Bartlett Flows and Mathematical Description of Motor Traffic Flows

Introduction: The class of mathematical traffic models is based on the theory of queuing. In these models, the application entering the service system corresponds to the vehicle. When developing a traffic model formulated in terms of queuing, it is necessary to specify a random flow that is incoming to the queuing system. The purpose of the study: Traditional queuing systems with recurrent incoming flow under appropriate conditions do not reflect the specific features of real traffic flows. Under certain conditions, for example, it may be appropriate to use a Markov-type flow in the model, the intensity of which depends on the state of a mathematical object called the control device. In the general case, such a flow can be specified as non-uniform, and with such a task, each request is assigned a type that also depends on the state of the control device. Setting the qualitative structure and parameters of a random flow depends on the assessment of the speed characteristics of the vehicles that form the flow, and, therefore, is related to the issues of studying the speed characteristics of real vehicles. Practical significance: At a sufficiently low density of the traffic flow, the incoming flow is close to the Poisson one. As traffic increases and road conditions worsen, the risk of overtaking increases and clusters are formed, consisting of a slow car moving in front and a group of fast cars that cannot overtake a slow one. In such cases, we can assume that the incoming flow is a Bartlett flow, which has the following form: clusters form a Poisson flow, and the cluster length distribution is a two-parameter Bartlett distribution. One of the parameters of this distribution is the probability of having a group of fast cars, and the second parameter characterizes the distribution of the number of cars in this group. Discussion: In this paper, we study the questions of setting a qualitative probabilistic structure and quantitative parameters of random flows, which are elements of queuing systems used as traffic models.