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TRAFFIC FLOWS MODELLING BASED ON PROBABILITY OF COINCIDENCE OF CHAOTIC IMPULSES OF RANDOM DURATION AND RANDOM INTERVALS BETWEEN THEM
А. Levterov, Professor, DSci., Kharkiv National Automobile and Highway University
Abstract. The features of applying the theory of chaotic electrical impulse flows in mathematical modelling of traffic flows have been considered. Both the moments of appearance of vehicles on the traffic lane and the moments of appearance of electrical impulse on the time axis are supposed to obey the Poisson law. It is assumed that the duration of an impulse is equivalent to the length of a vehicle, and the intervals between the impulses are equivalent to the intervals between the vehicles in the spatial-temporal axis.
Key words: traffic flow, chaotic electrical impulse flow, coincidence of impulses.
МОДЕЛЮВАННЯ ТРАНСПОРТНИХ ПОТОК1В НА ОСНОВ1 ЙМОВ1РНОСТ1 ЗБ1ГУ 1МПУЛЬС1В, ЩО ПРЯМУЮТЬ ХАОТИЧНО, ВИПАДКОВО1 ТРИВАЛОСТ1 Й ВИПАДКОВИХ 1НТЕРВАЛ1В М1Ж НИМИ
А.1. Левтеров, проф., к.т.н., Харкчвський нацюнальний автомобшьно-дорожнш ушверситет
Анотаця. У наш час при моделюванм транспортних потоюв розглядаетъся, в тому числ!, й клас макроскотчних моделей, коли рух транспортних 3aco6ie уподiбнюeтъся будъ-якому фiзичному потоку гдро- i газодинамiчнi моделi). Як фiзичний аналог транспортного потоку запропоновано використовувати теорiю хаотичних електричних iмпулъсних потоюв.
Ключов1 слова: транспортний потт, транспортний зааб, хаотичний електричний iмпулъсний потщ збiгання транспортних засобiв.
МОДЕЛИРОВАНИЕ ТРАНСПОРТНЫХ ПОТОКОВ НА ОСНОВЕ ВЕРОЯТНОСТИ СОВПАДЕНИЯ ХАОТИЧНЫХ ИМПУЛЬСОВ СЛУЧАЙНОЙ ДЛИТЕЛЬНОСТИ И СЛУЧАЙНЫХ ИНТЕРВАЛОВ МЕЖДУ НИМИ
А.И. Левтеров, проф., к.т.н., Харьковский национальный автомобильно-дорожный университет
Аннотация. В настоящее время при моделировании транспортных потоков рассматривается, в том числе, и класс макроскопических моделей, когда движение транспортных средств уподобляется любому физическому потоку (гидро- и газодинамические модели). В качестве физического аналога транспортного потока предложено исполъзоватъ теорию хаотических электрических импулъсных потоков.
Ключевые слова: транспортный поток, транспортное средство, хаотичный электрический импулъсный поток, совпадение транспортных средств.
Introduction
Computer simulation of traffic plays an important role in improving the transport situation
on the roads. The application of various mathematical methods and modelling systems enables to improve the traffic flow capacity, reduce accident rate on the roads, and preserve the envi-
ronment with minimal financial expenses.Road traffic is closely connected with the welfare and life of people. Therefore, the major road accidents are often covered in media, on television, on the Internet.
Analysis of Publications
According to the information of State Motor Vehicle Inspectorate of Ukraine, the most frequent reason for traffic accidents is related to the actions of road traffic participants, for which reason more than 90 % of accidents from their total number take place (Fig. 1) [1, 2]. Some of the most frequent violations connected with vehicles (except those involving pedestrians) are: exceeding the safe speed, violation of manoeuvring rules, non-observance of the distance and driving onto the oncoming lane of the road.
Fig. 1. The diagram of traffic accidents in Ukraine
Currently, in modelling traffic flows, the class of macroscopic models, when the motion of vehicles is likened to any physical flow (hydro- and gas-dynamic models) is also considered [3].
Purpose and Tasks
The aim of the work is mathematical modelling of transport flows on the basis of choosing the theory of chaotic electric impulse flows as a physical analogue of the transport flow.We will proceed from the assumption that violation of the manoeuvring rules, including driving onto the adjacent lane of traffic and onto the oncoming lane, will result in a collision of vehicles when they are in the cross section for a moment before the convergence or at the time of the convergence, that is, in the zone of compatible parallel movement in the cross section of the road (Fig. 2). We will assume that the moments
of the appearance of vehicles on the lane of motion obey the Poisson law, that is, they meet the criteria of ordinarinessand not controlling the after-effect [4].
