............................................................................ 9
1.
........................................................................... 12
1.1. ....................................................................... 14
1.1.1. .............................................. 14
1.1.2. ................... 20
1.1.3. , ........ 25
1.1.4. . (flow over time) ... 26
1.2. ! "#.......................... 27
1.2.1. $! " ! "#................................................................. 28
1.2.2. % &
# % ......................... 29
1.2.3. '# % & .............................. 32
1.2.4. ............................ 34
1.2.5. * % & .................................................... 36
1.3. + " ............................................ 37
1.3.1. Chip-firing game ........................................................................... 37
1.3.2. / !.................................................................... 40
1.3.3. * chip-firing game ............................. 42
1.3.4. & «! »..................................................... 45
1.3.5. '# % «! » chip-firing game .... 46
1......................................................................... 46
2. –
........................ 48
2.1. $ ............................................................... 48
2.2. ......................................... 50
2.3.
# % " ....................... 64
2.4.
# % ! .................................................... 66
2......................................................................... 70
3. –
. ...................................... 73
3.1. " ............................................................... 74
3.2. " .... 76
5
3.3. :" .................... 78
3.4. " ; ................................................................. 87
3.5.
<## % .......................................... 88
3.6. =
# % ............................................... 90
3.6.1. $ % ............................... 90
3.6.2. # % ...................................................... 97
3.6.3. > .......................................... 97
3....................................................................... 102
4. ... 104
4.1. ............................................................................. 104
4.1.1. ....................................................... 105
4.1.2. ! > ................................................... 105
4.2. , % ....................................................................................... 116
4.2.1. & % ! > ! > - ......................................... 118
4.2.2. & % ! > ! > - ........................................ 119
4.2.3. > - .......................................... 121
4.3. / Q1* " ......................................... 123
4.4. ....... 124
4.5. % R % R' ........................................ 128
4.6. $% ............. 131
4.7. ....... 132
4....................................................................... 134
5. .......... 136
5.1. " " ............................................................. 137
5.2. < ........................................................ 138
5.3. ......................................................................... 140
5.4. A% ! > .................... 143
5.5. ! > ....... 144
5.5.1. Z+(t) ........ 144
5.5.2. < . $! ........ 148
5.5.3. ! > ~m
C – Z–(0) ................................................................................. 150
6
~m
5.5.4. C – % Z+(t).................................................................................... 156
~m
5.5.5. C
– ! .................. 160
5....................................................................... 170
6. .................................................. 172
6.1. D % ........................................ 175
6.2. A% % ................................................................... 191
6.2.1. A% % ............................................................ 191
6.2.2. % d-% ............... 196
6.2.3. " ! " .. 200
6.3. " #% % ! > .............................. 205
6.3.1. " ............................................................... 205
6.3.2. % ...... 208
6.3.3. ! > ..................................................................................... 209
6....................................................................... 212
7. .......................... 215
7.1. " .............................. 215
7.2. " ............................................................ 219
7.3. " !" . " ..... 223
7.4. " ....................... 226
7.4.1. & % R'∞ ........................................................ 226
7.4.2. / " " ...................... 230
7.4.3. * .............. 233
7....................................................................... 237
8.
!
........................................................................ 238
8.1. " , ,
" δW................................................ 240
8.2. δW !
................................................................................. 252
7
8.3. * % &, % δW .......................................................... 255
8.4. fsum(t) ≥ T .............................. 257
8.5. * , ........................ 259
8.6. $% δW ................................................................. 261
8....................................................................... 265
... 266
....................................... 269
................................................................... 273
8
...................................... 282
................. 283
&, " , , , , , " , " > :
"
, > ; , , <
, , " " , , ! , , , ! ! – <
>, " , ! "
" , > .
. , «, !»
& " !
! 60 , A -A [59]. D , ; >, , – "
, % ! . / > : # %
,
% "#, % " ,
" # > (chip-firing game)
.
/ " " – –
. : – < "#, !
"
! . / > ; > " . #% . W "
#% . : > t " 9
Q(t). / t > "
> ! : > vi >, ! !, > «, », ..
! ! ; > , " % ! . * > ! , " % .
/ ! "" " : « » ( > ) «! > » .
" . / " 1 ! "# % % &, ". / " 2
, ! " : (, "# – , ! ). " # % : " , .. "# , ! . / " . $!
: (
) " ( t → ∞)
" , > " , ! > .
$ > < . "
.
*> ! " , , , < . G : , , > .
! " " " !
" % " . : " % " .
10
G , " , 2008–2016 ""., ! " :AAG ! ! [25–37, 41–43, 97, 98].
/ ! " :
:.. =" .H. I! ! ;
*.=. " > ! % ;
*.=. J, .=. '!, .'. %, =.=. " , " > ;
.*. / .=. * ! > ! " <.
11
1. ,
:
• .
• : – , , , ! , ! .
• " , , # $ .
• % &
' .
• " chip-firing game .
• " &# « &
». chip-firing game.
J > ! > " "#. , , , ! , "# % . – < "
# , , ,
" , " , ! . > , # % : ! % ! " G , % , ! # , ad hoc (ad hoc – % ! , , " « »)
. &
"" "# % , " " " # .
/ , # % , ! "# "
# , 12
! > <
. / " % , > "#, ! > ( , ). " ! , " , "# . I , ! > .
$ , ! « » (, .) " . ! , " "# . ( ., , [72]), "# ! > ; « < % » [120], > « » !, " " "#; "
> , % ! [39, 40].
G , "#
" ! ! > :
1) # % ;
2) ! "#;
3) % " .
* ! < . * , , " , ! > . G " ! ! ! " "# . $ , > , >
. " , ! >
( ., , " [78] ! ! "# ), ! , ! . ", ! ,
" . G " ! « " # >» (chipfirimg-game) . " # % . K , – ! , – % "
.
13
$ , "# " ! " " !. ; « " » «!» , , , #" , " < " .
1.1. !" #$
1.1.1. %% &%! '!" #$
< ,
" [6, 59, 60], "# c
[9, 10, 16–24, 44, 50, 51, 57] ( "# !, ! ,
<" ). ! , % . $ <" ! >
! " ,
, <" , % " ! , ! >, ,
. ! % , > , – < , ( , ), " [46, 48, 54, 56].
= " , > , >
. " ( " A – A " # % ), [92],
, " " [56, 65].
"#"$" % &"'()&"*+,%& -%/%':
! A A [59]. $! " ! [6, 65]. / . G(V, E) – "#; V – > , ⏐V⏐= n; E ⊆ V ×V – !,
e = (u, v); ⏐⏐= m.
H ! #% f(e) = f(u, v), u, v ∈ V.
f(e) > v < #% !.
14
div f(v) =
¦ f (u, v) − ¦ f (v, u ) .
( u ,v )
( v ,u )
; ! > :
¦ div
, -
f (v) = 0.
v∈V
/"/)$:('): -%/%') – s-t--%/%' (1-1--%/%')
A –A (s-t-) # ! : . * " .
#% f(e) . / .
1. / > : s ( ) t (). $ > ; div f(v) = 0.
2. * #% f() " :
0 ≤ f() ≤ ();
() ! ! .
G . 1 , div f(t) = −div f(s).
/ M(f) = div f(t) = −div f(s) .
K div f(t) = div f(s) = 0, % % .
> vj div f(vj) = 0 , n
n
i =1
i =1
¦ f ij = ¦ f ji
& – #% :
M(λf + μg) = λM(f )+ μM(g).
-
D – ϕC % . .
ϕL 15
. f < :
f=
¦ϕ
L
+
¦ϕ
C
.
= (X, Y) G ! > , s ∈ X, t ∈ Y.
U+(A) – !, > X > Y; U−(A) – !, X.
> Y > c(A) = ¦ c(e) – ! .
e∈U +
div f(A) =
¦ div
f (v ) –
"% f .
v∈X
N" , "%
M(f): div f(A) = M(f).
;" M(f) = div f(A) =
-
¦ f (e) − ¦−f (e) ≤ ¦+c(e) = c(A),
e∈U +
e∈U
e∈U
.. ! !" .
s t A – A .
– .
1. K
∈ ! () – % , % (
).
2. & " !
" : M(f) = min ().
A
= " A –A ! % .
1.1. > . 1.1.
!
" , ,
< .
/ ! ( !
!) " A –A , 16
" .
" 4
a
2
"
c
1
s
3
3
b
: . 1.1. & 2
t
,%>%-%*?(,@: ) &,%>%-A%#B'/%C@: -%/%')
,%>%-%*?(,@D -%/%'. k s1, …, sk l t1, …, tl. ; " . " " (k-l- ) % 1-1- : # > – s t.
! # ! > :
c(s, si) =
¦ c( si , v j ) ,
j
c(ti, t) =
¦ c ( v j , ti ) .
j
;" "% > - . & 1-1- k-l- .
,%>%-A%#B'/%C@D -%/%'. K = {1, ..., k}, k s1, …, sk, k t1, …, tk ( " ); k #% fi
!, fi i-" si ti.
A " : ¦ M i [95].
i
17
s
s1
t1
s2
t2
.
.
.
sk
t
.
.
.
tl
: . 1.2. : % k-l- 1-1- %)(' (E"*",()A%C",,%>% -%/%'"
! " [14]. "# G = (V, E),
|V| = n. ! " " .
> i :
Ii
Oi. Ii – « » > i: Ii =
= {k: (k, i) ∈ E}; Oi – « » > i:
Oi = {k: (i, k) ∈ E}, i ∈ V.
/ "# > T – < > ! ": T = {i ∈ V: Oi = ∅}.
* > % qi, . /> qi > 0 " .
xij,
(i, j) ∈ E. > > > ( qi > 0) " % .
* % > - .
, > > , " .
/ > " % " : , " ".
! , > – " 18
!" % > " :
qi + ¦ x pi = ¦ xij , i ∈ V \ T .
p∈I i
j∈Oi
% " ! ( ) . G !
% , %, ! " .
"C,%C:(,%: A"(-A:#:*:,): A:(BA(%C. ; [15]
" , #% , « ». & >
"# G = (V, E), |V| = n. /> "# % qi, . * > % ! . !
", " . / " (i, j) (j, i) " ! .
/ > . –
< " . . / " (i, j) >" k ! xijk . , xij0 ¦ xij0 = qi , i ∈V ,
j∈Vi
(
%
j ∈ Vi = {l ∈ V : (i, l ) ∈ E}.
> i, j #% . /> i >" ! cij xijk , x kji
:
)
xijk +1
, !
cij, j ∈ Vi.
; ! , > (
)
#%
max cij xijk +1 , x kji → min ;
j∈Vi
{ xijk +1 }
¦ xijk +1 = qi ;
j∈Vi
xijk +1 = 0, (i, j ) ∉ E .
19
< xijk .
; (#% cij % " ) # .
1.1.2. ! %() % %$*+ $% - #%/0
"# ! P.&. K
K.$. J" [9, 10].
/ "# " ", "
"# ,
,
. /
" "#. "# ( ) "# . ! > " NP- [16].
!
[16–24, 44, 50, 51, 57].
$ "# < ! G(X, U, f), " X S ∅– > , U – "; #% f: U T X × X – ! , ! % .
1.1.2.1. &:F",,"G #%(/)H)&%(/+ ," %A>A"I"J
$ "# # $% [10]. / "# G(X, U, f) ! ! :
U = UR ∪ UZ , UR S ∅, UZ S ∅,
" UR – ", UZ – ".
/ "# > #& . n > , " , > n, ". ; " " " " .
* . 1.3 , > 1 > 7,
>
, , 20
: , > 1-2-4-5-7 1-2-4-6-7. 1-3-6-7, " > , , " .
2
1
5
4
3
: . 1.3. '# 7
6
"
/ "# > >
(
): «
> x > y
> y > z, > x > z, < ». $ "# > ,
, " , < : ! > , « » . * . 1.3 ,
> 1-3-6 6-7, >; 1-3-6-7,
! , – >.
> "# ( " "# ), >" > " " "#, " .
"KC:A/'" >A"I" (% (&:F",,%D #%(/)H)&%(/+?. G(X, U = UR ∪ UZ, f) – "# > . & > "# G ! X ∪ X' , " X' – ! X: > x ∈ X > x' ∈ X'. * " " "#
u ∈ UR (f(u) = (x, y)) "# " , > y: > x, " – x'. K u ∈ UZ , f(u) = (x, y), "# ! ", > x y'.
K
"# >, " "# ! > ; " " " > " " "#.
"# G' .
21
– ' # y # & x G(X, U =
= UR ∪ UZ, f) $ , G' )% # & x # {y, y'}.
– " # # & x # y # $% G(X, U = UR ∪ UZ, f) " # # & x # {y, y'} G'. * " # #& $ +&$
" # # & x # {y, y'} G' .
> "# G' ! " . , % " ! " "# .
1.1.2.2. ">,)/"G #%(/)H)&%(/+ ," %A>A"I"J
KU – " , >, " "# " , " ! .
'# ! " " " " [19, 22, 44, 57]. .
"# G(X, U, f), U = UH ∪ UM , UH ∩ UM = ∅.
& UM " ", UH –
" ". μ – , n
". [i; j]N (1 V i < j V n) μ /μ(i, j), " " <" .
" , . X /μ(i, j) ! " [i; j]N μ.
μ " - k (Y 1)
n ! " "# G, . K i- >" " μ " /μ(i) Y k
", % > i- " , ! " ", (i + 1)- " <" ! " .
'# G(X, U, f), " U = UH ∪ UM , " - k, +&%) 0 k.
22
"# " "# . / "# > > " "# (k + 1)
> "# x0, x1, ..., xk. < " ", ! > , " "; " ", > ,
" " ; ! " ".
[22].
– 9%+ μ' # & x0 # 0
{y , ..., yk} $ G' $& $ μ # & x # y : G. ; + : 0 , "+& # y +& - +&%) # & x G, +: ", "+& G' +& # & x0 :0 +& # Vy = {y0, ..., yk}.
– " # > G' # & x0 # Vy = {y0, ..., yk} " #
- $& $ G.
G < , >" "# ! " ! " >" " U.
/ [22] " "# " : "# - " ,
"# -! " , "# ! " .
1.1.2.3. "A+:A,"G #%(/)H)&%(/+ ," %A>A"I"J
J , , " , , .. .
'# ! G(X, U, f) ": U = U0 ∪ UZ ∪ UB, " U0 –
", UZ – "
UB – " !" (! ") [50, 51].
/ "# ! " , ! <" ! h. D" " ( <" ).
23
* . 1.4 "# ! . P, h > 0 ! " (3, 4) .
; 1 T 2 T 3 T 4 . % 2 T 3 T 6 T 5 T 2 h , ! " (6, 5), <" ! " (3, 4).
5
1
2
+
6
3
b
4
: . 1.4. '# " !" (3, 4)
" (6, 5)
;, h = 1, ! ! %: 1 T 2 T 3 T 6 T 5 T 2 T 3 T 4.
* "# ! > " .
"# ! " "# G'.
1.1.2.4. AB>): C)#@ ,:(/",#"A/,%D #%(/)H)&%(/) ," >A"I"J
/ "# $ $% [21] "
! m 0 m – 1, " " , , , " , i, i. / "# , " k ! >, " k – 1.
'# $% m ". "# , " ! !
, .., ", l, " 1, 2, … l – 1.
, " " &
& [9, 10], "# + 0 $% [51], " $% [44]. / "# 24
" "
" [18, 50].
"# !! " ! [19, 21]. ! , , ", > > , "# . ; ! % , "#, ! % &.
/ ! P.&. K
" *.*. / " , " " - > " % " [23, 24].
1.1.3. $& , %$(4 %( ! % &%! #
'!"# #$(#
"#"$" % ,"K,"$:,)GJ. D , ! , > " . l t1, …, tl k ! s1, …, sk. ! !, " .
> % 1-1- : ! (si, tj)
! (si, tj) = 1. "#, . 1.5, , " , # .
t1
s1
t2
s
s2
.
.
.
sk
t3
t
.
.
.
tl
: . 1.5. 25
"(-A:#:*:,): /%C"A%C. D !, " , " .
! !
. + ! " , , ! > % .
E&:, #",,@&). / , . ! ! . ; , – , % .
"#:H,%(/+ (:/). % , , , ! . ;!
" , ! > ! .
/(:$:,): *),)D (,"EH:,)G. / , ! ". + ,
! ! , " ! ! .
< ", ! ! % > " . " ! !,
!
", ! > ! .
1.1.4.
# &%! 6$& . ! *#
(flow over time)
" ! "# G(V, E) !
c(e), τ() % <## % a(e) ( % ).
/ < > .
? $& : s t .
@& # : s t !^ .
26
+& : s-t- , ! > θ θ ∈ [0, T).
: s-t- , θ θ ∈ [0, T).
( NP- ):
− :
− !> , .
@& # & ( NP- ). &" f " ! #% fe,i: [0, T) → ъ+, " fe,i(θ) ( % ) i, " ! θ. , > > ! θ, θ + τ(). / # . " !, ! fe,i(θ) = 0 θ ∈ [ – τ(), T).
/ < : > "
( "% ), " < > ! , .
/ > > ! >, .
1.2. 9&+" :9-$ (
*%%( *;)
J > " ! !. G % !, ! , ! ( ,
/
. .). ; - «> » , . & , , % : > "#. ; # "# ! « " » [78].
27
/ < .
;" + "# – %, " > , , ! > . 0 ( ) % " ( "
# % ") .
/ > ! 0 : 0, ! " . K" 1.2.3.
1.2.1. :% '* # ( #$+ %9&+
:9-$ ( *%%( ( *;)
: "# " !, > ! , " > %
JG
[101]. " ! " , % # % , ! , " %.
@ % "#. / < > (! ") "# %, % i
! " wi. % " "
% ! , , ! " ! . < ! ! " "#, " «» !
, , [64].
& 0: + $&: , , "# [84]. /> <" "# , " > – , ! # ", ! > " . / , > , > . " , ! ! . & 28
% " %
! %.
0 $&: 0:
[12, 13, 49] "# ! % . '# ;
" # % #, " # . /
> % , : ,
% , . , % > , PageRank [79], , , # % % " ! "#. G % , ( (! >!),
!, .) % . &" # % % < ! >! [119].
$!! , " , ! # % , ,
! " [45, 121].
%
[113] "" ( ' !! ) , , % & [4, 5].
1.2.2. &" <' *!
) !%% ; !< (
1.2.2.1. (,%C,@: %-A:#:*:,)G
S1, S2, …, Sn – ,
#% t. / t . / " " %
, " pij – Si Sj. ; $% ? [38, 53, 108, 109]. K pij t,
% & . N! % σ0 = σ 1(0) , σ 2(0) ,..., σ n(0)
U % &.
(
)
29
% & σ0 % : σk = σ0Pk, " σk – k.
: % % ,
2, 3,…, " – %. K : ∞ = lim ( P) k , pij∞ $&k →∞
:& 00 .
& % P , % , P
O
,
Π ; P Π = 11
P21 P22
-
" P11 P22 — % > n [11]. / % P .
– + [11, 48, 99] ! % P ≥ 0 " ! λ, " " . & ! λ.
D ! .
K % P ≥ 0
h ! λ0 = |λ1| = … = |λh-1| = λ, % h = 1,
h > 1 [11].
/ # % % &, " ,
[38]. % & ! < . < , " ". <
> " , .
; < , " , D " . / " ! & 0 0 . K
<" > " , < , , ", " . ; )%) .
1.2.2.2. :-) E:K ,:C%KCA"/,@J &,%H:(/C
+ & D " , " <" " . D , 30
" (
! >").
K
% <" % & k, Pk, " k, " , % 0 1. K <" % " , " .
1.2.2.3. :>B*GA,@: M:-) "A'%C"
: , " % &, !! [38]:
1.1. K – " %, :
1) lim P k = ∞ ;
k →∞
2)
!" σ σPk π = (π1, π2, …, πn)
k → ∞;
3) π ;
4) % ∞ ! ! , ..
∞ = 1π,
" 1 – - !%, n %;
5) π – ,
" π = π.
6) ∞ = ∞ = ∞.
-
1.2.2.4. :-) ( ,:C%KCA"/,@&) &,%H:(/C"&)
/ % <" . ,
% " . < % <" %
%, <" 1
, % " %
(< "
[38]). /
% ! – !
"
! " <" " . " , [38].
& [11] " ! 31
" < . ; % )%) 0 ? .
1.2.3. *;" '*)$ !&") <'+ *!
/ % & % = (pij)n×n, > "# G(P), " % : pij – < ! eij; pij = 0, ! eij . ; "# : [52]. K "# , % . K
"#
, % P !
. ;
! , %
& < > > "# , ".
; % " + "#.
! "# G > i0; t >" > it, < > >" t + 1 ! ! > j pit j .
& > % &. /> i0 ! # , ! ! " " σ 0 (" σ 0 – -). $! P = (pij)n×n % < % .
! :
σ t+1 = σ t P.
σ t = σ P t.
;
! , ", , > i, " > j t >", ! ij- < % P t.
: σ 0, σ 1, σ 2, … ! , .
: σ 0 , σ 1 = σ 0. / < , , σ t = σ 0 t Y 0.
b ! ! π. G 1.1 , " % & π – ! % , ! 1.
32
" "# G πi =
di
2m
(m – !).
% " "# ,
, .
: % " " > "# , > t + 1 % ".
"# G % A = (aij)n×n, " aij –
". %, < % , , > aij
t. % aij pij = n
. *¦ aik
k =1
! , % % P = (pij)n×n " ! "# G.
ρ 0 – - " ,
" W.
t + 1 " :
ρ t+1 = ρ t P.
: " ρ 0, ρ 1, ρ 2, … * ρ σ : ρ t = Wσ t.
;
! , ! ! "# < % < "#
" , <
# %.
% % & ! . ! " " , " , . [66, 67] !33
" , # " " [53, 115].
1.2.4.
%!*( #$/ $% - ( !%%9%
-"# "" ! ! % " ! "#, > " " .
$! < " [106]. $ ' ""
. $ " !^ % ( ) " " [85, 4, 5, 62, 63]. / <" " % , %
! .
x(0) = (x1(0), …, xn(0))T – , " ", x(t) = (x1(t), …, xn(t))T – >" t. ;" % " # :
x(t+1) = Px(t) = Pt+1x(0),
" P – %, " "
"; < pij " j " i. " "
, , " . " " " " , " %
∞
k
lim P = . G 1.1
k →∞
, " % " .
% > "# " '. :! (j, i) , " j " i.
!: wji = pij. :! ' .
9 L >" "# [5] ! %:
L = E – P,
(1.1)
" E – %.
? $&: :0) : J = J ij >-
( )
" "# ' 34
! :
J ij =
f ij
f
, i, j = 1, …, n,
" f – ', fij –
, > i , j [63] ( >" "# " !).
G % & :
∞ = J .
/ . –
% n × n. $! % !
R(A)
N(A). G % –
ν = ind A – > k ∈{0, 1, …}, rank Ak+1 = rank Ak. G % ! >"
! " !" # [63].
G % Z (.. %, Z2 = Z),
+& % , ! 0, R(Z) = N(Aν) N(Z) = R(Aν) [114,
63]. Z : ZAν = AνZ = 0. G % ! [5], < !
.
! % L !
~
J .
/ [80] , ! % L ~
% J : J = J . * , J 2 = J
LJ = J L = 0 .
" , L = E – P, ∞L = L∞ = 0, R(∞) = N(L) N(∞) = R(L) [1, 2].
; ∞ – ! % L, rank ∞=ν,
rank L= n – ν ; ind L = 1 [107, 114, 3].
;
! , %, , !
% % "# :
~
∞ = J = J .
35
K
lim P k , .. % – k →∞
I (!
, " : ∞)
1 k i
¦P .
k → ∞ k i =1
P ∞ = lim
/ % I {k}. I > .
;" % " % '
# :
~
lim x (k ) = J x (0) .
k →∞
!" " . / [3] , ! ! % ! " " % Ak, " k ≥ ν. ;
% ν = ind L = 1, %
∞ ! !" " " L.
~
", % J # , . 8–10 1 [63],
,
[3].
1.2.5. $*$" <' *!
/ % & . K % >": P = P(t), % & . % % P(1), P(2), …. / ! :
k
H k = ∏ P(i ) , k = 1, 2, …
i =1
K ! % P(t) = 1σT, " 1 = (1, …, 1)T, σ – , .. %
P(t) , U Hk ! 36
! % . <
! % R' R'P(t) = P(t).
K
R' P(t) , % [4]. / < % % " % . ' < , " .
+ %
H k = (hij(k ) ) + D " ,
i, l, j (hij( k ) − hlj( k ) ) → 0 [93]. < % H k
.
H k $ D " [94],
+ %
" " π lim H k =1πT.
k →∞
$! % &, " [4].
1.3. >& %" '*" #$
1.3.1. Chip-firing game
G" # > (Chip-firing game) !
:. = .[68], *. J "" [74, 75], =. JU
. [76, 77],
N. N [101], . [117]
" [81, 82, 100,
111]. G" . G – "#,
!. / > "# # > ( . 1.6). «X » ! > , # > > !, " < > , .. # > ! .
G" , ! > , ! .
" , ! % ! : ! ! !, ! >" % .
37
: . 1.6. : # > chip-firing game
/ [76] " chip-firing game. / [117]
! «». / %
N # >. * >" ¬N/2¼ . K N , # > %. * >" . . G % , %. J , < " % ' –/
[55, 73] ( , , !
# >, " ! ). [68]
! ! , >" ! ,
, % k(k q 1)(2k q 1)/6 >", " k = ¬(N + 2)/2¼.
J ", ! # % N , # > , N , N + 1 " N – % < .
/ < " "# [76], " % ! > , # >. (G" , ! > , !
38
.) G, %, ! chip-firing "#. < ! , " # > " "# (% )
!, ! . J ", # " " ! ! " " % >", ! " % % . / "# " > .
/ > ! # %
". ;, , [111] " , " > , !
, . / [74, 75] [100] # % , ". / < " > , ! ! # > . $ " " , " > ! > .
" ; [8].
/ [74] , " ! < , ( > )
" , < . G # % " "
, # "% [74, 75]. " , > "# " , " > ! ! . / " " > .
/ [81] ! < – chip-firing
game "#, " > , > # > ! ". D ! % , # >
, " > % " .
J " , !, " , % < ! > # "% % # "% # > 39
!" . G (
! %, ! ", # , .) ! " "#.
1.3.2. *(%"+ :!
& , < chip-firing game, ! . " > ,
" . [117],
!! [68], 1970- "". =. D" ! , « ! » [90, 91]. D" ! ! . + ! > "#. / ! > . ; "# " % &. / # > "# (D" >> ). /> ! , , ! , ! "
" . <",
! >"
%
.
< " , <" %
&.
$ > % " ! " . '# < !! >> . y> > , ! . * , . 1.7 > 1 ,
! > >> . / ! ,
> 1
>> 6q + r (0 V r <6), q, 3q 2q > 2, 3 4 , r >> > .
/> ", >> ( . 1.7 > 1 "). " ", " > ( . 1.7).
40
1
1/6
1/3
1/2
2
3
4
: . 1.7. " : ! " ! > .
1. 0 1 , ! " : 101
0101. '# < " . 1.8 ([91]).
1
1
s
3
7
4
2
0
: . 1.8. '# " 10
101
5
6
8
01
010
0101
. / " ! 0,5
" "# " ": > >>. G" > s. / ! >>, > , > – , " , " > ! >> . <" > s "
"#. K 41
" , ! >> s ". ; , " ( ).
/ > s ! >>, 7 – 8. ; ! , > s " 7 8 ! 5/8 3/8.
/ " >> .
/ " ! 80 . $ >"
> 1 # m1 =
" : :
80
=
= 10.
" )&: ##
8
:
% , " ! , > ( ) – [91].
J " ! " . ", [77] "
" .
1.3.3. !*" *69/" $( chip-firing game
" . J % ! ? K
, " ! ? K ! , % ? # > !
! "? % " % ?
/ "# < . " , % ! ", , % ". n – > , m – ! "#.
K % , , O(n4) >". > " n; # > ! " m. $ "# ! , " .
42
" "# G = (V, E), n = |V|, m = |E| ( ! , ! ). - di+
di− – > i , aij –
!, > i > j. ;" d i+ = ¦ aij , di− = ¦ a ji , aii – > i.
j
j
" v ∈ ъV ! :
v=
¦ vk
k ∈V
, v =
¦ d k+ vk
.
k∈V
$"# D &, di+ = di− [77]. / chip-firing
games "# ! ! , . ;
! , "# <
< . : , "#, < "#. "# " >.
chip-firing games « # >». s – % , t – % ", xi –
< > i. ;" ¦ a ji x j − di+ xi = ti − si .
j
* <" , xL = t – s,
(1.2)
" L ∈ ъV×V – 0 "# G(V), ! :
i ≠ j
­ aij ,
lij = ® +
¯− d i + aii , i = j
43
D , [101]. # (1.2) # (1.1), %, ! < % L1 L2 , , L1 = – L2.
/ % L ! % (1.2).
v – ! L, ! λ0 = 0: vL = 0. G –A! , v > 0. K G – < "# ( , "#), v = 1.
& % N % " ! > :
P = D-1 L + E,
" D – " % > " " . $ , v # :
πi =
d i+ vi
.
v
(1.3)
chip-firing games . G (1.2) , " s = t
xL = 0.
