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2010

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519.1:519.2:519.6:519.8(075) 22.173 24

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9

. . ( ) . . . ( -), . , , (-) . , -.

, . , . ( . . . ), , , Computer Science. , -.

, state of the art . - , . , - , . : Transpor-tation Research B, Physical Review E, Review of modern physics, Transpor-

tation Science. , , , http://arxiv.org/. - : Traffic and granular flow, Springer. , 2011 . . - - - . . . . http://kozlov-traffic-ras.ru/, http://wtran.dvo.ru/ - . , () 2010 . . . . . , , ,

10

, () - () state of the art.

. . . , . . . , . . . , .-. . . -. , -, - . . . . . . . , , -. - . . ( 6- ). , , : . . , . . , . . , . . , . . , . . , . . , . . , . . , . , . . , . . , . . , . . -, . . , . . , . . , . . -, . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . , . . -, . , . . , . . , . . , . . , . . , . . , . . , . . -, . . , . . , . . , . . , . . . . . . .

. () . . . . . . . . . , , , .

- 20092013 ( 1.2.1, -15, 949; 1.3.1, -215, 1490; 14.740.11.0397) - 08-07-00158-, 10-07-00620-, 08-02-00347, 3, -14.

. . (avgasnikov@gmail.com) .

12 2010 .

11

6

50- , , , ( , - ). - () , (. . , . ), ( ), - (. , . .). - () (1955) - ( ) . - ( ) (= -) . - ( , ). , .

- - . (. . , 2000) ( -) (. , . , . . ). -, .

, - , () (. , 1999). -, - (). , - . . . - ,

6 , , -

. . . ( 4, 2010 .) ().

12

( ) - - (, ).

, . ( . -, . .) - ( - ). , - .

, - , , , , , . , , - , -, -, : , , ( -), () - . .

, - , - (. , . , . .), ( ). , . . , , , - . , , . , , - , , - : (. . .), , (. . .). 3, -, .. - -, -

13

,7 -. , - , - , - () .

, , - , . . . . - . (1994) = - + + -, . , http://pems.eecs.berkeley.edu . , . - , ( , ) , , . , . (, - ) , - .

2 3. - ( -

), , - - . ( ) ( , , ) . , -

7 , , , -- . ( . ) (, ).

14

( ), - ( ) - .

(, -) . , . . . . ( ). (), , . - . . , (- , web-). , , ( , ). (-) -. , , ( = ) - , . . , , ( ).

, , - - .

, - - - . , , . 1 . . - () . . (1967) ( ) (- , i, j). , - -, , -

15

, - . , - (-) () ( - ) . - , - . ( ) - ( - ) ( , ), - . ( ) . , , - , : . -, - .

, (), - ( , (1952)). , , , - . , . , ( - ). , , - , -, . , - , , 1 , . . . . ( ). , - GPS- ( ) - .

, , , . , -

16

() () .

. . - 8 (, - , ). ( ) ( ) (-). - (. , . , . , . .). -, -, . . , . (): ( -) . , , - ( ), - (, - , , -; . . . ). - : ( -), ( ), (- , ) .

8 , 1,

. . , , .

1. 1

1.1. ..................................... 181.1.1.

...................................... 181.1.2. .................................................. 201.1.3. ............... 221.1.4. .......... 301.1.5.

...................................... 331.1.6.

.................................... 361.2. ............................. 44

1.2.1. .......................................... 441.2.2. ............................................... 461.2.3. -

..................................................................... 511.3. .............................. 53

1.3.1. ................................................... 531.3.2. - ................. 56

1.4. ......................................................... 61.................................................................................. 65

1 . . . . .

17

1.1.

- -, - - (). - . - , - .

1.1.1.

- , . [48], .

1. - , - .

2. .

- .

- -, - - , (, , ..) . - (user optimization).

18

, . -, -. , , - . -, - . - , - , .

. (system optimization). , , .

, , . [36] . [45], , , , , ( ) .

, - , - , -: . , - , - - . - , . , - - . - .

19

- Rn xy x = xx, x, y Rn. -, , .

1.1.2.

, - , - .

(V,E) , V , E . - -. , -. . - .

- V : S V , , S -; D V , , - D . - , , , . :

W = {w = (i, j) : i S, j D}.

w = (i, j) W w , i j. {w : w W} - . w , w = (uw), uw w, - . -

20

, .

() , i j, - e1 = (i k1), e2 = (k1 k2), . . . ,el = (kl1 kl), el+1 = (kl j), et E t = 1, . . . , l+1. . - Pw , - w = (i, j) W i j. - P =

wW Pw.

xp , p P . - . w xp, p Pw, -

Xw = {xp 0 : p Pw,pPw

xp = w}.

xp x = (xp : p P ). - x , Xw:

X =wW

Xw = {x 0 :pPw

xp = w, w W}. (1)

p P - (, , , , ..). . -, . Gp p. , Gp , Gp = Gp(x).

. - , p Pw w , -

21

,

xp > 0, Gp(x) = min

qPwGq(x

) = uw(x), (2)

uw(x) ,

w W , , - x. x X, (2), . x X - () .

1.1.3.

- (2) [17, 30, 38, 40], - [3, 16, 20, 26], - .

- Gp(x) - G(x) = (Gp(x) : p P ). .

1.1.3.1.

w W w - . -.

1. x X (2) ,

G(x)(x x) 0, x X. (3). x = (xp : p P ) X

(3). , x - (2). , ,

w p Pw, , xp > 0 Gp(x) > Gq(x) q Pw, q 6= p. x = (xp : p P ),,

xp =

xp, p 6= p, p 6= q,xp , p = p,xq + , p = q,

22

> 0 x 0. , x X,

G(x)(x x) = Gp(x)(xp xp) +Gq(x)(xq xq) == (Gq(x

)Gp(x)) < 0, , x - (3). , x (2) .

x X (2). p Pw w W

Gp(x) uw(x) 0, (Gp(x) uw(x))xp = 0,

(Gp(x) uw(x))xp 0, x X,

,

0 wW

pPw

(Gp(x) uw(x))(xp xp) =

=wW

pPw

Gp(x)(xp xp)

wW

pPw

uw(x)(xp xp) =

= G(x)(x x)wW

uw(x)(pPw

xp pPw

xp

)= G(x)(x x),

x (3).

(., , - [32, 37, 38, 40]). , , - . .

1. y Rn X Rn piX(y) = argmin{y x : x X}.

, p y Rn X,

(p y)(x p) 0, x X. (4) (3) -

H(x) = piX(xG(x)), > 0 .

23

1. X X (3) - H(x), X = {x X : x = H(x)}.

. x X > 0, -

(x (x G(x))(x x) 0, x X,, (4) x = piX(x

G(x) == H(x).

x = H(x), (4) x X -

0 (x x)(x (x G(x)) = G(x)(x x), x X.

2. - G -, X , . X, (3) .

. X H(x) : X X . X , , - (., , [1, 8, 18]) H(x) x = H(x), 1 (3).

, - w X, (1), - ., Gp(x) - x X, . - - G(x).

2. - G : X Rn X, x, y X, , x 6= y, (G(x)G(y))(x y) > 0. 3. - G(x) , - (3) .

24

. , , - x1, x2 X, x1 6= x2, (3).,

G(x1)(x2 x1) 0, G(x2)(x1 x2) 0,

,

(G(x1)G(x2)(x2 x1) 0,

G(x).

3 , , -, - .

1.1.3.2.

, - , w = w(uw). , - p P Gp(x) -, w W w(uw) .

uw u = (uw : w W ), w(uw) (u) = (w(uw) : w W ).

z =

(x

u

), F (z) =

(G(x) uTx (u)

),

= (pw : p P, w W ) :

pw =

{1, p w,0 .

z - Z = {z : z 0}.

25

2. z = (x, u) 0

F (z)(z z) 0, z Z, (5) , x - (3) X = X(u) = {x 0 : Tx == (u)}.

. x X - (3) u = (uw(x

) : w W ), uw(x) = minqPw

Gq(x) > 0.

x 0 u 0 (G(x) u)x = 0, (G(x) u)x 0,(Tx (u))u = 0, (Tx (u))u = 0.

,

0 (G(x) u)(x x) + (Tx (u))(u u) = F (z)(z z). . z = (x, u) 0 -

(5), z 0 F (z)z F (z)z.

z = z 0 0.

F (z)z

0 = 0,

F (z)z

0 +.

, F (z)z = 0 F (z)z 0. l, , -

F (z) , Fl(z) < 0,

, zl +, F (z)z 0. F (z) 0. , z = (x, u) 0 -

F (z) 0, z 0, F (z)z = 0, (6) (., , [17, 32]).

