Assessing the Possibility of Presenting a Semi-Stochastic Speed-Density Function

. Underlying a fundamental diagram is a relation between traffic speed and density which roughly corresponds to drivers’ speed choices under varying car -following distances. Stochastic and deterministic models are mainly two different categories of speed-density models. The advantages of deterministic models are their mathematical simplicity and analytical tractability though their results will show just the average parameters. Although the stochastic models may represent more accurate results taking uncertainty into account, they are often hard to use and analytically not tractable. The aim of this paper is to investigate the possibility of presenting a model which is neither completely deterministic nor completely stochastic but easy to use and understand which incorporates different traffic conditions and speed distributions. Monte Carlo Method has been used to generate different speed distributions based on different traffic conditions and consequently generating their relevant densities. Surveying the relation between the mentioned speed distributions and the obtained densities kept the chance of presenting a model which is neither completely deterministic nor completely stochastic but easy to use and understand which incorporates different traffic conditions and speed distributions.


Introduction
Flying birds, running water, electric current, internet packets, and moving vehicles can all be considered as flows, yet each exhibits distinct characteristics. One of the attributes that distinguishes vehicular traffic flow from other kinds of flow is the so-called fundamental diagram -the relationships among traffic flow characteristics (e.g. flow q, speed v, and density k) which are typically represented graphically [1].
Fundamental relations of traffic flow have historically been established either empirically or derived from carfollowing models. Generally, they come in two flavors: speed-density relations and flow-density relations, which, respectively, express the mean speed and the mean flow rate as functions of traffic density. The speed-density relationship serves as the basis to understand system dynamics in various disciplines [2]. It can be used as a tool to study on moving objects (or particles) in many scientific areas: pedestrians [3,4], conveyors, network information packages [5], crowd dynamics [6], molecular motors, and biological systems [7]. It has been almost 75 years since Greenshields' seminal paper Study of Traffic Capacity in 1935 [8]. Attaching empirically derived curves to a fitted linear model of the speed-density relationship started a new era of transportation science and engineering. Due to its strong empirical nature, the efforts to find a perfect theory to explain these particular shapes mathematically never cease, but they always achieve limited success. There is a fairly large amount of effort devoted to revising or improving such an oversimplified relationship. These efforts include singleregime models: Greenberg's Model [9], the Underwood Model [10], Northwestern [11], Drew [12], and the Pipes-Munjal Generalized Model [13].There are also multiregime models which include: two-regime models such as Edie Model [14], multi-regime model by cluster analysis [15], two-regime model [16], modified Greenberg, and three-regime models [16,11].
A generalized deterministic form of speed-density relationship (so-called regression models or fitting models) is written in Eq. 1.
where, v(k) is the deterministic speed-density relationship in which given density k there is one fixed corresponding speed value and Ɛ is the measurement error [2]. Deterministic models are justified on the basis that they describe the average value of the dependent variable given the independent variable inputs [17]. The advantages and drawbacks of deterministic single-and multi-regime speed-density models have been well-documented in a multitude of publications [16,18,19]. The advantages of deterministic models are their mathematical simplicity and analytical tractability. The main drawback of deterministic models is that they describe average system behaviors. However, the mean alone may not be capable of describing the entire traffic DOI: 10.1051/ 0 (2016) matecconf/201 MATEC Web of Conferences conditions. It should be underlined that traditional deterministic families of models which essentially describe a 'pair-wise' relation between speed and density are not sufficient to include the traffic speed variance.
In the stochastic modeling framework of speeddensity relationship, traffic speed v is a random function of density k and random variable ω. In this type of modeling, the random process will be discretized by an appropriate expansion technique which characterizes second-order statistics of the random process using uncorrelated random variables and the deterministic orthogonal set of functions. A simple illustration of the deterministic and stochastic model is shown in Fig. 1.

Figure 1.
A simple illustration of deterministic and stochastic model [20] To sum up, stochastic models are hard to use and deterministic models are not enough accurate. The aim of this paper is to investigate the possibility of presenting a model which is neither completely deterministic nor completely stochastic but easy to use and understand which incorporates different traffic conditions and speed distributions. Practically this paper describes the possibility of presenting a model that has deterministic input parameters (desired deterministic density) and stochastic output parameters (related speed distribution) considering different traffic conditions (Referring to Fig.  1.). The main contribution of this paper is to survey on the relation of speed-density parameters to see whether it is possible to derive speed parameter capable of presenting behaviour of traffic dynamics rather than presenting just the average speed of the entire traffic flow.