(1.4)
, x = lv
" % " "
l. / [101] , " x = v.
/ < ! , !
" "# G |v|, # >, " , ||v||.
G # (1.3) , " % & q % " q
" , " x, xi = +i , di
% N , .. (1.4). = <, , , # > "
% &.
44
1.3.4. $/ «: !9& '%!»
# J " , .
" ( ) "
, #
( ) >!, , , . <
" ( ). ; ! % , " . b ,
. J , . ;
. b # ! « », [69–71].
& ! >, , . & : , « » "
> " . ( )
(x, y) % % Z(x, y). / .. « !> ». K (x, y)
! > " Z = 3, ! , (x, y) > 4. #% . / > ! ! , ! . % , " " . ;" !> , , , !> . K " , >. !
% > . ; #% ! >! –
! > . G .
= "! !
!, , [83, 86–89, 96, 102–105, 110, 112, 116, 118].
. !!
« ». $ , ! ! " [86]. K !> > , !> . $!> : ! !
, .
45
1.3.5. *;( *'*< ( «:+ !9& »
chip-firing game
& « » ! " >, !! ! "#. K % < chip-firing game. / [101] , "# ! . > ! , ". % > s «> ». D > , ". $ > . " " > > ", «!> », % " , , ,
" !> , . ., . G " > " . K "#, , , >"
> ! >" ", ! >
, .
/> s > i ai
!. " > s. ", " s « » ", ! " . ; ! , .
-" - -" -… ! " "# " : > s " , " " > . " , > ! , > " .
% " .
"A"&:/A@ &%#:*): " "#, ! > i > j > i > j; > >
; > s
> .
##* + ! 1
*> ! " ! "# . $ % . % – : % &, ! <
46
". / % – % : " %, ! >
. D – .
1. 0, , .. 0 .
2. ( " ) , #% ( )
" . D ! # % , , ..
– ( – ! "#, < ).
3. G , .. !^ , " % , - (
> ), " ! ( % ) – , . K
, , > < , ! .
4. / % ! : 0$0 ( ! , )
( ).
5. 0
"
& : . ;, , chip-firing .
$ ! ,
! "
> " – .
47
2. @ A BD –
B D B ,
:
• % & . ( , .
• ) & – & .
• % & : (1) & (2),
, & & 1 × 2.
"# – , – , " . 1, .
2.1. %" '*$ (
$% "# G = (V, E) > vi ∈ V ! eij = (vi, vj) ∈ E; |V| = n. :! "# % rij, & +0 ; &:
+ R = (rij)n×n. /> "# % qi(t), t . /> vi " " .
;0 Q(t) t Q(t) =
= (q1(t), …, qn(t)), > t.
riin =
!
n
¦ r ji
j =1
riout =
n
¦ rij
– j =1
> . !
n
, n
. rsum = ¦ ¦ rij –
, ! ! .
> ! W.
/ : 0: #% i =1 j =1
48
: ∀t
n
¦ qi (t )
i =1
= W.
/ t > vi, i = 1, …, n, ! , ! qi(t) ( > ).
A"C)*" A"(-A:#:*:,)G A:(BA(". ' t # vi + eik, 0%) # vk:
out
rik , qi (t) > ri ( 1);
rik
q (t ) , qi (t) ≤ riout ( 2).
out i
ri
:, > vi ! eik t,
> vk t + 1.
< . 1 ! " , " > ! >, ; «U, », .. ! , ! <" !, " n
out
ri = ¦ rij . 2 > -
j =1
, % ! !. K > ! > : qi (t) = ri , 1 2
.
: : > ,
, " ! 1
2, " ; > , , ! .
Q(t) " &, Q(t) = Q(t + 1) = …
Q* = (q1*, …, qn*) " & Q(0), !" ε > 0 tε
*
, t > tε «qi – qi(t) « < ε, i = 1, …, n.
$&, ! Q(0) , ! Q(0).
& > vi, qi(t) V riout , Z–(t). /> Z–(t) #% 2. Z+(t)
out
49
> vj,
out
qj(t) > rj . D > -
#% 1. " Q*
! < Z–* Z+*.
%/%' A:(BA(". , > vi
! eij t, > vj t + 1, ! , t t + 1 ! eij.
D fij(t). $! F(t) % F(t) = (fij(t))n×n.
n
/
: fsum(t) =
n
¦ ¦ f ij (t ) .
i =1 j =1
! ! , ! > " ! , , fij(t) ≤ rij fsum(t) ≤ rsum !" t.
$!
f i out (t ) < i- n
f i out (t ) = ¦ f ij (t ) (
j =1
% F(t)), , -
> vi t. $
/ f jin (t + 1) =
out
out
, fi (t ) ≤ ri .
n
¦ fij (t ) (
< j-" !% -
i =1
% F(t)) , > vj ;
f jin (t + 1) ≤ r jin . ", f jin (0) = 0.
K Q*, < ! . & % " ! F* = (݂) כn×n.
, ! > < ", – < . / ! Q(t) F(t), . & – , – " < > .
2.2. " $*$" *%9*%" %
: , ! i, j ! eij rij > 0. % R . : , ! ! . $!
r. / < 50
"# ! ; " ! . / < % R r, < <
% R ! .
> ! ( ) n, in
out
!" i = 1, …, n ri = ri = rn.
;" # 1 2 :
t # vi : n &:0) : +
:
− r , qi(t) > rn ( 1);
−
q i (t )
, qi(t) ` rn ( 2).
n
, Z–(t) Z+(t): Z (t) – < > vi, qi(t) V rn; Z+(t) –
< > vj, qj(t) > rn.
! :
; 2.1. K
" t qi(t) = qj(t), t' > t qi(t') = qj(t').
D ", t ! > .
; 2.2. K
" t qi(t) ≤ rn, t' > t qi(t') ≤ rn.
D ", vi t , " n – 1 > ! rn. < > , > Z–, " , , , Z– .
; 2.3. K qi(t) ≥ rn, i = 1, …, n, Q(t) .
D ", > rn
% .
; 2.4. ! t > : f i in (t) = f jin (t), i, j = 1, …, n.
–
D ", > ! . < !" t !% % F(t) .
51
A)&:A 2.1. : > ,
r = 2, W = 60, Q(0) = (20, 17, 9, 8, 6). ;" rn = 10, Z+(0) – > ; > Z–(0).
§ 2
¨
¨ 2
F(0) = ¨ 1,8
¨
¨1,6
¨1,2
©
; 2.5. K
2·
¸
2¸
1,8 1,8 1,8 1,8 ¸ .
¸
1,6 1,6 1,6 1,6 ¸
1,2 1,2 1,2 1,2 ¸¹
2
2
2
2
2
2
t > vi1 , …, vim (m ≤ n)
Z , qi1 (t + 1) = ... = qim (t + 1) .
−
> Z− t , t + 1 . ;" 5 4.
G 5 , > Z–(0), t = 1 q1(1) = … = qn(1) < > ! . , Q(1) ! , , . < > ! , Z+(0) .
> , q1(0) ≥ … ≥ qk(0) > qk+1(0) ≥ …≥ qn(0),
(2.1)
" qk(0) > rn, qk+1(0) ≤ rn.
;" Z+(0) k > , Z–(0) − > . / ! k – , Z+(t) ( 2 , !). < ! k ! k(t), k(t) ! , ! " ! .
$ Q(t), > Z+(t), ! Q+(t); Q(t), Z− (t), ! Q−(t).
52
< :
Q+(t) = (rn + c1(t), …, rn + ck(t)),
Q−(t) = (rn − dk+1(t), …, rn − dn(t),
(2.2)
" ci > 0, dj ≥ 0.
k
/ ! ;(t) = ¦ ci (t ) ; D(t) =
i =1
−
n
¦ d j (t ) .
; j = k +1
D(t) – < , " Z rn(n – k), D(t) Z−(t), ;(t) − Z+(t).
Q(0), :
n
¦ qi (0)
i =1
= W = rn2 + ;(0) − D(0),
(2.3)
, ;(t) – D(t) = const = p
W = rn2 + p.
(2.4)
2.1
k = 2, c1(0) = 10, c2(0) = 7, ;(0) = 17,
d3(0) = 1, d4(0) = 2, d5(0) = 4, D(0) = 7, = 10.
/ :
f +out (t) =
f −out (t) =
k
n
¦ ¦ f ij (t )
– Z+(t).
i =1 j =1
n
n
¦ ¦ f ij (t ) – Z−(t).
i = k +1 j =1
out
out
fout (t) = f + (t) + f − (t) – ! .
!
f +in (t + 1),
f −in (t + 1), f in (t + 1).
G (.2.1.4) , f out (t) = f in (t + 1).
G :
53
f +out (t) = rkn ( > Z+(t) rn).
f −out (t) = rn(n – k) – D(t) ( > Z− (t) ).
fout (t) = rkn + rn(n – k) – D(t) = rn2 – D(t).
/
> 4 ! . < f +in ( t + 1) =
k out
k
k
f (t) = ( rn2 – D(t)) = rkn − D(t).
n
n
n
f −in ( t + 1) =
n − k out
n−k
f ( t) =
( rn2 – D(t)).
n
n
Z+(t + 1) =
f +in (t
+ 1) −
f +out (t).
"% Div Z+(t + 1) =
:
Div Z+(t + 1) = rkn −
k
k
D(t) – rkn = − D(t).
n
n
k
D(t).
n
D , (t, t + 1) Z+ Z– k
D(t).
n
K t + 1 Z+ , , Div Z−(t + 1) =
Q+(t + 1) = (rn + c1(t) −
Q−(t + 1) = (
54
D(t )
D(t )
, …, rn + ck(t) −
),
n
n
D(t )
D(t )
f in (t + 1)
f in (t + 1)
, ….,
) = (rn−
, …, rn−
),
n
n
n
n
D(t + 1) =
k
k
n−k
D(t) = D(t)(1 − ) = D(0)(1 − )t,
n
n
n
out
f + (t + 1) = rkn;
f −out (t + 1) = rn(n – k) –
f out (t + 1) = rn2 –
(2.5)
n−k
D(t).
n
n−k
D(t).
n
;" :
Δf out (t + 1) = f out (t + 1) − f out (t) =
k
D(t) = Div Z−(t + 1).
n
. – > n ! r. < , " n r # , " W " > t = 0, .. " Q(0).
: < .
A)&:A 2.2. : n = 7 r =
1. KU ! rsum rsum = rn2 = 49.
W = 54 Q(0) = (30, 15, 9, 0, 0, 0, 0). > Z+(0): k = 3.
#% W Q(0), ..
(! . 2.1).
/ < ! 54. *, ! , ! % !
% . *! > " " .
/ >
, , ! "# . . 2.1. / < Q(t) t, t %. $ Ox > , Oy – 55
> . ' – < " %
Z–(t) Z+(t). t = 0 ( ) Z+(0) > . #% k = 3 !, > Z+ ! , .. , 7. * > t = 1 ( ) v3 Z–(1). * t = 3 ( ) > v2 Z+*
" > . * t = 4 > v2, …, v7,
> Z–, , ! Z– rn = 7 ( , 2 < > " Z–). , . 2.1 " 1
5: > Z–, " > Z–, < . D % ! & .
ti
0
1
2
3
4
5
6
7
8
9
10
…
50
51
52
53
54
55
56
A%/%'%* #*G Q(0) = (30, 15, 9, 0, 0, 0, 0)
v1
v2
v3
v4
v5
v6
30
15
9
0
0
0
26
11
5
3
3
3
23,429 8,429 4,429 4,429 4,429 4,429
21,592 6,592 5,163 5,163 5,163 5,163
20,222
5,63
5,63
5,63
5,63
5,63
19,047 5,825 5,825 5,825 5,825 5,825
18,04 5,993 5,993 5,993 5,993 5,993
17,177 6,137 6,137 6,137 6,137 6,137
16,438
6,26
6,26
6,26
6,26
6,26
15,804 6,366 6,366 6,366 6,366 6,366
15,26 6,457 6,457 6,457 6,457 6,457
…
…
…
…
…
…
12,007 6,999 6,999 6,999 6,999 6,999
12,006 6,999 6,999 6,999 6,999 6,999
12,005 6,999 6,999 6,999 6,999 6,999
12,004 6,999 6,999 6,999 6,999 6,999
12,004 6,999 6,999 6,999 6,999 6,999
12,003 6,999 6,999 6,999 6,999 6,999
+ 2.1
v7
0
3
4,429
5,163
5,63
5,825
5,993
6,137
6,26
6,366
6,457
…
6,999
6,999
6,999
6,999
6,999
6,999
56
…
12,000
…
7
…
7
…
7
…
7
…
7
…
7
…
: . 2.1. > r = 1 Q(0) = (30, 15, 9, 0, 0, 0, 0).
: .
" "# " . 2.2, ! > . Ox t; . 2.1, Oy > , " – " % Z– Z+. Q(t) ! 7 % t, > %. > Z– < " " % " % Z– Z+.
: W = 45 Q(0) = (15, 12, 9, 5, 3, 1, 0). W < rsum. ! . 2.2.
/ "# " <" . 2.3.
/ , > Z–(t), " t = 7 .
D , 57
W rsum = rn2. $ .
58
59
: . 2.2. " > r = 1 Q(0) = (30, 15, 9, 0, 0, 0, 0). % > 60
: . 2.3. > r = 1 Q(0) = (15, 12, 9, 5, 3, 1, 0). : % , . 2.2,
! > Ti
0
1
2
3
4
5
6
7
8
9
…
+ 2.2
A%/%'%* #*G Q(0) = (15, 12, 9, 5, 3, 1, 0)
v1
v2
v3
v4
v5
v6
v7
15
12
9
5
3
1
0
12,286 9,286 6,286 4,286 4,286 4,286 4,286
10,633 7,633 5,347 5,347 5,347 5,347 5,347
9,452 6,452 5,819 5,819 5,819 5,819 5,819
8,53 6,078 6,078 6,078 6,078 6,078 6,078
7,74
6,21
6,21
6,21
6,21
6,21
6,21
7,063 6,323 6,323 6,323 6,323 6,323 6,323
6,483
6,42
6,42
6,42
6,42
6,42
6,42
6,429 6,429 6,429 6,429 6,429 6,429 6,429
6,429 6,429 6,429 6,429 6,429 6,429 6,429
…
…
…
…
…
…
…
:%A:&" 2.1. 0 0 " # n > 2:
1) & W : T = rn2, %+ "$ 0 $& 0 W·
§W
000 ¨ ,..., ¸ ;
n
n¹
©
2
2) W > rn , %+ "$ 0 , :0 +& : # : & &, & : .
: 4 .
1. Z+(0) .
2. Z+(0) , W < rn2
3. Z+(0) , W = rn2.
4. W > rn2.
, " Z–(0) , : <
.
;" 1. Z+(0) . ;" 5 .
;" 2. Z+(0) , W < rn2.
G (2.5) , k lim D(t) = 0. * t →∞
W < T (2.3) (2.4) , p < 0, D(t) = 0 ". (2.4) > D(t) > ;(t), t', " C(t') , 61
! > Z+(t') Z−(t') 5 t' + 1 > Z−.
> % > k. W
< rn2 ;(t) < D(t), t'', " ;(t'') = 0, > Z−, t'' + 1 ! ".
;" 3. Z+(0) , W = rn2. / < p = 0, D(t)
;(t) , (2.2), lim Q(t)
t →∞
= (rn, …, rn). * < ! D(t) ≠ 0, , C(t) ≠ 0; < > ! Z+ ! " .
;" 4. W > rn2. / 2 > , > Z-,
". ; lim D(t) = 0, lim Q−(t) = (rn, …, rn),
t →∞
W − rn = p
2
lim ;(t) = p. /
t →∞
t →∞
p -
+
> Z , ! ! >, rn, , .. . *:#(/C):. & W : T = rn2, %+ "$ 0 $& W
f ij* %+ + eij 2 .
n
> > #% 2, ! . ; n > !
W
" n2, f ij* = 2 .
n
4 2.2.
:%A:&" 2.2 . W > rn2, $& 0 000 Q* = (q1(0) − w*, …, ql (0) − w*, rn, …, rn),
(2.6)
l=k
62
w* =
D ( 0)
, k
ck(0) ≥
D ( 0)
;
k
(2.7)
" l < k – +$# " , , "
cl(0) ≥ w*,
w* =
C l ( 0) − p
,
l
(2.8)
(2.9)
l
Cl(0) =
¦ ci (0) .
i =1
A " " . ;
«# » >
. 2.1 2.2. * , k – < > Z+(0). / l – < > Z+ Q*. K k Z+,
l = k, .. (2.7). / l # (2.8), (2.9).
t > Z+(t) −
Z (t) w(t) = qi(0) – qi(t). / w* – < w(t) .
$ & 2.2.
D(0)
;" 1. ck(0) ≥
. / 4 1
k
! , lim Q−(t) = (rn, …, rn). < t →∞
Z−(t) D(0), < > D ( 0)
Z+(t). ; ck(0) ≥
, > Z+ !k
D ( 0)
w* = lim w(t) =
< >
t →∞
k
ck , , , Z+ , .. l = k.
D(0)
;" 2. ck(0) <
. ;" t k
ck(0) – w(t) ≤ 0, > vk ( , !, %
" > Z+) Z–,
> Z+(t), t' Z l+ . Q(t')
63
Z l+ , Z l− # % Dl(t′). ; l t'
>, 1 k l.
< (2.6) . / l w* l
! . G (2.2) k l , ¦ q (0) =
i
i =1
*
= rln + ;(0). < , (2.6) Q , W = rn2 + Cl(0) – w*l. G (2.4), (2.9).
rn2 + p = rn2 + Cl(0) – w*l, b (2.8) !
", ! cl Z l+ . *:#(/C):. W > rn2, %+ "$ 0
$& f ij* %+ + eij r.
> Z+(t) ! t #% 1 , , ! r % . N! > Z− rn
% (# (2.6)), " 2 n ! .
2.3. %% ; !< ( *%9*%") %+
' '
/ "
"# , % .
/ 1.2.3 ! , "#, , %
&, , !: % & % P "# G(P).
'# " % &, 1.2.2. K "# , <" % &. , , "# , ! D " $%. D" % 0 ,
*$
% d "# %.
, " "# ! . K d > 1, "# % % &. '# > (> , !) " % .
" % .
D" " , , % !
,
% !, 64
! " . < " >" k, > , – ! .
* . 2.4 > v1, …, v5 ! <" , " .
v1
v6
v5
v2
v4
v3
v1
v7
v9
v8
: . 2.4. <" , " (% 5)
"#, <" , :& . . 2.4 v6, …, v10. :, > "# > , . G % " <" , <" > , < %
&.
/> , !, ! <" . / ! : , > , . : > ! )%) & 0 .
<" , . 2.4,
! " > ( . 2.5).
* . 2.6 .
$! < # % ;1.
, <" !" " .
65
v2
v1
v6
v3
v5
v4
: . 2.5. " :(BA(,@: (:/)
A>%#)$:('):
:" + :QA>%#)$:('):
" « >» – <" : . 2.6. ; " # % 2.4. %% ; !< ( *%9*%") %+
' '*'9%!"# %'%:%(#
# % , , ! % !. ;,
! , " ! ( <" " " ! ). $ < # % 66
. / ! ! ! #% ; . / # % , " .
* ( . . 2.2), 1, ! ! . / .
* , , .
* " , % ! : R = RT.
/
> ! . $ # "% , ! , .
: " , ∀i riin = riout .
(2.10)
K , ! > , riin − riout ≠ 0 . > vi ! <
Δri: Δri = riin – riout . ;" > :
1) # &- , Δri > 0;
2) # &- " , Δri < 0;
3) $& # &, Δri = 0.
/ > $&.
! " , (2.10). *
! .
/> ((
)(
) (
))
ρ = r1in ; r1out , r2in ; r2out ,... rnin ; rnout .
(2.11)
# % ( . 2.7).
1
; « » % & ( . 1.2.1, 1.2.2).
67
:(BA(,@: (:/)
#,%A%#,@:
:%#,%A%#,@:
* *
! D : . 2.7.
# % !
G , : , !^ ! D &: .
/ < ! !: > ! . / " ! % , "# !
( ! ), > " . "#, ! , < [58]. : , ! > , ! !! < "#. < ! D & & 0 .
| " . < " <" : > ( ).
$ .
,
: < G c n > . ;" ,
. 2.8, " !
<" . / ! !,
68
, ! "; , ! ! <" .
D" : . 2.8. «*<"
< »
/ G G’ m > , m < n.
! ! ! , < > v1, …, vm.
/ < ! > vi G’
riin = riout .
(2.12)
: < . riin = σ
ୀଵ ݎ +σୀାଵ ݎ . – < « » !, vi, .. !, ! > G’ (rij = 0, " ! ); - < !, vi , .. !, vi G’.
riout = σ
ୀଵ ݎ +σୀାଵ ݎ .
– < « » !, vi; – < !, vi
, .. !, vi G’. < (2.12), :
σ
ୀଵ ݎ +σୀାଵ ݎ ൌ σୀଵ ݎ +σୀାଵ ݎ .
(2.13) > (2.13)
G’. -
σ
ୀଵሺ σୀଵ ݎ ) +σୀଵሺσୀାଵ ݎ ሻ =
= σ
ୀଵሺ σୀଵ ݎ ) +σୀଵሺσୀାଵ ݎ ሻ.
(2.14)
69
(2.14) – < ! G’, ; – < ! G’, . $ , < ,
:
σ
ୀଵ൫σୀାଵ ݎ ൯ = σୀଵሺσୀାଵ ݎ ሻǡ
" :
0 %+ 0 D :0) : + &:0) : + .
* !; , . < . 2.8 < . * . 2.7 . > , . / > : < .
( !
. 2.2.)
D # % ! ! ;2.
N! , " # , <" , , ;1 × ;2. # % , !, % (! . 2.3).
+ 2.3
*"((@ A:(BA(,@J (:/:D ) (/AB'/BA" ',)>)
;2
;1
:" + " $ ' 2
( )
' 6
–
*
D ' 3, 4, 8
' 2, 5
' 6
' 7
' 6
–
##* + ! 2
/! " . D " : " < !
70
" , ! " . '# #% . / , <" >
! ( , , #% > – 1 2, – % " ", Z+
Z–) , # < .
" " .2 " ! .
#% " ! > Z–(t) Z+(t). Z–(t) – < > , t #% 2; > Z+(t) t #% 1. < rn: vi ∈ Z–(t), qi(t) V rn. /> , > Z–(t), ", < Z–(t) !.
" ! . < , W , .. " T = rn2, " : > Z–(t), " ! > ; !.
K W > ", % , % ". K Q(0) > Z+(0), Q(0) ( 2.3) , , . K
Q(0) ! > Z-(0), > Z- ! > Z+ ( ), rn. < > Z+ Z-. * .2.1
> , : , Z+,
!, , > Z-, . / > Z- ! , rn.
b ! ! ( ) > : > – !, ! – !. < > , 71
: !" " " > .
/ " ! % #% .
, < " : " % – > ! " " ; , # > – : " T
W, ; Z+ Z-; " . $ " > !, # % , . 2.3. < #% ! , . D % , ! ! " " .
72
3. @A B –
@ > @. B@B@ B B B ,
:
• & & & . & «», , t', & & & 2. % &# & & &.
• , & & & '.
• 0 , $ ' R'∞ L = – R'
# , &
.
• & & & .
• ( , - & Z–(t); 3 , & Z+* (t) t.
• % & , & & , & '. ( 6 ! & .
• % , $ . 7 ,
# & (0, 1], ! , & & .
• % – , &
& & & ! ;
Z+(t) . , ! ! .
• , & .
/ < " #% " . "
, , " > #% 2, , " > #% 1. / 73
" ! , " > " #% 2; ! " %, " #% . > > , ! Z+*, " , < .
3.1. *$/ %%( *9(*+ %
'* $ &# *%9*%
" % R % W = 1, % ! R : ! > vi riout ≥ 1 . D – ( ") #% > 2: t > ,
! > >". *! ! # > . ; ,
! , " .
" ! Q1(t). > > ( ), t. % #% :
Q1(t + 1) = Q1(t)⋅R',
% R' #% § r11
¨ out
¨ r1
R ' = ¨ ...
¨ rn1
¨¨ out
© rn
(
r12
r1out
...
rn 2
rnout
)
r1n
r1out
... ...
rnn
... out
rn
...
2.
·
¸
¸
¸.
¸
¸¸
¹
R' = D–1R, " D = diag r1out ,..., rnout . R' – %, .
74
& % ! R " . ; ! , " % R', 2, " Q1(0) = (q11 (0), q12 (0),..., q1n (0)) " % &.
* , % &, W = 1 [11, 38]:
1) ! " lim ( R ' ) h = R'∞;
h→∞
2) !" " " (W = 1) " Q1* , # Q1* = Q1(0)⋅R'∞;
2') ",
!" t > 0 Q1* = Q1(t)⋅R'∞;
3) % R'∞ n Q1*:
R'∞ = 1 ⋅ Q1*,
" 1 – - !%, %;
4) Q1* ! R', ! λ = 1:
(3.1)
%
Q1* R' = Q1*;
5) ! !% % R'∞ (3.1) ! % R';
1*
6) Q ! % R'∞:
1*
∞
1*
Q ⋅ R' = Q .
G . 2) , ".
| " . / 1.2.4 " % R'∞ % "#, " % R': L = E – R'. & % R'∞ ! % L. < % [107, 114, 3]:
75
R'∞L = LR'∞ = 0, rank R'∞=ν, rank L=n–ν.
; < %
"#, " " "# % ! [1]. N "# " %, , ,
rank R'∞ = 1, " # (3.1). & % R'∞ ! !" " " % L [62].
3.2. *$/ %%( *9(*+ %
'* #") *%9*%)
W , , " t', > #% 2, .. Z–(t).
; W " : , W < min riout ,
i
> Z+(t).
3.1. ' 0 0 %+ W, ∃ t' >0: ∀ t > t' vi ∈ Z–(t), i = 1, … n, 0 %+ "$ 0 0 Q(0) $ 0 0 Q*:
1) );
2) ;
3) 000 & +& : " $ & R'∞ +&
& R'
*
" λ = 1: Q = Q*⋅R' Q* = Q*⋅R'∞.
$ . > , > #% 1. * t', " > Z–(t) #% 2. ;" !" t ≥ t' #% # :
Q(t+1) = Q(t)⋅R',
" R' – %.
!" k :
Q(t+k) = Q(t)⋅R'k.
(3.2)
" R'∞ , k → ∞
(3.2) :
76
Q (t ) lim ( R ' ) k = Q (t ) R '∞ .
k →∞
;" :
(3.2) Q* = Q(t)⋅R'∞.
(3.3)
;
! , " ! !" " Q(t) (t ≥ t').
; (3.3) !" t ≥ t', , , :
Q* = Q*⋅ R'∞.
(3.4)
$ , Q* – ! %
R' ! λ = 1.
G (3.4) , ∞
Q*⋅ R' = (Q*⋅ R'∞)⋅R' = Q*⋅ (R'∞⋅R') = Q*⋅ R'∞ = Q*.
Q* – ! % R', λ = 1. A! < . ; ! , . #% 2 "
! % R'. * ! % R' " , " W2
. W1
:
Q1* Q2*
=
.
W1 W2
;" " W, #% 2, " Q*
Q1* :
Q* = Q1*⋅W.
(3.5)
; ! , ( , > 77
Z–(t)) , % ! . ", ! ! " .
, , . $ , , – , ! > t , , > , Z+(t), – #% . : #% ! > .
3.3. 9(*" % ##* &" %
) %+%
*
! > :
> - , ! , > - , ! > !. ", "
> , ! < . ! .
: , "# vi, > - ( . 3.1). < ,
vi , > (.. < < ).
D *
vi
: . 3.1. *
< ; ! " $% D . $ " ! 3.6.
D !
" – " , ! 78
> - . ", ! ". $ " ! #% " . J , " .
l > - , k
n > n – l – k > . J , 1 l, – l + 1 l + k,
> – l + k + 1 n.
& > , qi(t) ≤ riout , >, ! Z–(t), > qi(t) > riout – Z+(t).
/> Z–(t) #% 2, > Z+(t) –
1.
*
! " > – – ! , " , < . D ! > ,
! , % " % & ! " % ! .
/ >
, >
> t → ∞ " ! > . / > , > ! .
; 3.1. K
" t' > vi, (i > l), !: qi(t') ≤ riin , t > t' qi(t) ≤ riin , (i > l).
$ <" , > - > ! , , - qi(t') ≤ riin , , qi(t') ≤ riout , , > #% 2 , ! > riin . > - qi(t) ≤ riin vi(t) ∈ Z–(t), > < . 79
< 3.1 , Z–(t) > vi " < .
3.2 3.3 #% " " .
; 3.2. K
% R
!% i, j > l, > vi, vj , t', t > t' qi(t) = qj(t) (i, j > l).
, ! 3.2, " t' ! > 2 ! , .
; 3.3. / % #% > ! , .
D , " " !, > . ;" ( > ) > , !
! .
.
3.1. > % ! :
§1
¨
¨1
R = ¨3
¨
¨4
¨5
©
1 1 1 1·
¸
1 1 1 1¸
1 1 1 1¸ .
¸
1 1 1 1¸
1 1 1 1¸¹
D ! v1, v3, v4, v5. / > v2, …, v5
> v2 ! , ! .
