Estimation of the minimum required bus capacity between the hinterland and the centre of a functional urban region – a comparative study of the five largest Serbian cities

ABSTRACT This study presents a simply applicable method for comparing the capacity and potential of suburban bus services. The starting point for the analysis of the capacity is raw data from individual transport companies on the frequency of bus connections and the connectivity of centres with other parts of urban regions. A minimum seat capacity constant, based on the definition of a bus as a means of public transport, enables getting estimation of minimal required capacity. The concept of demographic gravitation serves as a comparative element of gravitational and capacity potential between settlements within urban regions, namely, demographic force, which measures the population of two settlements with the square of the distance between them with adequately made calibration based on real traffic flows. The results provide a comprehensive picture of the distribution of the suburban bus network, with detailed values of the required capacity in the lowest administrative units.


Introduction
The level of accessibility and connectivity of peripheral areas to the centre plays a crucial role in the functional delimitation of regions.Public transport, in particular, provides the necessary connections in more densely populated areas and especially in urban regions.If a region is created somewhat arbitrarily, without considering the catchment area's natural limits, there may be considerable disparities in the quality of transport services in the more peripheral parts; in extreme cases, even the absence of any direct connection to the centre.However, even a dense transport network does not necessarily mean that the demand for capacity is sufficiently filled.Passenger interchange serves as an important factor affecting network connectivity and capacity (Božović & Gladović, 2018).Two dominant gravitational factors, population mass and distance, must be considered to ensure adequate service between the core and the periphery.The stronger the gravitational relationship between two settlements, the greater the need for their integration.In general transport terms, this requires faster, more direct and more frequent connections, and in public transport, an increase in the number of direct connections and, where possible, the construction of new modes.The characterization of the mobility of the population within the intercity bus network concerning variable calibrations of gravity models is addressed in the works of Hong and Jung (2016) or Kwon and Jung (2012).The distance can also be understood in an economic sense instead of a strictly geographical sense (Zhao et al., 2021).An alternative behavioural approach takes individual differences in attractiveness and accessibility more seriously (Kalaanidhi & Gunasekaran, 2013).The estimation of passenger transport potential is most often done through a modified gravity model, which does not include a time dimension.Birr et al. (2014) and Oskarbski et al. (2015) analyse the speeds of selected vehicles and thus the total travel time.The classical use of a gravity model with the distance decay factor function in an environment without available information about real flows is practically excluded.Therefore, this study works with an original concept from social physics, demographic gravity, from which several essential equations determining the degree of gravitational attraction between two settlements are derived, including the so-called demographic force Stewart (1947Stewart ( , 1948)).The purpose is not to approximate model flows to reality but to compare the basic gravitational potential with the minimum capacity of a direct bus link between two settlements.An important factor here is the Serbian system of administrative division and, more specifically, the delimitation of urban regions, consisting of individual settlements (in Serbian called 'naselja)'.The suburban bus system should provide these settlements the best possible transport service.However, the administrative delimitation may differ from the various types of functionally delimited regions based primarily on the gravitational links between the settlement and the centre of the region.A closer look at the inconsistency of administrative divisions in the territory of Serbia is given in Tošić (2009) and Živanović et al. (2010).Thus, it is important to be aware of some discrepancies between the target territory of transport service and the gravitational catchment area when comparing the basic gravitational potential with the minimum required bus capacity.Several various papers have addressed the effectiveness of accessibility and related characteristics of bus services in the territory of the former Yugoslavia.The travel time aspect of accessibility using schedule data in GTFS format is analysed by Koblar (2021) for the Koroška region in Slovenia.A set of indicators for modelling supply and demand of transport services, including capacity, examines (Rajsman et al., 2013) using Zagreb as a case study.Filipović et al., 2009 evaluates the quality of mass passenger public transport service in Belgrade from the perspective of their users.According to this study, they are more satisfied with the suburban transport system than the urban one.In economic terms, according to Miličić et al., 2020, three main parameters have the greatest influence on the income and profit of the transport company: the coefficient of static utilization of bus capacity, the coefficient of dynamic utilization of bus capacity and the capacity of the bus.

