Modal Split and Bus Fare Allocation for a Congested Bottleneck: Autonomous Vehicles and Buses

Abstract

With the goal of reducing traffic congestion for urban commuters, and in alignment with the global shift toward smarter, more sustainable, and more equitable cities, the transportation field is faced with the challenge of improving and optimizing its systems. This paper will tackle the morning commute problem of a congested bottleneck shared between buses and autonomous vehicles (AVs). The objective of this study is to determine the optimal bus fare, departure time, and mode split simultaneously by minimizing the total system social travel cost using numerical techniques. In addition, the paper examines the sensitivity of the system social cost to parameters such as bus capacity and frequency. Using a simple numerical example, the relationships between mode split, bus fare, total cost of a bus trip, dispatching frequency, bus capacity, total cost of an AV trip, and AV fare to the system social cost are examined, and optimal values are presented and discussed. The distribution of trips and the influence of AV capacity are also explored. The findings suggest that, within a reasonable range, the optimal bus fare was the absolute minimum, and the optimal mode split was one with fewer bus passengers but no boundary value. Other interesting findings were observed assuming a predetermined and fixed bus fare. Overall, the methodology and results of this research paper offer valuable insights for traffic and transit authorities, which could aid in planning and operational decisions that ultimately reduce commuter delays and enhance the sustainability of the morning commute.

Introduction

In most North American cities, cars are the primary mode of transportation, which has caused traffic congestion, especially during peak periods, to become a major issue. With new emerging modes of transportation and global initiatives aimed at developing smarter and more sustainable cities, optimizing aspects of the transportation network is very timely. The problem that this paper will address is the issue of traffic congestion along a congested corridor, often referred to as the “morning commute problem.” Congestion theory was first introduced by Vickrey, in 1969 ( 1 ), for modeling this problem, in which all commuters travel from home to work in the morning with the same desired arrival time. In this paper the bottleneck model is used, which assumes that only one route option is available, as shown in the schematic representation of the problem in Figure 1.

Figure 1.

Schematic of the morning commute problem.

The objective of this research is to determine the optimal bus fare and mode split that minimizes the total system social cost (SSC) on a congested bottleneck shared by autonomous vehicles (AVs) and buses and to explore the effects of varying transit variables such as the bus capacity and dispatching frequency. AVs and buses were selected as the only two modes for this study, because it presents a possible future scenario in the context of smart cities integrating AVs into existing transportation systems. Although, a future with several additional mode choices may be likely, this study is limited to only two modes to reduce the complexity of the model and to make it easier to identify trends and relationships between variables. This choice of modes necessitates consideration about shared road space, as traffic congestion would affect users of both modes, as well as having an impact on crowding and the discomfort of traveling with strangers on public transit. The two modes are also different enough to warrant discussion about the unit costs of travel time and fares associated with the use of each.

For the purposes of this study, AVs are defined as “autonomous vehicles that are not privately owned but fleet operated.” The AV service would be similar to that of a current ridesharing service offered by a transportation network company (TNC), but without the need for a driver, and for the majority of this paper, it is assumed that each vehicle only serves one passenger per trip. Fleet-operated AVs offer potential benefits compared with privately owned AVs such as reducing the need for parking, especially in downtown or the central business district, and reducing the kilometers traveled by AVs without passengers, which is another reason for considering this mode. Buses considered in this study are public transit biarticulated buses with a higher capacity than most typical transit vehicles, since they are serving passengers in the peak period traveling along a corridor with high travel demand. It is assumed that bus operators have perfect driving capabilities, making them equivalent to AVs with respect to following distance.

Minimizing the SSC was chosen as the objective of this study, as both travel modes are fleet operated. Further, determining the modal split could provide traffic authorities with better information about the demand level for each mode, which, in conjunction with other information such as road capacity, would allow them to predict the level of congestion and future trends. Authorities could utilize this information to appropriately allocate AV traffic to the network since they can be programmed to create a system that achieves a social optimum. This optimized system would see reduced congestion, energy consumption, and delay and would thus benefit all commuters. Determining the optimal bus fare could also help transit authorities with budgetary decisions, aid in providing revenue necessary for operations, and ideally, entice commuters to use transit.

Despite the intensive literature using Vickrey’s bottleneck model, few studies address multimodal systems with shared roads/corridors, heterogeneous users/mode-dependent values of time, and consider the inconvenience cost of crowding in transit concurrently. To the best of our knowledge, the AV/bus multimodal system has not previously been studied in this capacity. Furthermore, knowledge gaps in the influence of city- or network-specific parameters such as bus frequency and capacity have been identified.

This study contributes to the existing literature by examining a multimodal system with shared roads/corridors, heterogeneous users/mode-dependent values of time, and considering the inconvenience cost of crowding in transit simultaneously. Capturing all these factors at once better represents the transportation system, along with the behavior and preferences of commuters, such that numerical modeling more accurately reflects real-world conditions. Further, considering an AV/bus system is an important contribution to the existing body of literature, as the majority of research on Vickrey’s bottleneck model has focused on traditional modes. Resultingly, this paper can help transportation authorities to develop new regulations surrounding AVs as they are introduced, such as fleet size restrictions, AV and bus fares, and other guidelines for TNCs that will be essential for the smooth integration of AVs. This study also adds to the existing body of work that has been done primarily on very crowded transit systems with short headways by considering a range of bus capacities and dispatching frequencies. This paper is thus applicable to a wider breadth of transportation systems and could help guide transit agencies’ scheduling and vehicle allocation. Finally, briefly exploring the effects of high-occupancy (double-occupant) AVs on traffic patterns could provide valuable information for traffic authorities and may inspire guidelines or incentives for ridesharing in strained transportation networks.

This paper determines the optimal bus fare and mode split simultaneously by minimizing the SSC function, without the need for additional functions such as utility functions. Further, a sensitivity analysis on the SSC is conducted for various parameters, and the effect of allowing multiple passengers per AV on the distribution of trips is also explored.

The remainder of the paper is structured as follows. Section 2 consists of a literature review that discusses considerations for a multimodal system and pricing regimes when addressing the morning commute problem. Section 3 outlines the methodology for determining time- and trip-independent cost functions for AV and bus trips, respectively, as well as the methodology for optimizing the bus fare and modal split using the objective function. Section 4 presents the limitations of the study, discusses the results of a numerical example, and explores the influence of key transit parameters on the system cost. Finally, Section 5 concludes the paper by highlighting the contributions and identifying areas for future work.