Let there be a transport flow with s random lengths of vehicles and random intervals of d between them and known densities of probability P (d) for the intervals between the vehicles and Q( z) for their lengths.
The distance between successive vehicles may be different. In some cases, it can slightly exceed the length of the vehicle, while in others it can be quite large. Similarly, the time interval between successive vehicles varies. This variability in time and space is the most notable characteristic of the traffic flow [5].
To know the intervals of time and distance between successive vehicles is sometimes more important than knowing the intensity or density of the flow, as it reflects the true nature of the traffic flow more fully. The intervals of time and distance between successive vehicles are those "bricks" on which the entire traffic flow is built [5]. Since vehicles are dynamic objects, which depending on the speed at each moment of time take a new position in the spatial-temporal plane [4] (traffic lane), this process can be represented as a random electric impulse flow on the time axis [6]. Therefore, it can be assumed that the duration of an impulse is equivalent to the length of a vehicle, and the intervals between the impulses are equivalent to the intervals between the vehicles.
To do this, we use the theory of random impulse flows. Let us consider the n probability of coincidence of vehicles of random lengths that go chaotic, and random intervals between them and the distribution of their coincidence [7].
Substantiation of mathematical modelling of
traffic flows on the basis of choosing the theory of chaotic electric impulse flows as a physical analogue of the transport flow
Let the lengths of the vehicles xi (i = 1,2) in two lanes of motion (n = 2) be distributed according to independent laws f (xi )in intervals (0, xim), where xim is the largest length of a vehicle on the lane. The distribution of the moments of the appearance of vehicles on each lane of motion is subject to the Poisson law, which is character-
ized by average density X « 1 / xim .We find the three-dimensional distribution law
W(3)( x1, x2, z), where x1 and x2 are the lengths of
the vehicles, which make a coincidence of with z duration.
As we know
W{3) (xj, x2, z) = 9! ( xj ) • 921 (X2 / X1 ) •
x2 m
•wu (z / xi, x2 )
(1)
where 9, (x,) is the law of the distribution of the duration of coincidence of vehicles, which are on the first lane of motion; 921 (x2 / x,) is the conditional law of distribution of length x2 of coincidence of vehicles of the second traffic lane to the length ... of vehicles of the first lane;
Ai iii
W12 (z / x,, x2) is the conditional law for the distribution of z duration of coincidence of the vehicles to the lengths of x1 and x2. The law of
distribution 9 (xi) differs from the law f (xi), as the probability p = A • (x1 + x2) of coincidence
of vehicles turns out to be higher with longer vehicles [4] and therefore the latter coincide on average more often than shorter vehicles. With the unchanged length x2 of the vehicles on the second lane, the distribution law is
9i (xi) = C • f1(xi) •^•(xi + X2) , (2)
where on the condition of normalization
C = [A-(X1 + x2 )J In the case of random
magnitude, the law expressed by formula (2) should be considered as a conditional law of distribution
9 (X1 ) Ф21 (X2 / X1 ) = (Xl ) • /2 (X2) X1 + X 2
912 (X1 / X2 ) . So
912 ( Xi/ X2 ) = iX^ • fi( Xi). (3)
X i + X2
An unconditional distribution law is found by averaging in the (2) probability
p = A • (xi + x2) by length x2
9i (xi) = Ci • fi(xi) • j A-(Xi + X2)• f2(x2)dx2 =
0
= Ci - A-fi(Xi) where Ci is the normalizing factor. Thus
( xi + X2) f ( )
9i(xi)= Xi+X2•fi(xi);
. . (X, + x2) 92 (x2 ) = X X • f2(x2). (4)
X1 + X 2
To determine the law W,2 (z / x,, x2 ), let us consider two vehicles with the lengths of x, and x2 correspondingly, (Fig. 2), and for certainty we will accept x2 > x,.
t Î
-•-!-1-
Tsr— ¡4 ,
-x2 1 z Xj
JUL
Fig. 2. The diagram of two vehicles coincidence
Let us mark as f a random magnitude which fixes the moment t of appearance of vehicles on the second traffic lane relative to the beginning of vehicles on the first traffic lane. When t changes in the interval (-x2, xl ), the random magnitude Z of coincidence duration changes according to trapezoid law z(t) taking the value
0 < z < x < x2 (Fig. 3)
CC 2 t ^C, ^C 2
z = x2 + t
with xj - x2 < t < 0 z = xj = const (5)
0 < t < x2
z = X, -1
So, some value ofz, where 0 < z < xj, corresponds to two values of t : tj Ta t2.
