Project – Mathematical Modelling of Urban Traffic Congestion and Optimal Signal Timing Under Uncertain Traffic Flow. A Study of Selected road intersections along the Abuja city transport network.

CHAPTER ONE

INTRODUCTION

1.1 Background to the Study

Urban transportation is an essential component of economic development, social interaction, and the efficient movement of people and goods. Road networks connect residential areas, commercial centres, educational institutions, government offices, industrial facilities, and other urban activities. As cities expand, the demand for road transportation often increases, placing pressure on existing infrastructure and traffic-management systems. When the number of vehicles approaching a road intersection exceeds the rate at which the intersection can discharge them, queues develop and delays increase. Traffic congestion can consequently result in longer travel times, increased fuel consumption, unreliable journeys, environmental pollution, and reduced productivity. The relationship between traffic demand, road capacity, and delay is therefore central to understanding urban congestion. In their analysis of congestion in the Federal Capital Territory, Abdul, Gani, and Umaru examined traffic congestion through a demand-and-supply perspective and discussed the consequences of growing automobile use and inadequate capacity for urban mobility (Abdul et al., n.d.). These concerns establish the importance of developing analytical methods that can describe traffic conditions and support effective management of urban road networks.

Mathematical modelling provides a systematic approach to understanding the complex relationships between traffic volume, vehicle speed, road capacity, queue formation, and travel delay. Instead of relying exclusively on general observations, mathematical models represent traffic conditions through variables, equations, parameters, and constraints that can be analysed using quantitative techniques. Traffic-flow theory commonly distinguishes between macroscopic approaches, which describe aggregate characteristics such as traffic density and average speed, and microscopic approaches, which represent the behaviour and movement of individual vehicles. Lighthill and Whitham (1955) developed a foundational theory of traffic flow that treats traffic as a continuum, while Richards (1956) independently developed a related formulation for analysing traffic movement and the propagation of disturbances. These foundational contributions demonstrate how mathematical relationships can be used to explain the development and movement of traffic waves. In an urban setting such as Abuja, mathematical modelling can help identify how variations in traffic demand, road capacity, and intersection control contribute to congestion, while also providing a basis for comparing alternative traffic-management strategies.

Road intersections are particularly important in urban traffic analysis because they create points where vehicle movements from different approaches compete for limited road space and signal time. At signalised intersections, the allocation of green time, cycle length, phase sequence, and clearance intervals influences how efficiently vehicles pass through the junction. Poorly allocated signal time may cause unnecessary waiting on heavily used approaches while providing excessive green time to approaches with lower demand. Webster (1958) established a widely recognised analytical method for determining traffic signal settings, including the calculation of cycle length and green-time allocation to reduce average delay under specified traffic conditions. Akçelik (1981) further developed methods for analysing signal capacity, traffic delays, queue lengths, and timing requirements. These approaches provide an important foundation for the mathematical optimisation of signal timing. However, the performance of a signal plan depends on the accuracy of the traffic-flow assumptions used in its development. Where traffic demand varies substantially across time, a timing plan that performs adequately during one period may become inefficient during another. Consequently, intersection modelling must account for both the operational characteristics of the road and the changing nature of traffic demand.

Uncertainty in traffic flow is a major consideration in modern urban transportation planning. Traffic volumes are rarely constant throughout the day, as they are influenced by commuting patterns, weather conditions, road incidents, public events, school schedules, vehicle breakdowns, roadworks, and changes in driver behaviour. These factors can cause traffic demand to fluctuate over short periods, making it difficult to determine an optimal signal-timing plan from a single traffic count or average flow estimate. When actual traffic demand differs from the demand assumed in a model, the resulting signal settings may produce excessive queues, longer delays, or spillback into neighbouring intersections. Research on robust traffic signal optimisation has examined methods for managing variations in traffic demand and improving signal performance under uncertain conditions (Gartner et al., 1991; Robust Optimal Traffic Signal Timing, 2009). Such approaches recognise that an effective signal plan should not merely perform well under an expected traffic scenario but should also remain reasonably effective when actual conditions differ from forecasts. Incorporating uncertainty into traffic models is therefore important for developing signal-timing strategies that are more responsive to the changing conditions of urban road networks.