Literature Review

The problem of peak-period congestion has been researched extensively, with one of the original papers published by Vickrey in 1969 ( 1 ), which introduced congestion theory and presented the first model for the morning commute problem. The basis of the model is to determine the total trip cost for commuters by summing the schedule delay and the travel time, with the objective of balancing these two components to reach an equilibrium in the distribution of departure time, where the trip price over the peak period is uniform, that is, no commuters can reduce their trip cost by changing their departure time. This original model considered cars on a single bottleneck where all commuters are homogenous.

Vickrey’s model motivated several subsequent research studies that have relaxed some of the assumptions and expanded the original model to address other questions about peak-period commuting. Several extensions have been explored, including route choice, mode choice, time-varying scheduling preferences, heterogeneous users with distinct value of time, single- and high-occupancy vehicles, tradable credits, redistribution of tolls, congestion/emission pricing, and parking pricing ( 2 ).

Multimodality

Since most urban areas have public transit systems available, and with the rise of alternative travel modes in addition to the traditional automobile, mode choice has been an area of focus for many studies. For multimodal systems, various additional factors become relevant, including whether the modes share space within the corridor, the homogeneity of commuters in relation to preferences and value of time, and the onboard congestion cost for transit vehicles. Table 1 summarizes the factors that have been considered in the literature.

Table 1.

Literature Review: Multimodality Considerations

Reference	Travel mode	Onboard crowding	Shared space	Heterogeneous commuters
Vickrey ( 1 )	Auto	na	na	N
Arnott et al. ( 12 )	Auto	na	na	N
Tabuchi ( 3 )	Auto and train	N	na	N
Huang et al. ( 16 )	Auto and bus	Y	Y	N
Qian and Zhang ( 8 )	Auto, carpool, and train	Y	Y	N
Gonzales and Daganzo ( 6 )	Auto and bus	N	Partly	Y
Wu and Huang ( 4 )	Auto and train	Y	na	N
Monchambert et al. ( 5 )	Auto and train	Y	na	Y
van den Berg and Verhoef ( 11 )	Auto and AV	na	Y	Y
Tian et al. ( 7 )	Auto and SAV	na	NA	NA
Tian ( 9 )	Auto and AV	na	NA	N
Yu et al. ( 10 )	Auto and AV	na	NA	NA

Note: Y = yes; N = no; na = not applicable; NA = not available; AV = autonomous vehicle; SAV = shared autonomous vehicle.

Tabuchi determined the optimal number of road users under different fare regimes for a multimodal system, where commuters could travel either by auto on a congested bottleneck or by train on a separated corridor, but did not consider the inconvenience cost of onboard crowding ( 3 ). Both Wu and Huang ( 4 ) and Monchambert et al. ( 5 ) incorporated the cost of crowding into their cost functions for the train, and Monchambert et al. further classified the users into two groups based on their travel preferences ( 5 ).

Auto and train systems are simple multimodal systems: as rail systems are typically separate from the road, they do not need to account for shared space, therefore, it is assumed that rail systems do not experience any schedule delay attributed to queuing, unlike cars. In most other multimodal systems, such as one comprised of cars and buses, the road space is usually shared between modes and would thus be subject to the same queuing delays; this aspect has not yet been studied extensively. Gonzales and Daganzo only partially addressed this challenge, assuming that transit-only lanes would be available, thus reducing the capacity available to cars, but allowing the buses to avoid queuing delays ( 6 ). Huang et al. did account for this by reducing the capacity of the road based on the frequency and size of buses on the bottleneck and including a time-dependent queuing delay cost, but later used a utility function to determine the mode split ( 7 ). Qian and Zhang added the dimension of shared costs between multiple passengers by comparing carpooling, single occupancy vehicles, and trains, and assumed that carpoolers had a dedicated high-occupancy-vehicle lane but could also use the regular lanes if desired ( 8 ). More recently, studies have examined the morning commute problem for a multimodal system with a mix of regular vehicles and AVs ( 7 , 9 – 11 ), in which they suggested that in-vehicle activities would influence the value of time (VOT) parameters. Van den Berg and Verhoef delved into the effect of differences in capacity and VOT between the two modes on the equilibrium queuing times for a congested bottleneck ( 11 ).

Pricing Regimes

Previous studies have explored different fare and congestion-pricing regimes, mainly, transit fare and tolls. The pricing model depends on the objective of the study. Typical objective functions are aimed at minimizing the SSC, maximizing revenue, or shifting peak-period congestion to off peak, a couple examples of each objective are presented in the following.

Minimizing System Social Cost

Considering only cars, Arnott et al. determined equilibrium under four different pricing regimes: no toll, optimal uniform toll, optimal step (coarse) toll, and optimal time-varying (fine) toll ( 12 ). Tabuchi determined the optimality of different fare-pricing strategies, including average cost pricing, marginal cost pricing, and optimal uniform fare, and fine and coarse tolls for a train and automobile system ( 3 ). Wu and Huang additionally evaluated three road-use pricing strategies: government pricing with financial subsidy given to drivers for changing their departure times, government pricing without subsidization, and private-organization pricing, for an auto and rail transit system with a fixed transit fare ( 4 ).

Maximizing Revenue

With the goal of maximizing revenue, Meng et al. considered a presale period divided into five price-varying time periods and developed a joint decision model where the operator chose the price and ticket allocation to maximize revenue ( 13 ), and Wu and Huang used a private-organization model to maximize its revenue from tolls ( 4 ). Meng explored the effects of pricing on demand using a demand elasticity model, and was also able to draw insights about relationships between the time of the ticket purchase and the sensitivity to price ( 13 ).

Shifting Peak-Period Congestion to Off-peak Periods

Yin et al. discussed several pricing methods: (i) the incentive method, in which the fare is increased during the peak period and decreased during off peak; (ii) the physical-barrier model, which uses barriers to control the flow of passengers entering during the peak period; and (iii) trip reservation, whereby travelers submit their travel information in advance through a reservation system ( 14 ). Yang and Tang proposed a fare reward scheme for rail transit, which would offer a free ride to commuters during the off-peak period after a certain number of peak-hour paid trips had been made ( 15 ). Both studies were conducted on a unimodal train system and may not be easily extendable to a multimodal system, since the objective is typically to shift commuters from a more congested mode to a less congested one, rather than changing their departure time.