Section [-x2, (xj - x2 )) of the graph presented on Fig. 2 characterizes the coincidence of two vehicles, and segment (0, xj] characterizes their
removal from each other. Segment [(xj - x2 ) ,0] characterizes the complete coincidence of two
vehicles and the value of this segment is determined by a longer vehicle, and its duration is determined by the speed of the vehicles convergence. The slope of the characteristics on segments [-x2, (x - x2)) and (0, x1 ] depends on the speed of convergence and drifting of vehicles from each other.
Fig. 3. The graph of changing duration of coincidence of two vehicles
Taking into account that the probability of coincidence dp = Xdt, shown in Fig. 2, we find an integral distribution law
- t1 - x
F ( z ) = P (Z < z ) = - J Idt + - Jldt,
P t
x2 Г t2
where(0 < z < x1).
Integrating and taking into account the fact that according to (5) t1 = z - x2 and t1 = z - x2,
and p = X • (x1 + x2), we find
F ( z ) =
(ti + X2) + (х- -12) _ 2
(xi + x2 ) (xi + x2 ) (0 < z < x1).
At the point z = x function F (z) receives a finite increment
AF = P (Z = x, ) = - f Xdt =(X2 - Xj) .
Px,J X2 (X1 + X2 )
Thus,
F ( z ) = • H ( z - x )
(*1 + X2 ) (X! + X2 )
(z > 0), (6)
where H (z) is a solitary function; with z < 0 F (z ) = 0.
Function (6) can be considered as a conditional integral function of distribution Z relative to X1 and X 2, that is,
(
\
, x2
V xi
=P
F ( z ) = FX1 (Z < z/X1 = x1,X2 = x2) . Hence the conditional distribution law is
W,2 (z / x,, x2) = dF = 12 ( 1 2) dz
7—2—г +(X2 Xl)^8(z-х,) (0< z < х,) (x, + x2) (x, + x2)
I, (7)
(z < 0, z > x,)
where S (z) is the delta function. A similar expression is found at x1 > x2.
Substituting expressions (3), (4) and (7) into formula (1), we obtain
W(3) (xl5 x2, z) =
f ( x, )• f2_( x2 ) X, + X 2
[2 + (x2 -x!)^5(z-x,)]
f (Х+Х 2 x2 ) [2 +( x, - x2 Ж z - x2 )]
Г (0 < z < x,) I [ (z < 0, z > x,) J.
(8)
Integrating, we define a two-dimensional distribution law
W{2)(x1,z) = j f1 (X^ fX(x2) • [2 + (x -x2)•
z X1 + X 2
•S ( z - x2) dx2] +
+yAMlf^) ^ + (x2 -x)-8(z-x1)dx2] =
X, + X
2
= 4Щ-[22 (z) + (x -z)• f2 (z) +
X, + X 2
x2m
+8(z-x,) J x2 • f2(x2)dx2 -8(z-x,)• x 2(x,)]
where accepted is
^ (z ) =! -Jf (У) dy .
(9)
0
Integrating again, we find a one-dimensional law of distribution of duration of vehicles' coincidence on two (n = 2) lanes of motion
W (z) = W2 (z) = j W(2)(xl, z) dxx
1
j + X 2
x2 m
[2 .^1 (z)¥2 (z) + fi (z) 2 (x)dx +
+f2 (z(x)dx]. (10)
z
When fi (x ) = f (x2 ) = f (x),
W(z) = W2 (z) = X j Иz)]2 + f (z) JY(x)dx j.(11)
The task of determining one-dimensional law W (z) of the distribution of the duration of vehicles' coincidence by n lanes (n > 2) is equivalent to the task of determining the law of distribution of the duration of the vehicles' coincidence by two lanes, one of which is the first given lane, and the other one is the equivalent lane, the length of the vehicle on which is equal to the length z = zn_, the coincidence of vehicles on n _ lgiven lanes. So, knowing distribution law Wn_ (z) for the n _ l lanes, we can find distribution law Wn (z) using formula (10).
Let us find the probability p2 of coincidence of vehicles for two lanes of motion, the lengths of which are distributed according to laws f (xi) (i = 1,2), provided that vehicles appear on one of the lanes.Let us assume that a vehicle of a certain length x1 moves on the first lane of the road. Then the probability of a case A / x2,
consisting in the coincidence of the vehicle with the length x2 that appeared in the second lane
with the vehicle of length x1, is expressed by
the equality [7]
P (V x2 ) = И x1 + x2 ) .
