This section provides the scope and context of the utilised data to inform the study’s level of generalisability. The data represents the traffic flow data collected from the N6 road at the dual lane segment between the Amasaman and Pokuase catchments, and in the north-south direction. Figure 1 provides the map of the test site for data collection. This segment of N6 is situated in the Ga-West Districts of the Greater Accra region of Ghana. The chosen segment was used for the standard random experiment to collect traffic data consistent with recurrent traffic and among locations in the district reportedly predisposed to severe uncertainty in the traffic flow states, including the morning rush-hour traffic. The experiment was therefore conducted between the hours of 6 am to 11 am daily for four continuous weeks during the period July-August over four weeks. This period coincided with the warm, dry season typical of the country’s tropical climate. The investigation involved counting the daily number $n$ of heterogeneous traffic flow across the test site every 60 seconds $t$ across encompassing six junctions to the road catchments. The heterogeneity of the traffic includes different types of road vehicles by size, wheels, and axle load, such as bikes, cars, lorries, and trucks.

Given that vehicles enter and exit on the segment, like any typical urban road in Ghana and many countries alike around the globe, the observation was that the junctions serve as sources and sinks to traffic build-up on the segment, per the respective studies of [44] - [46] . The simultaneous recordings from the six sources and sinks, as well as the main entry and exit to the road stretch were tabulated at $t=60$ seconds. The flux $Q\left(t\right)$ of the homogenous traffic was then calculated as the net sum of the sources and sinks. The main roadway was not signalised, so the only factor to the traffic builds up and composition is the contributions from the sources and sinks. The assumption here is that every vehicle that enters the segment exits it at some access point and that the observable traffic build-up is the net aggregate of the sources and sinks onto the main road segment and is consistent with the macroscopic conserved traffic flow [47] [48] .

The stochastic process, a widely used approach for describing the behaviour of random systems, was utilised in this study. In particular, the Birth and Death Poisson process [41] - [43] , and the intermediate traffic flow modelling approach [15] [17] [18] were used to develop the MEED model, which was then simulated. A statistical experiment was conducted to collect empirical data on traffic flow dynamics at the test site and validate the model.

$\frac{\partial \rho \left(x,t\right)}{\partial t}+\frac{\partial q\left(x,t\right)}{\partial x}=\{\begin{array}{ll}0;\hfill & \text{for homogeneous flow}\hfill \\ g\left(\rho \left(x,t\right),x,t\right)\hfill & \text{for non-homogeneous flow}\hfill \end{array}$ (1)

For comparative purposes, the classical deterministic MEED model structure is reproduced in Equation (1) as a first-order, quasi-linear partial differential equation (PDE). The LWR describes a conserved macroscopic traffic flow in a continuum, hydrodynamic, and speed-density relationships [24] [26] [45] [49] [50] . The essential variables of the LWR model are the distance $x$ and time $t$ for the road segment A to B of length L, such that $x\in \left[0,L\right]$; and $t\in \left[0,T\right]$ for time elapsed during which the macroscopic traffic flow was observed. The flow rate $q\left(x,t\right)$ is the observed net number of entry and exit vehicles that traversed $x$ in time $t$. Also, flow density $\rho \left(x,t\right)$ describes the number of vehicles within the segment AB at time $t$ and that $\rho \left(x,0\right)=0$ for the homogeneous and non-homogeneous cases. However, the available literature reports that both the homogeneous and the non-homogeneous forms in Equation (1) show the defects of diffusivity and inertial. The defects are at variance with actual vehicular traffic flow, as observed on any road [19] [21] [51] . Meanwhile, extensive studies have been done to improve the LWR, and its higher orders variants thereof, and this paper also contributes to the positive direction, however from the perspective of stochastic process.

The novelty of this paper is the proposed MEED framework. Consider a heterogeneous stream of vehicular traffic flowing on a segment AB of urban road with multiple access points, such as junctions, for vehicles to enter and exit the road segment. Also, suppose AB is on a one-dimensional straight line with the access points numbered $0,1,2,\cdots ,m$ on the line. Such configuration of the road segment A to B length L, such that 0 and $m$ correspond to the extreme A and B; and numbers $1,2,\cdots ,m-1$ correspond to the other access points. Given that each access point serves as a source and sink to traffic build-up on AB, and A is a source-only and B is sink-only to the total traffic on AB, then within the time interval $t$, the $k$ access point contributed to the traffic build-up. Now suppose that $E_k\left(t\right)$ and $Y_k\left(t\right)$ are the counted vehicles that:

