel costs, which include early/late arrival penalties. An infinite-dimensional variational inequality (VI) is formulated to model the DUE-SC. Based on this VI formulation, we establish an existence result for the DUE-SC by showing that the VI admits at least one solution. To analyze the necessary condition for the DUE-SC, we restate the VI as an equivalent optimal control problem. The Lagrange multipliers associated with the side constraints as derived from the optimality condition of the DUE-SC provide the traffic volume control scheme. The control scheme can be interpreted as additional travel delays (either tolls or access delays) imposed upon drivers for using the controlled links. This additional delay term derived from the Lagrange multiplier is compared with its counterpart in a static user equilibrium assignment model. If the side constraint is chosen as the storage capacity of a link, the additional delay can be viewed as the effort needed to prevent the link from spillback. Under this circumstance, it is found that the flow is incompressible when the link traffic volume is equal to its storage capacity. An algorithm based on Euler’s discretization scheme and nonlinear programming is proposed to solve the DUE-SC. Numerical examples are presented to illustrate the mechanism of the proposed traffic volume control scheme.     

Speed variation during peak and off-peak hours on urban arterials in Shanghai

Increased speed variation on urban arterials is associated with reductions in both operational performance and safety. Traffic flow, mean speed, traffic control parameters and geometric design features are known to affect speed variation. An exploratory study of the relationships among these variables could provide a foundation for improving the operational and safety performance of urban arterials, however, such a study has been hampered by problems in measuring speeds. The measurement of speed has traditionally been accomplished using spot speed collection methods such as radar, laser and loop detectors. These methods can cover only limited locations, and consequently are not able to capture speed distributions along an entire network, or even throughout any single road segment. In Shanghai, it is possible to acquire the speed distribution of any roadway segment, over any period of interest, by capturing data from Shanghai’s 50,000+ taxis equipped with Global Positional Systems (GPS). These data, hereafter called Floating Car Data, were used to calculate mean speed and speed variation on 234 road segments from eight urban arterials in downtown Shanghai. Hierarchical models with random variables were developed to account for spatial correlations among segments within each arterial and heterogeneities among arterials. Considering that traffic demand changes throughout the day, AM peak, Noon off-peak, and PM peak hours were studied separately. Results showed that increases in number of lanes and number of access points, the presence of bus stops and increases in mean speed were all associated with increased speed variation, and that increases in traffic volume and traffic signal green times were associated with reduced speed variation. These findings can be used by engineers to minimize speed differences during the road network planning stage and continuing through the traffic management phase.     

First-best marginal cost toll for a traffic network with stochastic demand

First-best marginal cost pricing (MCP) in traffic networks has been extensively studied with the assumption of deterministic travel demand. However, this assumption may not be realistic as a transportation network is exposed to various uncertainties. This paper investigates MCP in a traffic network under stochastic travel demand. Cases of both fixed and elastic demand are considered. In the fixed demand case, travel demand is represented as a random variable, whereas in the elastic demand case, a pre-specified random variable is introduced into the demand function. The paper also considers a set of assumptions of traveler behavior. In the first case, it is assumed that the traveler considers only the mean travel time in the route choice decision (risk-neutral behavior), and in the second, both the mean and the variance of travel time are introduced into the route choice model (risk-averse behavior). A closed-form formulation of the true marginal cost toll for the stochastic network (SN-MCP) is derived from the variational inequality conditions of the system optimum and user equilibrium assignments. The key finding is that the calculation of the SN-MCP model cannot be made by simply substituting related terms in the original MCP model by their expected values. The paper provides a general function of SN-MCP and derives the closed-form SN-MCP formulation for specific cases with lognormal and normal stochastic travel demand. Four numerical examples are explored to compare network performance under the SN-MCP and other toll regimes.     

Computational precision of traffic equilibria sensitivities in automatic network design and road pricing

Recent studies demonstrate the importance of computational precision of user equilibrium traffic assignment solutions for scenario comparisons. When traffic assignment is hierarchically embedded in a model for network design and/or road pricing, not only the precision of the solution itself becomes more important, but also the precision of its derivatives with respect to the design parameters should be considered.The main purpose of this paper is to present a method for precise computations of equilibrium derivatives. Numerical experiments are used for two evaluations: (1) precision of computed equilibrium derivatives for a medium-size network (Anaheim); and (2) the impact of precise derivatives on capacity-expansion solution quality for a small network (Sioux Falls).