26

(6)

G(x) u 0, x 0, (G(x) u)x = 0, (7)Tx (u) 0, u 0, (Tx (u))u = 0. (8)

(7) , u , x. (8) -, Tx (u) = 0, (5)

G(x)(x x) u(Tx (u)) = 0, x X(u).

2 , - (5), (6). Z (5) , z - - F (z).

- , , - , , - . . R > 0, , - B = {z : z R} Z, ZR = Z B 6= . 2 zR ZR, ,

F (zR)(z zR) 0, z ZR. (9) 4. - F -, Z , . - R > 0, , ZR 6= , zR ZR (9) zR < R, (3) .

. z Z (0, 1],, z = zR + (z zR) ZR. 0 F (zR)(z zR) = F (zR)(zR + (z zR) zR) = F (zR)(z zR),

27

zR - (3).

4 (., , [17]).

1. - F , Z , . - F (z) Z,

limz, zZ

F (z)(z z)z z Z, (10)

(5) .

. (10) C > 0 RC > 0, ,

F (z)(z z) Cz, z Z, z = RC , - z ZRC , C RC .

2

F (zRC )(z zRC

) 0, z ZRC .

zC < RC , 4 zRC (5).

zRC = RC ,

F (zRC )(z zRC

) CzRC = CR < 0,

zRC .

(5) - F (z) ( 3).

1.1.3.3.

,

f(x)(x x) 0, x X, (11)

28

f(x) min, x X, (12) f(x) - X.

, x = argmin{f(x) : x X}. x = x

+(xx) X, (0, 1) . :

0 f(x) f(x)

=

f(x) + f(x)(x x) + o() f(x)

.

0, (11). , -

f : X R, , f(x) = G(x) (- G ), - (3) (12).

, , [42]. . , - - , . f(x). - .

L, x0 G(x) x L.

L L = {x(t) : t [0, 1]}, x(t) -, x(0) = x0, x(1) = x.

I =x

x0

G(x(t))d(x(t)) =

10

G(x(t))xt(t)dt =

10

f(x(t))xt(t)dt =

=

10

df(x(t)) = f(x(t))

1

0

= f(x(1)) f(x(0)) = f(x) f(x0).

, I L. :

29

x(t) = x0+ t(xx0), G(x) = f(x) - (3) :

f(x) = f(x0) +

10

G(x0 + t(x x0))(x x0)dt min, x X. (13)

, -G : X Rn G(x) =

(Gp(x)xq

: p, q P) x X. -

(2), (13), .

, , - . -: Gp(x), Pw, - w.

1.1.4.

Gp(x). - , , - . , - , - .

ye e E. - , :

ye =pP

epxp, ep =

{1, p a;0 .

(14) = (ep : e E, p P ) , y = (ye : e E) ,

30

. y = x.

- . , - X, (1), x y, .

e e. e ye, . , - , - .. , e = e(y). - (y) = (e(y) : e E).

- - G(x) (y), , - p P , [29, 30, 40]:

Gp(x) =eE

epe(y). (15)

- G(x) (3)

G(x) = T (y), y = x. (16)

, e(y) ye, e(y) e(ye). p, q P , p 6= q,

Gpxq

=eE

epeye

yexq

=eE

epeqeye

=Gqxp

.

, G(x) x X, - G(x) - (13). (16),

31

f(x)

f(x) =

10

pP

Gp(x0 + t(x x0))(xp x0p)dt =

=10

pP

(eE

epe(y0e + t(ye y0e)))(xp x0p)dt =

=

10

pP

eE

ep(xp x0p)e(y0e + t(ye y0e))dt =

=10

eE

e(y0e + t(ye y0e))

pP

ep(xp x0p)dt =

=

10

eE

e(y0e + t(ye y0e))(ye y0e)dt =

=10

eE

e(y0e + t(ye y0e))d(y0e + t(ye y0e)) =

=eE

yey0e

e(z)dz.

, e(y) e(ye) (13)

eE

ye0

e(z)dz min, y = x, x X. (17)

e(y) BPR- (Bureau of Public Road), - :

e(y) = 0e (1 + (ye/ce)

n),

0e e, ce - e, n -. BPR- - (17).

e(y) -, . -

32

, ,, -, .

, Gp(x) . - [23, 34, 39]. , -, , - , . , [34] - , , :

Gp(x) = p

(eE

epe(y)

)+p(x) +

eE

epe(y),

e() , e, p() -, p , p() , p, , > 0 . [23] - :

Gp(x) = Uw

(eE

epe(y) + gp(p)

), p Pw,

p , p, gp() , , Uw() () w W .

, - , (3) ( (5)) - Gp(x) .

1.1.5.

, , -

33

.

. - - , , ( ). - , , ( ). , - , .

, - (17).

- [33], , , - . - , - - [21, 28, 35] , -, -.

- , -, , [25, 44]. - , , - , - - , . - . ,

34

- . - - , , - -, , [21, 25], -, - .

(3) , , - / - G(x) = (Gp(x) : p P ). - - , [32, 37], - -G(x) (3) .

, - :

xk+1 = piX(xk kG(xk)), k > 0, k = 0, 1, 2, . . . , (18)

piX(y) = argmin{||y x|| : x X} y - X. (, -, , ..) -, , . k 0,

k = ,

, .

(18) - , [9, 10] - [11]. - . X (1) - Hw

35

H+:

X =wW

Hw

H+,

Hw = {xp :

pPwxp = w}, H+ = {xp 0 : p P}. -

Hw H+ H = {H+, Hw :

w W} = {Hl : l = 1, 2, . . . |W | + 1}. piHl() Hl . - (18), [15]:

xk+1 = xk + kvk, vk = (piHl(x

k) xk)/k,Hl H, xk = xk kG(xk) / Hl, k > 0, k = 0, 1, 2, . . . .

(19)

(19) - k. V (k,m) = conv{vk, vk+1, . . . , vm} - vk, vk+1, . . . , vm B = {x : ||x|| 1} . t +0 t {kt}. - k .

1. t = 0 kt = 0, 0 > 0 , q (0, 1).2. t kt kt+1, ,

0 / V (kt, s) + tB, s = kt , kt s < kt+1, 0 V (kt, kt+1) + tB.3. kt+1 = qkt .

4. t = t + 1 (19) k.

, . 2 kt+1 kt , 0 conv{vkt , vkt+1, . . . , vkt+1}+tB, q < 1 (. 3).

1.1.6. -

, . -

36

1(y1) = 1

2(y2) = y2

A B

. 1.

, - , , , , , - .

- . [45], , , , , - A - B (. . 1). A , . y1 y2 , - . - . , A B . 1, , , , 1(y1) = 1. , . , , 2 y2 2(y2) = y2. - () (y1, y

2),

1(y1) = 2(y

2), y

1 + y

2 = 1, y

1, y

2 0,

37

, y1 = 0, y2 = 1,

c(y1, y2) = 1 y1 + y2 y2 = 1.

- ( ) - :

min y1 + y22 , y1 + y2 = 1, y1, y2 0, (20)

y?1 = y?2 = 0.5,

c(y?1 , y?2) = 0.75, 25%

. ,

4/3 - . , - - . - , - .

(2) x X : w W :

xp > 0, Gp(x) = uw Gq(x) q Pw.

, , , xp > 0 xq > 0

p, q Pw, Gp(x) = Gq(x) = uw, , , - .

( c(x)) x xp > 0,

w uw, c(x)

:

c(x) =pP

Gp(x)xp =

wW

pPw

Gp(x)xp =

=wW

uwpPw

xp =wW

uww.(21)

38

x? - c(x?) :

c(x) =pP

Gp(x)xp min, x X. (22)

cp(x) = Gp(x)xp , p P cp(x) -. .

5. x? (22), - . w W , x?p > 0, p Pw,

c(x?)

xp c(x

?)

xq q Pw.

. , ,

w p Pw, , x?p > 0 c(x?)

xp>

c(x?)

xq q Pw, q 6= p. x = (xp : p P ),,

xp =

x?p, p 6= p, p 6= q,x?p , p = p,x?q + , p = q,

> 0 x 0. , x X, c(x)

c(x) c(x?) c(x)(x x?) = (c(x)

xq c(x

)

xp

)< 0,

x?.

, Gp(x) Gp(xp), 5 - , x?

w W , x?p > 0, p Pw, cp(x

?)

xp cq(x

?)

xq

q Pw. , , -

cp(x?)

xp=

cq(x?)

xq

p, q Pw, x?p > 0, x?q > 0. (., , [7]).

