A Macroscopic Traffic Model based on Pavement Condition Index

A Macroscopic Traffic Model based on Pavement Condition Index | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article A Macroscopic Traffic Model based on Pavement Condition Index Shan Ul Haq, Zawar Hussian Khan, Inamullah Khan, Khurram Shahzad Khattak, and 5 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4800495/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The prevalence of poor road conditions makes urban traffic gridlock, leading to increased travel time and disruptions in urban mobility in developing countries. In this research the Payne-Whitham (PW) model which is second-order macroscopic traffic flow model was modified by replacing the speed constant ( \(\:{C}_{^\circ\:}\) ) with a novel parameter, the Pavement Condition Index (PCI) and its derivative with respect to PCI. By integrating PCI, drivers' responses are adjusted based on the road condition, potentially addressing drawbacks related to the lack of physical interpretation of ( \(\:{C}_{^\circ\:}\) ), and parameter sensitivity. The performance of the PW and Proposed model is simulated in MATLAB, over 3500m circular road, considering the PCI. The results shows that the proposed model provides realistic representation of traffic flow behavior, where density and speed sharp change patterns smoothen and exhibit inverse relationships as expected. Physical sciences/Engineering/Civil engineering Earth and environmental sciences/Environmental sciences Macroscopic Traffic Flow model Pavement Condition Index (PCI) Payne–Whitham (PW) model Speed Constant Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1. Introduction An established transportation system advances the financial and social development of a country. The essence of transportation systems is degraded with congestion occurrence as the travel time is increased. Congestion occurrence is due to the quick rise in vehicles on the roads in the urban regions which reduces the service quality of road network. The growth in vehicles not only fades the traffic markings on the road 1–8 , but also causes a quicker degradation in the road surface 9–10 . According to the department of transportation (TxDOT) of Texas, the deteriorated pavement conditions are one of the major causes of accidents. The severity of collision at a bad pavement is higher while on failed pavement accident rate decreases as the traffic speed is reduced 11 . Traffic density is the quantity of vehicles per unit distance, while flow is the result of density and velocity. The traffic density is higher on the decaying pavement, as the drivers slow down on the unpleasant pavement to avoid damage to the vehicle. This causes recurrent bottlenecks and reduces the capacity of the road infrastructure 12 . On a poor road surface condition, traffic is reduced by 20%, while the speed reduces by 50% 13 . The Pavement Condition Index (PCI) is a standardized metric for the road surface assessment, which ranges from 0 for failed conditions to 100 for good pavement condition 14 . This is a widely accepted tool for road surface assessment. The PCI rating is given in Fig. 1 . Pavements with minimal distress, that is, pavement condition index exceeding 40, minor changes occur in traffic speed. However, as the PCI decreases below 40, the speed changes significantly 15 . With the reduction in speed, headway is also impacted. In other words, the pavement condition significantly affects the distance between the vehicles. Time headway varies inversely with the pavement condition, while the speed holds a direct relationship. On a good rating road surface condition, that is, PCI is 96, headway recorded is 1.5 s for 14.6 m/s. While 4.2 s headway for 7.2 m/s are recorded on serious rating road with a PCI of 17. That is, road surface with a value of 17 PCI has a seriously deteriorated condition 16 . 55% speed reduction was recorded on poor road conditions having PCI of 19 as compared to good road conditions having PCI of 100. With distressed road surface, larger headway and slower speed was recorded 17 . Another study revealed that as speed reduces by 0.0037 m/s as the surface roughness increases 18 . The result eight road segments were used to study the impact of surface roughness on the capacity of two-lane roadways in India. The findings indicate that a vehicle's free flow speed drops as the road surface gets bumpy. Roughness has a greater impact on passenger car speed than it does on big trucks. The speed-volume correlations plotted for several two-lane rural road sections show that capacity declines as surface roughness increases 19 . Haj-Ismail 20 examined the impact of highway surface conditions on traffic speed. According to this study, no statistical changes in speed were observed at the same deterioration level during the day and night. In other words, the increased density on the road during the day has the same impact on speed as for a reduced density during the night. The traffic during the night has free flow conditions and is not impacted by the density. These studies highlight the importance of maintaining smooth roads for efficient traffic flow, reduced fuel consumption, and increased safety to mitigate the traffic conditions hazards. It is significant to characterize the impact of the PCI on the macroscopic traffic flow, to predict the traffic temporal and spatial evolution more realistically for the effective dissemination of urban. The increase in disturbance of traffic evolution has drawn the notice of traffic scientists. Different traffic systems are developed to comprehend the traffic disturbance in different conditions 22–32 . The most common modelling approaches used are fluid-dynamic models, car following models, and gas-kinetic models. These models can predict traffic behavior and help in identifying the traffic characteristics. Different scientists have considered the backward traffic effect 33 , lateral effect of the lane width on the vehicles 34 , driver presumptions and traffic alignment 21,35 , and physiological response of drivers 36–37 . Traffic flow is usually sorted by road conditions and can be portrayed as homogeneous or heterogeneous, and equilibrium or non-equilibrium. In homogeneous traffic, elements, for example, velocity and headway do not vary spatially 38 and vehicles follow lane discipline. Heterogeneous traffic comprises of motorized and non-motorized vehicles and lane discipline isn't fundamentally followed 39 . In an equilibrium flow, velocity is an element of density, so it happens when there is no change in velocity and there is spatial homogeneity. In a non-equilibrium flow, there will be changes in velocities and spatial homogeneity 40 . Three types of models have been utilized to portray traffic flow: microscopic, macroscopic, and mesoscopic. Macroscopic models manage total traffic behavior and are utilized to determine average density, velocity, and flow 21 . Microscopic models are about individual vehicle behavior and are dependent on the driver’s physical and psychological reactions 41 . They utilize vehicle features like distance headways, position, velocity, and time 42 . Mesoscopic models consider both individual and joined vehicle behavior 43 , so vehicles are portrayed both exclusively and collectively. Consequently, the issues should be integrated into traffic models to characterize traffic behavior precisely and realistically. The macroscopic models are also known as continuum models of vehicular traffic. These models are the simplest to characterize the temporal and spatial traffic evolution. The macroscopic models predict the traffic conditions at a road location x and time t, while considering the density ρ, speed v, and flow rate q. The fundamental relation is \(\:\text{q}\:=\:{\rho\:}\:\varvec{v}\) at x and t. Traffic conservation is in Eq. 1 . $$\:\frac{\partial\:\rho\:}{\partial\:t}+\:\frac{\partial\:q}{\partial\:x}=0$$ 1 , Lighthill, Whitham and Richards proposed, Eq. 1 as the principal macroscopic traffic model and is the well-known LWR model 44,45 . According to this model, traffic is always at equilibrium, which is an ideal condition and is one of the major shortcomings of this model. The equilibrium velocity decreases with increase in density, that is, the derivative of the equilibrium is negative with a large density \(\:{(\varvec{v}}^{{\prime\:}}\left({\rho\:}\right)\:<\:0)\) . In other words, the fundamental diagram is a concave function of density. Mostly, traffic is at non-equilibrium for larger traffic density. Therefore, Payne-Whitham (PW) characterized the non-equilibrium temporal and spatial traffic evolution model and laid the foundation of second order traffic systems. Isotropic behavior is independent of the direction of flow. Daganzo 46 severely criticized the PW model for the isotropic traffic behavior predicted by the PW models. According to 46 , traffic flow has a unique direction, that is vehicles move in a single direction and traffic is anisotropic in nature of traffic. Payne-Witham (PW) model enhances the LWR model by including driver behavior 47,48 . This model has two equations. The first depends on vehicle conservation and is given by Eq. 1 . Driver behavior is characterized by the anticipation term in the second equation. Anticipation is driver presumption of ahead changes in traffic and conditions occurs on road. The changes due to velocity are characterized by a relaxation term in this equation. The relaxation term is the time required for traffic velocity alignment and react to absorb road condition. The second equation in the PW model is given as Eq. 2, 21 . $$\:\frac{\partial\:\left(\rho\:v\right)}{\partial\:t}+\frac{\partial\:\left(\rho\:{v}^{2}\right)}{\partial\:x}+\:\frac{\partial\:{(C}_{^\circ\:}^{2}\rho\:)}{\partial\:x}=\:\rho\:\left(\frac{v\left(\rho\:\right)-v}{\tau\:}\right)$$ 2 , \(\:{C}_{^\circ\:}\:\) is a constant which characterizes driver behavior with units in m/s which is called sonic velocity 49 . That is the changes in traffic travel at the speed \(\:{C}_{^\circ\:}\) . The driver anticipation is \(\:{\left({C}_{^\circ\:}^{2}\rho\:\right)}_{x}\) , while relaxation is \(\:\rho\:\left(\frac{v\left(\rho\:\right)-v}{\tau\:}\right)\) 50 which accounts for the fact that traffic adjusts to changes in density over time. The anticipation term characterizes adjustments in flow because of a driver’s action according to road condition and the relaxation term characterizes adjustments in velocity. The anticipation causes changes in traffic flow and is accounted for by relaxation. In other words, the driver’s presumption is the microscopic change occurred due to the driver personality, while relaxation is the macroscopic alignment of the traffic changes caused by the driver anticipation. The alignment is quicker with smaller relaxation time τ and is a drastic change. This relaxation is one of the most sensitive variables of the PW model, and the changes in the traffic grow for smaller relaxation time. Traffic alignment occurs based on the equilibrium velocity distribution v(ρ). The PW model assumes that drivers have similar behavior for all road conditions and only small adjustments in velocity and density occur 51 . Consequently, the traffic alignment with the PW at larger changes in density 21 results in abrupt adjustments in traffic over short distances 52 , which is insufficient and unrealistic. A few methodologies have been considered to address the PW model shortcomings. By including driver anticipation based on changes in the equilibrium velocity, Zhang 40 enhanced the PW model. Although this model is anisotropic, and a motorist instantly adapts to the traffic density, this model did not take driver physiology into account 53 . Ross 54 improved the PW model by eliminating the anticipation term with the goal that driver presumption is disregarded 55 . This model can't describe traffic flow when there are huge changes or traffic jams. An anisotropic model that considers the driver's presupposition of forward circumstances based on density and spatial changes in velocity was proposed by Jiang et al. 56 , and which is given as Eq. 3 . $$\:\frac{\partial\:{\rho\:}\:}{\partial\:\text{t}\:}+\:\frac{\partial\:{\rho\:}\varvec{v}}{\partial\:\text{x}}=\:\text{g}(\text{x},\text{t})$$ 3 , Where g(x,t) is the egress and ingress of the vehicles to a straight road. $$\:\frac{\partial\:u}{\partial\:t}+u\frac{\partial\:u}{\partial\:x}=\frac{{\text{u}}_{\text{e}}-\text{u}}{\text{T}}+{C}_{^\circ\:}\frac{\partial\:u}{\partial\:x}$$ 4 , In Eq. 4 , \(\:{C}_{^\circ\:}\:\) represents the constant of rearward motion. As a result, for any transition dynamics, changes in upstream traffic propagate at a constant velocity, which is not feasible. Additionally, this constant may produce behavior that is irrational and has wide transitions. The Jiang model was enhanced by Zheng et al. 57 by adding a new relaxation term that provides. $$\:\frac{\partial\:\rho\:\:}{\partial\:t\:}+\:\frac{\partial\:\rho\:\varvec{v}}{\partial\:x}=\:0$$ 5 , $$\:\frac{\partial\:v}{\partial\:t}+\left(v-{C}_{^\circ\:}\right)\frac{\partial\:\varvec{v}}{\partial\:x}=\zeta\:\:[\frac{1}{\rho\:}-\frac{1}{{\rho\:}_{e}\left(v\right)}]\:\:,$$ 6 In Eq. 6 , ρe(v) is the equilibrium density distribution and ζ is the driver sensitivity coefficient. However, \(\:{C}_{^\circ\:}\:\) and ζ are constants, which are the fitting parameters for traffic situations, rather than being based on realistic traffic flow factors. Consequently, these parameters are not based on the physics of traffic flow. Michalopoulos 58 proposed a model while not considering the equilibrium velocity distributions. This model can provide unrealistic traffic evolution by ignoring the density dependent characteristics on the road Khan et al. 59 proposed a model that accounts for snow, ice, and compacted snow by introducing a weather-dependent transition velocity distribution. This allows the model to capture changes in driver behavior due to reduced traction. However, this model did not consider the pavement conditions. Despite these advancements, limitations still exist in characterization of the impact of road surface conditions on macroscopic traffic models. Most surface-aware models rely on simplified representations of microscopic complex interactions between surface conditions, vehicle dynamics, and driver behavior. The availability and accuracy of real-time surface data is also a challenge. By overcoming these challenges, road condition aware model can significantly improve understanding and management of traffic flow, particularly in dynamic environments with varying surface conditions. In this paper, the impact of road condition on macroscopic traffic flow has been characterized. Pavement Condition Index (PCI) is incorporated in a second order traffic system. The driver presumption is based on the changes in the PCI, that is a driver adjusts to the traffic changes in speed based on the PCI. The PW and proposed model are analyzed over 3500 m circular road for large changes in density. The proposed model performed better than the PW model. Section 1 is an introduction; the rest of the paper is organized as follows. In Section 2, the Proposed model is presented by improving the PW model. In Section 3, traffic models’ decomposition is performed for the Proposed and PW models. Section 4 presents the simulation results for Proposed and PW models which support the Proposed model by results. Finally, Section 5 presents the conclusions. 