Open AccessArticle

A Predictive Compact Model of Effective Travel Time Considering the Implementation of First-Mile Autonomous Mini-Buses in Smart Suburbs

Department of Software Science, Tallinn University of Technology, 12618 Tallinn, Estonia

Department of Mechanical and Industrial Engineering, Tallinn University of Technology, 19086 Tallinn, Estonia

FinEst Centre for Smart Cities, Tallinn University of Technology, 19086 Tallinn, Estonia

Author to whom correspondence should be addressed.

Smart Cities 2024, 7(6), 3914-3935; https://doi.org/10.3390/smartcities7060151

Submission received: 4 October 2024 / Revised: 4 December 2024 / Accepted: 6 December 2024 / Published: 11 December 2024

(This article belongs to the Special Issue Cost-Effective Transportation Planning for Smart Cities)

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Figure 1
Growth of annual number of publications dedicated to application of autonomous vehicles in future transportation. "> Figure 2
General structure of the calculation model. The upper corner numbers of the blocks correspond to the subsections in the paper text. "> Figure 3
Explanation of example suburban transport task: (a) Location of Järveküla residential area (purple rectangle) in Rae municipality beyond the southern border of Tallinn city (red line). The blue line marks the major public transportation bus line 132 to Tallinn center; (b) Current development stage of Järveküla residential area of approx. 200 houses; (c) Pilot AV shuttle minibus designed for first-mile transport service in residential area. "> Figure 4
Selection of two reference areas within the city limits of Tallinn (Mõigu and Kakumäe-Tiskre), for which the trip length distribution functions were found. "> Figure 5
Summary of trip distance statistics of daily outbound trips for two example residential areas of Tallinn city on basis of the synthetic population database of Tallinn: (a1) Differential distributions with 1 km step for Mõigu area; (b1) The integrated cumulative distributions for Mõigu area; (a2) Differential distributions with 1 km step for Kakumäe-Tiskre area; (b2) The integrated cumulative distributions for Kakumäe-Tiskre area. "> Figure 6
Results of RMS-fitting of the statistics of forenoon outbound trips by the 2-parameter sigmoid curves for Kakumäe-Tiskre and Mõigu districts. "> Figure 7
The constructed 3-parameter model of distribution of trip distances combining the initial short-distance contribution and the smooth sigmoid step for lengthier distances. Parameter values are estimated to represent the Järveküla example area. "> Figure 8
One-dimensional distances-based spatial model of transportation task: (a) an abstract map of residential area housing with local institutions, transport artery, and public transport stops on one edge; (b) distances-based concept of destination districts in metropolitan areas; (c) the simplified one-dimensional spatial scheme of origin zones and destination districts. "> Figure 9
Explanation of concept of three-dimensional modality-origin-destination matrix used to sum up the daily transport times. Matrix defines 5 origin zones, 6 destinations districts, and 5 + 2 transportation modes. Each cell of MOD matrix is characterized by transport time with optional psych-physiological and economical extra terms and weight factors of distance and transport mode. "> Figure 10
Explanation of the two-stage concept of outbound trips and input parameter set for calculation of effective transportation time costs. "> Figure 11
Explanation of 2-stage effective trip times methodology with actual numerical values of input parameters. "> Figure 12
The main output of the model: daily effective transportation times of an average suburban resident versus the aggregated parameter of autonomous vehicle acceptance <math display="inline"><semantics> <mi>α</mi> </semantics></math>. ">

Versions Notes

Abstract

Highlights

What are the main findings?

A general mathematical methodology and calculation model have been proposed, which allow taking into account most of the important factors that determine the impact of the introduction of first-mile autonomous vehicles on the daily time use of suburban residents.
Following the compact modelling approach, an easily understandable and definable set of source data with a minimum volume has been proposed, which allows to reach the desired result.

What is the implication of the main finding?

A practical tool for shaping transport solutions and local government decisions: Thanks to the transparency of the model and the easy-to-understand input data, the transport planners and local governments can perform estimation calculations without long and complex scientific studies.
Predictive capability of the model: a minimalistic and easy-to-understand input data set enables preliminary assessments of the implementation of autonomous vehicles for various local governments and suburbs located near metropolitan centres.

Abstract

An important development task for the suburbs of smart cities is the transition from rigid and economically inefficient public transport to the flexible order-based service with autonomous vehicles. The article proposes a compact model with a minimal input data set to estimate the effective daily travel time (EDTT) of an average resident of a suburban area considering the availability of the first-mile autonomous vehicles (AVs). Our example case is the Järveküla residential area beyond the Tallinn city border. In the model, the transport times of the whole day are estimated on the basis of the forenoon outbound trips. The one-dimensional distance-based spatial model with 5 residential origin zones and 6 destination districts in the city is applied. A crucial simplification is the 3-parameter sub-model of the distribution of distances on the basis of the real mobility statistics. Effective travel times, optionally completed with psycho-physiological stress factors and psychologically perceived financial costs, are calculated for all distances and transportation modes using the characteristic speeds of each mode of transport. A sub-model of switching from 5 traditional transport modes to two AV-assisted modes is defined by an aggregated AV acceptance parameter ‘a’ based on resident surveys. The main output of the model is the EDTT, dependent on the value of the parameter a. Thanks to the compact and easily adjustable set of input data, the main values of the presented model are its generalizability, predictive ability, and transferability to other similar suburban use cases.

Keywords:

autonomous vehicle acceptance; autonomous shuttle vehicle; compact model; distance distribution function; effective travel time; first-mile transport; psycho-physiological stress factor; smart city suburb; travel time model

1. Introduction

Transportation planning is an engineering field with a very long history [1]. In the past 10–15 years, autonomous vehicles (AVs) have acquired a game-changing role in the visions of future transport development [2,3,4,5,6]. Figure 1 illustrates the rapid growth in the number of scientific publications that discuss the application of AVs in solving future transportation issues.

To solve the ever-increasing environmental problems and parking problems in city centers, privately owned electric AVs can provide solutions for both the car owners and the cities. However, real high-value win-win solutions to complex socio-economic tasks are still expected from shared AVs and especially from autonomous shuttle (AS) minibuses [7,8,9]. To achieve the biggest positive effect for both the residents and the municipalities, the AS buses are brought into service to solve the first-mile and last-mile transport tasks in sparsely populated suburban areas where the maintenance of public transport bus lines is not economically feasible [7,9,10]. In this paper, we construct a model and estimate numerically the impact of the implementation of autonomous first/last-mile minibuses on the time usage of the average resident of a suburban area who has to make both local trips and longer trips to the adjacent city every day.

Transport analysis and planning is a complex multidisciplinary socio-techno-economical problem [11,12,13]. For example, the psychological pre-setting of people to switch to new modes of transport is an essential key factor to be considered [14,15,16]. The corresponding computational models tend to become very multi-dimensional and multi-level, with several feedback loops and thereby with a very large amount of heterogeneous input data. The latter fact often makes it difficult to obtain practically usable computational tools to solve actual transport planning tasks.

