## 1 Introduction
In another study, two stochastic fundamental diagram (SFD) models with lognormal and skew-normal distributions were developed to account for the scattering effect during traffic state variations [12]. The MFD describes the behavior of a network in terms of its density, speed, and flow; if a network has a MDF (i.e., the relationships between traffic indicators present well-defined shapes), it can be used to identify the current conditions, to project future conditions, and to implement (control) actions that benefit the network's traffic performance. Nevertheless, a network does not always present a well-defined MFD, and a hysteresis effect may be observed. An event that leads drivers to use unfamiliar routes can cause a hysteresis loop in the MFD [13]; during congestion recovery, a lesser flow is experienced compared to that observed during the congestion stage, since the more congested areas recover more slowly than those that are less congested. The effects of the hysteresis loop can be mitigated if drivers adaptively select routes. An incident originating a non-homogeneous congestion evolution in the network space can produce a hysteresis-like effect [14]. If an incident occurs in the centermost part of the surface streets, the hysteresis effect will not be visible in the MFD, but incidents that occur at other locations in the surface streets will be visible. Therefore, it was found to be convenient to separate surface streets (which were separated by the centermost streets and the rest) and highways (which were separated by penetrating roads and ring roads) in order to obtain MFDs with low scatter. The hysteresis phenomenon is explained as a consequence of departure time choices and user equilibrium in [15]. The MFD shape has been described using different methods. The MFDs obtained using the analytical method, the production method (based on the vehicles' trajectories), and using loop detectors were compared in [16]. An algorithm that combines the data obtained with loop detector data (LDD) and floating car data (FCD) to estimate the MFD is presented in [17]. The problem of defining the MFD with probe vehicles when the penetration market rate of these varies regionally within a network was solved in [18]. Methods to estimate the MFD were compared and presented in [19]. The analytical method is suitable for an urban corridor and homogeneous loadings, and the typical route method was found to be adequate to obtain the network MFD in a grid network. A comparison between the method of cuts (MoC) and the stochastic approximation (SA) was implemented to analytically obtain the MFD in [20]. The MoC (a deterministic method) accurately found the upper bound of the MFD, whereas the SA (a stochastic method) provided a more accurate approximation of the MFD free-flow branch. A validity domain (and the inaccuracies) of the accumulation-based model when demand varied rapidly is described in [21]. The production with the accumulation and the spatial spread of density was modeled in [22]. The model describes the so-called generalized macroscopic fundamental diagram (GMFD). A functional form of the MFD called the uMFD (a smooth approximation of an upper bound of technologically feasible traffic states) was introduced in [23]; the uMFD fits the MFD's empirical data through a trapezoidal shape, with four parameters considered to be physically meaningful. A novelty of this method is its use of the smoothing parameter, which allows quantification of the flow-reducing factors (infrastructure effects, vehicles interaction, and traffic heterogeneity). A two-step MFD calibration framework was used to select the form and to estimate parameters in [24]. Recently, the study of the MFD has been complemented by identifying the variables that modify the shape of the MFD. Interesting conclusions about the factors that affect the MFD shape were drawn in [25], which stated that ramp metering affects the MFD.