2. PROPOSED MODEL The macroscopic PW traffic system uses speed v and density 𝜌 at a road location x and time t to characterize the flow. The first equation is the traffic conservation system, that is given as $$\:\frac{\partial\:{\rho\:}}{\partial\:\text{t}}+\:\frac{\partial\:\left({\rho\:}\text{v}\right)}{\partial\:\text{x}}=0$$ 7 , This predicts small traffic changes over a road section. In other words, the traffic in-flux and egress from a road section does not significantly change and the traffic over a section remains conserved. Further, the traffic speed in Eq. 7 is only density dependent, which is incorrect. Apart from the traffic density, different personalities are involved in controlling vehicles behavior. For example, the speed of traffic at same density depends on driver response. Therefore, Payne and Whitham 21 complement Eq. 7 , with driver response. The first equation of the Payne and Whitham model (PW) remains the same as Eq. 7 , while the second equation is $$\:\frac{\partial\:\left({\rho\:}\text{v}\right)}{\partial\:\text{t}}+\frac{\partial\:\left({\rho\:}{\text{v}}^{2}\right)}{\partial\:\text{x}}+\:\frac{\partial\:{(\text{C}}_{0}^{2}{\rho\:})}{\partial\:\text{x}}=\:{\rho\:}\left(\frac{\text{v}\left({\rho\:}\right)-\text{v}}{{\tau\:}}\right)$$ 8 , \(\:\frac{\partial\:{(\text{C}}_{^\circ\:}^{2}{\rho\:})}{\partial\:\text{x}}\:\) is the driver’s response. According to this model, the driver response for negligible changes is uniform and is complemented with \(\:{C}_{0}\) , which is a speed constant. The driver response is personality dependent, while in PW model, it is characterized with constant. The role of a driver’s personality in traffic evolution is ignored \(\:\:\rho\:\left(\frac{\varvec{v}\left(\varvec{\rho\:}\right)-\varvec{v}}{\varvec{\tau\:}}\right)\) in Eq. 8 is the relaxation term and represents the driver adjustment process to the speed changes. That is, traffic yields the equilibrium traffic speed \(\:v\left({\rho\:}\right),\) which is density dependent. According to \(\:v\left({\rho\:}\right),\) the speed is slower for larger density and vice versa for smaller density. τ affects the traffic adaptation to variations in density. Traffic adjusts more rapidly when the relaxation time is shorter, and more slowly when the relaxation time is longer. The anticipation term has the tendency of traffic adjustment to forward vehicles based on driver response. For example, speed decays as a user notices congestion ahead. In a familiar road environment and uniform flow, a driver response quickly and traffic alignment is smoother. While the relaxation term mitigates the changes occurred in traffic due to anticipation. In this paper, PW model is utilized to contain road surface condition. One of the influencing factors for a driver to respond is the pavement conditions. The response is quicker for larger road surface irregularities due to quicker driver reactions, and slower for best road surface conditions as driver reactions are small. Significant variations in traffic occur at larger surface irregularities or for worst road conditions. The road surface irregularities are quantified as pavement condition index. Therefore, in this paper, the pavement condition index (PCI) is employed in the anticipation term as driver response to comprehend the traffic changes based on the surface irregularities. This is significant velocity variations worsen the traffic flow. A driver adjusts the vehicle speed as notices a change in speed with changes in the PCI. That is for larger changes in PCI, the response is large and small for smaller changes noticed. The speed due to surface irregularities is a function PCI, that is \(\:\text{v}\left(\text{P}\text{C}\text{I}\right)\) . Then changes observed in speed by a driver is \(\:\frac{\text{d}\varvec{v}\left(\text{P}\text{C}\text{I}\right)}{\text{d}\left(\text{P}\text{C}\text{I}\right)}\) . For a larger \(\:\frac{\text{d}\varvec{v}\left(\text{P}\text{C}\text{I}\right)}{\text{d}\left(\text{P}\text{C}\text{I}\right)}\) , driver response is large, while driver response is small for a smaller is \(\:\frac{\text{d}\varvec{v}\left(\text{P}\text{C}\text{I}\right)}{\text{d}\left(\text{P}\text{C}\text{I}\right)}\) . The driver response is also impacted by the running conditions of surface irregularities that is driver response is slower for a smaller PCI due to its slower speed, and vice versa for a larger PCI. That is, vehicles are slower on a deteriorated road, while for a good quality of road, vehicles are at a faster speed and flow is smooth. Reduced acceleration and deceleration occur. A driver response is considerate of the PCI, and changes in velocity due to PCI in ahead conditions. Therefore, the driver’s response is. $$\:\text{P}\text{C}\text{I}\:\frac{\text{d}\text{v}\left(\text{P}\text{C}\text{I}\right)}{\text{d}\left(\text{P}\text{C}\text{I}\right)}$$ 9 , By substituting Eq. 9 as \(\:{C}_{^\circ\:}\) , Eq. 10 is. $$\:\frac{\partial\:\left({\rho\:}v\right)}{\partial\:\text{t}}+\frac{\partial\:\left({\rho\:}{v}^{2}\right)}{\partial\:\text{x}}+\:\frac{\partial\:\left({\left(\text{P}\text{C}\text{I}\:\frac{\text{d}v\left(\text{P}\text{C}\text{I}\right)}{\text{d}\left(\text{P}\text{C}\text{I}\right)}\right)}^{2}{\rho\:}\right)}{\partial\:\text{x}}=\:{\rho\:}\left(\frac{v\left({\rho\:}\right)-v}{{\tau\:}}\right)$$ 10 , which reflects the impact of the pavement condition index (PCI) on vehicles speed and density. \(\:\frac{\partial\:\left({\left(\mathbf{P}\mathbf{C}\mathbf{I}\:\frac{\mathbf{d}\varvec{v}\left(\mathbf{P}\mathbf{C}\mathbf{I}\right)}{\mathbf{d}\left(\mathbf{P}\mathbf{C}\mathbf{I}\right)}\right)}^{2}\varvec{\rho\:}\right)}{\partial\:\mathbf{x}}\) is the anticipation which predicts the spatial changes in density based on the driver response to PCI. This can be utilized to assess the impact of traffic density and pavement condition index on the quality service of road network. The proposed model consists of Eq. 7 and Eq. 10 , and is given as $$\:\frac{\partial\:\rho\:}{\partial\:t}+\:\frac{\partial\:\left(\rho\:v\right)}{\partial\:x}=0$$ 11 , $$\:\frac{\partial\:\left({\rho\:}v\right)}{\partial\:\text{t}}+\frac{\partial\:\left({\rho\:}{v}^{2}\right)}{\partial\:\text{x}}+\:\frac{\partial\:\left({\left(\text{P}\text{C}\text{I}\:\frac{\text{d}v\left(\text{P}\text{C}\text{I}\right)}{\text{d}\left(\text{P}\text{C}\text{I}\right)}\right)}^{2}{\rho\:}\right)}{\partial\:\text{x}}=\:{\rho\:}\left(\frac{v\left({\rho\:}\right)-v}{{\tau\:}}\right)$$ 12 , Equation 11 of the proposed model considers the conservation of vehicles like the PW type models. While Eq. 12 is the second equation of the proposed model which illustrates the evolution of traffic density and speed based on the surface irregularities depicted as PCI. This proposed model provides a comprehensive framework for studying the complex interplay between traffic dynamics and road pavement conditions. In this paper, the speed based on PCI is used from the study 60 . The speed relation is developed from surveying a 7-kilometer, two-lane segment of the Grand Trunk Road connecting Peshawar's Chamkani Bus Rapid Transit (BRT) station to Pabbi. The velocity relation impacted by PCI 60 is. $$\:\varvec{v}\left(\text{P}\text{C}\text{I}\right)=\:-1.659\text{P}\text{C}\text{I}\:+\:75.62\:$$ 13 , The correlation coefficient R² = 0.995 represents the degree to which the data fits the models. It ranges from 0 to 1, with a value close to 1 indicating a strong fit. From Eq. 13 $$\:\frac{\text{d}\text{v}\left(\text{P}\text{C}\text{I}\right)}{\text{d}\left(\text{P}\text{C}\text{I}\right)}=-1.65$$ 14 , This is the change occurring in speed with changes in PCI. In other words, the driver presumptions of the proposed model are based on the changes given in ( 14 ) on the road chosen 60 . 3. Traffic Models Decomposition In this paper, the first order centered (FORCE) numerical method is used to decompose the proposed and PW traffic models. The FORCE scheme is employed to approximate the larger traffic changes more accurately. The subscripts x and t, respectively, stand for the spatial and temporal partial derivatives. A traffic system in conserved form is $$\:{\psi\:}_{t}\:+\:f\:{\left(\psi\:\right)}_{x}\:=\:S\left(\psi\:\right)$$ 15 , This is the change occurring in speed with changes in PCI. In other words, the driver presumptions of the proposed model are based on the changes given in ( 14 ) on the road chosen 60 . $$\:\psi\:\:=\left(\genfrac{}{}{0pt}{}{\rho\:}{\rho\:v}\right)\:,\:f\left(\psi\:\right)={\left(\genfrac{}{}{0pt}{}{\rho\:v}{{\frac{{\left(\rho\:v\right)}^{2}}{\rho\:}+C}_{^\circ\:}^{2}\rho\:}\right)}_{x},\:S\left(\psi\:\right)=\left(\genfrac{}{}{0pt}{}{0}{\rho\:\left(\frac{v\left(\rho\:\right)-v}{\tau\:}\right)}\right),$$ 16 While for the Proposed model from ( 11 ) and ( 12 ) $$\:\psi\:\:=\left(\genfrac{}{}{0pt}{}{\rho\:}{\rho\:v}\right)\:,\:f\left(\psi\:\right)={\left(\genfrac{}{}{0pt}{}{\rho\:v}{\frac{{\left(\rho\:v\right)}^{2}}{\rho\:}\:+\:\rho\:{\left(PCI\:\frac{dv\left(PCI\right)}{d\left(PCI\right)}\right)}^{2}}\right)}_{x},\:S\left(\psi\:\right)=\left(\genfrac{}{}{0pt}{}{0}{\rho\:\left(\frac{\varvec{v}\left(\varvec{\rho\:}\right)-\varvec{v}}{\varvec{\tau\:}}\right)}\right)$$ 17 To approximate the larger traffic changes more accurately and realistically, ( 15 ) is linearized over smaller temporal and spatial steps. The resulted linearized system is the quasilinear form and is $$\:{\psi\:}_{t}\:+\:A\left(\psi\:\right){\psi\:}_{x}\:=\:0$$ 18 , where A(ψ) is the Jacobian matrix, which presents the gradients of the functions of variables. The gradients in A(ψ) are essential to obtain the behavior of the functions of the data variables with changes in the variables itself. For a road length \(\:{\text{x}}_{\text{M}}\) , with M number of equal sized road segments, the segment size is \(\:{\delta\:}\text{x}\:=\:{x}_{M}/\text{M}\) . Whereas a time step is \(\:{t}_{N}\:=\:{t}_{N}/\text{N}\) , where t_N is the total time of traffic evolution and N is the number of equal sized time steps. tn is the nth time step then a time step is \(\:(\text{t}\text{n}+1,\:\text{t}\text{n})\) . The traffic data variables are approximated at \(\:(\text{t}\text{n}+1,\:\text{t}\text{n})\) , and road segment \(\:(\text{x}\text{i}+\:{\delta\:}\text{x}/\:2,\:\text{x}\text{i}\:-\:{\delta\:}\text{x}\:/2)\) . The FORCE scheme 61 , combines the first order Lax-Friedrichs scheme 62 and the second order Richtmyer scheme 63 , to approximate traffic at the boundaries of the road segments. This can precisely obtain the solution of the hyperbolic traffic systems of the proposed and PW models. Let ψ represent the average of the data variables in a road segment. The functions of the data variables are approximated at the road segment boundaries. At the boundary of road segments \(\:i\) and \(\:i+1\) , \(\:f\left({\psi\:}\right)\) at the nth time step can be approximated by using the Lax-Friedrichs scheme, and is given as $$\:{\left({f}_{i+\frac{1}{2}}^{n}\:\left({\psi\:}_{i}^{n},{\psi\:}_{i+1}^{n}\right)\right)}^{l}=\frac{1}{2}\left(f\left({\psi\:}_{i}^{n}\right)+f({\psi\:}_{i+1}^{n}\right))+\:\frac{1}{2}\frac{\delta\:t}{\delta\:x}\left({\psi\:}_{i}^{n}-{\psi\:}_{i+1}^{n}\right)$$ 19 , The Lax-Friedrichs scheme 62 is indicated by the superscript l. The Richtmyer scheme can approximate the data variables as 63 $$\:{\psi\:}_{i+1}^{n}=\:\frac{1}{2}\left({\psi\:}_{i}^{n}+{\psi\:}_{i+1}^{n}\right)+\:\frac{1}{2}\frac{\delta\:t}{\delta\:x}\left({f(\psi\:}_{i}^{n})-{f(\psi\:}_{i+1}^{n})\right),$$ 20 , and the corresponding \(\:f\left({\psi\:}\right)\:\) is obtained as $$\:{\left({f}_{i+\frac{1}{2}}^{n}\:\left({\psi\:}_{i}^{n},{\psi\:}_{i+1}^{n}\right)\right)}^{\text{r}}=\:f\left({\psi\:}_{i+\frac{1}{2}}^{n}\right)$$ 21 , where the Richtmyer scheme is indicated by the superscript r. For more accuracy \(\:f\left({\psi\:}\right)\) obtained from ( 20 ) and ( 21 ) at the segment boundaries are averaged, that is $$\:{f}_{i}^{n+1}=\:\frac{1}{2}\left({\left({f}_{i+\frac{1}{2}}^{n}\right)}^{r}+\:{\left({f}_{i+\frac{1}{2}}^{n}\right)}^{l}\right)$$ 22 , The source terms of the proposed and PW models respectively in ( 16 ) and ( 17 ) are obtained as $$\:S\left({\psi\:}_{i}^{n}\right)=\left(\frac{v\left({\rho\:}_{i}^{n}\right)-{v}_{i}^{n}}{\tau\:}\right)$$ 23 , By including the source term, which provides, the updated data variables at the ith road segment and nth time step are $$\:{\psi\:}_{i}^{n+1}={\psi\:}_{i}^{n}-\frac{\delta\:t}{\delta\:x}\left({f}_{i+\frac{1}{2}}^{n}-{f}_{i-\frac{1}{2}}^{n}\right)+\:\delta\:tS\left({\psi\:}_{i}^{n}\right)$$ 24 , 3.1 Traffic Models Hyperbolicity The traffic models are hyperbolic if changes in flow occur with a finite velocity. As a result, the changes impact during congestion reduces with time. The conditions of the strict hyperbolicity of the traffic systems are that the eigenvalues are real and distinct [64]. The eigenvalues of the traffic variables can be approximated from the Jacobian matrix. The Jacobian matrix of the PW model is. $$\:S\left(\psi\:\right)=\left(\begin{array}{cc}0&\:1\\\:{-v}^{2}+{C}_{^\circ\:}^{2}&\:2v\end{array}\right),$$ 25 and the corresponding eigenvalues from ( 25 ) are. $$\:{\lambda\:}_{1}\:=\:v+{C}_{^\circ\:}\:\:\:\:,\:\:\:\:{\lambda\:}_{2}\:=\:v-{C}_{^\circ\:}$$ 26 , The eigenvalues are real and distinct, and therefore the PW model is hyperbolic. These eigenvalues stand the characteristic speeds of the PW model. The shocks, that is the regions of high traffic density and low velocity, are characterized by \(\:{\lambda\:}_{2}\) , which is the slower speed characteristic. The rarefactions, or regions with low traffic density and high velocity, are characterized by \(\:{\lambda\:}_{1}\) , which is the faster speed characteristic. The Jacobian matrix of the Proposed model is. $$\:S\left(\psi\:\right)=\left(\begin{array}{cc}0&\:1\\\:{-v}^{2}+{\left(PCI\:\frac{dv\left(PCI\right)}{d\left(PCI\right)}\right)}^{2}&\:2v\end{array}\right)$$ 27 , and the corresponding eigenvalues are. $$\:{\lambda\:}_{1}\:=\:v+\left(PCI\:\frac{dv\left(PCI\right)}{d\left(PCI\right)}\right)\:\:,\:\:\:\:{\lambda\:}_{2}\:=\:\:v-\left(PCI\:\frac{dv\left(PCI\right)}{d\left(PCI\right)}\right)$$ 28 , The eigenvalues are real and distinct, and therefore the proposed model is hyperbolic. The traffic shock is characterized by \(\:{\lambda\:}_{2}\) , while the rarefaction is characterized by \(\:{\lambda\:}_{1}\) . The changes in shocks and rarefactions are due to deteriorated road conditions. In other words, the speed characteristics changes due to pavement condition index (PCI) that is traffic moves from good to bad road conditions and vice versa. section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn. 4. Performance Result Table 1 Simulation Parameter. Description Value Total simulation time for proposed and PW models. 1000 s Length of the circular road 3500 m Maximum Speed 20.14 m/s Maximum density 1 Time step for proposed and PW models δt = 0.01 Road step for proposed and PW models δx = 15 Relaxation time for proposed model τ = 0.50 Headway for proposed & PW model. h = 10 m Equilibrium velocity distribution v(ρ) = Greenshields Pavement Condition Index PCI = 10 In this section, the performance of the proposed model and PW models on a Circular (ring) road, of length 3500 m is evaluated. The simulation parameters, as shown in Table 1 , ensure stability by adhering to the Courant, Friedrich, and Lewy (CFL) stability conditions [65]. The road step size is 15 m, and the time step size is 0.01 s for both models. The total simulation time is 1000 s. The maximum speed is 20.14 m/s, following the Greenshields equilibrium velocity distribution [66]. The maximum density is 1, indicating that the road is fully occupied. The relaxation time for the proposed model is 0.50 s and is within the typical range of 0 s to 0.50 s [51,52]. While for the PW model, the relaxation time is 2 s, and is chosen greater than the relaxation time of the proposed model. At a smaller headway of 0.5 s, the PW model with the given conditions in Table 1 cannot characterize the traffic flow. The initial density at time t = 0 for the traffic simulation over the circular road for both the models are $$\:{\rho\:}_{^\circ\:}=\:\left\{\begin{array}{c}0.01,\:\:\:\:\:\:\:\:\:\:\:\:\:for\:x<500\:\\\:0.3,for\:510\le\:x<1500\\\:0.6,\:for\:1500\le\:x<2400\\\:0.01,\:\:\:\:\:\:\:for\:x\:\ge\:2415\:\:\:\:\:\\\:\:\:\:\:\end{array}\right.