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ISSN No. 2454-6186 | DOI: 10.47772/IJRISS | Volume IX Issue XXVI October 2025 | Special Issue on Education
Mathematical Modelling and MPC-Based Optimisation of Urban
Traffic Flow in the Yaba and Sabo Areas of Lagos Metropolis
Ogiugo Mike E
*
, Omikunle Oluwafisayo, Nwagwo Alexander, Ayeni Olayinka, Amusa Sesan, Olusan
Bolanle
Department of Mathematics, Yaba College of Technology, Lagos, Nigeria
*Corresponding author
DOI: https://dx.doi.org/10.47772/IJRISS.2025.903SEDU0686
Received: 28 October 2025; Accepted: 04 November 2025; Published: 23 November 2025
ABSTRACT
Traffic jams in Lagos Metropolis are at critical levels, causing annual economic losses estimated in billions of
naira and significantly impacting the standard of living. Using Model Predictive Control (MPC), this paper
provides a detailed mathematical approach to simulate and enhance metropolitan traffic flow. By adapting the
Cell Transmission Model (CTM) to the unique characteristics of Lagos' road networks, we develop a
macroscopic traffic flow model. The proposed MPC system integrates real-time traffic data, predictive
capabilities, and constraint management to optimise signal timing and traffic routes. The results indicate that the
MPC-based method reduces queues by 24%, increases network throughput by 28%, and decreases average travel
time by 32%, compared to traditional fixed-time control methods. The model accounts for Yaba and Sabo Areas
specific challenges such as mixed traffic composition, informal public transportation networks, and
infrastructural constraints.
Keywords: Model Predictive Control, Traffic Flow Optimisation, Cell Transmission, Model Urban
Transportation, Yaba and Sabo Areas, Intelligent Transportation Systems
INTRODUCTION
Background and Motivation
Lagos State, with an estimated population exceeding 20 million people, faces severe traffic congestion challenges
that impede economic productivity and quality of life. The Lagos Metropolis serves as Nigeria’s economic hub,
generating approximately 30% of the nation’s GDP, yet traffic congestion costs the economy an estimated $2.5
billion annually [1, 12, 13]. Average commute times in Lagos exceed 3 hours daily, with some corridors
experiencing near-permanent gridlock during peak hours.
The traffic management challenges in Lagos are multifaceted:
1. Rapid urbanisation with vehicle population growth outpacing infrastructure development
2. Mixed traffic composition including private vehicles, commercial buses (danfo), motorcycles (okada), and
tricycles (keke)
3. Inadequate traffic signal systems with many intersections lacking functional signals
4. Limited enforcement of traffic regulations
5. Informal transportation networks operating outside formal planning frameworks
Traditional traffic control methods, such as fixed-time signal control and pre-timed coordination, have proven
insufficient for managing the dynamic and unpredictable nature of Lagos traffic. This necessitates the
development of intelligent, adaptive control strategies capable of responding to real-time conditions.
This research aims to develop a mathematical model of traffic flow dynamics specific to Lagos Metropolis
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particularly Yaba and Sabo areas, incorporating local transportation characteristics, and design an MPC
framework for real-time traffic signal optimisation and network flow management.
The remainder of this paper is organised as follows: Section 2 reviews related work in traffic modelling and MPC
applications. Section 3 presents the mathematical formulation of the traffic flow model. Section 4 details the
MPC framework design. Section 5 describes the simulation setup and case study. Section 6 presents results and
analysis. Section 7 discusses implementation considerations, and Section 8 concludes with future research
directions.
LITERATURE REVIEW
Traffic Flow Modelling
Traffic flow modelling has evolved through three primary paradigms: microscopic, mesoscopic, and macroscopic
models. Microscopic models, such as car-following models and cellular automata, track individual vehicle
behaviour but become computationally prohibitive for large networks [2]. Macroscopic models, inspired by fluid
dynamics, treat traffic as a continuous flow and are well-suited for network-level optimization.
The Lighthill-Whitham-Richards (LWR) model [3] forms the foundation of macroscopic traffic theory,
describing traffic flow as a conservation equation:
𝜕𝜌 𝜕(𝜌𝑣)
+ = 0 (1)
𝜕𝑡 𝜕𝑥
where ρ is traffic density and v is velocity.
Daganzo’s Cell Transmission Model (CTM) [4] discretizes the LWR model into cells, making it computationally
tractable for real-time applications. The CTM has been successfully applied in various urban contexts, though
its application to African megacities with unique traffic characteristics remains limited.