To provide a solid framework for versatile transport tasks, the concepts of travel time savings (TTS) and multi-functional value of travel times savings (VTTS) have been introduced and widely applied, e.g., [17,18,19], In recent years, a significant amount of research has addressed the potential of AVs to reduce travel time with the implementation either in the form of private AVs or shared AVs [17,18,19,20]. However, most of these studies consider the potential applications of AVs for longer trips than just suburban regions, e.g., [21,22,23]. Although time savings from the application of AVs in suburban first/last-mile transportation problems have also been touched on in a few works, e.g., [3,8,20,24]. Following the central idea of transport tasks that travel time in some form should be used as the summarizing complex output parameter, in the present study, the effective daily travel time (EDTT) of an average suburban resident is introduced. As the advanced feature of the proposed EDTT formulation, we have here accounted within EDTT the psycho-physiological travel stress

φ

-factors and the financial cost perception

ψ

-factors that account for extra time saved by the travelers using the faster travel modes like car or taxi.

One general problem in transport research is large volumes of input data, which makes the models less transparent and tied to specific cases, thus reducing the generalizability and predictive ability of the models. In this study, in contrast, we have created a maximally compact EDTT model with a minimalist input data set that still describes all the important mechanisms that determine the positive and negative effects of implementing AVs. In other words, in this work we have tried to follow Einstein’s principle of simplicity—make the model or theory as simple as possible, but not simpler. It is expected that by keeping the set of initial data minimal and easy to understand, but at the same time reserving the possibility to detail several sub-models (e.g., sub-models for AV waiting times and for people’s readiness to switch from traditional transport modalities to AV-enabled modes), it is possible to propose a practical and sufficiently universal calculation model that allows proactively evaluating the beneficial effect of minibus AVs for different suburbs.

In summary, the advanced features of the proposed travel time model include:

Reasonably wide set of considered transportation modes (5 traditional and 2 AV-assisted);
Definition of a simplified but rather general one-dimensional spatial layout (5 residential zones and 6 city destination zones);
Using relatively well-defined statistics of pre-lunch outbound trips to forecast the daily trips;
Introduction of an aggregated AV acceptance parameter $α$ as the key input variable of the model;
Introduction of a practical 3-parameter distribution function for trip distances;
Introduction of a 2-stage trip scheme (local and city), allowing for the description of the different combined modes of transport;
Introduction of a minimalistic 2-parameter AV wait time model including dependence on the number of AVs in the service area;
Expressing the main output of the model via easily understandable and testable daily travel time of the average resident;
Expanding the concept of travel times with additional perceived extra time terms to take into account psycho-physiological stress and financial costs.

In the voluminous transport research literature, a partly similar approach to effective time components of main transport modes is discussed in [22,25,26]. People’s attitudes towards AVs (which we describe with one aggregated parameter) have been studied in great detail in many works, e.g., [6,14,15,16,23,27]. The important special question of AV waiting times that we describe in present research with a minimalistic 2-parameter model has been studied with a high degree of detail (for example, by the means of agent-based modeling) in several works [4,9].

2. Description of the Model

2.1. Concept, Sub-Models and Approximations

To arrive at an estimate of average daily transport time, it is necessary to look at a sufficiently adequate set of trip length options with statistical weighting for each option. In doing so, it is mandatory to add a weighted summation across modes of transport. In addition, if sufficient data are available, it would be desirable to add weighted averaging across population groups. With this, we arrive at the estimate that the imaginable 4-dimensional matrix of variants (trip origin zones, trip destination districts, mode of transport, population groups) has the size order of a thousand cells, all of which require input data. This is even if we apply averaging over hours (i.e., omit the 5th dimension of hours). The conclusion is that many simplifications are needed for a practically usable transport time model. And especially for preliminary assessments, when there is still little data from measurements and surveys.

In order to arrive at a practically usable calculation model for predictive estimates with a reasonably compact set of initial data, in this study the following simplifications have been introduced:

Detailed distribution over daily hours is omitted. An exception is the preparation of the distribution functions of travel distances for pre-lunch and all-day outbound trips based on the actual time-dependent statistics of the movement of the Tallinn city residents. In the final formulation of the EDTT model, all-day trips are calculated using pre-lunch outbound trips (6:00–12:00) with an empirical factor.
The spatial situation is reduced to a one-dimensional distance-based spatial model. The travel times are calculated by using estimated average speeds of different transport modes that add only a few input parameters to the model.
The differentiation by population groups is only implicit in order to avoid an additional dimension in the summation scheme. The calculation of groups of residents is indirectly included in the factor of non-moving residents N and in the weight parameter of local trips $D_{0}$ (mainly students in local schools).
To model the transition from traditional transport modes to AV-assisted modes, an aggregated single parameter of AV acceptance $α$ , based on the population survey summary, is introduced. If necessary, the relevant sub-model can be refined with additional studies that take into account the attitudes of residents using different modes of transport in more detail.
In order to estimate the waiting time for ordering AVs, a simple 2-parameter empirical sub-model with minimal complexity that accounts for the number of available AVs is used in the present study. With additional measurements in a real-world situation and with agent-based modeling, this sub-model can be relatively easily refined.

On the other hand, the task of assessment of AV-supported local transport has also required the introduction of several complexities, such as:

Formal framework of summation of trip times should be presented not on the basis of traditional two-dimensional Origin-Destination matrices but based on three-dimensional Modality-Origin-Destination (MOD) matrix, the size of which in the present study is 7 × 5 × 6. A relatively wide set of transportation modes (5 traditional and 2 AV-supported) has been necessary to assess the impact of AVs.
Introduction of 2 stages of outbound trips—local and remote. This is necessary complexity to account for residential use of AVs in combination with public transport (PT). Transfer occurs in origin zone $Z 0$ containing the transportation artery, along which the PT stops are located (see Figure 2 below).
Optional psycho-physiological stress $φ$ -factors to account for the increase in perceived travel time due to stress of private car driving during rush hours, bicycling in bad weather, walking difficulties, and walking with heavy hand luggage (AV using reasons indicated by residents in the surveys).
Adding waiting times to driving times for all modes of transport and for both stages of trips.
Accounting for optional financial cost terms for private cars, taxis, and PT $E_{c a r}, E_{t a x i} .$ $E_{b u s}$ converted to travel time on the basis of the national average hourly wage $A_{h}$ . In general, the cost of using AVs should be taken into account as well (currently zero in the case of the discussed pilot project in Rae municipality).
Accounting for optional perceived time saving $ψ$ -factors that take into account the time gains when using the faster modes of transport (private car and taxi).

Figure 2 summarizes the general structure of the proposed computational model and indicates the numbers of sections of the paper text that describe the construction of different sub-models.