$$ 29 , Figure 2(a) displays the density evolution according to the proposed model over a 3500 m road at five different time points: 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is presented in Table 2 . Starting at 1 s, the density is recorded consistently low as 0.01 from 1 m to 495 m. From 510 m to 1395 m, it increases to 0.29. However, it increases more to 0.6 which is the maximum from 1410 m to 2400 m and then drops to 0.01 which is the lowest from 2415 m to 3500 m over the road length. At 250 s, the density is 0.01 at 1 m. It starts to rise from 0.04 to 0.30 at 360 m and 1020 m respectively. Further increases to 0.6 at 1905 m. However, between 2820 m and 3500 m, it experiences a reduction from 0.017 to 0.013. At 500 s, the density is lowest as 0.01 at 1 m, which subsequently increases from 0.06 to 0.30 at 360 m and 1020 m respectively. It then increases to the maximum 0.58 at 1905 m and returns from 0.02 to 0.01 at 2970 m and 3500 m respectively. At 750 s, the density starts at 0.01 at 1 m. It then increases to 0.06 and 0.30 at 360 m and 1020 m respectively and further increases to the largest as 0.56 at 1905 m. However, at 2970 m to 3500 m it experiences a reduction from 0.04 to 0.01. Finally, at 1000 s, the density is 0.01 at 1 m and gradually increases to 0.06 at 360 m. It then increases to 0.29 at 1020 m. At 1845 m, the density increases to maximum of 0.54 and then reduces from 0.02 to 0.01 at 3210 m and 3500 m respectively. Figure 2(b) illustrates the Speed evolution with the proposed model over a 3500 m road at five different time intervals: 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is provided in Table 2 . At 1 s, the speed is 19.94 m/s from 1 m to 495 m, which is the highest, and then decreases to 14.10 m/s from 510 m to 1395 m. The Speed decreases more to 08.05 m/s from 1410 m to 2400 m. Subsequently, it rises to as high as 19.94 m/s from 2415 m to 3500 m. At 250 s, the Speed is 19.90 m/s at 1 m and gradually drops to 19.11 m/s at 360 m. It then reduces to 14.05 m/s and 08.08 m/s, which is lowest at 1020 m and 1905 m/s. Subsequently, it rises to 19.77 m/s and rises to maximum of 19.92 m/s at 3500 m. At 500 s, the Speed is 19.83 m/s at 1 m and reduces to 18.91 m/s and 13.99 m/s at 360 m and 1020 m respectively. At 1905 m, the Speed is lowest of 08.35 m/s, but it subsequently increases to 19.68 m/s and 19.91 m/s at 2970 m and 3500 m respectively. At 750 s, the Speed is 19.80 m/s at 1 m and gradually drops to 18.86 m/s and 14.04 m/s at 360 m and 1020 m respectively. It then reduces to the lowest of 08.77 m/s at 1905 m. Subsequently, it rises to 19.71 m/s and 19.87 m/s at 2970 m and 3500 m respectively. At 1000 s, the Speed is 19.77 m/s at 1 m and smoothly decreases to 18.78 m/s and 14.16 m/s at 360 m and 1020 m respectively. Between 1845 m and 3210 m, the speed rises from 9.13 m/s to 19.60 m/s, and finally traffic speed achieves 19.79 m/s at 3500 m. Speed appropriately adjusts in response to changes in Density. As density increases, the speed decreases and vice versa as shown in Figs. 2(a) and 2(b). That is, the speed adjusts as expected in response to changes in density. This shows the correctness of the proposed model. The observed behavior of density and speed in the proposed model is realistic, and it becomes progressively smoother over time. Table 2 Density and Speed with the proposed model at 1 s, 250 s, 500 s, 750 s and 1000. Time (s) Distance (m) Density (ρ) Speed (m/s) 1 1- 495 0.01 19.94 510–1395 0.29 14.10 1410–2400 2415–3500 0.60 0.01 08.05 19.94 250 1 360 1020 1905 2820 0.01 0.04 0.30 0.60 0.017 19.90 19.11 14.05 08.08 19.77 3500 0.013 19.92 500 1 360 1020 1905 2970 3500 0.01 0.06 0.30 0.58 0.02 0.01 19.83 18.91 13.99 08.35 19.68 19.91 750 1 0.01 19.80 360 0.06 18.86 1020 1905 2970 0.30 0.56 0.04 14.04 08.77 19.71 3500 0.01 19.87 1000 1 360 1020 1845 3210 0.01 0.06 0.29 0.54 0.02 19.77 18.78 14.16 09.13 19.60 3500 0.01 19.79 Figure 3(a) displays the density evolution with the PW model over a 3500 m road at 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is presented in Table 3 . At 1 s, the density is recorded as low as 0.01 from 1 m to 495 m. From 510 m to 1485 m, it increases to 0.30. However, density is 0.60 from 1530 m to 2190 m and is 0.61 from 2205 m to 2400 m. It then drops to 0.01, which is the lowest, from 2415 m to 3500 m. At 250 s, the density is 0.01 at 1 m. It then increases to 0.05 at 360 m. While it is 0.29 at 1020 m, and further increases to 0.59 at 1905 m. However, between 2820 m and 3500 m, it experiences a reduction to 0.01. At 500 s, the density is as lowest as 0.01 at 1 m, which subsequently increases to 0.06 at 360 m. It then increases to 0.29 at 1020 following the maximum 0.57 at 1905 m and returns to 0.02 at 2970 m. The density further reduces to 0.01 at 3500 m. At 750 s, the density is 0.01 at 1 m. It then increases to 0.06 at 360 m, which changes to 0.29 at 1020 m. The largest change 0.54 is observed at 1905 m. However, at 3090 m, the density changes from 0.02 to 0.01 at 3500 m. Finally, at 1000 s, the density is 0.02 at 1 m and gradually increases to 0.07 at 360 m. This further increases to 0.28 at 1020 m. At 1845 m, the density increases to 0.52 and then reduces to 0.2 at 3210 m. It further reduces to 0.01 at 3500 m. Figure 3(b) illustrates the speed evolution with the PW model over a 3500 m road at 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is provided in Table 3 . The speed is 19.8 m/s from 1 m to 495 m, which is the highest, and then decreases to 14.00 m/s at 510 m. The speed is uniform from 510 m to 1485 m. The speed is 8.00 m/s between 1530 m to 2190 m, while 7.80 m/s from 2205 m to 2400 m. Subsequently, it rises to as high as 19.8 m/s from 2415 m to 3500 m. At 250 s, the speed is 19.51 m/s at 1 m and gradually drops to 13.66 m/s at 360 m. It then drops to the lowest 08.02 m/s at 1905 m. Subsequently, it rises to 20.71 m/s at 2820 m which exceeds maximum speed. which then slightly drops to 19.72 m/s at 3500 m. At 500 s, the speed is 18.23 m/s at 1 m and reduces to 14.70 m/s at 360 m and slightly falls more to 13.63 m/s at 1020 m until reaches the lowest 08.73 at 1905 m. At 2970 m, the speed is 21.64 m/s that surpass maximum value, but it subsequently falls to 18.64 m/s at 3500 m. At 750 s, the speed is 18.23 m/s at 1 m and gradually reduces to 15.43 m/s at 360 m. It then experiences a consecutive drop to 13.63 m/s at 1020 m, which further reduces to 08.90 m/s at 1905 m. Subsequently, it rises to 21.72 m/s at 3090 m which outdoes maximum speed, which then reduces to 18.04 m/s at 3500 m. At 1000 s, the speed is 18.04 m/s at 1 m and smoothly decreases to 15.91 m/s at 360 m. Between 1020 m and 1845 m, the speed decreases from 13.72 m/s to 09.37 m/s, and finally it transcends maximum value and achieves 21.41 m/s at 3210 m. While the speed at 3500 m is 18.64. The relationship between density and speed in PW model exhibits limitations. While it reflects that speed decreases as density increases and vice versa, the model fails to capture realistic variations effectively. Particularly, abrupt changes in speed or surpass from maximum speed occur with changes in density, indicating poor road conditions in high-density areas where speeds drop significantly, and conversely, sharp increases in speed as density decreases, indicating smoother pavement areas. These discrepancies highlight the shortcomings of the PW model in accurately representing real-world scenario. Table 3 Density and Speed with the PW model at 1 s, 250 s, 500 s, 750 s and 1000. Time (s) Distance (m) Density (ρ) Speed (m/s) 1 1- 495 0.01 19.80 510–1485 0.30 14.00 1530–2190 2205–2400 2415–3500 0.60 0.61 0.01 08.00 07.80 19.80 250 1 360 1020 1905 2820 0.01 0.05 0.29 0.59 0.01 19.51 13.66 13.97 08.02 20.71 3500 0.01 19.72 500 1 360 1020 1905 2970 3500 0.01 0.06 0.29 0.57 0.02 0.01 18.23 14.70 13.63 08.73 21.64 18.64 750 1 0.01 18.23 360 0.06 15.43 1020 1905 3090 0.29 0.54 0.02 13.63 08.90 21.72 3500 0.01 18.64 1000 1 360 1020 1845 3210 0.02 0.07 0.28 0.52 0.02 18.04 15.91 13.72 09.37 21.41 3500 0.01 18.64 Figure 4(a) illustrates the spatial and temporal density evolution using the proposed model over a 3500 m road for a duration of 1000 s. The results demonstrate that the density continues to evolve smoothly over time. Importantly, the density generated by the proposed model remains within the required range. Specifically, at 1 s, the maximum observed density is 0.6 which occurs between 1410 m to 2400 m. As time progresses, the density exhibits a smooth and consistent behavior, indicating the effectiveness of the proposed model. Corresponding to the density evolution, Fig. 4(b) provides the speed variation with the proposed model. The findings indicate a smooth and continuous evolution of speed over time, while adhering to the upper limit of 19.94 m/s and the lower limit of 08.05 m/s. At 1 s, when the density reaches 0.6, the speed is 08.05 m/s at 1410–2370 m. The Proposed model outperforms the PW model by providing more realistic and plausible behavior in terms of density and speed patterns. Overall, the results emphasize that the proposed model ensures both smooth density and speed evolution over time. The proposed model maintains realistic ranges and exhibits improved performance compared to the PW model as shown in Figs. 5(a) and 5(b). Figures 5(a) and 5(b) display the traffic density and speed respectively with the PW model over a 3500 m road. The results indicate evolution of density and speed over time. The speed generated by the PW model exceeds required maximum 20.14 m/s. Having density of 0.02 between 2820 m to 3210 m at 250 s ,500 s, 750 s and 1000 s, speed exceeds from maximum speed to 20.71 m/s, 21.64 m/s, 21.72 m/s and 21.41 m/s respectively. At 3500 m having density of 0.01 the speed gets back within range. The maximum observed density is 0.61 which occurs between 2205 m to 2400 m. As time progresses, the density exhibits a smooth behavior with the PW model, but speed variation as shown in Fig. 5(b) indicates the drawback of the PW model. It is concluded that as density increases speed gets lowered as expected. With the proposed model, the speed and density stay within the required range. The speed changes as the density changes. Speed is larger for a lower density and smaller for larger density as expected. The proposed model behaves more realistic than the PW model as shown in the results for the given conditions. Figure 6 displays the traffic flow evolution with the Proposed model over a 3500 m road at 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is presented in Table 4 . At 1 s, the traffic flow is as low as 0.199 from 1 m to 495 m. From 510 m to 1495 m, it increases to 4.229. However, it further increases to 4.834 from 1530 m to 2400 m. Then it decreases to 0.199 from 2415 m to 3500 m. At 250 s, the flow is 0.199 at 1 m. It then increases to 0.946 at 360 m and further increases to 4.203 at 1020 m followed by maximum of 4.838 at 1905. It drops to 0.351 to 0.219 from 2820 m to 3500 m respectively. At 500 s, flow is as 0.244 at 1 m, which subsequently increases to 0.980 and 4.272 at 360 m and 1020 m respectively. It then increases to 4.888 at 1905 m and returns to 0.440 at 2970 m. 0.219 occurs at 3500 m. At 750 s, flow is 0.290 at 1 m. It then increases to 1.057at 360 m and further increases to 4.251 at 1020 m till it reaches 4.951 at 1905 m. At 1000 s, the traffic flow is 0.358 at 1 m and gradually increases to 0.189 at 360 m. It is 4.251 at 1020 m and reaches maximum of 4.991 at 1845 m then drops to 0.523 at 3210 m. While 0.329 occurs at 3500 m. Table 4 Traffic flow with the Proposed model at 1 s, 250 s, 500 s, 750 s and 1000s. Time (s) Distance (m) Traffic Flow (Veh/s) 1 1- 495 0.199 510–1485 4.229 1530–2400 2415–3500 4.834 0.199 250 1 360 1020 1905 2820 0.199 0.946 4.203 4.838 0.351 3500 0.219 500 1 360 1020 1905 2970 3500 0.244 0.980 4.272 4.888 0.440 0.219 750 3500 0.290 360 1.057 1020 1905 3090 4.251 4.951 0.501 3500 0.263 1000 1 360 1020 1845 3210 0.358 1.189 4.203 4.991 0.523 3500 0.329 Figure 7 displays the traffic Flow evolution with the PW model over a 3500 m road at 1 s, 250 s, 500 s, 750 s, and 1000 s with PCI of 10. The corresponding data is presented in Table 5 . At 1 s, the traffic flow is as low as 0.198 from 1 m to 495 m. From 510 m to 1485 m, it increases to 4.200. It is 4.800 at 1530 m to 2190 m. then it remains 4.758 from 2205 m to 2400 m. It drops to 0.198 at 2415 m, and remains the same until 3500 m. At 250 s, the traffic flow is 0.203 at 1 m. It then increases to 0.723 at 360 m and further increases to 4.200 and 4.819 at 1020 and 1905 respectively which is maximum flow region. It decreases to 0.359 at 2820 and further to 0.220 till 3500 m. At 500 s, it is 0.241 at 1 m, which increases to 0.949 at 360 m. It is 4.085 at 1020 m and reaches 4.819 at 1905 m. At 2820 m it starts decreasing from 0.484 to 0.220 at 3500 m. At 750 s, flow is 0.292 at 1 m. It then increases to 0.949 at 360 m and then reaches 3.955 and 4.891 at 1020 m and 1905 m, respectively. It reaches 0.567 at 3090 m. Subsequently, at 3500 m it experiences a drop to 0.268. At 1000 s, the traffic flow is 0.373 at 1 m and gradually increases to 1.082 at 360 m. And continues to increase to 3.839 and 4.907 at 1020 m and 1845 m respectively. It is 0.600 at 3210 and 0.345 at 3500 m. The expected traffic flow behavior of the proposed model as shown in Fig. 6 is consistent and exhibits superior accuracy in comparison with the PW model. Overall, the behavior of traffic flow in the proposed model is realistic and becomes smoother over time. Table 5 Traffic flow with the PW at 1 s, 250 s, 500 s, 750 s and 1000s. Time (s) Distance (m) Traffic Flow (Veh/s) 1 1- 495 0.198 510–1485 4.200 1530–2190 2205–2400 2415–3500 4.800 4.758 0.198 250 1 360 1020 1905 2820 0.203 0.723 4.200 4.819 0.359 3500 0.220 500 1 360 1020 1905 2970 3500 0.241 0.949 4.085 4.819 0.484 0.220 750 1 0.292 360 0.949 1020 1905 3090 3.955 4.891 0.567 3500 0.268 1000 1 360 1020 1845 3210 0.373 1.082 3.839 4.907 0.600 3500 0.354 5. Conclusions The traditional traffic flow model, called the PW model, uses a speed constant( \(\:{C}_{0}\) ) to adjust density. While Proposed model takes into account the influence of the Pavement Condition Index (PCI) on traffic flow. As the PCI decreases, indicating a worse pavement condition, the density of vehicles increases which causes the traffic flow to decrease. This relationship between the PCI, density, and velocity is determined. By incorporating the PCI into the determination of density changes, the Proposed model ensures that velocity and density values remain within acceptable limits. This eliminates the unrealistic behavior seen in the PW model and produces more realistic results. Overall, the novel macroscopic traffic flow model with the inclusion of the PCI offers improved performance compared to PW model. It provides a more accurate representation of real-world traffic flow conditions. This advancement can be beneficial for transportation planners and engineers in optimizing traffic flow and road maintenance strategies. Moreover, a Research window is open to develop Individual models for each surface distress. Declarations Funding Financial support was received from Universidad de Santiago de Chile, USACH, through project N°092218SF_POSTDOC, Dirección de Investigación Científica y Tecnológica, Dicyt. E.I.S.F. acknowledges funding from the Chilean National Research and Development Agency, ANID, research project Fondecyt Regular 1211767. Author Contribution Conceptualization, Zawar H. Khan and Shan Ul Haq; methodology, Zawar H. Khan, Inam Ullah Khan; software, Shan Ul Haq, Khurram S. Khattak; validation, Zawar H. Khan, Khurram S. Khattak and Khan Shahzada, Mujahid Ali, Krishna Prakash Arunachalam, Erick Saavedra Flores, Siva Avudaiappan; formal analysis, Shan Ul Haq; investigation, Shan Ul Haq, Inam Ullah Khan; resources, Zawar H. Khan; data curation, Khurram S. Khattak, Shan Ul Haq.; writing—original draft preparation, Shan Ul Haq, Inamullah Khan; writing—review and editing, Zawar H. Khan, Khan Shahzada Mujahid Ali, Krishna Prakash Arunachalam, Erick Saavedra Flores, Siva Avudaiappan; visualization, Khurram S. Khattak, Inam Ullah Khan; supervision, Khan Shahzada; project administration, Khan Shahzada; All authors have read and agreed to the published version of the manuscript. Acknowledgement Financial support was received from Universidad de Santiago de Chile, USACH, through project N°092218SF_POSTDOC, Dirección de Investigación Científica y Tecnológica, Dicyt. E.I.S.F. acknowledges funding from the Chilean National Research and Development Agency, ANID, research project Fondecyt Regular 1211767. Data Availability The data used to support the findings of this study are available from the corresponding author upon request. References Bhargab Maitra et al., "Micro-simulation based evaluation of Queue Jump Lane at isolated urban intersections: an experience in Kolkata," Journal of Transport Literature, vol. 9, pp. 10–14, 2015. F. A. Armah, D. O. Yawson, and A. 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Daganzo, C.F., Requiem for second-order fluid approximations of traffic flow. Transportation Research Part B: Methodological, vol. 29, no. 4, pp. 277–286, 1995. Whitham, G.B. Linear and Nonlinear Waves. Wiley, New York, 1974. Zhang, H.M. A non-equilibrium traffic model devoid of gas-like behavior. Transpn. Res. B, vol. 36, no. 3, pp. 275–290, 2002. Hashim, I. H., Badawy, R. M., & Heneash, U. (2023). Impact of Pavement Defects on Traffic Operational Performance. Sustainability (Basel, Switzerland), vol. 15, no. 10, pp. 8293–. https://doi.org/10.3390/su15108293 W. Imran, Z.H. Khan, T.A. Gulliver, K.S. Khattak, H. Nasir, A macroscopic traffic model for heterogeneous flow, Chin. J. Phys., vol. 63, pp. 419–435, 2020. Daganzo, C.F. Requiem for second-order fluid approximations of traffic flow. Transpn. Res. B, vol. 29, no. 4, pp. 277–286, 1995. Zhang, H.M. A theory of non-equilibrium traffic flow. Transpn. Res. B, vol. 32, no. 7, pp. 485–498, 1998. Z. H. Khan, W. Imran, S. Azeem, K. S. Khattak, T. A. Gulliver, and M. S. Aslam, ‘‘A macroscopic traffic model based on driver reaction and traffic stimuli,’’ Appl. Sci., vol. 9, no. 14, p. 2848, 2019. Ross, P. Traffic dynamics. Transp. Res. B: Methodological, vol. 22, no. 6, pp. 421–435, 1988. Newell, G.F. Comments on traffic dynamics. Transp. Res. B: Methodological, vol. 23, no. 5, pp. 386–389, 1989. Jiang, R., Wu, Q.