Optimal signal timing involves selecting the combination of cycle length, green-time allocation, and signal phases that best satisfies defined operational objectives while respecting engineering and safety constraints. Depending on the objectives of a study, an optimisation model may seek to minimise average vehicle delay, reduce queue lengths, limit the number of stops, improve intersection throughput, or balance performance among competing approaches. These objectives may sometimes conflict. For example, allocating additional green time to a heavily congested approach may reduce its queue but increase waiting time for traffic on another approach. Webster’s (1958) method provides a foundational approach for estimating efficient signal settings, while Akçelik (1981) provides more detailed procedures for analysing capacity and delay under different intersection conditions. Later research has also considered the minimisation of delay variability and the probability of queue spillback, rather than focusing exclusively on average delay (Analytical Derivation of the Optimal Traffic Signal Timing, 2018). This wider perspective is important because a signal plan that minimises average delay may still perform poorly during unusual traffic surges. Accordingly, a mathematical model for Abuja’s road intersections should identify appropriate performance measures and constraints while evaluating the consequences of uncertain traffic demand.

The problem of traffic congestion has particular relevance to Abuja, Nigeria’s Federal Capital Territory, where the movement of commuters, commercial vehicles, public transport, and government-related traffic places demands on the city’s transport network. Traffic congestion in the FCT has been associated with growing automobile use, travel-time unreliability, fuel consumption, and environmental and social costs (Abdul et al., n.d.). Research on the Abuja–Keffi corridor has also investigated congestion and travel-time variability, highlighting the importance of understanding local traffic conditions when considering mitigation measures (Ibitoye & Abosede, 2012). More recently, a study of the Federal Medical Centre intersection along Jabi Bye-pass, Abuja, assessed traffic volumes and signal operations and used intersection analysis and simulation to investigate possible improvements to signal timing and intersection performance (Ohiomah et al., 2025). These studies indicate that quantitative traffic assessment and signal optimisation are relevant to Abuja’s transport challenges. Nevertheless, the traffic characteristics of individual intersections may differ according to road geometry, turning movements, traffic composition, signal operation, and surrounding land use. This makes it necessary to examine selected intersections individually rather than assume that one signal-timing arrangement will be equally effective throughout the city.

Despite the availability of established traffic-flow theories and signal-optimisation techniques, there remains a practical need to develop and evaluate mathematical models that reflect the variable traffic conditions experienced at selected intersections along the Abuja city transport network. A model calibrated using observed traffic volumes, turning movements, queue lengths, and signal timings could be used to estimate congestion, evaluate existing operational performance, and compare alternative timing strategies. Where traffic-flow uncertainty is incorporated through probability distributions, scenario analysis, sensitivity analysis, or stochastic optimisation, the model may also reveal how robust a proposed signal plan is under changing demand. The study will therefore focus on developing a mathematical framework for modelling urban traffic congestion and determining optimal signal timing under uncertain traffic flow at selected road intersections in Abuja. By comparing existing signal operations with model-derived alternatives using suitable measures such as average delay, queue length, intersection capacity, and traffic throughput, the study seeks to provide an evidence-based approach to improving intersection performance. The findings may assist traffic engineers, transport planners, and relevant authorities in identifying feasible signal-timing improvements while recognising the operational uncertainty that characterises urban traffic systems.

1.2 Statement of the Problem

Urban traffic congestion remains a significant transportation challenge because it reduces the efficiency with which road networks move people and goods. At road intersections, congestion develops when vehicle arrivals exceed the available discharge capacity, when turning movements conflict, or when signal timing does not adequately reflect traffic demand. These conditions may lead to extended queues, increased travel time, fuel wastage, and unreliable journeys. Evidence from research on traffic congestion in the Federal Capital Territory identifies the imbalance between transport demand and available capacity as an important contributor to congestion and its associated consequences (Abdul et al., n.d.). Although road expansion and other infrastructure improvements can address some capacity constraints, they may be expensive, require substantial planning, and take considerable time to implement. Improving the operational performance of existing intersections through appropriate mathematical modelling and signal timing therefore represents an important alternative that requires systematic investigation.

A further problem is that traffic signal timing may not adequately respond to variations in traffic demand across different periods of the day. Fixed signal plans are generally developed using selected traffic observations or design-period estimates. When actual traffic volumes differ from the values used to establish those plans, some approaches may receive insufficient green time while others receive more time than their prevailing demand requires. The result can be increased delay, longer queues, and inefficient use of intersection capacity. Although established methods such as Webster’s (1958) signal-timing procedure provide useful analytical foundations, traffic demand that varies over time creates additional challenges for determining and maintaining efficient signal settings. Research on robust signal optimisation has consequently considered the effects of demand fluctuations and the need for timing plans that perform satisfactorily across multiple traffic scenarios (Robust Optimal Traffic Signal Timing, 2009). The problem is therefore how to develop signal-timing strategies that account for uncertain traffic conditions rather than relying exclusively on a single estimate of expected traffic flow.

