3. Results

We analysed the main external and internal factors influencing the operation of the street and road network and divided them into the following groups: road factors; urban planning factors; operational factors; natural and climatic factors; human factors; and socio-economic factors.

Road factors comprise the following:

1. (a) The category of the street or road, which makes it possible to determine their designation; the nature of the development, which implies the size of the roadway, sidewalks, green zones, and other planning elements of the street and road network; and the permitted speed and traffic capacity of the section;

2. (b) The length of the urban street and road network;

3. (c) The number of intersections and junctions. Each intersection should have sufficient traffic capacity since traffic flows connect or intersect here. It follows that the traffic capacity of a particular section of a street depends on the traffic capacity of its intersections;

4. (d) Means and methods of traffic management. Technical means of traffic management and methods for introducing one-way or contraflow traffic should improve the conditions for organising, regulating, and controlling the traffic and pedestrian flows and reduce the number of RTA;

5. (e) Type and condition of road pavement. The public street façade and the quality of transportation services or pedestrian traffic largely depend on the quality and condition of the road pavement. In turn, the type and condition of road pavement significantly affects traffic capacity, speed, and the number of RTA;

6. (f) Elements of landscaping and maintenance of the street and road network. Street maintenance includes traffic service system (external street lighting, public transport stops, bus stations, parking, etc.) and landscaping (green zones, monuments, road infrastructure).

According to Lutsyk and Stepanchuk (Reference Lutsyk and Stepanchuk2013), Bezlyubchenko et al. (Reference Bezlyubchenko, Gordienko and Zavalnyi2021), and Yang et al. (Reference Yang, Bieliatynskyi, Pershakov, Shao and Ta2022), the urban planning factors that affect the street and road network operation are as follows:

1. (a) Area of the city. The area of the city is the size of the territory occupied by the city. It significantly affects such indicators as road density and the average distance travelled on foot or by vehicles;

2. (b) Population. This is the main criterion for city classification. It influences the size of the territory, the planning structure, the choice of transport modes, the number of vehicles, the traffic volume, etc.

3. (c) Form of the city plan. Depending on the city plan, its form can be compact (the entire territory of a city represents a single array), dissected (the city territory is divided into several parts by a river or railway), or dispersed (approximately equal-sized residential areas are remote from each other). Dissected and dispersed forms complicate transport links within the city and its planning and increase trip distance;

4. (d) Planning scheme of the street and road network. The general design of the city plan is inseparably connected with zoning. The city plan involves developing the system of main streets and roads along the main routes of public transport and pedestrians;

5. (e) Functional and administrative subordination of the city. This factor is essential for specific land use and is functional in nature (industrial, port, resort, scientific, etc.). It affects urban planning and the location of industrial enterprises, cultural and community services, railway stations, ports, airports, bus stations, freight stations, access roads, etc. The administrative subordination of a city affects the size of its external relations, which entails greater or lower numbers of visitors;

6. (f) Access to places of attraction. The characteristics of this factor are the time expenditures and distances to the main places of attraction (workplaces, public centres, railway stations, etc.);

7. (g) The number of ‘bottlenecks’. A bottleneck denotes the discrepancy between the maximum traffic needs and the traffic capacity of an element of a street and road network, e.g., a bridge, overpass, intersection, or section of a street. A ‘bottleneck’ appears when several vehicles use the corresponding element of the street and road network and exceed its traffic capacity in their number. The reason for the formation of ‘bottlenecks’ is the lack of alternative routes to a given location. Therefore, the number of bottlenecks on the street and road network indicates the possibility of laying a certain number of routes between the places of attraction.

According to Lipyanin and Milash (Reference Lipyanin and Milash2022) and Lobashov (Reference Lobashov2011), operating factors are as follows:

1. (a) Traffic flow characteristics: traffic volume, traffic composition, traffic speed, traffic density, congestion level;

2. (b) Vehicle performance specifications, which are determined by the performance characteristics of each type of vehicle, such as the length, width, and height of the vehicle; its maximum speed, carrying capacity, maximum number of passengers, external and internal turning radius, and braking distance.

