A measure of the importance of roads based on topography and traffic intensity
AA measure of the importance of roads basedon topography and traffic intensity (cid:63)
Krzysztof J. Szajowski and Kinga Włodarczyk Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics,Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
[email protected]://szajowski.wordpress.com/ Wrocław University of Science and Technology [email protected]
Abstract.
Mathematical models of street traffic allowing assessment of the im-portance of their individual segments for the functionality of the street systemis considering. Based on methods of cooperative games and the reliability the-ory the suitable measure is constructed. The main goal is to analyze methodsfor assessing the importance (rank) of road fragments, including their functions.A relevance of these elements for effective accessibility for the entire system willbe considered.
Keywords: component importance · coherent system · road classification · graphmodels · traffic modelling.
Subject Classifications :MSC 68Q80 · (90B20; 90D80) The function of a road network is to facilitate movement from one area to another. Assuch, it has an important role to play in the urban environment to facilitate mobility.It furthermore determines the accessibility of an (urban) area (together with publictransport options). In many studies on the design and maintenance of roads, the au-thors raise the problem of alternative connections needed to ensure efficient transportbetween strategic places (cf. Lin (2010), Tacnet et al. (2012)). It is known that individ-ual segments of the road structure are exposed to various types of threats, resulting intemporary disconnection of such couplings. As a result, the road network determinesthe quality of life in the analyzed area. Therefore, it is worth trying to define mea-surable parameters, the quality of road connections, the road system constituting theinfrastructure used for transport. Further considerations focus on road systems for roadtransport. However, the proposed approach can be successfully applied to other similarstructures.When designing, it is worth conducting an analysis of the effects of excluding individ-ual segments and determining the measures that allow for the identification of criticalones. However, the difficulty with this kind of economic appraisal is first of all that itis not easy to measure the valuation of travel time. Different people and organizationsvalue travel time in different ways, depending on many factors such as income, goal ofthe trip, social background, etc (cf. Cherlow (1981)). It is relatively easier to measure thevalue travel time than the highway security measure (v. Sharpe (2012)). The purpose (cid:63)
Presented at
Bernoulli-IMS One World Symposium 2020, August 24-28, 2020 . Recordedpresentation is here. a r X i v : . [ m a t h . O C ] J a n K. Szajowski and K. Włodarczyk of the work is to determine the importance of roads segment in road traffic. Consider-ation will be commonly known measures of significance used to evaluate componentsof binary systems. Road topography is a long-term process that cannot be changedin a short time. Therefore, it is important to ensure safe road traffic when planningcommunication infrastructure. To this end, it is important to introduce objective meth-ods for assessing weak links in the road system. Using methods of stochastic processesand game theory, a quantitative approach to the importance of various elements of in-frastructure will be proposed. The introduced connection assessment proposals will beillustrated using information about the actual local road network in the selected city(see Example 1.2).
In the presented work, the network of streets ensuring access from point A to point B in Zduńska Wola will be treated as a system. The diagram of the streets analyzed canbe seen in Figure 1a. Let us emphasize that the purpose of modeling is not to reflect thecurrent traffic on the network, as shown in Figure 1b, but to establish the importanceof network elements due to their objective importance for the functioning of the roadsystem. (a) Road segments from A to B. (b) Google Maps presentation of the typicaltraffic load on the streets. Fig. 1: Analysed traffic network.In research, there are many measures that allow assessing the importance of individ-ual components, based on the system structures, lifetimes and reliability of individualcomponents or methods of estimating significance based on the methods of turningon and off. The most classic methods based on reliability theory will be used in the pa-per. To this end, the street network will be presented in the form of a system, where eachroad is presented as a separate component. Then, based on the construction of the sys-tem, the structure function will be determined, thanks to which it will be possible to cal-culate the meaning of individual components and the corresponding streets. The nextstage will be defining the theory of traffic in the context of significance measures. Thisarea will be examined in relation to the satisfaction and comfort of drivers. Driverssatisfaction means that the system works properly, if not the system is failed. So as reli-ability of particular road we consider probability of driver’s satisfaction from the journey.A road connection system in a given area should allow transport in a predictable time Source: Google Maps. measure of the importance of roads 3 between different points. Extending this time has a negative effect on drivers. As a re-sult, their right ride quality is compromised and they are more likely to fail to complywith the rules. Therefore, providing drivers with driving comfort and satisfaction is alsoimportant for general road safety. The use of this approach can, therefore, be a guidefor both drivers and road builders planning road infrastructure.
The purpose of the research presented here is to implement of various importance mea-sures introduced in the reliability theory to analyze the impact of elements of roadnetworks. The theory related to significance measures and their use in traffic theoryis described in the section 2. There is a close relationship between road delays andthe construction and function of both the road and the intersection that forms partof it. For this purpose, simulations of vehicle traffic on the analyzed roads were per-formed. The theory related to the method of modeling vehicle traffic and their behaviorat intersections is described in section 3. Section 4 describes the real traffic network, itstransfer to the simulation model, and the results obtained in this way. Then, on thisbasis, the importance of individual fragments was calculated depending on the intensityof traffic on these roads.Considering this work is a look at the impact on the comfort of communicationof the road structure in connection with traffic without directly referring to the behav-ior of drivers, which was devoted the paper Szajowski and Włodarczyk (2020). In theseprevious works, significant dependence on traffic quality on drivers’ compliance with ap-plicable rules was shown. Here, a similar approach was applied to the condition of chang-ing behavior to incorrect, which may further result in a deterioration in traffic quality.Therefore, the results obtained show important elements of the road network that havean impact on road safety and properly functioning.
The operation of most systems depends on the functioning of its individual components.It is important to ensure the proper running of the entire system. To this end, it is im-portant to assess the contribution of individual components. In road networks, networkcurves model road segments, intersections, and special places on the road that have a sig-nificant impact on the flow of traffic, such as railway crossings, tunnels, bridges, viaductsor road narrowing. In order to estimate the importance of particular elements, the con-cept of importance measures was introduced (for detailed description of the conceptand its extension to multilevel elements and systems see review paper by Amrutkar andKamalja (2017)). Since 1969 researchers offer various numerical representations to deter-mine which components are the most significant for system reliability. It is obvious thatthe greater are these values, the more this element have on the functioning of the en-tire system. The significance of individual elements depends on the system structureas well as the specificity and failure rate of individual elements. There are three basicclasses of measures of importance (v. Amrutkar and Kamalja (2017), Birnbaum (1969),Średnicka (2020)):i
Reliability importance measures subordinate changes in the reliability of the sys-tem depending on the change in the reliability of individual elements over a givenperiod of time and in depend on the structure of the system.ii
Structural importance measures are using when just the structure of the systemis known. Depending on the position of the components in the system, their relativeimportance is measured.
K. Szajowski and K. Włodarczyk iii
Lifetime importance measures focus on both, components position in the systemand lifetime distribution of each element. According to Kuo and Zhu if it is a functionof the time it can be classified as Time-Depend Lifetime(TDL) importance andif it is not a function of time we have Time Independent Lifetime(TIL) importance.Moreover, depending on the number of states, systems can be divided into two types:i
Binary systems — comprised of n components, where each of them can haveprecisely one of two states. State when the component is damaged and state when is working.ii Multistate systems (MSS) — comprised of n components, which can undergoa partial failure, but they do not cease to perform their functions and do not causedamage to the entire system. Establishing the hierarchy of components of a complex system has been reduced tomeasuring the influence of the element state on the status of the entire system. Theconcept of an element (system) state depends on the context. For the needs of the roadnetwork analysis, we assume, similarly to the reliability theory, a binary descriptionof both elements and the system (v. also Ramamurthy (1990)). For this purpose, wewill use the known results on significance measures obtained in research on this subjectdeveloped in recent years. Importance measures have been developed in many directionsand under many definitions. However, one of the most popular areas of development andapplication are: – theory of cooperative games (in simple game) – reliability theory (in coherent and semi-coherent structure)Many methods have been developed to combine and standardize the terminology associ-ated with both applications. Therefore, to begin with, we must briefly mention the rela-tionship between importance measure theory in the context of both concepts. The firstattempt to define it was made by Ramamurthy (1990), so the following notation wasproposed:(1) ∅ ∈ P , where P is set of subset of N ;(2) N ∈ P , where N is finite, nonempty set;(3) S ⊆ T ⊆ N and S ∈ P imply T ∈ P .The concepts related to game theory and reliability theory were compared with eachother and on this basis, it was possible to define the relationship between these concepts.To begin with, it is easy to see the relationship between players and components. Accord-ing to game theory, we have a set of players N = { , , , . . . , n } and a family of coalitions N . In the theory of reliability, we have a set of components N = { , , , . . . , n } , wherethe components and the entire system can be in two states, state 1 for functioning andstate 0 for failed. Similarly is in game theory, where λ : 2 N → { , } , which is appliedin simple game if on set N form of characteristic function fulfils(1) λ ( ∅ ) = 0 ;(2) λ ( N ) = 1 ;(3) S ⊆ T ⊆ N implies λ ( S ) ≤ λ ( T ) .Here this characteristic function has its counterpart as a structure function, and simplegames as semi-coherent structures. In addition, also winning and blocking coalitions arecomparable to path and cut sets. measure of the importance of roads 5 In this paper, the reliability concept of application of Importance measures willbe considered, and the traffic network will be shown as a system. That way, in the restof this paper we will use reliability terminology.