Methodology

This paper looks at a bottleneck system with two available modes for commuters to choose from during the morning peak period: AVs and transit (buses). The process involves formulating cost functions for each mode separately, and then determining cost expressions for the time equilibrium for AV commuters and the trip equilibrium for bus commuters. Finally, a function for the SSC of the entire system is formulated and minimized, which will indicate the optimal bus fare and modal split.

Cost Function Formulation

We considered that both modes share the same space on the road (no transit-only lanes), since buses operate in mixed traffic in many urban transit systems. As such, the commuting time, $T\left(t\right)$ , for either mode can be expressed as follows:

$$T\left(t\right)=T_f+Q(t)/s$$

(1)

where

• $T\left(t\right)$ = commuting time for a departure at time $t$ (h),

• $T_f$ = free-flow commuting time in uncongested conditions (h),

• $t$ = departure time from home,

• $Q\left(t\right)$ = queue length (vehicles), and

• $s$ = bottleneck capacity (vehicles per hour).

Note that, in the numerical implementation, mode-specific free-flow travel times are used, denoted as $T_{f,\mathrm{AV}}$ and $T_{f,\mathrm{Bus}}$ . This refinement does not alter the analytical structure of the model; it simply differentiates between the operating characteristics of AVs and buses.

The queue length can be expressed as,

$$Q\left(t\right)=\{\begin{array}{c}{\int }_{t_0}^t[r\left(t\right)+\lambda f]\mathrm{dt}-s(t-{\tau }_0)\mathrm{for}t\in [{\tau }_0,{\tau }_1]\hfill \\ \multicolumn{1}{c}{0\mathrm{otherwise}}\end{array}$$

(2)

where

• $r\left(t\right)$ = departure rate of vehicles at time $t$ (AV units per hour);

• $f=$ frequency of bus departures (buses per hour);

• $\lambda =$ ratio of AV passenger car units to buses (compares vehicle sizes, i.e., $\lambda =3$ means that a bus takes up the same length of road as three AVs);

• ${\tau }_0=$ the most recent time at which there was no queue (i.e., time of start of queue); and

• ${\tau }_1=$ time at which the queue ends.

An expression for work start time is given by Equation 3,

$$t^{*}=\tilde{t}+T(\tilde{t})$$

(3)

where $\tilde{t}$ is departure time at which an individual can leave home and arrive at work on time; and

• $t^{*}$ is work start time

AV Commuters

Again, AVs are defined as AVs owned by TNCs, much like existing ridesharing, but without the need for a driver. Initially, this paper assumes that each vehicle only serves one passenger per trip. A few important considerations and assumptions that distinguish TNC-owned AVs from other vehicles are

• Lower unit cost of travel time, $\alpha ,$ than a regular human-driven vehicle, since passengers can perform other tasks while in the vehicle.

• Higher bottleneck capacity than a regular human-driven vehicle, since vehicles can be packed together more closely.

• Out-of-pocket cost is solely associated with the fare for the use of the vehicle; energy consumption, parking fees, and other expenses are not included, since the fares collected by the TNC would cover these costs.

• Free-flow commuting time is equivalent to the design speed or speed limit.

• The vehicles are programmed to “behave” according to social optimum rather than user equilibrium, as a transportation authority or TNC would be able to program and dispatch vehicles to reduce congestion and minimize travel time across the entire system.

The generalized cost function for an AV commuter, as shown in Equation 4, is composed of a travel time cost component, a cost penalty for early arrival or a cost penalty for late departure, and a fare.

$$C_{\mathrm{AV}}\left(t\right)=\{\begin{array}{c}{\alpha }_{\mathrm{AV}}T\left(t\right)+\beta [t^{*}-t-T\left(t\right)]+k_{\mathrm{AV}}\mathrm{for}t\in [{\tau }_0,\tilde{t}]\hfill \\ \multicolumn{1}{c}{{\alpha }_{\mathrm{AV}}T\left(t\right)+\gamma [t-t^{*}+T\left(t\right)]+k_{\mathrm{AV}}\mathrm{for}t\in [\tilde{t},{\tau }_1]}\end{array}$$

(4)

where

• $C_{\mathrm{AV}}\left(t\right)=$ time-dependent cost function for the morning commute by AV ($),

• ${\alpha }_{\mathrm{AV}}=$ unit cost of AV travel ($/time),

• $\beta =$ unit cost of early arrival ($/time),

• $\gamma =$ unit cost of late arrival ($/time), and

• $k_{\mathrm{AV}}=$ fare for AV ($).

Bus Passengers

A few important considerations and assumptions that distinguish buses from other vehicles include

• Lower unit cost of travel time, $\alpha ,$ than a regular human-driven vehicle, since passengers can perform other tasks while in the vehicle. However, the cost of travel time for a bus is slightly higher than a TNC-owned AV, primarily because of privacy and safety concerns ( 14 ).

• Primarily passengers seated in uncrowded vehicles will be able to perform other tasks, and are generally more comfortable, so a unit cost of onboard congestion is introduced to account for the disutility to passengers in crowded vehicles.

• During periods of high demand, higher bottleneck capacity than with single occupancy vehicles, since buses have much higher carrying capacities.

• Out-of-pocket cost is for the transit fare, and is typically lower than most other motorized vehicles, since transit typically receives external funding, and the operational costs are divided between users across the entire system.

• Free-flow commuting time is longer than AVs owing to their slower acceleration and tendency to travel at lower speeds than smaller vehicles.

• Bus passengers’ departure time depends on the frequency of the bus service and is restricted to the departure time of a bus trip.

The expression for bus departure time is

$$t_j=\tilde{t}-\frac{j}{f}$$

(5)

where $t_j$ is the departure time for bus $j$ ; and $j$ is the integer value assigned to each bus trip, defined for. Where the commuters arrive at work exactly on time, $j=0$ , $a$ is the first bus trip with commuters and $b$ is the last bus trip with commuters.