39

, - p P , , Gp(x) (15), e(y) - e(y) = aeye + be, ae be e E.

c(x) =pP

eE

epe(y)xp =eE

e(y)ye =eE

(aey2e + beye),

c(x)xp

=eE

ep(2aeye + be).

y = (ye : e E) y? = (y?e : e E) , x x? . x - x? , :

: xp > 0, p Pw, q Pw eE

ep(aeye + be)

eE

eq(aeye + be); (23)

: x?p > 0, p Pw, q Pw -

eE

ep(2aey?e + be)

eE

eq(2aey?e + be). (24)

, e(y) = aeye (23) (24) - .

[46], [, ,G(x)] - , , - = (w : w W ) G(x) = (Gp(x) : p P ).

Gp(x) =eE

epe(y) =eE

ep(aeye + be). (25)

.

40

1. x [, ,G(x)]. 12x

-

[, 12,G(x)].

. x - [, ,G(x)], , , 12x

[, 12,G(x)], (24) 12x (23).

, p Pw, , xp > 0,

c( 12x)

xp=eE

ep{aeye + be} = uw(x).

2. x? , - [(, ,G(x)]. - x [, (1 + ),G(x)]

c(x) c(x?) + wW

vw(x?)w, (26)

0, vw(x?) = minpPw

c(x?)

xp.

. [, (1+),G(x)] x. Gp(x), (25), ae 0, c(x) ,

c(x) c(x?) + c(x?)

xp(x x?) = c(x?) +

pP

c(x?)

xp(xp x?p) =

= c(x?) +wW

pPw

c(x?)

xp(xp x?p) =

= c(x?) +wW

(pPw

c(x?)

xpxp

pPw

c(x?)

xpx?p

).

p, , x?p > 0, c(x?)

xp

:

c(x?)

xp= min

qPw

c(x?)

xq= vw(x

?),

41

pPW

c(x?)

xpx?p =

pPw

vw(x?)x?p.

, c(x),

c(x) c(x?) +wW

(pPw

vw(x?)xp

pPw

vw(x?)x?p) =

= c(x?) +wW

vw(x?)(

pPw

xp pPw

x?p) =

= c(x?) +wW

vw(x?)((1 + )w w) = c(x?) +

wW

vw(x?)w.

.

6. [, ,G(x)] (25) x? x

c(x)/c(x?) 4/3.

. C (21) - x

c(x) =wW

uw(x)w,

1 12x [, 12,G(x)],

vw(12x) = uw(x

). = 1 (26). x,

[, 2 12,G(x)] = [, ,G(x)],

c(x) c(12x)+wW

1

2vw

(12x)w =

= c(12x)+

1

2

wW

uw(x)w = c

(12x)+

1

2c(x).

(27)

42

c( 12x) c(x),

, :

c(12x)=eE

1

2ye

(12aey

e + be

) 1

4

eE

ye(aeye + be) =

1

4c(x),

- (ye, e E), - x. , 12x

( 12y

e, e E).

, (27),

c(x) 14c(x) +

1

2c(x) =

3

4c(x).

x, [, ,G(x)],

c(x)/c(x?) 4/3.

43

1.2.

- w, w = (i, j) , i S j D. w ij , i j.

- = (ij : i S, j D), - . , , -. , , [35; 13, 14, 20, 31].

1.2.1.

-, , , -

. - -, / , / , , .

ij = sidjc2ij

, i S, j D. (28)

si i S, dj j D, cij - i j, > 0 .

(28) . -, si dj , , , - (28) ij , . - (28) , (28) , -, . , () c2ij -

44

f(cij), (i, j) .

ij =sidjf(cij)

,

nj=1

ij = si,

mi=1

ij = dj , ij 0, i S, j D,

, :

ij = ijsidjf(cij), i S, j D, (29)

i j

i =

jD

jdjf(cij)

1

, j =

[iS

isif(cij)

]1. (30)

, , -

iS

si =jD

dj .

f - - , - . : f(cij) = exp(cij), 0.065, 1 (., , [20] ).

, i j si dj , , ij .

i j - . - [19, 22]. - . , [24] :

0ij = sidjf(cij)

[lD

dlf(cil)

]1,

45

:

%kij =

kijdj

[iS

kij

]1,

iS

kij > dj ,

kij

qi = si jD

%kij , rj = dj iS

%kij ;

k+1ij = %kij + qirjf(cij)

[lD

rlf(cil)

]1.

(31)

- . [12] ( 638 638) - (31) 4 .

1.2.2.

, - , -, , - , . - [47].

- . ,, , ( , - , ) .., . , ij - .

- , , , S, , - ,

46

D. , , -, . - . , , , . - .

. - ij , i S j D. -, , = {ij : i S, j D}. - , ij , P (), , - .

() - , Q() , .

P () = ()Q(). (32)

n m .

R =ni=1

mj=1

ij ,

ij > 0 ij .

() -

() = 1111 1212 . . . nmnm =mi=1

nj=1

ijij .

Q(). - 1 1 11, - C11R . , - 12 C

12R11

, 13 C13R1112

.

47

Q():

Q() = C11R C12R11 C13R1112

. . . CnmRm1

i=1

n1j=1

ij=

= R!(R11)!11! (R11)!

(R1112)!12!

(R1112)!(R111213)!13!

. . .

R

m1i=1

n1j=1

ij

!

mn!=

= R!mi=1

nj=1

ij !

.

, , ij - .

() Q() (32), :

P () = R!mi=1

nj=1

ijij

ij ! max . (33)

P () ij , , . . i S - si, j D dj . ij , :

nj=1

ij = si,

mi=1

ij = dj , ij 0, i S, j D. (34)

, - :

mi=1

si =

nj=1

dj = R. (35)

48

(34) :

mi=1

nj=1

cijij = C, (36)

cij i j, C .

, = (ij : i S, j D) (33), (34), (36).

, (33) P () - . P () , , P () (33)

lnP () = lnR! +mi=1

nj=1

(ij ln ij ln ij !) max . (37)

-, , , ij - . - (37) ln z! == z ln z z, z.

lnP () R lnR+mi=1

nj=1

ij lnijij

.

si dj (35) R lnR .

- , (34), (36)

mi=1

nj=1

ij lnijij

max . (38)

49

(34), (36), (38) -, - . , (i, j) ij ij =

1mn

, (38)

mi=1

nj=1

ij ln1

ij max . (39)

, (34), (36), - . (38) . , - (38) mn mn c { 1

xij}. -

m, n xij 0. , (34), (36), (38) . - . -, (34), (36), (38) , - . , , . [12] 800 800 . 22 29 -, , (34), (36), (38) 407 044 , 1277 (34), (36).

(34), (36), (38) - [19, 22]: {ij = iij} :

%kij = kijsi

jD

kij

1

, k+1ij = %kijdj

[iS

%kij

]1. (40)

[2] (40) - (34), (36), (38). - (., , [6, 31]).

50

, -, . . -. .

1.2.3. -

(34), (36), (38), , . - , :

L(, , , ) =mi=1

nj=1

[ij ln

ijij

+ i(si ij)+

+j(dj ij) + (C cijij)],

= (i : i S) , - (34) , = (j :j D) , - (34) , , (36).

L(, , , ) - (34), (36)

lnijij

1 i j cij = 0, i S, j D. (41)

(41)

ij = ij exp(1 i j cij). (42)

, ij 0 - ij , . , , ij = 0 - (i, j), , ij = 0

51

- .

i =exp(1 i)

si, j =

exp()dj

.

(42)

ij = ijsidjij exp(cij). (43)

(43) (34) i j :

i =

jD

jdjij exp(cij)1

, j =

[iS

isiij exp(cij)]1

.

, C , , , (36). -.

(43) (29), -, - f(cij). f(cij) = ij exp(cij) - (29) (34), (36), (38) . - , , si, dj , cij , - C - (i, j) , .

52

1.3.

, - .

1.3.1.

(. 1.1.6) - , , , - . , - , - .

- . , . 2. , 6 -. . 2 , .

): , . , - , . , . 2 ), y.

, - . - y = 3. 90, 540.

): - ,

53

.

. ()

10y

10y20y

20y

.

10y

10yy+50

20y

) )

.

10y

10yy+50

y+50

.

10y

10yy+50

y+50

y+10

)

)

. 2.

54

, - . - . (3.17), 2.83. 84.88, 84.88 6 = 509.28.

): - , - 83, 498, .

): - , - . - , - .

: , - , , 2 , - 92, 552 !

, a) , -, .

, ) - . 70 - . , , .

55

() , .

1.3.2. -

[41], -, - . , , (). - , - : -, , . -, , , , . , , .

1.3.2.1.