In the classical approach the system and their elements are binary (v. Birnbaum(1969),Birnbaum et al.(1961)). Let the system comprised of n components can be denotedby c = ( c , c , ..., c n ) . The description of the vector of component states (in the short state vector ) x = ( x , x , ..., x n ) , where each x i = χ W ( c i ) , c i ∈ { W, F } ( W - means theelement is functioning; F - means the element is failed). For state vector, we can usebelow notations [5] (cid:126)x ≤ (cid:126)y if x i ≤ y i for i ∈ { , . . . , n } (cid:126)x = (cid:126)y if x i = y i for ∀ i ∈{ ,...,n } (cid:126)x < (cid:126)y if (cid:126)x ≤ (cid:126)y and x (cid:54) = y (1 i , x ) = ( x , x , x , . . . , x i − , , x i +1 , ..., x n ) = (1 , x − i )(0 i , x ) = ( x , x , x , . . . , x i − , , x i +1 , ..., x n ) = (0 , x − i ) (cid:126) , , . . . , (cid:126) , , . . . , .If the structure of the system is known, we can define the state of the system φ ( (cid:126)x ) as Boolean function ( structure function ) of the state vector.If from x i ≤ y i for i ∈ { , . . . , n } results φ ( (cid:126)x ) ≤ φ ( (cid:126)y ) , and φ ( (cid:126)
1) = 1 , φ ( (cid:126)
0) = 0 , then wecall the system coherent . It is known (v. Birnbaum (1969)) that for every i = 1 , , . . . , n structure function can be decomposed as follows: φ ( (cid:126)x ) = x i · δ i ( (cid:126)x ) + µ i ( (cid:126)x ) , (1)where δ i ( (cid:126)x ) = φ (1 i , (cid:126)x ) − φ (0 i , (cid:126)x ) , µ i ( (cid:126)x ) = φ (0 i , (cid:126)x ) are independent of the state x i of the component c i .In addition, we can observe situations where the system can be functioning evenif some components are failed. The smallest set of functioning elements that ensuresthe operation of the entire system is called minimal path . The opposite situation is ob-served in the case of minimal cut set , which is the minimum set of components whosefailure cause the whole system to fail. We can define the structure function as a parallelstructure of minimal paths. According to the definition, this structure is damaged if,and only if all of the components are failed. So the system consists of n minimal pathsseries, denoted by ρ i ( · ) , for i = 1 , , . . . , n , can be presented as: φ ( (cid:126)x ) = n (cid:97) i =1 ρ i ( (cid:126)x ) = 1 − n (cid:89) i =1 (cid:2) − ρ i ( (cid:126)x ) (cid:3) . (2)Similarly, the structure function can be presented as series of minimal cut sets. So for n minimal cut parallel structures, marked by κ i ( · ) , for i = 1 , , . . . , n , structure functionlooks as follows: φ ( (cid:126)x ) = n (cid:89) i =1 κ i ( (cid:126)x ) . (3)If we simply replace the minimum paths and minimum cut sets with components, the for-mulas (2) and (3) apply for serial and parallel components.In most of the considerations about the functioning of systems, it is assumed thatelements work independently. Then the state of i -th element is a binary random variable X i and the reliability that the element i is unimpaired will be denoted by p i , where p i = P ( X i = 1) = 1 − P ( X i = 0) . (4) K. Szajowski and K. Włodarczyk
We also define the vector of reliabilities for n elements by (cid:126)p = ( p , p , . . . , p n ) . (5)Based on reliabilities vector and structure function we can define the probability of the sys-tem functioning P ( φ ( x ) = 1 | (cid:126)p ) = E [ φ ( x ) | (cid:126)p ] = h φ ( (cid:126)p ) . (6)For the structure φ ( (cid:126)x ) function h φ ( (cid:126)x ) is called reliability function . As was introduce, reliability importance measures are based on changes in reliabilitiesof components and on the system structure. This measure first time was introducedby Birnbaum (1969). At the beginning, from formulas (4), (5) and (1) he express the re-liability function by h φ ( (cid:126)p ) = p i · E [ δ i ( X )] + E [ µ i ( X )] , where, for every i = 1 , , . . . , n and according to equation (1), we have ∂h φ ( (cid:126)p ) ∂p i = E [ δ i ( (cid:126)X )] = E (cid:34) ∂φ ( (cid:126)X ) ∂X i (cid:35) . According to Birnbaum (1969) the reliability importance of the component c i for struc-ture φ ( · ) is defined as I i ( φ ; p ) = I i ( φ, p ) + I i ( φ, p ) , where I i ( φ, (cid:126)p ) = P { φ ( X ) =1 | X i = 1; (cid:126)p } − P { φ ( (cid:126)X ) = 1; (cid:126)p } , and I i ( φ, p ) = P { φ ( X ) = 0 | X i = 0; p } − P { φ ( X ) =0; p } . We have the following useful identities I i ( φ ; 1; (cid:126)p ) = (1 − p i ) · ∂h ( (cid:126)p ) ∂p i = E [(1 − X i ) δ i ( X )] I i ( φ ; 0; (cid:126)p ) = p i · ∂h ( (cid:126)p ) ∂p i = E [ X i δ i ( X )] I i ( φ ; (cid:126)p ) = ∂h ( (cid:126)p ) ∂p i = E [ δ i ( X )] . The Birnbaum importance measures for i = 1 , , . . . , n have forms (the symbol φ isdroped for short) B ( i | (cid:126)p ) = ∂h ( (cid:126)p ) ∂p i = ∂ [1 − h ( (cid:126)p )] ∂ [1 − p i ] , (7)here B ( i | p ) is p dependent. In case when reliabilities vector (cid:126)p is unknown, we haveto consider structural importance defined for i = 1 , , ..., n in the following way B ( i ) = I i ( φ ) = ∂h ( (cid:126)p ) ∂p i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = ... = p n = , (8)this information will be useful in the next section. When we looking for relevant component c i for the structure φ ( · ) and the state vector (cid:126)x is known, we are going as follow definition δ i ( (cid:126)x ) = φ (1 i , (cid:126)x ) − φ (0 i , (cid:126)x ) = 1 . We canalso highlight definitions if the component c i is relevant for the functioning of structure I i ( φ, (cid:126)p ) and I i ( φ, (cid:126)p ) are the reliability importance of the c i component for functioningand failure of the structure, respectively. measure of the importance of roads 7 φ ( · ) at the state vector (cid:126)x if (1 − x i ) · δ i ( (cid:126)x ) = 1 , and, if the component c i is relevantfor the failure of structure φ ( · ) at the state vector (cid:126)x gives x i · δ i ( x ) = 1 . Distinctly,depends on if the coordinate x i of the vertex (cid:126)x is equal to or , then c i is relevant forfunctioning or failure of the system.Birnbaum (1969) defined structural importance measure of component c i for the func-tioning of the structure φ ( · ) as I i ( φ,
1) = 2 − n (cid:80) ( x ) (1 − x i ) · δ i ( x ) , where sum extendson all combinations n of vertices of the state vectors. In the similar way is defined structural importance measure of the component c i for the failure of structure φ ( · ) by I i ( φ,
0) = 2 − n (cid:80) ( x ) x i · δ i ( x ) . Finally, by summarizing, the structural importancemeasure of the component c i for the structure φ ( · ) is I i ( φ ) = I i ( φ,
1) + I i ( φ,
0) =2 − n (cid:80) ( x ) δ i ( x ) .Barlow and Proschan (1975) used a more extended approach to structural measures.Their point of view assumes that all components have a continuous lifetime distribution,denoted by F i , for i = 1 , , . . . , n . It is possible to calculate the probability of a systemfailure caused by the c i component. For the c i component, which is described by the dis-tribution F i and the density function f i , the probability that a system failure at time t was caused by the c i component can be described as follows [ h (1 i , ¯ F ( t )) − h (0 i , ¯ F ( t ))] f i ( t ) (cid:80) nk =1 [ h (1 k , ¯ F ( t )) − h (0 k , ¯ F ( t ))] f k ( t ) . (9)In the consequence of (9), it obvious to define the probability that failure of the systemin [0 , t ] was caused by the c i component is (cid:82) t [ h (1 i , ¯ F ( u )) − h (0 i , ¯ F ( u ))] dF i ( u ) (cid:82) t (cid:80) nk =1 [ h (1 k , ¯ F ( u )) − h (0 k , ¯ F ( u ))] dF k ( u ) . Here, if t → ∞ , then we obtain that the system finally failed it was caused by the com-ponent c i . In this case, we have to note that the denominator is equal to 1. This limitis taken as a definition of component importance .Importance measures according Barlow and Proschan definition we will denotedby I BSi ( φ ) . We have I BPi ( φ ) = (cid:90) [ h (1 i , p ) − h (0 i , p )] dp, (10)where (1 i , p ) and (0 i , p ) is a probability vector where i -th component has probabilityequal 1 or 0, relatively.For further calculations, let us remind quick note from Section 2.2, that minimal path is the minimal set of elements, which ensures the proper functioning of the system. Basedon this we can define critical path set for component c i as { i } ∪ { j | x j = 1 , i (cid:54) = j } . In thisway, information about the system is functioning or failed is determined by the c i com-ponent functions or fails. A critical path vector (or set ) for the component c i , and itssize r , we have (cid:80) i (cid:54) = j x j = r , for r = 1 , , . . . , n . The formula for counting the numberof vectors of critical paths for the component c i with size r is the following n r ( i ) = (cid:88) (cid:80) i (cid:54) = j x j = r − [ φ (1 j , x ) − φ (0 j , x )] . Finally, we can define the structural importance of the component c i using the numberof vectors of critical paths n r ( i ) as follows I BPi ( φ ) = n (cid:88) r =1 n r ( i ) · ( r − n − r )! n ! . (11) K. Szajowski and K. Włodarczyk
The equation (11) can be also presented in two more interesting expressions. The firstexpression is the following I BPi ( φ ) = 1 n n (cid:88) r =1 n r ( i ) (cid:0) n − r − (cid:1) − , where n r ( i ) describes the number of vectors of critical paths with size r . The denom-inator in the above equation represents the amount of results in which precisely r − components are in operation among the n − components without c i component. Secondadditional representation of equation (11) can be written as follows I BPi ( φ ) = (cid:90) (cid:104) n (cid:88) r =1 n r ( i ) · (cid:0) n − r − (cid:1) − (cid:0) n − r − (cid:1) · (1 − p ) n − r · p r − (cid:105) dp, here (cid:0) n − r − (cid:1) (1 − p ) n − r p r − means the probability that from the n − components with-out c i component, r − elements are functioning. What’s more, n r ( i ) (cid:0) n − r − (cid:1) − meansthe probability that r − functioning elements including c i component determine the crit-ical path set to the c i component. So multiplication of them means the probability thatcomponents c i is responsible for system failure and integral of it over p is that reliabilityfor the component c i is a uniform distribution p ∼ U (0 , .As it was written at the beginning in Section 2.1 there is a big connection betweenthe concepts related to game theory and the theory of reliability. The measure introducedby Barlow and Proschan is an example of this. This definition is reflected in cooperativegames as Shapley’s value, which informs what profit a given coalition player can expect,taking into account his contribution to any coalition. As was said in section 1, binary systems are considered. The analyzed system is a streetnetwork allowing access from A to B , it is possible in several ways. We assume thatdrivers drive only from A to B , straight, without unnecessary U-turns on the route.Streets were presented at the beginning in Fig. 1a and can be transform to the formof the system (a scheme) as on Fig. 2.Fig. 2: Analysed traffic network presented in system form. measure of the importance of roads 9 Based on the system representation of the streets network we can determine the struc-ture function. As we know, the structure function can be defined using either minimalpath set or minimal cut set . So for the given structure both sets are presented in thetables 1 and 2.Table 1: Minimal path set. Path Elements1 1 2 3 8 122 1 2 5 9 11 123 4 6 9 11 124 4 7 10 11 12
Table 2: Minimal cut set.