Model Predictive Control in Traffic Systems
Model Predictive Control has gained prominence in traffic management due to its ability to handle constraints,
optimise multiple objectives, and incorporate predictions. Bellemans et al. [5] demonstrated MPC for freeway
traffic control, achieving significant improvements in traffic throughput. Aboudolas et al. [6] developed a store-
and-forward model with MPC for urban networks, showing 15-20% improvements in total time spent.
Recent advances have explored distributed MPC architectures [7], robust MPC under uncertainty [8], and hybrid
MPC frameworks combining different traffic models [9]. However, most applications focus on developed
countries with homogeneous traffic and robust infrastructure.
Research on traffic management in developing countries has highlighted unique challenges: heterogeneous traffic
with weak lane discipline [10], informal transportation networks [11], and limited data infrastructure. Few studies
have applied advanced control techniques like MPC to these contexts, representing a significant research gap
this paper addresses.
Mathematical Modelling of Traffic Flow
Modified Model for Lagos Traffic Characteristics
To account for Lagos-specific conditions, we introduce modifications:
Mixed Traffic Adjustment
The flow capacity is adjusted for vehicle heterogeneity:
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(2)
where V is the set of vehicle types, α
v
is the proportion of vehicle type v, and PCE
v
is the passenger car equivalent.
For Lagos, typical values are:
• Cars: PCE = 1.0, α = 0.40
• Buses (danfo): PCE = 2.5, α = 0.25
• Tricycle (keke): PCE = 0.3, α = 0.20
• Trucks: PCE = 3.0, α = 0.15
Informal Stop Dynamics
Informal stops by commercial vehicles reduce effective capacity:
Qifinal= Qieff (1 − λi · Pstop) (3) where λ
i
is the proportion of commercial vehicles and P
stop
is the probability
of stop- ping (estimated at 0.15-0.25 for Lagos).
Network-Level Model
Aggregating all cells, the network state evolution is:
n(k + 1) = n(k) + B · q(k) + d(k) (4)
where n(k) ∈ R
Nc
is the state vector of all cells, q(k) ∈ R
Nf
is the flow vector, B is the incidence matrix,
and d(k) represents demand inputs.
Model Predictive Control Framework
MPC Formulation
The MPC optimisation problem at time step k is formulated as:
min _{𝑢(𝑘), … , 𝑢(𝑘 + 𝑁_𝑝 − 1)}
𝐽(𝑘) = ∑_{𝑗 = 𝑘}^{𝑘 + 𝑁_𝑝 − 1} 𝐿(𝑛(𝑗), 𝑢(𝑗))
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:
𝑛(𝑗 + 1) = 𝑓(𝑛(𝑗), 𝑢(𝑗), 𝑑(𝑗))
𝑛_𝑚𝑖𝑛 ≤ 𝑛(𝑗) ≤ 𝑛_𝑚𝑎𝑥
𝑢_𝑚𝑖𝑛 ≤ 𝑢(𝑗) ≤ 𝑢_𝑚𝑎𝑥
𝑢(𝑗) ∈ 𝑈_𝑠𝑖𝑔𝑛𝑎𝑙
𝑗 = 𝑘, … , 𝑘 + 𝑁_𝑝 − 1
where:
1. N
p
is the prediction horizon
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2. u(j) is the control input vector (signal timings)
3. L(·) is the stage cost function
4. f(·) is the system dynamics from Section 3
5. U
signal
represents signal timing constraints
Case Study: Yaba and Sabo Areas of Lagos Metropolis
Traffic data was collected and synthesised over 8 months (January-August 2025) for four strategic junctions in
the Yaba and Sabo areas of Lagos Metropolis. The dataset comprises:
1. 186,624 validated traffic records with 15-minute sampling intervals
2. Real-time speed, congestion factor, and delay measurements at each junction
3. GPS trajectory data from commercial vehicles (danfo, ride-hailing services)
4. Historical traffic patterns from Lagos State Traffic Management Authority (LASTMA)
5. Weather impact data during rainy season (April-July 2025)
The four study junctions are:
1. Herbert Macaulay Way / Murtala Mohammed Way Intersection (Yaba)
2. Tejuosho / St Finbarrs Road Intersection (Yaba)
3. Sabo / Ikorodu Road Junction
4. Queen Elizabeth / Sabo Road Intersection
Data quality control procedures removed 1.47% of records as outliers, yielding a final dataset quality of 98.5%.
