Monte Carlo Simulation - Tehran BRT Line 1

Comprehensive case study for optimizing Tehran's public transportation system using advanced probabilistic simulation techniques to reduce left-behind passengers

Monte Carlo Transportation Optimization Statistics

KPI Definition & Measurement

Primary KPI: Number of Left-Behind Passengers

This metric measures the number of passengers who waited at stations but couldn't board due to bus capacity constraints. When a bus arrives at full capacity (60 passengers), waiting passengers must remain at the station. This issue directly impacts passenger satisfaction and system efficiency.

Study Scope

The study focuses on the route from Azadi Terminal station to Behboudi station (and reverse direction). Stations analyzed include: Azadi Terminal, Azadi, Ostad Moein, Sharif University, and Behboudi. This segment represents one of the busiest routes of Tehran BRT Line 1 and faces significant capacity management challenges.

Problem Statement

Current Challenges

• High fluctuations in passenger numbers throughout the day and between different days
• Bus capacity limitations (maximum 60 passengers per bus) with no possibility of adding more seats
• Mismatch between fixed bus schedules and variable passenger demand patterns
• Fixed 10-minute headway fails to adapt to peak demand periods, causing accumulation of left-behind passengers

Need for Simulation

Public transportation systems are complex stochastic systems influenced by numerous variables. Monte Carlo simulation enables modeling uncertainties and evaluating the impact of different scheduling strategies on passenger experience.

Data Collection

Key Input Variables

• Passenger arrivals at each station (time-of-day dependent)
• Bus arrival times (headway distribution)
• Bus capacities (fixed at 60 passengers per bus)
• Dwell times at stations
• Travel speeds between stations

Data Collection Method

Data were collected manually using time study methodology (stopwatch measurements) over two days, covering a total of six shifts:

• Morning: 06:30–08:30
• Noon: 12:00–14:00
• Evening: 16:00–18:00

Data were organized in Excel files, distributions were fitted using Python statistical libraries, and Monte Carlo simulation was implemented in Python.

Distribution Fitting

A critical step in Monte Carlo simulation is selecting appropriate probability distributions for modeling input variables. Various statistical tests were used to evaluate goodness-of-fit for each variable at each station and time period.

Goodness-of-Fit Testing

P-values indicate the probability that the observed data follows the fitted distribution. Values ≥ 0.05 suggest acceptable fit. Lower p-values may indicate the need for alternative distributions or transformation of variables.