(12)
This probability can be considered as the probability of case A associated with the hypothesis
x2, whose probability is equal to f2 (x2) dx2. Therefore, the average (full) probability of coin-
cidence of the vehicle with the length x1 with a
vehicle of any length (no matter which) moving along the second lane of motion, is found by averaging probability (12) by x2
P (Ax1 )= j M x1 + x2 ). f2 (x2 ) dx2 =
0
= a•(x + X2) . (13)
In turn, the probability expressed by the formula (13) can be considered as a conditional probability P(A / Xj)of case A (the coincidence of two vehicles of the same or different length, moving along two adjacent lanes of the road) associated with the hypothesis xl (the one of occurrence of the vehicle with length x1 on the first lane), whose probability is equal to f1(xj)dx. Thus, the sought probability of coincidence of vehicles of random length on two lanes is
P2 = x + X2)• f (x)dx1 =
0
x1 m x2 m
j j (x + x2)• f (x1 )• f2(x2)dx1dx2 = = ЦХ + X2) . (14)
00
The given considerations can be extended to any number of n lanes (i = 1,2,...,n). For unchanged lengths of vehicles for each of the lanes, the probability of their coincidence (subject to the appearance of vehicles on one of the lanes) is expressed by equality [7]
n 1 n
Pn = M-1^-Пх.
(15)
The probability of coincidence of vehicles of random length along the n lanes is found by
averaging probability pn by lengths xi on all lanes of motion.
x1m xnm n 1 n
pn = M MX 1 П • f 0
0 0 i =1 i i=1 n 1 n
= M Z j nx.
i=1 Л i i=1
(16)
Thus, as to the probability of coincidence, the lane of motion of a vehicle of a random length is
equivalent to a lane of motion of a vehicle of a permanent length, which is equal to the mathematical expectation of the length of the vehicles on the lane of the road.
It can be said that the distribution of moments of occurrence of vehicles involved in the coincidence of vehicles is also subject to the Poisson law with average density
^„ = ^P".
(17)
Mathematical expectation Z = Z2 of the duration of the coincidence of vehicles on two lanes of motion (n = 2) can be found by formula (10). Let us look into more general considerations. By unchanged lengths of vehicles x1 and x2 the average duration of coincidence of vehicles on two lanes of motion is expressed by formula [3] Z = x1 • x2 / (x1 + x2). By random values of X1 and X2, the result expressed by the latter equality, must be averaged by lengths x1 and x2 of the vehicles involved in the coincidence of the two vehicles, that is
X1mX2 m x x
Z2 = f f J-!—H■9'(2)(Xj,X2)dXjdX2,(18)
0 0 (Xj + x2 )
Z" =
i " 1 V1
IX
V i=1 Xi )
(20)
We emphasize that relation (20) does not depend on the type of distribution laws.
Let us consider the case where the length of vehicles on the lanes is distributed according to the limited exponential law
f ( x ) = 4т ^
(0 < x < xm), (21)
where x* - is the parameter of the distribution lawand A - is the normalizing factor
UU x 1
Г x , . 1
Г—ex dx; A =-
j x 1 - e
; Ym = (22)
We note that with limited law (21), the average length is X ^ x*
лт Л _
X = j x • — • ex dx = A • x* [1 -(1 + 7m )• e~ym ]
= x
•[1 -v • e 7m -v • e 2/m
[ / m tm
(23)
where, according to equalities (3) and (4),
9(2) (^ x2) = q>j (X1) • 921 (x2 / X1 ) =
= (=L±x^• f1 (x1 )• f2(x2) . (19)
X + X 2
According to expressions (9) and (11), for the case n = 2 we find
W (z) = A2 • [2 • e"2Y + e-2Ym + e-Ym-Y • (y - Ym - 3)],
(24)
Substituting (19) into (18) and integrating, we find
Z2 = .
2 x1 + X2
When n = 3, it is possible to replace two lanes with one lane, on which there are vehicles of random length z2 , from where
— Z2 • Х-
f
Z3 = =
Z2 + X3
Y
_L _L _L
X1 X 2 X3)
where y = — |0 < Y < Ym = "m = I- (25) x V x x )
Let us consider the boundary case, when x* is so small that we can take xm (practically
xm » x*). In this case X ^ x*, A ^ 1 and
2
1
W2 (z) = W2 (z)x = — • e x = — • e x2 , (26)
where x* = x* / 2 .
Passing from p and p +1 lanes of motion, we find the average value of duration of coincidence of vehicles by n lanes
At that, the average duration of the coincidence is (Z2)x = x* = x* / 2 . Actually, when
*
x
*
z
z
2—
2
xm > 5 • x , one may ignore the restrictions on the duration of the distribution law (22). This is illustrated by the relation of mathematical expectations
Z 2
Z 2
1 "(1 + Y m )•
Ym
( Z 2) *
1 + e e
= 1 "Ym (
e Ym + e 2 Ym +
...). (27)
1
Assuming f (x) = —* • e x it is not difficult to
find the distribution law of the duration of the vehicles coincidence by n lanes.