The generalized cost function, as shown in Equation 6 for a bus commuter, is composed of a travel time cost component, a cost penalty for early arrival or a cost penalty for late departure, passenger fare, and an onboard congestion cost.

$$C_{\mathrm{bus}}\left(j\right)=\{\begin{array}{c}{\alpha }_{\mathrm{bus}}T\left(t_j\right)+\beta [t^{*}-t_j-T\left(t_j\right)]+k_{\mathrm{bus}}+g\left(n_j\right)T\left(t_j\right)\mathrm{for}t\in [t_a,\tilde{t}]\hfill \\ \multicolumn{1}{c}{{\alpha }_{\mathrm{bus}}T\left(t_j\right)+\gamma [t_j-t^{*}+T\left(t_j\right)]+k_{\mathrm{bus}}+g\left(n_j\right)T\left(t_j\right)\mathrm{for}t\in [\tilde{t},t_b]}\end{array}$$

(6)

where

• $C_{\mathrm{bus}}\left(j\right)=$ trip-dependent cost function for the morning commute by bus ($),

• ${\alpha }_{\mathrm{bus}}=$ unit cost of bus travel ($/time),

• $k_{\mathrm{bus}}=$ bus fare ($), and

• $g\left(n_j\right)=$ unit cost of onboard congestion ($/time).

The expression for onboard crowding cost, also referred to as body congestion, as defined by Huang et al. is expressed in Equation 7 ( 16 ). A higher $m$ value indicates a greater aversion to crowding, and $K$ depends on the vehicle size and type, but is typically larger than the number of seats, since standing is normally permitted.

$$g\left(n_j\right)=-m\times \ln (1-\sqrt[4]{\frac{n_j}{K}})$$

(7)

where

• $m=$ parameter for aversion to crowding,

• $n_j=$ number of passengers in the bus for trip $j$ , and

• $K=$ capacity of the transit vehicle.

Note that the unit cost of early arrival ( $\beta$ ) and late arrival ( $\gamma$ ) are assumed to be mode-independent. Following the classical bottleneck formulation by Small (1982), these values satisfy $\beta <\alpha <\gamma$ , indicating that commuters are generally more sensitive to late arrivals than to early arrivals or in-vehicle travel time ( 17 ).

Departure Time/Bus Trip Equilibrium

Next, we need to determine the time equilibrium for AV commuters and the trip equilibrium for bus commuters. This means that AV commuters cannot reduce their trip cost by changing their departure time and bus commuters cannot reduce their trip cost by taking a different bus trip. This equilibrium will be achieved by striking a balance between the cost of travel time (including time spent in a queue), a cost penalty incurred for early or late arrival, and for bus passengers, an onboard congestion cost to account for the discomfort of crowded vehicles.

AV Commuters

The departure time equilibrium cost function for AV commuters can be determined analytically. The procedure is as follows: since neither the first nor last commuter will face a queue, based on Equation 1, $T\left({\tau }_0\right)=T_f$ and $T\left({\tau }_1\right)=T_f$ . Combining this with Equation 4 results in the following expressions:

$$C_{\mathrm{AV}}=C_{\mathrm{AV}}(t_0)={\alpha }_{\mathrm{AV}}{T}_f+\beta [t^{*}-{\tau }_0-T_f]+k_{\mathrm{AV}}$$

(8a)

$$C_{\mathrm{AV}}=C_{\mathrm{AV}}(t_1)={\alpha }_{\mathrm{AV}}{T}_f+\gamma [{\tau }_1-t^{*}+T_f]+k_{\mathrm{AV}}$$

(8b)

Then, by the definition of equilibrium, all commuters have the same cost. So, Equations 8a and 8b can be combined and simplified to give

$$\beta [t^{*}-{\tau }_0-T_f]=\gamma [{\tau }_1-t^{*}+T_f]$$

(9)

Then we define a function for the number of AV commuters, $N_{\mathrm{AV}}$ ,

$$N_{\mathrm{AV}}=({\tau }_1-{\tau }_0)(s-\lambda f)$$

(10)

where $N_{\mathrm{AV}}$ is numbers of commuters choosing to use AV mode (which is equal to the number of AVs, since one passenger per vehicle is assumed).

Under the assumption that commuters departing at $\tilde{t}$ , ${\tau }_0,$ and ${\tau }_1$ all have the same cost, combining Equations 9 and 10 to solve for ${\tau }_0$ , ${\tau }_1$ , and $\tilde{t}$ gives

$${\tau }_0=t^{*}-T_f-\frac{\psi }{\beta }$$

(11a)

$${\tau }_1=t^{*}-T_f+\frac{\psi }{\gamma }$$

(11b)

$$\tilde{t}=t^{*}-T_f-\frac{\psi }{{\alpha }_{\mathrm{AV}}}$$

(11c)

where $\psi$ is a combined parameter given by

$$\psi =\frac{\beta \gamma }{(\beta +\gamma )}\frac{N_{\mathrm{AV}}}{(s-\lambda f)}$$

(12)

Minimizing Equation 4 (i.e., setting $\frac{{\mathrm{dC}}_{\mathrm{AV}}}{\mathrm{dt}}=0$ ) gives expressions for the departure rate, $r\left(t\right)$ ,

$$r\left(t\right)=\{\begin{array}{c}\frac{{\alpha }_{\mathrm{AV}}}{{\alpha }_{\mathrm{AV}}-\beta }s-\lambda f\mathrm{for}t\in [{\tau }_0,\tilde{t}]\hfill \\ \multicolumn{1}{c}{\frac{{\alpha }_{\mathrm{AV}}}{{\alpha }_{\mathrm{AV}}+\gamma }s-\lambda f\mathrm{for}t\in [\tilde{t},{\tau }_1]}\end{array}$$

(13)

Then combining Equations 1, 2, and 13 the travel time function can be derived as

$$T(t)=\{\begin{array}{c}\frac{\beta }{{\alpha }_{\mathrm{AV}}-\beta }(t-{\tau }_0)+T_f\mathrm{for}t\in [{\tau }_0,\tilde{t}]\hfill \\ \multicolumn{1}{c}{\frac{\gamma }{{\alpha }_{\mathrm{AV}}+\gamma }({\tau }_1-t)+T_f\mathrm{for}t\in [\tilde{t},{\tau }_1]}\end{array}$$

(14)

Then we combine either Equations 8a and 11a, or Equations 8b and 11b,

$$C_{\mathrm{AV}}={\alpha }_{\mathrm{AV}}{T}_f+k_{\mathrm{AV}}+\psi$$

(15)

Bus Passengers

The trip equilibrium cost function was solved numerically owing to the complexity introduced by the crowding cost being a function of the number of bus passengers served. The procedure is explained below.