2.2. Example Use Case

In 2022, a pilot project of future transport was initiated by the Finest Centre of Smart Cities at Tallinn University of Technology [28,29,30,31] in order to test the applicability of ISEAUTO autonomous minibuses on the first-/last-mile section connecting the suburban residential region Järveküla with Tallinn Harbour (see Figure 3a below). The real spatial situation of the task and initial version of the developed autonomous minibuses are illustrated in Figure 3 below.

The problem of the recently established suburban area of private houses (approximately 200 houses and 1000 inhabitants per half square kilometer will be expanded) is the excessive use of private cars (over 80%). Although the district is closer to the center of Tallinn than some old suburbs, the residents do not want to give up private cars due to the excessive travel times spent, especially on the first kilometers of trips. At the same time, the local government and bus companies, in turn, cannot tighten the bus connection because of too much car use and because of too sparse a population. The application goal of the model is to investigate how much the local self-driving shuttle buses can save the time spent on transportation by residents.

2.3. Sub-Model of Trip Distance Distribution

In order to perform weighted averaging of transport times over journey lengths, it is necessary to know the statistical distribution of daily journeys over distances [32,33,34,35]. In the case of a very detailed analysis, it would also be necessary to know these distributions for different modes of transport and population groups. Considering the need to minimize the amount of input data, in our use case, we proceed from the assumption that the same distance distribution function applies to all groups of residents and to all modes of transport oriented to longer distances (private car, taxi, public transport bus).

Due to the lack of detailed trip length data for the suburban residential area of Järveküla, in this study, we have applied the hypothesis of the similarity of the trip length distribution functions with the remote private housing districts within Tallinn city borders. Reliable distribution functions can be constructed on the basis of the available detailed database of the daily movements of the Tallinn synthetic population database [36]. Figure 4 presents the two depicted example areas within Tallinn city limits (Mõigu and Kakumäe-Tiskre), which could be similar to the residential area of Järveküla (red rectangle in Figure 4) in terms of percentage of private housing and residents’ movement profiles.

Figure 5 summarizes the daily outbound trip statistics for the mentioned two example districts within Tallinn limits. In relation to the given statistics, it can be noted that the mobility statistics of the Mõigu region are based on an overly small number of trips, which is reflected in large fluctuations in the differential distribution. In turn, in the case of Kakumäe-Tiskre, there may be a noticeable difference observed in the behavior of the residents in the morning and throughout the day. For both regions, a general conclusion can be made that the integral distribution functions are relatively smooth and similar and can be sufficiently well described by some basic parameters such as the median path length and the characteristic path length of the decline in the share of longer trips.

Results in Figure 5 demonstrate that while the differential distributions are very uneven, the normalized integral curves that grow from 0 to 100 percent demonstrate similar smooth behavior of morning movement from two depicted example areas. This kind of smooth step curve may be approximated mathematically by sigmoid curves, for example, by the logistic function [37]:

c (x) = \frac{1}{1 + exp (- \frac{x - x_{c}}{L_{t}})}

(1)

where

x_{c}

is the median distance and

L_{t a i l}

is the tail abruptness parameter that represents the characteristic length of exponential approaching of the final 100% level.

Figure 6 shows the result of RMS (Root Mean Square) fitting of statistics of morning outbound trips of Figure 5 by the 2-parameter sigmoid curves (1) for districts of Mõigu and Kakumäe-Tiskre. The results show that the median path length of established morning journeys for Tallinn residents is between 10 and 13 km, and the willingness to choose more distant places of work and study decreases exponentially with a characteristic path length of 3.4 km. The greater median distance

x_{c}

for the Mõigu district may be explained by the airport, which separates the settlement of Mõigu further from schools, shopping centers and workplaces.

However, it must be mentioned that the analysis of the two-parameter sigmoid Model (1) shows that the description of short distances is not adequate. The special feature of the Järveküla sample area is that it has a young population and two nearby schools within a 2–3 km radius, where many pupils (estimated 20% of the population) move every morning. Thus, from a modeling point of view, it makes sense to add a third adjustable empirical parameter that describes the morning movement of residents to local schools and institutions. A modified 3-parameter sigmoid model that includes a contribution factor of local trips

D_{0}

and subsequent growth from level

D_{0}

to 1 by the smooth sigmoid step may be specified by the following equations:

c (x) = C_{0} + \frac{1 - C_{0}}{1 + exp (- \frac{x - x_{c}}{L_{t}})}

(2)

and

C_{0} = D_{0} + \frac{1 - D_{0}}{1 + exp (\frac{x_{c}}{L_{t}})} .

(3)

Figure 7 summarizes the constructed 3-parameter distribution function used below in the present study to describe both the contribution of local trips and the distribution of lengthier trips over longer distances provided by private cars, taxis, and public transport buses.

2.4. One-Dimensional Spatial Situation Model

In order to model the potential positive impact of AVs on the time use of suburban residents, it is most important to consider those remote neighborhoods that are located more than 1 km away from transport arteries. Secondly, in order to adequately take into account the transport times of moving to different parts of a larger city, the city must also be divided into regions by distance. In addition to this, if a generalizable model is the goal, the scheme of source zones and destination districts should be left one-dimensional to allow calculation of transport times from distances and estimated speeds. Figure 8 describes the constructed one-dimensional spatial situation model for calculation of morning outbound trips from 5 origin zones in residential areas to 6 destination districts, both in the local municipality and in the larger city area.

2.5. Initial Usage of Transportation Modes

In connection with the pilot project for the implementation of AVs in the municipality of Rae [30], a survey was conducted in 2021 [38], in which 819 residents gave answers about their present usage of different transport modalities in winter and in summer. The summary results of this survey, as well as the deduced input parameters for the model of the present work, are explained in Table 1 below.

2.6. Acceptance of Autonomous Vehicles

User adoption of AVs has been the subject of numerous in-depth studies, e.g., [6,14,15,16,19,27,28,39]. For the modeling task discussed in this work, it is important to construct a sub-model with a reasonable level of complexity that describes the transition of residents from five conventional modes of transport (see Table 1) to new AV-supported modes of transport. For the construction of the mentioned sub-model, Table 2 summarizes the residents’ responses regarding the possible use of AVs. The answers of 819 residents are collected from the Supplementary Materials [38].

The data in the table confirm the high readiness of nearly 50% of the population to use AVs. In order to realize a maximally compact calculation model for pre-assessment of the impact of AVs, here is proposed a single aggregated parameter

α

to account for the transition of people to AVs. With the addition of more detailed statistical data, it is possible to refine the treatment of this sub-model of AV acceptance.

2.7. Framework of Three-Dimensional Modality-Origin-Destination Matrix

The application of two-dimensional origin-destination (OD) matrices is a well-known methodology in transportation research; see, for example, [40]. In the present study, it is necessary to add a third dimension of transport modality to account for changes in proportions between transport modes due to the availability of AVs. The formal three-dimensional framework of the modality-origin-destination matrix is explained by Figure 9 below.