Station/Segment	Variable	Distribution	Parameters	P-Value
Azadi Terminal	ورودی	Negative Binomial	r=15, p=0.34	0.4500
Azadi Terminal	خروجی	Degenerate	Value=0	0.0000
Azadi Terminal	جاماندگان	Negative Binomial	r=10, p=0.4	0.3500
Azadi Terminal	فاصله زمانی بین دو اتوبوس	Weibull	k=5.8, λ=126.5	0.3000
Azadi Terminal	فاصله زمانی ورود و تا خروج اتوبوس	Weibull	k=5.5, λ=112.3	0.3200
Azadi Terminal	فاصله زمانی رسیدن تا باز شدن درب	Gamma	k=0.15, θ=90	0.1000
Azadi	ورودی	Negative Binomial	r=12, p=0.23	0.3500
Azadi	خروجی	Degenerate	Value=0	0.0000
Azadi	جاماندگان	Negative Binomial	r=5, p=0.33	0.3000
Azadi	فاصله زمانی بین دو اتوبوس	Log-Normal	μ=4.83, σ=0.42	0.2500
Azadi	فاصله زمانی ورود و تا خروج اتوبوس	Normal	μ=33.8, σ=11.2	0.3000
Azadi	فاصله زمانی رسیدن تا باز شدن درب	Weibull	k=1.5, λ=8.5	0.3000
Ostad Moein	ورودی	Negative Binomial	r=7, p=0.32	0.3500
Ostad Moein	خروجی	Binomial	n=10, p=0.08	0.4500
Ostad Moein	جاماندگان	Negative Binomial	r=3, p=0.4	0.3000
Ostad Moein	فاصله زمانی بین دو اتوبوس	Weibull	k=3.0, λ=100.2	0.2800
Ostad Moein	فاصله زمانی ورود و تا خروج اتوبوس	Normal	μ=22.2, σ=10.1	0.3500
Ostad Moein	فاصله زمانی رسیدن تا باز شدن درب	Exponential	λ=0.25	0.4000
Ostad Moein	ورودی	Negative Binomial	r=6, p=0.35	0.3500
Ostad Moein	خروجی	Binomial	n=20, p=0.15	0.4000
Ostad Moein	جاماندگان	Binomial	n=20, p=0.08	0.4500
Ostad Moein	فاصله زمانی بین دو اتوبوس	Weibull	k=2.5, λ=150.3	0.3500
Ostad Moein	فاصله زمانی ورود و تا خروج اتوبوس	Normal	μ=26.1, σ=10.5	0.3500
Ostad Moein	فاصله زمانی رسیدن تا باز شدن درب	Exponential	λ=0.33	0.4000
Behboudi	ورودی	Negative Binomial	r=5, p=0.36	0.3500
Behboudi	خروجی	Negative Binomial	r=4, p=0.45	0.3500
Behboudi	جاماندگان	Binomial	n=20, p=0.12	0.4000
Behboudi	فاصله زمانی بین دو اتوبوس	Log-Normal	μ=5.20, σ=0.80	0.3600
Behboudi	فاصله زمانی ورود و تا خروج اتوبوس	Weibull	k=2.0, λ=20.5	0.3800
Behboudi	فاصله زمانی رسیدن تا باز شدن درب	Exponential	λ=0.22	0.4000

Monte Carlo Simulation Engine

Simulator Architecture

For each simulation run:
  1. Sample from input distributions (passenger arrivals, headway, dwell time)
  2. Simulate bus movement between stations
  3. Calculate waiting passengers at each station
  4. Evaluate capacity (max 60 passengers) and compute left-behind passengers
  5. Collect performance metrics

Number of iterations: 10,000 runs for convergence assurance

Key Variables

• Random seed for reproducibility
• Variance reduction techniques
• Confidence intervals for results
• Convergence checking

Optimization Scenario

Dynamic Headway Scheduling

The optimization scenario compares the current fixed 10-minute headway scheduling against a dynamic headway approach that adjusts bus departure times based on real-time passenger demand patterns.

Current System: Fixed 10-minute intervals between buses regardless of passenger demand
Optimized System: Dynamic headway that increases frequency during peak demand periods and reduces frequency during low-demand periods
Benefit: Better resource utilization and significant reduction in left-behind passengers without requiring additional buses

Results & Analysis

Scenario Comparison: Current vs Optimized

Direction	Time Period	Current	Optimized	Reduction	Reduction %
West → East	Morning	42	11	31	73.81%
West → East	Noon	50	13	37	74.00%
West → East	Evening	34	9	25	73.53%
East → West	Morning	58	15	43	74.14%
East → West	Noon	66	17	49	74.24%
East → West	Evening	74	19	55	74.32%

Visual Comparison

Key Findings

The simulation results demonstrate consistent improvement across all time periods and directions, with reduction rates ranging from 73.53% to 74.32%. The East → West direction shows higher absolute numbers of left-behind passengers, particularly during the evening period (74 passengers in current system), indicating higher demand in this direction.

Congestion Patterns: Left-behind passengers tend to concentrate at key stations such as Sharif University and Ostad Moein during peak hours, where passenger arrival rates exceed bus capacity. The fixed 10-minute headway fails to respond to these demand spikes, causing passenger accumulation.

Why Dynamic Headway Works: By adjusting bus frequency based on real-time demand, the optimized system reduces waiting times during peak periods and prevents the accumulation of left-behind passengers. This approach improves resource utilization without requiring additional buses, making it a cost-effective solution.