Combining Equation 11c and Equation 5 gives

$$t_j=t^{*}-\frac{\psi }{\alpha }-\frac{j}{f}-T_f$$

(16)

For trip equilibrium to be achieved, unlike AV trips, bus trips do not need to fall within $[t_0,t_1]$ . Instead, three time intervals are defined as follows:

• [ $t_a,t_{\overline{a}}]$ : time interval between the first bus trip with passengers and the first bus trip that encounters a queue;

• [ $t_{\overline{a}},t_{\overline{b}}]$ : time interval between the first and last bus trips that encounter a queue; and

• [ $t_{\overline{b}},t_b]$ : time interval between the last bus trip that encounters a queue and the last bus trip with passengers.

Since $\overline{a}$ and $\overline{b}$ are the first and last trips that encounter a queue, respectively, they can, by definition, be expressed as the integer number of trips that can be taken between the time the queue starts or ends ( $t_0\mathrm{or}{t}_1$ ) and the departure time to arrive at work on time, $\tilde{t}$ ,

$$\overline{a}=(\tilde{t}-{\tau }_0)f={\left[\frac{(\alpha -\beta )}{\alpha \beta }\psi \right]}^{-}$$

(17a)

$$\overline{b}=({\tau }_1-\tilde{t})f={\left[\frac{(\alpha +\gamma )}{\alpha \gamma }\psi \right]}^{-}$$

(17b)

Note that ${\left[x\right]}^{-}$ in Equations 17a and 17b, and ${\left[x\right]}^{+}$ in Equations 19a and 19b are the notations used to indicate that $x$ should be rounded up and down to the nearest integer value, respectively.

By definition, $C_{\mathrm{bus}}\left(j\right)=C_{\mathrm{bus}}$ at equilibrium, since equilibrium implies that all trips, $j$ , have the same cost. For bus trips when there is no queue, that is, $t_j<t_{\overline{a}}$ and $t_{\overline{b}}>t_j$ , $T\left(t_j\right)=T_f$ . Combining Equations 6 and 16 and simplifying gives the expressions shown in Equation 18a and 18c. Then combining Equations 6, 11, 14, 16, 17a, and 17b, gives the expression shown in Equation 18b.

$$\begin{array}{c}\multicolumn{1}{c}{C_{\mathrm{bus}}=}\\ \multicolumn{1}{c}{\{\begin{array}{c}{\alpha }_{\mathrm{bus}}{T}_f+\beta [\frac{\psi }{\alpha }+\frac{j}{f}]+k_{\mathrm{bus}}+g\left(n_j\right)T_f\mathrm{for}t\in [t_a,t_{\overline{a}}](18\mathrm{a})\hfill \\ \multicolumn{1}{c}{{\alpha }_{\mathrm{bus}}{T}_f+\psi +k_{\mathrm{bus}}+g\left(n_j\right)T(t_j)\mathrm{for}t\in [t_{\overline{a}},t_{\overline{b}}](18\mathrm{b})}\\ \multicolumn{1}{c}{{\alpha }_{\mathrm{bus}}{T}_f-\gamma [\frac{\psi }{\alpha }+\frac{j}{f}]+k_{\mathrm{bus}}+g\left(n_j\right)T_f\mathrm{for}t\in [t_{\overline{b}},t_b](18\mathrm{c})}\end{array}}\end{array}$$

Based on the definitions of $a$ and $b$ , no commuters would be on bus $(a+1)$ or $(b+1)$ , that is, $g\left(n_{a+1}\right)=g\left(n_{b+1}\right)=0$ . Applying this to Equations 18a and 18c gives

$$a={\left[(\frac{C_{\mathrm{bus}}-{\alpha }_{\mathrm{bus}}{T}_f-k_{\mathrm{bus}}}{\beta }-\frac{\psi }{\alpha })f\right]}^{+}-1$$

(19a)

$$b={\left[(\frac{C_{\mathrm{bus}}-{\alpha }_{\mathrm{bus}}{T}_f-k_{\mathrm{bus}}}{\gamma }+\frac{\psi }{\alpha })f\right]}^{+}-1$$

(19b)

Rearranging Equations 18a, 18b and 18c to obtain an expression for the number of bus commuters for each trip gives

$$n_j=\{\begin{array}{c}{g}^{-1}\left(\frac{C_{\mathrm{bus}}-k_{\mathrm{bus}}-{\alpha }_{\mathrm{bus}}{T}_f-\beta [\frac{\psi }{\alpha }+\frac{j}{f}]}{T_f}\right)\mathrm{for}j\in (\overline{a},a]\hfill \\ \multicolumn{1}{c}{g^{-1}\left(\frac{C_{\mathrm{bus}}-k_{\mathrm{bus}}-{\alpha }_{\mathrm{bus}}{T}_f-\psi }{T\left(t_j\right)}\right)\mathrm{for}j\in [-\overline{b},\overline{a}]}\\ \multicolumn{1}{c}{g^{-1}\left(\frac{C_{\mathrm{bus}}-k_{\mathrm{bus}}-{\alpha }_{\mathrm{bus}}{T}_f+\gamma [\frac{\psi }{\alpha }+\frac{j}{f}]}{T_f}\right)\mathrm{for}j\in [-b,-\overline{b})}\end{array}$$

(20)

Objective Function: Minimizing System Social Cost

The SSC of the AV and bus system for the bottleneck during the morning commute can be expressed as a function of the trip cost and number of passengers for each mode. The goal is to determine the optimal bus fare, $k_{\mathrm{bus}}$ , and mode split, $N_{\mathrm{AV}}$ / $N_{\mathrm{bus}}$ by minimizing $\mathrm{SSC}$ The objective function is the minimization of Equation 21.

$$\mathrm{SSC}=C_{\mathrm{AV}}{N}_{\mathrm{AV}}+C_{\mathrm{bus}}{N}_{\mathrm{bus}}$$

(21)

where $\mathrm{SSC}$ is total social cost of the system, and $N_{\mathrm{bus}}$ is number of bus commuters during the morning commute.