In the proposed MOD matrix, the following denotation of cell indexes may be agreed upon:

m—index of transportation mode (1–7);
z—index of zone of origin (0–4);
d—index of destination district (0–6).

In the MOD matrix framework, the next central issue of the EDTT model is to calculate the effective travel times

t_{m z d}

for all cells of the MOD matrix. As already commented in Section 2.1, the possible fourth dimension of the general task associated with population groups was here excluded to keep the model compact. If desired, the consideration of population groups can be added to the calculation of

t_{m z d}

values for matrix cells with a higher weight. Such are, for example, the cells corresponding to the use of a private car at medium distances (m = 1 and d = 3 or 4).

2.8. Central Summation Formula over Modes, Zones and Districts

In the three-dimensional mode-origin-destination framework of the transport times task, the central summation formula may be written in the following form:

T = (1 - N) H R \sum_{m = 1}^{7} M_{m} \sum_{z = 0}^{n_{z}} Z_{z} \sum_{d = 0}^{n_{d}} D_{m d} t_{m z d}

(4)

where

T is the average daily travel time of an average resident,
N is the fraction of non-moving residents before noon (e.g., small children),
H is the ratio of full-day outbound trips to forenoon outbound trips (≈1.6, see Figure 5),
R is the ratio of all sections of outbound and return trips together to outbound trips (≈2),
m is the index of the transportation mode (values 1–7, see Figure 9),
$M_{m}$ is the statistical weight of the transport mode m (see Section 2.9 below),
z is the index of the zone of the origin (values 0–4, see Figure 9),
$n_{z}$ is the maximal index of the zone of the origin (4 in present study, see Figure 8c),
$Z_{z}$ is the statistical weight of the zone of the origin ( $1 / (n_{z} + 1) = 0.2$ in present study),
d is the index of the district of the destination (values 0–6, see Figure 9),
$n_{d}$ is the maximal index of the district of the destination (6 in present study, see Figure 8c),
$D_{m d}$ is the (mode) m-dependent statistical weight of district d (see Section 2.10 below), and
$t_{m z d}$ is the effective trip time for mode m, zone z, and district d (see Section 2.11 below).

It should be noted that several simplifying assumptions have been used in Formulation (4), such as the assumption that the statistical weight of transport modes is independent of the origin zone and destination district. The practical value of the Model (4) is that it explicitly provides the average daily transportation time of an average municipality resident, and the dependence on the number of residents is only indirect (first of all via sub-model of AV waiting times).

2.9. Sub-Model of Transportation Mode Weights

In Figure 9, the concept of transportation modes was presented—a set of 5 conventional modes completed with 2 local AV-assisted modes to support the first-mile mobility of residents. An AV-dependent sub-model of transport mode weights

M_{m}

is needed to describe the transition from the conventional usage proportions described in Table 1 to the expected AV-supported proportions. Very detailed models can be found in the literature that examine the readiness to switch to AVs by population groups, gender, and other indicators (for example, [41]). Here, in order to arrive at reasonable preliminary estimates in a situation of limited information, a practical compact modeling approach could be to link the transition with the aggregated parameter

α

of AV acceptance (see Table 2). Considering this, a simple approach to describing the expected decrease in weights of conventional transport modes may be approximated by the following formulas:

M_{1} = (1 - α) m_{c a r},

(5)

M_{2} = (1 - α) m_{t a x i},

(6)

M_{3} = (1 - α) m_{P T},

(7)

M_{4} = (1 - α) m_{b i c},

(8)

M_{5} = (1 - α) m_{w a l k}

(9)

where weights of conventional modes

m_{c a r}, m_{t a x i}, m_{P T}, m_{b i c}, m_{w a l k}

are defined and evaluated in Table 1.

Formulas (5)–(9) are based on the assumptions that all 5 traditional transport modes lose users equally according to the parameter

α

. Here, of course, the dominant mode of transport is the private car, the users of which showed 45.5% interest in the use of AVs in the survey (see Table 2), which justifies linking the mode change with the general parameter

α

. Formulas (10) and (11) describe the transfer of residents from the five traditional modes of transport to AV-supported modes 6 and 7.

Modeling the growth in the use of new AV-enabled transport modes 6 and 7 requires an assessment of the proportion between the local AV transport (mode 7) and combined mode of AV with urban buses (mode 6). The simplest modeling approach could assume 50:50 division, but a more realistic proportion could be stated on the basis of weight parameter

D_{0}

of local trips of the trip distance Models (2) and (3):

M_{6} = (1 - D_{0}) α (m_{c a r} + m_{t a x i} + m_{P T} + m_{b i c} + m_{w a l k}),

(10)

M_{7} = D_{0} α (m_{c a r} + m_{t a x i} + m_{P T} + m_{b i c} + m_{w a l k}) .

(11)

The constructed sub-model (5)–(11) is based on strong simplifying assumptions that are justified to obtain preliminary estimates. More focused questionnaires and the collection of real-world usage data by AVs after their introduction will enable significant refinement of this sub-model. It should be noted that the approach proposed here is based on the conservative assumptions that the total volume of trips and the profile of distances will remain the same after the introduction of AVs. Taking into account the corresponding changes is possible with an additional feedback loop in model when additional data on changes in people’s behavioral profiles become available.

2.10. Sub-Model of Weight of Distances

Assuming an even population distribution in the residential area, it is possible to determine the statistical weight coefficients of the origin zones

Z_{z}

in the main summation Formula (4) simply as the inverse of the number of zones

n_{z} + 1

Z_{z} = 1 / (n_{z} + 1) .

(12)

The sub-model of statistical weights of destination districts

D_{m d}

in Formula (4) may be specified on the basis of the distribution function of trip lengths (2) for the transport modes that cover larger distances in the city by private car, taxi, or PT buses. Local transport modes (bicycle, walking, local AV), whose final regions are

d = 0, 1

or 2 (see Figure 9), can be described in a simpler way without applying a distribution function. With this sub-model, it has been assumed that there is no dependence on the origin zone z, as the zone lengths

L_{z}

are remarkably smaller than distances to destination districts in city

d L_{d}

In summary, the weight coefficients

D_{m d}

of the destination districts in cases of long-range transport modes

m = 1, 2, 3, 6

may be calculated on the basis of distribution function (2) by the following formulas:

D_{m d} = c (d L_{d}) - c ((d - 1) L_{d}) for 0 < d < n_{z},

(13)

D_{m d} = 1 - c ((d - 1) L_{d}) for d = n_{z},

(14)

D_{m d} = D_{0} for d = 0,

(15)

In cases of short-range bicycle mode m = 4, the equal weight of close districts

d =

0, 1 and 2 may be assumed:

D_{m d} = 1 / 3 for d \leq 2,

(16)

D_{m d} = 0 for d > 2 .

