Figures and data in Methodological Foundations of the Traffic Simulation Process

Table 1

Classification of traffic simulation models by scale

Scale	Level of detail	Region size	Mobility modeling	Assumptions	Modeling principles	Underlying algorithms	Software options
Microscopic	Each vehicle	Small city districts (e.g., freeway, arterial, corridor, etc.)	Individual dynamics of vehicles in intersections, specific roads, or small city zones.	Individual vehicles make decisions to get to their destination.	Individual mobility decisions in networks will represent the mobility system.	Longitudinal model, Lateral model, intersection model	Aimsun, Transmodeler, SUMO, PTV VISSIM, MATSIM, Cube Dynasim
Mesoscopic	Each vehicle. Movement is ruled by the average speed on the travel link.	City districts	Flows of vehicles between city zones by type of activity.	Vehicles moving in queues	Queues of individual vehicles that follow shockwave dynamics.	Longitudinal model, intersection model	Aimsun, Transmodeler, Mezzo Polaris, SUMO Meso
Macroscopic	Average vehicle dynamics, such as traffic density and average speed.	Entire cities	Flows of traffic between largely aggregated zones.	Aggregated vehicles flowing by main roads.	Aggregated vehicles flowing through roads based on fluid particle dynamics.	Flow model	Aimsun, Transmodeler, PTV VISSUM

Alexiadis et al., 2004; Algherbal and Ratrout, 2025; Azam et al., 2019; Lopez et al., 2018.

Table 2

Car-Following traffic models

Class	Models	Description
Gap acceptance	Gipps model (Gipps, 1981)	Based on the premise that each driver sets limits on their desired braking and acceleration rates. The speed is based on a safe following distance to avoid a possible collision with the vehicle ahead. The behavior is controlled with response times, break rates, and maximum desired speed. This model is widely preferred for simulation purposes (Ranjitkar and Kawamura, 2005).
Gap acceptance	Krauss model (Krauß, 1998)	Alike but simpler than Gipps’ model. A stochastic model based on the same assumption that vehicles move without colliding. The model is set up with the same parameters as in the Gipps model. It is implemented in the SUMO simulator (Kanagaraj et al., 2013; Ranjitkar and Kawamura, 2005).
Psycho-physiological driver behavior	Gazis-Hermann-Rothery (GHR) model	Referred to as the general car-following behavior. The relationship between a leader and a follower vehicle is a stimulus-response type of function where the follower vehicle’s acceleration is proportional to its speed, the speed difference with respect to the leader, and the space headway. This type of model assumes that the follower reacts to arbitrarily small changes in the relative speed (Janson Olstam and Tapani, 2004).
	Leutzbach and Wiedemann model (1986)	It takes into consideration the influence of the driver’s perception on velocity control. A driver can be in different driving modes, such as free driving, approaching, following, and braking. The driver switches from one mode to another as soon as they reach a certain threshold, which can be expressed as a function of speed difference, space headway, etc. Versions of this model are used in VISSIM. (Kanagaraj et al., 2013; Ranjitkar and Kawamura, 2005).
	Fritzsche model (1994)	This model also considers human perception in defining the model regimes. For example, there is a threshold for the minimum speed difference that the driver perceives. In addition, it incorporates four thresholds for the follower’s space headway to its leader: Desired distance, the risky distance, the safe distance, and the braking distance (Janson Olstam and Tapani, 2004).
Cell based model	Nagel-Schreckenberg model (cellular – automata) (Nagel et al., 1998)	A computationally efficient stochastic model that simulates the traffic flow using only integer operations. It incorporates imperfections of driving using a noise term in the update rules. The model deals with single-lane traffic flow of cars moving in a one-dimensional cellular chain under periodic boundary conditions, which considers the desired maximum speed, vehicle acceleration, deceleration, random delays, and update of vehicle location (Liu, 2012; Ranjitkar and Kawamura, 2005).
Trajectory based model	Newell model (2002)	Based on the premise that the driver of the following vehicle drives as a shifted space trajectory of the leader vehicle. The space trajectory of the following vehicle is the same as that of the leader vehicle except for a translation in space and in time. It is driven by a time lag and a distance lag (Ranjitkar and Kawamura, 2005).
Trajectory based model	Intelligent Driver Model (IDM)	IDM is a car-following model that describes the dynamics of the positions and velocities of single vehicles using ordinary differential equations (Gora et al., 2020).