In this paper it is assumed that the total demand, or number of morning commuters, $N$ , along the bottleneck is known. As such, Equation 22 must hold.

$$N=N_{\mathrm{AV}}+N_{\mathrm{bus}}$$

(22)

Because of the complexity introduced by the relationship between the cost of a bus trip and the number of bus passengers, the objective function must be solved numerically using an optimization technique while ensuring parameters are restricted to feasible values. Code was implemented in RStudio using $C_{\mathrm{bus}}$ and $k_{\mathrm{bus}}$ , within their respective ranges, as the inputs to be optimized. (These constraints are described in the section covering optimal bus fare and mode split.) The optimization function looped through combinations of $C_{\mathrm{bus}}$ and $k_{\mathrm{bus}}$ , while simultaneously calculating the mode split, by running a loop through all possible values of $N_{\mathrm{AV}}$ (i.e., 0 to $N$ ) to calculate the corresponding number of bus passengers for each trip, $n_j$ , and, ultimately, the total number of bus commuters, $N_{\mathrm{bus}}$ , determined by $N_{\mathrm{bus}}=\sum n_j$ , using Equations 17, 18, and 19. The output $N_{\mathrm{AV}}$ value for which $\sum n_j=N-N_{\mathrm{AV}}$ to satisfy the requirement from Equation 22 was then used to calculate $C_{\mathrm{AV}}$ based on Equation 15. $\mathrm{SSC}$ , for each set of input values was computed using Equation 21, where the optimal solution is the one that produces the lowest $\mathrm{SSC}$ .

Numerical Results and Discussion

First, several limitations of the study, particularly surrounding the mode choice assumptions, are discussed. Then, the values presented in Table 2 are used in the numerical example to determine the optimal bus fare and mode split using the methodology described in the previous section, and the results are presented and discussed below. These results are generated directly from the model developed in this study and reflect the equilibrium outcomes derived under the parameter settings presented in the methodology.

Table 2.

Parameter Values for Numerical Example

Parameter	Description	Value	Units
$N$	Total number of commuters	6,000	Passengers
$S$	Bottleneck capacity	2500	AV units/h
$T_{f_{\mathrm{AV}}}$	Free-flow travel time in uncongested conditions for AVs	0.25	h
$T_{f_{\mathrm{Bus}}}$	Free-flow travel time in uncongested conditions for buses	0.5	h
${\alpha }_{\mathrm{AV}}$	Unit cost of AV travel	18	$/h
${\alpha }_{\mathrm{Bus}}$	Unit cost of Bus travel	20	$/h
$\beta$	Unit cost of early arrival	16	$/h
$\gamma$	Unit cost of late arrival	30	$/h
$k_{\mathrm{AV}}$	Fare for AVs	15	$
$m$	Parameter for aversion to crowding	5	na
$K$	Capacity of the transit vehicle	240	Passengers
$f$	Frequency of bus departures	6	Buses/h
$\lambda$	Ratio of AV passenger car units to buses	3	na

Note: AV = autonomous vehicle; na = not applicable.

The parameters used in the numerical example were selected to represent a typical high-demand urban corridor during the morning peak period while remaining consistent with assumptions commonly adopted in the literature. The total demand, bus capacity, and service frequency were chosen to ensure that congestion develops at the bottleneck under peak conditions. The values of the time- and schedule-delay parameters were drawn from previous studies applying Vickrey-type models to similar contexts ( 7 , 16 ), and adjusted to maintain proportional relationships between early, on time, and late-arrival penalties. Free-flow travel times were differentiated between AVs and buses to account for differences in operating speed and dwell times. This adjustment preserves the general model structure but produces more realistic equilibrium patterns. Together, these parameters provide a coherent, illustrative scenario suitable for testing the theoretical model rather than reproducing any specific city case.

Although the model was not calibrated with empirical data, the parameter settings were within realistic ranges reported in previous bottleneck and congestion-pricing literature ( 6 ). This idealized approach aligned with the study’s objective to analytically explore the behavioral and operational implications of AV and bus interactions, serving as a conceptual foundation for future, data-driven applications.

Next, sensitivity analyses were conducted on the bus cost, frequency of bus service, bus capacity, AV fare, and AV cost to investigate their influence on the $\mathrm{SSC}$ of the system. Lastly, the distribution of commuters arriving at the bottleneck was explored and the assumption that each AV only carries one passenger per trip was relaxed.

Study Limitations

There are several limitations that must be acknowledged and should be addressed before the methodology and findings are used in practice. This study only considers fleet-based AVs and buses and assumes that there are no human-operated vehicles or personally owned vehicles in the system. Further, the paper assumes that AVs are all single-occupant vehicles for the majority of the study. These assumptions simplify the model so that it is more easily interpretable and can be relaxed depending on the actual system being considered. For example, if privately owned rather than TNC-operated AVs are considered, then the fare for AVs, $k_{\mathrm{AV}}$ , in the cost function presented in Equation 4 could be replaced with a term for operating and maintenance costs associated with vehicle ownership, such as fueling, insurance, and/or repairs. Though, since this term would most likely be dependent on time or mileage driven, it would add complexity to the model, which is avoided under the current model where all these costs are reasonably covered by a single term.

Considering a 100% penetration rate for AVs would require that all current human-operated vehicles are replaced by AVs, which would need to be a gradual transition. A system with a mix of human-operated and AVs and potentially a mix of fleet-based and privately owned AVs should certainly be considered for a system in transition. As briefly mentioned in the methodology, human-operated vehicles differ from AVs with respect to safe following distance and resulting road capacity, as well as in how the system operates. Human drivers do not have perfect information about traffic in the system and travel times, and their objective is to minimize their own travel time. In an optimized system of AVs, on the other hand, the vehicles could be programmed and controlled to achieve system optimum and minimize the total travel time of the system as a whole. These “behavioral” differences would mainly affect the numerical values of the parameters and might influence the choice of objective function. In this study, which considered AVs in an ideal system where system optimum is achieved, minimizing total social cost was chosen as the objective function since evaluating the entire system was a priority. This may offer limited value to traffic or transit authorities if their primary concerns are maximizing revenue or reducing congestion. In such cases, a different objective function such as one of the pricing regimes described in the literature review could be used.

A further limitation is the simplification of the morning commute problem to a simple bottleneck model. This assumption overlooks the complexities of route choice, specific passenger origins, and bus dwell time. All passengers are assumed to start and end their trip at the same location and are assumed not to be captive passengers (i.e., either mode is a viable choice), which is not realistic in practice. However, this simplification allowed for a better understanding of the congested bottleneck and queuing without too many external complicating factors. The longer free-flow travel time assigned to buses compared with AVs is indicative of the slower travel and longer trip time as a result of the dwell time at stops that bus passengers experience, however, the effects of boarding and alighting passengers at intermediate stops, such as specific dwell time and time-dependent crowding were not considered under this assumption.