-S., & Zhu, Z.-J. A new continuum model for traffic flow and numerical tests. Transp. Res. B, vol. 36, no. 5, pp. 405–419, 2002. Zheng, L., He, Z., & He, T. An anisotropic continuum model and its calibration with an improved monkey algorithm. Transportmetrica A, vol. 13, no. 6, pp. 519–543, 2017. Michalopoulos, P.G., Yi, P., & Lyrintzis, A.S. Continuum modelling of traffic dynamics for congested freeways. Transp. Res. B: Methodological, vol. 27, no. 4, pp. 315–332, 1993. Khan, Zawar H., Syed Abid Ali Shah, and T. Aaron Gulliver. "A macroscopic traffic model based on weather conditions." Chinese Physics B, vol. 27, no. 7, pp. 070202, 2018. Khan, Imran, et al. "Impact of Road Pavement Condition on Vehicular Free Flow Speed, Vibration and In-Vehicle Noise." Science, Engineering and Technology, vol. 3, no. 1, pp. 1–8, 2023. E. F. Toro, ‘‘On Glimm-related schemes for conservation laws,’’ Dept. Math. Phys., Manchester Metropolitan Univ., Manchester, U.K., Tech. Rep. MMU-9602, 1996. P. S. J. S. A. S. A. Kachroo Al-nasur Wadoo and A. Shende, Pedestrian Dynamics: Feedback Control Crowd Evacuation, New York, NY, USA: Springer, 2008. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, 2nd ed. New York, NY, USA: Wiley, 1967. Little, J.D.C. A proof for the queuing formula: L = kW. Oper. Res., vol. 9, no. 3, pp. 383–387, 1961. de Moura, C. A., & Kubrusly, C. S. The Courant-Friedrichs-Lewy (CFL) Condition: 80 Years After its Discovery, Berlin, Germany: Springer, 2013. Ni, D. Traffic Flow Theory: Characteristics, Experimental Methods, and Numerical Techniques. Kidlington, U.K: Butterworth-Heinemann, 2016, pp. 55–58 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4800495","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":345732551,"identity":"bb769a7c-e9f9-4242-8b6f-1cca6ec7ab3c","order_by":0,"name":"Shan Ul Haq","email":"","orcid":"","institution":"UET Peshawar","correspondingAuthor":false,"prefix":"","firstName":"Shan","middleName":"Ul","lastName":"Haq","suffix":""},{"id":345732552,"identity":"b15109c4-a69a-432e-8c75-5f2b552a90f0","order_by":1,"name":"Zawar Hussian Khan","email":"","orcid":"","institution":"University of Victoria","correspondingAuthor":false,"prefix":"","firstName":"Zawar","middleName":"Hussian","lastName":"Khan","suffix":""},{"id":345732553,"identity":"f383edc8-46e4-4e4b-86ce-849c5f11a564","order_by":2,"name":"Inamullah Khan","email":"data:image/png;base64,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","orcid":"","institution":"NUST NIT Campus","correspondingAuthor":true,"prefix":"","firstName":"Inamullah","middleName":"","lastName":"Khan","suffix":""},{"id":345732554,"identity":"a0ef4aab-bc9f-4b22-a997-917f40a8b8b3","order_by":3,"name":"Khurram Shahzad Khattak","email":"","orcid":"","institution":"UET Peshawar","correspondingAuthor":false,"prefix":"","firstName":"Khurram","middleName":"Shahzad","lastName":"Khattak","suffix":""},{"id":345732555,"identity":"c2d80d75-0fc3-4b1f-8428-62e6cd54aca9","order_by":4,"name":"Khan Shahzada","email":"","orcid":"","institution":"UET Peshawar","correspondingAuthor":false,"prefix":"","firstName":"Khan","middleName":"","lastName":"Shahzada","suffix":""},{"id":345732556,"identity":"718673ef-1018-4bf5-a911-672113ec9580","order_by":5,"name":"Mujahid Ali","email":"","orcid":"","institution":"Silesian University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Mujahid","middleName":"","lastName":"Ali","suffix":""},{"id":345732558,"identity":"7f528b69-a942-4452-ba69-2030d0c3cf15","order_by":6,"name":"Krishna Prakash Arunachalam","email":"","orcid":"","institution":"Universidad Tecnologica Metropolitana","correspondingAuthor":false,"prefix":"","firstName":"Krishna","middleName":"Prakash","lastName":"Arunachalam","suffix":""},{"id":345732564,"identity":"91973fcb-51a2-4efe-943b-a44dfcc298f0","order_by":7,"name":"Erick I Saavedra Flores","email":"","orcid":"","institution":"Universidad de Santiago de Chile, Estación Central","correspondingAuthor":false,"prefix":"","firstName":"Erick","middleName":"I Saavedra","lastName":"Flores","suffix":""},{"id":345732565,"identity":"f2967410-b43d-41a5-b960-b6b7679220fe","order_by":8,"name":"Siva Avudaiappan","email":"","orcid":"","institution":"Universidad de Santiago de Chile, Estación Central","correspondingAuthor":false,"prefix":"","firstName":"Siva","middleName":"","lastName":"Avudaiappan","suffix":""}],"badges":[],"createdAt":"2024-07-25 09:05:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4800495/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4800495/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":63567277,"identity":"e74891e3-075f-42e8-ac9a-0fa7ffcd1a7c","added_by":"auto","created_at":"2024-08-29 16:22:55","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":28941,"visible":true,"origin":"","legend":"\u003cp\u003ePavement Condition Index (PCI) rating scale\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4800495/v1/1e558356c33a41ad2f76bdea.png"},{"id":63567275,"identity":"f2da6340-cf36-4cc4-8840-1506b1257dd2","added_by":"auto","created_at":"2024-08-29 16:22:55","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":367819,"visible":true,"origin":"","legend":"\u003cp\u003e(\u003cstrong\u003ea\u003c/strong\u003e) Traffic density, ρ evolution with the proposed model on a 3500 m circular road with PCI = 10 at 1 s, 250 s, 500 s, 750 s and 1000 s.; (\u003cstrong\u003eb\u003c/strong\u003e) Speed evolution with the proposed model on a 3500 m circular road with PCI = 10 at 1 s, 250 s, 500 s, 750 s and 1000 s.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4800495/v1/885955ba12a9fb6dd60416fc.png"},{"id":63567281,"identity":"33e1735a-a4e1-4420-ac2e-ec3c4b1de6d2","added_by":"auto","created_at":"2024-08-29 16:22:55","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":386338,"visible":true,"origin":"","legend":"\u003cp\u003e(\u003cstrong\u003ea\u003c/strong\u003e) Traffic density evolution with the PW model on a 3500 m circular road with PCI = 10 at 1 s, 250 s, 500 s, 750 s and 1000 s; (\u003cstrong\u003eb\u003c/strong\u003e) Speed evolution with the PW model on a 3500 m circular road with PCI = 10 at 1 s, 250 s, 500 s, 750 s and 1000 s.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4800495/v1/a995a7940379ee040b19b04b.png"},{"id":63567665,"identity":"2a88d17e-b07d-46f2-8017-c6fb0b3f53fe","added_by":"auto","created_at":"2024-08-29 16:30:55","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":532430,"visible":true,"origin":"","legend":"\u003cp\u003e(\u003cstrong\u003ea\u003c/strong\u003e) Density behavior with the Proposed model on a 3500 m circular road for 1000 s with τ = 0.5 s and PCI =10; (\u003cstrong\u003eb\u003c/strong\u003e) Speed evolution with the proposed model on a 3500 m circular road for 1000 s with τ = 0.5 s and PCI = 10.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4800495/v1/05078e32401fc3e3821aea8f.png"},{"id":63567666,"identity":"91401d0a-7f87-4bcf-9808-9621ff078bcc","added_by":"auto","created_at":"2024-08-29 16:30:55","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":529300,"visible":true,"origin":"","legend":"\u003cp\u003e(\u003cstrong\u003ea\u003c/strong\u003e) Density behavior with the PW model on a 3500 m circular road for 1000 s with τ = 0.5 s; (\u003cstrong\u003eb\u003c/strong\u003e) Speed behavior with the PW model on a 3500 m circular road for 1000 s with τ = 0.5 s.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4800495/v1/08208c31d150d3f50791fcd8.png"},{"id":63567279,"identity":"add83794-6dfd-4201-8bb6-a9a06eb9513e","added_by":"auto","created_at":"2024-08-29 16:22:55","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":218937,"visible":true,"origin":"","legend":"\u003cp\u003eTraffic Flow (ρv), veh/s evolution with the Proposed model on a 3500 m circular road with PCI = 10 at 1 s, 250 s, 500 s, 750 s and 1000 s.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4800495/v1/f57912637071b5afe0b1bb4b.png"},{"id":63567667,"identity":"faadf6a2-4a05-41d3-90d3-3195ae6eb4c0","added_by":"auto","created_at":"2024-08-29 16:30:55","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":231612,"visible":true,"origin":"","legend":"\u003cp\u003eTraffic Flow (ρv), veh/s evolution with the Payne–Whitham (PW) model on a 3500 m circular road with PCI = 10 at 1 s, 250 s, 500 s, 750 s and 1000 s.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4800495/v1/84407e5913794c48c4923c4c.png"},{"id":76628108,"identity":"abd537e5-f44e-42dc-b3e9-773111e8f658","added_by":"auto","created_at":"2025-02-19 06:05:27","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3632847,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4800495/v1/38abdb17-f64c-4936-9774-f0ce6d7aa3b3.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Macroscopic Traffic Model based on Pavement Condition Index","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eAn established transportation system advances the financial and social development of a country. The essence of transportation systems is degraded with congestion occurrence as the travel time is increased. Congestion occurrence is due to the quick rise in vehicles on the roads in the urban regions which reduces the service quality of road network. The growth in vehicles not only fades the traffic markings on the road \u003csup\u003e1\u0026ndash;8\u003c/sup\u003e, but also causes a quicker degradation in the road surface \u003csup\u003e9\u0026ndash;10\u003c/sup\u003e. According to the department of transportation (TxDOT) of Texas, the deteriorated pavement conditions are one of the major causes of accidents. The severity of collision at a bad pavement is higher while on failed pavement accident rate decreases as the traffic speed is reduced \u003csup\u003e11\u003c/sup\u003e. Traffic density is the quantity of vehicles per unit distance, while flow is the result of density and velocity. The traffic density is higher on the decaying pavement, as the drivers slow down on the unpleasant pavement to avoid damage to the vehicle. This causes recurrent bottlenecks and reduces the capacity of the road infrastructure \u003csup\u003e12\u003c/sup\u003e. On a poor road surface condition, traffic is reduced by 20%, while the speed reduces by 50% \u003csup\u003e13\u003c/sup\u003e. The Pavement Condition Index (PCI) is a standardized metric for the road surface assessment, which ranges from 0 for failed conditions to 100 for good pavement condition \u003csup\u003e14\u003c/sup\u003e. This is a widely accepted tool for road surface assessment.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe PCI rating is given in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Pavements with minimal distress, that is, pavement condition index exceeding 40, minor changes occur in traffic speed. However, as the PCI decreases below 40, the speed changes significantly \u003csup\u003e15\u003c/sup\u003e. With the reduction in speed, headway is also impacted. In other words, the pavement condition significantly affects the distance between the vehicles. Time headway varies inversely with the pavement condition, while the speed holds a direct relationship. On a good rating road surface condition, that is, PCI is 96, headway recorded is 1.5 s for 14.6 m/s. While 4.2 s headway for 7.2 m/s are recorded on serious rating road with a PCI of 17. That is, road surface with a value of 17 PCI has a seriously deteriorated condition \u003csup\u003e16\u003c/sup\u003e. 55% speed reduction was recorded on poor road conditions having PCI of 19 as compared to good road conditions having PCI of 100. With distressed road surface, larger headway and slower speed was recorded \u003csup\u003e17\u003c/sup\u003e. Another study revealed that as speed reduces by 0.0037 m/s as the surface roughness increases \u003csup\u003e18\u003c/sup\u003e. The result eight road segments were used to study the impact of surface roughness on the capacity of two-lane roadways in India. The findings indicate that a vehicle's free flow speed drops as the road surface gets bumpy. Roughness has a greater impact on passenger car speed than it does on big trucks. The speed-volume correlations plotted for several two-lane rural road sections show that capacity declines as surface roughness increases \u003csup\u003e19\u003c/sup\u003e. Haj-Ismail \u003csup\u003e20\u003c/sup\u003e examined the impact of highway surface conditions on traffic speed. According to this study, no statistical changes in speed were observed at the same deterioration level during the day and night. In other words, the increased density on the road during the day has the same impact on speed as for a reduced density during the night. The traffic during the night has free flow conditions and is not impacted by the density. These studies highlight the importance of maintaining smooth roads for efficient traffic flow, reduced fuel consumption, and increased safety to mitigate the traffic conditions hazards. It is significant to characterize the impact of the PCI on the macroscopic traffic flow, to predict the traffic temporal and spatial evolution more realistically for the effective dissemination of urban.\u003c/p\u003e \u003cp\u003eThe increase in disturbance of traffic evolution has drawn the notice of traffic scientists. Different traffic systems are developed to comprehend the traffic disturbance in different conditions \u003csup\u003e22\u0026ndash;32\u003c/sup\u003e. The most common modelling approaches used are fluid-dynamic models, car following models, and gas-kinetic models. These models can predict traffic behavior and help in identifying the traffic characteristics. Different scientists have considered the backward traffic effect \u003csup\u003e33\u003c/sup\u003e, lateral effect of the lane width on the vehicles \u003csup\u003e34\u003c/sup\u003e, driver presumptions and traffic alignment \u003csup\u003e21,35\u003c/sup\u003e, and physiological response of drivers \u003csup\u003e36\u0026ndash;37\u003c/sup\u003e. Traffic flow is usually sorted by road conditions and can be portrayed as homogeneous or heterogeneous, and equilibrium or non-equilibrium. In homogeneous traffic, elements, for example, velocity and headway do not vary spatially \u003csup\u003e38\u003c/sup\u003e and vehicles follow lane discipline. Heterogeneous traffic comprises of motorized and non-motorized vehicles and lane discipline isn't fundamentally followed \u003csup\u003e39\u003c/sup\u003e. In an equilibrium flow, velocity is an element of density, so it happens when there is no change in velocity and there is spatial homogeneity. In a non-equilibrium flow, there will be changes in velocities and spatial homogeneity \u003csup\u003e40\u003c/sup\u003e. Three types of models have been utilized to portray traffic flow: microscopic, macroscopic, and mesoscopic. Macroscopic models manage total traffic behavior and are utilized to determine average density, velocity, and flow \u003csup\u003e21\u003c/sup\u003e. Microscopic models are about individual vehicle behavior and are dependent on the driver\u0026rsquo;s physical and psychological reactions \u003csup\u003e41\u003c/sup\u003e. They utilize vehicle features like distance headways, position, velocity, and time \u003csup\u003e42\u003c/sup\u003e. Mesoscopic models consider both individual and joined vehicle behavior \u003csup\u003e43\u003c/sup\u003e, so vehicles are portrayed both exclusively and collectively. Consequently, the issues should be integrated into traffic models to characterize traffic behavior precisely and realistically.