Cut Elements Cut Elements1 1 4 11 3 9 102 2 4 12 8 9 73 1 6 7 13 8 9 104 2 6 7 14 3 4 95 4 5 3 15 4 8 96 4 5 8 16 3 117 1 6 10 17 8 118 2 6 10 18 1 119 3 5 6 7 19 2 1110 3 9 7 20 12
Based on tables 1 and 2, it is possible to define minimal path series structuresrepresented by the following equations ρ ( x ) = (cid:89) { , , , , } x i = x · x · x · x · x ρ ( x ) = (cid:89) { , , , , , } x i ρ ( x ) = (cid:89) { , , , , } x i ρ ( x ) = (cid:89) { , , , , } x i and minimal cut parallel structures described as follows κ ( x ) = (cid:97) { , } x i = x (cid:113) x κ ( x ) = (cid:97) { , } x i κ ( x ) = (cid:97) { , , } x i κ ( x ) = (cid:97) { , , } x i = x (cid:113) x (cid:113) x κ ( x ) = (cid:97) { , , } x i κ ( x ) = (cid:97) { , , } x i κ ( x ) = (cid:97) { , , } x i = x (cid:113) x (cid:113) x κ ( x ) = (cid:97) { , , } x i κ ( x ) = (cid:97) { , , , } x i κ ( x ) = (cid:97) { , , } x i = x (cid:113) x (cid:113) x κ ( x ) = (cid:97) { , , } x i κ ( x ) = (cid:97) { , , } x i κ ( x ) = (cid:97) { , , } x i = x (cid:113) x (cid:113) x κ ( x ) = (cid:97) { , , } x i κ ( x ) = (cid:97) { , , } x i κ ( x ) = (cid:97) { , } x i = x (cid:113) x κ ( x ) = (cid:97) { , } x i κ ( x ) = (cid:97) { , } x i κ ( x ) = (cid:97) { , } x i = x (cid:113) x κ ( x ) = (cid:97) { } x i = x From the definition in equation (2) and based on the above equations, we can writethe structure function of the presented system as follows φ ( x ) = ρ ( x ) (cid:113) ρ ( x ) (cid:113) ρ ( x ) (cid:113) ρ ( x ) == 1 − (1 − ρ ( x ))(1 − ρ ( x ))(1 − ρ ( x ))(1 − ρ ( x )) In addition, our structure function can be also expressed by the series of minimal cutstructures φ ( x ) = (cid:89) i =1 κ i ( x ) . And finally, thanks to equation (6), we can write the reliability function of the analyzedsystem h φ ( p ) = 1 − (1 − (cid:89) { , , , , } p i )(1 − (cid:89) { , , , , , } p i )(1 − (cid:89) { , , , , } p i ) (12) × (1 − (cid:89) { , , , , } p i ) , where p i , for i = 1 , , . . . , , are some probabilities, which definition will be introducein next section. To consider reliability importance we need to define what exactly means that systemis functioning or failed. We assume that the condition of the system’s functioningis the comfort and satisfaction of drivers. The state will mean the driver’s satis-faction with a given road section or route, the state — dissatisfaction. For drivers,the measure of satisfaction is the travel time on a given section of the road, and moreprecisely, the realization of the road according to the planned travel time. Drivers wantto finish the journey in the shortest possible time. The excess of this time, i.e. the delayon a given section of the road after exceeding a certain critical level causes dissatis-faction of drivers with the journey. This critical level that causes dissatisfaction maybe different for each driver and is close to the lifetime. Weibull distribution is often usedto represent the lifetime of objects. A similar approach was used in a paper publishedby Fan et al. in 2014. The cited article considered a situation when, while waiting beforeentering the intersection, the waiting time for a given driver exceeded a certain criti-cal value, the driver stopped complying with traffic rules. Like here, this critical valuewas determined by Weibull distribution. The variable from the Weibull distribution canbe represented as the cumulative distribution function given by the following formula: F ( t ) = (cid:40) − exp (cid:110) − (cid:0) tλ (cid:1) k (cid:111) , for t > , , otherwise.Based on the cumulative distribution function, it is possible to calculate the reliabilityfunction, i.e. the function that tells the probability of correct functioning of an object.We parametrize the segments by the acceptable delay time t by the driver. The pop-ulation of the diver is not homogeneous. The acceptable delay is the random variablewith some distribution Π . The delay of travel τ is a consequence of various factors.Let us assume that its cumulative distribution is F ( t ) . We will say that the segmentis reliable or works for given driver with accepted delay t if τ ( ω ) ≥ t . The probability Q ( t ) of the event is the subjective driver reliability of the segment. Its expected proba-bility with respect to Π , p = (cid:82) ∞ Q ( u ) dΠ ( u ) is mean reliability of the segment. For thehomogeneous class of drivers the delay time ξ on given road section is common valuefor all drivers, so (mean) reliability will means probability that for assumed delay timedriver is satisfied from the journey. Therefore, according to the theory of importancemeasures , p i , which indicates the reliability of the segment will be determined as proba-bility of driver’s satisfaction and − p i will means probability that driver is dissatisfiedof journey for assumed delay time. It can be determine as following formula: p i = P ( X i = 1 | t = ξ ) = 1 − P ( X i = 0 | t = ξ ) = Q ( ξ ) , measure of the importance of roads 11 where ξ is the delay time, X = 1 means the driver is satisfied with the road, X = 0 – heis dissatisfied. The paper adopts the same Weibull distribution parameters as in paperby Fan et al. (2014), i.e. λ = 30, k = 2 . . When we know the relationship between streetreliability from the time of delays, we can define the reliability of these route fragmentsfor a given traffic intensity. Different road sections react differently to increasing trafficintensity, which is why reliabilities will be different. Using simulations we determinethe dependence of traffic intensity and delay times. We will now proceed to briefly introduce these definitions on a simple example. Letus assume that we have a shortened road network scheme limited to Piwna 1, Zlot-nickiego, Laska, Sieradzka 1 streets. This scheme is presented in the way shown in Fig-ure 3. Here we have the components c i c in series, and the components c and c Fig. 3: Short version of analysed scheme of traffic network.in parallel. So we can define this system as " k –out–of– n " structure, where n is numberof all components, k means number of components in series, and n − k is the num-ber of components in parallel. On this basis, we can define the structure function as φ ( (cid:126)x ) = x · (1 − (1 − x ) · (1 − x )) · x , and the system reliability function correspond-ing to the above h ( (cid:126)p ) = p · (1 − (1 − p ) · (1 − p )) · p .To begin with, we assume that the reliability of individual components is unknown,so only structural measures of significance can be calculated. They will be calculatedbased on the definitions introduced in Section 2.4. Two proposals for structural measureshave been introduced: first proposed by Birnbaum, which assume that each reliability p i of components c i , for i = 1 , , . . . , n are the same and equal to and second the Bar-low and Proschan Importance Measures, which is define for p ∈ [0 , . So using thistheory and definitions in equation (8) for Birnbaum Importance and in (10) for Barlowand Proschan Importance, we count the importance of analyzed components. Obtainedresults are presented in Table 3. We see that roads connected in series are more im-Table 3: Structural importance of roads in the analyzed system. Id Street Birnbaum Barlow-Proschanname Importance B ( i ; φ ) Importance I BPi ( φ ) portant than roads connected in a parallel way. This is consistent with the logic, if one of the parallel roads is blocked, you can always choose a different route that will allowyou to reach your destination. For streets in a serial connection, this is not possible.We also note the differences in the values of the importance measures calculated us-ing the Birnbaum and Barlow-Proschan definitions, this is because the first measureis calculated for the constant reliability of the elements equal to , so it only examinesthe relationship between element positions. The second measure takes into account,apart from the structure itself, also the variability of reliability of individual elements.Now we will examine the reliability importance measures for the simplified systemshown in Figure 3. Let us assume that for a given traffic intensity, we have certaindelay times, on this basis, we will calculate the probability that drivers are still satisfiedwith the travel along the road, i.e. the reliability of the road. Next, for these values,using the formula (7), we calculate the value of the measures of significance definedby Birnbaum. The assumed delay times, as well as the corresponding reliability andimportance, will be presented in Table 4. As we can see, with a road delay of aboutTable 4: Hypothetical calculations of importance measures in the example system. Id Street Delay Probability of Importancename ξ satisfaction Q ( ξ ) B ( i | p )
25 seconds, the likelihood of driver satisfaction is close to , and in the case of delaysof about 5 seconds, drivers do not experience almost the negative effects of a slowdownin traffic. With such reliability of streets and with such a scheme, it is easy to noticesome issues: streets in a parallel position have less contribution to potential nervousnessor driver satisfaction than in a serial connection, in addition, in the case of streetsin a series, those with less reliability are more important, so these should be paid greaterattention to maintain proper traffic quality. In the case of streets in parallel connection,streets with greater reliability are more important. It is logical that drivers knowingthat the road is a better way will choose it, so it is important to constantly maintainit in good condition because when it fails the whole connection will lose much reliability. As was presented in the previous sections, if the reliability of individual componentsis unknown, it is possible to use structural measures of significance. Therefore, we willbegin our considerations about the analyzed system by calculating the structural sig-nificance of individual roads. In the same way as in the previous section, the definitionof significance measures introduced by Birnbaum, and Barlow and Proschan presentedin section 2.3 by the equations (8) and (10), respectively, were used. The results obtainedare presented in Table 5. We see that the results of both measures are similar. As wasexpected, the most important for the entire route is street Sieradzka 2, because eachroute finally leads along this street, for B-P importance for these streets is bigger thanfor B-importance. Next, the most important part of the route is Sieradzka 1, we see that3 of 4 ways to obtain point B are going by this street. For this street, Birnbaum’s valueis smaller than the Barlow-Proschan’s value. The importance of Piwna 1 is the last valueof importance bigger than . , anyway similar to this value are importances of Dolna,Zlota, and Nyska 2. Surmise that the significance of the Zlota and Dolna will be close measure of the importance of roads 13 Table 5: Structural importance of roads in the analyzed system.
Id Street Birnbaum Barlow-Proschanname Importance B ( i ; φ ) Importance I BPi ( φ ) to the value calculated for Piwna 1 was not difficult. However, it is not so easy to guessthe similarity of the importance of Nyska 2 street to Dolna and Zlota. Mickiewicza,Zlotnickiego, Piwna 2, Jasna, Laska and Nyska 1 streets have the smallest contributionto the proper functioning of the entire connections between A and B . Traffic modeling is a particularly complex issue. There are both modeling of individ-ual phenomena occurring on roads and entire road networks. The first research intovehicle movement and traffic modeling theory began with the work of Bruce D. Green-shields(1935). On the basis of photographic measurement methods, he proposed basicand empirical relationships between flow, density, and speed occurring in vehicle traffic.Next, Lighthill and Whitham (1955) and Richards (1956) introduced the first theoryof movement flow. They presented a model based on the analogy of vehicles in trafficand fluid particles. Interest in this field has increased significantly since the nineties,mainly due to the high development of road traffic. As a result, many models were cre-ated describing various aspects of road traffic. As a result, many models were createddescribing various aspects of road traffic and focusing on different detail models, we candistinguish: – microscopic models – mesoscopic models – macroscopic modelsThe differences in the models are at the level of aggregation of modeled elements. Meso-scopic models based mainly on gas kinetic models. Macroscopic models based on first andsecond-order differential equations, derived from Lighthill-Whitham-Richards(LWR) the-ory. Microscopic models focus on the simulation of individual vehicles and their in-teractions. One of the most popular are car-following models and cellular automatamodels, the last is used in this paper. The most popular cellular automata trafficmodel is the Nagel-Schereckenberg (1992) model , but also very interesting modelis LAI model (cf. Lárraga and Alvarez-Icaza(2010)), which is more advance than NaSchmodel. LAI model is used in this paper, therefore, in the next section theory aboutcellular automata will be introduced and later will be a more detailed descriptionof LAI model. Janos von Neumann, a Hungarian scientist working at Princeton, is the creator of cel-lular automata theory. In addition, the development of this area was significantly influ-enced by the Lviv mathematician Stanislaw Ulam, who is responsible for discreditingthe time and space of automats and is considered to be the creator of the definition of cel-lular automats as "imaginary physics"[19]. According to a book written by Ilachinski,cellular automata can reliably reflect many complex phenomena with simple rules andlocal interactions. Cellular automata are a network of identical cells, each of which cantake one specific state, with the number of states being arbitrarily large and finite.The processes of changing the state of the cells run parallel and according to the rules.These rules usually depend on the current state of the cell or the state of neighboringcells. From the mathematical point of view, cellular automatas are defined by the fol-lowing parameters [22] [35]: – State space — a finite, k -element set of values defined for each individual cell. – Cell grid — discrete, D -dimensional space divided into identical cells, each of whichat a given time t h has one, strictly defined state of all possible k states. In the caseof the D network, the cell status at i , j is indicated by the symbol σ ij . – Neighborhood — parameter determining the states of the nearest neighbors of a givencell ij , marked with the symbol N ij . – Transition rules — rules determining the cell state in a discrete time t h +1 depend-ing on the current state of this cell and the states of neighboring cells. The stateof the cell in the next step is presented in the following relationship: σ ij ( t h +1 ) = F ( σ ij ( t h ) , N ij ( t h )) , where: σ ij ( t h +1 ) — cell state in position i , j in step t h +1 , σ ij ( t h ) — cell state in position i , j in step t h , N ij ( t h ) — cells in the neighborhood of a cell in position i , j in step t h .The way the cell neighborhood is defined has a significant impact on the calculationresults. The most common are two types: – Von Neumann neighborhood Each cell is surrounded by four neighbors, imme-diately adjacent to each side of the cell being analyzed. The neighborhood for i, j constructed in this way is as follows: N i,j ( t h ) = σ i − ,j ( t h ) σ i,j − ( t h ) σ i , j ( t h ) σ i,j +1 ( t h ) σ i +1 ,j ( t h ) – Moore neighborhood Each cell is surrounded by eight neighbors, four directlyadjacent to the sides of the analyzed cell, and four on the corners of the analyzedcell. The neighbor cell matrix for i , j looks like this: N i,j ( t h ) = σ i − ,j − ( t h ) σ i − ,j ( t h ) σ i − ,j +1 ( t h ) σ i,j − ( t h ) σ i , j ( t h ) σ i,j +1 ( t h ) σ i +1 ,j − ( t h ) σ i +1 ,j ( t h ) σ i +1 ,j +1 ( t h ) There are also modifications to the above types, such as the combined neighborhoodof Moore and von Neumann, as well as numerous modifications to the Moore neighbor-hood itself, and a different way defined by Margolus to simulate falling sand. measure of the importance of roads 15
In addition, boundary conditions are an important aspect of cellular automata the-ory. Since it is impossible to produce an infinite cellular automaton, some of the simu-lations would be impossible because with the end of the automaton’s grid the historyof a given object or group of objects would end. For this purpose, boundary conditionsat the ends of the grid were introduced. There are the following types of boundaryconditions: – periodic boundaries — cells at the edge of the grid behind neighbors have cellson the opposite side. In this way, the continuity of traffic and ongoing processesis ensured. – open boundaries — elements extending beyond the boundaries of the grid ceaseto exist. This is used when new objects are constantly generated, which preventstoo high density of objects on the grid. – reflective boundaries — on the edge of the automaton a border is created, from whichthe simulated objects are reflected, most often it serves to imitate the movementof particles in closed rooms.In the next section the model using cellular automata used in the simulation will be pre-sented. Open boundary conditions are used in our simulations. After leaving the street,vehicles disappear. This is in line with logic, new vehicles are constantly appearing anddisappearing on the roads. The applied neighborhood is a modified version of the pre-sented neighborhoods, because vehicles as their neighbors take those vehicles that arenearby, and more specifically the nearest vehicle on the road, even if it is not directlyadjacent to the analyzed vehicle, and also cars move by more cell. We can assume thatit is a more extended version of the Von Neumann neighborhood. In order to define vehicle traffic rules and simulate their movement, the model proposedby Lárraga and Alvarez-Icaza (2010) was used. The proposed model meets the generalbehavior of vehicles on the road. Drivers with free space ahead are traveling at maximumspeed. Approaching the second vehicle, drivers react to changes in its speed, providingthemselves with a constant space for collision-free braking. This model is often called
LAI model , from the authors’ names. This part of the work will include a descriptionof this model and also comments on possible assumptions.The model presents traffic flow at a single-lane road, where vehicles move from leftto right. The road is divided into 2.5-meters sections, and each is presented as a separatecell. The length of the car is taken as 5 meters what is represented as two cells. Eachcell can be empty or occupied only by part of one vehicle. The position of the vehicleis determined by the position of its front bumper. Vehicles run at speeds from 0 to v max ,which symbolize the number of cells a vehicle can move in one-time step t . The time stepcorresponds to one second. The speed conversion from simulation to real is presentedin the Table 6.Here in the first column, we have the velocity used in the model, next column presentshow distance is done in a one-time step (1 second), the next columns present real velocityin m/s and km/h for better imagine how the model works. In simulations we decideto used maximum speed equals to 45 km/h, because traffic flow in the city is considered,so drivers have not too much space to fast driving.The model takes into account the limited acceleration and braking capabilities of ve-hicles and also ensures appropriate distances between vehicles to guarantee safe driving.Three distances calculated for the car following to its predecessor are included. Thesevalues calculate the distance needed for safe driving in the event that the driver wantsto slow down ( d dec ), accelerate ( d acc ) or maintain the current speed ( d keep ), assuming Table 6: The relationship between real and simulation speeds in the model used.