Model Calibration
Key parameters calibrated using the empirical Yaba and Sabo Areas data:
Table 1: Calibrated Model Parameters from Yaba and Sabo Areas Data (Jan-Aug 2025)
Parameter
Value
Unit
Average network speed
29.57
km/h
Free-flow speed (urban)
45.0
km/h
Average congestion factor
1.48
-
Jam density
180
veh/km/lane
Wave speed
18
km/h
Saturation flow rate
1650
veh/h/lane
Average vehicle length
5.5
m
Lost time per phase
4
s
Mixed traffic adjustment
0.65
-
Average delay
202
seconds
Peak hour speed
15.6
km/h
Off-peak speed
32.4
km/h
The data reveals significant temporal variations: weekday average speed (23.70 km/h) versus weekend (44.07
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km/h), with Friday showing 25% higher congestion than other weekdays. Evening peak (5-7 PM) congestion
factor reaches 2.74, compared to 2.31 during morning peak (7-9 AM).
RESULTS AND ANALYSIS
Empirical Traffic Characterization
Analysis of 186,624 traffic records from January-August 2025 reveals severe congestion patterns across all four
junctions. Table 2 summarises key traffic metrics.
Table 2: Empirical Traffic Statistics (January-August 2025)
Metric
Mean
Min
Max
Speed (km/h)
29.57
5.00
60.00
Congestion Factor
1.48
0.30
4.00
Delay (seconds)
201.88
0.00
941.20
Travel Time Index
1.85
1.00
4.79
Figure 1: Daily Average Traffic and Daily Average Congestion Factors
Junction-Specific Performance
Junction performance analysis reveals significant spatial heterogeneity:
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Table 3: Performance Metrics by Junction
Junction
Avg Speed
Congestion
Avg Delay
(km/h)
Factor
(seconds)
J1: Herbert Macaulay-Yaba
30.55
1.48
202.87
J2: Tejuosho-Yaba
26.15
1.48
192.04
J3: Sabo-Ikorodu
35.14
1.48
206.14
J4: Sabo-Queen Elizabeth
26.43
1.48
206.46
Figure 2: Daily Time Series
Junction 3 (Sabo-Ikorodu) demonstrates the highest efficiency with 35.14 km/h average speed, while Junction 2
(Tejuosho-Yaba) shows the lowest at 26.15 km/h, representing a 34.4% performance gap.
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Temporal Analysis
Strong temporal patterns emerge from the data with statistically significant differences (p < 0.001):
Table 4: Traffic Performance by Time Period
Time Period
Avg Speed
Congestion
Avg Delay
(km/h)
Factor
(seconds)
Night (00:00-06:00)
50.11
0.58
25.63
Morning (06:00-10:00)
18.68
2.11
352.49
Midday (10:00-15:00)
18.12
1.64
191.70
Afternoon (15:00-19:00)
13.95
2.60
466.24
Evening (19:00-24:00)
34.45
1.11
108.06
Afternoon period (15:00-19:00) exhibits the worst performance with only 13.95 km/h average speed and
congestion factor of 2.60, compared to night-time free-flow conditions (50.11 km/h, CF=0.58).
Figure 3: Average Speed & Congestion Factors (Hourly Heatmaps)
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Figure 4: Distributions
Figure 5: 4 junctions Comparison
Statistical Hypothesis Testing
Multiple hypothesis tests validate significant traffic pattern differences:
1. Weekday vs Weekend: Weekday mean speed (23.70 km/h) significantly lower than weekend (44.07
km/h); t=-207.75, p < 0.001
2. Peak vs Off-Peak: Peak hour speed (15.59 km/h) versus off-peak (32.36 km/h); t=-132.72, p < 0.001
3. Friday Effect: Friday congestion (2.04) is significantly higher than other weekdays (1.75); t=36.30, p <
0.001
4. Junction Differences: ANOVA confirms significant speed variations across junctions; F=1896.03, p <
0.001
Level of Service Distribution
Traffic conditions categorised by Level of Service (LOS) standards:
Table 5: Level of Service Distribution
LOS Category
Percentage
Congestion Range
A (Free Flow)
47.3%
CF < 1.0
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B (Stable Flow)
5.9%
1.0 ≤ CF < 1.3
C (Stable but Restricted)
7.5%
1.3 ≤ CF < 1.6
D (Approaching Unstable)
11.1%
1.6 ≤ CF < 2.0
E (Unstable)
7.9%
2.0 ≤ CF < 2.5
F (Forced Flow)
20.3%
CF ≥ 2.5
Critically, 20.3% of traffic operates under Level F (forced flow) conditions, with an additional 19.0% in unstable
or near-unstable states (LOS D-E), indicating severe systemic congestion.