Table 3

Zoning methods

Type	Author	Description
Constraint based	(Openshaw, 1977)	Maximization of the statistical precision of the estimation of the OD matrix cells.
	(Baass, 1981)	Trip generation/attraction homogeneity Adjustment of TAZ boundaries to political, administrative, or statistical ones. Minimization of intra-zonal trips
	(O’Neill, 1991)	Trip generation/attraction homogeneity Contiguity and convexity zones Compactness of TAZ shapes Exclusiveness (no doughnuts or islands) of zones Equity in terms of trip generation (small standard deviation across zones). Adjustment of TAZ boundaries to political, administrative, or statistical ones. Respect for physical separators Decision-makers' preferences are considered when defining the number of TAZs.
Model based	(Prosperi et al., 2021)	Traffic Zones Discretization and Origin-Destination Matrix Estimation by means of Fusion.
Model based	(Luo and Zhang, 2021)	TAZ are defined by combining static zoning and dynamic zoning, by adjusting the zones based on real-time traffic, enabling a maximum flow control strategy.
Data-Driven	(Martínez et al., 2009)	TAZs are defined based on actual travel demand patterns, using a density-smoothing algorithm.
Data-Driven	(Yang et al., 2022)	Clustering and GIS-based spatial analysis is used for grouping spatial units with similar mobility characteristics.
Criteria Based	(Crevo, 1991)	Minimize prediction errors associated with the spatial aggregation.
Criteria Based	(Ortúzar and Willumsen, 2011)	The aggregation error caused by the assumption that all activities are concentrated at the centroid of the TAZ should not be too large. Each zone must be compatible with other administrative divisions, particularly with census zones. TAZ should be as homogeneous as possible in terms of land use or composition. TAZ boundaries must be compatible with cordons, screen lines, and those of previous zoning systems. Avoid using main roads as zone boundaries. The zone configuration should facilitate the clear definition of centroid connectors. Zones do not have to be of equal size; if anything, they could be of similar dimensions in travel time units (i.e., smaller zones in congested areas).

Table 4

Required input data

Type	Example	Description	Used in:
Type	Example	Description	TM	SM	DC
Network	Road geometry and capacity	Number of lanes, designated turning lanes, length of lanes, free flow speed.		X
	Traffic controls	Traffic signals’ location, signal timing, etc. Also known as geometric data, it consists of the number of lanes, designated turning lanes, length, and free flow speed. Also, it may include data on traffic control devices, such as the location of traffic signals and their signal timing.		X
	Level of service (LoS)	The measure of the quality of flow through a roadway segment or an intersection.		X
	Location of activity enters	Predefined TAZ (attraction and generation zones).	X
	Transit system	Public transport information (type, lines, stops, users, etc.)	X	X
Vehicles	Vehicle mix	Composition of the vehicle fleet in the study area.	X	X
	Maximum acceleration rate	The performance obtained from vehicle characteristics like weight and power varies according to the type of vehicle. In general, for lower speeds, the acceleration rates are greater.		X
	LoS of vehicles	Capacity (number of passengers), idle capacity,
	Vehicle lengths	It is the average length of the vehicles in the study area, which may vary according to the region and the needs of the population.		X	X
Traffic	Volume studies	The annual average daily traffic: Count location and duration using physical counting. It can be done with Inductive loop, magnetic, pneumatic road tubes, active or passive infrared, microwave radar, ultrasonic, passive acoustic or video image processing.		X
	Directional volume studies	Entry volumes, turning volumes, and turning movements at intersections.		X	X
	Demand	Traffic reports (modal preferences, travel habits, OD matrices, accident reports)	X	X	X
	Travel time and delays studies	Average time that it takes to go from point A to B through studies requiring a test vehicle (floating-car technique, average speed-technique, moving vehicle techniques) or not (license-plate method and the interview method).		X
	Capacity flow data	Maximum volume per segment, maximum entrance rate, maximum (saturation) flow.	X	X
People	Temporal location	GPS position at a given time	X
	Socio-demographics	Age, education, economic activity, income, family composition, etc.	X
Driving characteristics	Aggressivity level	Average speed, free speed, and other safety perception parameters.		X	X
	Modal split	Proportion of mode and type of cars used by drivers.	X	X	X
	Average occupancy	Use of HOVs, trip generating activities.		X	X
	Accident rates	Frequency, location, time of the day of incidents and accidents.		X	X