Optimal Bus Fare and Mode Split

First, given the parameters in Table 2, a range was defined for the possible cost of a bus trip. The maximum value was determined by considering that, since the objective is to minimize the SSC, the maximum cost of a bus trip should be less than the cost of an AV trip when all commuters choose that mode (i.e., $N_{\mathrm{AV}}=N$ ). Using Equations 12 and 15, this gave a maximum bus cost of $44.73. The minimum was determined using Equations 7 and 18, where the minimum number of passengers for the trip to run, $n_j>0$ , was observed ( $n_j=1),$ the bus traveled at free-flow speed, $T\left(t_j\right)=T_f$ , it was the last trip to arrive early, $j=1$ , and $C_{\mathrm{bus}}$ was the minimum value based on the restrictions imposed subsequently, which was $22.58. For this example, the possible fare values were initially restricted to a range of −$5 to $15, where a negative bus fare would indicate a fare subsidy for enticing commuters to take transit. The results from this analysis are presented in Figure 2, where any scenario for which the mode split could not be optimized, using the process outlined in the methodology section, was omitted.

Figure 2.

Relationship between system social cost, bus fare, and the number of bus commuters.

As expected Figure 2 shows that, generally, for any given fare, as the number of bus commuters increased, so did the SSC. Similarly, for any given number of bus commuters, as the bus fare increased, the SSC did as well. It can also be noted that for the highest fares, larger numbers of bus commuters were not observed at all.

The results (Table 3) found that a $5 subsidy for riding the bus would be optimal, rather than a paid fare. One limitation was that, although this results in the lowest SSC, it does not provide the transit authority with any revenue from ticket sales. Typically, the revenue from ticket sales is used to (at least partially) cover the cost of transit operations. So, if the bus fare is less than the unit operating costs then the true cost of the trip is not really captured; moreover, transit operations still require a large budget, so further investigation into how this could be realistically achieved would be necessary. Some possible solutions include a variable fare system, for which a fare could be charged to passengers during off-peak periods, or operating costs could be subsidized by the government or another organization, potentially shared by the AV providers. Another possible solution is to add in an additional constraint that would further restrict the minimum fare to a value that would be able to cover the unit operating costs of a bus trip: this would require knowledge about the operating costs of the bus system.

Table 3.

Optimal Bus Fare, Mode Split, and System Social Cost for Numerical Example

		Mode split
Restricted bus fare range ($)	Optimal bus fare ($)	Bus commuters	AV commuters	System social cost ($)
–5 to 15	–5	2,803	3,197	189,506
0 to 15	0	2,252	3,748	202,099

Note: AV = autonomous vehicle.

To begin to address this issue, the possible range for the bus fare could be restricted to $ $0\le k_{\mathrm{bus}}<\$15$ . The nonnegative restriction means that it is either free or there is a cost assigned to the passenger to make the trip by bus; the upper limit of less than $15 was selected to be lower than the fare for AV usage, to incentivize some commuters to choose the bus. Considering only the range of positive bus fares in Figure 2, the results show that the optimal fare was a free ($ 0) fare and the associated mode split and SSC are shown in Table 3. Once again, although this resulted in the lowest SSC, it still did not provide the transit authority with any revenue from ticket sales. As such, the same concerns are valid, and the potential solutions should once again be considered.

The incentive at which all travelers would choose the bus was also determined, although this scenario was not feasible owing to the extreme earliness and lateness required to serve all passengers by bus. This was determined by first calculating the cost of a single AV trip (i.e., $C_{\mathrm{AV}}$ when $N_{\mathrm{AV}}=1$ ) using Equation 15, then setting the cost of a bus trip, $C_{\mathrm{bus}}$ to the same value and calculating the corresponding $k_{\mathrm{bus}}$ value for $N_{\mathrm{bus}}=N$ based on Equations 18, 20, and $N_{\mathrm{bus}}=\sum n_j$ . The required subsidy/incentive for all passengers to use the bus would be $35.51.

It is worth noting that the finding of a minimum optimal bus fare resulted from the model’s analytical structure rather than its simplifying assumptions. Although the framework omits traveler heterogeneity and other modes for tractability, these simplifications do not alter the qualitative relationship between fares, congestion, and mode split. Instead, they make the theoretical mechanisms behind fare minimization more transparent, providing a foundation for future extensions using richer behavioral data.

Sensitivity Analyses

System Social Cost with Changing Bus Cost

Figure 3a shows the relationship between the cost of a bus trip, the bus fare, and the SSC for each level of bus fare. The relationship between the cost of a bus trip and the SSC appears to follow a somewhat parabolic shape, but with sharp discontinuities, likely attributed to the discontinuous nature of bus trips and a maximum bus occupancy, that causes a jump in cost when an additional trip is added. The optimal bus cost for each bus fare can be determined from the graph by the point at which the SSC is a minimum and, as explained previously, the absolute minimum occurs at the minimum bus fare.

Figure 3.

(a) System social cost as a function of bus cost for different bus fares and (b) relationship between system social cost and bus cost for $3.70 bus fare.

For the subsequent analysis the bus fare, $k_{\mathrm{bus}}$ , was set to $3.70 (comparable to the upper limit for bus fares in North American cities) and the bus trip cost, $C_{\mathrm{bus}}$ , was set to the optimal value based on the parameters in Table 2, which was $33.40 and corresponded to a mode split of 2,618 bus commuters and 3,382 AV commuters. Figure 3b shows the relationship between the SSC and the bus cost, which followed the same relationship as in Figure 3a. The optimal bus cost can be determined from the graph by the point at which the SSC was minimum ($201,477).

System Social Cost With Changing Frequency

Fixing all variables except for the frequency of the bus service and relaxing the requirement that all trips, $j,$ for $-b\le j\le a$ must have passengers, the relationship between bus frequency and SSC is shown in Figure 4. The minimum value is shown in red in Figure 4. From this graph we can see that there was an optimal bus frequency of 10 buses per hour, which minimized the SSC. This would be of interest to transit planners during the scheduling of transit routes in peak hours.

Figure 4.

Relationship between frequency of bus service and system social cost.