, -, . 3, - (y), e(y) y. , , - . -

E = 2 3 (0.2 + 0.1) = 1.8 . -

E = 4 0.2 + 2 0.1 + 2 0 + 4 0.2 + 2 0.1 = 2!

, , , .

56

A B

U

D

(y)=y+50

e(y)=0.1y

(y) =10y

e(y) =0.2y

(y)=y+50

e(y)=0.2y

(y) =10y

e(y) =0.1y

A B

U

D

(y)=y+50

e(y)=0.1y

(y) =10y

e(y) =0.2y

(y)=y+50

e(y)=0.2y

(y) =10y

e(y) =0.1y

(y)=

y+10

e(y)=0

) )

. 3.

1.3.2.2.

, - . 4. , , -

A B

C

(y)=

y+1

e(y)

=0

(y)=

y+4

e(y)=0.01y

(y) = y + 1

e(y) = 0.5y

. 4.

, (y) - e(y) y . - 2 C B C A. , 3 : p1 = C A,p2 = C A B p3 = C B. -

57

(x1 + x2) + 1 + x2 + 1 = x3 + 4,x2 + x3 = 2, x1 = 1,

xi, i = 1, 2, 3 pi, i = 1, 2, 3. x1 = x

2 = x

3 = 1

E = 0.51. , C

A 1/2. x1 = 1/2,x2 = 7/6 > 1, x

3 = 5/6 < 1

E = 0.5 7/6 + 0.01 5/6 0.591 > 0.51.,

. , p3, C A p2, , , , A B .

1.3.2.3.

, , , .

, . 5, )., -

A B

(y) = y1 + 10

e(y) = 0.1y1

(y) = 3(y1 + y2)

e(y) = 0.5y2

A B

(y) = y1 + 10

e(y) = 0.1y1

(y) = 3(y1 + y2)

e(y) = 0.5y2

(y) = y + 11

e(y) = 0

) )

. 5.

5 A

58

A B

(y) = y + 5

e(y) = 0.2y

(y) = f + 5

e(y) = 0.2y

C D

(y) = y + 10

e(y) = 0.4y

(y) = y + 5

e(y) = 0.1y

. 6.

B , . y1 , y2 . - .

-

y1 + y2 = 5, y1 + 10 = 3(y1 + y2),

y1 = 5, y2 = 0.

= 15, 0.5., . 5, -

- (y) = y+11,

y1 + y2 + y = 5, y1 + 10 = 3(y1 + y2) = y + 11,

: y1 = y2 = 2, y

= 1. = 12, 1.2 !

1.3.2.4.

, ( -) (, ) - . ,

59

, . 10 A B 5 C D. A C, - C B , .

4 , , - . , , .

:

. - y1 = y

2 = 5. E = 2.

. y3 = 0, y4 = 5. E = 0.5.

2.5. , 2.5

. :

. - y1 = y

2 = 3.75. E = 1.5.

. y3 = 0, y4 = 5. E = 1.125.

2.625, (!) .

, 2.5 , - 3 , 1, 2. - - , - , , 4.

60

1.4.

1. , . - , - , . :

1) - - ;

2) -, , - - .

- .

2. = (V,E)( [27]), 25 (|V | = 25) 40 (|E| = 40). . 7.

() -:

1) Eh = {(6 7), (8 9), (10 11), (12 13),(14 15), (17 18), (19 20), (21 22), (23 24),(25 16)};

2) Eex = {(16 1), (15 1), (24 2), (7 2),(22 3), (9 3), (20 4), (11 4), (18 5), (13 5)};

3) Een = {(1 6), (1 17), (2 25), (2 8),(3 23), (3 10), (4 21), (4 12), (5 19), (5 14)};

4) Es = {(15 6), (7 8), (9 10),(11 12), (13 14), (16 17).(18 19), (20 21), (22 23),(24 25)}.

61

12

3

4

5

16

17 18

19

2025

24

23 22

21

10

11

12

13

1415

6

7

8

9

. 7. = (V,E)

100 , 50 .

. - 0e e E 0e = ele, le e, - 1, e > 0 , , e:

e =

0.011, e Eh,0.025, e Eex Een,0.033, e Es.

S = {1, 2, 3, 4, 5} s = (69, 90, 10, 100, 53) D = {17, 19, 21, 23, 25} d = (128, 59, 34, 61, 40).

- , w w W = S D.

62

1. = (V,E)

6 7 4 16 1 6 1 6 38 9 10 15 1 9 1 17 710 11 3 24 2 3 2 25 612 13 3 7 2 8 2 8 214 15 5 22 3 5 3 23 617 18 1 9 3 1 3 10 519 20 2 20 4 10 4 21 621 22 6 11 4 8 4 12 823 24 9 18 5 5 5 19 725 16 2 13 5 3 5 14 415 6 1 7 8 6 9 10 411 12 3 13 14 9 16 17 1018 19 4 20 21 6 22 23 1024 25 1

, w f(cw) = exp(0.065cw), cw - w. , cw - w :cw = 0.05, ,

cw = minpPw

eE

ep0e , ep =

1,

p e;

0, ,

Pw - w.

- , . - , - ( ). - ,

63

( ).

w e, . e - , e E, e(ye) =

0e (1+(ye/ce)

4), ce e.

64

1. . . ..: - , 1980.

2. . . - // .1967. 1. C. 147156.

3. . ., . ., . . . .: , 1981.

4. . ., . ., . . - . .: -, 1987.

5. . . . .: , 1978.

6. . . - // . 2009. . 49. 3. C. 453464.

7. . - .: . , 1970.

8. . ..: , 1972.

9. . . - // . 2008. . 48. 12. . 21212128.

10. . . c - // . 2008. . 422. 5. . 601605.

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14. . . -. .: , 2008.

15. . . - // - " ". -: - . 2008. C. 618-624.

16. . . . .: -, 1961.

17. . . , - . : - .-, 2001.

18. . . . .: , 1983.

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68

69

2.

1

2.1. .......................................................... 70 2.1.1. (LWR) ................ 70 2.1.2. ................................................................ 82 2.1.3. ............................................................... 83 2.1.4. ....................................... 92 2.1.5. ..................................................... 97 2.1.6. ............................... 99

2.2. ....................................................... 102 2.2.1. ..................... 102 2.2.2. ...... 107 2.2.3. ..................... 109 2.2.4. ...................................... 112

2.3. ..................................................................... 119 2.3.1. , LWR ................................ 119 2.3.2. , LWR ................................ 136 2.3.3. ( ) ......................... 143

2.4. ................................................ 146

2.4.1. ............................................................. 147 2.4.2. .................................................................................. 151 2.4.3 J ............................................................. 154

........................................................................................ 157

1 2 . . . . . . . - . . ,

. . 2.12.3 () (, ; ).

70

2.1.

2.1 ( , ) . . , , , ( ). - , - . , , - ( ). Equation Section 1

2.1.1. (LWR)

40- 50- , (., , [1, 2]). , - ( ) . , [3, 4]. 1955 . [5, 6] (. [7]) , -, (-) 2 , - () (LWR), ( - ) - ( ).

LWR ,

1) - (

) v ,t x ()

,t x ; 2) ( ).

2 , ( -

), (), ().

71

,t x t x .

, v ,t x ( ) t x .

, , () , (.. ).

,t x , v ,t x , , (., , [5, 8]), 3, - ( (. . 2.2)) . , - , , - . , , -

,t x , v ,t x -. ( ).

:

v , ,t x V t x . (1)

V :

0V . (2)

Q V

3 , . . -

: -

f x , x , -

f . ,

.

72

( ,

). Q ( ) . ,

V

(., , . . ). [7]:

0Q .

: , , . ( , , ), , ,

Q , , .

. 1.

. 1, 2

. , (

73

2005 .)

,

. ,

V Q .

. 2.

Q 60 115 / , -, , . , . , ( 30 50 /), , ,

50 120 / , , [9]. 3 (. [10]) , 50 120 / . , () .

74

().

, ,b b

a a

t x dx t x dx

, ,t t

t t

Q b d Q a d

.

, - 0t , x , ( -, - ), :

, , 0t x dx Q t x dt

. (3)

,t x :

v Vt x t x

,

..

0Q

t x

. (4)

( )

0

, ,

0, , ,

, .

x x

x x x x x

x x

(5)

(4), (5) , , - ():

0 0x , max ,

max ( -

). , - . - , , : . , . ? ( , -

75

.) (4) 2- 3- [7] (. . 2.3).

(3). , -

,t x . ,t x -, . t

x ,

, 0t x , , 0t x .

, ;t x

L . ,t x L - ( , ) , . 3. , , , L , (. . 3).