Pindus and Goncharenko (Reference Pindus and Goncharenko2013) and Bieliatynskyi et al. (Reference Bieliatynskyi, Yang, Pershakov, Shao and Ta2022c) distinguish the following natural and climatic factors:

1. (a) Terrain, which affects the location of streets, roads, intersections, junctions, the longitudinal slopes of the roadways and sidewalks, and surface drainage;

2. (b) Water obstacles, characterised by the presence of water bodies in the city, such as rivers, canals, lakes, and seas. These natural bodies generate the need to build bridges, dams, embankments, etc.;

3. (c) Climatic and meteorological conditions, such as atmospheric temperature, atmospheric pressure, wind, rain, fog, snow, and icy roads, that affect the well-being of drivers, passengers, and pedestrians, and vehicle operating conditions.

We analysed the works by Boykiv (Reference Boykiv2016) and Khitrov (Reference Khitrov2021) and determined the following human factors:

1. (a) Physical and psychological condition of road users. A driver has to perceive large amounts of information about road users, traffic control devices, the condition of the road and the environment, and the operation of systems and units of a vehicle. Furthermore, a driver has to analyse this information continuously and make appropriate decisions, often under severe time constraints;

2. (b) Driver qualification. The driver qualification is the level of general and special training of drivers, which has a significant impact on traffic safety, based on each driver's choice of the optimal speed, their choice of distances, their choice of the minimum time for decision-making, and their ability to choose the appropriate actions and make the right decisions;

3. (c) Age category and driving experience. This factor affects the road user's choice of a more appropriate option for decision-making and their state of health, and, consequently, the physical and psychological condition of road users;

4. (d) A person's social status influences their choice of modes of transportation and type of vehicle;

5. (e) The habits of a person. A habit is a special form of human behaviour that manifests itself in the tendency to perform particular actions that have become entrenched due to multiple repetitions. In this context, a habit manifests itself in choosing a route, vehicle type, violation or observance of traffic rules, etc.;

6. (f) Existential needs. These are manifested in the choice of a safe speed, observance of a safe passing distance, the use of personal protective equipment, behaviour on the road, etc.

Stepanchuk (Reference Stepanchuk2017) and Bieliatynskyi et al. (Reference Bieliatynskyi, Yang, Pershakov, Shao and Ta2022d) distinguish the following socio-economic factors:

1. (a) Living conditions and standards. These influence population mobility, trip distance, increase in traffic volume (especially on weekends), the level of motorisation, the choice of the type of transport (personal or public), etc.;

2. (b) Environmental conditions. This factor involves implementing measures to reduce the negative impact of vehicles and road infrastructure facilities on human health and the environment. These measures include bans on the movement of vehicles, the construction and reconstruction of elements of the street and road network, the expansion of transport nodes, places for parking and storage of vehicles, etc.;

3. (c) Economic development of a city. This factor determines the pace of life of a city and its financial capacities that provide for improving the conditions for the development of its transport system and infrastructure.

Each of the factors discussed has an impact on the operation of the street and road network to a certain extent. However, the problem arises because several different factors can have a simultaneous retaliatory effect. Interacting with each other, they have significant internal connections that do not allow the impact to be divided into independent components. These internal connections do not change the factors when determining the characteristics of the street and road network.

The study by Polishchuk et al. (Reference Polishchuk, Krasylnikova and Dzyuba2014) has shown that traffic jams on the street and road network of the city of Kyiv typically occur at signal-controlled intersections (69%), bridges (13%), multilevel intersections (9%), roundabouts (3%), and pot-holed roads (6%). Since traffic jams resulting from non-compliance with road traffic regulations or RTA are irregular, we do not consider them further in this study. Places with regular traffic jams are referred to in this paper as ‘bottlenecks’. Bottlenecks include bridges, overpasses, signal-controlled intersections, and entry and exit roads. Such an approach makes it possible to consider bottlenecks as places of origination and absorption of traffic flows. They are places where traffic problems arise, such as traffic delays and traffic jams.