Velocity v Distance Real speed Real speed that the predecessor will want to suddenly start slowing down with the maximum forceM until to stop. They are calculated as follows: d acc = max (cid:16) , (cid:98) ( vn ( t )+ ∆v ) /M (cid:99) (cid:88) i =0 [( v n ( t ) + ∆v ) − i · M ] − (cid:98) ( vn +1( t ) − M ) /M (cid:99) (cid:88) i =0 [( v n +1 ( t ) − M ) − i · M ] (cid:17) (13a) d keep = max (cid:16) , (cid:98) vn ( t ) /M (cid:99) (cid:88) i =0 [ v n ( t ) − i · M ] − (cid:98) ( vn +1( t ) − M ) /M (cid:99) (cid:88) i =0 [( v n +1 ( t ) − M ) − i · M ] (cid:17) (13b) d dec = max (cid:16) , (cid:98) ( vn ( t ) − ∆v ) /M (cid:99) (cid:88) i =0 [( v n ( t ) − ∆v ) − i · M ] − (cid:98) ( vn +1( t ) − M ) /M (cid:99) (cid:88) i =0 [( v n +1 ( t ) − M ) − i · M ] (cid:17) (13c) Here, vehicle n is the follower, and n + 1 is the preceding car. v n ( t ) means the valueof the velocity of vehicle n in time t , ∆v is the ability to accelerate in one-time step and M is ability to emergency braking.Updating vehicle traffic takes place in four steps, which are done parallel for eachof the vehicles.I. Calculation of safe distances d dec n , d acc n , d keep n .II. Calculation of the probability of slow acceleration.III. Speed update.IV. Updating position. Safe distances.
According to formulas 13 safe distances are counted for each ve-hicles. The calculation of these values is based on the assumption that if the vehiclein the next time step t + 1 increases its speed (or maintains it or slows it down respec-tively) and the driver preceding from the moment t will constantly slow down to speed0 (with maximum ability to emergency braking), there will be no collision. The differentbetween these equation is just in first part, which define traveled distance by vehicle n if it decelerate ( v n ( t + 1) = v n ( t ) − ∆v ), keep velocity ( v n ( t + 1) = v n ( t ) ) or accelerate v n ( t + 1) = v n ( t ) + ∆v , in next time step, and next begins to brake rapidly. The secondpart of equation determines the distance traveled by the preceding vehicle if it startsto braking with maximum force M . Calculation of the probability of slow acceleration.
The second step in the ve-hicle movement procedure focuses on calculating the stochastic parameter R a respon-sible for slowing down vehicle acceleration. It is assumed that low-speed vehicles havemore troubles to accelerate. According to human nature and the mechanism of the car,it is true that is that the faster we go, the easier we manage to accelerate, and standingor driving very slowly cause slower acceleration. The limiting speed at which accelera-tion comes easier is assumed to be 3, which corresponds to 27 km/h. The value of R a measure of the importance of roads 17 parameter is calculated based on the formula R a = min( R d , R + v n ( t ) · ( R d − R ) /v s ) , (14)where R and R d are fixed stochastic parameters, mean respectively probability to ac-celerate when the speed is equal to 0, and probability to accelerate when the speedis equal or more than v s , and v s limit speed below which acceleration is harder.Easily can be seen, that the relationship between the probability of accelerationat speed 0 and at a speed greater than the limit is interpolated linearly, which is takenfrom the idea presented also by Lee et al. [16]. In the simulations, 0.8 and 1 wereadopted as R and R d parameters, respectively, which will not cause frequent difficultiesin accelerating vehicles, however, the stochastic nature of this process will be taken intoaccount. The graph of the R a parameter change for the other parameters adopted in thisway is presented in Figure 4. velocity p r obab ili t y Probability of slow acceleration
Fig. 4: Values of R a parameters for fixed R , R d and v s [15]. Speed update.
In the beginning, as mentioned before ∆v means speed increasein one time step, fixed for all vehicles. v n ( t ) and x n ( t ) determine the velocity andthe position of vehicle n in time t . Distance from vehicle n to vehicle n + 1 is countedby the following formula d n ( t ) = x n +1 ( t ) − x n ( t ) − l s , which exactly means the distance from front bumper of vehicle n pointed by x n ( t ) to rearbumper of the vehicle in the front, presented by the difference between the positionof the front bumper x n +1 ( t ) and the length of the vehicle l s (in cells). The speed updateis done in four steps, the order of which does not matter.1. Acceleration.
If the distance to the preceding vehicle is greater than d aac n thenthe vehicle n increase velocity by ∆v with probability R a , what is presented as fol-lows v n ( t + 1) = (cid:26) min( v n ( t ) + ∆v, v max ) , with prob. R a v n ( t ) , otherwiseIn this rule is assumed that all drivers strive to achieve the maximum velocityif it is possible. Here is include irregular ability to accelerate depends on the dis-tance to preceding vehicles, relevant velocities of both, and stochastic parameterresponsible for slower acceleration defined in Step II.2. Random slowing down.
This rule allows drivers to maintain the current speed, if it al-lows safe driving, it also takes into account traffic disturbances, which are an in-dispensable element of traffic flow. The probability of random events is determined by the R s parameter. If d acc n > d n ( t ) ≥ d keep n , then the updated speed is deter-mined according to the formula v n ( t + 1) = (cid:26) max( v n ( t ) − ∆v, , with prob. R s v n ( t ) , otherwise3. Braking.
This rule ensures that the drivers keep an adequate distance from frontvehicles. Rapid braking is not desirable, so in order to ensure a moderate brakingprocess for the driver, when the free space in front of the car is too small, the vehiclespeed is reduced by ∆v , which reflects optimal braking. v n ( t + 1) = max( v n ( t ) − ∆v, if d keep n > d n ( t ) ≥ d dec n Emergency braking.
As can be seen in real life, it is not always possible to brakecalmly. What is more, road situations often force more aggressive braking. Suchsituations are included in this rule. When the driver gets too close to the other car,or when the other car brakes too much, it forces emergency braking. If the distanceis at least d dec , this rule is not applied. According to the commonly accepted stan-dard proposed in the literature (v. Alvarez and Horowitz Lárraga and Alvarez-Icaza,the emergency braking force is set to − m/s . With respect to the assumed modelparameters the value of M is 2. This step is described by v n ( t + 1) = max( v n ( t ) − M, if d n ( t ) < d dec n Updating position
Finally, with updated vehicle speed, it is possible to actualizevehicle positions. The vehicles are moved by the number of cells according to their speed.This is described by means of x n ( t + 1) = x n ( t ) + v n ( t + 1) , where x n ( t + 1) is actualized position, v n ( t + 1) is the previously determined vehiclespeed, and x n ( t ) is last position of vehicle. Intersections are an inseparable element of road traffic, they are an intersection witha road at one level. All connections and crossroads also count in intersections. Thereare the following types of intersections: – uncontrolled intersections – intersections with traffic signs – crossings with controlled traffic (traffic lights or authorized person)Modeling of traffic at intersections is an important element of road traffic modeling,many models have been created on this subject, such as models simulating the move-ment of vehicles at intersections of type T [34], describing the movement at un-signalizedintersections as in the case of [28], [10] and those considering traffic at intersections withtraffic lights [4]. Typically, these models consist of two aspects, modeling vehicle traf-fic and modeling interactions at intersections. General rules are set for intersections,however, the behavior of drivers who may or may not comply with these rules is alsotaken into account. Modeling of such behavior is also different, which usually distin-guishes these models. This aspects was consider in my engineering thesis [31]. Helpfulin modeling interactions at intersections is game theory, which facilitates the decision measure of the importance of roads 19 about the right of way, where players are drivers in conflict at the intersection, exam-ples of such use can be seen in [21]. Additionally, signalized intersection models usingMarkov chain are often used, as in the case of [33].However, the purpose of the work is simple modeling of road traffic, therefore ad-vanced intersection modeling methods will not be considered. It is assumed that alldrivers comply with traffic regulations and follow road safety. The consequences of chang-ing behavior to incorrect and inconsistent with traffic rules are not investigated. The pur-pose of the work is to find elements that affect the potential threat affecting the reluc-tance of drivers to comply with traffic rules. Depending on the maneuver performedby the drivers and the type of intersection, the following situations need to be modeled: – turn right from the road without right of way – turn right from the road with right of way – turn left from the road with right of way – turn left from the road without right of way – go straight ahead at traffic lights – turn left at traffic lightsWhen modeling the above situations, two basic rules were used: Rule 1 a driver who wants to join the traffic on the main road can do, if and only if,during the whole process, until the maximum speed is reached, he does not disturbthe driving of other vehicles on the main road. This maneuver may be describedby the following formula: l x − v x − v max (cid:88) ∆v =2 min( v max , v x + ∆v −
1) + v max (cid:88) ∆v =2 ∆v > d keep x , where l x is the distance of the vehicle on the main road to the intersection, v x is hiscurrent vehicle speed. The first sum symbolizes the distance traveled by the vehi-cle on the main road until the passing vehicle reaches maximum speed. The secondsum represents the distance traveled by the vehicle joining the traffic until it reachesmaximum speed, assuming that in the first second the vehicle will be at an inter-section with a speed equal 1. Both vehicles increase their speed by 1 in each secondand they do not exceed the maximum speed. The value of the left side of the in-equality must be greater than the distance needed by the driver on the main roadto maintain his speed. Otherwise, the driver would be forced to brake which woulddisturb his movement. Rule 2
The driver wanting to cross the opposite direction road can do it if there is nocollision with the opposite direction during the time needed to complete it andthe opposite driver will not be forced to brake. The time it takes to complete the ma-neuver depends on the initial speed at the start of the maneuver. This relationshipis described in the Table 7.Table 7: Relationship between the time of crossing of the opposite road and the initialspeed.