. 3. RRH-

(3) ,

0 , ,t x dx Q t x dt

76

c Q t c Q t o t ,

c dx dt L ,t x , t t . 0t -

( , (1848)) c , ( ,

. [11]) :

,Q Q

c

(RRH).

, (4) (- (3) (5) ) (. [12]), , , , .. .

(. . [13]). (. [14]):

0t x

:

1, 0,

0,1, 0.

xx

x

(, ) . . 2.1.3.

LWR, Q ( ).

1q 0t

11, ,

2

1, 0,

2,

1, 0 ,

2

11,

2

q

qx t

qq t x

t xq

q x t

qt x

77

0t (4) (3) ( -

, RRH) - (5) (. . 4).

. 4. . . (1957)

, -, . . 1958 . [15, 16] ( -

() Q . . . (1957) [13], . (1957) [11]) . . 1959 . [17]4 () .

, RRH-, E-:

, , , , ;

, , , , . , - . . , E- . . . (- , . . 2.1.3) , E- ( RRH-) . 2.3.1. , - (-

4 [17] , -

, . . 19571958 .

78

) - (3)

,t x ( -

, - Q ). , 70- 80- XX , , - . 2.3.

- (. . 5) -

( ):

Q , ,

,Q ,Q . c .

Q , , [11; 1719] ( ).

. 5. E-

(. [18]).

79

1, 0,

0,1, 0.

xx

x

(4), (5):

1

1, ,

, , ,

1, ,

x t

xt x t x t

t

x t

21, 0,

,1, 0.

xt x

x

0x , ..

0, x , -

x 0, x , (, -, (. . 6), ),

10

lim ( , ) ,t x t x

.

. 6

.. 2 ,t x -

. , 2 ,t x -

80

E-, 2 ,t x . - , ,

2 ,t x -. , .

,

;t x

,dx

Q t xdt

.

,t x -

0Qd dx

Qdt t x dt t x t x

,

.. , constt x . ,

- (4), .. -

: ,t x , , ( ). -

, , , (5) [19]. - (4), (5).

, , (4), (., , [18, 19] . 2.3.2). .

. . ,q t x 1q - (4), (5), E- ( -). (4), (5) , - ,

,q t x 1q -, -. , , (4) - ( t t ),

81

Q . , -, (4), , - , , . .

, - (4) , .

(. . [17]). -

, ( 0, 0x ,

, , 0t x ):

2 20

t x

,

2 32 30

t x

.

0

. RRH :

2 2 ?

1

2 2 1

2C

2 2 3 3?

22 2

3 32

3 2 2C

,

( ). RRH, :

q qc

0

q

t x

,

0 . (. [20]).

0Q ( 0Q ).

k :

82

Q k Q ( Q k Q )

, -

(4), , k .

2.1.2.

V ,

1963 . . [8, 21] (-,

).

.

maxv .

1

vvd

,

21 2v v vd L c c ()

v ,5 L , 1c , -

, 2c

(. . 2.2.3). vd -

(1) V , (2).

2c , , .

[21]:

2 2 2 2v 5,7 0,504 v 0,0285 v d ,

[22]

2 2 2 2v 5,7 0,504 v 0,0570 v d ,

[22]

2 2 2 2v 5,7 0,504 v 0,1650 v d .

5 vd , -

vd , -.

83

LWR ,

V , . , - [8].

2.1.3.

( 1955 . 1974 . . [7]) :

, ,v , ,

,

D t x t xt x V t x

t x x

, 0D .

, :

v0

t x

,

( - ):

Q

Dt x x x

. (6)

( (1) (4)) - , - . () (2), (5), (6) .

(. (1934)), Q (-),

D ( 0 ) , (4) -

: t t , x x , - () ( - (., , . (1915), . (1934), . (1940), . (1950))):

2

2t x x

, 0 .

84

() [4, 7, 19] (. ):

2 ln 2 xw

wx w

;

(4), (5) ( ) :6

2

2

w w

t x

,

0

10, exp 0,

2

x

w x d

.

, . 1950 . - [7, 14]. , , ( 7) 0 - :

0t x

.

, (4) (4), (5) , , , -. , , , . , - . , ( . . , . . , . ) (4), (5) (- (4) (3)). (. . . . 2.1.1) -

6 , , [23]

, ( -

( )). , , , .

[24] [25]. , - ( )

( ) (. . . [26]). 7 , .

85

. ,

(4), (5) ,t x ( x 0t )

8

0 , :D D , 0D

(6), (5) ,t x :

1, , Lt t t ( . . (1975)).

,t x ,

0D , (4) -

(5). ,t x - (4), (5) (. . 2.3.2).

. (4) (6) ( ) -. - 50- XX (. , . . , . . . . , . , . . , . . .). . . 1960- . ( . [11; 2736], . 2.3.2).

, (6) , , () . - . - [37, 38] (. . , . . -, . . , . . .), - -

8 ,

, (6). , , (6)

2

2

QD

t x x

,

, , D . , - . . (. . 3.1.1). ,

1D , - -

0D .

86

. , : - - ( ) , - ( ). , , - , -. (6), (5) 0 , - (4), (5).

(6) - (). :

2

2

0, 0,

Q

t x x

x x

, (7)

2

2

0

,

0, 0, .

x

U U UQ

t x x

U x y dy

(8)

, ( , - (7)):

QD

t x x x

, 0D ,

0, 0,x x x

0t :

0

lim , ,t x t x

.

, ,

0

lim , ,U t x U t x

0t , x .

,U t x , - :

87

0U U

Qt x

(9)

00

0, 0,

x

U x U x y dy . (10)

,U t x () (9), (10).9 (7) (8) , :

,

,U t x

t xx

.

, x

0t :

0

lim , ,t x t x

, x

0t :

0

,lim ,

U t xt x

x

.

9 ()

(., , [39, 40]):

0

0 0, ; , , ,

t

t

J t x u L x u d t x ,

, ,x f x u , u , x t x .

,U t x

, inf , ;u

U t x J t x u

, 0 0, ,U t x t x .

(., , [41, 42]).

( ). ,U t x - ():

sup , , , , , 0u

U U U Uf t x u L t x u Q

t x t x

, 0 0, ,U t x t x .

.

,u t x ,

u , , . ,

, , - .

88

() Q 0U x , x 0t :

0

, ,lim ,

U t x U t xt x

x x

. (11)

( )

10, , ,U t x , . (1965), ():11

1) 0U x ,

*0, sups

U t x sx Q s t U s

,

*0 0supx

U s sx U x

;

2) Q ,

*0, inff

U t x U x tf Q f t

,

* sups

Q f Q s sf

.

Q 0U x . , , 1).

( 0U x ),

10

. [43, 44] ( . [44]). - , , [45, 46]. ( ) ,

60- - XX [43]. . , 60- XX . . . . -

- ( ) [39, 47]. 11

, , . . . ;

, . , . . ; -

(-) 1965 . ( ) . . [48]. . . . . [49].

89

**0 00,U x U x U x ( [45]). ( ) -

, ,U t x t Q s t x , , , ,U t x x t x s t x ,

*0, = argsups

s t x sx Q s t U s

,t x ,

s *0sx Q s t U s . ,

, ,, , 0

U t x U t xQ Q s t x Q s t x

t x

.

, - ( arg , Arg ). ,

,

, ,t x s t x . ,

*0, , Argsups

t x s t x sx Q s t U s

(4), (5) -

0, x , ,t x , - , .

- (11), -

(4), (5) ,t x (, -, 2), [11, 42]). , , - . 2.3. , . . . . 1982 . , - 1 . 2.3.1, 1).

( - [5052]).

2

2

w w

t x

90

: 2 2

2

1

2

U U U

t x x

.

, ( . -)

2 lnU w

. , , . . 1948 . ( . . ). 0

2 lnw w U , ..

0 0lim lim 2 ln

def def

w w w U

,

. . . . [50] 80- XX - ( , ) ( , -, ()) - , , :

1 1w U , 2 2w U ;

1 22 21 2

0lim 2 ln e e

U Uw w

1 22 2

0 0min lim 2 ln e , lim 2 ln e

U U

1 2 1 2min ,def

U U U U ;

1 22 21 2

0lim 2 ln e e

U Uw w

1 22 2

0 0lim 2 ln e lim 2 ln e

U U

1 2 1 2def

U U U U . -

, . ( - ()) . , - [45],

91

. , . , - (, ) (. . ), ,

, : 1U , 2U

, 1 , 2 1 1 2 2U U U

. , , -

(4), (5), , [33],

, ; 0, ,t x x Q

(4) 0, x

Q 0, Q :

1

1 1 2 2, ; 0, , , ; 0, ,L

t x Q t x Q

11 2

0, 0,L

1 2 1 2min T.V. 0, , T.V. 0, Lipt Q Q , T.V. total variation ( ),

1 2

1 2 supdef

Lip

Q QQ Q

.