$X_k\left(t\right)=E_k\left(t\right)-Y_k\left(t\right)$ (2)

respectively entered and exited the $k$ access point, then the net traffic $X_k\left(t\right)$ contributed by the $k$ access point to the traffic build up on $AB$ is given by Equation (2). For all values of $k$ then $X_k\left(t\right)>0$ for $E_k\left(t\right)>Y_k\left(t\right)$ and $X_k\left(t\right)<0$ for $E_k\left(t\right)<Y_k\left(t\right)$. Hence, $X_k\left(t\right)$ generates a random variable for each of the finite $m$ access points that may contribute simultaneously but with different counts of entries and exits at the different time intervals $t$. Table 1 illustrates the MEED framework for the road configuration at the various access points in the experiment at time interval $t\in \left[0,\infty \right]$, such that $X_k\left(t\right)$ provides a discrete random variable, for distinct individual values on $\mathbb{Z}$.

Definition 1 (Poisson Distribution) The discrete random variable X with non-negative integer support $R_X\in {\mathbb{Z}}^{+}$ and a mean $\lambda \in \left(0,\infty \right)$, has a Poisson distribution $X~Poisson\left(\lambda \right)$ if $X$ has a probability mass function that satisfies the following [42] :

$p_X\left(x\right)=\{\begin{array}{ll}\frac{{\text{e}}^{-\lambda }{\lambda }^x}{x!}\hfill & \text{if}x\in R_X\hfill \\ 0\hfill & \text{if}x\notin R_X\hfill \end{array}$ (3)

Well known for modelling problems involving random arrivals of the number of customers at a supermarket and telephone calls at a switch [42] , the Poisson distribution is also satisfied by the definitions of $E_k$ and $Y_k$ for the respective arrivals and departures of vehicles to and from AB. The constants $\lambda$ and $\nu$ such that $\lambda \ne \nu$ correspond to the arrival and departure rates. Given their basis in Poisson distribution, then $X_k\left(t\right)$ also possess a distribution, and the accumulation of all the $X_k\left(t\right)$ for $k=\left\{0,1,2,\cdots ,m\right\}$ observed across the $m$ finite access points within the segment $AB$ generated the traffic flux $Q\left(t\right)$ at the end of time $t\in \left[0,\infty \right]$, according to Equation (4).

$Q\left(t\right)=\underset{k=0}{\overset{m}{{\displaystyle \sum }}}{X}_k\left(t\right)$ (4)

Hence, the possible observable values of counts at time $t$ is $Q\left(t\right)=n_t$, where $n_0\ge 0$ is the initial counted flux at time $t=0$. The sum across the row in Table 1 gives $Q\left(t\right)$ that satisfies the statistical experiment of counting a population of $n$ vehicles by the end of time $t\in \left[0,\infty \right]$ within AB. For the flux $Q\left(t\right)=n_t$ at time $t$, let $p_i\left(t\right)=Pr\left(Q\left(t\right)\right)=n_i)$, be the corresponding probability, then the sequence of $\left\{Q\left(t_i\right)\right\}$ for $i\in 0,1,2,\cdots ,\infty$ has the Markov property given by Equation (5).

$\begin{array}{l}Pr\left(X\left(t_n\right)=j|X\left(t_1\right)=j_1,X\left(t_2\right)=j_2,\cdots ,X\left(t_n\right)=j_n\right)\\ =Pr\left(X\left(t_n\right)=j|X\left(t_{n-1}\right)\right)\end{array}$ (5)

and that the future values of $Q\left(t_i\right)$ depends on only the current counted values and no other [41] [43] .

The transition in the flux, $Q\left(t_i\right)$, is based on the observation that individual vehicles entered and exited the road segment at the rates of $\lambda$ and $\nu$. Respectively. Given that the entries and exits satisfy the Poisson distribution, then neither $\lambda$ nor $\nu$ show systematic trends in time and are therefore considered constants [42] [43] .

Definition 2 (Poisson Process) A counting process $N\left(t\right):t\ge 0$ with stationary and independent increments is said to be a Poisson process if $N\left(t\right)$ counts the total number of events within any interval of time $t$ and has a Poisson distribution with mean $\lambda t$ such that $\lambda \ge 0$, and $\forall s,t>0$. Therefore $\forall s,t>0$ then $N\left(t\right)$ satisfies the following equation according to [43] :

$P\left[X\left(s+t\right)-X\left(s\right)=n\right]=\frac{{\left(\lambda t\right)}^n}{n!}{\text{e}}^{-\lambda t}n=0,1,2,3,\cdots$

Now since the contributions $X_k\left(t\right)$ from the access point $k$ generated the flux $Q\left(t\right)$, then $Q\left(t\right)$ satisfies proposition 3.