Methodology
Since there is a distribution of traffic speed at a certain density level due to the stochastic nature of traffic flow, automatically 'pairwise' pattern from deterministic models will be in contrast with the nature of traffic flow. What if in deterministic speed-density models a certain level of density will be incorporated with one or more part of different speed distributions related to that specific traffic condition?! The development of mathematical tools focused on the modelling of the speed distribution in a traffic flow is widely reported in the scientific literature [21][22][23][24]. Many papers concern this problem since vehicles speed distribution is an important input parameter in lots of issues, such as kinematical traffic simulation model, road design, speed limit evaluation, road traffic noise prediction, traffic safety evaluation, bicycle performance evaluation, analysis of pedestrian walking, road transport emission estimation [25][26][27][28][29][30][31][32][33], [46] etc. Since 1940 majority of papers considered speed distribution as Gaussian or Normal distribution (Fig. 2) [34] and only extreme situations are considered as the traffic volume exceeds the practical road capacity, the speed distribution may become so heavily skewed toward the higher speeds that all semblance of normality is lost.

Figure 2.
A simple illustration of stochastic speed-density model [2] If the traffic conditions are not homogeneous, the normal distribution is not well suitable anymore. There is a lot of research that has examined distribution models for motorized vehicle speed data, such as normal distribution [35][36][37], lognormal distribution [38,39], skew-t distribution [40,41], gamma distribution [42], Beta distribution and Chi-Square distribution [25], as well as composite distribution [43]. Overall, roadway sections may have more than two modes, e.g., uncongested speed range, interim speed range which lies between uncongested and congested conditions, and congested speed range. Jun [43] mentioned if the characteristics of speed data are homogeneous (or unimodal distribution), the speed data of interstate freeways can be generally described by conventional probability density function (pdf) such as normal distribution, gamma distribution, lognormal distribution etc. However, if speed data follow a bimodal distribution (or a mixed distribution), there is no specific distribution function available [22].
The conditions that turn to a different speed distribution are quite often realized in non-highway or urban roads, where, in general, the traffic stream is much more complicated. This assumption is supported by Harmonoise [44], where different speed distributions are related to different traffic situations. This large spread of typologies, together with other parameters such as the presence of traffic lights and the road surface conditions, leads to a significant deviation of the speed distribution curve from the generally accepted unimodal normal distribution [25]. In the present research, the traffic condition has been assumed mixed containing the all possible traffic conditions and consequently the all possible speed distributions in order to cover all the possible scenarios.
In the current paper, Monte Carlo Method has been used to build different speed distributions and generate a link between them and their related density. Monte-Carlo methods (or Monte-Carlo simulations) are computational algorithms that rely on repeated random sampling [45]. This method generates suitable random numbers of parameters or inputs to explore the behaviour of a complex system or process. The random numbers generated follow a certain probability distribution function (PDF) that describes the occurrence probability of an event. Typically simulations run many times in order to obtain the distribution of an unknown probabilistic parameter. It should be underlined that the number of vehicles, their average speed, and travel distance are remain constant in all cases as similar inputs in Monte Carlo Simulation, but the outputs (speed and density) were totally different based on the nature of traffic flow. Generated speed distributions by Monte Carlo Method have been shown in Fig. 3.

Results
By surveying different densities, it was found that in some cases different speed distributions will describe better the traffic conditions and some speed distributions were roughly covered each other (e.g. Poisson and beta distribution). In this paper based on the international literature overview, the different traffic scenarios were identified and described. Schematic fitting of the mentioned correlation in Fig. 4. by generally accepted deterministic speed-density models may imply the possibility of capturing the chance of presenting the semistochastic speed density model which deals with different traffic conditions rather than the average speed (see  It should be highlighted that these schematic fitting (Fig.5.) is based on the relevant average speed and density of different speed distributions. It should be noted that these different speed distributions may have the intersection in some certain densities with each other with different weights. For instance, although the average of the exponential speed distribution is around 55 km/hr, its domain is between zero km/hr and around 110 km/hr with more weight toward zero km/hr (see Fig.4.). Since these different speed distributions are based on different traffic conditions, this fact implies that the output of the deterministic speed-density function cannot be a single average speed but should be the function of speed composed of different relevant speed distributions.

Conclusion
In this paper, the authors tried to provide a method in which traffic speed variance will be incorporated by traffic density. The aim of this paper was to investigate the possibility of presenting a model which is neither completely deterministic nor completely stochastic but easy to use and understand and incorporates different traffic conditions by determining different speed distributions. The main contribution of this paper is to survey on the relation of speed-density parameters to see whether it is possible to derive speed parameter capable of presenting behaviour of traffic dynamics rather than presenting just the average speed of the entire traffic flow. The results showed that different speed distribution based on different traffic conditions may have the intersection with each other-however their averages were fitted to the generally accepted deterministic speed-density modelsand this fact implies that discrete modelling is not viable in this case. The scope for future research of the authors will be applying the fuzzy theory to help the composition of the complex speed-density probability function taking intersections into account. The final complex semistochastic model will be validated by the traffic data of Budapest, Hungary compared to other density-velocity models.