Another problem concerns the limited usefulness of signal-timing recommendations when mathematical models do not adequately reflect the characteristics of the intersections under investigation. Traffic performance depends on factors such as approach geometry, lane configuration, turning movements, vehicle composition, saturation flow, pedestrian activity, and the timing of conflicting signal phases. A model that ignores these factors may generate timing recommendations that are mathematically feasible but operationally unsuitable. Research on Abuja’s transport network has examined congestion and travel-time variability along the Abuja–Keffi corridor (Ibitoye & Abosede, 2012), while an assessment of the Federal Medical Centre intersection along Jabi Bye-pass has examined existing signal operations and possible improvements using traffic analysis and simulation (Ohiomah et al., 2025). These studies provide relevant local context, but the conditions at one intersection cannot automatically be assumed to represent all other intersections. There is therefore a need to investigate selected road intersections using observed local traffic data and an explicit mathematical framework that can evaluate their distinct operational characteristics.

The central problem addressed by this study is the need to determine how mathematical modelling can support congestion analysis and more reliable signal timing at selected intersections along the Abuja city transport network when traffic flow is uncertain. Without an appropriate model, decision-makers may find it difficult to quantify the effect of traffic fluctuations, compare alternative signal plans, or identify the timing strategy that provides the best balance between delay, queue length, and throughput. The study will address this problem by modelling traffic conditions using relevant intersection data, representing uncertainty through suitable analytical or computational techniques, and evaluating alternative signal-timing strategies against defined performance measures. Its purpose is not to assume that an optimised model will eliminate congestion, but to establish whether the proposed approach can improve intersection performance compared with existing or baseline signal settings under the traffic scenarios considered.

1.3 Aim and Objectives of the Study

The aim of this study is to develop a mathematical model for analysing urban traffic congestion and determining optimal traffic signal timing under uncertain traffic flow at selected road intersections along the Abuja city transport network.

The specific objectives are to:

1. Develop a mathematical model for describing traffic flow, queue formation, and congestion at selected road intersections along the Abuja city transport network.

2. Analyse the effects of uncertain traffic demand on vehicle delay, queue length, and intersection performance at the selected intersections.

3. Determine optimal traffic signal timings that minimise traffic delay and improve intersection performance under varying traffic-flow conditions.

4. Evaluate the performance of the proposed optimal signal-timing model against existing or baseline signal-timing arrangements using appropriate traffic-performance indicators.

1.4 Research Questions

The study will be guided by the following research questions:

1. How can a mathematical model be developed to represent traffic flow, queue formation, and congestion at selected intersections along the Abuja city transport network?

2. How does uncertainty in traffic demand affect vehicle delay, queue length, and intersection performance at the selected intersections?

3. What signal-timing arrangements can minimise traffic delay and improve intersection performance under varying traffic-flow conditions?

4. How does the performance of the proposed optimal signal-timing model compare with existing or baseline signal-timing arrangements?

1.5 Research Hypothesis

The following null hypothesis will be tested at the 0.05 level of significance:

H₀: There is no statistically significant difference in traffic delay at the selected road intersections between the proposed optimised signal-timing model and the existing or baseline signal-timing arrangements under uncertain traffic-flow conditions.

The hypothesis can be tested by comparing delay estimates obtained from the baseline and optimised signal plans using observed traffic data, simulation, or repeated traffic-demand scenarios. The statistical test should reflect the structure of the data and the modelling approach used.

1.6 Significance of the Study

The findings of this study are expected to be useful to traffic engineers, transport planners, government agencies, road users, researchers, and other stakeholders involved in urban mobility and traffic management.

Traffic engineers: The study may provide a mathematical framework for estimating intersection performance and selecting signal timings under fluctuating traffic demand. It may help engineers assess cycle length, green-time allocation, queue development, and traffic delay using measured traffic conditions rather than relying exclusively on fixed assumptions.

Transport planners: The results may assist transport planners in identifying congestion-prone intersections and comparing operational improvements before considering more expensive infrastructure interventions. The model may also provide a basis for evaluating whether adjustments to signal timing could improve the use of existing road capacity.

Federal Capital Territory Administration and relevant traffic-management authorities: The findings may provide useful analytical evidence for reviewing signal operations at selected intersections. The model could support decisions concerning traffic-monitoring priorities, signal retiming, and the evaluation of alternative traffic-management scenarios. However, any implementation would require validation against actual intersection conditions and relevant safety requirements.

Road users and commuters: Improved signal timing may reduce waiting time, increase travel-time reliability, and reduce unnecessary stopping at intersections. If the proposed model produces measurable improvements, its application may contribute to more predictable journeys and more efficient use of urban road infrastructure.

Researchers and students: The study may contribute to research in applied mathematics, transportation engineering, operations research, and traffic-flow modelling. It may also provide a methodological foundation for subsequent studies involving stochastic modelling, robust optimisation, adaptive traffic control, and coordinated signal timing across multiple intersections.