„. / ч n * 1 Wn (z) = — • e 1 = — •
z
P ( F ) = P ( F A ) = p .
(29)
From where
P ( G ) = P ( A )P ( F A )+ P ( A2 )P ( F\À2 ) = = 0,5 0 + 0,5p . (30)
Obviously, expression (2) is valid for the adjacent (second) lane of one traffic direction for the road of category I. Therefore, expression (2) will be written in the form of
p ( F ) = P ( F1A2 ) = p.
(31)
Let us determine the probability of vehicles' collision on the third lane of motion in the same direction. Let's mark the case of collision of vehicles on the third lane of traffic with F2. Then
where x„ = x / n = Zn
F2 =( F n F ) .
(32)
As we can see, the exponential law of the distribution of the vehicles' length by the lanes has stability: the duration of the coincidence is also distributed according to the exponential law.
Let us determine the probability of a collision of vehicles for lanes of I-IV categories of roads by violation of the rules of manoeuvring and driving the vehicle onto the adjacent or oncoming lane of motion. When the vehicle moves onto the oncoming traffic lane, the probability of vehicles' collision is denoted by p which depends on a number of parameters, including: flow density, vehicles' motion speed, condition of road surface, mental and physical state of the driver, his behaviour on the road, time of day, day of the week and other random factors.
From where the probability of the collision will be equal to
p(F>A2) = P(pyP(F) = (1 _p)-p, (33)
and the probability of case G will be equal to
P (G ) = P (A2 )• P (FJA2 ) =
= 0,5-(p + (1 _p)-p) . (34)
Similarly, expressions (32) and (33) at the collision of vehicles on the fourth lane will be rewritten in the form of
F2 =( F n F n F ) ,
(35)
We will mark the following cases for categorie-sII-IV of roads. Let A1 be the case when a vehicle goes on the roadside with the control lost, and A2 is the case when a vehicle goes onto the oncoming traffic lane. Then the probability of a collision will be equal to
P (A1 ) = P ( A2 ) = 0,5. (28)
P ( F3IA2 ) = P ( F )• P ( F )• P ( F ) = = (1 _p)-(1 _p)-p . (36)
Then the probability of the case G will be equal to
P ( G ) = P ( A2 )• P ( F3IA2 ) =
= 0,5•(p + (1 _p)•(!_p)• p) . (37)
x
e
We will mark with F the case of a collision of vehicles at vehicle moving onto the oncoming traffic lane, and G _ will be the case of vehicles collision for the II-IV categories of roads. Then
Let us mark the collision of vehicles on any of four lanes for the first category road with V. Then
V = (F U F2 u F3) , (38)
From where the probability of the collision will be equal to
P (VIA2 ) = P( F1IA2) + P (F2IA2) + P (F3IA2 ) = = p + (1 _p)• p + (1 _p)\1 _p)• p . (39)
Then the probability of case G for the first category road will be equal to
P ( G ) = P ( A2 )• P (VIA2 ) = = 0,5 •(p + (1 _p)• p + (1 _p)2 • p) . (40)
Taking into account the traffic in both directions for roads of I-IV categories with different number of lanes in each direction, with the loss of controllability of the vehicle and its collision with another vehicle on one of the adjacent lanes, the probability of case G can be written in general as
P (G) = 0,5 • p •g(1 _ p) . (41)
k=0
Conclusion
The probability of n coincidence of vehicles of random length, whose moments of appearance on the lanes of motion are subject to the Poisson law, is determined only by the average value of the lengths of the vehicles on the lanes and the average frequency of their repetition.
The average value of duration ofn vehicles coincidence under the specified conditions is determined only by the average value of the lengths of the vehicles on the lanes, regardless of the laws of distribution of the vehicles' lengths by the lanes.
The law of the distribution Wn (z) of duration of
n vehicles' coincidence, generally speaking, significantly depends on the laws of distribution f (xi) of lengths xi of vehicles on the lanes. However, with the increase in the number of n lanes, the law Wn (z) approaches the exponential one. At the exponential law of the distribution of vehicles lengths on the lanes, the duration of n vehicles coincidence is also distributed according to the exponential law.
Taking into account the traffic in both directions for roads of categoriesI-IV with different num-
ber of lanes in each direction, with the loss of controllability of the vehicle on one of the lanes, the probability of case G (collision of vehicles on one of the lanes) is determined by expression
(41).
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Рецензент: Е. В. Нагорный, профессор, д.т.н., ХНАДУ.
