## 2 Problem Definition and Basic Idea  
### Formal Definition of the Problem  
Consider there is an expressway with \(n\) different exits, there are in total \(m\) drivers on the expressway at a specific time \(T_{k}\). For each exit \(E_{i}\), there is an observation station (e.g., toll station or video surveillance camera) \(F_{ik}\), which is the total number of vehicles leaving the expressway through the exit \(E_{i}\) at time \(T_{k}\) with a sampling time interval \(\Delta T_{k}\). Generally, we assumed all the \(i,j,k\), which are the indexes of exit, driver, and time beginning from 1, and \(i=\{1,2,\cdots,n\},j=\{1,2,\cdots,m\}\), and \(k=\{1,2,\cdots,s\}\). In the real world, the only observable variable is \(F_{ik}\), which is also one of the most important factors to indicate the characteristics of the expressway conditions. Obviously, \(F_{ik}\) is continuously changing over time, and we hoped to simulate the dynamics of \(F_{ik}\) as accurately as possible. We assumed that \(F_{ik}\) consists of two fundamental parts: the deterministic part \(F_{ik}\) and the random part \(F_{ikr}\). We assumed that \(F_{ikd}\) is a combination of completely predictable components, such as the mean value and the regular cycles of the traffic flow. Thenis the unobservable random variation caused by random exit selection. So, we have the fundamental equation of traffic flow in each exit \(E_{i}\) as  
\[F_{ik}=F_{ikd}+F_{ikr} \tag{1}\]  
As \(F_{ikd}\) is totally predictable, which means we can use a simple model, such as the autoregressive moving average model, to approximate it. The major issue is how to simulate the dynamics of random exit selection to improve the predictability of \(F_{ikr}\). There were three basic procedures that were required to simulate \(F_{ikr}\) by considering random exit selection. The first was how to model all the possible choices for exits as a random process, which we called the decision space. The second was modeling the probability distribution and its evolution to express the dynamic decision process on exit selection. The third was how to construct the relationship between the probability distribution and \(F_{ikr}\). In the following sections, we will present our solution in modeling \(F_{ikr}\) in three procedures. We first formulated the problem associated with each of these procedures and then described our basic idea in addressing these problems. Procedure 1: Modeling all the possible choices of exits as the decision space. The essence of exit selection is that a driver chooses an exit to leave the expressway traffic at a specific time. Therefore, the key representation of the choice for a random exit is the presentation of a driver choosing which exit at what time with how much probability. Assuming that the decision process of any driver \(D_{j}\) expected to exit the highway at exit \(E_{i}\) at time \(T_{k}\) is a node \(V_{(i,k)}\), and the complete set of exit selections is \(E_{i}\times T_{k}\). Therefore, in time and space, all possible exit selections of any driver \(D_{j}\) can be combined into a tree-style network. Any node \(V_{(i,k)}\) on the network is a state that a chosen exit, and any edge \(E\) represents the spatiotemporal evolution of an exit selection of a given driver. As both the traffic direction and time of expressway traffic is in one direction, the path on the network can only be generated sequentially, therefore, we can reformulate the \(n\times s\) network into a tree-style network as is shown in Figure 1. We named this tree-style network the exit selection decision space. For each node of this decision space, a probability vector can be assigned to express the driver's preference for the decision. Generally, an n-dimensional probability vector \(P(i,j,k)=(p_{1},p_{2},\cdots,p_{n})\), where \((p_{1},p_{2},\cdots,p_{n})\) can be used to indicate the probability of a driver at any time \(T_{k}\) to choose an exit \((E_{1},E_{2},\cdots,E_{i})\) to exit traffic. Clearly, the probability vector \(P(i,j,k)\) is unobservable.