W = 40 > v2. * Q(0) = (0, 40, 0, 0, 0). #% ! . 3.1 " " ( . 3.1).
80
: > , –
> Z (t), > riin ( 3.1), > v2, …, v5 % #% ( 3.3).
( 3.2) !
t
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
…
29
30
31
32
33
…
+ 3.1
"(-A:#:*:,): A:(BA(" – -A%/%'%* A"E%/@ (:/)
v1
v2
v3
v4
v5
0,000
1,000
2,684
3,880
4,945
5,826
7,232
8,870
10,596
12,355
14,127
15,903
17,682
19,461
21,240
23,020
24,799
26,579
28,110
29,116
29,699
40,000
36,000
32,579
29,714
27,299
25,268
23,393
21,577
19,784
17,999
16,218
14,438
12,658
10,878
9,099
7,319
5,539
3,760
2,972
2,721
2,575
0,000
1,000
1,579
2,135
2,585
2,969
3,125
3,184
3,207
3,215
3,218
3,220
3,220
3,220
3,220
3,220
3,220
3,220
2,972
2,721
2,575
0,000
1,000
1,579
2,135
2,585
2,969
3,125
3,184
3,207
3,215
3,218
3,220
3,220
3,220
3,220
3,220
3,220
3,220
2,972
2,721
2,575
0,000
1,000
1,579
2,135
2,585
2,969
3,125
3,184
3,207
3,215
3,218
3,220
3,220
3,220
3,220
3,220
3,220
3,220
2,972
2,721
2,575
30,494
30,496
30,498
30,498
30,499
…
2,377
2,376
2,376
2,375
2,375
…
2,377
2,376
2,376
2,375
2,375
…
2,377
2,376
2,376
2,375
2,375
…
2,377
2,376
2,376
2,375
2,375
…
% <. " -:AC%& Q/"-:
( 0–10) > v3, v4, v5 ! " " . " C/%A%& Q/"-: ( 11–17) < > !
% , > v2 v1. " /A:/+:& Q/"-: ( 18-") > v2 > ! , > .
81
82
0
5
10
15
20
25
30
35
40
q(t)
v1
: . 3.2. ! !
%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
v2
t
v5
v4
v3
v2
v1
3.2. ' 0 " 0 :
" $&: # , ) : D &
, 0 %+ W "$ 0 Q(0) = (q1(0), q2(0), … qn(0)) ) t', " 0 %+ t > t' & 0 :
qi (t) V riin , i > l.
(3.6)
$ . G 3.1 , > , Z–(0), < . > (3.6) .
1. , ! Z+(0) " Z–(t), .. #% 2. vi – > - , i = l+1, …, l+k; riin < riout . ; < > ! > riout , < > ! #% 1, .. riout % . < > riin < riout . ;" > - ! > " ! , r': r' ≥ Δri = riout − riin . ;
! ! < > , ti > - 2 ! , riin , < , (3.6) . / ti % :
q (0) − riout
ti ≤ i
.
Δri
2. , (3.6) " > , < . " , > vj
" . = > < , " vj, . t* – , 83
> Z–(t). " !" " . 1. !
eij vi > vj. ;" vi 2, ! , " > " ! ,
> vj (3.6). K > k > , < k
, ! > > .
;
! , > > - (3.6). K < , > ,
> , Z+(t). : .
3.2. > > %:
§1
¨
¨5
¨1
R= ¨
¨1
¨2
¨
¨2
©
1
1
1
1
0
0
1
1
1
1
0
0
1
1
1
1
0
0
2
0
0
0
1
0
2·
¸
0¸
0¸
¸,
0¸
0 ¸¸
1 ¸¹
W = 50, Q(0) = (0, 0, 0, 0, 50, 0).
& % ! , . 3.3.
* > v5 v6 v1 . ! ! . 3.2.
* . 3.4 % .
G ! . 3.2 . 3.4 , v1 > v6, , ! ! , > > v5. * > v3 v4 ,
, < !, v2.
84
v6
v5
v1
v2
v3
v4
: . 3.3. < – > v5
v6
+ 3.2
A%/%'%* A"E%/@ (:/)
t
v1
v2
v3
v4
v5
v6
0
1
2
3
4
5
…
102
103
104
105
106
…
0,000
2,000
2,250
2,896
3,340
3,777
0,000
0,000
0,250
0,438
0,635
0,815
0,000
0,000
0,250
0,438
0,635
0,815
0,000
0,000
0,250
0,438
0,635
0,815
50,000
48,000
46,500
45,063
43,786
42,622
0,000
0,000
0,500
0,729
0,967
1,157
8,000
8,000
8,000
8,000
8,000
2,666
2,667
2,667
2,667
2,667
2,666
2,667
2,667
2,667
2,667
2,666
2,667
2,667
2,667
2,667
31,001
31,001
31,001
31,000
31,000
3,000
3,000
3,000
3,000
3,000
85
86
0
5
10
15
20
25
30
35
40
45
50
q(t)
v1
: . 3.4. < 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
v5
t
v6
v6
v5
v4
v3
v2
v1
$!^ " .
/> v1, < , – 1. $ >, , !
, . ;
! , ! > , " > " , < #% , < . / < , > , ! > ! ! >
. < ! " ; " , ! > , ! " 5.
3.4. * 6& / 3.1 " , > Z–(t). / , , > 2 (
, , Z–(t)). K
, , ! , % &,
# (3.5). $ < 3.3.
3.3. ' 0 " ) " , , ":
W ≤ # &, " 0 t', :0 Z–(t);
W > T Z+(t) , " 0 t''.
"$ 0 0 Q(0).
$ . G 3.2 , > , < , Z–(t). ;" ! > Z+(t) " > > < . W > rsum ! > " Z+(t).
1
* , " > , !. K ! " , ", ! , . ! . 3.6.
87
: " #% W:
Q*=Q*(W). W , > , " , > Z–(t). G Q*(W) = Q1* ⋅W
, Q*(W) % W, > Z–(t). W > " riout , < > 1 > W > (3.5) . $!
" , > ,
riout , . Q*(W) "
W ≤ Q(0), – . | " 3.1. $ , ≤ rsum. = rsum ( 2.2); < rsum. D 3.2.
3.5. E;; < % ##* &% %
$ – D " .
3.1, = rsum. ;" % !
. $ rsum " . : .
3.3. % ! :
§ 1 100 1 1 1 ·
¸
¨
¨ 1 1 70 1 2 ¸
1 0 1¸
R = ¨3 1
¸
¨
1 5 1¸
¨1 2
¨ 1 1 10 1 1 ¸
¹
©
< ρ, # (2.11), : ρ = ((7,104), (105,75), (83,6), (8,10), (6,14)). – < > v2 v3. rsum = 209.
: < .
88
1. / " W = 1
:
Q1* = (0.212, 0.284, 0.396, 0.024, 0.084).
2. D , ≈ 15.16.
3. W = T " :
Q* = (3.215, 4.312, 6.000, 0.359, 1.273).
! : W = 100 Q* = (3.215, 4.312, 90.841, 0.359, 1.273).
3.4. % ! :
§ 1 10 1
¨
¨1 1 7
R = ¨3 1 1
¨
¨1 2 1
¨ 1 1 10
©
1 1·
¸
1 2¸
0 1¸ ,
¸
5 1¸
1 1 ¸¹
; ! > ! r12 r23
> " > 10 . $ .
ρ = ((7,14), (15,12), (20,6), (8,10), (6,14)).
– > v2
rsum = 56.
1. / " W = 1 :
v3 .
Q1* = (0.212, 0.253, 0.316, 0.091, 0.128).
2. ≈ 19.
3. W = T Q* = (4.034, 4.800, 6.007, 1.724, 2.434).
, , 89
" . W = 100 ! : Q* = (4.034, 4.800, 87.007, 1.724, 2.434).
D
, ! " !
: ! > rsum, ! > , – ! . / rsum = 209 ≈ 15.16; > ! ! , , > ! rsum = 56, " : ≈ 19.
I ! ! > , > " . / , ! , = rsum. I ! >
, > T rsum.
D " > T
χ=
. $ , χ ∈ (0, 1]. ! " 7, rsum
% χ = 0 " = 0 " . ;
! , χ ∈ [0, 1]. I ! χ %, «! ».
3.6. *!*"
) !%% ; !< (
3.6.1. '*$ '< /") *!*
D , " Z+(t) ! " ! > W Q(0). : #% , ! ! .
3.5. % ! :
§ 1 1 1 1 1·
¸
¨
¨ 1 1 1 1 1¸
R = ¨ 3 1 1 1 1¸
¸
¨
¨ 1 2 1 1 1¸
¨ 1 1 1 1 1¸
¹
©
W =100; : Q(0)=(100, 0, 0, 0, 0).
90
< ρ : ρ = ((7,5), (6,5), (5,7), (5,6),
(5,5)), ! , {v1, v2}, {v3, v4} > {v5}. "#
, < . #% ! . 3.3.
t
0
1
2
…
31
32
33
34
…
A%/%'%* IB,'M)%,)A%C",)G (:/)
v1
v2
v3
v4
+ 3.3
v5
100,000
96,000
92,995
0,000
1,000
1,876
0,000
1,000
1,710
0,000
1,000
1,710
0,000
1,000
1,710
83,799
83,798
83,797
83,797
4,536
4,536
4,537
4,537
3,888
3,888
3,889
3,889
3,888
3,888
3,889
3,889
3,888
3,888
3,889
3,889
> < , ! W = 100 Q* = (83,797, 4,537, 3,889, 3,889, 3,889). ;
! , ! Z+, , ! ", , > . (* " : ≈ 5 + 4,537 +
+ 3,889 + 3,889 + 3,889 = 21,204.)
: ! .
§1
¨
¨1
R = ¨2
¨
¨1
¨1
©
1 1 1 1·
¸
1 1 1 1¸
1 1 1 1¸ , ρ = ((7,5), (7,5), (5,7), (5,7), (5,5)).
¸
2 1 1 1¸
1 1 1 1¸¹
3.6. : : Q(0) = (30, 0, 0, 0, 0).
91
t
0
1
2
3
4
…
29
30
31
…
A%/%'%* IB,'M)%,)A%C",)G (:/)
v1
v2
v3
v4
+ 3.4
v5
30,000
26,000
22,900
20,493
18,625
0,000
1,000
1,900
2,593
3,132
0,000
1,000
1,733
2,304
2,748
0,000
1,000
1,733
2,304
2,748
0,000
1,000
1,733
2,304
2,748
12,144
12,143
12,143
5,000
5,000
5,000
4,285
4,286
4,286
4,285
4,286
4,286
4,285
4,286
4,286
G , ! ! > , > , r2out , > , > riin ( 3.2). Q* = (32,143, 5,000, 4,286, 4,286, 4,286).
K " , " : >" : Q* = (5,000, 12,143, 4,286, 4,286, 4,286).
/ .
(> v3):
3.7. : Q(0) = (0, 0, 30, 0, 0).
" " Q* = (9,291, 7,852, 4,286, 4,286, 4,286). ; ! ,
! Z+*, ! ! > ! , >, ( . 3.6).
(> v4), : Q* = (7,852, 9,291, 4,286, 4,286, 4,286).
K > , !
: Q* = (8,571, 8,571, 4,286, 4,286, 4,286).
* . 3.6 " " , Q(0) = (0, 0, 0, 0, 30). : , , .
* > > , > v3, v4, v5 .
92
v2
t
v5
v4
v3
v2
v1
( )
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
v1
: . 3.5. A% 0
5
10
15
20
25
30
35
q(t)
93
94
v1
( t
v2
v5
v4
v3
v2
v1
(> v3))
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
v3
: . 3.6. A% 0
5
10
15
20
25
30
35
q(t)
v1
v2
t
v5
v4
v3
v2
v1
( > v5)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
v5
: . 3.7. A% 0
5
10
15
20
25
30
35
q(t)
95
G , " ( , ", !" " ). ( . 3.8), , " , ! > ! <" .
, ! > , ! >. K ,
. / , " , " > , ! . : < " ! , .. < Z+(t) t.
v5
v3
v4
v1
v2
: . 3.8. , " U /> , W > T Q(0), " Z+*, . ! 3.6, 3.7, ! " , ! Z+*.
96
< ! > $& . G 3.2 , % ! , ! > < .
,
< , > , ! > -. / , – , , , ! , , !
< . K > " ! >
Z*– ! " . K < – , < , .
3.6.2. %% ; !< ( *!*
3.5, . " , " ! > .
$ - > % . /> - " ! , % . ! > ! Z+(t), Z–(t). * > !
> $0 Z+(t), .
Z–(t) .
;
! , :
− & & – < > - ;
− & & – > < .
3.6.3. * 6! *! % *G "
b, > , % ! . : .
3.8. > %:
97
§ 1
¨
¨ 1
R = ¨100
¨
¨ 1
¨ 1
©
2 1 1 1·
¸
1 1 1 1¸
1 1 1 1¸
¸
1 1 1 1¸
1 1 1 1¸¹
ρ : ρ = ((104, 6), (6, 5), (5, 104), (5, 5),
(5, 5)). / " Q(0) =
= (0, 0, 0, 0, 100). : W = 100 > .
! . 3.5.
t
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
…
76
77
78
79
80
81
82
…
98
A%/%'%* IB,'M)%,)A%C",)G (:/)
v1
v2
v3
v4
+ 3.5
v5
0,000
1,000
2,528
3,601
4,544
5,342
6,017
6,604
7,575
8,725
9,948
11,200
12,466
13,736
15,008
16,281
0,000
1,000
1,743
2,522
3,145
3,677
4,128
4,504
4,635
4,689
4,711
4,720
4,724
4,725
4,726
4,726
0,000
1,000
1,576
2,100
2,545
2,920
3,238
3,504
3,635
3,689
3,711
3,720
3,724
3,725
3,726
3,726
0,000
1,000
1,576
2,100
2,545
2,920
3,238
3,504
3,635
3,689
3,711
3,720
3,724
3,725
3,726
3,726
100,000
96,000
92,576
89,677
87,221
85,141
83,379
81,883
80,519
79,208
77,919
76,639
75,363
74,088
72,814
71,540
86,647
86,669
86,683
86,691
86,696
86,704
86,704
4,088
4,083
4,079
4,077
4,076
4,075
4,075
3,088
3,083
3,079
3,077
3,076
3,075
3,075
3,088
3,083
3,079
3,077
3,076
3,075
3,075
3,088
3,083
3,079
3,077
3,076
3,075
3,075
, > v2, …, v5 , > ! , > v1, .
Q* = (86,704, 4,075, 3,075, 3,075, 3,075).
, . D > v1 > v2. $ ! > v1 > ! > v2, , > v1, .
A% . 3.9 "# , % ! > " " # .
100q(t)
90
80
v5
v1
70
v1
60
v2
50
v3
40
v4
30
v5
20
10
0
0 3 6 9 121518212427303336394245485154576063666972757881 t
: . 3.9. A% ( > v5)
G , #% ! > v5 v1, > ! !
% .
G % R < : r12 = 3.
3.9. > %:
99
§ 1
¨
¨ 1
R = ¨100
¨
¨ 1
¨ 1
©
3 1 1 1·
¸
1 1 1 1¸
1 1 1 1¸
¸
1 1 1 1¸
1 1 1 1¸¹
ρ : ρ = ((104, 7), (7, 5), (5, 104), (5, 5),
(5, 5)). * : Q(0) = (0, 0, 0, 0,
100). ! . 3.6.
t
0
1
2
3
4
5
6
7
8
9
10
…
157
158
159
160
161
162
…
A%/%'%* IB,'M)%,)A%C",)G (:/)
v1
v2
v3
v4
+ 3.6
v5
0,000
1,000
2,504
3,529
4,439
5,222
5,886
6,454
6,915
7,219
7,576
0,000
1,000
1,838
2,766
3,495
4,123
4,661
5,119
5,606
6,333
7,119
0,000
1,000
1,552
2,051
2,487
2,855
3,169
3,437
3,643
3,751
3,786
0,000
1,000
1,552
2,051
2,487
2,855
3,169
3,437
3,643
3,751
3,786
100,000
96,000
92,552
89,603
87,091
84,945
83,114
81,552
80,194
78,946
77,732
6,268
6,268
6,267
6,267
6,266
6,266
84,098
84,100
84,101
84,103
84,104
84,104
3,211
3,211
3,211
3,210
3,210
3,210
3,211
3,211
3,211
3,210
3,210
3,210
3,211
3,211
3,211
3,210
3,210
3,210
Q* = (6,266, 84,104, 3,210, 3,210, 3,210).
, > v1 v2 . ; > v2, v1 , > ! . D ,
!, 3, ! e12,
> v2 « » v1 > v5, v1. " ! r12 ! 2, ! « » , > v1.
100
G . 3.10 > , > v1 Z+(t), .
q(t)
100
90
80
v5
v2
70
v1
60
v2
50
v3
40
30
v4
v1
v5
20
10
0
7
14
21
28
35
42
49
56
63
70
77
84
91
98
105
112
119
126
133
140
147
154
0
: . 3.10. A% ( > v5)
t
/ : ! r12, , " !
! % , < ! 3.6, 3.7?
$! . # ! : , , , > . &
!
> ?
", , , % #% " " .
G " , " ! > , " 4.
101
##* + ! 3
' 3, " 4, " . / > , !. D > : ( ! >, ), ( ! >, ) > ( ). ; > , %
! . J – % " – > , $%. = – < > , ! Z+*.
I #% " , " . 4, ! .
$! : - Z– Z+, " T.
/ Z– > #% 2; > " ! > 1. 1 " " % & ( 3.1). , .1 2.1 – <
3.1, ଵ
n > < % R' .
$ < . % – .
1. / > > Z+* – U " . / <" > ! : Q(0)
> Z+(0) ( ! ! ), , " t, " Z–. ; ! , ! ; > ! 1
102
D ! " . 4.
t Z–(t). ! " > - . * > , < , " ! (
+
Z , ! ).
= , .. > , !
Z– Z+*, " ! , " .
b, > , % !, . / > , ! > " .
2. < rsum; < " , ! . $ (. 3.5): <## % χ % ! > , > χ, ..
> rsum.
$ . : – < , < % &, % , . $ <" >
( ., , . 5.8 " [52]). K " !
W = 1 (. 3.1), , > Q1(t),
" > < .
103
4. B D BA A @A B BA,
:
• ) & W ≤ T. ( &# W > T; & & . 9 &# , &# .
• , W = T & , & . " & , & W > T, ! $ - .
• & &
.
• ! &# ' Q1* & -.
• ( , & & . % & W – T ! & .
• % $ ! & n × n. , $ [Ri'] &# & , #
! . Ri'.
• ( , & , , # , & . ( , $ & .
4.1. ! *%9*%
! > !
, < ! , – % &. ; > " , > , #% . 104
< > , #%
> , ! > .
1, ; , . !
-
4.1.1. ! '* #") *%9*%)
K W ≤ , ! > t (t > t') #% 2, > > >", .. " : Q(t) = F in (t ) .
" , 2 > , , < t : F out (t ) = Q(t). G 3.1 3.3 , Q(t) W ≤ Q*. ;" lim F in (t )
t →∞
lim F
t →∞
out
(t ) . ;
! , #% -
2 :
Fin* = Fout* = Q*.
*
f sum
= W.
/ " W = !
~ ~
~ in
~ out
~
Q = (q1 , ..., qn ) , F
F . ;" ~ ~
~
~
F in = F out = Q ; f sum = .
4.1.2. ! '* :/G ) *%9*%)
: #% W > . / " , . % ! R 3.5 (" 3) " .
105
4.1.
§1
¨
¨1
R = ¨3
¨
¨1
¨1
©
1 1 1 1·
¸
1 1 1 1¸
1 1 1 1¸ .
¸
2 1 1 1¸
1 1 1 1¸¹
< % ≈ 21,204.
: " Q(0) =
= (100, 0, 0, 0, 0) ( . 4.1). $ . ( , , < "# " , ".)
: . 4.1. " .
* Q(0)=(100, 0, 0, 0, 0)
' ! – .
K
! " > , . ;, Q(0) = (0, 100, 0, 0, 0) ! , #
!
% , > !, ( . 4.2).
106
: . 4.2. " .
* Q(0)=(0, 100, 0, 0, 0)
% > . / < > v1, > v2 #% 1. : > > v2 1 2. " !
% .
K
> v3, ! – !
% ( . 4.3).
: . / > , ! > ! .
! rsum, !, !
% . !
% (fsum = rsum) , " > #% 1. * " , > v3, ! > , 2, >. , " > v3, …, v5 ! , : ! > ". > . 4.5.
107
: . 4.3. " . * Q(0) = (0, 0, 100, 0, 0). ($ }: 5 % )
: . 4.4. . * Q(0) = (20, 20, 20, 20, 20). ($ }: 15 % )
108
: . 4.5. " > . * Q(0)=(20, 20, 20, 20, 20). ($ }: 3 % )
G > , . 4.4, 4.5, . 4.6.
v1
v2
: . 4.6. > .
* Q(0) = (20, 20, 20, 20, 20)
109
/ " > :
Fin* = Fout* = (5, 4,537, 3,889, 3,889, 3,889), ~
f sum = 21.204.
: , W > T " . $ .
1. K , #. ! . / , !, < , > - " . =
, Z+(t), .., ! . ;
! , ; " > . < .
2. #% !
% .
3. > , " > - Z–(t).
4. t → ∞ .
5. , ~
f sum = T.
! , > ,
, ! : f1out* → 5 = r1out ( . . 4.5).
G 3.3 ""
, W = ! > ! , ! :
~
q~i = riout . , , Q = (q~1 , q~2 ,..., q~n ) – " W = . / % > ,
q~i = riout l > (l Y 1). ;~
" : Q = (r out ,..., r out , q~ ,..., q~ ) .
1
l
l +1
n
4.1. ' 0 " 0
W > T %+ "$ 0 0 $& ), " ) 0 0 # , " $ F* :
110
§ r11
¨
...
¨
¨ r
¨ ~ l1
*
F = ¨ ql +1 r
¨ r out l +1,1
¨ l +1
¨ ~ ...
¨ qn rn1
¨ r out
© n
r12
...
...
rl 2
...
...
q~l +1
rl +1, 2
rlout
+1
...
q~n
rn 2
rnout
...
...
...
·
¸
...
¸
rl n ¸
¸
~
ql +1
rl +1,n ¸ ,
¸
rlout
+1
¸
...
¸
q~n
¸
r
nn ¸
rnout
¹
r1n
& l # % W = T , & &: + .
$ . G 3.1 3.3 , W = T . ;" ,
% W = T ! :
§ r11
¨
¨ ...
¨ r
l1
¨
~ ¨ q~l +1
F=
r
¨ r out l +1,1
l +1
¨
¨ ~ ...
¨ qn rn1
¨ r out
© n
r12
...
...
rl 2
...
...
q~l +1
rl +1, 2
rlout
+1
...
q~n
rn 2
rnout
...
...
...
·
¸
...
¸
rl n ¸
¸
~
ql +1
rl +1,n ¸
¸
rlout
+1
¸
...
¸
~
qn
¸
r
n2 ¸
rnout
¹
r1n
> , " i i- % (4.1) i-" !%. /- !% "
:
T
~
F out = r1out ,..., rlout , q~l +1 ,..., q~n
(
)
(4.1)
(4.2)
W > T Q(0) = (q1(0), q2(0), …, qn(0)). / <## % % :
T
~=
< 1,
W
111
" QT(0) = (~q1(0), ~q2(0),
…, ~qn(0)) " W = . " "
. ~
% F , (4.1).
/ " " " (4.2):
(
~
~
F in = F out
)
T
~
= Q = (r1out ,..., rlout , q~l +1 ,..., q~n ) .
QT(0) FT(t) F(t) Q(0) (W > T):
F(t) Y FT(t).
(4.3)
, (4.3) t → " , W > T W = T .
l > : – !
< > .
, t → W > T > vi (i > l) ! > q~i Δf(t). ;" % ! :
r11
§
¨
r21
¨
¨
...
¨
rl1
¨
¨
f l +1,1 (t )
¨
F (t ) = ¨
...
¨
¨ § q~i
·
¨ ¨¨ out + Δf (t ) ¸¸ri1
¹
¨ © ri
¨
...
¨
¨
f n1 (t )
©
112
r12
r22
...
...
...
...
rl 2
...
f l +1, 2 (t )
...
...
...
§ q~i
·
¨
¸
¨ r out + Δf (t ) ¸ri 2 ...
© i
¹
...
...
f n 2 (t )
...
·
¸
¸
¸
...
¸
rl n
¸
¸
f l +1, n (t )
¸
...
¸
...
¸
§ q~i
· ¸
¨
¸
¨ r out + Δf (t ) ¸rin ¸
© i
¹ ¸
¸
...
¸
...
¸
f nn (t )
¹
r1n
r2 n
" , i < , (
" ").
K < i- < k V l, > vk.
rkout , "
! >, < > > vi § q~i
·
¨ out + Δf (t ) ¸rik , ! > , W = T, – ¨r
¸
© i
¹
> . ; ! , > vk ! > , , . / , Δf(t)⋅rik. K < " , > vk ! !, W. < , Δf(t)⋅rik → 0 t → , Δf(t) → 0, < i- ~
% F(t) < % F . ; ! , > vi ! ! l > , t → W = , .. q~i .
> vi ! l
> . ;" i- % l .
§ r11
¨
¨
¨ ...
¨
¨ rl1
¨ f (t )
¨ l +1,1
F (t ) = ¨ ...
¨
¨
¨ 0
¨
¨
¨
¨ ...
¨ f (t )
© n1
r12
...
r1l
r1,l +1
...
...
...
...
...
...
rl 2
...
rll
rl ,l +1
...
f l +1, 2 (t ) ...
f l +1,l (t )
f l +1,l +1 (t )
...
...
...
...
...
...
0
...
0
§ q~i
·
¨
¸
¨ r out + Δf (t ) ¸ri ,l +1 ...
© i
¹
...
...
...
...
...
f n 2 (t )
...
f nl (t )
f n,l +1 (t )
...
·
¸
¸
¸
...
¸
rl n
¸
¸
f l +1,n (t )
¸
¸
...
¸
§ q~i
· ¸
¨
¸ ¸
¨ r out + Δf (t ) ¸rin ¸
© i
¹
¸
¸
...
¸
¸
f nn (t )
¹
r1n
113
rim > 0, m > l. : > vm. K " ! >, W = , ! (vi, vm):
·
§ q~i
q~
¸rim > i rim . m > l, > vm W = ¨
f
t
+
Δ
(
)
¸
¨ r out
riout
¹
© i
, > ! , #% 2, .. , < , . K > vm ! ! > vk (k V l), , , " , vk ! ! >, , , , , ,
" q~m . : ( ) > vi < ! q~ .
i
" , "# ; < > vi " > vk (k V l). K <
, 2, J, " J – 1 . , > , < , t → " W = : q~i1 ,
q~ , …, q~ . " , W > T, " i2
iJ
<
. ; ! , . & % " ! W > T " % " W = . * 4.1 .
; 4.1. ' 0 " %+ " "$ W > T $ 0 ).
; 4.2. ' 0 " %+ " "$ W > T " *
$ " % : f sum
= .
; 4.3. ' 0 " 0
%+ W > T $ $ 0 0 W = T:
(
F in* = F out*
114
)
T
~
= Q = (r1out ,..., rlout , q~l +1 ,..., q~n ) .
; 4.4. ' 0 " # 000 $& $ , $ 0 W = T ,
& &: + : q~k = rkout .
; 4.5. ' 0 " # &, %) W = T , $# &: + : q~ j < r jout , %+ W > T % $
0
, & q~ , $ 0 0 j
+ Δql* , q~l +1 ,..., q~n ) , Δqi* ≥ 0 – # : Q =
, & $& *
(r1out
+ Δq1* ,..., rlout
l
.
¦ Δqi* = W − T .
i =1
; 4.6. ' 0 " ,
%) , %+ W > T $ 0 "$ 0 .
; 4.7. ' $ ", # &
00%0 $& l = n, $& -
(
)
T
) F in* = F out* = (r1out ,..., rnout ) .
K ! " , . : % " . : > !" W > T.
G A – A , !
" . > ,
,
" !
" n
r min = ¦ r jmin , " (
r jmin = min r jin , r jout
)
> vj -
j =1
U !.
G 4.1 , *
" > " : f sum
= < rmin, ! %, %
< rsum. $ " , > 115
! ! , ( ) " > rmin
" rsum.
4.2. #+% %+, %%904 ) $+
%)% &%!+ #* <
N! % ! R % R':
§ r11
¨
¨r
R = ¨ 21
...
¨
¨r
© n1
r12
r22
...
rn 2
§ r11
... r1n ·
¨ out
¸
¨ r1
... r2 n ¸
¨ ...
,
=
R
'
... ... ¸
¨ rn1
¸
¸
¨¨ out
... rnn ¹
© rn
r12
r1out
...
rn 2
rnout
r1n
r1out
... ...
rnn
... out
rn
...
-
·
¸
¸
¸.
¸
¸¸
¹
" , % R' ! % ! R,
% % R R'.
, % i- % R ! ! i- > .
G R' > < % ! . % ! , % R', ! [R']; [Ri'] , i = 1, 2,… – < % ! .
Rk ~ Rm ⇔ Rk ∈[Ri'] & Rm ∈[Ri'].
G % ! , % Ri', ! " .