,

1 2Q Q (6), - . , ( (6)) - ( x 0t ) -

(4), (5). ( (4) (6)).

92

2.1.4. 12

(1971) [7, 53] ( (. . 2.1.3)).

v0

t x

,

( , -). 13

v v 1

v v vDd

Vdt t x x

v

D

Vx

,

( 1 ) ( - ; - () -, [54]).

v 01

vv v vD Vt x

, (12)

(-

x ).

, , : - - , , , () . , - ( - ), () , ,

12

. . . 13

:

20 0D c .

93

2vv 1v

PV

t x

,

0

1P D d

.

. ( ), -. - (, . , . . , - () .), ., , [4, 11, 18, 33; 5456] .14

14

(

) [54, 55]. ( . 3.2.4), .

[11] (. [54, 55]), -

, . . 50- XX -

. 0, x -

0, kx , 1k x k ,

, k 0, x , 1k k , ..

1

10,

k

k

k

x dx

.

0, x . k (. . 3.3.1). ,

k 1k , , max 2t c ,

max maxc Q . , -

, .

max2c ( , - [2]

1

maxc

), -

. , - LWR .

94

[19, 29], - ( . (12) -, ) -, . -

D ( ) [18]. - 15 - (. [11; 2936]): - , - .15 , -, c ( - ):

0 , :D I , diag 1, ...,1I

, , . ( ) , [34]

(. ). (-, , ) , , [18]

[57]. - front tracking - [33, 34], (- ),

- ( Q ) - . ( -

) - [33, 34]. , LWR (. . 3.2.4).

, LWR- . . ( - (. . 3.2)). -

, ( , , ) , . 15

. (1965) [11, 34], -

( 0,k k , 0,1R

0,1 ), - , ( -

). , (-) (-) ( , - 2004 . . . ,

), . . . - http://www.mathnet.ru/php/presentation.phtml?option_lang=rus.

95

( ).

LWR . . , , () , , , , [54]. -, ( ) - (-) - , . : , .. - : . (1993), (1994) (. . 2.4) . [58] ( 100 ). - ( . ): . . , . . . () [5961]; . . . () [62].

( , ) . () (1995) [56, 58, 63] (. - . . [10], . 2.4 3). -, , - ( -). -, ( ). , , , , . XXI . . [64], . . [65], . . [66] , , . . :

v v

v v 0p pd

pdt t x

.

, 0p . ,

p :

p , 0 . . (2000)

p V . . (2001), . . (2002)

96

[64] , :

1

v vd

p Vdt

.

, - .

(. , . , . , . , . , . , . . , . () ., . http://arxiv.org/), . , [67], , - . . (1996) [10] - (. 3): , ( ) , ( -) , ( ). , -, - ( ) - , - .

: 0 ( ) LWR? , ( , . 2.1.3, , ). , , . , -, [30, 64].

(1995) [58, 68], ( ) () ( ). ( - ) , . , , - , , (-) , - ( ) . - 2000 . . -

97

, , ( ). (- ), , - , . -, -, . - . . [69] . . [70]. , [69] - : - -? , , , - (. . [30], [71]).

2.1.5.

, - . ( . . ) 1960 . - ( ) [56, 58, 72]. - . . - (1975), . (1995) . [56, 58].

( ) (-) , (), .. - - -

( ) ; ; vt x . , -, .16 ,

16

, -

() [73]. , - (. . . [74]), - . . .

. . , [75].

98

, - ( ), . - , ( () ), ( , - ). , ( - ), -, .. - () -, , , (). , ( ) - ( ) -, . , - ( ) [36].

, , [36], [75], - (). - . (, -), (. . 2.2.4), -. (, -, ) - (-, ), . , , , : .

, , - , - . , , , . . (. [76, 77]).

99

2.1.6.

, LWR ( -- ) . (, , ). - - (4) . - (., -, [78, 79]). LWR ( ) . , , , - (. . , . . , . , . . [80, 81]) -.

. 7

100

, ( , ) - , , .17 , (. . 7): - , - , - (, , ).

. 8.

(, -) , . - [82, 83] (CTM Cell Transmission Model, . . 2.2.4) . = (1959) (4) + - 18 (. . 8),

17

, () [34, 78, 79]. - (, -

) - . 18

. ,

. 2 -, .

101

-, ( ) - [80, 81].

, - , - (. , . , . .), . , . . [10] (. 3), , , -. , , . , , , , : LWR (. . [34, 78]; . . . [80, 81]), , (. . . [86, 87]). , - . ( . [63]) (, ). ( ) , - , , - (-, , ..).

102

2.2.

2.2 - . . , - . . - ( ) . ( , , - ), .

2.2.1.

. ns t n- 0t .

1n n nh t s t s t , vn nt s t .

. 9.

( 1961 .

103

[7, 88]) , ( , ):

v 1n nt V h t ,

, ,

0V .

,

Q V

max (

j (., , [7]))

, L [7] ( 5,7L ):

maxL Q ( 0,4 . 2).

, V , ,

1 L .

( ) max 1 L :

1 L

V

.

max

maxL

Q

.

L

7,5d ( . 2 , 6,5d ). [10] (. . 2.4). , , , - , ( ), (. 8).

.

,h t x , , 1 ,t x h t x , v ,t x ,

0t , x ,

104

1,

2

n n

n

s t s th t h t

, v , vn nt s t t .

nh t ,

v , , 1 2n n nt s t V t s t h t ,

1, 1 2 v , v ,n n n ndh t s t h t t s t t s t

dt ,

v , v , v , v ,n t n n x nt s t t s t t s t t s t

, , , 1 2 ,n n x n nV t s t V t s t t s t h t s t ,

, v , , v , ,t n n x n x n nh t s t t s t h t s t t s t h t s t .

2

( ,t x v ,t x 0t ),

v v vv2

t x x

VV

,

v 0t x .

, , [7], ,

2D V .

0 , .

1 2 xV h

V (, h ), LWR.

. (1780) ( ), ( ), ( ),

105

. ( u (.

. 10)) ( ) .

. 10. , (1780)

, - , ( , , , ) , . , ( 50- XX ), : ( (1895)) . [89] . , ( ) - ( , , , , ). 0 (, , ( ), , LWR ) [18, 33; 90111] (. . 2.3.1).

:

,t x , v ,t x V .

106

, [7], , , 1) ; 2) 19

2D V .

:

22 1V .

19

.

, ,t x r t x , v , ,t x V w t x ,

,r t x , ,w t x , , . : (

) r w .

, ,r t x , ,w t x

(, , , xr r rw , ). -

expz ikx i t

.

,

, exp 0A k z ikx i t

.

det , 0A k ( - , :

2k ( )):

2 2: 0V k i Q k D k .

,k ( )

, exp ,z k ikx i t d k

,

,k ,z k

, (., -

, . [112]). , ( ) ( ) -

, , ( ):

1 k , 2 k , k ,

Im 0 . -

.

107

: . .

. , 0 ( ) , [97101; 104106]. , ,

vn t . vn t

kF t

k n ,

. ,

. . (., , [18, 33, 54, 55], . 2.1.4). . , , , .

, 0 (. . 2.1.2). , ( )

vn t :

2

1 1 2v vn n n ns t s t L c t c t .

2.2.2.

( ) [8, 10, 56, 58, 113].

1959 . . , . , . [58, 113] (, -, . (1950) . (1953)) - ( , ),

108

. :

1

1

n n

n

n n

s t s ts t

s t s t

, 0 .

n- ns t :

1vn n nt s t s t

( v 0n t , 0ns t n- ; v 0n t -

; v 0n t ( )) () - :

1n n nh t s t s t . :

vln

n

n

d t dh t

dt dt ,

v v ln0

n

n n

n

h tt

h .

v 0n , max0 1nh . ( )

maxlnV

,

1

L .

. 1959 . -.

1961 . . , . . [56, 113]:

1

21

1

m

n

n n nm

n n

s ts t s t s t

s t s t

, 0 ,

1 1m , 2 1m ( 1 0,8m ,

2 2,8m ). ( -

), :

109

2 1

11 1

0

max

1

m m

V V

,

0V ( , -

). 1 0m , 2 2m

(. . 2.1.3). , 0

, : - .

2.2.3.

- (Intelligent Driver Model (IDM)):

1 1, ,n n n n n ns t F s t s t s t s t s t . , 20 . () (1999) [10, 56, 58, 114, 115]:

2*

1

0

1

,1

n n n nn

n n

n nn

d s t s t s ts ts t a

s t s tV

.