Velocity v n Need time τ n The condition ensuring the correct execution of the maneuver can be describedby the following inequality: l x − τ n (cid:88) ∆v =0 min( v max , v x + ∆v − > d dec x , where l x is the distance of the opposite car opposite to the intersection and the sumis responsible for calculating the distance traveled by this car in the time neededto complete the turn. The value of the left side of the inequality must be greater thanthe speed needed for safe braking by the vehicle. Otherwise, it would force the driverto emergency braking, which is not desirable, and in the event of a possible delayedreaction of the driver could lead to an accident.The above rules are the basis used to define behavior at intersections, more informationon where the rules were applied, and why, it will be described in Section 4.1.In addition, modelling of traffic lights was needed. The traffic light scheme was usedin accordance with the Polish regulations. The traffic light cycle follows the diagram 5.The duration and meaning of individual signals are as follows:Fig. 5: Traffic light cycle.1. Red light — no entry behind the signal light. The duration is 60 seconds.2.
Red and yellow light — means that in a moment will be a green signal. Accordingto the regulations, it lasts 1 s.3.
Green light — allows entry after the signal light if it is possible to continue drivingand this will not cause a road safety hazard. The duration is the same as for the redsignal, equal to 60 s.4.
Yellow light — does not allow entry behind the signal light, unless stopping the ve-hicle would cause an emergency brake. According to the regulations, it should lastat least 3 seconds.Such a traffic light cycle and the duration of each signal were adopted in the simulation.of course, there is also a relationship between the capacity of intersections and the timeof the traffic light cycle. However, the most standard signaling scheme was adoptedto ensure optimal intersection capacity. In addition, it was assumed that both directionsof travel are equivalent, which is why this cycle is the same on both roads.In accordance with the theory described for traffic modeling, as well as with the pro-posed method of traffic conditioning at intersections. For each street from the diagramin the drawing 1, traffic simulations were performed in the MATLAB package. Real andsimulated street sizes are presented in Section 4.1 in the next chapter. For each simu-lation, the time it took me from the beginning of the road to leaving the intersectionat its end was calculated for each vehicle. Simulations have been carried out many timesfor different probabilities of a new driver appearing on the road, which in the furtherunderstanding will be taken as traffic intensity. measure of the importance of roads 21
An important aspect in the case of traffic modeling, which we could not fail to mentionin this chapter, is the calibration of models. In general, this topic is part of a largerproblem, which is simulation optimization. The area of development of simulation opti-mization in recent years has enjoyed great interest among researchers and practitioners.Simulation optimization is the pursuit of the maximum performance of a simulatedreal system. The system performance is assessed based on the simulation results, andthe model parameters are the decision variables. The assessed performance in this caseis the model’s ability to recreate reality. Therefore, it is a very important topic in mod-eling traffic, which aims to enable the reconstruction of real vehicle traffic, so correctlychoose the model parameters so that the model used is a reliable model and correctlyshows the modeled behavior characteristics. Optimization in the context of motion sim-ulation models has evolved in many areas and the only ones were optimization and cal-ibration of motion, but they were not often combined with optimization theory, wheresome of the problems in motion modeling are well known. One of the most importantconclusions is that there is no algorithm that is suitable for all problems and needsand that the choice of the right algorithm depends on the example being examined (v.Spall et al.(2006)).Most studies focused on testing the performance of the optimizationalgorithm, where models are evaluated against actual traffic data, e.g. Hollander andLiu in 2008. However, based on real traffic data, it is not possible to evaluate the effec-tiveness of the algorithm and the entire calibration process. Another approach proposedin the literature is the use of synthetic measurements, i.e. data obtained from the modelitself. This approach was proposed e.g. by Ossen and Hoogendoorn(2008), and testedchanges in model calibration due to the use of errors in synthetic motion trajectoriesby Ciuffo et al.(2007), which used tests with synthetic data to configure the processof calibration of microscopic motion models, based on trial and error.
Using the models proposed in section 3, a simulation of vehicle movement was performedon each street presented in the Figure 1. The model of vehicle traffic along a straightroad is presented in Section 3.3. The modeling movement between streets was morecomplicated. Section 3.4 describes the general rules needed to define traffic at intersec-tions. There are, various maneuvers required simulation. In addition, the actual roadlengths have been converted into simulation values to best reflect the road traffic. Table8 describes real and simulation road lengths and maneuvers that should be performedon a given road section. In the case of Nyska 1, Sieradzka 1, and Sieradzka 2 streets,drivers go through given section with priority, driving straight ahead.For Dolna and Zlotnickiego roads, drivers join the traffic on the main road beingon a road without the right of way. Rule 1 was applied, assuming that when approachingan intersection, drivers must slow down to a speed of 0 or 1, and then decide accordingto the condition described.At Mickiewicza street, at the end of the road, the driver is forced to slow down to 2,which corresponds to the real speed of 18 km/h, we can assume that this is a reasonablespeed to make a turn. After decelerating, drivers can leave the intersection.For Zlota and Piwna 1 streets, drivers with probability turn left, otherwise theygo straight. Before turning, the drivers slow down to at least speed 2, if they cancross the opposite direction lane they continue driving if they do not slow down more.Therefore, drivers must give way to oncoming vehicles, Rule 2 applies. Table 8: Description and base information on analysed roads.
Id Street Length Intersections and turningname In meters In cells
In the case of Piwna 2 and Jasna streets, we assume that the drivers slow downbefore the intersection to 0 or 1 and with probability turn right or left. In bothsituations, it is necessary to apply Rule 1, because drivers must give way to vehiclesthat are on the road they are turning, in addition in the case of a left turn, Rule 2should be applied too because the vehicle will cross the opposite direction.The last two traffic situations relate to traffic at intersections with traffic lights. Whendriving along Laska Street, in the event of red light, drivers wait at the intersection, thenthey can leave it. The case where drivers would like to turn left is not being consideredbecause in real life a left lane is intended for a left turn. When leaving Nyska 2 Street,drivers may ride to the right, left, or straight. In the case of a right turn or straight ahead,the process goes without any problems, so we allow drivers to leave the intersection.When turning left, you must pass vehicles driving in the opposite direction, so Rule 2applies.According to the above assumptions, simulations were made, and repeated 1000 timesfor each traffic intensity to obtain the average values of delays depending on the inten-sity. The sample code and description of the program are at the end of the work inAppendix A. In accordance with the characteristics described in the previous section, simulationsof motion were made. The results of road delays are shown in graph 6. We see thatthe delay increase characteristics for different roads are different. It is easy to see thatone of the most difficult streets to travel are Jasna and Piwna 2, we see here a highsensitivity to traffic intensity. Another group of streets in terms of delays are Zlotnickiegoand Dolna, and also Laska Street is similar to them, although the growth characteristicsare different. Nyska 2 has a completely different behavior from the rest, but it is the onlystreet with such a complex intersection, including traffic lights. In this case, the delayincreases very quickly, reaching a critical level for this street, related to the capacityof the road. Therefore, despite the fact that the final result of the delay is not the largest,it can be considered that the efficiency of this intersection is the worst. The next, but measure of the importance of roads 23
Traffic intensity T i m e o f de l a y Delays on streets depend on traffic intensity
Piwna2JasnaZlotnickiegoDolnaLaskaPiwna1Nyska2ZlotaMickiewiczaSieradzka2Sieradzka1Nyska1
Fig. 6: Time of delays depend on traffic intensity for different roads.definitely more efficient streets are Piwna 1 and Zlota, and the most fluid traffic canbe seen on the last 4 streets, where there are no intersections and traffic disturbances.In addition, both Sieradzka streets have the same delay times, because they both arewithout intersections streets and they have the same length.Next, using the approach introduced in Section 2.6 and based on the calculated delaytimes, it is possible to determine the probability of driver satisfaction with a given sec-tion of the route. An undesirable phenomenon is exceeding a certain critical level of delaytime, which will cause dissatisfaction to the driver. The probability that the critical valuefor a given delay is not exceeded is described by the reliability function of the Weibulldistribution. Using this, the probability of driver satisfaction for a given delay on eachroad will be calculated depending on traffic intensity. These probabilities are presentedin Figure 7.We see that the reliability of individual streets is different. Ones of the streets
Traffic intensity R e li ab ili t y Probabilities of drivers satisfaction depend on traffic intensity
Piwna2JasnaZlotnickiegoDolnaLaskaPiwna1Nyska2ZlotaMickiewiczaSieradzka2Sieradzka1Nyska1
Fig. 7: Probabilities of drivers’ satisfaction vs. traffic intensity for different roads.already at a low traffic intensity reach a critical state, which will cause drivers’ dissatis-faction for sure (these are e.g. ulice Nyska 2, Piwna 2, Jasna, Laska, Dolna, Zlotnickiego).We can also observe streets such as Nyska 1, where traffic is constantly flowing and doesnot irritate drivers. This drawing shows us that it is true that individual streets reactdifferently to increasing traffic. Therefore, it is worth examining how these changesaffect the overall functioning of the traffic network and the importance of individualfragments.
To begin with, we calculate the reliability of the entire system depending on traffic andthe reliability of each route. In this way, we obtain the probability of finish the journeywith the satisfaction of all roads. In order to calculate the reliability values of individualroads p i , for i = 1 , , . . . , are substituted the appropriate values in the formula(12). In addition, we will calculate the reliability of individual routes that correspondto the minimum paths. The relationship between the elements of each route is in series.Individual routes include the following streets: Route 1:
Dolna, Zlota, Mickiewicza, Jasna, Sieradzka 2
Route 2:
Dolna, Zlota, Nyska 1, Nyska 2, Sieradzka 1, Sieradzka 2
Route 3:
Piwna 1, Zlotnickiego, Nyska 2, Sieradzka 1, Sieradzka 2
Route 4:
Piwna 1, Piwna 2, Laska, Sieradzka 1, Sieradzka 2Table 9: The probabilities of comfortable driving between points A and B vs. variousroutes and different traffic intensity. Traffic Probability of satisfaction fromintensity All routes Route 1 Route 2 Route 3 Route 40.050
The calculated reliability values are presented in Table 9. We can see that the systemis no longer efficient at a traffic intensity of 0.175. In addition, we see that the capacityof the system is always greater than the efficiency of individual roads. This is importantinformation regarding the critical value of traffic intensity that causes failure of the en-tire network. In addition, we can see that the capacity of Routes 1 and 4 is greater,which may suggest that with heavy traffic it is better to choose one of these two routesto ensure a better chance of a quiet ride.We will now proceed to calculate the importance of individual roads in the function-ing of the entire system. The calculated values are shown in Table 10. For each street,received values of measure of significance at a given traffic intensity were presented. Asmentioned before, these values are calculated on the basis of the structure function (12)and importance measures theory introduced by Birnbaum (7) using the received relia-bility for individual traffic intensities. As we can see, the most interesting results wereobtained for the traffic intensity of 0.125 and 0.150. At low traffic intensities, the re-liability of individual elements does not affect the functioning of the system, becausethe whole system works properly and the reliability of the roads are close to 1. Forthe intensity of 0.125, the contribution of individual streets begins to be noticeable. Wesee that, according to structural measures, the largest contribution to the functioningof the network has Sieradzka 2 street, the next streets have the value of importanceclose to 0.03, with the exception of Nyska 1, Zlotnickiego, and Nyska 2, which are measure of the importance of roads 25
Table 10: The importance of route elements for different traffic intensities.