Policy and infrastructure decision-makers: By estimating the potential benefits of improved signal timing, the study may help decision-makers compare operational traffic-management measures with other interventions. This can support more evidence-based planning, particularly where infrastructure expansion is financially or physically constrained.

1.7 Scope of the Study

The geographical scope of this study is limited to selected road intersections along the Abuja city transport network in the Federal Capital Territory, Nigeria. The specific intersections will be selected based on their relevance to urban traffic movement, the availability of traffic data, and the feasibility of observing or modelling their operational conditions.

The content scope covers mathematical modelling of traffic flow, queue formation, congestion, and signal timing under uncertain traffic demand. The principal variables may include traffic volume, turning movements, vehicle arrival rates, saturation flow, cycle length, green-time allocation, queue length, vehicle delay, and intersection throughput. The study will examine the relationship between traffic-flow uncertainty and intersection performance, with particular attention to how alternative signal-timing plans perform under different traffic scenarios.

Methodologically, the research may use field traffic counts, existing signal-timing information, intersection geometry, and other relevant operational data. A suitable mathematical model will be developed and solved using an appropriate optimisation or computational technique. Depending on data availability and model complexity, the proposed signal settings may be assessed using scenario analysis, numerical computation, or traffic simulation. The study will compare the proposed model against existing or baseline signal timings using clearly defined performance measures.

The study will not assume that optimising signal timing alone can resolve all causes of urban congestion. Road capacity constraints, illegal parking, pedestrian movements, roadside activities, incidents, and other operational factors may influence actual performance but may fall outside the model’s primary scope unless suitable data are available.

1.8 Limitations of the Study

One possible limitation is the availability and accuracy of traffic data. Traffic counts collected over a limited number of days may not capture all variations in vehicle demand, unusual events, or seasonal changes. To address this issue, the study should collect data across relevant peak and off-peak periods and clearly document the observation periods and assumptions used in the model.

Another limitation concerns the representation of uncertain traffic flow. Traffic arrivals may vary according to time of day, vehicle type, road conditions, and external events. A mathematical model may simplify some of these factors to remain computationally manageable. The researcher should therefore justify the selected probability distributions, uncertainty scenarios, or modelling assumptions and test the sensitivity of the results to plausible changes in input parameters.

The study may also be constrained by access to traffic-signal timing records, limitations in available simulation software, and the time required for field observations. If simulation is used, its outputs will depend on the quality of the input data and the extent to which the model represents real traffic behaviour. Consequently, proposed signal timings should be interpreted as model-based recommendations requiring appropriate validation before field implementation.

Finally, the study will examine selected intersections rather than every road junction in Abuja. Its findings may therefore be most directly applicable to the intersections and traffic conditions represented in the analysis. Wider application would require additional data and validation across other parts of the city transport network.

1.9 Operational Definition of Terms

Mathematical modelling: The use of mathematical equations, variables, parameters, and constraints to represent traffic conditions and analyse the behaviour of a transportation system.

Urban traffic congestion: A condition in which traffic demand approaches or exceeds the effective capacity of a road or intersection, resulting in reduced speed, queue formation, and increased travel delay.

Traffic flow: The movement of vehicles along a road or through an intersection, commonly measured as the number of vehicles passing a point within a specified period.

Uncertain traffic flow: Variation or unpredictability in traffic demand or vehicle arrivals that makes actual traffic conditions differ from expected or average conditions.

Traffic signal timing: The allocation of cycle length, green time, red time, phase sequence, and clearance intervals to regulate conflicting traffic movements at a signalised intersection.

Optimal signal timing: A signal-timing arrangement selected by an optimisation model to achieve a defined objective, such as reducing average vehicle delay, minimising queue length, or improving throughput, subject to operational and safety constraints.

Queue length: The number of vehicles waiting at an intersection approach or the physical distance occupied by the waiting vehicles.

Vehicle delay: The additional time a vehicle experiences compared with an appropriate reference condition, often expressed as average delay per vehicle.

Intersection capacity: The maximum rate at which vehicles can pass through an intersection under specified traffic, geometric, and control conditions.

Robust optimisation: An optimisation approach that seeks solutions that remain effective when uncertain parameters, such as traffic demand, differ from their expected values.

Traffic simulation: The computational representation of traffic operations used to examine the effects of different traffic volumes, road conditions, and signal-timing arrangements.

Abuja city transport network: The network of roads and intersections serving movement within Abuja, Federal Capital Territory, from which the study’s selected intersections will be identified.

Project – Mathematical Modelling of Urban Traffic Congestion and Optimal Signal Timing Under Uncertain Traffic Flow. A Study of Selected road intersections along the Abuja city transport network.