TM = Transport demand modeling; SMB = Simulation model building; DC = driving characteristics;

Table 5

Transport demand models

	Trip-based approach (TBA)	Activity-based approach (ABA)
Scale	City or large urban region	Urban district or smaller
Type of simulation	Macroscopic	Mesoscopic or microscopic
Assumption about trips	Production and attraction of trips by zone (Ortúzar and Willumsen, 2011)	Trips generated and constrained by the activities of individuals
Spatial assumptions	Each zone only contains the centroid.	Zones contain individual households and activity centers (Codeca, 2022)
Approach	Individual, not related trips (Ortúzar and Willumsen, 2011)	Interrelated tours and trip chains
Required data	Zone aggregated data	Individual data by household and individual attributes
Comments	Data must be complete for each zone; fewer approximations are acceptable	Approximations at the individual level can lead to a good representation by aggregation.

Table 6

Different formulations of OD matrices required in simulation models

Input for simulation software	Description	Transformation to OD matrix
# Trips from centroid to centroid once a day	Software packages may require input trips that go from one location to another.	This is a direct formulation of the OD matrix, which could result from the generation stage of the transportation model, and can be requested as an input for each location.
# Trips from centroid to centroid over periods of time (hourly, 15 minutes)	Simulation software may require assigning trips at different time periods as the simulation resolution increases.	This is a formulation that is extended into a 3^rd dimension of time intervals (Fellendorf and Vortisch, 2010).
# Trips from centroid to centroid of different modes over periods of time, etc.	As more detail is used in the simulation, additional disaggregated information could be included as input for simulation software packages.	This formulation extends the previous OD matrices and may include several factors that differentiate the traffic as intended.
Probability that a user has a specific location, activity chain, and mode of transport	Meso- and microscopic simulation software will require a broad set of highly disaggregated data.	This formulation of the OD is disaggregated and may be obtained from the previously explained multidimensional OD matrix. It may require calculating the conditional probabilities of the individual’s travel characteristics (Codeca, 2022; Eom et al., 2026).