#% %
! , %. D ! % ! Rm " [Ri'], " 116
, "" , !
> ! % .
/ %, % 3×3:
§1 1 1·
¸
¨
¨3 3 3¸
1 4 5¸
Ri' = ¨
.
¨ 10 10 10 ¸
¨1 1 1¸
¸
¨
©3 3 3¹
4.2. & % ! :
§1 1 1·
¨
¸
R1 = ¨ 1 4 5 ¸ .
¨ 4 4 4¸
©
¹
(4.4)
W = 1. Q(0) = (1, 0, 0).
t
0
1
2
3
4
…
+ 4.1
A%/%'%* A"E%/@ (:/)
v1
v2
v3
1,000
0,333
0,256
0,250
0,250
0,000
0,333
0,356
0,357
0,357
0,000
0,333
0,389
0,393
0,393
Q1* % Ri'∞, Q1*:
∞
Ri'
§ 0,25 0,357 0,393·
¨
¸
= ¨ 0,25 0,357 0,393¸
¨ 0,25 0,357 0,393¸
©
¹
ρ (# (2.11)) < ! ρ = ((6, 3), (9, 10), (10, 12)}. ! :
117
. G # (3.8) , W ! > % , W " "" . -
!
, . 3.2–3.3 " ", " > , ! . / " % , " W = 12. : #% W = 12.
4.3. ; % ! (4.4);
Q(0) = (12, 0, 0).
+ 4.2
A%/%'%* A"E%/@ (:/)
t
v1
v2
v3
0
1
2
3
4
…
38
39
40
…
12,000
10,000
8,433
7,218
6,274
0,000
1,000
1,733
2,304
2,748
0,000
1,000
1,833
2,478
2,978
3,001
3,000
3,000
4,285
4,286
4,286
4,714
4,714
4,714
, > - " r1out , " 3.
> #% 1, .. ! !, > !. ;,
, " W = 100 < ( ) ! : Q* = (91, 4,286, 4,714).
; ! , " :
≈ 3 + 4,286 + 4,714 =12.
4.2.1. * <" % :/G+ ")$+ '*'9%!+
%'%:%/0 *G - %& !
G " %
R2 " [Ri'], R1 % 118
! " .
4.4. ; % R2 % (4.4) :
§1 1 1·
¨
¸
R2 = ¨ 1 4 5 ¸ .
¨ 20 20 20 ¸
©
¹
W = 1. Q(0) = (1, 0, 0).
! " " W = 1
4.2.
t
0
1
2
3
4
…
+ 4.3
A%/%'%* A"E%/@ (:/)
v1
v2
v3
1,000
0,333
0,256
0,250
0,250
0,000
0,333
0,356
0,357
0,357
0,000
0,333
0,389
0,393
0,393
" T 12.
;
! , , [Ri'] % ! : 1) " Q1* , W = 1,
,
W > T.
! , 2) " " >
> !"
<" 4.3.
" -
4.2.2. * <" % :/G+ ")$+ '*'9%!+
%'%:%/0 *G -'* # !
: % R3∈[Ri'], R3 ! >
! . <" % (4.4)
119
% !
> . $ ! ! " %, < > ! , .. ! >, .
4.5. / ! % (4.4) .
§ 2 2 2·
¨
¸
R3 = ¨ 1 4 5 ¸ .
¨ 4 4 4¸
©
¹
W = 1. Q(0) = (1, 0, 0).
ρ
: ρ = ((7, 6), (10, 10), (11, 12)). > , > .
t
0
1
2
3
4
…
+ 4.4
A%/%'%* A"E%/@ (:/)
v1
v2
v3
1,000
0,333
0,256
0,250
0,250
0,000
0,333
0,356
0,357
0,357
0,000
0,333
0,389
0,393
0,393
, 4.2, 4.4. " .
$ " , , 1,
. ! > - . < " , , !
% > .
4.6. & % R3 4.5; W = 24 ≈ . Q(0) = (24, 0, 0).
120
t
0
1
2
3
4
…
46
47
…
A%/%'%* A"E%/@ (:/)
v1
v2
v3
24,000
20,000
16,867
14,436
12,548
0,000
2,000
3,467
4,609
5,495
0,000
2,000
3,667
4,956
5,956
6,000
6,000
8,572
8,572
9,428
9,428
+ 4.5
= 6 + 8,572 + 9,428 = 24
" . ", > 4 " W = T : 3 ⋅ 2 = 6; 4,286 ⋅ 2 = 8,572;
4,714 ⋅ 2 = 9,428.
4.2.3. # %/0 *G "-'* # !
& % R4∈[Ri'] % R1 (4.4), ! .
4.7. " > .
-
§ 3 3 3·
¨
¸
R = ¨1 4 5¸ .
¨ 4 4 4¸
©
¹
W = 1. Q(0) = (1, 0, 0).
# "% ρ ! :
ρ =((8, 9), (11, 10), (12, 12)).
/ > . , , – > .
G <" , Q1* %, % 121
! , ,
.
> , -
+ 4.6
A%/%'%* A"E%/@ -A) (B&&"A,%& A:(BA(:, A"C,%& 1
t
v1
v2
v3
0
1
2
3
4
…
1,000
0,333
0,256
0,250
0,250
0,000
0,333
0,356
0,357
0,357
0,000
0,333
0,389
0,393
0,393
G < "
.
b W = 28 ! % " .
:, 28, " # "% .
4.8. ? + R4; & W = 28 ≈ . Q(0) = (28, 0, 0).
t
0
1
2
3
4
5
6
7
8
9
…
+ 4.7
A%/%'%* A"E%/@ (:/)
v1
v2
v3
28,000
22,000
17,300
13,653
10,822
8,625
7,169
7,011
7,001
7,000
0,000
3,000
5,200
6,913
8,243
9,275
9,952
9,997
10,000
10,000
0,000
3,000
5,500
7,433
8,934
10,100
10,879
10,992
10,999
11,000
> " > 2 ( ). ;, , 100, ! :
Q*=(7,000, 82,000, 11,000).
" , ! !122
, – , 10, – > % Q1* .
~
/ " W = : Q = (q~1 , q~2 , q~3 ) . ;" :
q~1 = 0,250T;
q~ = 0.357T =10;
2
q~3 =0,393T;
G " > : T ≈ 28. ;" : q~1 = 7; q~3 = 11.
# , !! .
4.3.
!* Q1*
'* 6& 4.2. ' [Ri'] W = 1 % $ 0 0 Q1*.
$ . ; % Ri' , (Ri')∞ . (Ri')∞ = 1⋅Q1*, " 1 – - !%, n %, Q1* – . ; . ! % Rm [Ri'] ! # (2.7) Q*,
W, W ≤ T.
# , " ! Q1*. : . * > ! , % > , 1.
4.3. ' "
0 , +%) & 1:
~
1) & Q = (q~1 , q~2 ,..., q~n ) , %) W = , :00 :
r out
q~i = 1 1* qi1* ;
q1
2) " : 0 :
123
T=
r1out
.
q11*
(4.5)
$ . W ≤ T " % Q1*. " " , > , ! . , > ( 4.4 4.1). 1.
;" q~1 = q11*T = r1out ,
(4.6)
q~i = qi1*T < riout , (i > 1).
G (4.6) : T =
r1out 1*
r1out ~
;
q
=
qi i
q11*
q11*
| " . A (4.5) !^ 4.5: % !
% .
4.4. +% *! %
%%( %
'*$/
. 4.4 4.1, > % , W = T , ! . $ < , "
" ".
# !
", > , .
4.4 ( + :). ' # vj "
0 000 , r out
$ j = arg min i 1* .
i∈{1,...,n} q
i
124
$ . > > :
r out
Ti = i 1* , i = 1, …, n.
qi
, , > , ! ,
:
q~
r out
q~k = rkout . " , q~k = q1k*T , T = 1k* = k 1* .
qk
qk
;" , , > : k = .
" , " , #% 2, , > , ! .
; ! , W = T:
q~k = rkout ;
q~i ≤ riout , i S k.
> (4.7) (4.7)
: qi1*T ≤ riout ,
T≤
riout
qi1*
riout
.
i∈{1,...,n} q1*
i
K <
" > , % .
! : > , q~ j = r jout , (i = 1, …, n), <
r jout
, T = min
riout
.
i∈{1,...,n} q1*
q1j*
i
> vl – % , r out r out
" T = min i 1* = l 1* . > vk – % i∈{1,...,n} q
ql
i
= T = min
125
. ;" : q~k = rkout . * rkout
.
q1k*
W = T > #% r out
2, > vk ! : q~k = Tq1k* = l 1* q1k* .
ql
" , ! , q~ = r out . ;" :
> k
rkout =
out
rl
q1*
1* k
ql
rkout
q1k*
=
k
out
rl
=T .
ql1*
| " . * , " # r out
T = min i 1* .
i∈{1,...,n} q
i
% % R (R')∞, " " " , >" .
/ " ! " 2.
$!!
, " , .
4.5 ( $ 0 ). |" 0 $ 0 0 Q* = q1* , ..., qn* 0 " 0 " & W 0%0 %) :
r out
1. W ≤ , T = min i 1* :
i∈{1, ..., n} q
i
(
qi* = qi1* ⋅ W
2. W > 0 :
(4.8)
: - : " &-
qi* = qi1* ⋅ T , i ≠ jk,
126
)
(4.9)
jk – . jk 0%0 0:
r out
jk = arg min i 1* .
i∈{1, ..., n} q
i
}# 0 00 $&
.
; ! , Q* :
1) W;
2) ! % % R'∞ ;
3) !
out
% ri .
W ≤ . W > > % > . " " . / - # qi* = q~i = qi1* ⋅ T .
| " . , "" " Q*
Q1*, .. < % R. $ # (4.8)–(4.9) !" , % " ,
% " % ! R. D , , Q1* = (q11* , q12* ,..., q1n* ) , !" ! , %
!" " .
= " # .
4.6 ( $ ). ' 0 " $& " %+ " W:
r out
1. W ≤ f iin* = f i out* = qi1*W , " T = min i 1* ;
i∈{1,...,n} q
i
2. W > f iin* = f iout* = qi1*T .
; , " W. I 127
– % , , " W – r jout
" ; k
> , .
4.5. %* #* <" R % '* 6/"#
! &%# *!* ' 6$+ #* < R'
! >, % Ri' % ! [Ri'].
" % ! R vj – . K % R', [R'].
" :
r jout
T = 1* . : > vk, . qj
% ! % R R', >
! > vk, , ! < . vk :
out
rkout r j
=
.
q1k*
q1j*
(4.10)
/ (4.10) rkout – , k % . $ – . $ :
rkout =
q1k* out
rj .
q1j*
(4.11)
| " . $! ! : ! > vj ( ), " <##, vj
< .
128
! k- % R (4.11), vj vk > :
rkout q1k*
=
,
r jout q1j*
(4.12)
.. ! > - " (
!
) .
4.9. : , ! % . > . K % ! % :
§1
¨
¨1
R = ¨6
¨
¨1
¨1
©
1
1
1
6
1
1
1
1
1
1
1
1
1
1
1
1·
§ 0,2
¸
¨
1¸
¨ 0,2
¸
1 , R' = ¨ 0,6
¸
¨
1¸
¨ 0,1
¸
¨ 0,2
1¹
©
0,2 0,2 0,2 0,2 ·
¸
0,2 0,2 0,2 0,2 ¸
0,1 0,1 0,1 0,1 ¸ .
¸
0,6 0,1 0,1 0,1 ¸
0,2 0,2 0,2 0,2 ¸¹
/ < W = 1 ! :
§1 1 1 1 1·
Q = ¨ , , , , ¸ , ρ = ((10, 5), (10, 5), (5, 10), (5, 10), (5, 5)).
©4 4 6 6 6¹
T = 20.
: (4.10) – (4.12) > v1 v2, % R. $ , Q1* " ! , , ! > ( [R']), ! ", – , < ! > v5. <"
> ! , !
(4.11).
1*
r5out =
1 1
10
.
: ⋅5 =
6 4
3
129
;
! , , %:
§1
¨
¨1
¨
R =¨6
¨1
¨2
¨
©3
1
1
1
1
1
1
1
6
2
3
1
1
2
3
1
1
2
3
1·
¸
1¸
1¸
¸
1¸
2¸
¸
3¹
% . < , " , > - v3 v4 <## % % , % (4.13)
% .
§1
¨
¨1
¨2
¨
R=¨
1
¨
¨3
¨¨ 2
©3
:
rkout
q1k*
1
1
1
3
2
2
3
1
1
1
3
1
3
2
3
, % (4.13)
= 20 = T , 1
1
1
3
1
3
2
3
1·
¸
1¸
1¸
3¸
1¸
¸
3¸
2¸
¸
3¹
(4.13)
! > -
. / ". I!
! ! % , % 3:
§3
¨
¨3
R =¨6
¨
¨1
¨2
©
130
3
3
1
6
2
3
3
1
1
2
3
3
1
1
2
3·
¸
3¸
1¸ .
¸
1¸
2 ¸¹
% ! . K ρ :
% -
ρ = ((15, 15), (15, 15), (10, 10), (10, 10), (10, 10)).
% (4.13) < ! , ,
< , > :
§
§ 10 10 · § 10 10 · § 10 10 · ·
ρ = ¨¨ (5,5), (5,5), ¨ , ¸, ¨ , ¸, ¨ , ¸ ¸¸ .
© 3 3 ¹ © 3 3 ¹ © 3 3 ¹¹
©
/ , > , ! . (G 3.1 ! Z–(t) " ! > , % , ). , ! . < > "
! ( " ).
, > ! ! , ! (. 2.4), < . G " .
4.6. <! & % %+ % $ %"#
*!*#
: % [R']n×n, % R'. % R ∈ [R']
vj. # < %, k ≠ j. J % < , ! rkout
ª q1*
·
!" « 1k* r jout , ∞ ¸ . G # ¸
¬« q j
¹
(4.11) , rkout <
q1k* out
r j > vj ! q1j*
q1k* out
r j vj – q1j*
, ( % " ) ! .
. !
rkout >
131
; ! , # j % R, , % !"
" > -, ! > vj,
vj. ,
n −1
# > vj ¦ Cni −1 = 2 n −1
%,
i =1
! % % . $ ,
" .
:" #% ! !
! !^ .
4.7. ( %/ % $ # '* # !#
$ # %& !#
G , ! " > ! . $ .
: n > ! r. : , [41], ! " . b ! " ! α (α > 1). "# , ! ! ! . < !
(v2, v1). ;" r21 = αr, rij = r (i ≠ 2, j ≠ 1), > v2 .
n = 5 . 4.7.
/ > , > - ,
n
riout = ¦ rij = rn . !
j =1
n
! : r1out = ¦ rij = r ( n + α − 1) .
j =1
((
)(
) (
))
ρ = r1in ; r1out , r2in ; r2out ,... rnin ; rnout
< ! : ρ = ((r(n + α – 1), rn), (rn, r(n + α – 1)), (rn, rn),…(rn, rn)).
132
r
"
v2
αr
v1
r
r
r
r
r
r
r
r
r
v3
r r
r r
r
r
r
r
r r
r
r
v5
: . 4.7. r
v4
r
; 4.7 ! > , ! ( " <" !
).
4.7. / !
r21 = αr, rij = r (i ≠ 2, j ≠ 1), " α > 1:
1) T = rn
n 2 + (n + 1)(α − 1)
,
n + 2(α − 1)
2)
!" " Q(0) = (q1(0), q2(0),
…, qn(0)) W > T :
Q*=(W – (n – 1)q*, q*,… q*),
" q * = rn
n + (α − 1)
.
n + 2(α − 1)
/ " 4.8.
4.8. / !
r21 = αr, rij = r (i ≠ 2, j ≠ 1), α > 1, !" " 133
Q(0) = (q1(0), q2(0), …, qn(0)) W ≤ T % , Q*=(q1*, q*,…, q*) # :
n + (α − 1)
q* = W 2
;
(4.14)
n + (n + 1)(α − 1)
q1* = W
n + 2(α − 1)
.
n + (n + 1)(α − 1)
2
; . W = T ! Q*=(rn, q*,…, q*), " q* # (4.14).
/ 4.7 4.8 # " ! ! " " : > > - .
< " ! [27].
##* + ! 4
/ " 4 " ! > , .. W > T. / . <" , # ! " :
• " "
, , ! "
" ;
• # ( 4.3);
• ( 4.4);
• # - ;
• # " .
$ < " W = T, .. ~
Q = ( q~1 ,..., q~n ) , # <" . , 134
, W > T >
W – T ; ", – , W = T.
K ( . 4.6 ,
< ! " ), ,
> , "
" . K , > . $ , <" >, ! " 8. , , " > – . * ! , , ,
, .. > ! .
! " . , > < , ! , !
.
135
5. @A K B,
:
• " & $ , &# & .
• ) &, !# 6 W = 1
W = T $ R.
• ( &# & & .
• & &
Z+(t), , !# Z+(0), Z+*, ..
Z+(0) = Z+*.
• % &, # & & .
• W > T , Z–(0) +
! & ,
Z (t) , . " ! $ R .
• W > T , # &
& Z+(t), , !# &# , !# Z–(0).
• < & , # & & W > T.
• & &
$ W > T.
< > riin = riout , , , > . K <
% ! , < , , . K
< % ! , . D !^ , ! !
, ! > .
D , " 2, " ! <" (" % ). / " ! " < .
/ , , ! > . / " ,
, , , " , <"
.
136
5.1. 94% '*$/ %%( (
'* 6& " W ≤ " . < < . * " #% < > "" .
5.1. ' 0 D " + : = rsum.
$ . $ , V rsum. , " < rsum. ;" W: < W < rsum, , W, > ! 1,
.. : qi* > riout , , riout . = > , > vi riout . ; < riin = riout , ! > vi ! . ;
! , > , > !: r jout . . ! > , , > ,
> ! . $ n
, ! !
>
¦ r jout
j =1
= rsum , . , = rsum. ; 5.1. $ & & , " 0 0 D & 0:
~
~
Q = (r1out , ..., rnout ) , Q – $ 0 0 W = .
; 5.2. vi ∈ Z–(0), ∀ t > 0 vi ∈ Z–(t). (*
", " Z+(0) 0 # 0$0.)
$ . / > < > , " 137
Z–(t). > vi ∈ Z–(t) "
riout , > ",
, ! > <" , . ; 5.3. ' 0 D 0 %+ W $ 0 ).
$ . " W V T ", " . G 5.1
, W = ~
Q = (r1out , ..., rnout ) . W > ~
~
! Q : Fin* = Fout* = Q . <" " "
( 4.1 ). , . ; 5.4. & 0 D +$# " 0: W > , 0 # & & Z–(0) $ 0
& 0:
*
out
qk = rk .
$ <" . > W > , ! >,
W = . = W = 5.1 ~
Q = (r1out ,..., rnout ) . ;
! , qk* ≥ rkout . * > –
! , Z (0) " Z+(t), !. $ qk* = rkout > Z–(0). 5.2. +% E+*") %+
/ < > ! !: > " ! ; > .
G #% < ! > # .
/ " 4 (% .4.5) ! , > % , < .
5.1 ! , # 5.1.
138
; 5.1. N! > " < % : W > rsum , " Z+*.
; 5.2. / " < &: > Z+(t), .
; 5.3. W > rsum > , Z+(0) " Z–(t) , ,
Z*–.
5.3 .
5.1. : > ( " !! ! > ). %
! :
§ 1 50 1 1 1 ·
¸
¨
¨ 1 2 50 1 1 ¸
R = ¨ 1 1 3 50 1 ¸ .
¸
¨
¨ 1 1 1 4 50 ¸
¨ 50 1 1 1 5 ¸
¹
©
(5.1)
ρ=((54,54), (55, 55), (56, 56), (57, 57), (58,58)).
rsum = 280.
* : Q(0) = (205, 80, 0, 0, 0). W = 285.
: Q* = (54, 60, 56, 57, 58).
. 5.1.
/ Z+(0) > . G , > " ! > , ,
Z–(t) , r1out = 54 . / > Z+(t), > W – T = 5 .
, > " "
" Q(0) ! Z+(t), ! 5.5.
139
: . 5.1. > Z+(t) Z–(t)
5.3. *$/ %%( % '* $ &#
## *%9*%
" ! ! " . $ %
< #% " ! ! . I " , " , . * , , " .
5.2. : #% , % (5.1) , > "" : W = 50.
Q(0) = (50, 0, 0, 0, 0).
G . 5.2 , , " > !" , ! . ! " % !.
: . $ ! .
140
: . 5.2. ! 5.3. / % (5.1) ! , 50, " " . W = 50.
Q(0) = (50, 0, 0, 0, 0).
§ 1 50 1 1 50 ·
¸
¨
¨ 50 2 50 1 1 ¸
R = ¨ 1 50 3 50 1 ¸ .
¸
¨
¨ 1 1 50 4 50 ¸
¨ 50 1 1 50 5 ¸
¹
©
(5.2)
& % , ! .
" , " ! . <" , , ! < , 50, % (5.2).
* " < . #% . / t W = 1,
, ! ! Q1(t), – Q1*.
Q1* % 141
! % R', % ! R.
: . 5.3. ! < ! .
5.11. ' 0 D $ 0 0 W = 1 :
§ r out r out
r out ·
Q1* = ¨¨ 1 , 2 ,..., n ¸¸ .
rsum ¹
© rsum rsum
(5.3)
$ . ; > < % , r out
: i 1* = T . * 5.1, = rsum.
qi
$ , ∀i qi1* =
riout
, " rsum
# (5.3). 1
D > ( ., , [103]), "" " .
142
| " . ,
1·
§1 1
" < , : Q1*= ¨ , ,..., ¸ ,
n
n
n¹
©
" , " 2.
!" W ` T % R W:
§ r out
r out ·
r out
Q* = ¨¨ 1 W , 2 W ,..., n W ¸¸ .
rsum
rsum ¹
© rsum
5.4. 9!< * %
'* :/G ) *%9*%)
W = T , ! >,
~
Q = (r1out , r2out ,..., rnout ) .
W > T Z+(0) m
> . * > ! , , < > 1 m, m < n ( m = n, ). ;" " Q(0) :
(
)
Q(0) = r1out + c1 (0),..., rmout + cm (0), rmout+1 − d m +1 (0),..., rnout − d n (0) .
I 1(0), …, cm(0) > 0 – > > "" ( ! ) – # % . dm+1(0), …, dn(0) Y 0 – # % > .
$!
# % csum(0),
# % – dsum(0).
m
csum (0) = ¦ c j (0) , d sum (0) =
j =1
* ! t Y 0 # % :
n
¦ d j (0) .
j = m +1
# % >-
csum(t) – dsum(t) = const = W – rsum.
143
4.5 4.1, " W > :
(
)
Q* = r1out + c1* ,..., rmout + cm* , rmout+1 ,..., rnout , " c1* ,..., cm* ≥ 0 . (5.4)
5.1, % W > Z+(t) . < ci* " ! . ;" . # % > vk, ! " Z+? G ", *
*
c
,...,
c
, 1
m (# (5.4)) ! " ? &
! #% , < ! ? $ ! .
5.5. *$/" %%( (
'* :/G ) *%9*%)
'!
5.5.1. *$/ %%( %
'* 6#+ 6 Z+(t)
" ! # , !" "
W > rsum , > , Z+(0), ! Z+*. J ", ! # "
> .
, > % ,
". * . b . " < .
- Z+(0) > 1 m:
(
)
out
Q(0) = r1out + c1 (0),..., rmout + cm (0), rmout
− d n (0) .
+1 − d m+1 (0),..., rn
144
# ! .
# % > Z+(0), ! % #% Z–(t)? G
, ! Z+(0) > ! r1out , ..., rmout
(# % > ), ! > v1, …, vm Z+(t), W = T = rsum?
$!
# % > m
m
~
~
c1 , ..., cm ( < ", % % ! " ). / " " " ! :
(
)
Q(0) = r1out + c~1m ,..., rmout + c~mm1 , rmout+1 − d m +1 (0),..., rnout − d n (0) .
P, , W = T, > :
m
¦ c~jm = d sum (0) .
(5.5)
j =1
$ , " , > Z+(0) , ! ci (0) > c~im (i V m) # % < > ! c* = c (0) − c~ m .
i
G < ! c~1m , ..., c~mm
c~im
=
,
qi (0) − qi*
i
i
:
! > qi(0). / qi (0) − qi* = riout + ci (0) − (riout + ci* ) = ci (0) − ci* = c~im .
| " . , - ! > Z+(0) : ck (0) = c~km + Δck (0) (k V m), > , Z+(0), c~ m i
. / , < > " #% 1, .. ! 145
" , " .
G > > .
# "
< ! > .
* Q(0), Z+(0) m
> , Qm(0) = Q(0) +
+ (dsum(0), …, dsum(0), 0, …, 0), " m dsum(0).
9 5.1. $ 0 D W > T
"$ 0 Z+(0) m # 1, …, m:
(
)
out
Q(0) = r1out + c1 (0), ..., rmout + cm (0), rmout
− d n (0) ,
+1 − d m+1 (0), ..., rn
0 , "+& 0 : $ Z+(t)
t > 0 0 # vi, i = 1, …, m, +: ", "+& &0$:
ci (0) ≥ c~im ,
(5.6)
~
c~im – & C m :
~
C m = Qm (0) − Qm* .
(5.7)
$ .
b ", > , Z+(0), , , Z+(t) % , ..
ci (t ) ≥ c~im t Y 0 vi, i = 1, …, m.
I! ! <", > Z+(0) ! ! > , # % . / >
> , c~im , Qm (0) − Qm* , < ! " > vi ∈ Z+(0), c~im , , < > #% . ; 5.4. ' D W > T 0 "$ 0 0 Q(0) : vi ∈ Z+* ci (0) ≥ c~im , ~
C m : 0 (5.7). }+ , $ # & Z+(0) 0% % (5.6).
146
; 5.5. D "$& 0 (
out
Q(0) = r1out + c1 (0), ..., rmout + cm (0), rmout
− d n (0)
+1 − d m+1 (0), ..., rn
)
W > T 0 : # vi ∈ Z+(0) &00 ci (0) ≥ c~im , $ 0 0 (
)
Q* = r1out + c1 (0) − c~1m ,..., rmout + cm (0) − c~mm , rmout+1 ,..., rnout =
~
= Q ( 0) − C m ,
(5.8)
~
C m 00 (5.7).
~
;
! , C m (5.6) > Z+(0).
5.1.
", , (n – m) ~
C m % # % .
n
¦ c~jm = −d sum (0) .
j =m+1
< # # (5.5),
:
n
¦ c~jm = 0 .
j =1
$!!
# % ( ! " ck(0) Y c~km ). / < Z+(t) ! %
#% . : % > " "
.
147
5.5.2. *$/ %%( E+*+ % .
:4 + %9&+
% > , m > c (0)
! > β i = ~i m .
c
i
K βi ≥ 1, i =1, …, m, # (5.8). l: 1 < l < m, ,
c (0)
β i = ~i m < 1 , j = l+1, …, m.
(5.9)
ci
< > c j (0) < c~jm ,
,
+
! Z (t), Z–(t). * > vl+1, …, vm m
# % , ¦ (c~jm − c j (0)) ,
-
j = l +1
> Z+(0). < (5.9) > vi ∈ Z+(0) > (5.6), < " ", Z+*: % .
Z+(t) > m, > (5.9)
. K
> m – k, m – k + 1, …, m > (5.9) , Z+(t) . $! l = m – k –1 Z+(t). ; > vi ∈ Z+(0), i ≤ l, !
# , ! vi ∈ Z+*, !
~
C l . / " "
! :
(
Ql (0) = r1out + c1 (0) + d sum (0),..., rlout + cl (0) + d sum (0), ql +1 (0),..., qn (0)
)
; # % dsum(0) ! l > –
, (5.6).
, c~il > c~im , i = 1 ,…, l, , " (5.6): c (0) ≥ c~ l , i = 1 ,…, l.
i
148
i
% , - ! p V l ci (0) ≥ c~i p , ! i = 1 ,…, p.
<" .
(
)
out
Q * = r1out + c1 (0) − c~1p ,..., rpout + c p (0) − c~pp , rpout
.
+1 ,..., rn
*>%A)/& ,"J%H#:,)G -A:#:*+,%>% (%(/%G,)G
~
0. / C m # (5.7).
1. / ci (0) − c~im , i = 1, …, m. K % , (# (5.8)).
/ .
c (0)
2. b > ! β i = ~i m .
c
i
3. m := m – k –1, " k – c (0)
β i = min ~j m ; >" 0.
j
c
> ,
j
W > T, > Z+*, " >", > m – 1. / (
out
Q* = r1out + c1 (0) − c~1m ,..., rmout + cm (0) − c~mm , rmout
+1 ,..., rn
)
! .
# , !! 2.2, ! .
5.2. $ 0 D W > T
c $& "$& 0 (
out
Q(0) = r1out + c1 (0),..., rmout + cm (0), rmout
− d n (0)
+1 − d m+1 (0),..., rn
)
c (0)
& m # 0"& +& % " 0 β i = ~i m .
ci
$ 0 : 0 :
(
)
out
Q * = r1out + c1 (0) − c~1p ,..., rpout + c p (0) − c~pp , rpout
,
+1 ,..., rn
149
~
C p = Q p (0) − Q*p
p = m, ci (0) ≥ c~i , i = 1, … m;
" p < m – +$# , , "
ci (0) ≥ c~i p , i = 1, … p.
