This paper extends the bottleneck model to study congestion behavior of morning commute with flexible work schedule. The proposed model assumes a stochastic bottleneck capacity which follows a uniform distribution and homogeneous commuters who have the same preferred arrival time interval. The commuters are fully aware of the stochastic properties of travel time and schedule delay distributions at all departure times that emerge from daytoday capacity variations. The commuters’ departure time choice follows user equilibrium (UE) principle in terms of the expected trip cost. Analytical and numerical solutions of this model are provided. The equilibrium departure time patterns are examined which show that the stochastic capacity increases the mean trip cost and lengthens the rush hour. The adoption of flexitime results in less congestion and more efficient use of bottleneck capacity than fixedtime work schedule. The longer the flexitime interval is, the more uniformly distributed the departure times are.
The bottleneck model was first proposed by Vickrey [
The study of decision making under risk (and uncertainty) has a long history in the fields of economics, psychology, transport, and beyond (Machina [
This paper applies the expected utility theories to capture the departure time choice behavior in morning commute problem under uncertainty. Morning commute plays an important role in a monocentric city and the traffic congestion in such network is caused by concentration of travel demand around the work start time. The introduction of flexible work schedule is one of the transport demand management measures for alleviating peak congestion. The paper provides useful insight into traveler’s decision making in contrast to fixedtime schedule pattern. Henderson [
Most literatures on morning commute have assumed that the capacity of the bottleneck is deterministic and the traffic demand is also deterministic or governed by a predetermined elastic demand function. In reality, not only does travel time increase as traffic volume increases towards capacity but also the travel time becomes increasingly random and unpredictable due to the chaotic behavior of traffic at the micro level. The source of variation in road capacity may occur due to physical and operational factors, such as road repairs, construction, accidents, and bad weather. The variations in road capacity from physical and operational reasons are what make the analysis of travel behavior so complex and yet interesting. As such, understanding travelers’ attitudes and their behavior in varying settings is key to developing sustainable transport polices. There has been recent attention to the stochastic nature of the bottleneck models (Siu and Lo [
The focus of this paper is to analyze the departure time choice behavior under uncertainty in the morning commuting problem with flexitime work schedule. It is expected that the stochastic capacity leads to uncertainty in queuing, travel time, and trip cost, which in turn influences the commuters’ travel choice behavior. We assume that travelers are fully aware of the stochastic properties of the travel time and schedule delay distributions throughout the morning peak period which emerges from their daytoday travel experience. Furthermore, we consider homogenous travelers have the same preferred arrival time interval (PATI). We formulate a stochastic bottleneck model for this flexitime commute problem and derive its analytical solution. The properties of the model are investigated.
The solution of the proposed model shows that the capacity variability of the bottleneck leads to significant changes in departure time patterns, which are different to those derived under deterministic conditions. In a deterministic bottleneck model with flexitime work schedule, an individual can choose either to depart in the tails of the rush hour when travel time is low and pay the penalty of arriving at work early or late, or to depart close to the PATI when travel time is high but schedule delay cost is low. In other words, under the deterministic equilibrium, schedule delay early/late and arrival on time cannot occur simultaneously for a given departure time (Henderson [
The rest of this paper is organized as follows. Section
We formulate the peak period congestion based on the bottleneck model developed by Vickrey [
The queue length that a trip maker departing at time
Suppose that every morning, a fixed number of
Some commuters may still arrive at the destination earlier or later than PATI, in order to avoid a long queue at the bottleneck. The cost for commuters traveling from home to the CBD consists of three components: the cost of travel time and the cost of schedule delay early or late. It can be formulated as follows:
Substituting (
Figure
Departure time distributions in the deterministic case.
The deterministic case models a singleday departure time equilibrium. In the real world, road capacities may vary from day to day due to unexpected events such as incidents and weather conditions. Because of the capacity fluctuations, both commuters’ travel time and their schedule delays are stochastic. In this section, we hypothesize that a constant longterm departure time pattern may emerge given the responses of the travelers to the daytoday capacity variation. Each commuter chooses an optimal departure time which minimizes his/her longterm expected trip cost. We call this pattern, if it exists, a longterm equilibrium pattern.
The following assumptions are made in the model formulation.
Commuters are homogeneous with the same
The capacity of the bottleneck is constant within a day but fluctuates from day to day. The uncertainty of capacity is completely exogenous and independent of departures.
The capacity is a nonnegative stochastic variable changing around a certain mean capacity. Following Li et al. [
Commuters are aware of the capacity degeneration probability, and their departure time choice follows the user equilibrium (UE) principle in terms of mean trip cost.