Features of the operation of the street and road network, such as traffic load, have their regularities. The operation of the street and road network largely depends on the zone of the city, the time of day, the day of the week, the functional designation of the street, etc. Therefore, it is possible to observe sections of streets and roads with gridlocks although they have a traffic load that is only 10–15% of their designed traffic capacity. If the problem with the traffic congestion on these sections is resolved first, they can serve as alternative routes (relief roads) for vehicles to be redirected along and unload complex sections of the street and road network. Hence, the indicator of the availability of alternative routes (relief roads) for overloaded main roads can be determined by the following formula:

(1)\begin{equation}{K_R} = \frac{{1 + {B_R}}}{{{B_{\textrm{MR}}}}}\end{equation}

where

• K_R is an indicator of the availability of relief roads for overloaded main roads;

• B_R refers to the number of alternative routes (relief roads) for overloaded main roads;

• B _MR means the total number of main roads of city and district designations with heavy traffic (their traffic capacity is close to the maximum traffic volume during peak hours).

The graphical model of the street and road network of the left-bank part of Kyiv is shown in Figure 1. According to the results of our research, the vertices shown in Figure 1 can be divided into three groups, depending on their load (Gavrilov et al., Reference Gavrilov, Dmytrychenko and Dolya2007):

1. (1) flow-generating: a significant number of vehicles are concentrated here, and their movement is directed in a particular direction;

2. (2) flow-absorbing: vehicles are distributed in several directions here, or it is the endpoint for a vehicle;

3. (3) with uniform flow: a feature of such a transport node is its ability to ensure the movement of the same number of vehicles in one direction.

Figure 1. Graphical model of the street and road network of the left-bank part of Kyiv

An analysis of the graphical model of Kyiv shows that drivers do not always use all routes to bypass traffic jams; therefore, these unused routes do not play a significant role in the distribution of the main flow and can be removed from this model. In order to reduce the size of the graphical model, transit vertices (intersections and junctions) that are not flow-generating or flow-absorbing were also removed since they are not related to the origin-destination movement. As a result of the removal of transit streets (arcs) and intersections (vertices), a simplified network is adopted for the study that provides traffic for the maximum number of possible traffic routes on the urban street and road network.

Between the various districts of the city, therefore, there are bottlenecks in addition to the places that provide transport links. Bottlenecks influence the generation of traffic flows within and outside a district. Our approach makes it possible to exclude a certain number of streets and intersections from the management system since they do not significantly affect the generation of traffic flows. In order to study street and road network operation, it is necessary to consider only the nodes, where the traffic direction changes depending on the increased number of vehicles, and the elements of the street and road network to which main traffic flows are directed. Therefore, we suggest considering the given situation for Kyiv, highlighting only bottlenecks and elements of the street and road network that ensure the connection between districts. Thus, we distinguished 99 places on the street and road network of Kyiv that are bottlenecks, and their corresponding elements are as follows:

• – 5 pedestrian bridges;

• – 36 overpasses across a railway, metro line, or high-speed tram line;

• – 24 multilevel interchanges;

• – 23 signal-controlled intersections (located on the border of administrative districts or close to them);

• – 11 entry and exit roads.

Based on the scheme of the Kyiv street and road network thus obtained, it is possible to design a geometrical graph. This graph is the source material for creating an algebraic image of the bottlenecks of the given street and road network, which can be used directly in mathematical models (Figure 2). A compressed graph represents the Kyiv street and road network, where each vertex is characterised as a bottleneck, including all adjacent nodes through which vehicles are directed to the given vertex. This model indicates a particular number of alternative routes and possible relief roads that help to redistribute traffic flows in the event of a difficult situation in one of the directions of traffic.

Figure 2. Model of interconnected bottlenecks on the Kyiv street and road network

This model of bottlenecks on the urban street and road network is based on the selection of optimal sections for the passage of transit traffic flows to the relevant area with minimal time and financial expenditures. Each edge of the graph of this model can be characterised by a weight (t _ij) corresponding to the total time spent on movement over a given section of the street and road network. Stepanchuk (Reference Stepanchuk2012) recommends using the value of the reduced length of the edge, which depends on the trip time along it, and also suggests considering the delays associated with the street and road network overload and the passage of intersections. Given the research results, the indicator of edge weight can be determined by the following formula:

(2)\begin{equation}{t_{ij}} = {L_{ij}}\frac{{({{l_{ij}} \times {{k_1}} )} }}{{{V_{ij}}}} + {b_c}\end{equation}

where

• l _ij is the length of the edge, m;

• V _ij is the permitted speed on the edge, m/s;

• k ₁ is the coefficient of the edge load;

• b _c is the time spent on passing intersections located on the given edge, s.