System Social Cost with Changing Bus Capacity, AV Fare, and AV Cost

Once again using the parameter values shown in Table 2 as well as $k_{\mathrm{bus}}=\$3.70$ and $C_{\mathrm{bus}}=\$33.40$ as before, the relationships between bus capacity and SSC, the AV fare and SSC, and unit cost of AV travel can be seen in Figure 5, a to c , respectively.

Figure 5.

Relationship between (a) bus capacity and system social cost, (b) autonomous vehicle (AV) fare and system social cost, and (c) unit cost of AV travel and system social cost.

In Figure 5a, the SSC was approximately inversely proportional to the bus capacity. This result was expected since during the morning peak period, most trips see buses filled almost to capacity, which indicates a high demand, so increasing the capacity would allow more passengers to be served. Figure 5, b and c , both show linear relationships. This indicates that as either the AV fare, $k_{\mathrm{bus}}$ , or the unit cost of travel, ${\alpha }_{\mathrm{bus}}$ , increased the SSC increased proportionally.

Trip Distribution of Commuters

Next, the commuting pattern during the peak period for a 9:00 a.m. desired arrival time is explored. Figure 6a shows the distribution of AV and bus commuters joining the bottleneck across the entire morning peak period. Although, AVs joined the bottleneck continuously, it is shown in 1-min intervals here, and bus trips only occurred every 10 min (i.e., six buses per hour as in Table 2). The first bus arrived at the bottleneck at 7:47 a.m. and the last trip happened at 9:37 a.m. The queue began at 7:51 a.m. and finally dissipated at 9:13 a.m., whereas 7:57 a.m. was the latest time a commuter could depart to arrive at work on time. Any AV commuter that departed before 7:57 a.m. arrived early; and since the cost of a late arrival is much higher than that of an early arrival, most AV commuters departed during this period at a rate of 375 commuters per minute before dropping to 15 commuters per minute until the queue fully dissipated.

Figure 6.

Trip distribution of commuters: (a) one passenger per vehicle, and (b) two passengers per vehicle.

The pattern of bus commuters was less straight-forward because of the crowding cost, which depends on the number of passengers on board. Many of the bus trips operate at near or full capacity (240 passengers, as per the capacity reported in Transport for Cape Town’s 2015 report [ 18 ]), whereas the dip in the number of bus passengers in the trips at 8:17 and 8:27 a.m. could be explained by the high cost of a late-arrival time paired with the cost of a long travel time to traverse the bottleneck. The emptier buses for the first and last bus trips were mainly explained by the cost of very early or very late arrivals.

Next, the initial assumption that all AVs only have one passenger was relaxed. Equation 12 had to be adjusted, as shown in Equation 23, because the number of AVs was no longer equal to the number of commuters using the AVs.

$$\psi =\frac{\beta \gamma }{(\beta +\gamma )}\frac{N_{\mathrm{AV}}/o_{\mathrm{AV}}}{(s-\lambda f)}$$

(23)

where $o_{\mathrm{AV}}$ is occupancy of the AV (number of commuters sharing the AV).

Further, the expression for the cost of an AV from Equation 15 also had to be adjusted, as shown in Equation 24, to account for the fare, $k_{\mathrm{AV}}$ , now being shared between AV occupants.

$$C_{\mathrm{AV}}={\alpha }_{\mathrm{AV}}{T}_f+k_{\mathrm{AV}}/o_{\mathrm{AV}}+\psi$$

(24)

Still assuming a fare of $3.70, the optimal mode split to minimize the SSC was determined. It was assumed that the unit cost of AV travel, ${\alpha }_{\mathrm{AV}}$ , increased slightly compared with the previous section, since the space in the AV was now shared, and a value of $19 per hour was used. The SSC was greatly reduced and the trip distribution, shown in Figure 6b, was significantly different from that of Figure 6a. For the higher occupancy bus trips, fewer bus trips were needed to serve all commuters and the buses were slightly less full. The morning commute period was also reduced from approximately 2 h to 1-h 15 min. Overall, more commuters used AVs than in the scenario with one passenger per vehicle, but there were fewer AVs on the road.

Conclusion

This paper looked at congestion on a bottleneck system with two modes: AVs and buses. These modes were chosen for this study because they represent a possible reality, where transit systems are more well-developed and AVs are deployed. This paper determined the optimal bus fare and mode split that minimized the SSC on a congested bottleneck shared between AVs and buses. It also explored the influence of bus capacity and frequency on the SSC of the system. Additionally, the study considered space sharing between the two modes, the disutility cost of riding in a crowded bus, and the heterogeneity between commuters of each mode, which have often been overlooked in prior studies.

Owing to the complex relationship between terms in the objective function, numerical techniques were required for optimization. In the numerical example provided, the optimal fare was the minimum of whatever range it was restricted to, and was associated with lower bus trip costs and fewer passengers. Later, by fixing the bus fare, an approximately parabolic relationship between the bus trip cost and SSC was observed. The plot of the bus dispatching frequency and SSC also demonstrated a vaguely parabolic shape, whereas the plot of capacity and SSC indicated an inversely proportional relationship. Because of the careful balance between the travel costs, early/late-arrival costs, AV fares, bus fares, and crowding costs embedded into the objective function, the choice of parameters can significantly influence the results and should be selected carefully to accurately represent the system.

A possible extension of this work could be to formulate a time-dependent fare for AV commuters, similar to taxis or other ridesharing services, where they charge per minute or kilometer in addition to a flat rate for usage. It would also be interesting to do a similar analysis on the frequency and capacity of buses by optimizing the objective without fixing the bus fare and trip cost. Another extension could be restricting the flexibility of commuters’ arrival times by assigning nonlinear penalties, which is very realistic in a setting where employees are expected to start work within a few minutes of their scheduled work hours. Finally, removing the assumption that all passengers start and end their trip at the same location would also provide more realistic results with respect to both bus and AV passengers.

In conclusion, although several limitations should be addressed before the methodology and findings are used in practice, the process explained in this paper will be able to assist traffic and transit authorities and, ultimately, the commuters. The modal split can help traffic authorities forecast future trends, allocate an appropriately sized fleet of AVs to the network, and predict congestion by providing an estimate of AV demand compared with the system capacity. Determining the optimal bus fare can help transit authorities with budgetary decisions, aid in providing revenue necessary for operations, and attract riders. Overall, an optimized system would see reduced congestion and delay, benefit commuters, and reduce energy consumption from idling vehicles.