Table 7

Methods for travel demand modelling

Category	Method	Approach		Model Stage				Level			Details	Sources
Category	Method	TBA	ABA	G	D	M	A	Mi	Me	Ma	Details	Sources
Simple methods	Growth factor	X		X						X	Updating an old matrix using the actual number of trips.	(Furnes, 1965)
Simple methods	Tri- proportional	X		X						X	Growth factor considering cost distribution.	(Bierlaire, 1997)
Theoretical methods	Gravitational	X			X			X		X	Analogy with Newton’s gravitational law.	(Casey, 1955)
	Entropy maximization	X				X	X	X	X		Derived from the second law of thermodynamics.	(Ortúzar and Willumsen, 2011)
	Agent-based		X	X	X	X	X	X	X	X	Derived from cellular automata theory, using individual beha-vioral rules to define aggregate transportation.	(Zhang and Levinson, 2004)
Counting-based methods	Proportional assignment	X		X	X			X	X	X	Consider that the flow proportion using a specific link is inde-pendent of the complete traffic flow.	(Dial, 1971)
Counting-based methods	Restricted capacity assignment	X			X			X	X		Based on Wardrop equilibrium (1952), taking congestion into account. The flow proportion depends on the complete traffic flow.	(Beckman et al., 1956)
Estimation methods	Linear least squares	X		X	X			X	X	X	Fit the parameters of a model that minimizes the difference between estimated and real flows	(Bierlaire, 1997)
Estimation methods	Maximum Likelihood estimation		X		X	X		X	X	X	With a probability model and statistical hypotheses to make statistical inference.	(Ben-Akiva and Bierlaire, 1999)
Machine Learning methods	Classification		X			X		X	X	X	Algorithms based on decision trees that optimize the modeling of mode choice based on decision rules.	(Hillel et al., 2021)
	Clustering		X	X	X			X	X	X	Unsupervised algorithms that group zones in the study region. Based on individual characteristics.	(Hafezi et al., 2019)
	Regression	X	X	X	X		X	X	X	X	Supervised algorithms that forecast the demand for travel, the distribution of activities of trips, and can forecast.	(Rocha et al., 2021)

TBA: Travel-based approach, ABA: Activity-based approach, G: Generation stage, D: distribution stage, M: Mode choice stage, A: Trip assignment stage, I: Microscopic simulation level, E: Mesoscopic simulation level, and A: Macroscopic simulation level.

Table 8

Most common simulation software tools

Software	Cost & License	User-Friendliness	Modeling Approach	Visualization	Strengths (Power)	Limitations
Aimsun	Commercial, expensive; academic licenses available	User-friendly GUI, steep learning curve	Microscopic, mesoscopic, hybrid	Strong 2D/3D visualization	Flexible (multi-res), good for large urban projects	Costly, proprietary
TransModeler	Commercial (Caliper Corp.)	GUI is okay but less intuitive than Aimsun/VISSIM	Micro + meso + dynamic traffic assignment	3D visualization (moderate quality)	Strong for planning + operations integration	Less global user base, fewer academic users
SUMO	Free, open-source	Moderate (steep learning initially), script-based	Micro + meso	2D GUI, limited native 3D (extendable via Unity/ Blender)	Scalable (city-wide), great for AI, CAV, RL	Graphics not polished; scripting needed
PTV VISSIM	Commercial, very expensive	Very user-friendly GUI	Microscopic only	Excellent 2D/3D visuals	Most realistic driver behavior (Wiedemann), industry standard	High cost, slower on very large networks
PTV VISUM	Commercial, costly	GUI-focused, planner-friendly	Macroscopic (strategic planning)	Visualization is more schematic than realistic	Excellent for demand modeling, network assignment	Not for micro-level traffic operations
MATSim	Free, open-source	Code-based, not very beginner-friendly	Agent-based, mesoscopic	Limited built-in; external tools for visuals	Huge-scale simulations (whole regions)	Requires coding skills, less visual
Cube	Commercial (Bentley)	Planner-oriented GUI	Macroscopic (demand forecasting)	Limited visualization	Widely used for travel demand modeling	Not for microscopic operations
Dynasim	Commercial (PTV legacy)	Less common now	Microscopic	Decent visualization	Earlier detailed micro tool	Mostly replaced by VISSIM
Mezzo	Free (academic, KTH Sweden)	Basic, not very user-friendly	Mesoscopic (stochastic simulation)	Very limited	Efficient meso-sim, academic research	Not widely used, poor visuals
Polaris	Free (open source, Argonne National Lab)	Requires coding	Agent-based, mesoscopic	Limited	Strong research tool for large networks	Steep learning, low adoption

Table 9

Metrics used for calibrating traffic models.