\u003c/p\u003e \u003cp\u003eThe macroscopic models are also known as continuum models of vehicular traffic. These models are the simplest to characterize the temporal and spatial traffic evolution. The macroscopic models predict the traffic conditions at a road location x and time t, while considering the density ρ, speed v, and flow rate q. The fundamental relation is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{q}\\:=\\:{\\rho\\:}\\:\\varvec{v}\\)\u003c/span\u003e\u003c/span\u003e at x and t. Traffic conservation is in Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\rho\\:}{\\partial\\:t}+\\:\\frac{\\partial\\:q}{\\partial\\:x}=0$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eLighthill, Whitham and Richards proposed, Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e as the principal macroscopic traffic model and is the well-known LWR model \u003csup\u003e44,45\u003c/sup\u003e. According to this model, traffic is always at equilibrium, which is an ideal condition and is one of the major shortcomings of this model. The equilibrium velocity decreases with increase in density, that is, the derivative of the equilibrium is negative with a large density \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{(\\varvec{v}}^{{\\prime\\:}}\\left({\\rho\\:}\\right)\\:\u0026lt;\\:0)\\)\u003c/span\u003e\u003c/span\u003e. In other words, the fundamental diagram is a concave function of density. Mostly, traffic is at non-equilibrium for larger traffic density. Therefore, Payne-Whitham (PW) characterized the non-equilibrium temporal and spatial traffic evolution model and laid the foundation of second order traffic systems. Isotropic behavior is independent of the direction of flow. Daganzo \u003csup\u003e46\u003c/sup\u003e severely criticized the PW model for the isotropic traffic behavior predicted by the PW models. According to \u003csup\u003e46\u003c/sup\u003e, traffic flow has a unique direction, that is vehicles move in a single direction and traffic is anisotropic in nature of traffic. Payne-Witham (PW) model enhances the LWR model by including driver behavior \u003csup\u003e47,48\u003c/sup\u003e. This model has two equations. The first depends on vehicle conservation and is given by Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Driver behavior is characterized by the anticipation term in the second equation. Anticipation is driver presumption of ahead changes in traffic and conditions occurs on road. The changes due to velocity are characterized by a relaxation term in this equation. The relaxation term is the time required for traffic velocity alignment and react to absorb road condition. The second equation in the PW model is given as Eq.\u0026nbsp;2, \u003csup\u003e21\u003c/sup\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\left(\\rho\\:v\\right)}{\\partial\\:t}+\\frac{\\partial\\:\\left(\\rho\\:{v}^{2}\\right)}{\\partial\\:x}+\\:\\frac{\\partial\\:{(C}_{^\\circ\\:}^{2}\\rho\\:)}{\\partial\\:x}=\\:\\rho\\:\\left(\\frac{v\\left(\\rho\\:\\right)-v}{\\tau\\:}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{^\\circ\\:}\\:\\)\u003c/span\u003e \u003c/span\u003e is a constant which characterizes driver behavior with units in m/s which is called sonic velocity \u003csup\u003e49\u003c/sup\u003e. That is the changes in traffic travel at the speed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{^\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e. The driver anticipation is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\left({C}_{^\\circ\\:}^{2}\\rho\\:\\right)}_{x}\\)\u003c/span\u003e\u003c/span\u003e, while relaxation is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\rho\\:\\left(\\frac{v\\left(\\rho\\:\\right)-v}{\\tau\\:}\\right)\\)\u003c/span\u003e\u003c/span\u003e \u003csup\u003e50\u003c/sup\u003e which accounts for the fact that traffic adjusts to changes in density over time. The anticipation term characterizes adjustments in flow because of a driver\u0026rsquo;s action according to road condition and the relaxation term characterizes adjustments in velocity. The anticipation causes changes in traffic flow and is accounted for by relaxation. In other words, the driver\u0026rsquo;s presumption is the microscopic change occurred due to the driver personality, while relaxation is the macroscopic alignment of the traffic changes caused by the driver anticipation. The alignment is quicker with smaller relaxation time τ and is a drastic change. This relaxation is one of the most sensitive variables of the PW model, and the changes in the traffic grow for smaller relaxation time. Traffic alignment occurs based on the equilibrium velocity distribution v(ρ). The PW model assumes that drivers have similar behavior for all road conditions and only small adjustments in velocity and density occur \u003csup\u003e51\u003c/sup\u003e. Consequently, the traffic alignment with the PW at larger changes in density \u003csup\u003e21\u003c/sup\u003e results in abrupt adjustments in traffic over short distances \u003csup\u003e52\u003c/sup\u003e, which is insufficient and unrealistic.\u003c/p\u003e \u003cp\u003eA few methodologies have been considered to address the PW model shortcomings. By including driver anticipation based on changes in the equilibrium velocity, Zhang \u003csup\u003e40\u003c/sup\u003e enhanced the PW model. Although this model is anisotropic, and a motorist instantly adapts to the traffic density, this model did not take driver physiology into account \u003csup\u003e53\u003c/sup\u003e. Ross \u003csup\u003e54\u003c/sup\u003e improved the PW model by eliminating the anticipation term with the goal that driver presumption is disregarded \u003csup\u003e55\u003c/sup\u003e. This model can't describe traffic flow when there are huge changes or traffic jams. An anisotropic model that considers the driver's presupposition of forward circumstances based on density and spatial changes in velocity was proposed by Jiang et al. \u003csup\u003e56\u003c/sup\u003e, and which is given as Eq.\u0026nbsp;\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ3\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\rho\\:}\\:}{\\partial\\:\\text{t}\\:}+\\:\\frac{\\partial\\:{\\rho\\:}\\varvec{v}}{\\partial\\:\\text{x}}=\\:\\text{g}(\\text{x},\\text{t})$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eWhere g(x,t) is the egress and ingress of the vehicles to a straight road.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ4\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:u}{\\partial\\:t}+u\\frac{\\partial\\:u}{\\partial\\:x}=\\frac{{\\text{u}}_{\\text{e}}-\\text{u}}{\\text{T}}+{C}_{^\\circ\\:}\\frac{\\partial\\:u}{\\partial\\:x}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eIn Eq.\u0026nbsp;\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{^\\circ\\:}\\:\\)\u003c/span\u003e\u003c/span\u003e represents the constant of rearward motion. As a result, for any transition dynamics, changes in upstream traffic propagate at a constant velocity, which is not feasible. Additionally, this constant may produce behavior that is irrational and has wide transitions. The Jiang model was enhanced by Zheng et al. \u003csup\u003e57\u003c/sup\u003e by adding a new relaxation term that provides.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ5\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\rho\\:\\:}{\\partial\\:t\\:}+\\:\\frac{\\partial\\:\\rho\\:\\varvec{v}}{\\partial\\:x}=\\:0$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:v}{\\partial\\:t}+\\left(v-{C}_{^\\circ\\:}\\right)\\frac{\\partial\\:\\varvec{v}}{\\partial\\:x}=\\zeta\\:\\:[\\frac{1}{\\rho\\:}-\\frac{1}{{\\rho\\:}_{e}\\left(v\\right)}]\\:\\:,$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eIn Eq.\u0026nbsp;\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, ρe(v) is the equilibrium density distribution and ζ is the driver sensitivity coefficient. However, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{^\\circ\\:}\\:\\)\u003c/span\u003e\u003c/span\u003e and ζ are constants, which are the fitting parameters for traffic situations, rather than being based on realistic traffic flow factors. Consequently, these parameters are not based on the physics of traffic flow. Michalopoulos \u003csup\u003e58\u003c/sup\u003e proposed a model while not considering the equilibrium velocity distributions. This model can provide unrealistic traffic evolution by ignoring the density dependent characteristics on the road Khan et al. \u003csup\u003e59\u003c/sup\u003e proposed a model that accounts for snow, ice, and compacted snow by introducing a weather-dependent transition velocity distribution. This allows the model to capture changes in driver behavior due to reduced traction. However, this model did not consider the pavement conditions.\u003c/p\u003e \u003cp\u003eDespite these advancements, limitations still exist in characterization of the impact of road surface conditions on macroscopic traffic models. Most surface-aware models rely on simplified representations of microscopic complex interactions between surface conditions, vehicle dynamics, and driver behavior. The availability and accuracy of real-time surface data is also a challenge. By overcoming these challenges, road condition aware model can significantly improve understanding and management of traffic flow, particularly in dynamic environments with varying surface conditions. In this paper, the impact of road condition on macroscopic traffic flow has been characterized. Pavement Condition Index (PCI) is incorporated in a second order traffic system. The driver presumption is based on the changes in the PCI, that is a driver adjusts to the traffic changes in speed based on the PCI. The PW and proposed model are analyzed over 3500 m circular road for large changes in density. The proposed model performed better than the PW model.\u003c/p\u003e \u003cp\u003eSection 1 is an introduction; the rest of the paper is organized as follows. In Section 2, the Proposed model is presented by improving the PW model. In Section 3, traffic models\u0026rsquo; decomposition is performed for the Proposed and PW models. Section 4 presents the simulation results for Proposed and PW models which support the Proposed model by results. Finally, Section 5 presents the conclusions.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e"},{"header":"2. PROPOSED MODEL","content":"\u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe macroscopic PW traffic system uses speed v and density \u0026#120588; at a road location x and time t to characterize the flow. The first equation is the traffic conservation system, that is given as\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ7\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:{\\rho\\:}}{\\partial\\:\\text{t}}+\\:\\frac{\\partial\\:\\left({\\rho\\:}\\text{v}\\right)}{\\partial\\:\\text{x}}=0$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThis predicts small traffic changes over a road section. In other words, the traffic in-flux and egress from a road section does not significantly change and the traffic over a section remains conserved. Further, the traffic speed in Eq.\u0026nbsp;\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e is only density dependent, which is incorrect. Apart from the traffic density, different personalities are involved in controlling vehicles behavior. For example, the speed of traffic at same density depends on driver response. Therefore, Payne and Whitham \u003csup\u003e21\u003c/sup\u003e complement Eq.\u0026nbsp;\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, with driver response. The first equation of the Payne and Whitham model (PW) remains the same as Eq.\u0026nbsp;\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, while the second equation is\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ8\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\left({\\rho\\:}\\text{v}\\right)}{\\partial\\:\\text{t}}+\\frac{\\partial\\:\\left({\\rho\\:}{\\text{v}}^{2}\\right)}{\\partial\\:\\text{x}}+\\:\\frac{\\partial\\:{(\\text{C}}_{0}^{2}{\\rho\\:})}{\\partial\\:\\text{x}}=\\:{\\rho\\:}\\left(\\frac{\\text{v}\\left({\\rho\\:}\\right)-\\text{v}}{{\\tau\\:}}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\partial\\:{(\\text{C}}_{^\\circ\\:}^{2}{\\rho\\:})}{\\partial\\:\\text{x}}\\:\\)\u003c/span\u003e \u003c/span\u003e is the driver\u0026rsquo;s response. According to this model, the driver response for negligible changes is uniform and is complemented with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{0}\\)\u003c/span\u003e\u003c/span\u003e, which is a speed constant. The driver response is personality dependent, while in PW model, it is characterized with constant. The role of a driver\u0026rsquo;s personality in traffic evolution is ignored\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:\\rho\\:\\left(\\frac{\\varvec{v}\\left(\\varvec{\\rho\\:}\\right)-\\varvec{v}}{\\varvec{\\tau\\:}}\\right)\\)\u003c/span\u003e\u003c/span\u003e in Eq.\u0026nbsp;\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e is the relaxation term and represents the driver adjustment process to the speed changes. That is, traffic yields the equilibrium traffic speed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:v\\left({\\rho\\:}\\right),\\)\u003c/span\u003e\u003c/span\u003e which is density dependent. According to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:v\\left({\\rho\\:}\\right),\\)\u003c/span\u003e\u003c/span\u003e the speed is slower for larger density and vice versa for smaller density. τ affects the traffic adaptation to variations in density. Traffic adjusts more rapidly when the relaxation time is shorter, and more slowly when the relaxation time is longer. The anticipation term has the tendency of traffic adjustment to forward vehicles based on driver response. For example, speed decays as a user notices congestion ahead. In a familiar road environment and uniform flow, a driver response quickly and traffic alignment is smoother. While the relaxation term mitigates the changes occurred in traffic due to anticipation. In this paper, PW model is utilized to contain road surface condition.\u003c/p\u003e \u003cp\u003eOne of the influencing factors for a driver to respond is the pavement conditions. The response is quicker for larger road surface irregularities due to quicker driver reactions, and slower for best road surface conditions as driver reactions are small. Significant variations in traffic occur at larger surface irregularities or for worst road conditions. The road surface irregularities are quantified as pavement condition index. Therefore, in this paper, the pavement condition index (PCI) is employed in the anticipation term as driver response to comprehend the traffic changes based on the surface irregularities. This is significant velocity variations worsen the traffic flow. A driver adjusts the vehicle speed as notices a change in speed with changes in the PCI. That is for larger changes in PCI, the response is large and small for smaller changes noticed. The speed due to surface irregularities is a function PCI, that is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\text{v}\\left(\\text{P}\\text{C}\\text{I}\\right)\\)\u003c/span\u003e\u003c/span\u003e. Then changes observed in speed by a driver is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\text{d}\\varvec{v}\\left(\\text{P}\\text{C}\\text{I}\\right)}{\\text{d}\\left(\\text{P}\\text{C}\\text{I}\\right)}\\)\u003c/span\u003e\u003c/span\u003e. For a larger \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\text{d}\\varvec{v}\\left(\\text{P}\\text{C}\\text{I}\\right)}{\\text{d}\\left(\\text{P}\\text{C}\\text{I}\\right)}\\)\u003c/span\u003e\u003c/span\u003e, driver response is large, while driver response is small for a smaller is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\text{d}\\varvec{v}\\left(\\text{P}\\text{C}\\text{I}\\right)}{\\text{d}\\left(\\text{P}\\text{C}\\text{I}\\right)}\\)\u003c/span\u003e\u003c/span\u003e. The driver response is also impacted by the running conditions of surface irregularities that is driver response is slower for a smaller PCI due to its slower speed, and vice versa for a larger PCI. That is, vehicles are slower on a deteriorated road, while for a good quality of road, vehicles are at a faster speed and flow is smooth. Reduced acceleration and deceleration occur. A driver response is considerate of the PCI, and changes in velocity due to PCI in ahead conditions. Therefore, the driver\u0026rsquo;s response is.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ9\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:\\text{P}\\text{C}\\text{I}\\:\\frac{\\text{d}\\text{v}\\left(\\text{P}\\text{C}\\text{I}\\right)}{\\text{d}\\left(\\text{P}\\text{C}\\text{I}\\right)}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eBy substituting Eq.\u0026nbsp;\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e9\u003c/span\u003e as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{^\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e, Eq.\u0026nbsp;\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e is.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ10\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\left({\\rho\\:}v\\right)}{\\partial\\:\\text{t}}+\\frac{\\partial\\:\\left({\\rho\\:}{v}^{2}\\right)}{\\partial\\:\\text{x}}+\\:\\frac{\\partial\\:\\left({\\left(\\text{P}\\text{C}\\text{I}\\:\\frac{\\text{d}v\\left(\\text{P}\\text{C}\\text{I}\\right)}{\\text{d}\\left(\\text{P}\\text{C}\\text{I}\\right)}\\right)}^{2}{\\rho\\:}\\right)}{\\partial\\:\\text{x}}=\\:{\\rho\\:}\\left(\\frac{v\\left({\\rho\\:}\\right)-v}{{\\tau\\:}}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003ewhich reflects the impact of the pavement condition index (PCI) on vehicles speed and density. