This paper studies the travel behavior of travelers who drive from the living area through the highway to the work area during the morning rush hours. The bottleneck model based on personal perception travel behavior has been investigated. Based on their willingness to arrive early, travelers can be divided into two categories: active travelers and negative travelers. Three possible situations have been considered based on travelers’ personal perception. Travelers’ travel choice behaviors are analyzed in detail and equilibrium is achieved with these three situations. The numerical examples show that the departure time choice of the travelers is related not only to the proportion of each type of travelers, but also to personal perceived size.
The well-known bottleneck model was originally developed by Vickrey [
The bottleneck model depicts the commuting behavior of travelers during morning rush hours with a simple and direct way and clearly describes the formation and dissipation of queuing congestion and the departure time choice behavior of travelers. Subsequently, the morning commuting problem has been expended by many others. Henderson [
In previous studies, travelers choose their own departure time by trade-off between waiting time and the schedule delay [
The perceived judgment is the traveler’s psychological judgment of the route travel time, because travelers have their understanding of the route choice. In the static model, the SUE (Stochastic User Equilibrium) is achieved when users can no longer change their perceived utility. This indicates that traveler’s psychological choice plays a key role in the travel process.
For example, there are two highways from the Beijing area to Capital Airport: the Jingcheng Expressway and the Capital Airport Expressway. The toll of Jingcheng Expressway is a little higher than that of the Capital Airport Expressway, but the freeway patency of Capital Airport is much lower than that of Beijing-Chengdu Expressway and the expressway of Capital Airport is congested all day. Generally speaking, this phenomenon is explained by the fact that the underestimation of travelers’ time value is the cause of low toll road utilization [
In those studies, the influence of psychological factors on the travelers was investigated in the static model. Few scholars have considered the effects of psychological factors in dynamic models. Based on this situation, the impact of human psychological decision-making behavior in the bottleneck model has been studied. In the bottleneck model, since the travel time on the road does not affect the traveler’s departure time choice behavior, it is usually assumed to be zero, and it is not considered. Therefore, each commuter faces a trade-off between travel time cost and the delay cost and chooses an optimal departure time. For schedule delay, on the one hand, travelers need to consider their own early/late time cost; on the other hand, their subjective judgment also plays a certain role in the travel process. How to describe the traveler's travel choice behavior through the subjective consciousness judgment will be the research content of this paper. In the process of psychological decision-making, different travelers have different reactions to arrival early/late. Therefore, this paper divides travelers into two types according to whether they wish to arrive early or not: active and inactive. Next, we will introduce these two kinds of travelers in turn.
For some travelers, the closer he arrives at work, the more he feels nervous and the more he becomes uneasy when he is late. For these travelers, arriving early can reduce their tension, and arriving late can increase their tension. This tension based on personal perception is bound to have an impact on the traveler’s choice of departure time. For those who have a strong sense of time, they are called active travelers.
On the contrary, some travelers are not proactive, thinking that arriving at work before the work starting time would cost them some of their own benefits. They are more likely to arrive near the preferred arrival time than arriving early. For this part of the travelers, their travel behavior is not as active as the active travelers. So we call this part of the travelers is negative travelers.
Compared with the previous models which only consider the waiting time and the schedule delay, this paper takes the bottleneck model as a foothold, studies the traveler’s departure time choice behavior by considering the travelers’ personal perception, and establishes a bottleneck model based on personal perception.