Id Street Traffic intensityname 0.050 0.075 0.100 0.125 0.150 0.175 0.200 . . . ≈ ≈ ≈ . . . ≈ ≈ ≈ . . . ≈ ≈ ≈ . . . ≈ ≈ ≈ . . . ≈ ≈ ≈ . . . ≈ ≈ ≈ . . . ≈ ≈ ≈ . . . ≈ ≈ ≈ , . . . ≈ ≈ ≈ , . . .
10 Laska ≈ ≈ ≈ . . .
11 Sieradzka 1 ≈ ≈ ≈ . . .
12 Sieradzka 2 ≈ ≈ . . . smaller. For the intensity of 0.150, we can see that there are difficulties in movement.The first thing that draws our attention is the importance of Nyska 2 Street, which wasone of the smallest before, now it has become the most significant element. Anotherimportant component of the system is again Sieradzka 2, which is obvious. However,the streets that are worth paying attention to are Piwna 2 and Jasna, whose significancehas also risen dramatically. With subsequent increases in intensity, we see that only these3 streets really affect the quality of traffic, and of them the most street Nyska 2. Based on the analysis made in the previous section, the streets Nyska 2, Piwna 2, andJasna have the greatest importance for the appropriate functioning of the entire systemat high traffic intensity. The analyzed traffic system is a real traffic network, which is whywe know what traffic really looks like on individual roads. The presented scheme of travelfrom A to B shows the travel from two strategic positions in the city. The main streetsin the city are Laska and Sieradzka, they pass through the center of the city. The traffic"on top" of Laska Street is greater than on Sieradzka Street because here we are alreadyapproaching the exit from the city. The results obtained are in line with expectations.One of the most important points in the city is the Nyska–Laska–Sieradzka intersection.In fact, this intersection is more extensive and we can see that a lot of work has beenput into its proper functioning. Many simplifications are used there, which would alsoslightly change the results obtained from the simulation. For example, vehicles turningleft into Sieradzka Street have more space so, when they are waiting for a turn, theydo not obstruct the traffic of other vehicles going straight or turning right. In addition,time counters are used on the traffic lights that increase drivers’ watchfulness and theirstart when the green light comes on.The intersection of Piwna and Laska streets was critical enough that it was impos-sible to turn left there, the sign ’right to turn right’ was in force. This was a majorimpediment to general traffic as well as to the routes presented in the paper. Thatis why a roundabout intersection has recently been built here. This decision certainlyrequired a lot of consideration by the city authorities, because there is not enough spacefor a full-size roundabout here, so it has a slightly flattened one side. However, as canbe seen from the results obtained, it was one of the critical parts of traffic in the city,so this decision seems sensible.The last problematic street is Jasna, but here in real traffic, there is no such intensityof vehicles, both on Jasna Street and the "bottom" part of Sieradzka Street. Assuming that traffic in this part of the city is smaller and that turning into Mickiewicza streetis not very problematic gives important information to drivers who considered whichof these two roads is better.
A quantitative approach to road quality assessment is proposed. The measures of signif-icance defined for the reliability systems were used as a tool to calculate the importanceof individual road fragments. An actual traffic network diagram was analyzed, ensur-ing access from point A to point B . To begin with, assuming that only the structureof the analyzed road network is known, the structural significance of individual road frag-ments was calculated. For this purpose, two measures were used: proposed by Birnbaum,assuming constant reliability of individual road fragments, and Barlow and Proschanmeasure, which takes into account the variability of individual element reliability. Therewere noticeable differences between the received values, but the final result was similarin both cases. The most important for maintaining the efficiency of the analyzed roadnetwork are the streets that occur in the largest number of possible routes to the point B as a serial connection. This confirmed our expectations, but also helped to locate someof the roads that at the first consideration were not potentially important routes. Whencomparing the differences between Birnbaum and Barlow-Proschan measures, the sec-ond one was considered more appropriate for use in the context of road traffic, becausethe reliability of individual roads are not the same, many factors affect on them.Then a method of assessing the reliability of street elements was proposed. Forthis purpose, it was assumed that the quality of roads is the satisfaction of driverswith the route traveled, and the delay time on individual roads was used as a measureof this. It was assumed that drivers have limited patience, which is close to a lifetime andwas presented as a variable from the Weibull distribution. Having calculated the delaytimes on individual roads, it was possible to determine the probability of upset the driverat such a delay. However, in order for the obtained value to be able to be used in the the-ory of measures of importance, it had to be transformed so that it was responsible forthe reliability of a given element. Therefore, the Weibull distribution reliability functionwas used, which in our example reflected the probability that with a given road delay,the driver would still be satisfied. Alternate method of reliability assessing to the netof roads has been used by Pilch and Szybka(2009).In the paper by Szajowski and Włodarczyk (2020)), it was shown that if driversare dissatisfied with driving then they can stop complying with traffic rules. And oneof the factors influencing their change and negative behavior on the road is delays.Therefore, the measures defined in this way are a guide for both drivers and trafficmanagers. For drivers, it shows which road is better to avoid because there is a chanceof potential nervousness, and gives road drivers information about dangerous pointsin the city and points that have a negative impact on drivers. In addition, the largedelay time on individual roads indicates the failure of the fragments concerned. Basedon the simulations performed, the delay times on each road were calculated. Then it wasshown which road fragments are the most important. The obtained results were con-fronted with the actual feelings regarding the given fragments. And they were consideredlikely because with the network as defined it was used as the most significant elementsthat were improved in real traffic. Which proves the real importance of these elements. Author Contributions: Author Contributions:
Development of the application of theimportant measures to element of road networks, Krzysztof Szajowski(KSz) and Kinga Wło-darczyk(KW); implementation of the algorithm and the example, numerical simulations, KW;writing and editing, KSz and KW. measure of the importance of roads 27
Funding:
Supported by Wrocław University of Science and Technology, Faculty of Pure andApplied Mathematics, under the project 049U/0051/19(KSz).
Conflicts of Interest:
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this article:
TDL
Time-Dependent Lifetime (p. 4)
TIL
Time Independent Lifetime (p. 4)
LAI model
Model of general behavior of vehicles on the road (p. 15; v. [15]) ibliography [1] L. Alvarez and R. Horowitz. Safe platooning in automated highway systemspart i: Safety regions design.
Vehicle System Dynamics , 32(1):23–55, 1999.https://doi.org/10.1076/vesd.32.1.23.4228. Cited on p. 18.[2] K. P. Amrutkar and K. K. Kamalja. An overview of various importance measuresof reliability system.
International Journal of Mathematical, Engineering and Man-agement Sciences , 2(3):150–171, 2017. Cited on p. 3.[3] R. E. Barlow and F. Proschan. Importance of system components and fault treeevents.
Stochastic Processes and their Applications , 3(2):153–173, 1975. ISSN 0304-4149. https://doi.org/10.1016/0304-4149(75)90013-7. Cited on p. 7.[4] S. Belbasi and M. E. Foulaadvand. Simulation of traffic flow at a signalized intersec-tion.
Journal of Statistical Mechanics: Theory and Experiment , 2008(07):P07021,jul 2008. https://doi.org/10.1088/1742-5468/2008/07/p07021. Cited on p. 18.[5] Z. W. Birnbaum. On the importance of different components in a multicomponentsystem. In P. Krishnaiah, editor,
Multivariate Analysis, II (Proc. Second Internat.Sympos., Dayton, Ohio, 1968) , pages 581–592. Academic Press, New York, 1969.Cited on pp. 3, 5, 6, and 7.[6] Z. W. Birnbaum, J. D. Esary, and S. C. Saunders. Multi-component systems andstructures and their reliability.
Technometrics , 3(1):55–77, 1961. ISSN 0040-1706.Cited on p. 5.[7] J. R. Cherlow. Measuring values of travel time savings.
Journalof Consumer Research , 7(4):360–371, 1981. ISSN 0093–5301, 1537–5277.https://doi.org/10.2307/2488690. Cited on p. 1.[8] B. F. Ciuffo, V. Punzo, and V. Torrieri. A framework for the calibration of mi-croscopic traffic flow models.
Transportation Research Board 86th Annual Meetingof the Transportation Research Board, Washington, D.C. , 2007. Cited on p. 21.[9] H. Fan, B. Jia, J. Tian, and L. Yun. Characteristics of traffic flowat a non-signalized intersection in the framework of game theory.
Phys-ica A: Statistical Mechanics and its Applications , 415(C):172–180, 2014.https://doi.org/10.1016/j.physa.2014.07.0. URL https://ideas.repec.org/a/eee/phsmap/v415y2014icp172-180.html . Cited on pp. 10 and 11.[10] M. E. Foulaadvand and S. Belbasi. Vehicular traffic flow at a non-signalized in-tersection.
Journal of Physics A: Mathematical and Theoretical , 40(29):8289–8297,jul 2007. https://doi.org/10.1088/1751-8113/40/29/006. Cited on p. 18.[11] B. Greenshields. A study of traffic capacity.
Proc. of the Highway Research Board ,14:448–477, 1935. Cited on p. 13.[12] Y. Hollander and R. Liu. Estimation of the distribution of travel timesby repeated simulation.
Transportation Research Part C , 16(2):212–231, 2008.https://doi.org/10.1016/j.trc.2007.07.005. Cited on p. 21.[13] A. Ilachinski.
Cellular Automata: a Discrete Universe . World Scientific, 2001.ISBN 9789813102569. URL https://books.google.pl/books?id=BPY7DQAAQBAJ .Cited on p. 14.[14] W. Kuo and X. Zhu. Relations and generalizations of importance measures in re-liability.
IEEE Transactions on Reliability , 61(3):659–674, 2012. Cited on p. 4.[15] M. Lárraga and L. Alvarez-Icaza. Cellular automaton model for traf-fic flow based on safe driving policies and human reactions.
Phys-ica A: Statistical Mechanics and its Applications , 389(23):5425–5438, 2010.https://doi.org/10.1016/j.physa.2010.08.0. Cited on pp. 13, 15, 17, 18, and 27.[16] H. K. Lee, R. Barlovic, M. Schreckenberg, and D. Kim. Mechanical restrictionversus human overreaction triggering congested traffic states.
Phys. Rev. Lett. , 92: measure of the importance of roads 29
Proc.of the Royal Society of London. Series A. Math. and Physical Sci. , 229(1178):281–345, 1955. Cited on p. 13.[18] Y.-K. Lin. System reliability for quickest path problems under time threshold andbudget.
Computers & Mathematics with Applications , 60(8):2326 – 2332, 2010.ISSN 0898-1221. https://doi.org/10.1016/j.camwa.2010.08.026. Cited on p. 1.[19] K. Małecki and K. Szmajdziński. Symulator do mikroskopowej analizy ruchudrogowego.
Logistyka , 3:8, 2013. URL . Bib-liogr. 13 poz., rys., wykr., pełen tekst na CD. Cited on p. 14.[20] K. Nagel and M. Schreckenberg. A cellular automaton model forfreeway traffic.
Journal de Physique I France , 2(12):2221–2229, 1992.https://doi.org/10.1051/jp1:1992277. Cited on p. 13.[21] M. Nakata, A. Yamauchi, J. Tanimoto, and A. Hagishima. Dilemma game struc-ture hidden in traffic flow at a bottleneck due to a 2 into 1 lane junction.
Physica A: Statistical Mechanics and its Applications , 389:5353–5361, 12 2010.https://doi.org/10.1016/j.physa.2010.08.005. Cited on p. 19.[22] J. Opara. Metoda automatów komórkowych - zastosowanie w modelowaniu pro-cesów przemian fazowych.
Prace Instytutu Metalurgii Żelaza , T. 62, nr 4:21–34,2010. Cited on p. 14.[23] S. Ossen and S. P. Hoogendoorn. Validity of trajectory-based calibration approachof car-following models in presence of measurement errors.