0

1n

n

n

s ta

V

, - - ( ). -

20

, - ;

.

110

0

1n

n n

n

s ts t a

V

, , , , , , ,

2*

1

1

,1

n n n n

n n

n n

d s t s t s ts t a

s t s t

. ,

( 1 , -

na 0

nV ).

*nd ( -

) :

1n n nh t s t s t , :

1

*

1,2

n n n

n n n n n n n

n n

s t s t s td s t s t s t d T s t

a b

,

nd ( n - 1n -) (,

nd L , 5,7L , , , -

, 7,5nd ), nb -

( 22 n na b ), nT . .

n- ,

n nT s t . , , , ,

( 1n ns t s t ), -

( nb ) ,

1n - , ,

n- 1n - ,

111

12

n n n

n n

s t s t s t

a b

.

, ( 1n ns t s t ), . , - ( )

2 n na b .

, ()

nT (

). , , (, ) L 5 / (.. 18 /). ,

5,7 [] 5 1,1nT , , (. . 2.1.4) (. . 2.2.1) .

,

0ns t , 1 0n ns t s t , ns t V :

1 2

* 0

1 ,0 1def

n nd V s t s t d V V V

.

(, v 1 d V ) -

V , Q . (. . 8):

01

min ,d

Q VT

.

( )

7,5d 5,7L (., , . 2.2.1). - , 1 , 0nd (, - ), -

V .

112

2.2.4. 21

(Cellular Automata (CA)) , . ( [8083], . ) , . , -. , - .

. - () 50- [116] . [117]. - . . [118] ( . [119, 120]).

(1992). CA-- 1m m -

. 1. ( -

, ):

maxv 1 min v 1, vn nm m . 2. (

):

1v 1 min v ,n n n nm m s m s m d , 7,5d (. . 2.2.3).

3. ( ):

max v 1, 0 , ,v 1

v , 1 .

n

n

n

m pm

m p

4. :

1 vn n ns m s m m .

. , , 3 -

21

. . .

113

.

[121, 122] ( , ) . [123] - .

, . . [124126] (. . . ),

V , - CA-.

, n ().

. () . ,

n . , , ,

(), n -

( , ., , [127] - ) n , -

. - . 0 1 .

, - , , -, 0 1p . .

:

1

1 ndefSi

i

V m V mn

.

(. - . . ) 0 1p -

( 1p . [124])

.. ..

1

1lim lim

NdefT S

i i iN m

m

V V m V V mN

.

V

. 11, , , . 11, . - . 11 -.

114

. 11. 1 2

CA- . [128]. , - [119, 126, 128], , . , , -, (. [10] 3).

( -) . . . ( ) [129131]. (. . 2.1.2), , - - , . , (. 2) .

, , , -

115

. . 2.1.6 (CTM ). , . - ( , ) . () - . ( ), (. . 12).

11i i i i i in m n m r m q m s m q m , i i is m q m ,

in m i- m .

. 12

, [84, 85] ( )

1

1i i i iq m Q n m

LWR. -

( 0i 0ir m ). , ( ). ( ), ( ). - , .. . , . , , , -, .

0iq m , .

116

() - .

[8083] ( - (. . 8), -

max,i in Q )

max max max1 1 1 1min 1 v , , ,i i i i i i i i iq m n m Q Q n n m w (. . 2.1.4) LWR (CTM- . (1994)). - :

max

v iii

Q

n

,

max

1

1 max

1 1

i

i

i i

Qw

n n

,

.. . ,

( , 1 1i in m n ),

1iw

t 1iw t . -

, , , . , -

, , max,i in Q i .

CTM- - , . -, . , ( - ) - ( ). , , : -. A - C, B C. , C 4000 / (- ). A B , , , 4000 /. , . , -

117

, , , - ( ). , - -, , , . - , - . - . , 1936 . . , , , [132]. -, 2010 - . , - . , .

, , CTM- - (., , [8083] ).

(- LWR-), , -, - - LWR-, .. , [84, 85]. ( ) -, - (. . 2.1.4) - , LWR ( - ), .

: ( , ) - , - LWR- ( - ). 2004 . . . - . [84] (, - ..). 2006 . . . [85] . , , - - . 2007 . . . - [80] (-

118

) ( - ) CTM- ( + ) - (.. - ).

() ,22 [81, 83]:

max max max:

min min v , , min ,ji i j j j i i i ij j i

dQ w Q

dt

max max

max

max:

:

min ,min v , min 1,

min v ,

k k k ki

k i i i lk i k k l l l

l l k

w QQ

Q

,

-

, i i- ,

j

i

j , i . ,

,ji t

.

, , , -

, , , :

, 1ikk i k

t

. -

(., , [34, 78, 79]).

** ( [80, 84, 85]). ( ) , - () - .

22

, , - , , , .

119

2.3.

2.3 .

(, ) , LWR, , , . . , , , , , (, ) , , . ( LWR) : ?

2.3.1. , - LWR

, Q - , - .

( LWR):

0Q

t x

, (13)

, 0

0,, 0

xx

x

( ), (14)

0

,

0, ,

,

x x

x x x x x

x x

( ), (15)

0 x (. . 13, 14).

(14), (15) .

120

( :Q Q ). (13), (14) () () .

. 13.

. 14.

121

. 15

. 16.

122

, (13), (14) (, , , . 2.1.1), , :

(13) (0x

)

(. . 15):

00

, ,

,

x x ctx ct

x x ct

, RRH- E-;

(13) ( t t ;

x x a )

(. . 16):

1

, ,

, ,

,

x Q t

x xg Q Q t x Q t

t t

x Q t

(16)

( 1Q Q )

, 0Q , , -, , ,

(, ).

. ., , [1, 133, 134], . , , ( ) , (.. t x ).

, , (- (.. t x ))

, . . , , , ( -), .. -

123

. , , -, . - , -, ( ) . , -, , , , .

, (13) , - . , (13) :

t t ; x x a ; t kt , x kx . , x ct ( c ) :

t t , x x c ,

x t :

t kt , x kx ,

, , (13)

x ct x

gt

.

( ) ( ) [1; 8997; 100111; 133144]. ( , ) , , ( ) (. 1, 2 . 2.3.1).

( ) , . 2.1.1, . . 50- (. [17, 28]) , .. (13), (14). (. . 2.1.3): . , , RRH- E- (13) .

124

. . , , [28], . . 60- 70- XX .

**Q F , ,F Q I ,

,0, , ,

, , ,I

* supy

F x xy F y

, ( ) F y [45].

, Q

, : , ,q q Q .

0Q

t x

(17)

, :x

t xt

(, x t :

x kx , t kt , (17)), -

(14). x t (17)

0Q . (14) ,

1,x

t x Qt

( 1Q Q ).

- . (. . 17)

0 0 1 1, : , , ... ,n nQ Q ,

125

0 0 0c Q Q , n n nc Q Q ,

k k kc Q Q (, 1k kc c ), 1, ..., 1k n ,

0 0 1 1 2 1... n n n .

. 17. . . (1958)

( ), - [106]:

0Q 1,k k , 0, ..., 1k n ;

0 0Q , 0 0 0 0Q c ;

0nQ , n n n nQ c .

(. . 18 0kd )

0

1

1

, ,

, , , 1, ..., ,

, .

k k

n

x c t

t x Q x t c t x c t k n

x c t

(18)

, - :

126

0Q 1,k k , 0, ..., 1k n ;

0 0Q , 0 0 0 0Q c ;

0nQ , n n n nQ c .

. 18.

, - ( ), (. [110, 111] ).

, ,t x x 0t (. . 2.1.3).

(18) ,t x . (18) ,

1Q x t , 1k kc t x c t , 1, ...,k n

( , 1k kc c ,

1k k ).

127

, (14), (13) . , , (13), (15) (18). , [14, 90; 9396; 106, 111] (. . 18).

1. 0

n

k kd

(

1k kc c , 1k kd d ), (13), (15)

t 1 xL

0 0

1

1 10

, ,

, ; , , 1, ..., ,

, .

n

k k k k kk

n n

x c t d

t x d Q x t c t d x c t d k n

x c t d

(19)

, [90, 95, 96]

kd .

:

Q

Dt x x x

, 0D , 0 . (20)

(20), (15). (13), (15) (20), (15) ( ) () , , (20), (15) (13), (15) , , (19).

, ( ) (13), (15), (20). [17, 90]:

(20) -

( 0x )

(. . 19):

0x ct x , ,

lims

s

, lims

s

,

, RRH-

128

Q Qc

E- (, )

, , , ;

(20) - ( - (20) (

21 t ), ) (16) , 0Q

, , , , , .