Hence, k₁ is the coefficient that considers the time loss caused by overloading the street and road network with vehicles. It is the primary indicator of the interrelation between the traffic volume on a section of the street and that section's capacity. Zablotsky invented this coefficient to characterise the ‘traffic conditions’ of transport flows along the street and road network. For each edge of the graph, this coefficient is determined by the following formula:

(3)\begin{equation}{k_1} = {e^{({({N_{ij}}/{P_{ij}}) - 1} )}}\end{equation}

where

• e is a natural logarithm (Euler's number);

• N _ij is the actual traffic volume on the edge, car/s;

• P_ij is the traffic capacity on the edge, car/s.

It is possible to study the operation of the urban street and road network based on the assumption that the bottleneck is a node, and the main traffic flows are directed to or from it, according to the theory of networks (Ilchuk, Reference Ilchuk2010). Such an approach makes it possible to exclude a particular number of streets and intersections from the management system since they hardly influence the movement of vehicles. In order to study street and road network operation, it is necessary to consider only the nodes where the traffic direction changes depending on the increased number of vehicles, and the elements of the street and road network to which the main traffic flows are directed.

In modern network theory, the number of links of a node is called its degree. The degree of a node is a quantitative measure of its importance based on the number of links belonging to that node. The degrees of vertices characterise the number of possible directions of routes that can be laid through the corresponding vertex. Hence, the following formula is used to calculate the number of directions for routes that can pass through the corresponding bottleneck:

(4)\begin{equation}{n_r} = ({{k_j} - 1} )\end{equation}

where

• n_r is the number of directions for routes, passing through the corresponding bottleneck;

• k_i is the number of sections of streets and roads that approach the corresponding bottleneck or the degree of the vertex.

Based on the above, all bottlenecks can be distributed according to the appropriate levels of the hierarchy, which will characterise their functional values within the system of the urban street and road network. The hierarchy levels of bottlenecks indicate their importance in ensuring transport links between the planning elements of the city. In other words, it shows the degree to which the failure of a bottleneck deteriorates the transport situation and how many alternative routes remain under certain conditions (Figure 3).

Figure 3. Graph of connections of bottlenecks on the street and road network of Dniprovskyi District. II-Darnytskyi District; III-Desnianskyi District; IV-Dniprovskyi District; VII-Pecherskyi District

It is also possible to represent the minimum distances between the bottlenecks of the Kyiv street and road network in the form of a graph. A weighted graph shows the actual distance between bottlenecks for the Dniprovskyi District of Kyiv (Figure 4). A similar graph can be designed for the entire street and road network of any city, regardless of its planning.

Figure 4. Minimum distances between bottlenecks in the Dniprovskyi District

The remoteness of bottlenecks can be represented as a distance matrix, where the minimum distance from one vertex of the graph (bottleneck) to another is clearly traced. Thus, the presented matrix, which determines the remoteness of all bottlenecks in the city, should be taken as the basis for ensuring the efficient distribution of traffic flows on the street and road network. The appropriate distance matrix, based on Dijkstra's algorithm (Barrat et al., Reference Barrat, Barthélemy, Pastor-Satorras and Vespignani2004), can help to find the shortest path and consider the basic indicator of the minimum time spent moving between the corresponding points of attraction.

When the individual edges are loaded, the throughput capacity of each edge is considered, and the Ford-Fulkerson algorithm is applied. It is then possible to determine the maximum possible flow between the corresponding start and end nodes (bottlenecks). The Ford-Fulkerson algorithm is based on the important ‘max-flow min-cut’ theorem, which determines the maximum possible flow with the help of cuts to the transport network. For this purpose, we compiled a matrix of distances between the bottlenecks of the district under study (Table 1).

Table 1. Matrix of distances between the bottlenecks of the Dniprovskyi District of Kyiv

We studied the data on commuting trips of Kyiv residents by personal car and established the loading of bottlenecks. Based on the graphs of bottlenecks for each city district, it is possible to determine the polynomial rank graph using the technique. The polynomial rank graph makes it possible to analyse many spanning subgraphs, thereby creating conditions for considering enlarged transport districts, which can be reduced in size, for example, to the existing administrative city districts. The polynomial rank graph G has the following formula:

(5)\begin{equation}{P_\upsilon }({x,\, y} )= \mathrm{\Sigma }{x^{\upsilon _G^\ast{-} \upsilon _H^\ast }}{y^{{\upsilon _H}}}\end{equation}

where

• v _H is the corank of the spanning graph, which includes all vertices of the graph, subgraph H;

• v ^∗_H is the rank of subgraph H;

• v*_G is the rank of graph G.