Methods and data
A common problem when dealing with socio-economic geographic data in Serbia is either its outdatedness or the general lack of reliable sources.Specifically, in the case of traffic analysis, one has to be very cautious and not automatically refer to any public data, which can be quite outdated.Given these negatives, the attempt to create an original, topical, and applicable method becomes a necessary compromise between satisfactory data accuracy and the complexity of the chosen parameters.In this study, the author attempts an effective synthesis of available comparable data from proven sources, intending to present a simple variant for estimating public transport potential and the associated minimum capacity limit in areas where more sophisticated statistics are practically unavailable.
In terms of sources, the publicly available timetables of the transport companies of each city region were primarily used.Except for Belgrade, public transport is always managed by one company.Novi Sad is represented by GSPNS (2020), Niš (JGPNIS, 2020), Kragujevac (KGBUS, 2020) and Subotica (SUTRANS, 2020).In the case of the capital Belgrade, the situation is more complicated, where the bus network is divided among several transport companies, only some of which provide public data, see GSP (2020), LASTA (2020) and LASTRA (2020).Aggregate data for all transport companies in the Belgrade city region maintains the Secretariat for public transport, City administration of Belgrade (BGPREVOZ, 2020).A suitable alternative with additional continuously updated information offers (Busevi; 2020).Besides the publicly available platforms of the city transport companies, the related primary sources include the state railway, which is the second most important transport mode with verifiable data after buses.According to SRBVOZ (2020) and calculations from the synthesis of other transport companies, bus transport accounts for a completely dominant share of all public transport connections compared to rail (Belgrade 98,3%; Novi Sad 97,6%; Niš 97,2%; Kragujevac 98,2%; Subotica 94,4%).Another indispensable source, the Serbian National Statistical Office, extends the otherwise purely transport analysis with population data for all administrative units (STATGOV, 2020).In 2012, the Serbian Statistical Office published detailed results from a census conducted a year earlier.This census is still the most up-to-date database of demographic data for all administrative units.A demographic yearbook with partially updated data for higher administrative units and population projections for the lowest territorial units is also published annually.The analysis used population data for municipalities for all years 2011-2019.Due to the absence of an official forecast for 2020, extrapolation was calculated using a simple linear regression in the form of: a+ bx, where: The third significant component of the underlying data is the distances in km between each of the 315 village-centre sessions of the region.The source of all distances is a mapping platform 'Mapy.cz,'within which the variant of the shortest road connection between two centres of settlements (main bus stops) was selected.
The study deals with urban and suburban bus transport operating on the routes between the hinterland and the centre of the functionally defined territory of the five largest Serbian cities in terms of population (Belgrade, Novi Sad, Niš, Kragujevac, Subotica) (Figure 1)).In total, it includes 320 territorial units (5 urban centres) + (15 lower administrative units in Novi Sad, 156 in Belgrade, 70 in Niš, 56 in Kragujevac and 18 in Subotica).For each of the five city regions, data on the number of all bus connections for both directions (periphery-core, core-periphery) were compiled.Only direct connections between the centres of both settlements are included in the calculations.Weekdays, Saturdays and Sundays were analysed separately.The resulting value is then calculated as the weighted average of the days of the week (weight weekday = 5, Saturday = 1, Sunday = 1) and also the average for both directions.The difference between the two directions was not found significant enough to be further used in the map output.However, in the overall average, the frequency of connections from the periphery to the core slightly outweighs the core-periphery direction.All traffic data are relative to June 2020, the period affected by the pandemic.Analysing the archived traffic data for 2019, the data as of June 2021 and comparing them to each other, the overall variation in the number of connections for all cities is found to be less than 0.5%, so the year-to-year updates can be considered significantly unchanged.
The essence of the estimation of the minimum required capacity for bus transport service is based on a simple determination of the minimum number of seats that the relevant carrier must guarantee to be a public bus service.On the territory of Serbia, this minimum number is set by Serbian law on the transport of passengers in road transport available from MGSI (2019).According to this law, the minimum capacity corresponds to 8 seats.This value of 8 seats further serves as the main comparable benchmark for bus capacity.It can be understood as the minimum number of 8 persons for which a given transport company must provide public transport.Unlike the opposite indicator of maximum bus capacity, it does not provide an accurate picture of the quality of transport service in the form of the capacity of the available fleet.However, each transport undertaking uses its management system and does not provide specific data on its fleet.From empirical observation, even a cursory estimate of maximum capacity cannot be relied upon, as there is no fixed rule for deploying the same types of buses on a single intercity route.Thus, the reason why the whole study is based on the idea that a minimum number of seats provided is the only valid legislative rule ensuring at least some reliability of the estimation in all cities studied.
The resulting estimate of the bus seat capacity for any direct connection between two settlements is obtained by multiplying the weighted average of the number (frequency) of connections and the minimum number of seats.One of the concepts of John Stewart's so-called social physics, 'demographic force,' which measures the relationship between two settlements in terms of the product of their two population masses divided by the square of their distance, was taken as the basic comparative aspect of the analysis (Stewart, 1948).This formula is strikingly reminiscent of the basic form of the gravity model, see for example Rodrigue (2020) or Erlander and Stewart (1990).Due to the limited availability of public data in the Serbian statistical environment, a single indicator of real traffic flows between settlements was used to calibrate the gravity constant and the weight of the distance exponent.The indicator, named Daily migrations of the active population performing an occupation, is originally based on the 2011 national census and, similarly to the population data, data for the most recent period were extrapolated.Basic calibration of the model was carried out by estimating the gravitational constant and exponent weights by finding the highest possible correlation between the calculated minimum capacity and the actual extrapolated census data.
Thus, the final part of the methodology consists of two basic calculations.The first one, based on absolute values of the minimum required link capacity, gives the minimum number of passengers that can be carried by a direct bus service between two settlements within an urban functional region on an average day of the week.The second calculation allows relative comparisons between settlements concerning their population and distance from the centre of the region.It is the ratio of the result of the first calculation and the value of demographic force that indicates the ratio of the minimum possible bus capacity and gravitational linkage between two settlements, so-called CFP (capacity- force-potential).The main methodological procedure can be summarized in a simple workflow (Figure 2).