(17)

In the case of short-range walking mode m = 5 and local AV mode m = 7, the equal weight of the remaining two local districts

d =

0 and 1 in the same municipality may be assumed:

D_{m d} = 1 / 2 for d \leq 1,

(18)

D_{m d} = 0 for d > 1 .

(19)

It can be noted that since the use of bicycles and walking has a small share (4.2% and 5.1%, respectively, see Table 1), the accuracy of modeling these modes of movement is not critical in the present use case.

2.11. Sub-Model of Transport Times

In addition to the formulation of sub-models for weight factors for modes

M_{m}

, zones

Z_{z}

, and districts

D_{m d}

in the general summation Formula (4), the key question of modeling is the estimation of travel times

t_{m z d}

for every transportation mode m, origin zone z and destination district d. An important aspect that follows from the logic of transportation modes that rely on public transport is that the modeling of trips is reasonable to divide into two stages: stage 1 for local movement towards local transportation artery zone

Z_{0} = D_{0}

and stage 2 for later movement to destination districts

D_{0}

–

D_{6}

. This formal scheme includes a special case of local destination district d = 0 that corresponds to zero distance of stage 2.

A significant reduction of the amount of input data can be achieved via the calculation of travel times on the basis of average speed estimates for cars (private and taxi), public transport buses, AVs, bicycles, and human walking. Figure 10 explains the 2-stage concept and the introduced set of waiting and driving time parameters with optional psycho-physiological [42] and economical additional parameters for the calculation of effective transport time costs.

Based on the one-dimensional spatial structure of the task in Figure 8c, the trip lengths for stages 1 and 2 are calculated as follows:

l_{1} = z L_{z} + l_{e n d w a l k},

(20)

l_{2} = (d - 1 / 2) L_{d} for d > 1,

(21)

l_{2} = 0 for d = 0

(22)

where

L_{z}

and

L_{d}

denote the size of origin zones and destination districts, respectively, and the nonzero

l_{e n d w a l k}

term describes the walk from the PT end stop to the destination in the case of modes 3 and 6, which include public transportation.

The presented 2-stage concept of trip distances makes it possible to construct a universal sub-model for effective transport times with the inclusion of optional terms of psycho-physiological stress factors and psychologically perceived financial costs:

t_{m z d} = w_{1} + \frac{l_{1} (1 + φ_{1})}{v_{1}} + \frac{E_{1} l_{1} ψ_{1}}{A_{h}} + w_{2} + \frac{l_{2} (1 + φ_{2})}{v_{2}} + \frac{E_{2} l_{2} ψ_{2}}{A_{h}}

(23)

where

$w_{1}, w_{2}$ are the transport waiting times for stages 1 and 2,
$l_{1}, l_{2}$ are the travel distances for stages 1 and 2,
$v_{1}, v_{2}$ are the estimated travel speeds for stages 1 and 2,
$φ_{1}, φ_{2}$ are the psycho-physiological stress factors for stages 1 and 2,
$E_{1}, E_{2}$ are the financial costs per distance unit for stages 1 and 2,
$ψ_{1}, ψ_{2}$ are the psychological factors of perception of the financial costs for stages 1 and 2,
$A_{h}$ is the national average hourly wage.

Figure 11 illustrates the 2-stage effective trip time concept with actual values of parameters for the present use case. Denotations have the following meanings:

φ

—psycho-physiological stress factor;

ψ E

—financial cost with psychological perception factor of time usage;

n_{A V}

—number of AV shuttles in residential service area.

2.12. Sub-Model of AV Waiting Time

When implementing AVs as first-mile vehicles in a residential area, the waiting time

w_{1}

can become the key efficiency parameter. This parameter primarily depends on the number of AVs in the service area

n_{A} V

and on the spatial extent of the service area, as well as on the number of residents who want to use AVs simultaneously in the morning hours, the number of seats in AV shuttles, the quality of the online system of ordering AVs etc. It is possible to develop detailed, sophisticated models to investigate this problem, e.g., [4,9]. In conditions where input data is not yet sufficient or the goal is to obtain preliminary estimates for the impact of AVs, it is practical to use the methodology of a compact empirical model based on a minimum number of easily estimated input parameters in the present study. In the sample task of this work, we have used a maximally simplified 2-parameter model, which still contains a significant dependence on the number of AVs:

w_{A V} = \frac{w_{A V 1}}{n_{A V}}

(24)

where

$w_{A V 1}$ is the average AV waiting time in the case of one AV in service area,
$n_{A V}$ is the number of AVs in the service area.

A possible estimate for the parameter

w_{A V 1}

is 6 min, which corresponds to the arrival of an AV from a distance of 2 km at a speed of 20 km/h.

3. Summary of Input Data

Transport studies usually deal with large amounts of data, due to which the studies become tied to a specific case, and the models are not easily transferable to other cases. In this study, the goal has been to create the most compact model possible, which would be able to describe the effect of introducing AVs in solving first-mile transport problems, not only in one sample case but also more generally. In doing so, it is important to design such a set of input data that would be easy to understand and easily specified. The complete list of primary input data.

The complete list of input parameters of the compact model proposed in this study is summarized below in Table 3. The following Table 4 presents the cross-use of these input parameters for trip stages 1 and 2 in the case of all 7 modes of transport.

4. Simulation Results

The simulation results of Models (2)–(24) with input data values listed in Table 3 and Table 4 are discussed below. Figure 3 illustrates the spatial situation of the example task of the suburban area. Since the central summation Formula (4) of the model is defined using weight coefficients and transport times of MOD matrix cells, the model directly provides the average daily transport time of an average suburban resident. The defined factor

α

of acceptance of AVs, describing the reorientation of residents from traditional modes of transport to new AV-supported modes according to the sub-model (5)–(11), allows all results to be presented in a generalized form depending on this parameter.

Below in Figure 12 are summarized the main results of the daily transport times model for 3 levels of complexity of inclusion of different transport times costs:

Only waiting and driving time terms of sub-model (23) are included;
Psycho-physiological stress $φ$ -factors added;
Psycho-physiological stress factors and financial costs with perception $ψ$ -factors added.

The results from Figure 12 show, firstly, that the obtained people’s daily transport times are of the order of 50 min, which are very realistic estimates that confirm the basic adequacy of the model. The easy-to-verify model output in the form of the averaged transport time provides good opportunities for further refinements of the model when more accurate monitoring data are received. If considering only the wait and pure driving times, then the positive effect of the reduction in time consumption predicted from the implementation of AVs is only in the order of a few percent, which can be justified by the fact that the use of AVs together with public transport buses remains slow compared to private cars and taxis. However, the model opens new possibilities for municipality officials to estimate, for example, improvements from the increasing frequency of public transport buses.