The appropriate technique makes it possible to build a polynomial for any graph that reproduces the relationship between the elements on the urban street and road network. The following formula is used to build the polynomial graph of bottleneck connections of the street and road network of the Dniprovskyi District of Kyiv, which has 20 vertices and 30 arcs:

(6)\begin{align} & {({x + 1} )^5}({x^{14}} + 25{x^{13}} + 8{x^{12}}y + 6{x^{11}}{y^2} + 5{x^{10}}{y^3} + 4{x^9}{y^4} + 3{x^8}{y^5} + 2{x^7}{y^6} + {x^6}{y^7} + 300{x^{12}} + 185{x^{11}}y\nonumber\\ & \quad +\, 156{x^{10}}{y^2} + 131{x^9}{y^3} + 102{x^8}{y^4} + 71{x^7}{y^5} + 41{x^6}{y^6} + 15{x^5}{y^7} + 4{x^4}{y^8} + {x^3}{y^9} + 2292{x^{11}} + 2028{x^{10}}y\nonumber\\ & \quad +\, 1864{x^9}{y^2} + 1576{x^8}{y^3} + 1186{x^7}{y^4} + 766{x^6}{y^5} + 392{x^5}{y^6} + 156{x^4}{y^7} + 52{x^3}{y^8} + 13{x^2}{y^9} + 4x{y^{10}}\nonumber\\ & \quad +\, {y^{11}} + 12465{x^{10}} + 13952{x^9}y + 13631{x^8}{y^2} + 11511{x^7}{y^3} + 8330{x^6}{y^4} + 5022{x^5}{y^5} + 2463{x^4}{y^6}\nonumber\\ & \quad +\, 998{x^3}{y^7} + 335{x^2}{y^8} + 106x{y^9} + 25{y^{10}} + 51096{x^9} + 67199{x^8}y + 68234{x^7}{y^2} + 56833{x^6}{y^3}\nonumber\\ & \quad +\, 39319{x^5}{y^4} + 22346{x^4}{y^5} + 10429{x^3}{y^6} + 4011{x^2}{y^7} + 1306x{y^8} + 296{y^9} + 162992{x^8} + 239201{x^7}y\nonumber\\ & \quad +\, 246954{x^6}{y^2} + 200153{x^5}{y^3}+ 130680{x^4}{y^4} + 68967{x^3}{y^5} + 29317{x^2}{y^6} + 9895x{y^7} + 2194{y^8}\nonumber\\ & \quad +\, 411632{x^7} + 646893{x^6}y + 663284{x^5}{y^2} + 511904{x^4}{y^3} + 308023{x^3}{y^4} + 144566{x^2}{y^5}\nonumber\\ & \quad +\, 51255x{y^6} + 11331{y^7} + 828612{x^6} + 1343485{x^5}y + 1330403{x^4}{y^2} + 947313{x^3}{y^3} + 500778{x^2}{y^4}\nonumber\\ & \quad +\, 190399x{y^5} + 42899{y^6} + 1326268{x^5} + 2135044{x^4}y + 1966584{x^3}{y^2} + 1225264{x^2}{y^3} + 515876x{y^4}\nonumber\\ & \quad +\, 121782{y^5} + 1666708{x^4} + 2544505{x^3}y + 2056904{x^2}{y^2} + 1011792x{y^3} + 259982{y^4} + 1601000{x^3}\nonumber\\ & \quad +\, 2172014{x^2}y + 1383644x{y^2} + 410548{y^3} + 1116838{x^2} + 1201668xy + 458960{y^2} + 509711x\nonumber\\ & \quad +\, 329090y + 115772) \end{align}

Based on the model of connections between bottlenecks, polynomials were determined for each district of Kyiv. A similar graph can be designed for each district of any city. These graphs indicate the possible traffic connections between districts and existing main routes within each district, which makes it possible to redistribute traffic flows efficiently throughout the urban street and road network with the help of an intelligent traffic management system. Then, a model of the street and road network of the city is created based on the number of its nodal points (bottlenecks) to which traffic flows are naturally drawn. When deciding on the redistribution of traffic flows along the street and road network of the city, the primary focus should be on the capacity and the number of bottlenecks.