Traffic model	Software	Parameter	Description
Gap acceptance	Based on Gipps (AIMSUM)	Look ahead distance or Safety gap or safety distance or Headway or min distance between vehicles	Necessary distance to avoid a collision if the leader decelerates heavily
Psycho-physiological driver behavior	Based on GHR models (MITSIM) SUMO VISSIM Based on Fritzsche (PARAMICS) Based on Wiedeman model	(Driver’s) Reaction time or Time headway	Can be the same for all drivers (macro) or specific per type of vehicle, per level of congestion (micro), for example. It influences density and flow.
		(Driver’s) magnitude of the reaction (acceleration, deceleration, or retardation rates)	Can influence travel time delay and average speed
		Car-following parameters (α, β, γ)	Proportionality rations between acceleration and speed, speed difference between follower and leader, and space headway.
		Max waiting time, Standstill distance, Speed model, critical gap, follow up gap.	The gaps can influence the maximum capacity of a subordinate flow within a node. The max waiting time is the longest time a vehicle of subordinate flow can wait to enter a node.
		Action points	Thresholds where the driver changes his/her behavior.
		Threshold for perception of negative and positive speed differences	Below these thresholds, the follower doesn’t perceive the speed differences.
		Desired time gap, risky time gap Safety time gap	Required to compute the thresholds: Desired distance, risky distance, safe distance, and braking distance.
		Several fix and random numbers for different thresholds.	Required to compute the thresholds: Desired distance between stationary vehicles, desired min following distance at low-speed differences, max following distance, approaching point, and decreasing, and increasing speed differences.
Cell based model	MATSIM	Length of the road, length of the cell, probability of decrease speed, max speed, probability of switch lane, initial density, number of cars, acceleration rate.	Parameters required to simulate one-lane highway traffic model with the Cellular automaton model.
Trajectory based model		Time lag, Distance lag Desired speed, Desired time gap, Minimum gap, Max acceleration, Comfortable deceleration	Required by the IDM model.

(Olstam and Tapani, 2004; Sun et al., 2020)

Table 10

Recommended error functions in traffic simulation

Function	Description	Formula	Recommendation / Assumption/Strength
R² Coefficient of determination	Indicates how well the simulated variable explains the real one.	$R^{2} = 1 - \frac{\sum {(y^{o} - y^{s})}^{2}}{\sum {(y^{o} - \bar{y})}^{2}}$	Assumes linear correlation. Easy to understand, and common [0,1]
GEH statistic (Geoffrey E. Havers)	Normalizes the relative differences with respect to the magnitude of the compared flows.	$G E H = \sqrt{\frac{2 \sum {(y^{o} - y^{s})}^{2}}{y^{o} + y^{s}}}$	Mitigates the misleading effects that relative differences may produce. <5: good fit, > 10: poor fit
MAPE Mean absolute percentage error	Indicates the average percentage difference between observed and predicted values.	$M A P E = \frac{100}{n} \sum^{n} \| \frac{y^{o} - y^{s}}{y^{o}} \|$	Easy to interpret as a percentage [0 – 100] <10%: good fit
RMSE Root Mean Square Error	Statistical measure that estimates the standard deviation of the errors distribution.	$R M S E = \sqrt{\frac{\sum {(y^{o} - y^{s})}^{2}}{n}}$	In the same units as the variable. Not a standard range. Penalizes large errors.
Relative differences with Cosine similarity	Similarity between two vectors or tensors by means of the dot product between them.	$R D = 1 - \frac{\| y^{o} \cdot y^{s} \|}{‖ y^{o} ‖ ‖ y^{s} ‖}$	For multi-dimensional variables. Easy to read. Range [0,1] >0.9: good fit