. 19.

, - -,

0x ct x

129

(20)

lims

s

:

d sQ s c s Q c D s

ds

.

, RRH-

lims

s

.

E- ( -

s ) (, 0s ) - , -

0s ( ).

* ,s ,

1

Q c Q c d

*s . ,

*s *s .

* ,s .

RRH- E- - (13) :

000 1 00

, ,

.

x ct xx ct xx ct x

x ct x

* (. [19, 28]). E- - 0 E-?

( ). , (. . 2.1.3),

130

22

, ,1v , ,

,

t x t xt x V t x

t x x x

,

,V t x , (

0 , 0 ), , :

v0

t x

,

() ():

2 3

2 3

Q

t x x x

. (21)

, , - (20) (21). , -, E-, c

RRH. (. [138]) 23

2

,

4 supQ Q

c

,

(21)

, 0x ct x . , (21) 0Q ,

, (16). , - - (. . [30], [34, 36]).

( (-

[110]) , 1k kc c ,

23

, ( ) -

( sin x x ),

. . . (., , [54, 145] - ). ,

, . ( , 15. - . 2011. 1 ( )).

131

1, ...,k n (, , 1k k , 1,...,k n )) [14, 90, 92; 102

110] (. . 20).

. 20.

2.

0

ln 1n

k kd t t

,

(20),(15) t xL ( x )

0

0 1

1

, ,

, ,

0,..., ,, ;

, ,

1,..., ,

, ,

k

k k k k

n

k k

k k

n

x c t t

x c t d t c t t x c t t

k nt x d t

Q x t c t t x c t t

k n

x c t t

132

k kx c t (20)

lim k ks

s

, lim k ks

s

.

(. . , . . (1960)). - (. [90, 91, 107, 110]),

0 0 0 0Q c , 0 0Q c ,

0 0limt

d t d

,

0d (20) ( 0t ):

00 0 0; , 0I t d t x x c t d dx

.

. . . .

0 0 0, ,x

w t x t s s c t d ds

,

, ,t x , -

, lim , 0x

w t x

.

, , - 1.

(. . , . . (2004)). - (. [104106]),

0

ln 1n

k kd t t

[104] - ( ):

; , 0k

k

c t t

k

k k k k

c t t

I t d t t x x c t d t dx

.

;k kI t d t ,

(20), kd t (

, , 1 2kd t t ). :

, kk kt c t t t d t , kx k x kt c t t t d t .

133

, -

. , t

, . ,

1D [103; 105106] (-) . . . , - . , , , .

. (. [108110]), 2 -

24 31 t , ..

0 31

, , ;n

k kt x t x d t

t

.

1 (. [35, 90, 95, 96]) ( ,

) 1 t , - [90] .

** (. . 2.1.3). (20) ?

kd t () . , [14] (. [7]) ( - . [103]).

. , (. ), , , 2, (21),

(15), 2 ( ,

Q , (15)). ,

24

(. [90]), ,

1 t . ( - ) ( ), 2

1 t .

134

Q , - ( , ) - . . . . (1991) [139, 140]. . . ( - . . ) [141].

, Q , - . . (2007) [142].

. , ( , ,

lnkd t t ) ( . [97107]). (., , [7]) - -. , - RRH.

(.-. . (1997)). (20), (15) - (20) ( ., , [143, 144]).

- 1, 2 . -

( , ,t x ) [37, 38, 136]:

1 20, 0,x x 1 2, ,t x t x , 0t

1 2 1 20, 0, 0, 0, ,x x x x x x

1 2 1 2, , , , ,t x t x t x t x x x , 0t . , - ( ) . ( [37, 38], , ) (

135

t x ).

( , () ( ) ). , ( , , ) . , , , . , . , , ( ) , ( ), .. , . , . . , , [146].

(, ), (, ) .

: ( ) . x ( ,

)

0 0 1 1, , ,..., , ,n n n

, , - .

, - 1, 2 ( ) -

136

( , -, ). , 1937 . . . , . . -, . . [137] - . [136, 138] .

[136], 2 . [133], - , 1, 2, - ( (. 2), ( ), ).

2.3.2. , - LWR

, - ( ) - ( ). ( -!) ( ,

). h t /

l t t (. . 21). ( - ),

02h t l t S ,

0S ( ) .

- , , , (-, , ). , ( ) -

( , . 7, , 0 0Q ) - .

, -

0 0c Q ,

137

0 . -

RRH:

0

0 0

0

1

2

Q t Ql t c Q h t

t

,

0t h t t . -

,

0 0Q S

l tl t

0 02l t Q S t

0

0

2 1Sh t

Q t .

.

. 21.

- N-- (. . , .-. , . , . , . . . . [2, 7, 11, 35, 93, 96]).

(. [35, 42, 96]). , ,

0Q .

138

(. . 2.1.3), . , -

, Q .

, (20)

0 1, x xt x L L ( )25 x :

2

2

0

,1, ,

2

t xdt x dx D t x dx

dt x

2

min

,0

t xD dx

x

,

max

2

min0,min 0D D

.

, [19, 107], , - . ,

2

0

1, ,

2V t t x dx

[135] (20) -

,t x , 0 1, x xt x L L . -,

, 0dV t

dt

2

0,xL

t x

0 0dV

dt ;

, 0V t 2

0,xL

t x

0 0V .

, ( -

25

, (. . 3.1.3, 3.3.1). , :

lim 0,x

x

0 lim ,x

t t x

;

,t x , (20), .

, , - (. ).

139

2n , 1k , 2p q r (. )) -

0K ,

2,,

dV tKV t

dt

2

0

1, 2 ,

Lt V t

t

.

(, , [107, 110]) -

1L ( -

)26 L ( ), , LWR:

0

1,

Lt

t

.

. :27

26

, (. . 21) . 27

. . 30- -

XX . K , , , ( ). ( , - ) 40-

. ( - ) 1950- . . ( ) . ( -

). , 1960- , . . . . ( ) - [39, 47], (

) ( - ). : , -

, , , (-) . ,

: , . (., , [45]) - .

( ) ( ), ( -

( ) - ). -

140

,0 : ,pq r

k nn

p r LL LK z x W z K z z

0 k n , 0 , ,p q r , , 0 , ,n

p rW -

pz x L , 1n - - n-

n rz x L . K n , k , p , q , r . :

1 1

1 1

n k r q

n r p

, 1 .

. . [147, 148] ,

q p 0k ,

, , , ,K n k p q r n k p k r n q , 1r . , () K , , , [45, 149].

. 2.1.3 LWR . , - . , (-) ( ),

, , , , - ( ): - ( )

, .. , , - ,

. . ( -, , ),

, , . -, () - , . -

(, , - ) . , , .

, - ( ) - - . -

.

141

( )

( ):

, , 02

Q QdV t Q t x dx

dt

.

, , - [19].

, - , LWR ( ) , .

(. . , . . (1969, 1989, 1994)).

,

, 0f t x

, : 0,t x t x .

(20) , ,t x f t x , - (

1D ( ., , [20, 35])):

2

0 , , , ,x tk

t x t x f t x dt dx f t x d

, , ,x xxk

f t x Q d f t x t x dtdx

,

k . 0 , -

, , 0t xk k

f t x d f t x Q d dtdx

,

k . , , , . - :

k

d

,

142

k

q Q d

.

- ( ) [29, 35]:

0

q

t x

,

( ). - , -, . XX (., , [29, 54]). (. . . . . . 2.1.1), (13), , , , (13)

(13), , , 0 , , . [150], (13) 28, - . 1960- . . - ( , - (. . 2.1.3)) (13) [27].29

28

, ( ) (13) -

, , -. 29

. ,

-, . , , ,

.

k

k

,

. , -

. , , - (13) , RRH- E- (

, , [19]). 1980-, -

, . . [151]: - ? . . [152]. , -

, Q () , - (

). Q -

() , - [152].

143

1970- ( -) . [11, 29, 35].

2.3.3. ( )

1955 . [5] (. [7]) . . - ( LWR) -:

0k , (

: ) ,

T T k .

, -

i m ( iq ), m ,

. (. 22). -

30

maxmax max

i i

i i

Q Q qc

.

, ,

maxmax

i

i i

i

qT qT

.

(. 23, 24). , ,

mq .

- (. . 2.3.1). , ,

m i i q q T qT i

m i

qk

q q

.

30

, , -

, max , ..

.

144

. 22.

. 23

145

. 24.

, - . , , , , , ( - ), : , -

i qT , ,

, :

m i q q T . , - -

mq , -

iq .

146

2.4.

31

2.12.3 - , ( ) - . - - , , 10 [10]. - - [153]. - , - () . , .