Results
Based on the collection of raw public data (population statistics from the national census, a detailed administrative division of the country into the most basic parts of settlements, numbers of bus connections from the databases of private transport companies of the surveyed cities), calculations and analyses of the minimum bus capacity of connections and seats (according to the legislation in force) between the settlements and the centres of the regions were carried out.In addition, a  relative comparison was made using the minimum bus capacity and demographic force (calibrated gravity model) ratio.Finally, according to the results of direct bus accessibility, it was possible to delimit the new five largest functional Serbian regions (Figure 3).
Out of 315 settlements in the five largest Serbian regions, direct accessibility of the settlement to the centre of the region was found in 242 of them, i.e. 76.8%.In terms of specific urban regions, the final selection is as follows, Belgrade (91 out of 156 settlements = 58.3%),Niš (66 out of 70 settlements = 94.3%),Kragujevac (52 out of 56 settlements = 92.9%),urban regions Novi Sad and Subotica remain unchanged.These 242 selected settlements thus form a new definition of urban functional regions based on the function of direct bus accessibility of the settlement to the centre.The worst accessibility appears to be in the Belgrade region, differing from the rest of the cities in the area and especially in population.However, the Novi Sad region comes out worst on average in terms of capacity and its ratio with demographic force (CFP).The gravity and capacity potential here points to the lack of frequency and thus the capacity of connections compared to the gravitational pull of population size of settlements versus centres.The best combination of all factors abounds in the smallest region, Subotica, with the amplitude of the average CFP between Novi Sad and Subotica more than eight times.
Further results of the average values for each region are presented in Table 1.All values listed except the Total number of settlements incl.the centre were obtained by our own calculations, by analysing raw publicly available data.The differences can largely be explained by the distribution of values by individual factors.However, when evaluating the average for whole regions, it is also important to remember the initial delimitation of the primary administrative regions, which determines how many territorial units with variable distance from the centre and population enter the comparison.Kragujevac and Niš are characterized by a higher density of smaller settlements, where population mass has less impact on the CFP result.On the other hand, Novi Sad is surrounded by more population-size settlements with short distances.Subotica, in particular, evinces low gravitational forces and above-average population size.Eventually, the high proportion of settlements without direct bus connection most affects the Belgrade urban region.Yet, there is no obvious regularity in the overall comparison of the three main parameters examined (population, minimum required bus capacity and ratio of capacity and demographic force), which is also demonstrated by the relatively surprising results in the individual territorial units (Main Map).