We assume that the capacity of the single bottleneck is stochastic but the commuters’ departure time choice is deterministic. The calculation of the mean trip cost relies on the calculations of the mean travel time, the mean schedule delay early and late. For simplicity, we set the
The equilibrium condition for commuters’ departure time choice in a single bottleneck with stochastic capacity is that no commuter can reduce his/her mean trip cost by unilaterally altering his/her departure time. This condition implies that the commuters’ mean trip cost is fixed with respect to the time instant with positive departure rate. That is
Due to the stochastic capacity over days, the travel time experienced by a traveler departing at the same time
As we assume that stochastic capacity is completely exogenous and independent of departure flows, the expectation of travel time and schedule delay cost with respect to different situations can be derived respectively as follows:
No commuters experience schedule delay later subject to all possible values of the bottleneck capacity. We get the departure rate
The boundary condition for this situation is
If the capacity of the bottleneck is large enough, only schedule delay early will occur. On the contrary, no schedule delay occurs when the capacity is small. The watershed capacity satisfies
The boundary condition for this case is
No commuters experience schedule delay subject to all possible values of the bottleneck capacity. Therefore, the departure rate is
The boundary condition for this case is
If the capacity of the bottleneck is large enough, individuals arrive on time. On the contrary, schedule delay late occurs when the capacity is small. The watershed capacity satisfies
The boundary condition for this case is
Similar to Situation
The boundary condition for this case is
Similar to Situation
The boundary condition for this case is
Since the departure rate
With the resulting stochastic departure pattern at longterm equilibrium, the experienced daytoday travel times and number of travelers experiencing queues change according to varied capacities over days. Given the boundary conditions of Situations
Departure time distributions in the stochastic case.
In this subsection, we investigate the theoretical properties of the equilibrium solution of the proposed stochastic bottleneck model with PATI.
At equilibrium, the expected trip cost for all commuters is a monotonically increasing function of traffic demand and a monotonically decreasing function of
Since
At equilibrium, the expected trip cost for all commuters is a monotonically decreasing function of parameter
Submitting
At equilibrium, the departure rate is a monotonically decreasing function of the departure time throughout the whole peak period; that is,
According to (
Equations (
In summary, the departure rate
When parameter
According to the L’Hôpital’s rule, we have
When the number of commuters is given, increasing the value of parameter
According to (
The above proof also shows that the length of peak period is not affected by the
The input parameters of our numerical example are
The influence of parameter











1.00  3.62  −1.10  −0.74  −0.74  −0.40  −0.40  0.40  0.40  1.50 
0.95  3.78  −1.14  −0.77  −0.73  −0.46  −0.31  0.36  0.39  1.52 
0.90  3.95  −1.18  −0.80  −0.73  −0.52  −0.21  0.31  0.37  1.55 
0.85  4.13  −1.23  −0.85  −0.72  −0.59  −0.09  0.26  0.35  1.57 
0.80  4.33  −1.28  −0.89  −0.71  −0.66  0.05  0.21  0.32  1.60 
The mean equilibrium trip cost and other components.
It is interesting to investigate the impact of the fraction parameter
The departure rates against different
The influence of parameter
Table
The influence of parameter











0  4.98  −1.28  −0.80  −0.80  −0.80  −0.55  0.21  0.27  1.55 
5  4.46  −1.23  −0.80  −0.72  −0.68  −0.38  0.26  0.32  1.55 
10  3.95  −1.18  −0.80  −0.73  −0.52  −0.21  0.31  0.37  1.55 
15  3.43  −1.13  −0.80  −0.74  −0.36  −0.04  0.36  0.42  1.55 
20  2.91  −1.08  −0.80  −0.75  −0.20  0.13  0.41  0.47  1.55 
Figure
The influence of parameter
Figure
Travel time with different
Figure
The equilibrium trip cost
This paper investigated the travel choice behavior under uncertainty on morning commute problem by considering the capacity variability of a highway bottleneck. The bottleneck model was applied to analyze the departure time pattern of a group of homogeneous commuters with the same preferred arrival time interval. The capacity of the bottleneck is assumed to follow a uniform distribution and the commuters’ departure time choice to follow the UE principle in terms of the mean trip cost. The analytical solution of the stochastic bottleneck model was derived. Both analytical and numerical results show that increasing the capacity variation results in longer peak period and higher commuters’ mean trip cost. In addition, it is shown that with longer flexitime interval, the departure time distributions become flatter. This suggests that flexitime is an effective demand management measure for alleviating peak congestion. For future research, we will further improve the model with consideration of heterogeneous commuters and travel risk and apply the model in analyzing such policy measures as congestion pricing, metering, and flexible work scheme.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors acknowledge thank the financial support from the National Basic Research Program of China (2012CB725401), the PhD Student Innovation Fund of Beihang University (302976), and the China Scholarship Council.