Table 11

Model performance metrics

Metric	Units	Description	Detail			Scale		Purpose
Metric	Units	Description	Mi	Me	Ma	N	I	I	O
Travel time	min/veh sec/veh	Average travel time of all vehicles of the simulation or in a specific facility or trajectory.	x	x	x	x		x	x
Travel Speed	km/h	Rate of motion (expressed in distance per unit of time)	x	x	x	x		x	x
Travel Distance	km/veh	Average extent of the space between the trip origin and the destination, measured along a vehicular route.	x	x	x	x		x	x
Delay	Sec or min	Additional travel time experienced by travelers at speeds less than the free flow (posted) speed.	x	x		x			x
Speed ratio	-	Degree of traffic flow through intersections (speed adjacent to the access point over speed along the corresponding road section).	x	x	x	x			x
Stopping frequency	% Stops per time-space bin	Average relative frequency of stops per time, space, or segment bin. Shown in histograms.	x	x		x			x
Volume of Traffic or Traffic Flow	% Vehicles or persons per time-space bin	Number of persons or vehicles passing a point (or intersection) on a roadway per time interval. Shown in histograms.			x	x		x	x
Congestion of streets	sec or min	Time lost per trips, level of congestion per road.			x	x		x	x
Saturation (flow) rate	% Vehicles per time bin	Maximum rate of flow of traffic in a road segment per time interval.	x	x	x	x		x	x
Density	veh/km or veh/km-lane or % Capacity of lane/road	Number of vehicles on the roadway segment averaged over space.	x	x	x	x		x	x
Queue Length	km or # Vehicles	Length of queued vehicles waiting to be served by the system.	x	x		x			x
Mode Split	% Persons	Percentage of travelers using each travel mode (SOV, HOV, transit, bicycle, pedestrian, etc.).				x		x	x
Time-Space Diagrams	km/h per (time-space bin)	3D graphs or surface response plots showing the speed at various times and spaces (positions).				x			x
Time-Flow Diagrams	% of maximum flow	Graphs show the flow behavior at various times with different densities.				x			x
Cost of travel (due to delays)	Time	The sum of the volume delay, turn penalty and junction delay functions.	x		x	x			x
Cost of travel	$ or time	Dynamic cost functions depend on the experienced travel time.	x	x		x	x		x

Mi: Microscopic or specific, Me: Mesoscopic, Ma: Macroscopic or network-wide, I: Input, O: Output, N: Network-wide, I: vehicle or user specific.

Table 12

Examples of expected results from different scenarios

Simulated scenario element modified	Examples	Expected results	Examples of Sources
Network	Add/remove stoplights, lanes, entrances, intersection designs, etc. Change of lane uses (bikes per vehicles) Different modal orientations of integrated urban transportation systems	Changes in congestion, in travel times, in transferring time, integration effectiveness.	(Fierek and Zak, 2012)
Vehicle	Change buses for electric buses Penetration of connected and automated vehicles	Changes in emissions Changes in congestion and network capacity Changes in accident rate and road safety	(Raju and Farah, 2021)
Traffic flow management	Lane sorting strategies Converting HOVs into CACC (Cooperative Adaptive Cruise Control) lanes Cooperative ITS (Intelligent Transportation Systems)	Changes in traffic streams, operational traffic performance	(Codeca and Härri, 2018; Raju and Farah, 2021)
Policy implementation	Floating cars versus local taxis fleets	Differences in passenger wait time, pickup trip time	(Maciejewski et al., 2016)

Figures

Proposed methodology for implementing traffic simulation at any scale, aiming at evaluating strategies to improve mobility in a given region.

Zoning examples of the same area: District Tec in Monterrey, Mexico.

Examples of OD matrices representations (a) single trips, one mode, (b) activity chains.

Comparison of available traffic simulation software capabilities

Validation example based on traffic volume profiles using five goodness-of-fit measures.

Examples of improvement implementations in the network.

Schemes of activity-based trips.

Tables

Classification of traffic simulation models by scale

Car-Following traffic models

Zoning methods

Required input data

Transport demand models

Different formulations of OD matrices required in simulation models

Methods for travel demand modelling

Most common simulation software tools

Metrics used for calibrating traffic models.

Recommended error functions in traffic simulation

Model performance metrics

Examples of expected results from different scenarios

Data and code availability

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