Transportation ResearchRecord , 2088(1):117–125, 2008. https://doi.org/10.3141/2088-13. Cited on p. 21.[24] R. Pilch and J. Szybka. Estimation reliability of road net.
Journal of Machine Con-struction and Maintenance–Problemy Eksploatacji , 1:157–165, 2009. ISSN 1232-9312. oryg.title in Polish ”Ocena niezawodności sieci komunikacyjnych”. Cited onp. 26.[25] K. G. Ramamurthy.
Coherent structures and simple games , volume 6 of
Theoryand Decision Library. Series C: Game Theory, Mathematical Programming andOperations Research . Kluwer Academic Publishers Group, Dordrecht, 1990. ISBN0-7923-0869-7. https://doi.org/10.1007/978-94-009-2099-6. Cited on p. 4.[26] M. Średnicka. Importance measures in multistate systems reliability. Technicalreport, Faculty of Pure and Applied Mathematics, Wrocław University of Scienceand Technology, Wrocław, 2020. 38p. Master’s Thesis. Cited on p. 3.[27] P. I. Richards. Shock waves on the highway.
Operations Research , 4(1):42–51, 1956.https://doi.org/10.1287/opre.4.1.42. Cited on p. 13.[28] H. J. Ruskin and R. Wang. Modeling traffic flow at an urban unsignalized inter-section. In P. M. A. Sloot, A. G. Hoekstra, C. J. K. Tan, and J. J. Dongarra,editors,
Computational Science — ICCS 2002 , pages 381–390, Berlin, Heidelberg,2002. Springer Berlin Heidelberg. ISBN 978-3-540-46043-5. Cited on p. 18.[29] L. Sharpe. Highway security measures ’are hardly ever cost-effective’.
Engineering& Technology , 7(10):13–14, 2012. ISSN 1750-9637. Cited on p. 1.[30] J. Spall, S. Hill, and D. Stark. Theoretical framework for comparing several stochas-tic optimization approaches. In G. Calafiore and F. Dabbene, editors,
Probabilisticand Randomized Methods for Design under Uncertainty , pages 99–110. SpringerLondon, 2006. https://doi.org/10.1007/1-84628-095-8_3. Cited on p. 21.[31] K. J. Szajowski and K. Włodarczyk. Drivers’ skills and behavior vs.traffic at intersections.
Mathematics , 8(3):paper:433, pages:20, mar 2020.https://doi.org/10.3390/math8030433. Cited on pp. 3, 18, and 26. [32] J.-M. Tacnet, E. Mermet, and S. Maneerat. Analysis of importance of roadnetworksexposed to natural hazards. In J. Gensel, D. Josselin, and D. Vandenbroucke, ed-itors,
Multidisciplinary Research on Geographical Information in Europe and Be-yond. Proc. of the AGILE’2012 Int. Conf. on Geographic Information Science,Avignon, April, 24-27, 2012. , pages 375–392. Academic Press, New York, 2012.ISBN 978-90-816960-0-5. Cited on p. 1.[33] I. A. Villalobos, A. S. Poznyak, and A. M. Tamayo. Game theory applied to the ba-sic traffic control problem.
IFAC Proc. Volumes (IFAC-PapersOnline) , 39(12):319– 324, 2006. ISSN 1474-6670. https://doi.org/10.3182/20060829-3-NL-2908.00056.11th IFAC Symposium on Control in Transportation Systems. Cited on p. 19.[34] Q. Wu, X. Li, M.-B. Hu, and R. Jiang. Study of traffic flow at an unsignalizedt-shaped intersection by cellular automata model.
Physics of Condensed Matter ,48:265–269, 01 2005. https://doi.org/10.1140/epjb/e2005-00398-5. Cited on p. 18.[35] J. Żygierewicz. Automaty komórkowe, 2019. URL . strona domowa, dostęp: 08.01.2019r. Cited on p. 14.
A Code of modelling and simulation
This work uses the LAI model presented in Section 3.3. Which was then modified toadd restrictions at intersections. Each of the streets included in the intersection wassimulated individually and the interactions at the intersection were examined. As itwas mentioned before, streets and intersections can be divided into several types. Inthis appendix, we will present the most advanced in terms of regulations used, i.e. theintersection on Piwna 2 and Jasna streets. Cars there may go to the right joining thetraffic in this direction or turn left crossing the second direction of travel in addition.Both rules apply at intersections. Code presentations will start with the most basic onesand then we will go to the main code. To calculate the distance to the vehicles preceding,which will provide the possibility of acceleration, maintain speed or deceleration wascalculated using three functions d_acc.m , d_keep.m , d_dec.m , listed below f u n c t i o n r e s u l t = d_acc (n , speeds ,M, dv ) % n − id o f car % speeds − v e c t o r o f speeds % M − max a b i l i t y to d e c e l e r a t e % dv − a b i l i t y do a c c e l e r a t e r e s u l t = max(0 , sum ( ( speeds ( n ) + dv ) − ( 0 : ( f l o o r ( ( speeds ( n )+dv ) /M) ) ) ∗M) − . . . sum ( ( speeds ( n+1) − M) − ( 0 : ( f l o o r ( ( speeds ( n+1)−M) /M) ) ) ∗M) ) ; end f u n c t i o n r e s u l t = d_keep (n , speeds ,M, dv ) r e s u l t = max(0 , sum( speeds ( n ) − ( 0 : ( f l o o r ( speeds ( n ) /M) ) ) ∗M) − . . . sum ( ( speeds ( n+1) − M) − ( 0 : ( f l o o r ( ( speeds ( n+1)−M) /M) ) ) ∗M) ) ; end f u n c t i o n r e s u l t = d_dec (n , speeds ,M, dv ) r e s u l t = max(0 , sum ( ( speeds ( n ) − dv ) − ( 0 : ( f l o o r ( ( speeds ( n )−dv ) /M) ) ) ∗M) − . . . sum ( ( speeds ( n+1) − M) − ( 0 : ( f l o o r ( ( speeds ( n+1)−M) /M) ) ) ∗M) ) ; end Another important element is adding a new vehicle on the road, each vehicle has itsown index, speed, position, and information where it goes. Depending on the simulationbeing performed, the probability of route selection can be set to others, if it is notneeded, the where parameter is not given. measure of the importance of roads 31 f u n c t i o n [ pos , speeds , ids , where ] = new_car ( pos , speeds , ids , v_min , prob_new_car ,M, dv , l s , L , where ) % where − where the v e h i c l e are going − 1 right , 0 − l e f t i f nargin < 10 where = [ ] ; end % Generating a new car , with some p r o b a b i l i t y . New c a r s are added with % v_min v e l o c i t y i f isempty ( pos ) i f rand ( ) <= prob_new_car pos = [ 2 , pos ] ; speeds = [ v_min , speeds ] ; where = [ rand ( ) >1/2, where ] ; i d s = 1 ; end e l s e temp = 1 : ( L/2) ; i f pos ( 1 ) ~= 2 speeds_temp = [ v_min , speeds ] ; d_ke = d_keep (1 , speeds_temp ,M, dv ) ; % space s u f f i c i e n t to maintain c u r r e n t speed i f pos ( 1 ) − l s >= d_ke % i f the d i s t a n c e to the p r e v i o u s one i s s u f f i c i e n t % to maintain the c u r r e n t speed then you can e n t e r i f rand ( ) <= prob_new_car pos = [ 2 , pos ] ; speeds = [ v_min , speeds ] ; where = [ rand ( ) >1/2, where ] ; % 1 right , 0 l e f t i d s = [ min ( temp(~ ismember ( 1 : ( L/2) , i d s ) ) ) , i d s ] ; end end end end end Then, a speed update is performed for each vehicle according to the diagram describedin chapter 3.3. In addition, parameters are used to say whether the vehicle can leavethe intersection or not, and to what speed it should slow down before the intersection. f u n c t i o n [ speeds , can_go ] = velocity_updade (n , speeds , pos , l s , dv , Rd, R0 , Rs , vs , M, vmax , L , can_go , dec_to ) % V e l o c i t y update f o r v e h i c l e ’n ’ % can_go −−− % 0 − the car cannot l e a v e the i n t e r s e c t i o n , % 1 − the car can l e a v e the i n t e r s e c t i o n % dec_to −−− v e l o c i t y do d e c e l e r a t e b e f o r e i n t e r s e c t i o n i f n ~= numel ( pos ) % f o r l a s t car no dependence o f p r e v i o u s car d_ac = d_acc (n , speeds ,M, dv ) ; d_ke = d_keep (n , speeds ,M, dv ) ; d_de = d_dec (n , speeds ,M, dv ) ; % A c c e l e r a t i o n i f ( pos ( n+1) − pos ( n ) − l s ) >= d_ac new_speeds = speeds ( n ) ; i f rand ( )<= min (Rd, R0+speeds ( n ) ∗(Rd−R0) / vs ) new_speeds = min ( speeds ( n )+dv , vmax) ; end speeds ( n ) = new_speeds ; end % Random slowing down i f d_ac > ( pos ( n+1) − pos ( n ) − l s ) && ( pos ( n+1) − pos ( n ) − l s ) >= d_ke new_speeds = speeds ( n ) ; i f rand ( ) <= Rs new_speeds = max( speeds ( n )−dv , 0 ) ; end speeds ( n ) = new_speeds ; end % Braking i f d_ke > ( pos ( n+1) − pos ( n ) − l s ) && ( pos ( n+1) − pos ( n ) − l s ) >= d_de new_speeds = max( speeds ( n )−dv , 0 ) ; speeds ( n ) = new_speeds ; end % Emergency braking i f ( pos ( n+1) − pos ( n ) − l s ) < d_de new_speeds = max( speeds ( n )−M, 0 ) ; speeds ( n ) = new_speeds ; end e l s e speeds_temp = [ speeds , dec_to ] ; d_ke = d_keep (n , speeds_temp ,M, dv ) ; d_de = d_dec (n , speeds_temp ,M, dv ) ; i f d_ke < (L − pos ( n ) ) | | can_go % A c c e l e r a t i o n new_speeds = speeds ( n ) ; i f rand ( )<= min (Rd, R0+speeds ( n ) ∗(Rd−R0) / vs ) new_speeds = min ( speeds ( n )+dv , vmax) ; end speeds ( n ) = new_speeds ; % Random slowing down new_speeds = speeds ( n ) ; i f rand ( ) <= Rs new_speeds = max( speeds ( n )−dv , 0 ) ; end speeds ( n ) = new_speeds ; can_go = 0 ; % swich o f f o r next car end % Slow b e f o r e end od road i f d_ke > (L − pos ( n ) ) && (L − pos ( n ) ) >= d_de new_speeds = max( speeds ( n )−dv , 0 ) ; speeds ( n ) = new_speeds ; end % Emergency braking i f (L − pos ( n ) ) < d_de new_speeds = max( speeds ( n )−M, 0 ) ; speeds ( n ) = new_speeds ; end end end Finally, we go to the main program codes. Depending on the exact type of intersection,the program looks slightly different, but the overall characteristics and construction arepreserved. At the beginning we define the variables used in the model, then we haveloops after repetitions for different probabilities of a new vehicle. We define new emptyroads in each loop. Then we add the first vehicle on each road and go on to furtherprocesses. In the original program, before starting the loop after repeating the updateon the road, the road was filled with vehicles. At each step, we update speeds and adda new vehicle on the road, according to the probability. Then update the speeds andremove those vehicles whose position has exceeded the length of the road. Finally, weanalyze the interactions of drivers at intersections, resulting in a change in the parametersaying whether the vehicle can leave the intersection or not. This is done in accordancewith the previously described assumptions. On the posted program we have an examplefor Piwna Street 2, where the driver turning right gives way to other vehicles on thisroad and turning left gives way to vehicles driving in the opposite direction than heplans because he crosses their lane. Below is the code. MC = 1000; % Monte Carlo dx = 2 . 