Discussion and conclusion
The use of traffic and population data to estimate available link capacity and its inter-territorial comparison brings many challenges and potential problems with no easy solution.The question can be asked whether it makes sense to even explore such a complex issue in an area without sufficient relevant data.The structure and operation of public bus transport in Serbia, both in cities and suburban regions, is not one of the strengths of the Serbian infrastructure.The data basis is limited exclusively to private transport companies with different publishing and updating timetables approach.Gladović et al. (2021) mentions one of the fundamental problems of the Public urban passenger transport system (JGTP) in Serbia on Kragujevac and Niš the division of public transport among many carriers, which makes the control and management of the system complex.Another related problem is the overall lack of integration of the various components of public transport (Milenković et al., 2017).Population data for the lowest administrative units can be obtained either from a national census conducted once every ten years or by extrapolation in linear regression.The need for at least a partial knowledge of the quality and comparison of public transport services, to which many other sectors are linked and, in some cases, the only economically viable link between peripheral settlements and the centre, is a logical answer to the question of potential meaningfulness.The methodology of this paper, involving a combination of minimum capacity and demographic force calculations, thus serves as an alternative solution in the circumstances outlined above, which at least partially compensates for the lack of resources for a more in-depth analysis of transport accessibility and other more sophisticated characteristics.In the Western Balkan area, which besides Serbia also includes most of the former Yugoslav states, this is the first study within the field of suburban bus capacity, seeking a compromise between the application potential of the values obtained, the lack of adequately detailed data and a wholly original methodology.The idea of applying the legal definition of minimum bus capacity to estimate the number of minimum possible mobility between a settlement and the centre of a catchment area says practically nothing about the real demand for connections.However, universally in any part of Serbia reflects the highest value of maximum guaranteed capacity.Of course, it can be argued that the minimum capacity of a conventional bus type will be considerably higher.Ćiraković et al. (2014) uses accurate data on Belgrade's vehicle fleet in their public transport analysis.Petrović et al. (2014) in turn, provides sophisticated calculations of vehicle characteristics and the total number of possible transport places for the territory of the city of Niš.Manojlović et al. (2011) even estimates the statistics of urban bus fleet renewal for total Serbia.However, a fundamental problem with the credibility of the data arises when comparing fleet characteristics which vary considerably in quantity and quality across the selected cities.While the results of the individual calculations cannot be used unambiguously to show a comprehensive picture of accessibility and subsequent transferability to other areas, they nevertheless allow a basic comparison of transport services concerning gravity and capacity potential.Absolute values may differ from reality by up to a factor of several times, but differences in the relative proportions can be assumed to be relatively minimal due to the uniform format.Apart from the main negative of the study, the limited explanatory value of the results, the simple form of the methodology with the minimum of data required for processing and consistent use throughout Serbia remains a significant asset.

Software
Quantum GIS 3.16 Hannover was used for all map outputs.All other editings, including the final layout design, were made within Microsoft Powerpoint.

Figure 1 .
Figure 1.Location of the five largest urban regions in Serbia.

Figure 2 .
Figure 2. Workflow diagram of the methodology.

Figure 3 .
Figure 3. Administrative vs functional urban regions based on direct bus connection.

Table 1 .
Main characteristics of total and average results per each urban functioning region.
Source: own calculations based on raw publicly available data.