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\frac{\\partial\\:\\left({\\left(\\mathbf{P}\\mathbf{C}\\mathbf{I}\\:\\frac{\\mathbf{d}\\varvec{v}\\left(\\mathbf{P}\\mathbf{C}\\mathbf{I}\\right)}{\\mathbf{d}\\left(\\mathbf{P}\\mathbf{C}\\mathbf{I}\\right)}\\right)}^{2}\\varvec{\\rho\\:}\\right)}{\\partial\\:\\mathbf{x}}\\)\u003c/span\u003e\u003c/span\u003e is the anticipation which predicts the spatial changes in density based on the driver response to PCI. This can be utilized to assess the impact of traffic density and pavement condition index on the quality service of road network. The proposed model consists of Eq.\u0026nbsp;\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e and Eq.\u0026nbsp;\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e, and is given as\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ11\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\rho\\:}{\\partial\\:t}+\\:\\frac{\\partial\\:\\left(\\rho\\:v\\right)}{\\partial\\:x}=0$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e,\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\partial\\:\\left({\\rho\\:}v\\right)}{\\partial\\:\\text{t}}+\\frac{\\partial\\:\\left({\\rho\\:}{v}^{2}\\right)}{\\partial\\:\\text{x}}+\\:\\frac{\\partial\\:\\left({\\left(\\text{P}\\text{C}\\text{I}\\:\\frac{\\text{d}v\\left(\\text{P}\\text{C}\\text{I}\\right)}{\\text{d}\\left(\\text{P}\\text{C}\\text{I}\\right)}\\right)}^{2}{\\rho\\:}\\right)}{\\partial\\:\\text{x}}=\\:{\\rho\\:}\\left(\\frac{v\\left({\\rho\\:}\\right)-v}{{\\tau\\:}}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eEquation \u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e of the proposed model considers the conservation of vehicles like the PW type models. While Eq.\u0026nbsp;\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e12\u003c/span\u003e is the second equation of the proposed model which illustrates the evolution of traffic density and speed based on the surface irregularities depicted as PCI. This proposed model provides a comprehensive framework for studying the complex interplay between traffic dynamics and road pavement conditions.\u003c/p\u003e \u003cp\u003eIn this paper, the speed based on PCI is used from the study \u003csup\u003e60\u003c/sup\u003e. The speed relation is developed from surveying a 7-kilometer, two-lane segment of the Grand Trunk Road connecting Peshawar's Chamkani Bus Rapid Transit (BRT) station to Pabbi. The velocity relation impacted by PCI \u003csup\u003e60\u003c/sup\u003e is.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ13\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\:\\varvec{v}\\left(\\text{P}\\text{C}\\text{I}\\right)=\\:-1.659\\text{P}\\text{C}\\text{I}\\:+\\:75.62\\:$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe correlation coefficient R\u0026sup2; = 0.995 represents the degree to which the data fits the models. It ranges from 0 to 1, with a value close to 1 indicating a strong fit. From Eq.\u0026nbsp;\u003cspan refid=\"Equ13\" class=\"InternalRef\"\u003e13\u003c/span\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ14\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$$\\:\\frac{\\text{d}\\text{v}\\left(\\text{P}\\text{C}\\text{I}\\right)}{\\text{d}\\left(\\text{P}\\text{C}\\text{I}\\right)}=-1.65$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThis is the change occurring in speed with changes in PCI. In other words, the driver presumptions of the proposed model are based on the changes given in (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e) on the road chosen \u003csup\u003e60\u003c/sup\u003e.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e"},{"header":"3. Traffic Models Decomposition","content":"\u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eIn this paper, the first order centered (FORCE) numerical method is used to decompose the proposed and PW traffic models. The FORCE scheme is employed to approximate the larger traffic changes more accurately. The subscripts x and t, respectively, stand for the spatial and temporal partial derivatives. A traffic system in conserved form is\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ15\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$$\\:{\\psi\\:}_{t}\\:+\\:f\\:{\\left(\\psi\\:\\right)}_{x}\\:=\\:S\\left(\\psi\\:\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThis is the change occurring in speed with changes in PCI. In other words, the driver presumptions of the proposed model are based on the changes given in (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e) on the road chosen \u003csup\u003e60\u003c/sup\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ16\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$$\\:\\psi\\:\\:=\\left(\\genfrac{}{}{0pt}{}{\\rho\\:}{\\rho\\:v}\\right)\\:,\\:f\\left(\\psi\\:\\right)={\\left(\\genfrac{}{}{0pt}{}{\\rho\\:v}{{\\frac{{\\left(\\rho\\:v\\right)}^{2}}{\\rho\\:}+C}_{^\\circ\\:}^{2}\\rho\\:}\\right)}_{x},\\:S\\left(\\psi\\:\\right)=\\left(\\genfrac{}{}{0pt}{}{0}{\\rho\\:\\left(\\frac{v\\left(\\rho\\:\\right)-v}{\\tau\\:}\\right)}\\right),$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWhile for the Proposed model from (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e) and (\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e)\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$$\\:\\psi\\:\\:=\\left(\\genfrac{}{}{0pt}{}{\\rho\\:}{\\rho\\:v}\\right)\\:,\\:f\\left(\\psi\\:\\right)={\\left(\\genfrac{}{}{0pt}{}{\\rho\\:v}{\\frac{{\\left(\\rho\\:v\\right)}^{2}}{\\rho\\:}\\:+\\:\\rho\\:{\\left(PCI\\:\\frac{dv\\left(PCI\\right)}{d\\left(PCI\\right)}\\right)}^{2}}\\right)}_{x},\\:S\\left(\\psi\\:\\right)=\\left(\\genfrac{}{}{0pt}{}{0}{\\rho\\:\\left(\\frac{\\varvec{v}\\left(\\varvec{\\rho\\:}\\right)-\\varvec{v}}{\\varvec{\\tau\\:}}\\right)}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTo approximate the larger traffic changes more accurately and realistically, (\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e) is linearized over smaller temporal and spatial steps. The resulted linearized system is the quasilinear form and is\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e\n$$\\:{\\psi\\:}_{t}\\:+\\:A\\left(\\psi\\:\\right){\\psi\\:}_{x}\\:=\\:0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003ewhere A(ψ) is the Jacobian matrix, which presents the gradients of the functions of variables. The gradients in A(ψ) are essential to obtain the behavior of the functions of the data variables with changes in the variables itself. For a road length \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\text{x}}_{\\text{M}}\\)\u003c/span\u003e\u003c/span\u003e, with M number of equal sized road segments, the segment size is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\delta\\:}\\text{x}\\:=\\:{x}_{M}/\\text{M}\\)\u003c/span\u003e\u003c/span\u003e. Whereas a time step is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{t}_{N}\\:=\\:{t}_{N}/\\text{N}\\)\u003c/span\u003e\u003c/span\u003e, where t_N is the total time of traffic evolution and N is the number of equal sized time steps. tn is the nth time step then a time step is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(\\text{t}\\text{n}+1,\\:\\text{t}\\text{n})\\)\u003c/span\u003e\u003c/span\u003e. The traffic data variables are approximated at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(\\text{t}\\text{n}+1,\\:\\text{t}\\text{n})\\)\u003c/span\u003e\u003c/span\u003e, and road segment \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:(\\text{x}\\text{i}+\\:{\\delta\\:}\\text{x}/\\:2,\\:\\text{x}\\text{i}\\:-\\:{\\delta\\:}\\text{x}\\:/2)\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe FORCE scheme \u003csup\u003e61\u003c/sup\u003e, combines the first order Lax-Friedrichs scheme \u003csup\u003e62\u003c/sup\u003e and the second order Richtmyer scheme \u003csup\u003e63\u003c/sup\u003e, to approximate traffic at the boundaries of the road segments. This can precisely obtain the solution of the hyperbolic traffic systems of the proposed and PW models. Let ψ represent the average of the data variables in a road segment. The functions of the data variables are approximated at the road segment boundaries. At the boundary of road segments \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:i\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:i+1\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:f\\left({\\psi\\:}\\right)\\)\u003c/span\u003e\u003c/span\u003eat the nth time step can be approximated by using the Lax-Friedrichs scheme, and is given as\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e\n$$\\:{\\left({f}_{i+\\frac{1}{2}}^{n}\\:\\left({\\psi\\:}_{i}^{n},{\\psi\\:}_{i+1}^{n}\\right)\\right)}^{l}=\\frac{1}{2}\\left(f\\left({\\psi\\:}_{i}^{n}\\right)+f({\\psi\\:}_{i+1}^{n}\\right))+\\:\\frac{1}{2}\\frac{\\delta\\:t}{\\delta\\:x}\\left({\\psi\\:}_{i}^{n}-{\\psi\\:}_{i+1}^{n}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe Lax-Friedrichs scheme 62 is indicated by the superscript l. The Richtmyer scheme can approximate the data variables as 63\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ20\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e\n$$\\:{\\psi\\:}_{i+1}^{n}=\\:\\frac{1}{2}\\left({\\psi\\:}_{i}^{n}+{\\psi\\:}_{i+1}^{n}\\right)+\\:\\frac{1}{2}\\frac{\\delta\\:t}{\\delta\\:x}\\left({f(\\psi\\:}_{i}^{n})-{f(\\psi\\:}_{i+1}^{n})\\right),$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eand the corresponding \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:f\\left({\\psi\\:}\\right)\\:\\)\u003c/span\u003e\u003c/span\u003eis obtained as\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ21\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e\n$$\\:{\\left({f}_{i+\\frac{1}{2}}^{n}\\:\\left({\\psi\\:}_{i}^{n},{\\psi\\:}_{i+1}^{n}\\right)\\right)}^{\\text{r}}=\\:f\\left({\\psi\\:}_{i+\\frac{1}{2}}^{n}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003ewhere the Richtmyer scheme is indicated by the superscript r. For more accuracy \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:f\\left({\\psi\\:}\\right)\\)\u003c/span\u003e\u003c/span\u003e obtained from (\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e) and (\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e) at the segment boundaries are averaged, that is\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ22\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ22\" name=\"EquationSource\"\u003e\n$$\\:{f}_{i}^{n+1}=\\:\\frac{1}{2}\\left({\\left({f}_{i+\\frac{1}{2}}^{n}\\right)}^{r}+\\:{\\left({f}_{i+\\frac{1}{2}}^{n}\\right)}^{l}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe source terms of the proposed and PW models respectively in (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e) and (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e) are obtained as\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ23\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ23\" name=\"EquationSource\"\u003e\n$$\\:S\\left({\\psi\\:}_{i}^{n}\\right)=\\left(\\frac{v\\left({\\rho\\:}_{i}^{n}\\right)-{v}_{i}^{n}}{\\tau\\:}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e23\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eBy including the source term, which provides, the updated data variables at the ith road segment and nth time step are\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ24\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ24\" name=\"EquationSource\"\u003e\n$$\\:{\\psi\\:}_{i}^{n+1}={\\psi\\:}_{i}^{n}-\\frac{\\delta\\:t}{\\delta\\:x}\\left({f}_{i+\\frac{1}{2}}^{n}-{f}_{i-\\frac{1}{2}}^{n}\\right)+\\:\\delta\\:tS\\left({\\psi\\:}_{i}^{n}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e24\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Traffic Models Hyperbolicity\u003c/h2\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe traffic models are hyperbolic if changes in flow occur with a finite velocity. As a result, the changes impact during congestion reduces with time. The conditions of the strict hyperbolicity of the traffic systems are that the eigenvalues are real and distinct [64]. The eigenvalues of the traffic variables can be approximated from the Jacobian matrix.\u003c/p\u003e \u003cp\u003eThe Jacobian matrix of the PW model is.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ25\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ25\" name=\"EquationSource\"\u003e\n$$\\:S\\left(\\psi\\:\\right)=\\left(\\begin{array}{cc}0\u0026amp;\\:1\\\\\\:{-v}^{2}+{C}_{^\\circ\\:}^{2}\u0026amp;\\:2v\\end{array}\\right),$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e25\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eand the corresponding eigenvalues from (\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e) are.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ26\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ26\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}_{1}\\:=\\:v+{C}_{^\\circ\\:}\\:\\:\\:\\:,\\:\\:\\:\\:{\\lambda\\:}_{2}\\:=\\:v-{C}_{^\\circ\\:}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe eigenvalues are real and distinct, and therefore the PW model is hyperbolic. These eigenvalues stand the characteristic speeds of the PW model. The shocks, that is the regions of high traffic density and low velocity, are characterized by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{2}\\)\u003c/span\u003e\u003c/span\u003e, which is the slower speed characteristic. The rarefactions, or regions with low traffic density and high velocity, are characterized by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{1}\\)\u003c/span\u003e\u003c/span\u003e, which is the faster speed characteristic.\u003c/p\u003e \u003cp\u003eThe Jacobian matrix of the Proposed model is.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ27\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ27\" name=\"EquationSource\"\u003e\n$$\\:S\\left(\\psi\\:\\right)=\\left(\\begin{array}{cc}0\u0026amp;\\:1\\\\\\:{-v}^{2}+{\\left(PCI\\:\\frac{dv\\left(PCI\\right)}{d\\left(PCI\\right)}\\right)}^{2}\u0026amp;\\:2v\\end{array}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e27\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eand the corresponding eigenvalues are.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ28\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ28\" name=\"EquationSource\"\u003e\n$$\\:{\\lambda\\:}_{1}\\:=\\:v+\\left(PCI\\:\\frac{dv\\left(PCI\\right)}{d\\left(PCI\\right)}\\right)\\:\\:,\\:\\:\\:\\:{\\lambda\\:}_{2}\\:=\\:\\:v-\\left(PCI\\:\\frac{dv\\left(PCI\\right)}{d\\left(PCI\\right)}\\right)$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e28\u003c/div\u003e\u003c/div\u003e,\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe eigenvalues are real and distinct, and therefore the proposed model is hyperbolic. The traffic shock is characterized by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{2}\\)\u003c/span\u003e\u003c/span\u003e, while the rarefaction is characterized by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\lambda\\:}_{1}\\)\u003c/span\u003e\u003c/span\u003e. The changes in shocks and rarefactions are due to deteriorated road conditions. In other words, the speed characteristics changes due to pavement condition index (PCI) that is traffic moves from good to bad road conditions and vice versa. section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Performance Result","content":"\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eSimulation Parameter.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDescription\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eValue\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eTotal simulation time for proposed and PW models.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1000 s\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eLength of the circular road\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500 m\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eMaximum Speed\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e20.14 m/s\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eMaximum density\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eTime step for proposed and PW models\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026delta;t\u0026thinsp;=\u0026thinsp;0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eRoad step for proposed and PW models\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026delta;x\u0026thinsp;=\u0026thinsp;15\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eRelaxation time for proposed model\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026tau;\u0026thinsp;=\u0026thinsp;0.50\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eHeadway for proposed \u0026amp; PW model.