It is possible to analyse the capacity of the street and road network of the district and the whole city, considering bottlenecks as sections of the main transport links, separating them as a transport node, and comparing the total number of traffic lanes directly included in this node. When the relevant information is considered, nodes can be distributed by the number of lanes, which ranks the bottlenecks of the street and road network according to their traffic capacity. Each bottleneck has geometric coordinates and the total number of traffic lanes that directly enter and exit it. Hence, the capacity of the street and road network can be determined according to the technique for distributing its elements by multiplicity factor (Stoilova and Stoev, Reference Stoilova and Stoev2015).

When the capacity of elements of the street and road network is established, the relation between their traffic capacity and traffic demand can be determined. This helps to identify the need to construct additional elements of the street and road network, or the existence of alternative traffic routes with less traffic load, which is the main thing to consider when designing intelligent traffic control systems. The efficiency of the operation of the urban street and road network also depends on its reliability, which is an important indicator of the operating condition of the urban street and road network. The reliability of the street and road network as an urban engineering structure is a system of probabilities, such as its failure-free operation, the speed of vehicles corresponding to the road traffic regulations, and the safe speed of the traffic flow (Stepanchuk et al., Reference Stepanchuk, Bieliatynskyi, Pylypenko, Popovic, Manakov and Breskich2020). The concept of reliability of the street and road network includes the reliability of its individual elements.

Reliability is closely related to the failure probability or accident risk. The problem of network reliability is well studied nowadays. As a rule, a graph is used as a model of the network, the elements of which are subject to failure. A graph considers various clarifications of this system, such as elements subject to communication failures or nodes, the network nodes that require to connect one element with another. However, the key factor for ensuring the reliability of the operation of city nodes is their connection. Therefore, the appropriate total number of traffic lanes in bottlenecks that provide communication between city districts in a particular direction is required to ensure reliable conditions for street and road network operation. In order to determine the optimal routes for the redistribution of vehicles, it is necessary to consider the probability of occurrences (RTA, road maintenance, vehicle breakdown, etc.) that can lead to the partial or complete exclusion of certain elements and sections of the street and road network on possible routes. However, when a vehicle stops moving over some lanes, there is a risk of a lack of the necessary number of lanes. In this paper, we examine the lack of traffic lanes precisely in the bottlenecks of the urban street and road network. Therefore, the efficient operation of the street and road network requires ensuring the necessary traffic capacity of traffic lanes during peak hours. The traffic capacity is a qualitative characteristic of the roadway operation of all street and road network elements. Thus, the effective operation of the street and road network is ensured when the traffic capacity of its section or element is greater than or equal to the maximum traffic volume (P _c ≥ N _max).

Traffic jams in the bottlenecks of the urban street and road network can be considered as the probability of an undesirable event that poses a risk of potential danger and has a significant impact on its operation.

(7)\begin{equation}R = \frac{{{K_{\textrm{BN}}}}}{{{K_{\textrm{total}}}}}\end{equation}

where

• R is the probability of an undesirable event that poses the risk of a potential traffic jam;

• K _BN is the number of traffic jams caused by insufficient traffic capacity of the bottlenecks;

• K _total is the total number of traffic jams on the street and road network of the city. Risk refers to the probability of an event caused by increased traffic, poor road conditions, etc.

The mathematical relationship of the risk theory is based on the summation of the density of the distribution of physically defined or experimentally obtained values. It is a theoretically probabilistic model for comparing the average value (A) and root-mean-square deviation (GÆ) of the dangerous parameter of a transport construction with the same characteristics of this parameter (A _r, G _r), which are in a critical state with a risk of damage of 50%. When summing the normal, lognormal, and Gram-Charlier distributions, the Laplace function leads to a probability of 0 ⋅ 5 in the structure of calculation formulas of the risk theory. Using risk theory, the probability of traffic jams due to inconsistency in traffic lane capacity can be determined by the following formula:

(8)\begin{equation}r = 0 \ \cdot\ 5\mathrm{\Phi }\left( {\frac{{{P_a} - {P_1}}}{{\sqrt {G_{{P_a}}^2 - G_{{P_1}}^2} }}} \right)\end{equation}

where

• P _a is the actual traffic capacity of bottlenecks between city districts;

• $G_{{P_a}}^2 = C_V^{{P_a}} \times {P_a}$ is the root-mean-square deviation of actual traffic capacity of one traffic lane;

• C_V is the coefficient of variation.