5 ; % c e l l s i z e l s = 2 ; % car length , 5 m vmax = 5 ; % 5 −> 12.5 m/ s −> 45 km/h lub 6 −> 15 m/ s −> 54 km/h Rs = 0 . 0 1 ; % prob . o f emergency Rd = 1 ; % max . prob o f a c c e l e r a t i o n R0 = 0 . 8 ; % min . prob . o f a c c e l e r a t i o n vs = 5/dx + 1 ; % min v e l o c i t y f o r f a s t e r d r i v i n g M = 2 ; % a b i l i t y to emergency d r i v i n g 5m/ s2 dv = 2.5/ dx ; % a b i l i t y to a c c e l e r a t e dec_to = 0 ; % v e l o c i t y to slow down b e f o r e i n t e r s e c t i o n same_probs = 0 . 0 5 : 0 . 0 2 5 : 0 . 6 ; % d i f f e r e n t t r a f f i c i n t e s i t y i t = 1 ; f o r same_prob = same_probs disp ( same_prob ) SAVE = z e r o s (1 ,MC) ; f o r M = 1 :MC % I n t e r s e c t i o n Piwna2 i Laska , z prawd . 1/2 turn r i g h t or l e f t % Inputs f o r Piwna2 L_Piwna2 = 180/dx ; % road length prob_new_car_Piwna2 = same_prob ; pos_Piwna2 = [ ] ; speeds_Piwna2 = [ ] ; measure of the importance of roads 33 v_min_Piwna2 = 4 ; can_go_Piwna2 = 0 ; % Inputs f o r Laska1 , from r i g h t L_Laska1 = 100/dx ; prob_new_car_Laska1 = same_prob ; pos_Laska1 = [ ] ; speeds_Laska1 = [ ] ; v_min_Laska1 = 4 ; can_go_Laska1 = 1 ; % Inputs f o r Laska2 , from l e f t L_Laska2 = 100/dx ; prob_new_car_Laska2 = same_prob ; pos_Laska2 = [ ] ; speeds_Laska2 = [ ] ; v_min_Laska2 = 4 ; can_go_Laska2 = 1 ; % F i r s t car on Piwna2 , Laska1 and Laska2 i f isempty ( pos_Piwna2 ) % i f on road t h e r e i s no car pos_Piwna2 ( 1 ) = 2 ; speeds_Piwna2 ( 1 ) = v_min_Piwna2 ; ids_Piwna2 = 1 ; where_Piwna2 = [ 1 ] ; % 1 − right , 0 − l e f t end i f isempty ( pos_Laska1 ) % i f on road t h e r e i s no car pos_Laska1 ( 1 ) = 2 ; speeds_Laska1 ( 1 ) = v_min_Laska1 ; ids_Laska1 = 1 ; end i f isempty ( pos_Laska2 ) % i f on road t h e r e i s no car pos_Laska2 ( 1 ) = 2 ; speeds_Laska2 ( 1 ) = v_min_Laska2 ; ids_Laska2 = 1 ; end point = L_Laska1 ; % i n t e r s e c t i o n point on Laska1 i Laska 2 , at the end od roads stop = z e r o s ( 1 , 3 ) ; licz_czas_Piwna2 = z e r o s ( s i z e ( 1 : ( L_Piwna2/2) ) ) ; l i c z = 1 ; saved_czas_Piwna2 = [ ] ; can_go_Piwna2 = 0 ; % must always stop b e f o r e the i n t e r s e c t i o n f o r i = 1:1000 % Adding new car with v_min v e l o c i t y on Piwna2 [ pos_Piwna2 , speeds_Piwna2 , ids_Piwna2 , where_Piwna2 ] = new_car ( pos_Piwna2 , speeds_Piwna2 , ids_Piwna2 , v_min_Piwna2 , prob_new_car_Piwna2 ,M, dv , l s ,L_Piwna2 , where_Piwna2 ) ; % V e l o c i t y updating f o r each v e h i c l e on road Piwna2 f o r n = 1 : numel ( pos_Piwna2 ) [ speeds_Piwna2 , ~ ] = velocity_updade (n , speeds_Piwna2 , pos_Piwna2 , l s , dv , Rd, R0 , Rs , vs , M, vmax , L_Piwna2 , can_go_Piwna2 , dec_to ) ; end % Adding new car with v_min v e l o c i t y on Laska1 [ pos_Laska1 , speeds_Laska1 , ids_Laska1 ] = new_car ( pos_Laska1 , speeds_Laska1 , ids_Laska1 , v_min_Laska1 , prob_new_car_Laska1 ,M , dv , l s , L_Laska1 ) ; % V e l o c i t y updating f o r each v e h i c l e on road Laska1 f o r n = 1 : numel ( pos_Laska1 ) speeds_Laska1 = velocity_updade (n , speeds_Laska1 , pos_Laska1 , l s , dv , Rd, R0 , Rs , vs , M, vmax , L_Laska1 , can_go_Laska1 , dec_to ) ; end % Adding new car with v_min v e l o c i t y on Laska2 [ pos_Laska2 , speeds_Laska2 , ids_Laska2 ] = new_car ( pos_Laska2 , speeds_Laska2 , ids_Laska2 , v_min_Laska2 , prob_new_car_Laska2 ,M , dv , l s , L_Laska2 ) ; % V e l o c i t y updating f o r each v e h i c l e on road Laska2 f o r n = 1 : numel ( pos_Laska2 ) speeds_Laska2 = velocity_updade (n , speeds_Laska2 , pos_Laska2 , l s , dv , Rd, R0 , Rs , vs , M, vmax , L_Laska2 , can_go_Laska2 , dec_to ) ; end % P o s i t i o n update pos_Piwna2 = pos_Piwna2 + speeds_Piwna2 ; pos_Laska1 = pos_Laska1 + speeds_Laska1 ; pos_Laska2 = pos_Laska2 + speeds_Laska2 ; % Removing c a r s i f sum( pos_Piwna2 > L_Piwna2) speeds_Piwna2 ( pos_Piwna2 > L_Piwna2) = [ ] ; where_Piwna2 ( pos_Piwna2 > L_Piwna2) = [ ] ; ids_Piwna2 ( pos_Piwna2 > L_Piwna2) = [ ] ; saved_czas_Piwna2 ( l i c z ) = max( licz_czas_Piwna2 ) ; l i c z = l i c z + 1 ; licz_czas_Piwna2 ( licz_czas_Piwna2 == . . . max( licz_czas_Piwna2 ) ) = 0 ; pos_Piwna2 ( pos_Piwna2 > L_Piwna2) = [ ] ; end i f sum( pos_Laska1 > L_Laska1 ) speeds_Laska1 ( pos_Laska1 > L_Laska1 ) = [ ] ; pos_Laska1 ( pos_Laska1 > L_Laska1 ) = [ ] ; end i f sum( pos_Laska2 > L_Laska2 ) speeds_Laska2 ( pos_Laska2 > L_Laska2 ) = [ ] ; pos_Laska2 ( pos_Laska2 > L_Laska2 ) = [ ] ; end licz_czas_Piwna2 ( ids_Piwna2 ) = licz_czas_Piwna2 ( ids_Piwna2 ) + 1 ; % I n t e r s e c t i o n i n t e r a c t i o n i f ~isempty ( speeds_Piwna2 ) i f sum( pos_Piwna2 == L_Piwna2) % some v e h i c l e i s at the end o f the road % D i f f e r e n t c o n d i t i o n s f o r c a r s which go in r i g h t and in l e f t i f where_Piwna2 ( end ) == 1 % in r i g h t to_point_on_left = pos_Laska2 − point ; finds_on_left = f i n d ( to_point_on_left <0) ; i f ~isempty ( finds_on_left ) found_on_left = f i n d ( to_point_on_left ( finds_on_left ) == . . . max( to_point_on_left ( finds_on_left ) ) ) ; % founded car on Zlota i s the c l o s e s t to i n t e r s e c t i o n s dist_to_point_on_left = abs ( pos_Laska2 ( found_on_left ) − point ) ; d_keep_on_left = max(0 , sum(vmax − ( 0 : ( f l o o r (vmax/M) ) ) ∗M) − . . . sum ( ( vmax − M) − ( 0 : ( f l o o r ( ( vmax−M) /M) ) ) ∗M) ) ; % d i s t a n c e which w i l l not f o r c e the d r i v e r % to be r e l e a s e d on the main road at the maximum speeds o f both v e h i c l e s i f dist_to_point_on_left − speeds_Laska2 ( found_on_left ) − . . . sum( min (vmax , speeds_Laska2 ( found_on_left ) +(2:vmax) −1) ) + . . . sum ( 2 : vmax) >= d_keep_on_left % i f the c o n d i t i o n i s met i t can e n t e r can_go_Piwna2 = 1 ; % the v e h i c l e b e f o r e the i n t e r s e c t i o n can already go e l s e can_go_Piwna2 = 0 ; end e l s e can_go_Piwna2 = 1 ; end e l s e % where=0, in l e f t cond = 0 ; % Checks i f i t i s f r e e on the r i g h t to_point_on_right = pos_Laska1 − point ; finds_on_right = f i n d ( to_point_on_right <0) ; i f ~isempty ( finds_on_right ) found_on_right = f i n d ( to_point_on_right ( finds_on_right ) == . . . max( to_point_on_right ( finds_on_right ) ) ) ; % founded car on Zlota i s the c l o s e s t to i n t e r s e c t i o n s dist_to_point_on_right = abs ( pos_Laska1 ( found_on_right ) − point ) ; d_keep_on_right = max(0 , sum(vmax − ( 0 : ( f l o o r (vmax/M) ) ) ∗M) − . . . sum ( ( vmax − M) − ( 0 : ( f l o o r ( ( vmax−M) /M) ) ) ∗M) ) ; % d i s t a n c e which w i l l not f o r c e the d r i v e r % to be r e l e a s e d on the main road at the maximum speeds o f both v e h i c l e s i f sum( pos_Piwna2 == L_Piwna2) % some v e h i c l e i s at the end o f the road i f dist_to_point_on_right − . . . sum( min (vmax , speeds_Laska1 ( found_on_right ) +(0:2) ) ) − . . . sum( min (vmax , speeds_Laska1 ( found_on_right ) +(2:vmax) −1) ) + . . . measure of the importance of roads 35 sum ( 2 : vmax) >= d_keep_on_right % i f t h i s c o n d i t i o n i s met on the r i g h t we have % the opportunity to e n t e r the i n t e r s e c t i o n cond = cond + 1 ; end end e l s e cond = cond + 1 ; end % Checking i f i t i s f r e e on the l e f t to_point_on_left = pos_Laska2 − point ; finds_on_left = f i n d ( to_point_on_left <0) ; i f ~isempty ( finds_on_left ) found_on_left = f i n d ( to_point_on_left ( finds_on_left ) == . . . max( to_point_on_left ( finds_on_left ) ) ) ; dist_to_point_on_left = abs ( pos_Laska2 ( found_on_left ) − point ) ; speeds_temp = [ speeds_Laska2 ( found_on_left ) , 0 ] ; d_ke_on_left = d_keep (1 , speeds_temp ,M, dv ) ; d_de_on_left = d_dec (1 , speeds_temp ,M, dv ) ; needed_time = 3 ; i f dist_to_point_on_left − speeds_Laska2 ( found_on_left ) − . . . min ( speeds_Laska2 ( found_on_left ) +1,vmax) − . . . min ( speeds_Laska2 ( found_on_left ) +2,vmax) >= d_de_on_left cond = cond + 1 ; % second c o n d i t i o n i s met end e l s e cond = cond + 1 ; % second c o n d i t i o n i s met end i f cond == 2 % i f both c o n d i t i o n s were met v e h i c l e can go can_go_Piwna2 = 1 ; e l s e can_go_Piwna2 = 0 ; end end end end end SAVE(M) = nanmean ( saved_czas_Piwna2 ( 5 : end ) ) ; end
SAVE_MC( i t ) = nanmean (SAVE(SAVE>0) ) ; i t = i t +1; endend