Next, if psycho-physiological stress factors are included in the study, the picture changes significantly more in favor of AVs. A nearly 19% reduction effect of AVs on the effective transportation time is obtained. This is explained by the fact that the local autonomous shuttle buses can significantly reduce the needed amount of first-mile walking in the residential area. Also, using public transport buses does not cause psycho-physiological stress, but with private cars, a 25 percent increase in perceived driving time is estimated due to psycho-physiological stress (see parameter

φ_{c a r}

in Table 3). Note that it is relatively easy to obtain more accurate estimates of the impact of factors like psycho-physiological stress, financial costs and perception of time gains due to faster transportation with additional focused surveys among suburban residents after the implementation of first AVs.

Furthermore, the results from Figure 12 show that including in the calculation the

ψ E

-terms of perceived financial cost increases the positive effect of implementing AVs, but not as much as adding the

φ

-terms of psycho-physiological stress to the pure waiting and driving times.

Including the

ψ E

-terms of perceived financial cost in the calculation further increases the positive effect of implementing AVs but not as much as adding the

φ

-terms of psycho-physiological stress to the pure wait and driving times.

5. Discussion

In this work, a practical, complete model of transport times affected by AVs is proposed, which comprehensively allows model users to reach from a set of transparently defined input data with minimal necessary volume to a concise output, which is the effective daily travel time EDTT of an average suburban resident. The model takes into account, either precisely or simplified, most of the important mechanisms that shape the impact of AVs in the role of first-mile vehicles.It should be emphasized that, following the compact modeling methodology, in order to preserve the model’s predictive power and generality, the inclusion of overly detailed sub-models has been deliberately avoided, and only the most important aspects that affect EDTT have been taken into account. Thus, the paper focuses on the formulation of the mathematical computational system rather than the details of the system components (submodels), for which many high-level detailed publications are available.

The newly introduced moments, which have been used in this work to reach the desired final results, are as follows:

Forenoon outbound trips are taken as a basis for evaluating the movements of the whole day;
A sufficiently complete set of 5 traditional transport modes has been considered;
An aggregated parameter $α$ is introduced to characterize people’s willingness to adopt AV;
A simplified sub-model is proposed to describe the transition of people from 5 traditional modes of transport to the extended set of 7 modes augmented by AVs;
For a compact description of trip lengths, a one-dimensional spatial model with origin zones and destination districts is applied;
To describe combined movements such as local AVs combined with PT buses, a 2-stage trip description is introduced;
A 3-parameter empirical distribution function of trip distances is constructed on the basis of real mobility statistics;
A provisional AV wait time sub-model that includes practically important dependence on the number of AVs is proposed;
The psycho-physiological stress factors of different transportation modes are introduced;
A methodology for converting kilometer prices into effective transport times based on the country’s average hourly wage has been proposed;
Perception factors of financial cost are introduced to describe people’s time gain in faster modes of transport.

Based on the set of the input data, the present work is partially similar to the studies [22,25,26], where characteristic parameters of transport modes have been used and people’s attitudes towards different types of AVs have been studied.

Although this work has been devoted to the formulation of the mathematical calculation scheme, leaving aside the details of the sub-models, it makes sense here to comment on the applicability of the model for the analysis of some current special issues of smart city traffic management.

Positive effect of improved crowdsourcing [43]. In any case, the deployment of on-demand AVs requires a cloud-based or on-board computer system that collects and takes into account data from real traffic. Well-organized crowdsourcing can reduce the waiting time of AVs

w_{A V}

in the present computational model.

Impact of parking problems in city [44]. The proposed model here takes into account parking problems for private cars in at least three parameters: waiting time for private cars

w_{c a r}

(i.e., necessary extra time to park the car near the destination), psycho-physiological stress of private car driving

φ_{c a r}

, and cost per kilometer

E_{c a r}

(see Table 3).

Impact of improved walkability in suburban area [45]. Improved conditions for walking can be taken into account by increasing the weight parameter

m_{w a l k}

and decreasing the walking stress parameter

φ_{w a l k}

in the model (see Table 3).

6. Conclusions

In conclusion, this work has proposed a compact but sufficiently complete and general model of daily travel times with a minimal set of input parameters to proactively assess the impact of implementing AVs in first and last mile transportation in different suburbs.

The main output of the model is the effective daily transport time of an average suburban resident as a function of the aggregate AV adoption parameter. In addition to actual trip times, the model also provides effective travel times augmented with psycho-physiological stress and mentally perceived financial cost terms.

In order to achieve compactness and concrete numerical estimations, several sub-models of the proposed calculation scheme as model of transition from traditional transportation modes to AV-supported modes and the AV wait time model have been implemented as simply as possible. Combined with more detailed monitoring data and focused model-based surveys of a population, these sub-models can be easily refined.

The theoretical importance of the present work for transport research could be that it shows the possibility of reducing a complex multi-dimensional summation task to a three-dimensional one by means of several reasonable simplifying assumptions. At the same time, the construction of a compact computational model has made it possible to define a minimal sample set of initial data, which allows them, taking into account the most important influencing factors, to obtain realistic estimates of changes in transport times due to the introduction of AVs. The practical importance of the proposed compact calculation model lies in the fact that urban planners and local government officials have a tool at their disposal that, thanks to an easily understandable and relatively easily determined set of initial data, allows one to obtain, without long-term complex scientific studies, quick preliminary estimates of the impact of autonomous first-mile minibuses on the quality of life of suburban residents.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/smartcities7060151/s1, Rae municipality mobility analysis—Results of the mobility survey [38].

Author Contributions

Conceptualization, A.U., R.S., K.K. and D.A.; methodology, A.U. and D.A.; data collection, A.U., D.A. and K.K.; software, A.U.; writing—original draft preparation, A.U. and R.S.; writing—review and editing, all authors. All authors have read and agreed to the published version of the manuscript.

Funding

This research has received funding from two sources: the European Union’s Horizon 2020 Research and Innovation Programme under the grant agreement No. 856602, and the European Regional Development Fund, co-funded by the Estonian Ministry of Education and Research, under grant agreement No. 2014-2020.4.01.20-0289.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors express their gratitude to the Serio Angelo Maria Agriesti from Aalto University, Finland for his valuable comments and detailed extracts of the mobility data of the residents of the city of Tallinn.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

AV	Autonomous Vehicle
EDTT	Effective Daily Travel Time
MOD	Modality-Origin-Destination (matrix)
OD	Origin-Destination (matrix)
PT	Public Transport
TTS	Travel Time Savings
VTTS	Value of Travel Time Savings

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Figure 1. Growth of annual number of publications dedicated to application of autonomous vehicles in future transportation.

Figure 2. General structure of the calculation model. The upper corner numbers of the blocks correspond to the subsections in the paper text.