\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eh\u0026thinsp;=\u0026thinsp;10 m\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eEquilibrium velocity distribution\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003ev(\u0026rho;)\u0026thinsp;=\u0026thinsp;Greenshields\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003ePavement Condition Index\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003ePCI\u0026thinsp;=\u0026thinsp;10\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003eIn this section, the performance of the proposed model and PW models on a Circular (ring) road, of length 3500 m is evaluated. The simulation parameters, as shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, ensure stability by adhering to the Courant, Friedrich, and Lewy (CFL) stability conditions [65]. The road step size is 15 m, and the time step size is 0.01 s for both models. The total simulation time is 1000 s. The maximum speed is 20.14 m/s, following the Greenshields equilibrium velocity distribution [66]. The maximum density is 1, indicating that the road is fully occupied. The relaxation time for the proposed model is 0.50 s and is within the typical range of 0 s to 0.50 s [51,52]. While for the PW model, the relaxation time is 2 s, and is chosen greater than the relaxation time of the proposed model. At a smaller headway of 0.5 s, the PW model with the given conditions in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e cannot characterize the traffic flow. The initial density at time t\u0026thinsp;=\u0026thinsp;0 for the traffic simulation over the circular road for both the models are\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ29\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ29\" class=\"mathdisplay\"\u003e$$\\:{\\rho\\:}_{^\\circ\\:}=\\:\\left\\{\\begin{array}{c}0.01,\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:for\\:x\u0026lt;500\\:\\\\\\:0.3,for\\:510\\le\\:x\u0026lt;1500\\\\\\:0.6,\\:for\\:1500\\le\\:x\u0026lt;2400\\\\\\:0.01,\\:\\:\\:\\:\\:\\:\\:for\\:x\\:\\ge\\:2415\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\end{array}\\right.$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e29\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e,\u003c/p\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003eFigure 2(a) displays the density evolution according to the proposed model over a 3500 m road at five different time points: 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is presented in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. Starting at 1 s, the density is recorded consistently low as 0.01 from 1 m to 495 m. From 510 m to 1395 m, it increases to 0.29. However, it increases more to 0.6 which is the maximum from 1410 m to 2400 m and then drops to 0.01 which is the lowest from 2415 m to 3500 m over the road length. At 250 s, the density is 0.01 at 1 m. It starts to rise from 0.04 to 0.30 at 360 m and 1020 m respectively. Further increases to 0.6 at 1905 m. However, between 2820 m and 3500 m, it experiences a reduction from 0.017 to 0.013. At 500 s, the density is lowest as 0.01 at 1 m, which subsequently increases from 0.06 to 0.30 at 360 m and 1020 m respectively. It then increases to the maximum 0.58 at 1905 m and returns from 0.02 to 0.01 at 2970 m and 3500 m respectively. At 750 s, the density starts at 0.01 at 1 m. It then increases to 0.06 and 0.30 at 360 m and 1020 m respectively and further increases to the largest as 0.56 at 1905 m. However, at 2970 m to 3500 m it experiences a reduction from 0.04 to 0.01. Finally, at 1000 s, the density is 0.01 at 1 m and gradually increases to 0.06 at 360 m. It then increases to 0.29 at 1020 m. At 1845 m, the density increases to maximum of 0.54 and then reduces from 0.02 to 0.01 at 3210 m and 3500 m respectively.\u003c/p\u003e\n\u003cp\u003eFigure 2(b) illustrates the Speed evolution with the proposed model over a 3500 m road at five different time intervals: 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is provided in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. At 1 s, the speed is 19.94 m/s from 1 m to 495 m, which is the highest, and then decreases to 14.10 m/s from 510 m to 1395 m. The Speed decreases more to 08.05 m/s from 1410 m to 2400 m. Subsequently, it rises to as high as 19.94 m/s from 2415 m to 3500 m. At 250 s, the Speed is 19.90 m/s at 1 m and gradually drops to 19.11 m/s at 360 m. It then reduces to 14.05 m/s and 08.08 m/s, which is lowest at 1020 m and 1905 m/s. Subsequently, it rises to 19.77 m/s and rises to maximum of 19.92 m/s at 3500 m. At 500 s, the Speed is 19.83 m/s at 1 m and reduces to 18.91 m/s and 13.99 m/s at 360 m and 1020 m respectively. At 1905 m, the Speed is lowest of 08.35 m/s, but it subsequently increases to 19.68 m/s and 19.91 m/s at 2970 m and 3500 m respectively. At 750 s, the Speed is 19.80 m/s at 1 m and gradually drops to 18.86 m/s and 14.04 m/s at 360 m and 1020 m respectively. It then reduces to the lowest of 08.77 m/s at 1905 m. Subsequently, it rises to 19.71 m/s and 19.87 m/s at 2970 m and 3500 m respectively. At 1000 s, the Speed is 19.77 m/s at 1 m and smoothly decreases to 18.78 m/s and 14.16 m/s at 360 m and 1020 m respectively. Between 1845 m and 3210 m, the speed rises from 9.13 m/s to 19.60 m/s, and finally traffic speed achieves 19.79 m/s at 3500 m.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cp\u003eSpeed appropriately adjusts in response to changes in Density. As density increases, the speed decreases and vice versa as shown in Figs.\u0026nbsp;2(a) and 2(b). That is, the speed adjusts as expected in response to changes in density. This shows the correctness of the proposed model. The observed behavior of density and speed in the proposed model is realistic, and it becomes progressively smoother over time.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab2\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eDensity and Speed with the proposed model at 1 s, 250 s, 500 s, 750 s and 1000.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTime (s)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDistance (m)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDensity (\u0026rho;)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSpeed (m/s)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"3\" align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1- 495\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.94\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e510\u0026ndash;1395\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.29\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e14.10\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1410\u0026ndash;2400\u003c/p\u003e\n\u003cp\u003e2415\u0026ndash;3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.60\u003c/p\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e08.05\u003c/p\u003e\n\u003cp\u003e19.94\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e250\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2820\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003cp\u003e0.04\u003c/p\u003e\n\u003cp\u003e0.30\u003c/p\u003e\n\u003cp\u003e0.60\u003c/p\u003e\n\u003cp\u003e0.017\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.90\u003c/p\u003e\n\u003cp\u003e19.11\u003c/p\u003e\n\u003cp\u003e14.05\u003c/p\u003e\n\u003cp\u003e08.08\u003c/p\u003e\n\u003cp\u003e19.77\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.013\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.92\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2970\u003c/p\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003cp\u003e0.06\u003c/p\u003e\n\u003cp\u003e0.30\u003c/p\u003e\n\u003cp\u003e0.58\u003c/p\u003e\n\u003cp\u003e0.02\u003c/p\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.83\u003c/p\u003e\n\u003cp\u003e18.91\u003c/p\u003e\n\u003cp\u003e13.99\u003c/p\u003e\n\u003cp\u003e08.35\u003c/p\u003e\n\u003cp\u003e19.68\u003c/p\u003e\n\u003cp\u003e19.91\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"4\" align=\"left\"\u003e\n\u003cp\u003e750\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.80\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.06\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18.86\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2970\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.30\u003c/p\u003e\n\u003cp\u003e0.56\u003c/p\u003e\n\u003cp\u003e0.04\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e14.04\u003c/p\u003e\n\u003cp\u003e08.77\u003c/p\u003e\n\u003cp\u003e19.71\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.87\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e1000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1845\u003c/p\u003e\n\u003cp\u003e3210\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003cp\u003e0.06\u003c/p\u003e\n\u003cp\u003e0.29\u003c/p\u003e\n\u003cp\u003e0.54\u003c/p\u003e\n\u003cp\u003e0.02\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.77\u003c/p\u003e\n\u003cp\u003e18.78\u003c/p\u003e\n\u003cp\u003e14.16\u003c/p\u003e\n\u003cp\u003e09.13\u003c/p\u003e\n\u003cp\u003e19.60\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.79\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003eFigure 3(a) displays the density evolution with the PW model over a 3500 m road at 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is presented in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. At 1 s, the density is recorded as low as 0.01 from 1 m to 495 m. From 510 m to 1485 m, it increases to 0.30. However, density is 0.60 from 1530 m to 2190 m and is 0.61 from 2205 m to 2400 m. It then drops to 0.01, which is the lowest, from 2415 m to 3500 m. At 250 s, the density is 0.01 at 1 m. It then increases to 0.05 at 360 m. While it is 0.29 at 1020 m, and further increases to 0.59 at 1905 m. However, between 2820 m and 3500 m, it experiences a reduction to 0.01. At 500 s, the density is as lowest as 0.01 at 1 m, which subsequently increases to 0.06 at 360 m. It then increases to 0.29 at 1020 following the maximum 0.57 at 1905 m and returns to 0.02 at 2970 m. The density further reduces to 0.01 at 3500 m. At 750 s, the density is 0.01 at 1 m. It then increases to 0.06 at 360 m, which changes to 0.29 at 1020 m. The largest change 0.54 is observed at 1905 m. However, at 3090 m, the density changes from 0.02 to 0.01 at 3500 m. Finally, at 1000 s, the density is 0.02 at 1 m and gradually increases to 0.07 at 360 m. This further increases to 0.28 at 1020 m. At 1845 m, the density increases to 0.52 and then reduces to 0.2 at 3210 m. It further reduces to 0.01 at 3500 m.\u003c/p\u003e\n\u003cp\u003eFigure 3(b) illustrates the speed evolution with the PW model over a 3500 m road at 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is provided in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. The speed is 19.8 m/s from 1 m to 495 m, which is the highest, and then decreases to 14.00 m/s at 510 m. The speed is uniform from 510 m to 1485 m. The speed is 8.00 m/s between 1530 m to 2190 m, while 7.80 m/s from 2205 m to 2400 m. Subsequently, it rises to as high as 19.8 m/s from 2415 m to 3500 m. At 250 s, the speed is 19.51 m/s at 1 m and gradually drops to 13.66 m/s at 360 m. It then drops to the lowest 08.02 m/s at 1905 m. Subsequently, it rises to 20.71 m/s at 2820 m which exceeds maximum speed. which then slightly drops to 19.72 m/s at 3500 m. At 500 s, the speed is 18.23 m/s at 1 m and reduces to 14.70 m/s at 360 m and slightly falls more to 13.63 m/s at 1020 m until reaches the lowest 08.73 at 1905 m. At 2970 m, the speed is 21.64 m/s that surpass maximum value, but it subsequently falls to 18.64 m/s at 3500 m. At 750 s, the speed is 18.23 m/s at 1 m and gradually reduces to 15.43 m/s at 360 m. It then experiences a consecutive drop to 13.63 m/s at 1020 m, which further reduces to 08.90 m/s at 1905 m. Subsequently, it rises to 21.72 m/s at 3090 m which outdoes maximum speed, which then reduces to 18.04 m/s at 3500 m. At 1000 s, the speed is 18.04 m/s at 1 m and smoothly decreases to 15.91 m/s at 360 m. Between 1020 m and 1845 m, the speed decreases from 13.72 m/s to 09.37 m/s, and finally it transcends maximum value and achieves 21.41 m/s at 3210 m. While the speed at 3500 m is 18.64.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003eThe relationship between density and speed in PW model exhibits limitations. While it reflects that speed decreases as density increases and vice versa, the model fails to capture realistic variations effectively. Particularly, abrupt changes in speed or surpass from maximum speed occur with changes in density, indicating poor road conditions in high-density areas where speeds drop significantly, and conversely, sharp increases in speed as density decreases, indicating smoother pavement areas. These discrepancies highlight the shortcomings of the PW model in accurately representing real-world scenario.\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab3\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eDensity and Speed with the PW model at 1 s, 250 s, 500 s, 750 s and 1000.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTime (s)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDistance (m)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDensity (\u0026rho;)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSpeed (m/s)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"3\" align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1- 495\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.80\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e510\u0026ndash;1485\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.30\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e14.00\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1530\u0026ndash;2190\u003c/p\u003e\n\u003cp\u003e2205\u0026ndash;2400\u003c/p\u003e\n\u003cp\u003e2415\u0026ndash;3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.60\u003c/p\u003e\n\u003cp\u003e0.61\u003c/p\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e08.00\u003c/p\u003e\n\u003cp\u003e07.80\u003c/p\u003e\n\u003cp\u003e19.80\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e250\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2820\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003cp\u003e0.05\u003c/p\u003e\n\u003cp\u003e0.29\u003c/p\u003e\n\u003cp\u003e0.59\u003c/p\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.51\u003c/p\u003e\n\u003cp\u003e13.66\u003c/p\u003e\n\u003cp\u003e13.97\u003c/p\u003e\n\u003cp\u003e08.02\u003c/p\u003e\n\u003cp\u003e20.71\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19.72\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2970\u003c/p\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003cp\u003e0.06\u003c/p\u003e\n\u003cp\u003e0.29\u003c/p\u003e\n\u003cp\u003e0.57\u003c/p\u003e\n\u003cp\u003e0.02\u003c/p\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18.23\u003c/p\u003e\n\u003cp\u003e14.70\u003c/p\u003e\n\u003cp\u003e13.63\u003c/p\u003e\n\u003cp\u003e08.73\u003c/p\u003e\n\u003cp\u003e21.64\u003c/p\u003e\n\u003cp\u003e18.64\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"4\" align=\"left\"\u003e\n\u003cp\u003e750\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18.23\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.06\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e15.43\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e3090\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.29\u003c/p\u003e\n\u003cp\u003e0.54\u003c/p\u003e\n\u003cp\u003e0.02\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e13.63\u003c/p\u003e\n\u003cp\u003e08.90\u003c/p\u003e\n\u003cp\u003e21.72\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18.64\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e1000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1845\u003c/p\u003e\n\u003cp\u003e3210\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.02\u003c/p\u003e\n\u003cp\u003e0.07\u003c/p\u003e\n\u003cp\u003e0.28\u003c/p\u003e\n\u003cp\u003e0.52\u003c/p\u003e\n\u003cp\u003e0.02\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18.04\u003c/p\u003e\n\u003cp\u003e15.91\u003c/p\u003e\n\u003cp\u003e13.72\u003c/p\u003e\n\u003cp\u003e09.37\u003c/p\u003e\n\u003cp\u003e21.41\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18.64\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003eFigure 4(a) illustrates the spatial and temporal density evolution using the proposed model over a 3500 m road for a duration of 1000 s. The results demonstrate that the density continues to evolve smoothly over time. Importantly, the density generated by the proposed model remains within the required range. Specifically, at 1 s, the maximum observed density is 0.6 which occurs between 1410 m to 2400 m. As time progresses, the density exhibits a smooth and consistent behavior, indicating the effectiveness of the proposed model. Corresponding to the density evolution, Fig.