Proposition 3 (Poisson Process) The stochastic process $Q\left(t\right):t\ge 0$ is a Poisson process given that $Q\left(t\right)$ counted the total net volume of events of vehicles that accumulated within the road segment in a fixed time when the events occurrences were observed to have a Poisson distribution with a constant rate.

Now since $Q\left(t\right)$ describes a process for the point of events on the real line due to $t$ and that $Q\left(t,t+\delta t\right)$ provides the number of events in the half-opened time interval $\left(t,t+\delta t\right]$.

$Pr\left\{Q\left(t,t+\delta t\right)=0\right\}=1-\lambda \delta t+o\left(\delta t\right)$ (6)

$Pr\left\{Q\left(t,t+\delta t\right)=1\right\}=\lambda \delta t+o\left(\delta t\right)$ (7)

$Pr\left\{Q\left(t,t+\delta t\right)>1\right\}>o\left(\delta t\right)$ (8)

Then given the positive constants $\lambda$ and $\nu$ such that $\delta t\to 0$, then equation (6) to (8) establishes the random variable $Q\left(t\right)$ as a Poisson process, while $\lambda \ne \nu$, the following Poisson process conditions governed the transition of vehicle population [42] [43] for both the entries and exits in an infinitesimal time $\delta t>0$:

1) The flux can increase or decrease at any time $t$ but not both simultaneously, since each access point serves the dual purpose of entry and exits.

2) When the population of vehicles in the flux has $n$ number of vehicles at time $t$, then:

a) $Pr\left\{\text{a unit vehicle increase occurs in the time interval}\left(t,t+\delta t\right)\right\}$

$={\lambda }_n\delta t+o\left(\delta t\right)$

b) $Pr\left\{\text{a unit vehicle decreases occurs in the time interval}\left(t,t+\delta t\right)\right\}$

$={\nu }_n\delta t+o\left(\delta t\right)$

c) $Pr\left\{\text{more than a unit change occurs in the time interval}\left(t,t+\delta t\right)\right\}=o\left(\delta t\right)$

3) Given conditions (2a) to (2c) above, then when the flux has size of $n$ vehicles:

4) $Pr\left\{\text{no change in the flux in the time interval}\left(t,t+\delta t\right)\right\}$

$=1-\left({\lambda }_n+{\nu }_n\right)\delta t+o\left(\delta t\right)$

By conditions (1) to (4), the occurrence of the points of events of vehicle entries and exits, captured by the flux, and the corresponding probabilities are provided in Equation (9).

$Pr\left\{X\left(t+\delta t\right)=j|X\left(t\right)=n\right\}=\{\begin{array}{ll}{\lambda }_n\delta t+o\left(\delta t\right)\hfill & j=n+1\hfill \\ {\nu }_n\delta t+o\left(\delta t\right)\hfill & j=n-1\hfill \\ 1-\left({\lambda }_n+{\nu }_n\right)\delta t+o\left(\delta t\right)\hfill & j=n\hfill \\ o\left(\delta t\right)\hfill & \left|j-1\right|>n\hfill \end{array}$ (9)

Consequently, for the counting process $N\left(t\right)$, the number of individual vehicles in the flux $Q\left(t\right)$ at time $t$, have the probability probability $Pr\left\{N\left(t\right)=n\right\}$. As such the possible values of $j$ in Equation (9) leads to Equation (10) as the forward equation associated with the counting process.

${p^{\prime }}_n\left(t\right)=-\left(\lambda +n\nu \right)p_i\left(t\right)+\left(n+1\right)\nu p_{n+1}\left(t\right)+\lambda p_{n-1}\left(t\right)\left(n=0,1,2,\cdots \right)$ (10)

Observe that for $n=0$ the function $p_{n-1}$ is identically equal to zero [41] [43] , and rewriting Equation (10) resulted in Equation (11), where $G\left(z,t\right)$ is the probability generated function that describes the population $Q\left(t\right)$ at time $t$.

$\frac{\partial G\left(z,t\right)}{\partial t}+\nu \left(z-1\right)\frac{\partial G\left(z,t\right)}{\partial z}=\lambda \left(z-1\right)G\left(z,t\right)$ (11)