Figure 3. Explanation of example suburban transport task: (a) Location of Järveküla residential area (purple rectangle) in Rae municipality beyond the southern border of Tallinn city (red line). The blue line marks the major public transportation bus line 132 to Tallinn center; (b) Current development stage of Järveküla residential area of approx. 200 houses; (c) Pilot AV shuttle minibus designed for first-mile transport service in residential area.

Figure 4. Selection of two reference areas within the city limits of Tallinn (Mõigu and Kakumäe-Tiskre), for which the trip length distribution functions were found.

Figure 5. Summary of trip distance statistics of daily outbound trips for two example residential areas of Tallinn city on basis of the synthetic population database of Tallinn: (a1) Differential distributions with 1 km step for Mõigu area; (b1) The integrated cumulative distributions for Mõigu area; (a2) Differential distributions with 1 km step for Kakumäe-Tiskre area; (b2) The integrated cumulative distributions for Kakumäe-Tiskre area.

Figure 6. Results of RMS-fitting of the statistics of forenoon outbound trips by the 2-parameter sigmoid curves for Kakumäe-Tiskre and Mõigu districts.

Figure 7. The constructed 3-parameter model of distribution of trip distances combining the initial short-distance contribution and the smooth sigmoid step for lengthier distances. Parameter values are estimated to represent the Järveküla example area.

Figure 8. One-dimensional distances-based spatial model of transportation task: (a) an abstract map of residential area housing with local institutions, transport artery, and public transport stops on one edge; (b) distances-based concept of destination districts in metropolitan areas; (c) the simplified one-dimensional spatial scheme of origin zones and destination districts.

Figure 9. Explanation of concept of three-dimensional modality-origin-destination matrix used to sum up the daily transport times. Matrix defines 5 origin zones, 6 destinations districts, and 5 + 2 transportation modes. Each cell of MOD matrix is characterized by transport time with optional psych-physiological and economical extra terms and weight factors of distance and transport mode.

Figure 10. Explanation of the two-stage concept of outbound trips and input parameter set for calculation of effective transportation time costs.

Figure 11. Explanation of 2-stage effective trip times methodology with actual numerical values of input parameters.

Figure 12. The main output of the model: daily effective transportation times of an average suburban resident versus the aggregated parameter of autonomous vehicle acceptance

α

Figure 12. The main output of the model: daily effective transportation times of an average suburban resident versus the aggregated parameter of autonomous vehicle acceptance

α

Table 1. Actual usage of transportation modes in Rae municipality by 2021 survey and the deduced parameters for modeling example of the present study. Answers from 819 residents.

2021 Survey in Rae Municipality				Adopted Model Parameters
Transportation Mode	Winter	Summer	Average	Parameter	Value	Symbol
1. Passenger car	80.5%	72.9%	76.7%	1. Private car	69.0%	$m_{c a r}$
				2. Taxi *	7.7%	$m_{t a x i}$
2. Pyblic transport (walk+bus)	15.1%	11.1%	13.1%	3. Public transport (walk+bus)	13.1%	$m_{P T}$
3. Bicycle	0.7%	7.7%	4.2%	4. Bicycle	4.2%	$m_{b i c}$
4. Walking	2.8%	7.4%	5.1%	5. Walking	5.1%	$m_{w a l k}$

* Estimated 10% of passenger cars.

Table 2. Formulation of the aggregated autonomous vehicle acceptance factor on the basis of Rae municipality 2021 survey (answers from 819 residents).

Question/Parameter	Positive Answers
Should AV become a part of public transport?	72.8%
Would you use AV for everyday needs?	63.0% *
Would you use AV in case of on-site travel assistant?	48.8%
Would you use AV in case of remote travel assistant?	49.9%
Would you use fully automatic AV?	35.2%
Allow children to use AV in case of on-site travel assistant?	62.3%
Allow children to use AV in case of remote travel assistant?	38.8%
Allow children to use fully automatic AV?	23.1%
Consider AV safe on the street?	63.6%
Could AV replace your travelling with car?	45.5%
Aggregated average acceptance of AVs $α$	50.3% **

* ‘Yes’ answers and half of ‘May be’ answers summarized. ** Estimated maximal value of parameter

α

in present study.

Table 3. The complete list of input parameters of the compact model of effective transport time.

Parameter Description	Denotation	Value	Comment
Fraction of non-moving residents	N	0.1	Residents staying at home before noon
Ratio of all-day 24 h and morning 6–12 a.m. outbound trips	H	1.6	Used statistics of Mõigu area, see Figure 4
Return trips accounting factor	R	2.0	One return trip for every outbound trip
Farthest origin zone number	$n_{z}$	4	Zones 0–4 accounted for residential area
Size of origin zone	$L_{z}$	0.5 km	Up to 2 km if 4 zones
Number of destination districts	$n_{d}$	6	District 1 in local municipality
Size of destination district	$L_{d}$	4 km	Last district assumed large (up to 30 km)
Empirical parameter of fraction of local morning trips for 3-parameter distance distribution model	$D_{0}$	0.20	Local before-noon trips, e.g., children to local schools, see Model (2)
Median distance of city trips for 3-parameter distance distribution Model (2)	$x_{c}$	11.5 km	Average from the Mõigu and Kakumäe-Tiskre statistics, see Figure 5
Tail decay parameter for 3-parameter distance distribution Model (2)	$L_{t a i l}$	3.4 km	Fitting result of Mõigu and Kakumäe-Tiskre statistics, see Figure 5
Private car usage (before implementation of AVs)	$m_{c a r}$	69.0%	On the basis of 2021 survey, assumed 90% of passenger cars, see Table 1
Taxi usage (before implementation of AVs)	$m_{t a x i}$	7.7%	10 % of passenger cars assumed to be taxis in 2021 survey, see Table 1
Public transport usage (before implementation of AVs)	$m_{P T}$	13.1%	Winter and summer average, includes walk to local transport artery, see Table 1
Bicycle usage (before implementation of AVs)	$m_{b i c}$	4.2%	Winter and summer average, see Table 1
Walking percentage (before implementation of AVs)	$m_{w a l k}$	5.1%	Winter and summer average, see Table 1
Aggregated acceptance of AVs	$α$	0–0.5	Main input variable of model, maximum 50.3% from 2021 survey, see Table 2
Average speed of passenger cars	$v_{c a r}$	35 km/h	Same for private cars and taxi, averaged estimation from `Google Maps directions’
Average effective speed of PT busses	$v_{b u s}$	25 km/h	Data from before noon timetables of Tallinn city (e.g., bus line 132)
Average estimated speed of bicycle	$v_{b i c}$	15 km/h	Local mobility until district $D_{2}$ (8 km)
Average estimated speed of walking	$v_{w a l k}$	5 km/h	Local mobility until district $D_{1}$
Average speed of AVs	$v_{A V}$	25 km/h	Local mobility until district $D_{1}$
Average waiting time of private car	$w_{c a r}$	8 min	Car initial warming (and end location parking)
Average waiting time of taxi	$w_{t a x i}$	10 min	Estimate for arrival of Bolt system taxis
Average preparation (waiting) time of bicycle	$w_{b i c}$	2 min	Estimated preparation time of bicycle
Average waiting time of PT buses	$w_{b u s}$	11 min	Half-interval towards city in morning 6:00–12:00 (from Tallinn city and Harju county timetables)
Estimated AV waiting time (single AV case)	$w_{A V 1}$	6 min	Empirical parameter, depends on size of service area, see Model (24)
Number of AVs in local service area	$n_{A V}$	2	Planned 2 AVs in present use case, see Model (24)
Average walk length from end PT stop to destination	$l_{e n d w a l k}$	0.5 km	Important addition to realistic transport situation, see Figure 10
Psycho-physiological stress factor of private car driving	$φ_{c a r}$	0.25	Accounts for driving stress in rush hours, see Model (23)
Psycho-physiological stress factor of bicycling	$φ_{b i c}$	0.5	Accounts for fatigue and weather stress, see Model (23)
Psycho-physiological stress factor of walking	$φ_{w a l k}$	1.0	Accounts for fatigue due to baggage, physical difficulties of elderly people etc.
Private car kilometer cost due to price for full mileage, maintenance and fuel	$E_{c a r}$	0.3 EUR/km	May be reduced due to car sharing (improvement of present model)
Taxi car kilometer cost	$E_{t a x i}$	0.8 EUR/km	Estimated average value on basis of ordering system of Bolt
Public transportation bus kilometer cost	$E_{b u s}$	0.04 EUR/km	Estimation based on price of typical 30-day tickets (≈1 EUR/day) and daily trip distances
Estimated hourly wage in the country	$A_{h}$	9 EUR/h	Conversion coefficient of financial costs to travel time (=0.15 EUR/min), see Model (23)
Psychological cost perception coefficient of private car	$ψ_{c a r}$	0.33	Reduction factor of cost perception due to working and rest time savings, see Model (23)
Psychological cost perception coefficient of taxi	$ψ_{t a x i}$	0.33	Reduction factor of cost perception due to working and rest time savings, see Model (23)
Psychological cost perception coefficient of PT buses	$ψ_{b u s}$	1.0	Reduction irrelevant due to low speed of PT buses, see Model (23)