In order to optimise the efficiency of the urban street and road network, it is necessary to optimise the number of traffic lanes on each main street (especially in bottlenecks) since the number of traffic lanes on all streets does not correspond to the traffic volume or will not be optimal. An optimisation tool can be an analysis of the risk of lack or overabundance of traffic lanes. The increased number of traffic lanes makes it possible to eliminate traffic jams and increase the efficiency of the street and road network. When analysing the efficiency of the urban street and road network, the question is always raised of how it is possible to effectively distribute traffic flows along it, subject to the failure of some sections of the streets, or rather, to ensure the reliability of its functioning in difficult operating conditions.

The optimal route is characterised by the movement of the vehicle bypassing the places of transport congestion according to the criterion of minimising time. It is almost impossible for drivers to choose the optimal route on their own. A driver cannot objectively assess the traffic situation that has developed on the street and road network of the corresponding transport area since there is a lack of complete information about the state of other sections of the network that can be used as alternative routes.

Relying on their considerations and interests in choosing a route, drivers become hostages of circumstances that do not allow them to change driving direction and force them to move only in the direction where the traffic flow is directed. Therefore, the management of the distribution of traffic flows along the urban street and road network should be reduced to a solution to the problem of minimising the total travel along the route between the two transport nodes. The efficiency of the street and road network is determined by indicators of time savings when moving over it under particular conditions of its loading, characterised by minimising the time spent on a certain route $({t_{\textrm{route}}} \to \min )$. Under difficult road conditions, the efficiency of the route does not mean the shortest path per distance but the minimum time of movement over a particular route.

Since modern conditions are characterised by an intensive increase in the number of cars on city streets, there is a need to distribute the traffic flows that predominantly consist of cars. The task set in this study can be solved by distributing traffic flows in different driving directions based on indicators of vehicle engine volume. This can help to redirect the transit vehicles, the engine volume of which is larger than an established indicator, along other parallel streets, thereby reducing the traffic load on the problem sections of the street and road network.

The traffic flow consisting of cars should be distributed at the intersection since the traffic becomes complicated straight after it. Therefore, a multi-position road sign is installed at the entrance to the intersection, prohibiting the movement of vehicles whose engine volume exceeds the established indicator. This road sign indicates that (for this section of the street) transit vehicles, the engine volume of which exceeds the established distribution indicator, must pull off the main traffic flow direction and move over a different (alternative) route. Moving over a separate traffic flow should be facilitated by road signs and information boards on further movement over this route.

A feature of this traffic flow distribution method is the ability to distribute the traffic flow, consisting of 70–95% of light vehicles, into two flows, etc. We consider the approach to traffic flow distribution and the direction of vehicles to alternative routes. For example, there are two possible routes between points A and B (Figure 5). A distribution indicator (regulator) is installed in point A that decides the path for each driver based on optimisation principles and information about the density of the traffic flow q(t) at point A.

Figure 5. Distribution of flow (q) at vertex A into two alternative routes directed to vertex B

In this case, a traffic flow with density q ₁(t) will move over the first route, and a traffic flow with density q ₂(t) will take the second path. It can be assumed that vehicles whose engine volume is less than the established distribution indicator should move over the first route, and vehicles with a larger engine volume should take the second path, respectively. Thus, the forced distribution of vehicles is performed, characterised by such conditions as q ₁(t) = K _n≤W and q ₂(t) = K _n>W. K _n≤W is the number of vehicles in the flow, the engine volume of which is less than or equal to the established distribution indicator. K _n>W is the number of vehicles in the flow, the engine volume of which exceeds the established distribution indicator. The total number of vehicles (K _n≤W and K _n>W) that move over to the first and second alternative routes should not exceed the traffic capacity of those routes. In other words, this condition can be represented by the formulas: Kn ≤ W ≤ P ₁ and Kn > W ≤ P ₂.