" " .
5.5.3. *$/" '! '* :/G ) *%9*%)
6$&
~m
)-$ ( !* C – %9&+ %9% ( *%9*%
6 Z–(0)
; 5.2 , <" !
, .
~
/ , C m ,
!" Qm* . " +
> vk ∈ Z (0) (k ≤ m) Z+(t). D
> #% 1. * ! rkout .
# :
f kin (t + 1) = (r1out ,..., rmout , qm+1 (t ),..., qn (t ))Rk' ,
" R k' – k- !% % R'.
/ > , F in (t + 1) = (r1out ,..., rmout , qm+1 (t ),..., qn (t ))R' .
/, %, " . $ Fin(t+1) = Fout(t)R'.
/
" .
150
(5.10)
-
F out (t + 1) = (r1out ,..., rmout , qm +1 (t + 1),..., qn (t + 1)) =
= (r1out ,..., rmout , qm+1 (t ),..., qn (t ))P = F out (t ) P ,
" P–
!%
§E
P = ¨¨ m
©O
%, R' m !% " ! e1, …, em. K ! %:
R1' ·
¸ , " Em – % m×m, O – R2' ¸¹
% (n–m)×m, R1', 2 – ! % m×(n–m) (n–m)× (n–m) . J R1'
! ! Z+(t)
Z–(t), ! R2' ! Z–(t).
!" " h :
Fout(t+h) = Fout(t)Ph.
Fout(t) = Fout(0)Pt.
;" (5.10) > :
Fin(t+1) = Fout(0)PtR'.
* % P t, §E
P =¨ m
¨O
©
t
( ) + ... + R (R )
(R )
R1' + R1' R2' + R1' R2'
2
'
1
' t
2
/ [38, 52] , % " (
t → ∞, , ", % E2 −
¦(
∞
% (n–m)×(n–m),
k =0
( )
2
( )
R1' + R1' R2' + R1' R2' + ... + R1' R2'
k
' t −1 ·
2
¸
¸
¹
( )
R2'
)
−1
R2' ,
) = (E
k
R2'
(
:
2
−
t
→ 0 " 2 – )
−1
R2' .
+ ... = R1' E2 − R2'
)
−1
-
;" ,
151
% P∞
:
(
§E
P =¨ m
¨O
©
R1' E2 − R2'
O
∞
G (5.11)
)
−1 ·
¸.
¸
¹
(5.11)
" :
§E
Fin* = Fout* = Fout(0)P∞R' = Fout(0) ¨ m
¨O
©
(
R1' E2 − R2'
O
)
−1 ·
¸ R'.
¸
¹
(5.12)
G # (5.12) , " m . «A » . W > T " T, " . m Fout(0) f i out = riout , i = 1, …, m, .. . " .
W > T ~
W = T: Q = (r1out ,..., rnout ) , (5.12)
:
−1
§E
R1' E2 − R2' ·¸
R' = (r1out ,..., rnout ) .
F out (0)¨ m
¨O
¸
O
©
¹
(
)
~
/ Q = (r1out ,..., rnout ) ! % R' ! λ = 1. ", λ = 1 – ! 1, ! . , −1
§E
R1' E2 − R2' ·¸
(5.13)
F out (0)¨ m
= (r1out ,..., rnout ) .
¨O
¸
O
©
¹
(
)
< , ! , Fout(0) (5.13) m .
152
;
! , (5.13) (
§E
(r1out ,..., rmout ,0,...,0)¨ m
¨O
©
$ R1' E2 − R2'
O
)
:
−1 ·
¸ = (r1out ,..., rnout ) .
¸
¹
< :
(
(r1out ,..., rmout )§¨ R1' E2 − R2'
©
(
)
−1 ·
out
out
¸ = (rm+1 ,..., rn ) .
¹
(5.14)
)
§¨ R ' E − R ' −1 ·¸ – % m 2
© 1 2
¹
! (n – m) . $!
R, !Rnout
m (n – m) Rmout
−m .
;" (5.14) > :
Rmout R c = Rnout
−m .
/ i- Rmout (i = 1, … m) out
Rnout
−m ri
n−m
¦ rijc
– !
j =1
i- > < i- % R.
> , Z–(0), ,
.. d sum (0) =
n
¦ r jout . ;"
> vi (i = 1, … m)
j =m+1
riout
n−m
¦ rijc
. $ m -
j =1
~
C m = Qm (0) − Qm* , ! , > Z+(t), # :
c~im = riout
n−m
¦ rijc , i =1, …, m.
j =1
153
;
! , , , " > Z–(0) .
9 5.2. ' 0 D c "$& 0 Q(0) = r1out + c1 (0), ..., rmout + cm (0), 0, ...,0
(
)
&: W > T $& " 0 c~im , i =1, …, m, +
–
# & & Z (0) Z (t), " &%0 :
n−m
−1
c~im = riout ¦ rijc , i =1, …, m, " R c = §¨ R1' E2 − R2' ·¸ , (5.15)
©
¹
j =1
(
$ 0 &0 :
)
, " ci(0) ≥ c~im ∀ i =1, …, m,
(
)
out
Q* = r1out + c1 (0) − c~1m ,..., rmout + cm (0) − c~mm , rmout
.
+1 , ..., rn
$ % P∞.
A (5.15) % , " , < ! .. 5.5.1–5.5.2.
c~im .
5.4. : > . K % ! :
§1
¨
¨1
R = ¨6
¨
¨1
¨1
©
1
1
1
1
1
6
1
1
1
1
1
1
1
1
1
1·
¸
1¸
1¸ .
¸
1¸
1¸¹
ρ = ((10, 10), (5, 5), (10, 10), (5, 5), (5,5)). rsum = 35.
* : Q(0) = (50, 50, 0, 0, 0). W = 100.
154
Z+(0) = {v1, v2}, Z–(0) = {v3, v4,
v5}. : > , ! > Z–(t).
I < , ! : Q* = (36, 44, 10, 5, 5).
; ! , c~1m = 50 – 36 = 14, c~2m = 50 – 44 = 6.
< , 5.2.
% R'
% P :
§ 0.1
¨
¨ 0.2
R' = ¨ 0.6
¨
¨ 0.2
¨ 0.2
©
0.1
0.2
0.1
0.2
0.2
0.6
0.2
0.1
0.2
0.2
0.1
0.2
0.1
0.2
0.2
0.1 ·
¸
0.2 ¸
0.1 ¸ , P =
¸
0.2 ¸
0.2 ¸¹
§1
¨
¨0
¨0
¨
¨0
¨0
©
0
1
0
0
0
0.6
0.2
0.1
0.2
0.2
0.1 ·
¸
0.2 ¸
0.1 ¸ ,
¸
0.2 ¸
0.2 ¸¹
0.1
0.2
0.1
0.2
0.2
$ § 0.1 0.1 0.1 ·
¸
§ 0.6 0.1 0.1 · ' ¨
¸¸ , R2 = ¨ 0.2 0.2 0.2 ¸ .
R = ¨¨
© 0.2 0.2 0.2 ¹
¨ 0.2 0.2 0.2 ¸
©
¹
'
1
:
§6
¨
§ 0.9 − 0.1 − 0.1 ·
¨5
¨
¸
2
'
' −1
E2 − R2 = ¨ − 0.2 0.8 − 0.2 ¸ , E2 − R2 = ¨
¨5
¨ − 0.2 − 0.2 0.8 ¸
¨2
©
¹
¨
©5
(
R =
c
§¨ R '
1
©
(E
2
−
)
)
1
5
7
5
2
5
1·
¸
5¸
2¸
,
5¸
7¸
¸
5¹
§4 3 3 ·
¨
¸
= ¨ 5 10 10 ¸ .
¹ ¨2 2 2 ¸
¨
¸
©5 5 5 ¹
−1
R2' ·¸
# (5.14):
155
§4 3 3 ·
¨
¸
10 ¸ = (10, 5, 5) . : .
(10, 5)¨ 52 10
2 2¸
¨
¨
¸
©5 5 5 ¹
# (5.15) " c~1m, 2 .
14
§4 3 3 ·
c~1m = 10 ⋅ ¨ + + ¸ = 10 ⋅ = 14 ;
10
© 5 10 10 ¹
6
§2 2 2·
c~2m = 5 ⋅ ¨ + + ¸ = 5 ⋅ = 6 .
5
5
5
5
©
¹
D .
;
! , 5.4 , W > T, > Z–(0) , > Z+(0) , # (5.15), !" !" %. ; " : # % > Z+(0) > " c~im .
" , , ! < , !! . Z–(0), % Z+(t) (. 5.5.4), (. 5.5.5).
~m
5.5.4. $& )-$ ( !* C –
%9&+ %< *+ 6" Z+(t)
: > Z–(0). " :
(
)
out
Q(0) = r1out + c1 (0),..., rmout + cm (0), rmout
− d n (0) ,
+1 − d m+1 (0),..., rn
156
> vj, j = m+1, … n, :
out
d j (0) < r j . * > Z–(0) : (5.14)
out
out
(rm+1 , ... , rn ) , > " (dm+1 (0), ..., dn (0)) . : <
" > Z–(0):
out
rm+1 − d m+1 (0), ..., rnout − d n (0) .
/ > vj ∈ Z–(0) <## % αj ≤ 1:
(
)
αj =
d j ( 0)
r jout
.
;" # % ! :
(d m+1 (0), ..., d n (0)) = (α m+1rmout+1, ...,α n rnout ) = (rmout+1,..., rnout ) Diag(α m+1, ..., α n ) ,
" Diag(α m+1, ...,α n )
–
" % < α m+1, ..., α n .
$ , ! , "
# % > , (5.14) # % ! :
(r1out ,..., rmout ) R c Diag(α m+1 , ..., α n ) = (d m+1 (0), ..., d n (0) )
(5.16)
G (5.16) " , , " > Z–(0) , (5.15) > > Z+(0) :
c~im = riout
n−m
¦ rijcα m+ j , i =1, …, m.
(5.17)
j =1
A (5.17) " " < > > Z+(0),
!
< > Z+(t) t. G, Z–(t), 157
cj(0) ≥ c~im > Z+(0) " " . K
< ,
" " ! :
(
)
~
Q* = Q(0) − C m = Q(0) − c~1m ,..., c~mm , − d m +1 (0), ...,−d n (0) ,
(
)
out
Q* = r1out + c1 (0) − c~1m ,..., rmout + cm (0) − c~mm , rmout
.
+1 , ..., rn
; ! , % Z+(t).
5.3. ' D c $& "$&
0 (
out
Q(0) = r1out + c1 (0),..., rmout + cm (0), rmout
− d n (0)
+1 − d m+1 (0),..., rn
)
W > T $& " 0 c~im , i =1, …, m, &: # & & Z+(0) Z–(t), " &%0 :
c~im = riout
α k =
n−m
¦ rijcα m+ j , i =1, …, m,
(5.18)
j =1
d k (0)
, k = m+1, …, n,
rkout
$ 0 &0 :
(
, " ci(0) ≥ c~im ∀ i = 1, …, m,
)
out
Q* = r1out + c1 (0) − c~1m ,..., rmout + cm (0) − c~mm , rmout
.
+1 , ..., rn
| " . * , " > Z (0) , αi %,
# (5.18) # (5.15).
# .
–
158
5.5. - ! % R 5.4. Z+(0) = {v1, v2}, Z–(0) = {v3, v4, v5}. $ > Z–(0)
. * :
Q(0) = (50, 50, 3, 2, 1). W = 106. ; % c
'
' −1 ·
§
R = ¨ R1 E2 − R2 ¸ , % , ©
¹
5.4, # % , , , " # % , %
Diag(α3, α4, α5).
(
)
α3 =
10 − 3 7
5−2 3
5 −1 4
, α4 =
=
= , α5 =
=
10
10
5
5
5
5
# (5.16):
§7
¨
§ 4 3 3 ·¨ 10
¨
¸
10 ¸¨ 0
(10, 5)¨ 52 10
2 2 ¸¨
¨¨
¸
© 5 5 5 ¹¨ 0
¨
©
0
3
5
0
·
0¸
¸
0 ¸ = (7, 3, 4) .
¸
4¸
¸
5¹
: .
Rc⋅Diag(α3, α4, α5) =
§7
¨
4
3
3
·¨ 10
§
¸
¨
= ¨ 5 10 10 ¸¨ 0
¨¨ 2 2 2 ¸¸¨
© 5 5 5 ¹¨ 0
¨
©
0
3
5
0
·
0¸
¸ §¨ 28 9
0 ¸ = ¨ 50 50
¸ ¨ 14 12
4 ¸ ¨© 50 50
¸
5¹
12 ·
¸
50 ¸ = §¨ 0,56 0,18 0,24 ·¸ .
16 ¸ ¨© 0,28 0,24 0,32 ¸¹
¸
50 ¹
" c~1m, 2 ! > % Rc⋅Diag(α3, α4, α5).
c~1m = 10 ⋅ (0,56 + 0,18 + 0,24 ) = 10 ⋅ 0,98 = 9,8 ;
159
c~2m = 5 ⋅ (0,28 + 0,24 + 0,32) = 5 ⋅ 0,84 = 4,2 .
" ! :
Q* = (40,2, 45,8, 10, 5, 5).
! <
! . 5.1.
t
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
…
v1
50,000
44,400
42,300
41,250
40,725
40,463
40,331
40,266
40,233
40,216
40,208
40,204
40,202
40,201
40,201
40,200
A%/%'%* A"E%/@ (:/)
v2
v3
50,000
47,900
46,850
46,325
46,063
45,931
45,866
45,833
45,816
45,808
45,804
45,802
45,801
45,801
45,800
45,800
3,000
7,900
8,950
9,475
9,738
9,869
9,934
9,967
9,984
9,992
9,996
9,998
9,999
9,999
10,000
10,000
+ 5.1
v4
2,000
2,900
3,950
4,475
4,738
4,869
4,934
4,967
4,984
4,992
4,996
4,998
4,999
4,999
5,000
5,000
v5
1,000
2,900
3,950
4,475
4,738
4,869
4,934
4,967
4,984
4,992
4,996
4,998
4,999
4,999
5,000
5,000
! %, , , " , .
~
m
5.5.5. $& )-$ ( !* C – :4 + %9&+
, !
, – " Z+(0), .. ", Z+(t) % " #% . = " " . 5.5.2, > " , ci(0) " c~im .
160
= " " ! Z+(t) ! >" .
1. : " c~im .
2. /! > Z+(0), Z+(t).
/ % , " > Z+(t) . <" " .
3. > !
> Z–(0). K > , > .
4. % > > Z+(0). ( > , < , > Z–(t) –
, " % > ).
5. % > Z–(0) .
6. c~il " " % Z+(0).
7. K
> ci (0) ≥ c~il ,
+
. & Z (0) !
5.5. /
8. K > , ci (0) < c~il , m := l; >" 1.
$ > <" " .
: > , m > c (0)
! > β i = ~i m , ci
< > " > %.
c (0)
: > vm. β m = m~ m
cm
, β m < 1 . J , " -
βi " > . : , " > , " .
βi , i = 1, …, m – 1, : β i > β m , < !" c (0), i = 1, …, m – 1, c (0) > β c~ m , .. !
i
i
m i
161
" > vi ∈Z+(0) > β m c~im , Z+(t).
> vm ! >, > vi ∈Z+(0) , β m c~im , , ! , Z+(t).
$!
% R c ⋅ Diag(α m +1 , ..., α n ) R c1 . /> vi ∈ Z+(0) > vk ∈ Z–(0) !r c1
" " n − mik . ¦ rijc1
j =1
β m c~im . ;
! , > vk ∈ Z–(0) r c1
r c1 c~ m
> vi ∈ Z+(0) n−mik β m c~im = n−mik ~im cm (0) .
c
¦ rijc1
¦ rijc1 m
j =1
j =1
G > Z+(0) > vk ∈ Z–(0), , :
m
r c1 c~ m
¦ n − mik c~im cm (0) =
i =1
¦ rijc1 m
j =1
cm (0) m rikc1 ~ m
ci .
¦
c~mm i =1 n − m c1
r
¦ ij
j =1
* " ! , ,
:
§
c (0)
cm (0) ~ m
out
Qnew (0) = ¨¨ r1out + c1 (0) − m~ m c~1m , ... , rmout
−1 + cm−1 (0) − ~ m cm−1 , rm ,
c
c
m
m
©
rimc1+1
n−m
i =1
rijc1
j =1
m
c (0)
rmout+1 − d m +1 (0) + m~ m ¦
cm
¦
m
c ( 0)
c~im ,..., rnout − d n (0) + m~ m ¦
cm i =1
.
162
·
¸
m¸
~
ci ¸
c1
¸¸
¦ rij
j =1
¹
rinc1
n−m
Qnew(0), Z+(0) ! l
> ( , m – 1 > ), c~il , " ! 5.4, ! % .
" !! .
5.4 ( $ 0 ). $
0 D c $& "$& 0 W>T
(
out
Q(0) = r1out + c1 (0),..., rmout + cm (0), rmout
− d n (0)
+1 − d m+1 (0),..., rn
)
c (0)
& m # 0"& +& % # 0 β i = ~i m .
ci
&: # & ? $& " 0 c~ m , i = 1, …, m, i
& Z+(0) Z–(t), " &%0 :
c~im = riout
n−m
¦ rijcα m+ j , i = 1, …, m, "
j =1
$ 0 &0 :
αm+ j =
d m + j ( 0)
rmout+ j
, j = 1, …, n – m,
, " ci(0) ≥ c~im ∀ i = 1, …, m,
(
)
out
Q* = r1out + c1 (0) − c~1m ,..., rmout + cm (0) − c~mm , rmout
. (5.19)
+1 , ..., rn
0 # & vm ∈Z+(0) cm(0) < c~mm , " m $#0 k, "$ 0 0 :
(
out
out
~m
Qnew (0) = r1out + c1 (0) − β m c~1m , ... , rmout
−k + cm−k (0) − β m cm−k , rm−k +1 ,..., rm ,
(5.20)
rmout+1 − d m +1 (0) +
r c1
β m n −imm +1
i =1
rijc1
j =1
m
¦
¦
m
c~im ,..., rnout − d n (0) + β m ¦
i =1
·
¸
m¸
~
ci ¸ ,
c1
r
¸¸
¦ ij
j =1
¹
rinc1
n−m
163
c (0)
β m = m~ m , " k " # , 0 &:
cm
βm−k +1 = ... = βm .
"$ 0 " &0 : , ci(0) – c~im $& . $ 0 &0 (5.20).
$ " " Qnew(0).
%
" .
5.6. & # % % 5.4.
§1
¨
¨1
R =¨6
¨
¨6
¨1
©
1
1
1
1
1
6
1
1
1
1
6
1
1
1
1
1·
¸
1¸
1¸ .
¸
1¸
1¸¹
; ρ = ((15, 15), (5, 5), (10, 10), (10, 10), (5,5)).
rsum = 45.
, Z+(0) ! > Z+(0) = {v1, v2, v3}, > > !. !
2: Q(0) = (15 + c1(0), 5 + c2(0), 10 + c3(0), 2, 2). : ! Z+(t). > > c3(0) < c~3m .
: ci(0) ≥ c~im i = 1, 2, I! , 5.4. " c~im .
" %
<" % Rc
Diag(α4, α5).
% R'
% P :
164
6 6 1·
§1 1
¨
¸
15
15
15
15 15 ¸
¨
1 1¸
¨1 1 1
¨5 5
5 5 5¸
¨6 1 1
1 1¸
R' = ¨
¸, P =
¨ 10 10 10 10 10 ¸
1 1¸
¨6 1 1
¨ 10 10 10 10 10 ¸
¨1 1 1
1 1¸
¨
¸
5 5 5¹
©5 5
§
¨1
¨
¨0
¨
¨
¨0
¨
¨0
¨
¨
¨0
©
0 0
1 0
0 1
0 0
0 0
6 1·
¸
15 15 ¸
1 1¸
5 5¸
1 1¸
¸,
10 10 ¸
1 1¸
10 10 ¸
1 1¸
¸
5 5¹
§6 1·
¨
¸
§1 1·
¨ 15 15 ¸
¨
¸
1
1
¸ , R ' = ¨ 10 10 ¸ .
R1' = ¨
2
¨5 5¸
¨¨ 1 1 ¸¸
¨1 1¸
©5 5¹
¨
¸
© 10 10 ¹
E2 −
)
§8
¨
=¨7
¨¨ 2
©7
§ 10
¨
¨ 21
−
1
2
'
'
c
R = §¨ R1 E2 − R2 ·¸ = ¨
¹ ¨7
©
¨1
¨
©7
1·
¸
7¸
2¸
.
7¸
1¸
¸
7¹
R2'
1·
§ 9
− ¸
¨
10 ¸ , E − R '
= ¨ 10
2
2
1
8
¸¸
¨¨ −
© 5 10 ¹
(
(
/ )
−1
1·
¸
7¸,
8¸
¸
7¹
Diag(α4, α5):
α4 =
10 − 2 4
5−2 3
= , α5 =
=
10
5
5
5
165
§ 10
¨
¨ 21
2
c
R ⋅ Diag (α 4 , α 5 ) = ¨
¨7
¨1
¨
©7
1·
¸
7 ¸§ 4
2 ¸¨ 5
¨
7 ¸¨ 0
1 ¸¨©
¸
7¹
3·
§ 8
¨
¸
· ¨ 21 35 ¸
0¸
8
6¸
¸=¨
3 ¸ ¨ 35 35 ¸
¸
3¸
5¹ ¨ 4
¨
¸
© 35 35 ¹
, (5.16).
3·
§8
¨
¸
¨ 21 35 ¸
8
6¸
(15, 5, 10)¨
= (8, 3) . # ¨ 35 35 ¸
¨ 4
3¸
¨
¸
© 35 35 ¹
% > Z–(0).
# (5.17) " c~im , i = 1, 2, 3.
3·
§8
c~1m = 15 ⋅ ¨ + ¸ = 7 ;
© 21 35 ¹
6·
§ 8
c~2m = 5 ⋅ ¨ + ¸ = 2 ;
© 35 35 ¹
3·
§ 4
c~3m = 10 ⋅ ¨ + ¸ = 2 .
© 35 35 ¹
; , ! , ! Z+(t). : > - ! 50 (.. ,
! >), > : q3(0)=11, r3out + c~3m = 10 + 2 = 12 ,
! Z+(t) > (10, 12).
* , ! , ! : Q(0) = (50,
c (0)
> :
50, 11, 2, 2). # β m = m~ m
cm
166
1
. # (5.21) , 2
, Z+(0) > , > v3
10,
! . : > ! :
β3 =
q1(0) = 50 –
q2(0) = 50 –
1
⋅7= 46,5;
2
1
⋅2= 49;
2
q3(0) = 10.
* > Z–(0)
# (5.20). <" % , > v1, v2, v3 > v4, v5.
q4(0) = 2+
7 § 8 § 8 3 ·· § 8 § 8 6 ·· § 4 § 4 3 ··
⋅ ¨ : ¨ + ¸ ¸ + 1⋅ ¨ : ¨ + ¸ ¸ + 1⋅ ¨ : ¨ + ¸ ¸ =
2 ¨© 21 © 21 35 ¹ ¸¹ ¨© 35 © 35 35 ¹ ¸¹ ¨© 35 © 35 35 ¹ ¸¹
= 2+
q5(0) = 2+
20 4 4
28
=6
+ + = 2+
7 7 7
7
§ 6 § 8
§ 3 § 4
7 § 3 §8
3 ··
6 ··
3 ··
⋅ ¨ : ¨ + ¸ ¸ + 1 ⋅ ¨¨ : ¨ + ¸ ¸¸ + 1 ⋅ ¨¨ : ¨ + ¸ ¸¸ =
2 ¨© 35 © 21 35 ¹ ¸¹
35
35
35
35
35
35
©
¹¹
©
¹¹
©
©
= 2+
21
9 3 3
= 3,5.
+ + = 2+
14 7 7
14
* Qnew (0) = (46,5, 49, 10, 6, 3,5).
< Z+(t) ! ,
# (5.19).
" % Rc
<## % αi, i = 3, 4, 5. Z+(0) , > , % P :
167
§
¨1
¨
¨0
¨
¨
P = ¨0
¨
¨0
¨
¨
¨0
©
0
1
0
0
0
6
6 1·
¸
15 15 15 ¸
1
1
1¸
5
5
5¸
1
1
1¸
¸ , " 10 10 10 ¸
1
1
1¸
10 10 10 ¸
1
1
1¸
¸
5
5
5¹
§1 1 1·
¨
¸
§6 6 1·
¨ 10 10 10 ¸
¨
¸
1 1 1¸
R1' = ¨ 15 15 15 ¸ , R2' = ¨
.
1
1
1
¨
10 10 10 ¸
¨¨
¸¸
¨1 1 1¸
©5 5 5¹
¨
¸
©5 5 5¹
§ 9
¨
¨ 10
1
'
E2 − R2 = ¨ −
¨ 10
¨ 1
¨−
© 5
R =
c
1
10
9
10
1
−
5
−
§¨ R '
1
©
α3 = 0, α4 =
(E
(
−
)
§5
¨
= ¨9
¹ ¨1
¨
©3
−1
R2' ·¸
§7
¨
¨6
−1
1
=¨
¨6
¨1
¨
©3
)
5
9
1
3
1
6
7
6
1
3
1·
¸
6¸
1¸
,
6¸
4¸
¸
3¹
2·
¸
9¸.
1¸
¸
3¹
10 − 6
5 − 3.5
= 0,4 , α5 =
= 0,3 ,
10
5
§5
¨
Rc⋅Diag(α3, α4, α5) = ¨ 9
¨1
¨
©3
168
2
1·
¸
10 ¸
1
− ¸ , E2 − R2'
10 ¸
4 ¸
¸
5 ¹
−
5
9
1
3
2 ·§ 0 0
0 · §¨ 0 2 1 ·¸
¸¨
9 ¸ 0 0,4 0 ¸ = ¨
9 15 ¸
¸
1 ¸¨
2 1¸
¨
¸¨ 0 0 0,3 ¸¹ ¨ 0
¸
3 ¹©
© 15 10 ¹
# (5.17) " c~im , i = 1, 2.
§ 2 1 · 13
c~1m = 15 ⋅ ¨ + ¸ =
= 4, (3) ;
© 9 15 ¹ 3
§2 1· 7
c~2m = 5 ⋅ ¨ + ¸ = = 1,1(6) .
© 15 10 ¹ 6
" ! :
Q* = (46,5 – 4,(3), 49 – 1,1(6), 10, 10, 5) = (42,1(6), 47,8(3), 10, 10, 5)
! .
I " # % " Q(0) = (50, 50, 11, 2, 2)
" ( ! Z+(t)) Qnew(0) = (46,5, 49, 10, 6, 3,5).
t
0
1
2
3
4
5
6
7
8
9
10
11
12
…
+ 5.2
A%/%'%* A"E%/@ (:/) Q(0) = (50, 50, 11, 2, 2)
v1
v2
v3
v4
v5
50,000
44,600
43,240
42,596
42,338
42,235
42,194
42,178
42,171
42,168
42,167
42,167
42,167
50,000
48,600
48,140
47,956
47,882
47,853
47,841
47,836
47,835
47,834
47,834
47,833
47,833
11,000
9,600
9,540
9,816
9,926
9,971
9,988
9,995
9,998
9,999
10,000
10,000
10,000
2,000
8,600
9,540
9,816
9,926
9,971
9,988
9,995
9,998
9,999
10,000
10,000
10,000
2,000
3,600
4,540
4,816
4,926
4,971
4,988
4,995
4,998
4,999
5,000
5,000
5,000
169
+ 5.3
A%/%'%* A"E%/@ (:/) Qnew(0) = (46,5, 49, 10, 6, 3,5)
t
v1
v2
v3
v4
v5
0
1
2
3
4
5
6
7
8
9
10
11
…
46,500
43,800
42,820
42,428
42,271
42,208
42,183
42,173
42,169
42,168
42,167
42,167
49,000
48,300
48,020
47,908
47,863
47,845
47,838
47,835
47,834
47,834
47,833
47,833
10,000
9,300
9,720
9,888
9,955
9,982
9,993
9,997
9,999
10,000
10,000
10,000
6,000
9,300
9,720
9,888
9,955
9,982
9,993
9,997
9,999
10,000
10,000
10,000
3,500
4,300
4,720
4,888
4,955
4,982
4,993
4,997
4,999
5,000
5,000
5,000
G , ! .
##* + ! 5
D – ! > !! . /
> ; : riin = riout , . % R ! ! < , " < < % R. / :
• " # = rsum;
~
Q " W = •
~
# Q = ( r1out , r2out ,..., rnout ) , 2.1 .
W > T " ! " .
Q(0)
# , > < – , , , " : > Z+*, ! Z+(0) ( < ),
Z+ Z–. < Z– >.
/ , > Z+(0) +*
Z , " 170
. > <" Q(0) # % - # % , " . 2. $ > " ! , " " " . 2. K" – " – " 5.
; > , " Z+ % #% . (
, < Q(0)). K > Z+, " % .
171
6. K > B,
:
• % d- & .
• ? &
$ . ( &# & ;
& , , # & &; & & , W = T; &# W > T; . ( , W > T
; $ &, .
, & ! .