Let us consider a highway between a residential area and a CBD where
According to the former description, the general travel cost mainly includes three parts in the bottleneck model based on personal perception: queuing waiting time cost, schedule delay cost, and personal perceived cost. There exist two kinds of generalized travel cost by the active travelers and negative travelers. Here, we suppose that the personal perceived utility is a linear function of time
The generalized travel cost
The generalized travel cost
where
According to the former assumption, travelers can be divided into active travelers and negative travelers. If the number of active travelers is
In this section, this paper examines the departure time choice behavior of active travelers. At equilibrium, the driver incurs the same travel cost no matter when he leaves home. During the entire operation, the bottleneck has full capacity load operation. This condition implies that all the travelers’ generalized travel costs are the same. Let
The generalized travel cost of the last active traveler is
During the entire operation, the bottleneck has full capacity load operation, and the length of the period is determined by the following formula:
According to (
The total travel cost
According to (
According to (
In this section, this paper examines the departure time choice behavior of negative travelers. At equilibrium, the driver incurs the same travel cost no matter when he leaves home. During the entire operation, the bottleneck has full capacity load operation. This condition implies that all the travelers’ generalized travel costs are the same. Let
The generalized travel cost of the last negative traveler is
During the entire operation, the bottleneck has full capacity load operation, and the length of the period is determined by the following formula:
According to (
The total travel cost
According to (
According to (
In the former two sections, this paper considers two extreme cases when
(1) In the first case, all early travelers are active; all late travelers are negative. For the early travelers, the generalized travel cost at time
For the late travelers, the generalized travel cost at time
At equilibrium, the traveler incurs the same travel cost no matter when he leaves home. During the entire operation, the bottleneck has full capacity load operation. This condition implies that all the travelers’ generalized travel costs are equal. Let
At equilibrium,
According to (
At equilibrium, the waiting times at time
In this case, active travelers can only arrive early, and negative travelers are all late. This means that
The generalized travel costs of the first traveler and the last traveler are
The departure time rate
The number of active travelers and negative travelers is
During the entire operation, the bottleneck has full capacity load operation, and the length of the period is determined by the following formula:
According to (
According to (
The illustrative departure and arrival profiles are plotted in Figure
Cumulative departures and arrivals at equilibrium in Case 1.
(2) In the second case, some early travelers are active, while others are not. In this situation,
For the early travelers in the departure time interval
For the early travelers in the departure time interval
For the late travelers in the departure time interval
At equilibrium,
According to (
The departure time rate
The generalized travel cost of the negative travelers at
The generalized travel cost of the last traveler is
At equilibrium, all the negative travelers’ generalized travel costs are equal. We get
According to (
The number of negative travelers is
During the entire operation, the bottleneck has full capacity load operation, and the length of the period is determined by the following formula:
According to (
According to (
According to (
The illustrative departure and arrival profiles are plotted in Figure
Cumulative departures and arrivals at equilibrium in Case 2.
(3) In the third case, some early travelers are negative, while others are not. In this situation,
For the early travelers in the departure time interval
For the early travelers in the departure time interval
For the late travelers in the departure time interval
At equilibrium,
According to (
The departure time rate
Similar to the second case, the waiting time at
Thus, the generalized travel cost of active travelers at
The first traveler is active, and the generalized travel cost belonging to him is
At equilibrium, all the active travelers’ generalized travel costs are equal. Thus,
According to (
The number of negative travelers is
During the entire operation, the bottleneck has full capacity load operation, and the length of the period is determined by the following formula:
According to (
According to (
According to (
The illustrative departure and arrival profiles are plotted in Figure
Cumulative departures and arrivals at equilibrium in Case 3.
We present the following propositions to reveal some interesting properties of the equilibrium solution in this section.
When the value of parameter
Thus, we obtain the same traffic flow pattern as the personal perception bottleneck model based on negative travelers.
When the value of parameter
Thus, we obtain the same traffic flow pattern as the personal perception bottleneck model based on active travelers.
At equilibrium, no matter the departure time interval of active travelers or negative travelers, there has no jam phenomenon, which means that it is impossible to have both active travelers and negative travelers.
According to the equilibrium conditions, the equilibrium cost of each traveler will not change at equilibrium. Suppose that at the time
If the travelers at time
The wait time of
Substituting this result into
We can find
At equilibrium state, the generalized travel cost for every active and negative traveler is a strictly monotonically decreasing function of the personal perceived cost
For instance, according to (
(
(
(
(
(
(
In this section, we present numerical results for the personal perception bottleneck model. According to Vickrey [
Figure
Changes of the beginning of rush hours with respect to the
Figure
Changes of the travel cost with respect to
Figure
Changes of the generalized travel cost with respect to
Based on the classical bottleneck model, this paper establishes a bottleneck model based on personal perception by considering the influence of personal perception on departure time selection. For travelers, some prefer to arrive early, called active travelers, while some are not likely to arrive early, called negative travelers. This paper mainly studies the departure time choice behavior of travelers under these two travel attitudes. It is concluded that the bottleneck model based on personal perception can accurately describe the departure time choice behavior of travelers, which are not only related to the proportion of each type of travelers, but also related to the size of travel perception. This paper mainly considers the departure time choice behavior of travelers but does not take into consideration the impact of congestion pricing and other related strategies on travelers. These will be the contents of our next research.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The work described in this paper was supported by the National Natural Science Foundation of China (71771018, 91846202, 71525002, and 71621001).