Correlation Analysis
Strong correlations identified between key variables:
• Speed vs Congestion Factor: r = -0.857 (strong negative correlation)
• Congestion vs Delay: r = 0.972 (very strong positive correlation)
• Speed vs Delay: r = -0.750 (strong negative correlation)
• Hour of Day vs Congestion: r = 0.327 (moderate positive correlation)
The extremely strong correlation (r = 0.972) between congestion factor and delay validates the theoretical
relationship in the Cell Transmission Model and justifies the use of congestion as a primary control objective.
MPC Performance Comparison
Using the empirical baseline, MPC simulations demonstrate substantial improvements:
Table 6: Performance Comparison: Baseline vs MPC-Optimised
Metric
Baseline
Fixed-Time
MPC
Improvement
(Empirical)
Control
Optimized
(%)
Avg Speed (km/h)
29.57
28.30
42.15
42.5
Congestion Factor
1.48
1.52
0.98
33.8
Avg Delay (sec)
201.88
215.40
98.25
51.3
Peak Hour Speed
15.59
14.80
23.50
50.8
Travel Time Index
1.85
1.92
1.18
36.2
Queue Length (veh)
185
192
118
36.2
MPC achieves 42.5% improvement in average speed and 51.3% reduction in delays compared to the empirical
baseline, demonstrating substantial potential for traffic flow optimisation.
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Peak Hour Performance Analysis
Detailed analysis of critical peak periods reveals MPC’s effectiveness during the highest congestion:
Table 7: Morning Peak (7-9 AM) vs Evening Peak (5-7 PM) Performance
Metric
Morning Peak
Evening Peak
Difference
Baseline (Empirical):
Avg Speed (km/h)
17.98
13.85
-22.9%
Congestion Factor
2.31
2.74
+18.4%
Avg Delay (seconds)
414.87
506.18
+22.0%
MPC-Optimized:
Avg Speed (km/h)
26.45
21.80
-17.6%
Congestion Factor
1.42
1.68
+18.3%
Avg Delay (seconds)
178.20
248.50
+39.5%
Improvement (%):
Speed Improvement
47.1
57.4
-
Delay Reduction
57.0
50.9
-
Evening peak shows 57.4% speed improvement under MPC, compared to 47.1% in morning peak, indicating
greater optimisation potential during the most congested periods.
Figure 6: Speed and Congestion Relationship
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Figure 7: Peak Hour Analysis
Figure 8: Weekly Traffic Pattern
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Figure 9: Monthly Traffic Trends
DISCUSSION AND KEY FINDINGS
The research demonstrates several critical findings:
1. Successfully modelling mixed traffic (cars, danfo buses, Tricycles(keke), okada, trucks) with informal stop
patterns validates MPC flexibility beyond homogeneous developed-country traffic.
2. Established that MPC effectiveness scales with baseline congestion severity, making it particularly suitable
for developing cities with severe traffic challenges.
3. Validated the model using extensive empirical data comprising 186,624 traffic records collected over eight
months from four strategic junctions in Yaba and Sabo areas.
4. Designed an MPC framework that achieved significant improvements: 42.5% increase in average speed,
51.3% reduction in delays, and 33.8% improvement in congestion factors compared to baseline conditions.
Figure 11: Traffic Comprehensive Dashboard
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CONCLUSION AND RECOMMENDATIONS
This paper has developed and validated a Model Predictive Control framework for optimizing traffic flow in
Yaba and Sabo Areas of Lagos Metropolis, addressing one of Africa's most severe urban congestion challenges.
Through rigorous analysis of 186,624 traffic records collected over eight months across four strategic junctions
in the Yaba and Sabo areas, we have established both the magnitude of the problem and the potential of intelligent
control solutions. The results demonstrate that MPC-based control achieves dramatic improvements over
traditional fixed-time signal systems: 42.5% increase in average speed, 51.3% reduction in delays, and 33.8%
reduction in congestion factor. More generally, this paper supports the increasing awareness that African cities
don’t always have to just imitate developed countries’ transport problems and remedies. Using contemporary
optimization theory, universal sensors (smartphones, GPS), and cloud computing, cities such Lagos can have
sustainable, effective mobility networks that meets or outperform developed-country results.
ACKNOWLEDGMENTS
We thank Yaba College of Technology for computational resources and the community stakeholders who
participated in consultations. TETFund IBR partially funded this research.
Competing Interests
The authors declare no competing financial or non-financial interests related to this work.
Data Availability
Aggregated traffic data supporting this study’s findings are available from the corresponding author upon
reasonable request. Raw GPS trajectory data cannot be shared due to privacy restrictions.
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