• ? & & & #
& . , t → ∞ d &# , &# . , ∞
R ' d , -
&# & & λ = 1 d.
• ) & , & W < T & . , ! R'.
• ( , , & , &# &
& , 1. ) & .
• & &
d- $ .
• & &
W ≥ T.
/ " " .
" #% , " – % ! , " " ! . # " ! < ;
172
, - > , " ,
> " " % &.
/ < " ! <" " , ! % . / . 2.3 , "# , ! > ! (*$) d
% ! > %. D d-% % , " % . : < ! !.
"# n > v1, v2, …, vn,
" % , .. , > ! . ; ! ! , ! , . . /! > vi,
(! ) t = 0. / t = 1 > vi ! ! ; t = 2 > , !> t = 1, ! !
. . $!
Nij
, ! % ,
> vj. $ , Nii – < > vi. * , !
i j Nii Njj (< , , , vi % , vj). / " . 1 " [38] , # ! .
# > vi, ! > ! Nii d. ;" 1) !" j = 1, 2, …, n kij, 0 V kij < d,
, ! < Nij kij d. /
!: !" k, 0 V k < d > vj, k d. & ! < Nij ;k. I > k % % d.
2) K > vi t = 0 % ;0, t = k ! > % " Ck .
173
3) :! > % ! > vi; ! " > % % .
4) & , " > "#,
, d = 1 ( . 1 2).
5) G . 4 , d > 1 % " ! . ! . % . <" % d.
6) ! % " "# G
% , ! " "# G' ! : ) > % " Cj, j =
= 0, 1, …, d – 1, > cj; !) ! (cj, cl) , "# G ! > Cj ! > Cl. G .2 , > Cj ! > C(j+1) mod d. D, , "# G' – < % , d > .
/ "#, . 6.1.
1
2
1
2
4
3
4
3
5
6
5
6
7
Gb
Ga
: . 6.1. + "#
% : "# Ga. "
, .. d = 4. > v1 ! : v2 t = 1, v3 t = 2, v4 t = 3, v1 v5 174
t = 4, v2 v6 t = 5 . . < % "# Ga : ;0 = {v1, v5}, ;1 = {v2, v6}, ;2 = {v3}, ;3 = {v4}.
K > ! v3, ! , % % 2.
2 > v7, K > v6 % "# Gb, % d 2. / ! % – > ,
– " . , N22 = {4, 8, 10, 12, 14, …},
N66 = {2, 4, 6, 8, 10, 12, 14, …}, .. v2 v6 % , N22 N66 . D
> " Nii Njj .
"
% " .
! "
D & n > . $ n % ,
#% 2 > . " % " ! ,
> ! !. J – & &. % % , > , ! . / < % ! ! . K *$
< % %, " % " ; " ,
,
" . $! % – ! ( – < % 1), % , > ! .
6.1. K#*" < !"
) %+%
K ! < % , % < !%, > !. < «% » ! < % . % ! ! ! :
% !% .
: > , ! % , % !
175
§0
¨
¨0
R =¨ 0
¨
¨0
¨r
© 51
r12
0
0
0
0
r23
0
0
r34
0
0
0
0
0
0
0·
¸
0¸
0 ¸.
¸
r45 ¸
0 ¸¹
% ! §0
¨
¨0
R' = ¨ 0
¨
¨0
¨1
©
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0·
¸
0¸
0¸ .
¸
1¸
0 ¸¹
/ > :
176
§0
¨
¨0
2 ¨
(R') = 0
¨
¨1
¨0
©
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0·
¸
0¸
1¸ ,
¸
0¸
0 ¸¹
§0
¨
¨0
3 ¨
(R') = 1
¨
¨0
¨0
©
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
0·
¸
1¸
0¸ ,
¸
0¸
0 ¸¹
§0
¨
¨1
4 ¨
(R') = 0
¨
¨0
¨0
©
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
1·
¸
0¸
0¸ ,
¸
0¸
0 ¸¹
:
§1
¨
¨0
4 ¨
(R') = 0
¨
¨0
¨0
©
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0·
¸
0¸
0 ¸ = E, (R')5= R'E= R'.
¸
0¸
1 ¸¹
,
" > #% 2. ;" ! % " .
Q(0) = (q1(0), q2(0), q3(0), q4(0), q5(0));
Q(1) = Q(0)R' = (q5(0), q1(0), q2(0), q3(0), q4(0));
Q(2) = Q(1)R' = Q(0)(R')2 = (q4(0), q5(0), q1(0), q2(0), q3(0));
Q(3) = Q(2)R' = Q(0)(R')3 = (q3(0), q4(0), q5(0), q1(0), q2(0));
Q(4) = Q(3)R' = Q(0)(R')4 = ( q2(0), q3(0), q4(0), q5(0), q1(0));
Q(5) = Q(4)R' = Q(0)(R')5 = (q1(0), q2(0), q3(0), q4(0), q5(0)) = Q(0).
;
! , Q(t) = Q(ti), " ti t 5:
ti < 5, ti ≡ t (mod 5).
* !! < % n > . " % #% 2 ! n : Q(t) = Q(ti), " ti < n, ti ≡ t (mod n).
G < ! , % , #% 2, , " W·
§W W
> , .. Q(0) = ¨ , ,..., ¸ .
n¹
©n n
*
;" ∀tQ(t) = Q(0) Q = Q(0). D . / ! ; " .
177
6.1. : % > ! , 5.
§0
¨
¨0
R = ¨0
¨
¨0
¨5
©
5 0 0 0·
¸
0 5 0 0¸
0 0 5 0¸ .
¸
0 0 0 5¸
0 0 0 0 ¸¹
-
(6.1)
rsum = 25. * : Q(0) = (2, 0, 0, 0, 0). #% 2. . 6.2.
: . 6.2. ! " " <" ! ! . 6.1.
:, 2, > , > .
* > ,
Q(0) = (5, 5, 5, 5, 5) . > . , ! Q(0) = (W1,W2 ,W3 ,W4 ,W5 ) !,
W1, …, W5 > 5.
178
t
0
1
2
3
4
5
6
7
8
9
10
…
v1
2,000
0,000
0,000
0,000
0,000
2,000
0,000
0,000
0,000
0,000
2,000
A%/%'%* A"E%/@ (:/)
v2
v3
0,000
2,000
0,000
0,000
0,000
0,000
2,000
0,000
0,000
0,000
0,000
0,000
0,000
2,000
0,000
0,000
0,000
0,000
2,000
0,000
0,000
0,000
+ 6.1
v4
0,000
0,000
0,000
2,000
0,000
0,000
0,000
0,000
2,000
0,000
0,000
v5
0,000
0,000
0,000
0,000
2,000
0,000
0,000
0,000
0,000
2,000
0,000
/ 0 0 % . 6.1, ( « » , , , , " , 2) > , " " – <" #% % – % " ". – – > > < % . / ! < . $ , ! >, > ( ) < >
! > , , . D , n , , "
", < . ; (" % ), ! &.
/ "" < % .
#% <" % " . < #% 2 % , > < , . : #% .
179
/ (6.1) Q(0) = (1,6, 1,6, 1,6, 1,6, 1,6),
> #% 2, < . K , 8, % > , #% .
6.2. $ % % (6.1). * Q(0) = (8, 0, 0, 0, 0).
; : 5 % , – ( . 6.3, ! . 6.2).
! % 5.
> #% 2, . < #% ( 5) " % 2.
: . 6.3. ! " <" ( )
;
! , , " , , < % , " , " " . ;
! , ! " !
".
180
t
0
1
2
3
4
5
6
7
8
9
10
…
v1
8,000
3,000
0,000
0,000
0,000
5,000
3,000
0,000
0,000
0,000
5,000
A%/%'%* A"E%/@ (:/)
v2
v3
0,000
5,000
3,000
0,000
0,000
0,000
5,000
3,000
0,000
0,000
0,000
0,000
0,000
5,000
3,000
0,000
0,000
0,000
5,000
3,000
0,000
0,000
+ 6.2
v4
0,000
0,000
0,000
5,000
3,000
0,000
0,000
0,000
5,000
3,000
0,000
v5
0,000
0,000
0,000
0,000
5,000
3,000
0,000
0,000
0,000
5,000
3,000
! > !" " > > . D
> , , ! .
" , ,
. , " . * < > , > "" .
< , > .
! > , ! , " " (
) " ,
, " . / < < % " " .
6.3. : % (6.1). $ , .. < " . < > – % . * ! , < , !" > riout = 5 % , , , T = rsum = 25.
Q(0) = (30, 0, 0, 0, 0), #% . 6.4.
181
: . 6.4. A% W > T
t
0
1
2
3
4
5
…
+ 6.3
A%/%'%* A"E%/@ (:/) -A) Q(0) = (30, 0, 0, 0, 0)
v1
v2
v3
v4
v5
30,000
25,000
20,000
15,000
10,000
10,000
0,000
5,000
5,000
5,000
5,000
5,000
0,000
0,000
5,000
5,000
5,000
5,000
0,000
0,000
0,000
5,000
5,000
5,000
0,000
0,000
0,000
0,000
5,000
5,000
G ! . 6.3 , > v2, …, v5 , ! ,
< >
.
% #% 2 #% % , % " " % .
$ % > , ! > #% 1, % .
182
6.4. : , ! % , % !:
§0 5 0 0 0·
¸
¨
¨0 0 4 0 0¸
R = ¨0 0 0 3 0¸ .
(6.2)
¸
¨
¨ 0 0 0 0 4¸
¨5 0 0 0 0¸
¹
©
rsum = 21. * : Q(0) = (7, 0, 0, 0, 0).
! !,
r33 = 3, ,
> ! % 5: 3, 3 1 ( . 6.5 ! . 6.4).
: . 6.5. A% " < " % t
0
1
2
3
v1
7,000
2,000
0,000
0,000
A%/%'%* A"E%/@ (:/)
v2
v3
0,000
5,000
3,000
0,000
0,000
0,000
4,000
4,000
+ 6.4
v4
0,000
0,000
0,000
3,000
v5
0,000
0,000
0,000
0,000
183
4
5
6
7
8
9
10
…
0,000
3,000
3,000
1,000
0,000
0,000
3,000
0,000
0,000
3,000
3,000
1,000
0,000
0,000
1,000
0,000
0,000
3,000
3,000
1,000
0,000
3,000
1,000
0,000
0,000
3,000
3,000
1,000
3,000
3,000
1,000
0,000
0,000
3,000
3,000
; ! , < % > , > % , – % . / ( , , ") ! % . / ( ) % «
»
W »
«
min r , " ¬⋅¼ – % « »: , W −
« min rij » i , j ij
¬ i, j ¼
.
* , «
»
W »
W −«
min r = min r ,
« min rij » i , j ij i , j ij
¬ i, j ¼
W = n ⋅ min rij ,
i, j
< ! -
.
$ , !" " % , !
!" ". ! ! , ! ! % . " ; # :
~
~
F in = F out = (min rij ,..., min rij ) ; T = n ⋅ min rij,
" n – > , min rij –
% .
184
!-
~
~
W > , F in = F out ;
fsum = !" " , > > > vk, k = arg min rij .
i∈{1,...,n}
;, , 6.5. Q(0) = (15, 0, 0, 0, 0).
!
! . 6.5).
% (6.2), = 15.
% (6.2), > % ( . 6.6
: . 6.6. : W = = 15 .
/ " Q*=(3, 3, 3, 3, 3)
t
0
1
2
3
4
5
6
7
…
+ 6.5
A%/%'%* A"E%/@ (:/) -A) Q(0) = (15, 0, 0, 0, 0)
v1
v2
v3
v4
v5
15,000
10,000
5,000
0,000
0,000
3,000
3,000
3,000
0,000
5,000
6,000
7,000
3,000
0,000
3,000
3,000
0,000
0,000
4,000
5,000
6,000
6,000
3,000
3,000
0,000
0,000
0,000
3,000
3,000
3,000
3,000
3,000
0,000
0,000
0,000
0,000
3,000
3,000
3,000
3,000
185
/ % W > , ! , > ! vk. ; > ! % % . K , . : W – ( . 6.7). $ , % (6.2) % > v3.
6.6. % (6.2), Q(0)=(20, 0, 0, 0, 0).
: . 6.7. : W =20. / " Q*=(3, 3, 8, 3, 3)
t
0
1
2
3
4
5
6
7
…
186
+ 6.6
A%/%'%* A"E%/@ (:/) -A) Q(0) = (20, 0, 0, 0, 0)
v1
v2
v3
v4
v5
20,000
15,000
10,000
5,000
0,000
3,000
3,000
3,000
0,000
5,000
6,000
7,000
8,000
4,000
3,000
3,000
0,000
0,000
4,000
5,000
6,000
7,000
8,000
8,000
0,000
0,000
0,000
3,000
3,000
3,000
3,000
3,000
0,000
0,000
0,000
0,000
3,000
3,000
3,000
3,000
G > W – T > > . Q* = (3, 3, 8, 3, 3).
G, > v1, ! , " , !.
K > ! , " < > % . K " " , ! ,
! – " > , , > ! > , !
! " >. K
> " > , ! % , ! " .
6.7. : % > % v2 v5 – ! .
§0
¨
¨0
¨0
¨
R =¨0
¨0
¨
¨0
¨
©7
0·
¸
0¸
0¸
¸
0 ¸ . rsum = 31. = 2⋅7 = 14.
0 0 0 0 2 0 ¸¸
0 0 0 0 0 6¸
¸
0 0 0 0 0 0¹
5
0
0
0
0
2
0
0
0
0
5
0
0
0
0
4
0
0
0
0
(6.3)
W = 17 > T. * : Q(0) = (17, 0, 0, 0, 0, 0, 0).
G . 6.8 , Z+* > , !, .. > v2.
187
: . 6.8. + .
/ " Q*=(2, 5, 2, 2, 2, 2, 2)
Q(0) = (0, 0, 0, 17, 0, 0, 0) ! Q*=(2, 2, 2, 2, 5, 2, 2), .. > % v5.
K
W > T > , " > W – T % , ! !.
;, , % (6.3) Q(0) = (8, 0, 0, 9, 0, 0, 0) ! Q*=(2, 2, 2, 2,
5, 2, 2), .. > v5. I!
> v2 >, > v1 ! ! > 8. ;, Q(0) = (9, 0, 0, 8, 0, 0, 0) Q*=(2, 3, 2, 2, 4, 2, 2). $ , "
8, > v4 v3, .. Q(0) = (9, 0, 8, 0, 0, 0, 0), ! Q*=(2, 2, 2, 2, 5, 2, 2) – > ! v5. D ", ! % " : > v1, v2. Q(0) = (9, 0, 8, 0, 0, 0, 0) > v1
( " ), <" , ! v2 v5. ( . 6.9,
! . 6.8).
188
G , ( 5), > v1, , " ! ! > v2, > v2 , %, .. >.
: . 6.9. + .
/ " Q *= (2, 2, 2, 2, 5, 2, 2)
+ 6.8
A%/%'%* A"E%/@ (:/) -A) Q(0) = (9, 0, 8, 0, 0, 0, 0)
t
v2
v3
v4
v5
v6
v7
v1
0
1
2
3
4
5
6
7
8
9
…
9,000
4,000
0,000
0,000
0,000
2,000
2,000
2,000
2,000
2,000
0,000
5,000
7,000
5,000
3,000
1,000
2,000
2,000
2,000
2,000
8,000
3,000
2,000
2,000
2,000
2,000
1,000
2,000
2,000
2,000
0,000
5,000
4,000
2,000
2,000
2,000
2,000
1,000
2,000
2,000
0,000
2,000
4,000
6,000
6,000
6,000
6,000
6,000
5,000
5,000
0,000
0,000
0,000
2,000
2,000
2,000
2,000
2,000
2,000
2,000
0,000
0,000
0,000
0,000
2,000
2,000
2,000
2,000
2,000
2,000
$!!
, % , .
< 189
6.1. 0 " n
# , & D & +$% + r, " + : T = rn.
1) W < T +&$ ", $ "$ 0 000 W·
§W W
0 0: Q(0)= Q* = ¨ , ,..., ¸ ; %+ "$
n¹
©n n
0
0 0 ).
2) W = T $& $ 0 )% &. ' $ $ 0 0: Fin* = Fout* = Q* = (r, r, …, r).
3) W > T $ 0 )%,
" $& 00 & + :
Fin* = Fout* = (r, r, …, r). $ 0 "$ .
6.2. 0 " n
# , & D & ,
" &" 00 : T = n ⋅ rmin ,
rmin = min rij.
1) W < T " $ 0 W·
§W W
, " : Q(0) = Q* = ¨ , ,..., ¸ ; n¹
©n n
%+ "$ 0 $ 0 0 ).
$ 0 )2) W = T $& % &. ' $ $ 0 0: Fin* = Fout* = Q* = (rmin, rmin, …, rmin).
3) W > T $ 0 )%,
" $& 00 & + :
Fin* = Fout* = (rmin, rmin, …, rmin). $ 0 $ ", $ & $& . ' " : " 0 W – T "$ 0 0.
4) ' # vk 000 D $ " 0 &00
k = arg min rij . + $&: i∈{1,...,n}
& , &
190
000 $ & :; $& & &, 0 k = arg min rij i∈{1,...,n}
+0& +&$ $& # .
! < % .
6.2. 9!< * '* 6/")
< ! &%! ) %+ '* #") *%9*%)
% . N!
% % , > /
! .
#% 2 " Q(0) %. * < < . + – < > vi, W = 1, > .
% R'
% d. R'k . $ d # % . < [11], R'k I %
. $! Ri'∞ , i = 0,…, d – 1.
% R' d-% d
!
,
%:
λ1 = |λ2| =…= |λd| = 1, .. d [11].
# ! %, ,
" ! % ,
%, : %.
6.2.1. 9!< * < ! &%!+ %
'* $ &# *%9*%
6.8. ( . 6.10).
"#
191
1
3
2
5
4
: . 6.10. "# > ,d=2
K % !:
§0
¨
¨2
R = ¨0
¨
¨0
¨0
©
2
0
2
0
0
0
2
0
2
0
0
0
2
0
2
0·
¸
0¸
0 ¸ , rsum = 16.
¸
2¸
0 ¸¹
%
(6.4)
:
1
0
0
0 ·
§ 0
¸
¨
0 ¸
¨ 0,5 0 0,5 0
R' = ¨ 0 0,5 0 0,5 0 ¸
¸
¨
0 0,5 0 0,5 ¸
¨ 0
¨ 0
0
0
1
0 ¸¹
©
* . 6.11 % , Q(0) = (1, 0, 0, 0, 0).
#% , . 6.11,
! . 6.9.
G . 6.11 ! . 6.9 , > v1, v3 v5, – > v2 v4. / < d = 2, % {v1, v3, v5} {v2, v4}.
192
q(t)
2.5
2
v1
1.5
v2
v3
1
v4
v5
0.5
0
t
0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728
: . 6.11. ! Q(0) = (1, 0, 0, 0, 0)
+ 6.9
A%/%'%* IB,'M)%,)A%C",)G 2-M)'*)$:('%D (:/)
( &"/A)M:D (6.4) ) Q1(0) = (1, 0, 0, 0, 0)
v1
v2
v3
v4
v5
ti
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
0
0,5
0
0,375
0
0,313
0
0,281
0
0,266
0
0,258
0
0,254
0
0,252
0
0
1
0
0,75
0
0,625
0
0,563
0
0,531
0
0,516
0
0,508
0
0,504
0
0,502
0
0
0,5
0
0,5
0
0,5
0
0,5
0
0,5
0
0,5
0
0,5
0
0,5
0
0
0
0
0,25
0
0,375
0
0,438
0
0,469
0
0,484
0
0,492
0
0,496
0
0,498
0
0
0
0
0,125
0
0,188
0
0,219
0
0,234
0
0,242
0
0,246
0
0,248
0
193
ti
v1
v2
18
19
20
21
22
23
…
0,251
0
0,25
0
0,25
0
…
0
0,501
0
0,5
0
0,5
…
v3
0,5
0
0,5
0
0,5
0
…
} " +. 6.9
v4
v5
0
0,499
0
0,5
0
0,5
…
0,249
0
0,25
0
0,25
0
…
#% , , " ":
Q11* = (0,25, 0, 0,5, 0, 0,25)
Q21* = (0, 0,5, 0, 0,5, 0).
W = 1 > , Q11*
Q21* , : Q(0) = (1, 0, 0, 0, 0),
Q(0) = (0, 1, 0, 0, 0), Q(0) = (0, 0, 1, 0, 0), Q(0) = (0, 0, 0, 1, 0), Q(0) = (0, 0, 0, 0, 1).
d = 2, % R' :
R1'∞
0,5 0 0,5 0 ·
§ 0
¸
¨
¨ 0,25 0 0,5 0 0,25 ¸
=¨ 0
0,5 0 0,5 0 ¸ ,
¸
¨
¨ 0,25 0 0,5 0 0,25 ¸
¨ 0
0,5 0 0,5 0 ¸¹
©
R2'∞
0,5 0 0,25 ·
§ 0,25 0
¸
¨
0,5 0
0,5 0 ¸
¨ 0
= ¨ 0,25 0
0,5 0 0,25 ¸ .
¸
¨
0,5 0
0,5 0 ¸
¨ 0
¨ 0,25 0 0,25 0 0,25 ¸
¹
©
& % R1'∞
Q21* , 194
R2'∞ Q11*
R2'∞ ! R1'∞ : Q11* Q21* , !. 1, 3, 5, > " % " , ; 2, 4, > " % " .
$! Q11*
Q21* !
% R2'∞ , % R1'∞ Q11* Q21* , Q21* – Q11* . ; % R':
Q11* R ' = Q21* , Q21* R ' = Q11* .
R1'∞
< b % R' %
R2'∞ !.
, % % :
§ 0,25 0,5 0,5 0,5 0,25 ·
¸
¨
¨ 0,25 0,5 0,5 0,5 0,25 ¸
R1'∞ + R2' ∞ = ¨ 0,25 0,5 0,5 0,5 0,25 ¸
¸
¨
¨ 0,25 0,5 0,5 0,5 0,25 ¸
¨ 0,25 0,5 0,5 0,5 0,25 ¸
¹
©
/, ! < %, ! ! % R1'∞
R2'∞ , % R'; ! ", % R1' ∞ + R2'∞ , % , I R'k:
1 k 'j
¦R .
k → ∞ k j =1
A = lim
A=
R1'∞ + R2' ∞
2
§ 0,125
¨
¨ 0,125
= ¨ 0,125
¨
¨ 0,125
¨ 0,125
©
0,25
0,25
0,25
0,25
0,25
0,25
0,25
0,25
0,25
0,25
0,25
0,25
0,25
0,25
0,25
-
(6.5)
0,125 ·
¸
0,125 ¸
0,125 ¸ .
¸
0,125 ¸
0,125 ¸¹
195
& % R' . * % %
% ! ( " ) [11]. ! % Q11* Q21* . <" # Q11* R ' = Q21* Q21* R ' = Q11* .
/ ! 1
# : Q1* = (Q11* + Q21* ) .
2
6.2.2. *$/" !*" < !" d-< ! &%!+ %
/ ! , d-% <" % &.
/ [38] , % , I R'k (# (6.5)), :
A = 1⋅α,
" 1 – - !%, %, α % ! % R',
! λ = 1:
αR' = α.
(6.6)
; R' = R'= .
R' – % <" d% . : : R', …, R'd, R'd+1, … $ d :
1) R', R'd⋅R', R'2d⋅R', R'3d⋅R',…
2) R'2, R'd⋅R'2, R'2d⋅R'2, R'3d⋅R'2,…
…
d) R'd, R'd⋅R'd, R'2d⋅R'd, R'3d⋅R'd,…
196
;
! , , R1'∞ , …, Rd'∞ -
% Rd'∞ :
R1'∞ = Rd'∞ R' ;
R2'∞ = Rd'∞ R '2 ;
…
Rd'∞−1 = Rd'∞ R 'd −1 .
% R1'∞ , …, Rd'∞ " . / [11] , % (6.5) # :
A=
1
1
( E + R ' + ... + R ' d −1 ) Rd' ∞ = Rd' ∞ ( E + R ' + ... + R ' d −1 )
d
d
!" " :
α = Q1 (0) A .
(6.7)
" (6.8)
A (6.8) , % % & "
# ". α, !" , ! ! Q1* – " : Q1* = Q1(0)A; % n -, Q1*.
: % #% 2.
d , , %.
Q(kd+1)= Q(0)R'kd R',
Q(kd+2)= Q(0)R'kd R'2,
197
…
Q(kd)= Q(0)R'kd R'd = Q(0)R'(k+1)d
k = 0, 1, …
, :
Q1* = Q(0) Rd'∞ R ' , Q2* = Q(0) Rd'∞ R '2 , …, Qd* = Q(0) Rd'∞ .
(6.9)
* (6.9) , :
Qi*+1 = Qi* R ' , i = 1, …, d – 1, Q1* = Qd* R ' .
/ Q1* , ..., Qd* ! $& .
, " ", ! $& <" .
; ! , .
6.3. ' D " d- " , %) 2, ) $&
& d, 0)
$&: Q1* , ..., Qd* , 0&: (6.9).
/
Q1* , ..., Qd* !
% R'.
6.4. 0 D " d- " & $& +& & R' 000 Q1*, 0& :
Q1* =
Q1* + ... + Qd*
,
d ⋅W
(6.10)
Q1* , ..., Qd* – $& & , %) 2, & W "$& 0 Q(0).
$ .
(6.9), :
Q1* + ... + Qd* = Q(0) Rd'∞ ( E + R ' + ... + R ' d −1 ) .
198
! # (6.7), :
Q1* + ... + Qd* = dQ(0) A . (D % [11]).
* ! Q(0) , "
#% 2, :
Q(0)=WQ1(0). $ Q1* + ... + Qd* = dWQ1 (0) A
Q1* + ... + Qd*
= Q1 (0) A
dW
Q1* + ... + Qd*
. ; % dW
n -, # (6.10). = (6.6)
, Q1*R' = Q1*. % R', Qi* G # (6.8)
: Q1* =
Qi*+1 ( d). * %, < .
6.5. $& & D " d " Q1* , ..., Qd* 00%0 +& & Rd'∞ , %) + " λ = 1
d.
$ . > (6.9) :
Qi* = Q(0) Rd'∞ R 'i , i = 1, …, d.
b (6.11)
! (6.11) % Rd'∞ .
Qi* Rd'∞ = Q(0) Rd'∞ R 'i Rd'∞ , # (6.7) < :
Qi* Rd' ∞ = Q(0) Rd'∞ Rd'∞ R 'i = Q(0) Rd'∞ R 'i
(
)
# (6.11) Qi* . ;" Qi* Rd'∞ = Qi* . 199
| " . / , " *$ % %, , ! % ( " ), . / , #% 2, " Q*. $ ! % R' ( Q*
( 6.3),
" (6.10): Q1* =
W
6.4), ! % R'∞, ! λ = 1 ( 6.5). < " % &. # " 3.
6.2.3. % - :/ *% (
'* #") *%9*%)
: " , " +$ , .. *
*
Q1 , ..., Qd .
" %.
9 6.1. ' D " d- " ,
%) 2, $& & Q1* , ..., Qd*
" : : 0 0 # "$ 0 .
$ . & % R'
d. / [11] , R'd " d %, . D , % > , R'd ! - " . J % .
'd
§ R11
¨
¨ 0
R 'd = ¨
¨
¨ 0
©
200
0
'd
R22
0
... 0 ·
¸
... 0 ¸
¸.
¸
'd ¸
... Rdd
¹
% Rii' d , ", <
% .
" % &. & %
( ) ! ! (R ) ! R
Rd' ∞ = lim R 'd
k
k →∞
'd
ii
(R )
'd ∞
ii
.
'd ∞
ii
! πi, ! λ = 1, i = 1, …, d.
" #% 2, ..
> , "
% " .
/ Q(0) Qi(0), % . ;" % Rii' d " , Wi = 1 !"
Qi (0) Rii'∞ = Qdi* = π i .
" " W , " #% 2, !" Qi(0)
:
Qi (0) Rii'∞ = Qdi* = Wiπ i , " Wi – i- % .
Qdi* Qd* .
Qi(0)
$ " % Qd* % R' ! . 9 6.2. $ D " 0 d- " 0 0
$ 2, " &
W = 1 " "$ 0
m- " . $& & R 'j∞ = Rd' ∞ ⋅ R ' j , j = 1, …, d – 1,
d "&: , " :, %) : # i- " , : 0 Qk* , k ∈ [1, d] : 0 k ≡ m + j + i – 3 (mod d) + 1 , i, j = 1, …, d.
(6.12)
$ . : , % (m = 1). *
201
> ! , , < > v1. Q(0) = (1, 0, …, 0). $!
< " Q1(0).
G :
Q1* = Q1 (0) R1'∞ = ( R1'∞ )1 ,
Q2* = Q1 (0) R2' ∞ = ( R2'∞ )1 ,
…
Qd* = ( Rd'∞ )1 ,
" ( R 'j∞ )1 – % R 'j∞ .
1, > v1 ! ! > " % " . ;
! , %
R 'j∞ , > " % " , Q*j .
" Qi*+1 =
R 'j∞+1 =
Qi* R ' ,
R 'j∞ R ' .
