# Macroscopic Traffic-Flow Modelling Based on Gap-Filling Behavior of Heterogeneous Traffic  
[PERSON]  
1 National Institute of Urban Infrastructure Planning, University of Engineering & Technology, Peshawar 25000, Pakistan; [EMAIL_ADDRESS] (M.U.K. ); [EMAIL_ADDRESS] (S.S.); [EMAIL_ADDRESS] (R.R. )1  
[PERSON]  
1 National Institute of Urban Infrastructure Planning, University of Engineering & Technology, Peshawar 25000, Pakistan; [EMAIL_ADDRESS] (M.U.K. ); [EMAIL_ADDRESS] (S.S.); [EMAIL_ADDRESS] (R.R. )1  
[PERSON]  
2 Department of Civil and Environmental Engineering, Western University, London, ON N6A 5B9, Canada 2*Correspondence: [EMAIL_ADDRESS]; Tel: +1-519-661-2111 (ext. 88308)2  
[PERSON]  
1 National Institute of Urban Infrastructure Planning, University of Engineering & Technology, Peshawar 25000, Pakistan; [EMAIL_ADDRESS] (M.U.K. ); [EMAIL_ADDRESS] (S.S.); [EMAIL_ADDRESS] (R.R. )1  
Footnote 1: email: {khan, }  
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Footnote  
practical implementation of these systems in the near future seems far-fetched in developed countries, let alone in the developing countries where heterogeneous traffic conditions are more prevalent. In addition, researchers are now focusing on providing traffic planners with tools helping them in controlling traffic facilities in order to mitigate traffic congestion. In this regard, a PDDL+ approach was introduced by [PERSON] et al. [5], which controls traffic lights to aid in congestion reduction. Latest research interests include developing models to characterize the vehicular travel times. [PERSON] et al. [6] developed an equation that characterizes travel times and then classified urban roads on the basis of those travel times. However, despite what the future may bring, traffic congestion and its consequences must be addressed presently. These issues may be mitigated by analyzing traffic flows and then proposing effective solutions. For the analysis of traffic flow, the first step is to capture the behavior of traffic using mathematical equations, which is called traffic-flow modelling. Three approaches to traffic-flow modelling are usually adopted: microscopic, macroscopic, and mesoscopic modelling. Microscopic models, such as those in References [7; 8], deal with the behavior of individual vehicles. Each vehicle in the traffic stream is modelled by using a separate Ordinary Differential Equation (ODE). Conversely, macroscopic models [9; 10; 11; 12] focus on the aggregate behavior of the traffic stream and involve three parameters of prime interest: traffic density (number of vehicles per unit length per lane), speed, and flow (number of vehicles crossing a point per unit time). Lastly, mesoscopic models, for example [13], are a combination of both microscopic and macroscopic models and, hence, act as a bridge between the two approaches. This research focuses on macroscopic modelling because of its relatively simpler computations. Traffic flow is characterized either as homogeneous or heterogeneous flow. In homogeneous traffic flow, there is strict lane discipline with traffic mainly composed of motorized vehicles, the dimensions of which do not vary much from each other [14]. Conversely, in heterogeneous traffic flow, the presence of both motorized and non-motorized vehicles is accompanied by very loose or no lane discipline at all [15]. In such traffic conditions, the gap-filling behavior is prevalent in which the drivers tend to escape the traffic behind by occupying the gaps they see ahead of them. Because of this, different vehicle types follow different levels of service resulting in complex interactions between the vehicles and causing severe disturbances within the traffic stream [16]. The first macroscopic traffic-flow model was developed by [PERSON] and [PERSON] [9] and then independently by [PERSON] [10]. This model is referred to as the Lighthill-Witham-Richards (LWR) model and is composed of a single equation based on the law of conservation of mass.