Stability analysis of Activity-Based models

The Tel Aviv activity based model structure is similar to other activity based models described in the literature. The model run is supposed to converge to the equilibrium between generated tours and corresponding level of service (LOS) data. However, individual tour generation uses random draws for various choices (activity, time of day, destination, and mode). This introduces simulation errors, which combined with population sampling and limited precision of static traffic assignments, prevents the convergence of the model results. This paper analyses the above uncertainty sources on the basis of multiple model runs conducted for this study. Three averaging procedures are investigated and compared. Practical considerations regarding setting up the averaging procedures required for obtaining stable model results are discussed.


Introduction
Activity-Based Models (ABM) are disaggregate models that simulate individual decisions as random draw from choice sets, thus inserting random component to the results of model run. The Tel Aviv ABM structure is similar to other activity based models described in the literature. The model run is supposed to converge to the equilibrium between generated tours and corresponding level of service (LOS) data.

Goals of current study
Experience of working with Tel Aviv ABM has revealed the need in conducting comprehensive analysis of the randomness of the model results, and in developing practical methods for the model stability monitoring and control. Therefore, the goals of the current study are: -Analyze sources of randomness and their influence on model results -Produce practical recommendations for correct use of the ABM

Assignment errors
Three variants of traffic assignment implementation were compared:

Assignment errors: conclusions
The analysis of the assignment implementations has shown that usage of path-based algorithm may practically eliminate the assignment error. The time savings of the PB algorithm is further expanded due to extensive use of path analysis allowing to obtain various characteristics of assignment results very fast. The FW and FWP algorithms require conducting additional time consuming assignments.

Tour Generator -Simulation error
The starting point of the analysis is the demand matrices that result from the TG component. According to the flowchart presented in the previous slide, the individual random choices are aggregated to form the demand matrices. There are over 30 demand matrices generated by the model for different modes and periods of day. To analyse the TG randomness effects, we consider the car demand matrix for the AM period.

TOUR GENERATOR
The demand matrices are random The random draw of choices for each person in a sample (Activity, Destination, Period of Day, Mode, etc.) results in random matrices, for example: Although the overall distribution of trips in the matrix is quite stable, there are considerable changes at the cell level.

TOUR GENERATOR
• The convergence rate of all procedures follows the " " rule of thumb: • For MSA-R and MSA-M the VHT standard deviation decreases from the original 420 to 120-140 after 20 iterations, that is comparable to the expected 420/ 20 93.9; • The resulting VHT standard deviation of 35 for Quasi-aggregate procedure after 16 iterations with 6 inner iterations each is even closer to 420 / 16 ( 6 42.8 .

Run time [min]
Quasi-aggregate MSA-M MSA-R

Tour Generator (conclusions)
All three arrangements of model run with averaging results converge similarly. Note that the MSA-M procedure requires more time for assignments than MSA-R, since for path-based assignment used the run time depends on number of non-zero cells in demand matrix, and in MSA-M procedure the number of such cells increases with the iterations, whereas in MSA-R this number is almost constant. Further, Quasi-aggregate procedure has different proportion between number of TG runs and number of assignment runs, depending on amount of inner iterations.

Population Generator random component
PG generates list of individuals with random characteristics based on forecast of aggregate control variables.
In addition, in order to accelerate the model's run, a sample from the full population is often used.
-In this case, a sample is taken randomly, and trips of each person in a sample take proper weight to assure correct total number of trips in a system.
In this presentation only errors related to the population sampling will be addressed.

POPULATION GENERATOR
The results indicate that the " " rule of thumb works both for the number of iterations and for the sampling rate.

POPULATION GENERATOR
The sample size affects the stationary point of ABM results.
In addition, population sampling does not bring any significant savings in number of iterations, if the goal is to assure certain accuracy of ABM results. This is because the sampling would require more iterations to converge to the same accuracy in comparison to the full sample.

Conclusions
Three sources of the ABM results instability were analyzed: random population sampling, random tour generation, and assignment procedures.
In line with previous studies, the effect of assignment procedures may be practically eliminated when using path based assignment algorithms. The effect of randomness of tour generation may be decreased significantly by averaging the results of model run. The analysis of ABM stability allows developing practical measures for performing estimation of transportation projects with controlled accuracy 36

Further research
Population sampling increases the efficiency of the model run, but the relationship between model steady states with different samples required further analysis.
Two major issues emerged from the presented work: Analysis of the ABM steady states More profound study of errors related to Population generator: uncertainty of synthetic population created from limited set of aggregate control variables