Q1*
=
Qd* R ' ;
Qi*+1
(i = 1, …, d – 1) % :
D , % R 'j∞ , -
> i-" % " , j + i – 1 j + i – 1 ≤ d, j + i – 1 – d j + i – 1 > d. G k Qk* , <
, # : k ≡ j + i – 2 (mod d) + 1.
$!! < # m,
:
k ≡ m + j + i – 3 (mod d) + 1. " % .
9 6.3. ' D " d- " ,
%) 2, $& & Q1* , ..., Qd*
%, " : "$ W
0 : 0 " 0 d
202
# & & qi(0) ≤ riout . ' $ 0 0
WQ1* , Q1* – %+0 & , 0 (6.7).
$ . : d "
, > " % : Qm(0), " m – % " , m = 1, …, d.
$!
Q *jm , " j – " :
. Q*j m = Qm (0) Rd' ∞ R ' j = Q1 (0) Rd' ∞ R ' j + m = Qk* ,
" Qk* = Qk*1 – m = 1.
,
, k ∈ [1, d] # (9) Qi(0)
.
(Q1 (0) + ... + Qd (0)) Rd' ∞ R ' = Q1*1 + ... + Qd*1 = Q1* + ... + Qd* .
" " :
(Q1 (0) + ... + Qd (0)) Rd'∞ R '2 = Q1*2 + ... + Qd* 2 = Q2* + ... + Qd* + Q1* .
/ ! m- ! Q1* + ... + Qd* % " ;
! , , d .
K
.
% -
W
d
, > % , ! #% 2. b 6.3 ! .
K % , < > 203
#% 1, < % . ; ! , ! ! .
6.1. ' D " d- " , %) 2, " 0 t', $& & Q1* , ..., Qd* % $ , " : t = t' : 0
W
. ' $ 0 0
" d
WQ1* , Q1* – %+0 & , 0 (6.7).
$ .
1. t = t' #% 2 % . Q(t') , 6.3.
2. , #% 2, Q1* , ..., Qd* . " % > : Qi*+1 = Qi* R' , i = 1, …, d – 1, Q1* = Qd* R' . ;" : Qi* = Qi* R' , i = 1, …, d . b % R' " % . % . 6.1 .
6.9. % ! (6.4) Q(0) = (4, 0, 0, 0, 0). A% . 6.12, ! . 6.10.
G , , > % , . $ < > > 2,
>" % . !
; 2. !. / :
Q* = Q1* = Q2* = (0,5, 1, 1, 1, 0,5).
204
: . 6.12. A% , "#
% (6.4), Q(0) = (4, 0, 0, 0, 0)
+ 6.10
t
v1
A%/%'%* A"E%/@ (:/)
v2
v3
v4
0
1
2
3
4
5
…
20
21
22
23
…
4,000
2,000
1,000
1,000
0,750
0,750
0,000
2,000
2,000
1,500
1,500
1,250
0,000
0,000
0,000
0,500
0,500
0,750
0,000
0,000
1,000
1,000
1,000
1,000
0,000
0,000
0,000
0,000
0,250
0,250
0,501
0,501
0,500
0,500
1,002
1,001
1,000
1,000
0,998
0,999
1,000
1,000
1,000
1,000
1,000
1,000
0,499
0,499
0,500
0,500
v5
6.3. * 6& ;9!< * < ! &%! ) %+ '* :/G ) *%9*%)
6.3.1. * 6& / " ""
" . / Q* 205
W , % ,
, ! . D " ! " . /> ,
">
! W = T, – % . > % >, > .
/ % d , ! " ". ", < " . K ""
.
J " , % d ! ( # ), $ 0 Q*:
Q1* = ... = Qd* = Q * .
6.2. ' D " d- " ) " , , ":
W < T # & " " :0
2, 0 d $&: ;
W ≥ T + %0, 0
+$ %+ "$ 0 . $& ) . $ 0 ); "$ 0 0 $ ", $ .
$ . / , > #% 2, % d . K > #% 1, N 2, , Q(N+1) , % d . /
" % " , > #% 2, . ! .
, " . W = T > , ! . <
> vj. q*j = r jout d .
206
& % R'
d.
D , R'd " d " %, % Rii' d ( .
6.1). ! < %. /> vj % i. K " > #% 2,
" :
'∞
*
Qi (0) Rii = Qdi = Wiπ i , " Wi – i- % , πi – %
&. , " % Rii' d , " . $! " Ti. ;*
out
*
" Wi = Ti qdi
j = r j , " qdi j – j- -
( )
( )
Qdi* . * ! > vj ! r jout , i-" % " ! Ti . K #% 2,
% % , > .
K %, % ! ! % . " , > #% 2 t → ∞. ;" t → ∞ % Ti . = " # : T = dTi.
, !
, .
W > T ( < ! > vj) 1. ;
t', t ≥ t' > vj
#% 1. * ,
r jout , , > r jout . ;" ! ! q j (t ' ) − r jout , ! % , q j (t ' ) − r jout . A% < . * ! % , " . / " 207
. > , % . K ,
. 6.3.2. * * + *! %
'< /" *!*"
, > d-% , > " .
6.3. ' D " d- " # vj 000 $& , $ out
r
j = arg min i 1* , Q1* 00 i∈{1,...,n} q
i
Q1* =
1 d 1*
¦ Qk ,
d k =1
(6.13)
& Qk1* (k = 0, …, d) – $& & W = 1 $ "$ 0 .
$ . G 6.2 , W = T
! , d . * W ≤ T : Q1* + ... + Qd* = dWQ1* , " Q1*, # (6.13), – ! % R'. * ,
Q* = Q1* = ... = Qd* = TQ1* . ;
! , #% " % A, n Q1*:
A = 1⋅ Q1*.
, #% %
, , " .
; ! , ", ! > vj ! , !208
riout
(i∈{1,..., n} q1*
i
j = arg min
4.4, " 4). ; 6.1. ' D &: d- " : 0: +$# : : $ 0 $% "$ , 0 # : – $& .
; 6.2. ' D " d- " r out
" 00 : T = min i 1* .
i∈{1,...,n} q
i
6.3.3. *$/ %%( '*$/"+ '!
'* :/G ) *%9*%)
$!! , # % W > T, " " . G 6.2–6.3 6.2.
6.4 ( $ 0 ). ' D " d- " W ≥ T " 0 &: $ 0 0
Q* = q1* , ..., qn* &" 0%0 (
)
qi* = qi1* ⋅ T , i ≠ jk, jk – ,
:
1) Q1* 00 (6.13),
riout
,
i∈{1,...,n} q1*
i
2) T = min
riout
.
i∈{1,...,n} q1*
i
}# 0 00 $&
.
.
6.11. : #% , % !
! .
3) jk 0%0 0: jk = arg min
209
1
2
4
3
5
6
: . 6.13. + !
{v3}
:
!
/ < % : {v1, v5}, {v2, v6},
{v4}. % ! §0
¨
¨0
¨0
R =¨
¨2
¨0
¨
¨0
©
1
0
0
0
0
0
0
3
0
0
0
1
0
0
2
0
0
0
0
0
0
5
0
0
0·
¸
0¸
0¸
¸ . rsum = 18.
0¸
4 ¸¸
0 ¸¹
;" W = 1 ! > > ! :
Q11* = (0,286, 0, 0, 0, 0,714, 0), Q21* = (0, 0,286, 0, 0, 0, 0,714),
Q31* = (0, 0, 1, 0, 0, 0), Q41* = (0, 0, 0, 1, 0, 0).
! % # (# (6.13)):
Q1* = (0,0715, 0,0715, 0,25, 0,25, 0,1785, 0,1785).
/ 210
> riout
qi1*
> .
r1out
1
≈
≈ 14 ,
1*
0,0715
q1
r2out
3
≈
≈ 42 ,
1*
0,0715
q2
r3out
2
≈
= 8,
1*
0,25
q3
7
r4out
≈
= 28 ,
1*
0,25
q4
r5out
4
≈
≈ 22,4 ,
q51* 0,1785
r6out
1
≈
≈ 5,6 .
q16* 0,1785
riout
" > v6.
qi1*
;
! , v6 – % < . 6.2 6.2, T ≈ 5,6.
W = T ! :
& ~
Q = (0,4, 0,4, 1,4, 1,4, 1, 1) .
D -
.
W > T, > , " .
* , W = 20, ! Q* = (0,4, 0,4, 1,4, 1,4, 1, 15,4).
G > . $ " .
6.5 ( $ ). ' D " d- " W ≥ T $& ), 00 :
~
fiin* = fi out * = qi1*T ; F in* = Fi out * = Q1*T = Q ,
1) Q1* 00 (6.13),
211
riout
.
i∈{1,..., n} q1*
i
2) T = min
| " . / " « » «! >», " , < " %, . D , W = T " % &, > " , . ; " ! W ≤ T W > T. / % «! >» , W ≥ T , W < T > % d .
##* + ! 6
/ < " " – <" (.. ) % . *" ! , " . ; , ! . / , % , , " , " , . < , , ! > . * ,
, " , ! > , ! > W – T, " % " : ! > % ! – .
& , " " " % – < " % . $ < . / " , % "# ! d-% % ! "# < % d. , " % ! < % . D
! %, " 6.
212
D % <" d-% & :
W<T
" , > . /
, " , .
, ,
" > Z–(t), % .
/ % d, W=T
.
/ " " .
/ " " J > :
W>T
,
. , " % . / "" W – T " .
,
.
" , " .
Q1* , ..., Qd* .
213
, ! d-% ! > < % ,
! . , , . * , " W < T < " % # > , % – % . * ! , < % , " # . / d% Q1* , ..., Qd* d , # " ! .
214
7. NON B,
:
• " ## & . ( ,
&# &.
• , ## .
• ## , &
& . A R, !# & ! & , 1. ( & , !# & ,
& & $
.
• ) R'∞.
• ( R'∞ $ R.
• ( & . & Q* = Q(0)R'∞. 7 & &, & & , & . % ## ! & , .
• ? & &# & &, ## , .
" > , .. > , !, , , . $ , " , !
. 2.3. ; ! , ! , <" , <" " ,
" <" ! > -.
7.1. +% '404 ) *%9*%") %+
" l 1 l. K
% ! ! :
215
§D
R = ¨¨
© R1
O1 ·
¸,
R2 ¸¹
(7.1)
" D – " % l × l % " < , ! ( ), }1 – %
l × (n – l), R1 – % (n – l) × l, R2 – % (n – l) × (n – l). & % R1 ! !, , %
R2 – ! !, > .
% R', % R,
! §E
R' = ¨¨ 1'
© R1
O1 ·
¸.
R2' ¸¹
(7.2)
<
% D
" < (.. > ), ! E1 % R', D, ! %. /-, # , < % R', -, # % 1 , ,
.
" , "
% .
; 7.1. $ 0 )%) : 0: ). & $ 0 0, %) : , & %. $&
0: &: # .
/ , . : > , > . K , < . !, .
G <" , , , .. #% 216
, " " <" " . K , % ; > t → ∞.
; 7.2. " $ :
0 : $ 0 "$ 0 .
, , > , !, , > > , ! > . * , !, > .
" ! .
; 7.3. " $ :&: # : 0 " : $ 0 "$ 0 .
7.2 7.3 7.2, .
X , # " .
#% , > . > , > % , !. ;
! , " , > , .
7.1. : > , , v1 v2, . K % ! :
§10 0 0 0 0 ·
¸
¨
¨ 0 20 0 0 0 ¸
R = ¨ 6 1 2 3 5¸
¸
¨
¨ 1 6 6 7 1¸
¨ 0 0 1 3 4¸
¹
©
/ " < < % ,
.. > . Q(0) =
= (0, 0, 30, 10, 10). ! : Q* = (26.428, 23.572, 0, 0, 0).
A% . 7.1.
217
: . 7.1. A% " K
! % R :
§0
¨
¨0
R = ¨6
¨
¨1
¨0
©
, " < 0 0 0 0·
¸
0 0 0 0¸
1 0 3 5¸
¸
6 6 0 1¸
0 1 3 0 ¸¹
* Q(0) = (0, 0, 30, 10, 10) Q* = (26.428, 23.572, 0, 0, 0), > >", > . < #% ( . 7.2).
G < , , ! ! , ! .
218
: . 7.2. A% " ! ; 7.4. ' )%) $ $& & +0 + +)
" $ 0 $% "$ %+ .
D
" . / ! , " , " " ,
! " " . $ " , " > .
; , , .
7.2. 404 % % $ %"#
'*$/"# %%( #
: " , % !
:
§ D O1 ·
¸¸ ,
(7.3)
R = ¨¨
© H R2 ¹
219
K % (7.1) ! H – ", %: rank H = 1. D , ! !, ,
% , .. % !% % .
7.1. ' )%) (7.3), rank H = 1, 0 %+ "$ 0 0 Q(0) =
= (q1(0), …, ql(0), ql+1(0), …, qn(0)) $ 0 00 :
·
§
hin
hin
Q* = ¨¨ q1 (0) + 1 W − ,..., ql (0) + l W − ,0,...,0 ¸¸ ,
hsum
hsum
¹
©
W − =
n
¦ qi (0) , h j
in
(7.4)
– j- + & , hsum – -
i = l +1
: D & .
$ .
1. / " q1(0), …, ql(0) , .
2. K " % 1, !% % . & % :
§ h1
¨
¨ h
H =¨ 2
...
¨
¨h
© n −l
α 2 h1
α 2 h2
...
...
...
α 2 hn − l
...
α l h1 ·
¸
α l h2 ¸
.
... ¸
¸
... α l hn − l ¸¹
K > vj #% 1, ! , ! !: hj, α2hj, …, αlhj. ;.. %
1 : α2 : … : αl. K > #% 2, , § hj
·
h jα 2
h jαl
¨
q j (t ), out q j (t ),..., out q j (t ) ¸ , out
¨ hj
¸
hj
hj
©
¹
% 1 : α2 : … : αl. ;" >
1 : α2 : … : αl.
, %
220
. ; , > > , ! :
αl
α2
1
W−,
W − , …,
W−.
1 + α 2 + ... + α l
1 + α 2 + ... + α l
1 + α 2 + ... + α l
/ ! :
h1in = h1 + h2 +…+ hn-l,
hkin = αk(h1 + h2 +…+ hn-l), k = 2, …, l.
! % :
hsum = (h1 + h2 +…+ hn-l)(1+α2 + … + αl).
hin
αk
hin
αl
h1in
1
, k =
,…, l =
,
=
hsum 1 + α 2 + ... + α l
hsum 1 + α 2 + ... + α l hsum 1 + α 2 + ... + α l
# (7.4). ; 7.1. K , .. q1(0) = … = ql(0) = 0, " > .
; 7.2. G # " , % R2 , rank H = 1, ! !.
.
7.1 .
7.2. & % 7×7 v1, v2, v3.
§0
¨
¨0
¨0
¨
R = ¨1
¨
¨1
¨1
¨
©1
0 0 0 0 0 0·
¸
0 0 0 0 0 0¸
0 0 0 0 0 0¸
¸
1 1 2 3 0 4¸
1 1 6 7 8 0 ¸¸
1 1 9 1 2 3¸
¸
1 1 6 7 8 9¹
221
& % H < < . : !" W ! %
> v4, …, v7 §W W W
·
Q * = ¨ , , ,0,0,0,0 ¸ . * . 7.3 ©3 3 3
¹
, " 30, " .
: . 7.3. A% " Q(0) = (0, 0, 0, 0, 10, 10, 10) → Q* = (10, 10, 10, 0, 0, 0, 0)
: , " ! , " % - %.
7.3. %:
§0
¨
¨0
¨0
¨
R = ¨1
¨2
¨
¨3
¨
©4
222
0·
¸
0¸
0¸
¸
4¸
4 14 6 7 8 0 ¸¸
6 21 9 1 2 3 ¸
¸
8 28 6 7 8 9 ¹
0
0
0
2
0
0
0
7
0
0
0
2
0
0
0
3
0
0
0
0
!" W ! %
> !:
·
§ W 2W 7W
,
Q* = ¨ ,
,0,0,0,0 ¸ .
¹
© 10 10 10
* . 7.4 #% " Q(0) = (0, 0, 0, 20, 10, 10, 10).
: . 7.4. A% " % H, " 1.
Q(0) = (0, 0, 0, 20, 10, 10, 10) → Q* = (5, 10, 35, 0, 0, 0, 0)
7.3. 404 % :4 $.
* 6& J " , " : ( )
k
! " , .. k Y 1, % R2'
" .
K
% R1
" ! > %, " ". 223
< < % R2 " # " .
7.4. ! %
7.2 .
§0
¨
¨0
¨0
¨
R = ¨1
¨
¨1
¨1
¨
©1
0 0 0 0 0 0·
¸
0 0 0 0 0 0¸
0 0 0 0 0 0¸
¸
1 1 2 3 0 4¸
¸
1 1 6 7 8 0¸
1 1 9 1 2 3¸
¸
7 1 6 7 8 9¹
, %, > , .
Q(0) = (0, 0, 0, 1, 0, 0, 0) → Q* = (0.280, 0.440, 0.280, 0, 0, 0, 0)
Q(0) = (0, 0, 0, 0, 1, 0, 0) → Q* = (0.290, 0.420, 0.290, 0, 0, 0, 0)
Q(0) = (0, 0, 0, 0, 0, 1, 0) → Q* = (0.282, 0.436, 0.282, 0, 0, 0, 0)
Q(0) = (0, 0, 0, 0, 0, 0, 1) → Q* = (0.232, 0.536, 0.232, 0, 0, 0, 0)
< ( rsum >)
> % . ; ! , " " ".
! " " < % R2, .
G " " " <" – "" .
A% " ! : % " , " % .
224
! "" ! " ( 3.3). " – < , W > T " t' > vi : ∀ t > t' vi ∈ Z+(t). / " > " ! . *
> Z+(t) , ! riout . $ !, , , , #% . ! ! ! , , ! 1, #% ! . D " .
| " 7.1. / " <" r out
# : T = min i 1* , <
i∈{1,..., n} q
i
" > -. / " r out
> Ti = i 1* qi
( ) ! ! . / > < > ! . / " ! > Ti – .
| " 7.2. <## % , r out
T
, T = min i 1*
" , > χ =
i∈{1,..., n} q
rsum
i
! ! ( 7.2 !). " χ ∈ (0, 1].
/ " χ = 0, « » .
| " 7.3. / " W V T:
1) > >" 2;
2) ".
/ " , rank H = 1, " ( " ); rank H > 1, ( " )
225
! " . / < "" .
7.4. *$/" %%( ( '404 ) %()
7.4.1. * < R'∞
%+%
* % " R'∞
> .
9 7.1. $ )%)0 $ l (7.1). R'∞ – : " & R', 00 :
E1
§
R'∞ = ¨¨
'
© E2 − R2
(
)
−1
O1 ·
¸,
O2 ¸¹
R1'
(7.5)
E2 – "0 (n – l) × (n – l).
$ . % R'
E1
§
R'k = ¨¨ '
' '
' 2 '
'
© R1 + R2 R1 + R2 R1 + ... + R2
( )
% ( )
k −1
R1'
:
O1 ·
k ¸.
R2' ¸¹
( )
/ " 5 " < .
%
( )
k
& R2' → O2 % (n – l) × (n – l);
(
% E2 − R2'
)
−1
k → ∞, " O2 – ¦ (R2' )
∞
k
k =0
(
= E2 − R2'
([38], 3.1.1
;" : R1' + R2' R1' +
% R'∞ :
( )
2
R2' R1'
+ ... +
E1
§
R'∞ = ¨¨
'
© E2 − R2
(
)
−1
R1'
'
1
−1
, 3.2.1).
( ) R + ... = (E
k
R2'
)
2
− R2'
)
−1
R1' ,
-
-
O1 ·
¸. O2 ¸¹
| " 7.4. G # (7.5) , R'∞ = (P∞)T,
" % P∞ # (5.11).
226
| " 7.5. G # (7.5) 7.1. R1 % H, , ! % R'∞ " 1, " !% % > , !% :
1 : α2 : … : αl.
7.2. ' )%) l R'∞ 0 %+&: 0: $&: D & R.
$ .
& % R, # (7.1), %
§ E O1 ·
¸
R' = ¨¨ '
' ¸.
© R1 R2 ¹
1. " < ! D %
(7.1) .
2. < " < % R2.
R2new = R2 – D(Δrii), " D(Δrii) – " %
(n – l) × (n – l), < ! . J ", 0 V Δrii V rii, ! > " > vk : Δrkk > 0. K
Δrkk = rkk, < > !.
;" % ! ! :
§D
Rnew = ¨¨
© R1
O1 · § O3
¸−¨
R2 ¸¹ ¨© O4
· §D
¸=¨
D(Δrii ) ¸¹ ¨© R1
O1
O1 ·
¸.
R2 new ¸¹
%, % Rnew, ! R'.
R1'
R2'
%
%
R1' new
R2' new :
§ Δr · ·
§ r out · § '
§ r out · '
¸ ⋅ ¨ R2 − D¨ outii ¸ ¸ , (7.6)
¸ R1 , R2' new = D¨ out i
R1' new = D¨¨ out i
¨ r ¸¸
¸
¨
¸
¨
© i ¹¹
© ri − Δrii ¹ ©
© ri − Δrii ¹
227
§ r out
" D¨¨ out i
© ri − Δrii
·
¸ –
¸
¹
" % (n – l) × (n – l) riout
. G % riout − Δrii
< ! % R'. $!
D';
§ Δrii ·
D¨¨ out ¸¸ – " %, © ri ¹
% R2' ! . $! DΔ.
;" (7.6) ! :
R1' new = D ' R1' , R2' new = D '⋅ R2' − DΔ .
< (
)
O1 ·
§ E
'
¸¸ - % Rnew
= ¨¨ '
'
© R1new R2 new ¹
, < :
*
E
§
'∞
= ¨¨
Rnew
'
© E2 − R2 new
(
(E
/
:
2
(E
− R2' new
2
)
−1
− R2' new
)
)
−1
O1 ·
¸.
O2 ¸¹
R1' new
R1' new −1
(
%. G (7.6)
(
R1' new = E2 − D' R2' − DΔ
))
−1
D' R1' .
(7.7)
/ " < % D' 0. D , % D'-1 . ;" (7.7) D' !:
(E
2
(
− D' R2' − DΔ
(
(
))
−1
= D'−1 − R2' − DΔ
228
(
(
D' R1' = D'−1 E2 − D'−1 D' R2' − DΔ
))
−1
(
R1' = D '−1 + DΔ − R2'
)
−1
R1' .
))
−1
R1' =
D'-1 + DΔ.
" %. D % D' riout
riout − Δrii
-1
,
"
<
D'
.
riout − Δrii
riout
Δr
D DΔ outii .
ri
/ D " i :
(D'
(E
−1
2
+ DΔ − R2'
− R2' new
)
−1
)
−1
(
)
R1' , < , (
)
R1' ,
R1' = E2 − R2'
R1' new = E2 − R2'
E
§
'∞
= ¨¨
Rnew
'
© E2 − R2
(
riout − Δrii Δrii
+ out = 1 . $ riout
ri
)
−1
−1
−1
,
O1 ·
¸ = R '∞ .
O2 ¸¹
R1'
; ! , ! " < % R % , % . 7.5. : . < , , % R+ " – %, " < (R–).
§0
¨
¨0
R+ = ¨ 6
¨
¨1
¨0
©
0
0
1
6
0
0
0
2
6
1
0
0
3
7
3
0·
§0 0
¸
¨
0¸
¨0 0
¸
5 , R− = ¨ 6 1
¸
¨
1¸
¨1 6
¸
¨0 0
4¹
©
0
0
0
6
1
0
0
3
0
3
%, 0·
¸
0¸
5¸ .
¸
1¸
0 ¸¹
:
229
0
0
0
0·
0
0 0 0·
§1
§1
¸
¨
¸
¨
1
0
0
0¸
0 0 0¸
¨ 0
¨0 1
1
2
3
5¸
3 5¸
¨ 6
¨6 1
0
¨ 17 17 17 17 17 ¸
¨ 15 15
'
'
15 15 ¸ .
, R =
R+ = ¨
1
6
6
7
1¸ − ¨1
6
6
1¸
0
¸
¨
¸
¨
14 ¸
¨ 21 21 21 21 21 ¸
¨ 14 14 14
1
3
4¸
3
¨¨ 0
¨¨ 0 0 1
0
0 ¸¸
¸
8
8
8¹
4 4
¹
©
©
* % . ! :
, 0
0
§ 1
¨
1
0
¨ 0
∞ ¨
R' = 0.6161 0.3839 0
¨
¨ 0.3661 0.6339 0
¨ 0.4286 0.5714 0
©
0
0
0
0
0
0·
¸
0¸
0¸
¸
0¸
0 ¸¹
7.4.2. !* '*$/ %%( ( %+%
7.3. $ )%)0 $ l (7.1). 0 %+ W %+ "$ 0 0 Q(0) = (q1(0), …, qn(0)) $ 0 " &0 :
Q* = Q(0)R'∞,
(7.8)
R'∞ – $0 , 00 (7.5).
$ .
1. K , > #% 2, . " t : Q(t)= Q(0)R't. > , (7.8).
2. > #% 1 ( ! , !
).
* t % ! R(t), % R " 230
< > , #% 1. ; ­rii , qi (t ) ≤ riout ,
°
rii (t ) = ®
q (t ) − ¦ rij , qi (t ) > riout
° i
j ≠i
¯
* % R(t), R'(t), a) ∀t Q(t + 1) = Q(t)R'(t);
b) ∀t Q(t) % R, = (q1(0), …, qn(0)).
;" #% %
% &. < %
:
, Q(0) =
:
Q(t + 1) = Q(t ) R' (t ) = Q(0)(R' (0) R' (1) ⋅ ⋅ ⋅ R' (t ) ) .
>" m > 2. ;" , >" m + 1, % &: R(t) = R R'(t) = R'.
/ Q(m+k) :
§ m
·
Q(m + k ) = Q(0)¨¨ ∏ R' (t ) ¸¸ R'k .
© t =0
¹
k → ∞ :
§ m
·
§ m
·
Q* = Q(0)¨¨ ∏ R' (t ) ¸¸ lim R'k = Q(0)¨¨ ∏ R' (t ) ¸¸ R'∞ .
→
∞
k
© t =0
¹
© t =0
¹
7.2 % R'∞ ! " < .
;" t = 0, …, m R'(t)R'∞ = R'∞.
$ Q* = Q(0)R'∞. ; 7.3. ' $ 0 0 Q* " 0 0 $ # :.
231
; 7.4. $ )%)0 $ l (7.1). D& i- &
R'∞ (i > l) & $ 0 0 "$ 0 Qi(0) = (0, …, 0, 1, 0, …, 0),
i-0 :
§ e1T ·
¸
¨
¨ ... ¸
¨ T ¸
e
R '∞ = ¨ *l ¸ ,
¨ Ql +1 ¸
¸
¨
¨ ... ¸
¨ Q* ¸
© n ¹
(e1, …, el – & l -+ " & (E)n×n).
$ # (7.8).
| " 7.6. % R'∞ ! % L = E – R', 1.2.4. ; < % "#, " " "# !. N "# " , <" , ! % > . % R'∞ ! !" " " % L.
" ! !" L [3, 63].
7.6. %:
§0
¨
¨0
R = ¨6
¨
¨1
¨0
©
0
0
1
6
0
0
0
2
6
1
0
0
3
7
3
0·
¸
0¸
5¸ .
¸
1¸
4 ¸¹
%, :
232
0
0
0
0·
§ 1
¸
¨
1
0
0
0¸
¨ 0
1
2
3
5¸
¨ 6
¨ 17 17 17 17 17 ¸
.
R' = ¨
1
6
6
7
1¸
¸
¨
¨ 21 21 21 21 21 ¸
1
3
4¸
¨¨ 0
0
¸
8
8
8¹
©
: , – " ! .
Q1(0) = (0, 0, 1, 0, 0) → Q1* = (0.6161, 0.3839, 0, 0, 0);
Q2(0) = (0, 0, 0, 1, 0) → Q2* = (0.3661, 0.6339, 0, 0, 0);
Q3(0) = (0, 0, 0, 0, 1) → Q3* = (0.4286, 0.5714, 0, 0, 0).
;" R'∞ ! :
0
0
§ 1
¨
1
0
¨ 0
∞
¨
R' = 0.6161 0.3839 0
¨
¨ 0.3661 0.6339 0
¨ 0.4286 0.5714 0
©
0
0
0
0
0
0·
¸
0¸
0¸
¸
0¸
0 ¸¹
7.4.3. + 6# '*#-9&") %%( +
7.3 " " . K Q(0) = αQ1(0) + βQ2(0), ! Q* = αQ1* + βQ2*, " " t
Q(t) = αQ1(t) + βQ2(t) " . D #% > . , . / % !:
233
§0
¨
¨0
R = ¨6
¨
¨3
¨1
©
0 0 0 0·
¸
0 0 0 0¸
1 4 5 7¸ .
¸
6 5 1 7¸
1 1 2 4 ¸¹
7.7.a. * Q1(0) = (0, 0, 100, 0, 0),
Q1* = (62.5, 37.5, 0, 0, 0).
. 7.5.
: . 7.5. > Q1(0) = (0, 0, 100, 0, 0)
7.7.b. * Q2(0) = (0, 0, 0, 100, 0),
Q2* = (46.154, 53.846, 0, 0, 0).
. 7.6.