However, the principal condition is to ensure approximately the same time expenditures for moving over a route, i.e., t1 ≈ t2. Hence, it is possible to achieve the same time values at different path lengths (S ₁ < S ₂) only by providing an appropriate speed of flow along the first and second routes. It should be noted that the proposed method for the distribution of traffic flows, consisting of cars, helps to separate one-quarter, one-third, one-half, or two-thirds of the main flow and direct it to another alternative route. Based on the obtained indicator of the traffic volume of the bottleneck, it is possible to determine the priority and long-term measures to ensure its traffic capacity. Thus, the main measures for improving the efficiency of the street and road network operation in Ukraine have both local and global natures. Local measures involve the following: improving the parameters of traffic-light regulation; ensuring the efficient traffic control of vehicles and pedestrians at intersections; ensuring the compliance of traffic capacity of bottlenecks with the traffic demand and safety headway; optimal distribution of vehicles along lanes; allocating separate lanes for public transport; and constructing or reconstructing road sections, bridges, tunnels, overpasses, intersections, etc. Global measures imply the following: ensuring compliance between the number of available workplaces in the city and its demand; determining the optimal traffic route; ensuring the adequate number and optimal placement of elements of transport communication between the city districts; directing vehicles to alternative traffic routes; introducing an intelligent traffic control system; imposing tolls on overloaded sections of roads; and improving operating conditions of public transport.

It is worth noting that three main measures are applied in the largest cities of Ukraine that ensure the efficient operation of the urban street and road network (Stepanchuk et al., Reference Stepanchuk, Bieliatynskyi, Pylypenko, Murgul and Pukhkal2021). These measures differ significantly in their complexity and cost and involve the following:

1. (1) Increasing the traffic capacity of bottlenecks by expanding the roadway at signal-controlled and uncontrolled intersections, reconstructing existing obsolete overpasses, and constructing new multilevel interchanges, bridges, tunnels, overpasses, etc.;

2. (2) Designing and implementing an intelligent traffic control system, which instantly processes and rapidly adopts an appropriate decision on the optimal distribution of traffic flows; notifying drivers of the need to change their direction of movement and choose an alternative route with the help of multi-position road signs, electronic information boards, radio communication, and GPS navigation system;

3. (3) Expanding and developing the public transport network (city train, subway, high-speed tram, etc.) and ensuring its comfort conditions (Figure 6).

Figure 6. Measures to ensure the effective operation of the urban street and road network

The adoption of a reliable decision to ensure the effective operation of the street and road network lies in its economic justification. In order to confirm the decision, it is necessary to determine the economic feasibility of the construction or reconstruction of a particular element of the street and road network. The decision can be considered cost-effective if the benefit exceeds the construction or reconstruction cost (Britchenko et al., Reference Britchenko, Filyppova, Niekrasova, Chukurna and Vazov2022).

Empirical research shows that the main approach to the methods for improving the efficiency of the street and road network operation involves designing a traffic flow management system based on an operational solution to the problem of reducing traffic jams and the possibility of their occurrence. Furthermore, such a system obtains reliable information on the most efficient routes and calculates and determines the optimal routes in real-time mode. All the above makes it possible to distribute traffic flows rationally along the urban street and road network. Such an approach provides solutions to problems both local and global in nature (for the entire network) (Figure 7).

Figure 7. Methodology for the effective operation of the urban street and road network

The economic effect and type of optimisation should also be considered. The introduction of measures for increasing the efficiency of the street and road network operation opens up new opportunities for the economic development of the city, accompanied by the following socio-economic effects:

• – reduce the travel time for personal transport vehicles;

• – public transport will be able to adhere to its schedules;

• – increase road safety and decrease the number of RTA;

• – timely delivery of cargo will increase the efficiency of transportation and the work of enterprises;

• – emergency vehicles will be able to move faster and will not interfere with other road users;

• – less operating time of internal combustion engines will decrease emissions of harmful substances into the environment;

• – the absence of frequent changes in acceleration, braking, or parking modes will reduce the physical wear of vehicles;

• – decrease in psychological irritation (stress and transport fatigue) of drivers and passengers caused by prolonged downtime in traffic jams.

The values of indicators of the socio-economic effect show how much the costs of developing the street and road network are offset by the benefits that the population or enterprises of the city receive.