\u0026nbsp;4(b) provides the speed variation with the proposed model. The findings indicate a smooth and continuous evolution of speed over time, while adhering to the upper limit of 19.94 m/s and the lower limit of 08.05 m/s. At 1 s, when the density reaches 0.6, the speed is 08.05 m/s at 1410\u0026ndash;2370 m. The Proposed model outperforms the PW model by providing more realistic and plausible behavior in terms of density and speed patterns. Overall, the results emphasize that the proposed model ensures both smooth density and speed evolution over time. The proposed model maintains realistic ranges and exhibits improved performance compared to the PW model as shown in Figs.\u0026nbsp;5(a) and 5(b).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003eFigures 5(a) and 5(b) display the traffic density and speed respectively with the PW model over a 3500 m road. The results indicate evolution of density and speed over time. The speed generated by the PW model exceeds required maximum 20.14 m/s. Having density of 0.02 between 2820 m to 3210 m at 250 s ,500 s, 750 s and 1000 s, speed exceeds from maximum speed to 20.71 m/s, 21.64 m/s, 21.72 m/s and 21.41 m/s respectively. At 3500 m having density of 0.01 the speed gets back within range. The maximum observed density is 0.61 which occurs between 2205 m to 2400 m.\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eAs time progresses, the density exhibits a smooth behavior with the PW model, but speed variation as shown in Fig.\u0026nbsp;5(b) indicates the drawback of the PW model. It is concluded that as density increases speed gets lowered as expected. With the proposed model, the speed and density stay within the required range. The speed changes as the density changes. Speed is larger for a lower density and smaller for larger density as expected. The proposed model behaves more realistic than the PW model as shown in the results for the given conditions.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003eFigure \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e displays the traffic flow evolution with the Proposed model over a 3500 m road at 1 s, 250 s, 500 s, 750 s, and 1000 s. The corresponding data is presented in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e. At 1 s, the traffic flow is as low as 0.199 from 1 m to 495 m. From 510 m to 1495 m, it increases to 4.229. However, it further increases to 4.834 from 1530 m to 2400 m. Then it decreases to 0.199 from 2415 m to 3500 m. At 250 s, the flow is 0.199 at 1 m. It then increases to 0.946 at 360 m and further increases to 4.203 at 1020 m followed by maximum of 4.838 at 1905. It drops to 0.351 to 0.219 from 2820 m to 3500 m respectively. At 500 s, flow is as 0.244 at 1 m, which subsequently increases to 0.980 and 4.272 at 360 m and 1020 m respectively. It then increases to 4.888 at 1905 m and returns to 0.440 at 2970 m. 0.219 occurs at 3500 m. At 750 s, flow is 0.290 at 1 m. It then increases to 1.057at 360 m and further increases to 4.251 at 1020 m till it reaches 4.951 at 1905 m. At 1000 s, the traffic flow is 0.358 at 1 m and gradually increases to 0.189 at 360 m. It is 4.251 at 1020 m and reaches maximum of 4.991 at 1845 m then drops to 0.523 at 3210 m. While 0.329 occurs at 3500 m.\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab4\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eTraffic flow with the Proposed model at 1 s, 250 s, 500 s, 750 s and 1000s.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTime (s)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDistance (m)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTraffic Flow (Veh/s)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"3\" align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1- 495\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.199\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e510\u0026ndash;1485\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.229\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1530\u0026ndash;2400\u003c/p\u003e\n\u003cp\u003e2415\u0026ndash;3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.834\u003c/p\u003e\n\u003cp\u003e0.199\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e250\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2820\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.199\u003c/p\u003e\n\u003cp\u003e0.946\u003c/p\u003e\n\u003cp\u003e4.203\u003c/p\u003e\n\u003cp\u003e4.838\u003c/p\u003e\n\u003cp\u003e0.351\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.219\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2970\u003c/p\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.244\u003c/p\u003e\n\u003cp\u003e0.980\u003c/p\u003e\n\u003cp\u003e4.272\u003c/p\u003e\n\u003cp\u003e4.888\u003c/p\u003e\n\u003cp\u003e0.440\u003c/p\u003e\n\u003cp\u003e0.219\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"4\" align=\"left\"\u003e\n\u003cp\u003e750\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.290\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.057\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e3090\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.251\u003c/p\u003e\n\u003cp\u003e4.951\u003c/p\u003e\n\u003cp\u003e0.501\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.263\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e1000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1845\u003c/p\u003e\n\u003cp\u003e3210\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.358\u003c/p\u003e\n\u003cp\u003e1.189\u003c/p\u003e\n\u003cp\u003e4.203\u003c/p\u003e\n\u003cp\u003e4.991\u003c/p\u003e\n\u003cp\u003e0.523\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.329\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e displays the traffic Flow evolution with the PW model over a 3500 m road at 1 s, 250 s, 500 s, 750 s, and 1000 s with PCI of 10. The corresponding data is presented in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. At 1 s, the traffic flow is as low as 0.198 from 1 m to 495 m. From 510 m to 1485 m, it increases to 4.200. It is 4.800 at 1530 m to 2190 m. then it remains 4.758 from 2205 m to 2400 m. It drops to 0.198 at 2415 m, and remains the same until 3500 m. At 250 s, the traffic flow is 0.203 at 1 m. It then increases to 0.723 at 360 m and further increases to 4.200 and 4.819 at 1020 and 1905 respectively which is maximum flow region. It decreases to 0.359 at 2820 and further to 0.220 till 3500 m. At 500 s, it is 0.241 at 1 m, which increases to 0.949 at 360 m. It is 4.085 at 1020 m and reaches 4.819 at 1905 m. At 2820 m it starts decreasing from 0.484 to 0.220 at 3500 m. At 750 s, flow is 0.292 at 1 m. It then increases to 0.949 at 360 m and then reaches 3.955 and 4.891 at 1020 m and 1905 m, respectively. It reaches 0.567 at 3090 m. Subsequently, at 3500 m it experiences a drop to 0.268. At 1000 s, the traffic flow is 0.373 at 1 m and gradually increases to 1.082 at 360 m. And continues to increase to 3.839 and 4.907 at 1020 m and 1845 m respectively. It is 0.600 at 3210 and 0.345 at 3500 m. The expected traffic flow behavior of the proposed model as shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e is consistent and exhibits superior accuracy in comparison with the PW model. Overall, the behavior of traffic flow in the proposed model is realistic and becomes smoother over time.\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab5\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eTraffic flow with the PW at 1 s, 250 s, 500 s, 750 s and 1000s.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTime (s)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDistance (m)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTraffic Flow (Veh/s)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"3\" align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1- 495\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.198\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e510\u0026ndash;1485\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.200\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1530\u0026ndash;2190\u003c/p\u003e\n\u003cp\u003e2205\u0026ndash;2400\u003c/p\u003e\n\u003cp\u003e2415\u0026ndash;3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4.800\u003c/p\u003e\n\u003cp\u003e4.758\u003c/p\u003e\n\u003cp\u003e0.198\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e250\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2820\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.203\u003c/p\u003e\n\u003cp\u003e0.723\u003c/p\u003e\n\u003cp\u003e4.200\u003c/p\u003e\n\u003cp\u003e4.819\u003c/p\u003e\n\u003cp\u003e0.359\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.220\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e2970\u003c/p\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.241\u003c/p\u003e\n\u003cp\u003e0.949\u003c/p\u003e\n\u003cp\u003e4.085\u003c/p\u003e\n\u003cp\u003e4.819\u003c/p\u003e\n\u003cp\u003e0.484\u003c/p\u003e\n\u003cp\u003e0.220\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"4\" align=\"left\"\u003e\n\u003cp\u003e750\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.292\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.949\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1905\u003c/p\u003e\n\u003cp\u003e3090\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e3.955\u003c/p\u003e\n\u003cp\u003e4.891\u003c/p\u003e\n\u003cp\u003e0.567\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.268\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e1000\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003cp\u003e360\u003c/p\u003e\n\u003cp\u003e1020\u003c/p\u003e\n\u003cp\u003e1845\u003c/p\u003e\n\u003cp\u003e3210\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.373\u003c/p\u003e\n\u003cp\u003e1.082\u003c/p\u003e\n\u003cp\u003e3.839\u003c/p\u003e\n\u003cp\u003e4.907\u003c/p\u003e\n\u003cp\u003e0.600\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3500\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.354\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e"},{"header":"5. Conclusions","content":"\u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe traditional traffic flow model, called the PW model, uses a speed constant( \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{0}\\)\u003c/span\u003e\u003c/span\u003e ) to adjust density. While Proposed model takes into account the influence of the Pavement Condition Index (PCI) on traffic flow. As the PCI decreases, indicating a worse pavement condition, the density of vehicles increases which causes the traffic flow to decrease. This relationship between the PCI, density, and velocity is determined. By incorporating the PCI into the determination of density changes, the Proposed model ensures that velocity and density values remain within acceptable limits. This eliminates the unrealistic behavior seen in the PW model and produces more realistic results. Overall, the novel macroscopic traffic flow model with the inclusion of the PCI offers improved performance compared to PW model. It provides a more accurate representation of real-world traffic flow conditions. This advancement can be beneficial for transportation planners and engineers in optimizing traffic flow and road maintenance strategies. Moreover, a Research window is open to develop Individual models for each surface distress.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eFinancial support was received from Universidad de Santiago de Chile, USACH, through project N\u0026deg;092218SF_POSTDOC, Direcci\u0026oacute;n de Investigaci\u0026oacute;n Cient\u0026iacute;fica y Tecnol\u0026oacute;gica, Dicyt. E.I.S.F. acknowledges funding from the Chilean National Research and Development Agency, ANID, research project Fondecyt Regular 1211767.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eConceptualization, Zawar H. Khan and Shan Ul Haq; methodology, Zawar H. Khan, Inam Ullah Khan; software, Shan Ul Haq, Khurram S. Khattak; validation, Zawar H. Khan, Khurram S. Khattak and Khan Shahzada, Mujahid Ali, Krishna Prakash Arunachalam, Erick Saavedra Flores, Siva Avudaiappan; formal analysis, Shan Ul Haq; investigation, Shan Ul Haq, Inam Ullah Khan; resources, Zawar H. Khan; data curation, Khurram S. Khattak, Shan Ul Haq.; writing\u0026mdash;original draft preparation, Shan Ul Haq, Inamullah Khan; writing\u0026mdash;review and editing, Zawar H. Khan, Khan Shahzada Mujahid Ali, Krishna Prakash Arunachalam, Erick Saavedra Flores, Siva Avudaiappan; visualization, Khurram S. Khattak, Inam Ullah Khan; supervision, Khan Shahzada; project administration, Khan Shahzada; All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eFinancial support was received from Universidad de Santiago de Chile, USACH, through project N\u0026deg;092218SF_POSTDOC, Direcci\u0026oacute;n de Investigaci\u0026oacute;n Cient\u0026iacute;fica y Tecnol\u0026oacute;gica, Dicyt. E.I.S.F. acknowledges funding from the Chilean National Research and Development Agency, ANID, research project Fondecyt Regular 1211767.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe data used to support the findings of this study are available from the corresponding author upon request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBhargab Maitra et al., \"Micro-simulation based evaluation of Queue Jump Lane at isolated urban intersections: an experience in Kolkata,\" Journal of Transport Literature, vol. 9, pp. 10\u0026ndash;14, 2015.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eF. A. Armah, D. O. Yawson, and A. ANM Pappoe, \"A systems dynamics approach to explore traffic congestion and air pollution link in the city of Accra, Ghana,\" Sustainability, vol. 2, no. 1, pp. 252\u0026ndash;265, 2010.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJ. 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Kidlington, U.K: Butterworth-Heinemann, 2016, pp. 55\u0026ndash;58\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Macroscopic Traffic Flow model, Pavement Condition Index (PCI), Payne–Whitham (PW) model, Speed Constant","lastPublishedDoi":"10.21203/rs.3.rs-4800495/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4800495/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe prevalence of poor road conditions makes urban traffic gridlock, leading to increased travel time and disruptions in urban mobility in developing countries. In this research the Payne-Whitham (PW) model which is second-order macroscopic traffic flow model was modified by replacing the speed constant (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{^\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e) with a novel parameter, the Pavement Condition Index (PCI) and its derivative with respect to PCI. By integrating PCI, drivers' responses are adjusted based on the road condition, potentially addressing drawbacks related to the lack of physical interpretation of (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{^\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e), and parameter sensitivity. The performance of the PW and Proposed model is simulated in MATLAB, over 3500m circular road, considering the PCI. The results shows that the proposed model provides realistic representation of traffic flow behavior, where density and speed sharp change patterns smoothen and exhibit inverse relationships as expected.\u003c/p\u003e","manuscriptTitle":"A Macroscopic Traffic Model based on Pavement Condition Index","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-08-29 16:22:50","doi":"10.21203/rs.3.rs-4800495/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"5a314f94-ad20-4160-9249-e3888e44d42a","owner":[],"postedDate":"August 29th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":36650890,"name":"Physical sciences/Engineering/Civil engineering"},{"id":36650891,"name":"Earth and environmental sciences/Environmental sciences"}],"tags":[],"updatedAt":"2025-02-19T05:56:35+00:00","versionOfRecord":[],"versionCreatedAt":"2024-08-29 16:22:50","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4800495","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4800495","identity":"rs-4800495","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}