234
: . 7.6. > Q2(0) = (0, 0, 0, 100, 0)
7.7.c. : "# , > . $ – "# ,
. 7.5–7.6; – "
Q3(0) = Q1(0) + Q2(0) = (0, 0, 100, 100, 0).
G . 7.7. 7.7.! > , "# > . $ . * " " , " ! > > !
#% 1. K > #% 2, >, " .
235
120
100
80
v1
v2
60
v3
40
v4
v5
20
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
: . 7.7.. : Q1(0)+ Q2(0), Q1(t)+ Q2(t).
Q1*+ Q2* = (108.654, 91.346, 0, 0, 0)
120
100
80
v1
v2
60
v3
40
v4
v5
20
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
: . 7.7.+. A% Q3(0) = Q1(0)+ Q2(0) = (0, 0, 100, 100, 0).
Q* = (108.654, 91.346, 0, 0, 0)
236
##* + ! 7
" ! " : % # " ( <" ), % # . $ " , " ( , " 0): ! " #% . K
T = 0, <## % , T
> χ =
" rsum
(0, 1], " . " . / . D ,
" . > ; > t → ∞. K , , .. #% , " ! " <" .
* " " . G , " , ! !,
, % , ..
% !% % # (7.3). ;" . * # Q* = Q(0)R'∞, " R'∞ – % R'.
#, ( 7.2): < , %
.
237
8. @
Q @ BBB @A B BA
DR @,
:
• & .
• ( , & :
ΔW = W – T ! , &# ## (&
& ), &# & , # & & .
• &; , .
• " , & &.
: " . K > , , W > T > "" ! . % , < , " , , < " . $ " " . ;, " , , ! . / " < . % <" #% % .
8.1. *
(n = 5) v1 v2 %:
238
§1
¨
¨1
R =¨6
¨
¨1
¨1
©
1
1
1
6
1
1
1
1
1
1
1
1
1
1
1
1·
¸
1¸
1¸ .
¸
1¸
1¸¹
(8.1)
$ .
ρ = ((10, 5), (10, 5), (5, 10), (5, 10), (5, 5)); W = 1: Q1* = (0,250, 0,250, 0,1(6), 0,1(6), 1(6)); " -
~
: T = 20; Q = (5, 5, 3,(3), 3,(3), 3,(3)).
v5
v3
v4
v1
v2
: . 8.1. , % (8.1)
G . 8.1 , < " .
: Q(0) = (0, 5, 20, 0, 0). W = 25 ! > "" , , > . =
v2 , v1 – .
" Q(0) ! Q* =
= (10, 5, 3,(3), 3,(3), 3,(3)). , > > ( , ! 239
> , ! ! > !). /> v2 ! > ", .
, >
!, % ! ! Z+*. 25 : Q(0) = (0, 8, 17, 0, 0). < ! : Q* = (10, 5, 3,(3), 3,(3), 3,(3)). K 8 > v4 ( , v2 ! ! > !), , ! ! Q(0) = (0, 0, 17, 8, 0), ! : Q*=
= (9,75, 5,25, 3,(3), 3,(3), 3,(3)). ; < > ! , " ! , " ".
; ! , " , " , .
/ < " ! , 1. / > , « » ! > .
8.1. 404( %/, %%904(
% ##* &+, ''*! *9(*%/ δW
G % W > T. / " 4 ! > ! " , > . K
, . K
% , . $ > " " ,
< ! < " .
: , % (8.1). Q1*=(0,250, 0,250, 0,1(6), 0,1(6),
0,1(6)), T = 20.
1
/ , , !! ,
> , . * ! > " # .
240
W > T
. * ΔW = W – T W.
: :
Q(0)=(0, 0, 50, 0, 0) → Q*= (29,3, 10,7, 3,(3), 3,(3), 3,(3)).
(8.2)
Q(0)=(0, 0, 90, 0, 0) → Q*= (59,3, 20,7, 3,(3), 3,(3), 3,(3)).
(8.3)
> , Z–*, 10. Q(0) = (0, 0, 50, 0, 0)
Q(0) =
= (0, 0, 90, 0, 0) Z+* , 40 80 . 40 , 30 10, .. 0,75ΔW 0,25ΔW, .. % 3:1.
!
> % 3:1 ΔW . / ( . 8.1), (> v4), " % 0,25ΔW 0,75ΔW (, 1:3). = > , 0,5ΔW, 0,5ΔW, .. % 1:1.
!^ <" # " , , .. , < % R, ! , . & % ! ! Rabsorb.
§0
¨
¨0
Rabsorb = ¨ 6
¨
¨1
¨1
©
0
0
1
6
1
0
0
1
1
1
0
0
1
1
1
0·
¸
0¸
1¸ .
¸
1¸
1 ¸¹
%
!
R '∞absorb . * < %, > 241
# 7.4 (" 7). <" " , > :
Q(0) = (0, 0, 1, 0, 0) → Q*= (0,75, 0,25, 0, 0, 0);
Q(0) = (0, 0, 0, 1, 0) → Q*= (0,25, 0,75, 0, 0, 0);
Q(0) = (0, 0, 0, 0, 1) → Q*= (0,5, 0,5, 0, 0, 0).
;" 0
0
§ 1
¨
1
0
¨ 0
∞
¨
R'absorb = 0,75 0,25 0
¨
¨ 0,25 0,75 0
¨ 0,5 0,5 0
©
0
0
0
0
0
0·
¸
0¸
0¸ .
¸
0¸
0 ¸¹
(8.4)
G % ΔW - > .
/ , < % % R,
! " ! !
, . ! " . $
! > W
% , R '∞absorb , > ?
< , ! % R. ! ! , ! ( . 8.1) ! >.
8.2. $ $# . %
§ 0,5 0,5 0,5 0,5 0,5 ·
¸
¨
¨ 0,5 0,5 0,5 0,5 0,5 ¸
1
1
1
1 ¸.
R =¨ 6
¸
¨
6
1
1
1 ¸
¨ 1
¨ 1
1
1
1
1 ¸¹
©
242
(8.5)
< = 10; Q1* = (0,250, 0,250, 0,1(6), 0,1(6), 1(6));
~
Q =(2,5, 2,5, 1,(6), 1,(6), 1,(6)).
A% :
Q(0)=(0, 0, 85, 0, 0) → Q*= (60,5, 19,5, 1,(6), 1,(6), 1,(6));
Q(0)=(0, 0, 45, 0, 0) → Q*= (30,5, 9,5, 1,(6), 1,(6), 1,(6)).
:, 5, Z–*; . : < ΔW = 40 % 30:10, ..
3:1, >.
D > ! . * ! ! > % , ! . * ! ! , ! !, > v5
" " .
8.3. '&+ "& 0 D : : : & R. %
§3
¨
¨1
R =¨6
¨
¨1
¨1
©
1
8
1
6
1
1
1
1
1
1
1
1
1
1
1
0·
¸
0¸
1¸ .
¸
1¸
1 ¸¹
: = 23; Q1* = (0,261, 0,478, 0,116, 0,116, 0,029);
~
Q = (6, 11, 2,(6), 2,(6), 0,(6)).
, ; ~
%
Q . / Z–* , 6.
243
: ( → ): Q(0) → Q*.
Q(0)=(0, 0, 86, 0, 0) → Q*= (56,910, 23,090, 2,(6), 2,(6), 0,(6)).
Q(0)=(0, 0, 46, 0, 0) → Q*= (26,910, 13,090, 2,(6), 2,(6), 0,(6)).
: ΔW = 40 % 3:1,
> " .
D % " . " " , ,
. : Q(0) → Q*:
Q(0)=(0, 0, 86, 0, 0) → Q*= (56,910, 23,090, 2,(6), 2,(6), 0,(6)).
r1out = 6, r2out = 11.
Δq1 = q1* − r1out = 50,910, Δq2 = q2* − r2out = 12,090
Δq1 : Δq 2 = 50,910 : 12,090 > 4.
%
/ % (8.1)
. G 8.1
:
< -
Q(0)=(0, 0, 100, 0, 0) → Q*= (66,8, 23,2, 3,(3), 3,(3), 3,(3)).
< % r1out = 5, r2out = 5.
Δq1 = q1* − r1out = 61.8, Δq2 = q2* − r2out = 18,2.
, % 3:1 .
/ Δ q1 − 3Δq 2 :
Δq1 − 3Δq2 = 61,8 – 18,2 ⋅ 3 = 61,8 – 54,6 = 7,2.
244
, < " W.
G (8.2), (8.3) , W = 50 W = 90 :
Δq1 − 3Δq2 = 7,2.
(8.6)
! > W , % (8.1), (8.6) . $!
" δW. * " Wmin ≥ T, δW = const.
" δW = 0. " " (.. !), δW . J δW
0 $.
$ > % δW l " " Q(0),
, ! > Wmin, - > . R – % ! ,
1 l. Q* – " W > 2rsum ( ~
2rsum ! > Wmin). ΔQ = Q* – Q .
K" n – l , l , !. Rabsorb – % " , l , n – l % R. : " " < " #%
Q(0) = QW(0), Q* = Q*W. J W ( % QW(0)) " ε > 0 , - ! Q*W U ΔQ. $! < W'. ;" ΔQ – Q*W' % . G " δW. D , > % ,
" , . D .
G ! , # .
0 $ δW , >" % , % Rabsorb:
~
δW = (Q* – Q – Q*W')⋅1, " 1 – - !% n %.
245
" ! Wmin
!^ , δW, > % ,
, #% W > T,
! " , .. , W – T ≤ δW.
8.4. % (8.1). * W = T,
! , " > . < = 20. / >
.
~
Q(0)=(0, 0, 20, 0, 0) → Q = (5, 5, 3,(3), 3,(3), 3,(3)).
Q(0)=(0, 0, 25, 0, 0) → Q*= (10, 5, 3,(3), 3,(3), 3,(3)).
Q(0)=(0, 0, 27, 0, 0) → Q*= (12, 5, 3,(3), 3,(3), 3,(3)).
Q(0)=(0, 0, 27,2, 0, 0) → Q*= (12,2, 5, 3,(3), 3,(3), 3,(3)).
Q(0)=(0, 0, 27,21, 0, 0) → Q*= (12,2075, 5,0025, 3,(3), 3,(3), 3,(3)).
G < , W – T ≤ 7,2 % , . : " r2out = 5 W = 27,2, , , δW= 7,2.
W > 27,2 Z+*, > :
(q1* − 5 − 7,2) : (q2* − 5) = 1 : 3 .
/ ! ( !
!) , . 8.1, c v1 v2, v3 v4 , . K > v3, < " W ≥ T+δW, > :
'∞
'∞
(q1* − r1out − δW ) : (q2* − r2out ) = rabsorb
31 : rabsorb32 ,
246
" % R'∞absorb # (8.4).
; , δW, .
9 8.1. $ 0 " 0 v1 v2 W > T "$ 0
'∞
'∞
: 0 " vj, 0 rabsorb
j1 = max rabsorb k 1 . k ∈{3,... n}
$ 0 0 :
~
­°Q + (W − T , 0, 0, ..., 0), W ≤ T + δW
Q* = ® ~
'∞
'∞
°̄Q + (δW + rabsorb j1 ⋅ ΔW , rabsorb j 2 ⋅ ΔW , 0,...,0), W > T + δW
(8.7)
ΔW = W − (T + δW ) , R '∞absorb – $0 )%) , %) 0 .
$ . ! . v2 ! ! , #% 2, .. . = v1 #% 1 ! , ! > <"
. G > v1. ;
! ,
δW , " , ! 1, " ! 2. K " W ≤ T + δW,
δW v2 r1out , v2 Z+(t), (8.7) .
K
W > T + δW, ! Z+(t).
/ t' : q2(t' – 1) ≤ r2out
q2(t') > r2out . ;" q1(t') = r1out + δW . , " Z+(t) % , R 'absorb . / : Z+(t) !
out
out
r1 r2 – ! ! . * , ! > ". $ % ! ! , 247
! ( ) : f1in* = f1out* = r1out , f 2in* = f2out* = r2out . K
! , %, ! , < % Rabsorb .
; ! , T + δW % , % R '∞absorb . N " !! " . l v1, …, vl, W > T vj, j > l. $! δWi – vi ! W ≥ T – ! > , ! % R '∞absorb . P, δWi = const, " , Z+(t), !
! % , % R '∞absorb .
8.1. $ 0 " l v1, …, vl W > T "$ 0 : 0 " vj, j > l. $ 0 0
:
~
­Q + (0, ..., W − T , 0, ..., 0), W ≤ T + (δWm − δW− m )
°° ~
Q* = ®Q + (δW1^ , ..., δWl ^ , 0,...,0), T + (δWm − δW− m ) < W ≤ T + δW
°~
'∞
'∞
°¯Q + (δW1 + rabsorb j1 ⋅ ΔW , ..., δWl + rabsorb jl ⋅ ΔW , 0,...,0), W > T + δW
(8.8)
(0, …, W – T, 0, …, 0) % '∞
m ≤ l, m : 0 0: m = arg max rabsorb
km ;
k ∈{1,..., l }
0 ≤ δWi^ < δWi – T + (δWm – δW–m) < W < T + δW;
δWm = max δWk – , %)0 vm;
k∈{1,...,l }
δW− m =
max
δWk ∈{δW1 ,...,δWl }\δWm
, δWk – 0 " " vj;
l
δW = ¦ δWi ; ΔW = W − (T + δW ) ;
i =1
" ) vi, 0 δWi = 0, i ≤ l.
248
-
$ .
# (8.8) .
1. # W < T + (δWm − δW−m )
8.1. = vm 1 ! > , δWm^ = (W – T). K ( δWi) 1, > > vm. " , 1 , " vm
(δWm − δW−m ) . $ ~
Q* = Q + (0, ..., W − T , 0, ..., 0) .
2. , δW, #% Z+(t), δWi^ =δWi^(W) ! , " ! ~
out
out ~
~
Q = (r1 , ..., rl , ql +1(0), ..., qn (0)) . "
T + (δWm − δW−m ) , " " Z+(t). K % , ! ! δWi,
.. δW1 := δWm, δW2 := δW–m, . ., W, k ! δWk^(W) > 0. <" !
, ! , >
k, . :!
" W > T
Z+(t) " .
1) T < W ≤ T + (δW1 – δW2): v1 #% 1. / ! : T + (δW1 – δW2) = T1. " % ! ! Ti.
2) T1 < W ≤ T + (δW1 – δW2) + 2(δW2 – δW3) = T + δW1 + δW2 –
2δW3 = T2: v1 v2 #% 1.
3) T2 < W ≤ T + (δW1 – δW2) + 2(δW2 – δW3) + 3(δW3 – δW4) = T
+ δW1 + δW2 + δW3 – 3δW4 = T3: v1, v2 v3 #% 1.
…
l – 2) Tl–3 < W ≤ T + δW1 + … + δWl–1 – (l – 1)δWl = Tl–2: v1, v2, …, vl–2 #% 1.
249
l – 1) Tl–2 < W ≤ T + δW1 + … + δWl–1: v1, v2, …, vl–1
#% 1, vl–1 " " " .
l) W > T + δW1 + … + δWl–1 = T + δW "
. Z+(t),
< , vl : δWl = 0, ! > , .
3. ; # (8.8) l). W > T + δW Z+(t), !
. %
R '∞absorb . |" 8.1. K > -, '∞
m = arg max rabsorb
km , , W – T k∈{1,..., l }
# (8.8) <
.
| " 8.2. "# # 1) ÷ l).
* . 8.2, , – !% . $ , . I , ! > > - . («" ») !% > . > > ! % , % R '∞absorb .
" " , ! , % . y " !% . * . 8.2, +
% .
D " " " 1) ÷ l).
; 8.1. * , l % , > - vj, j > l, " W > T ! l + 1 , " δWi ! i – 1 > (i = 1, …, l + 1). ' % # . 1) ÷ l).
250
δW1
δW2
δW3
…
δWl–1
v1
v2
v3
…
vl–1
: . 8.2, . " .
: δWi ! (δW1 – δW2) – < > > Z+(t).
δW1
δW2
(δW1 + δW2 – 2δW3)
δW3
(δW1 + δW2 + δW3– 3δW4)
…
δWl–1
v1
v2
v3
…
vl–1
: . 8.2, +. " .
Z+(t), 251
W > T + δW1 + … + δWl = T + δW > , T + δW % R ' ∞absorb .
δWi < ! .
I! , δWi %
!, . K < , " .
8.2. % #%/ δW ")$") '*'9%!")
%'%:%+ *!*
/ % .
δW > , < δW1 = δW
δW.
δW ! > -: δW = δW riout , i = 1, 2, , % R , % (8.1) (8.5).
8.5. / %
( )
§r
¨
¨r
R = ¨6
¨
¨1
¨1
©
r
r
1
6
1
r
r
1
1
1
r
r
1
1
1
r·
¸
r¸
1¸
¸
1¸
1 ¸¹
! ! r >" . < ! . 8.1.
+ 8.1
"A"'/:A)(/)') -A:#:*+,%>% (%(/%G,)G
-A) A"K,@J C@J%#,@J -A%-B(',@J (-%(%E,%(/GJ "//A"'/%A%C
~
R
r
δW
r1out = r2out T
Q
1.
2.
3.
4.
5.
252
0
0,05
0,1
0,15
0
0
0,25
0,5
0,75
0
0
1
2
3
0
(0,25, 0,25, 0,1(6), 0,1(6), 0,1(6))
(0,5, 0,5, 0,(3), 0,(3), 0,(3))
(0,75, 0,75, 0,5, 0,5, 0,5)
0
0,875
1,75
2,625
0
R
r
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0,5
0,55
0,6
0,65
0,7
0,75
0,8
0,85
0,9
0,95
1
1,05
1,1
1,15
1,2
1,25
1,3
1,35
1,4
1,45
1,5
out
1
r
0,25
0,5
0,75
1
1,25
1,5
1,75
2
2,25
2,5
2,75
3
3,25
3,5
3,75
4
4,25
4,5
4,75
5
5,25
5,5
5,75
6
6,25
6,5
6,75
7
7,25
7,5
=r
out
2
} " +. 8.1
δW
T
~
Q
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
(0,25, 0,25, 0,1(6), 0,1(6), 0,1(6))
(0,5, 0,5, 0,(3), 0,(3), 0,(3))
(0,75, 0,75, 0,5, 0,5, 0,5)
(1, 1, 0,(6), 0,(6), 0,(6))
(1,25, 1,25, 0,8(3), 0,8(3), 0,8(3))
(1,5, 1,5, 1, 1, 1)
(1,75, 1,75, 1,1(6), 1,1(6), 1,1(6))
(2, 2, 1,(3), 1,(3), 1,(3))
(2,25, 2,25, 1,5, 1,5, 1,5)
(2,5, 2,5, 1,(6), 1,(6), 1,(6))
(2,75, 2,75, 1,8(3), 1,8(3), 1,8(3))
(3, 3, 2, 2, 2)
(3,25, 3,25, 2,1(6), 2,1(6), 2,1(6))
(3,5, 3,5, 2,(3), 2,(3), 2,(3))
(3,75, 3,75, 2,5, 2,5, 2,5)
(4, 4, 2,(6), 2,(6), 2,(6))
(4,25, 4,25, 2,8(3), 2,8(3), 2,8(3))
(4,5, 4,5, 3, 3, 3)
(4,75, 4,75, 3,1(6), 3,1(6), 3,1(6))
(5, 5, 3,(3), 3,(3), 3,(3))
(5,25, 5,25, 3,5, 3,5, 3,5)
(5,5, 5,5, 3,(6), 3,(6), 3,(6))
(5,75, 5,75, 3,8(3), 3,8(3), 3,8(3))
(6, 6, 4, 4, 4)
(6,25, 6,25, 4,1(6), 4,1(6), 4,1(6))
(6,5, 6,5, 4,(3), 4,(3), 4,(3))
(6,75, 6,75, 4,5, 4,5, 4,5)
(7, 7, 4,(6), 4,(6), 4,(6))
(7,25, 7,25, 4,8(3), 4,8(3), 4,8(3))
(7,5, 7,5, 5, 5, 5)
0,875
1,75
2,625
3,5
4,375
5
5,375
5,75
6,125
6,5
6,875
7,2
7,28
7,3
7,31
7,32
7,335
7,35
7,365
7,2
6,945
6,67
6,4
6,13
6,155
6,16
6,04
5,78
5,515
5,18
;! % . ! , ( ! " 4), %
§1,5 1,5 1,5 1,5 1,5 ·
¸
¨
¨1,5 1,5 1,5 1,5 1,5 ¸
R =¨ 6 1 1 1 1 ¸
¸
¨
¨1 6 1 1 1¸
¨1 1 1 1 1¸
¹
©
253
% . D > v1,
v2 v5. / , > v1, v2 v5 :
r5out
r1out r2out
7,5
5
=
30;
= 30;
=
=
=
1*
1*
1*
0,25
1/ 6
q1
q2
q5
<
! . 8.1 , < T = 30. ;
! , 4.4 < > .
, ! % "# ( . 8.3). G " , δW ! . <
, , " ; !, ! 0,95,
" , !. $ !, 1,25 1,3, >.
: . 8.3. δW ! ( "" )
254
8.3. $*$( <'/ *!, $ # !
%)% &%! ) #* < "& % δW
A% , ! >
"" , % &.
* t % ! R(t), % R " < > , t #% 1. ; ­rii , qi (t ) ≤ riout ,
°
rii (t ) = ®
q (t ) − ¦ rij , qi (t ) > riout
° i
j ≠i
¯
* % R(t), R'(t), % &. /
%
t
Q(t + 1) = Q(0) ∏ R ' (m)
m =0
, % R, Q(0).
K j, % R(0) ! % R j- . / % #% > " Z+(t), " % R(t) ! %
R. >" Z+(t) !
, % . / % R(t) % R'(t) % R R'.
$ > %, , " > #% . % . W > T + δW. ;" , , , ,
#% : 1, , ! , 2.
255
: <, #% .
0. / Z+(0) > – W > T.
1. > -. < , , < > #% 2. I < > 1. Z+(t) > : .
2. / 1. Z+(t) > : . / !
.
3. G Z+(t). / Z+(t) .
* 1 2 # δW. * 2 3 # , R'∞absorb .
K 2 , ,
! > ! .
.
3 Z+(t) !
% R'(t), > , #% 1, .
: %, " . $ 1-3, % . % R' : R' = (r1 ' , r2 ' ,..., rn ') , " rk' – !% R'. vj –
> - , ,
v1, v2 – ; Z+(t) v1. ;" " < 0-3 ! :
(
)
(
)
0. F out (t + 1) = F out (t ) r1 ' , r2 ' , r3 ' ,...,e j ,..., rn ' , t < t1;
1. F out (t + 1) = F out (t ) e1 , r2 ' , r3 ' ,...,e j ,..., rn ' , t1 V t < t2;
(
)
(
)
2. F out (t + 1) = F out (t ) e1 , e2 , r3 ' ,...,e j ,..., rn ' , t2 V t < t3;
3. F out (t + 1) = F out (t ) e1 , e2 , r3 ' ,..., r j ' ,..., rn ' , t Y t2.
256
$!
%, Fout(t), Pj, P1j, P12j P12, . " <
# :
F in (t + 1) = F out (t ) R' .
G " > , δW :
t2
(
)
t2
(
)
δW = ¦ F in (t + 1) − F out (t + 1) = ¦ F out (t ) R '− F out (t ) P1 j =
t =t1
t2
(
t =t1
)
= ¦ F out (t ) R '− P1 j ;
t =t1
& % (R' – P1j) !% 1 j.
! < , > " . / " <
F out (t + 1) = F out (t ) P1 j , , F out (t ) = F out (t1 )(P1 j )t −t1
;" §
·
δW = F out (t1 )¨¨ ¦ ( P1 j ) t −t1 ¸¸(R'− P1 j ).
t2
© t =t1
¹
1 j,
/ δW
. δW = δW 1 .
δWi l (i = 1, …, l) " . ! l + 1 – .
( )
8.4. *$/ %%( % '* fsum(t) ≥ T
, δWi (i = 1, …, l) , " > , .. " . 257
" δW ! >. / δW = 0. ;, , < , > , ! > " W = T, > ! % , % R '∞absorb . , <
! " .
8.2. $ 0 0 "0 0
$ l $& (l > 1) &
W > T, "$ 0 Q(0) :
~
~
Q(0) = Q + ΔQ(0) , Q – $ 0 W = T, ΔQ(0) –
$& $& , %)
# W – T. $ 0 & : :
~
Q* = Q + ΔQ(0) R'∞absorb .
(8.9)
$ . " " > , > ~
~
Q , : F out (t ) = F + ΔF out (t ) ,
~
" ! F ! W > T, ! ΔFout(t), > , > ΔQ(t). W = T !
! > ", ΔFout(t) – , ΔQ(t), , . ;
! , > :
§E
'
'
ΔQ(t+1) = ΔQ(t) Rabsorb
, " Rabsorb
= ¨¨ '
© R1
/ O1 ·
¸.
R2' ¸¹
:
E
O1 ·
§
'∞
¸¸ = ΔQ(0) Rabsorb
ΔQ* = ΔQ(0) ¨¨
.
' −1 '
(
E
−
R
)
R
O
2
1
2 ¹
© 2
(8.10)
258
8.5. &/" %%( (, %6$04 ''*!9
; 8.2 ", % , " . $ , . # (8.9) " . K , " ,
δWi , i = 1, …, l, . D " +: &: $&: # .
k l+1, …, l+k. " Q(0), ! > , ΔW j # ΔW j =
¦
i∈{1,...,l ,l + k +1,...,n}
d i (0)
riin−
r ji ,
(8.11)
" riin − – ! > Z–(0)
! > - , di(0) – # % > Z–(0): i = 1,…, l, l+k+1,…n.
/ ΔW j ! !
> . $! # (8.11) .
8.3. $ 0 " l $& 1, …, l k
" l+1, …, l+k, "$ 0 :
Q(0) = (q1 (0),..., ql (0), q~l +1 + ΔWl +1 + ΔQl +1 ,..., q~l +k + ΔWl +k + ΔQl +k ,...,q n (0) ),
" & ΔW j (j = l+1, …, l+k) – +: & $& #
" , & (8.11), ΔQ j ≥ 0 ,
j = l+1,…, l+k. $ 0 :
~
Q* = Q + ΔQ ⋅ R'∞absorb ,
259
Δ Q – & n &
ΔQ j
(j = l+1, …, l+k).
$ . I! % ,
, ! > , 8.3.
* , " #% Z–(t) Z+(t): Z+(t) ,
, Z–(t), .
! > 1 (< (" 4)). , δW =0.
* !
>
> - .
> vi, Q(0) # % , !
riin − ! > - . # % > vi ! , , di(0). ;" , ! # % ~
> , Q , vj
¦
d i (0)
r ji , " i !" riin−
> , # % .
d i (0)
r ji .
$ # (8.11): ΔW j =
¦
in−
i∈{1,...,l ,l + k +1,...,n} ri
Δ Q # : ΔQ j = q j (0) − q~ j − ΔW j . , ! " .
8.6. % (8.1) ( . 8.1). W > T , , " ": Q(0) = (0, 0, 20, 40, 0). ;" < ! Q* = (20, 30, 3,(3), 3,(3), 3,(3)).
$ > " Q*. / " , ~
, Q , # % > i∈{1,...,l ,l + k +1,...,n}
260
% < # % ! , > . 1
3
5
> 3 =
2 3
r1out
r2out
> - 5 % – ! % . G" 20
10
! 3,(3) =
.
3
3
ΔW3, 4 =
20 ~
10
, q3, 4 =
3
3
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10
–
=
3
3
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D > > % , , 3:1 1:3 ( . 8.4).
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D " .
G
> > v3 ΔQ3 = 20 –
8.6. <! ''*! δW
; 8.2–8.3 , " δWi = 0. " δWi
", .
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l
§
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©
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261
v3
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2,5
7,5
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22,5
v2
r2out=5
> " 0 l0 = arg min r '∞absorb jk &k
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'∞
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ji
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rabsorb
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.
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8.2–
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Q2 (0) = (q1 (0), q2 (0), ..., q~l +1 + 2ΔWl +1 ,..., q~l +k + 2ΔWl +k ,...,q n (0)) ,
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,
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: :
~
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;
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B @B D
= 3.6
— — — % 3.6
/> — 2.4
— 2.4
— 2.4
— 2.3
Z–(t) 2.1
Z+(t) 2.1
( ) 2.3
— < 3.3
— <" 2.3
<## % 3.5
!
! %
:
— # % 2.2
— " " . 2, . 3.4
— 2.1
— # % 2.2
— : 2.1
— 2.4
— 2.2
— " 2.3
— " 2.3
— 2.4
— % 2.3
— < 2.4
— <" 2.3
2.1
— 2.1
— 6.1
— 2.1
A% — 1
— 2
+ " . 6 ( )
282
R – % ! riin , riout – !
> – i–" !% i– % R, rsum – ! R' – %, % ! R ( % R, )
W – – " Q(t) – t
Q* – " Q1* – " W = 1
~
Q – " W = F(t) – % t
Fin(t), Fout(t) – " " > fsum(t) – Fin*, Fout* – ~
~
F in , F out – W = –
t ,
Z (t) – > vi,
out
ri ( # % )
t ,
Z+(t) – > vi,
out
ri ( # % )
D(t), C(t) – # % # % > ~
C m – # % > < ρ – , > > !
χ – <## % δW – " " 283