Table 4. Cross-use table of input data for 7 transport modes and 2 trip stages (local and remote) considered in transport times model (23). The referenced parameters are defined in the general input data Table 3.

Parameter of Model (23) for Stages 1 and 2		Mode 1 Private Car	Mode 2 Taxi	Mode 3 Walk + Bus	Mode 4 Bicycle	Mode 5 Walking	Mode 6 AV + Bus	Mode 7 Local AV
Waiting time, stage 1 (local)	$w_{1}$	$w_{c a r} / 2$	$w_{t a x i}$	0	$w_{b i c}$	0	Model (24)	Model (24)
Waiting time, stage 2 (city)	$w_{2}$	$w_{c a r} / 2$	0	$w_{b u s}$	0	0	$w_{b u s}$	0
Average speed, stage 1	$v_{1}$	$v_{c a r}$	$v_{c a r}$	$v_{w a l k}$	$v_{b i c}$	$v_{w a l k}$	$v_{A V}$	$v_{A V}$
Average speed, stage 2	$v_{2}$	$v_{c a r}$	$v_{c a r}$	$v_{b u s}$	$v_{b i c}$	$v_{w a l k}$	$v_{b u s}$	$v_{A V}$
Psycho-physiological stress factor, stage 1	$φ_{1}$	$φ_{c a r}$	0	$φ_{w a l k}$	$φ_{b i c}$	$φ_{w a l k}$	0	0
Psycho-physiological stress factor, stage 2	$φ_{2}$	$φ_{c a r}$	0	0	$φ_{b i c}$	$φ_{w a l k}$	0	0
Financial cost, stage 1	$E_{1}$	$E_{c a r}$	$E_{t a x i}$	0	0	0	0 *	0 *
Financial cost, stage 2	$E_{2}$	$E_{c a r}$	$E_{t a x i}$	$E_{b u s}$	0	0	$E_{b u s}$	0 *
Cost perception factor, stage 1	$ψ_{1}$	$ψ_{c a r}$	$ψ_{c a r}$	0	0	0	0	0
Cost perception factor, stage 2	$ψ_{2}$	$ψ_{c a r}$	$ψ_{c a r}$	$ψ_{b u s}$	0	0	$ψ_{b u s}$	0

* Free AV service in present use case.

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Udal, A.; Sell, R.; Kalda, K.; Antov, D. A Predictive Compact Model of Effective Travel Time Considering the Implementation of First-Mile Autonomous Mini-Buses in Smart Suburbs. Smart Cities 2024, 7, 3914-3935. https://doi.org/10.3390/smartcities7060151

AMA Style

Udal A, Sell R, Kalda K, Antov D. A Predictive Compact Model of Effective Travel Time Considering the Implementation of First-Mile Autonomous Mini-Buses in Smart Suburbs. Smart Cities. 2024; 7(6):3914-3935. https://doi.org/10.3390/smartcities7060151

Chicago/Turabian Style

Udal, Andres, Raivo Sell, Krister Kalda, and Dago Antov. 2024. "A Predictive Compact Model of Effective Travel Time Considering the Implementation of First-Mile Autonomous Mini-Buses in Smart Suburbs" Smart Cities 7, no. 6: 3914-3935. https://doi.org/10.3390/smartcities7060151

APA Style

Udal, A., Sell, R., Kalda, K., & Antov, D. (2024). A Predictive Compact Model of Effective Travel Time Considering the Implementation of First-Mile Autonomous Mini-Buses in Smart Suburbs. Smart Cities, 7(6), 3914-3935. https://doi.org/10.3390/smartcities7060151

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A Predictive Compact Model of Effective Travel Time Considering the Implementation of First-Mile Autonomous Mini-Buses in Smart Suburbs

Abstract

Highlights

Abstract

1. Introduction

2. Description of the Model

2.1. Concept, Sub-Models and Approximations

2.2. Example Use Case

2.3. Sub-Model of Trip Distance Distribution

2.4. One-Dimensional Spatial Situation Model

2.5. Initial Usage of Transportation Modes

2.6. Acceptance of Autonomous Vehicles

2.7. Framework of Three-Dimensional Modality-Origin-Destination Matrix

2.8. Central Summation Formula over Modes, Zones and Districts

2.9. Sub-Model of Transportation Mode Weights

2.10. Sub-Model of Weight of Distances

2.11. Sub-Model of Transport Times

2.12. Sub-Model of AV Waiting Time

3. Summary of Input Data

4. Simulation Results

5. Discussion

6. Conclusions

Supplementary Materials

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

Abbreviations

References

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