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Intellectual Technologies on Transport. 2015. ?2
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serega_svetl@mail.ru
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??????? ?????????? ???????????????? ?????????, ??????????? ?????? ?????????????? ??????? ????????????.
???????????? ?? ??? ?????? ???????? ???????? ??????????
?????? ?????????????. ? ??? ??????? ??????? ???? ??????????? ????? ??????? ?????? ?????????????? ??????
???????????? ??? ??????? ? L-?????????. ????? ??????????
????????? ???????? ?????????? ??????? ?????????????
??? ??????? ??????.
???????? ?????: ??????? ?????????????? ??????
????????????,
????????
?????????
???????
?????????????, ??????? ?????? ?????????????? ???????
????????????.
????????
????????????? ?????? ??????????? ???????????????????? ?????????? (???) ???????? ??, ??? ??? ???????? ???, ?????? ?????, ?????? ???? ?????????????
?? ???????????????? ?? ?????? ? ??????????, ?? ? ?
??????????? (?????????) ????????. ??? ??????????? ??????????? ?????????? ???? ????????, ????????? ? ??????????????? ?????? ??????????, ?? ???????? ?????????
????????? ????????? ?????????????? ?????????????.
?????????????? ????? ???????? ?????? ????????? ???????????? (???). ??????????? ??????? ??????????
?????? ??? ? ?????????????, ??? ??????? ?????? ??????????, ?????????? ???????????? ?????, ? ???????????
???????? ?? ????????? ??????? [1-4]. ?????? ?????????? ???????????? ? ????????????? ??????? ????????????
?????? ?????????????? ?????? ???????????? (???).
? [5] ?????? ???????? ????????????? ?????, ? ???????
???????? ? ???????? ??????? ???????????? ???????????????? ????????? (???), ??????????? ???, ??????????? ???????? ????????????????, ??? ??????? ???????
???? ?????, ????? ?????-?????. ? [6] ??????? ??? ??????????? ? ????? ?????? ? ????????????.
?????????? ?????????? ????????????? ???. ??
???? ??? ??????????????? ????????? N ???????? ?? ??-
???????. ????????????? ????????? ?????????? ?????
????????? ??????????? ???????? ??????????? ????????????????? ???????? ? ??????????????? {?1 , ?2 , ? , ?N },
??? ?i ????????????? i-? ??????????? ??????. ???????, ???
??????? ?? ????? ??????. ????? ????????????? ??????
???????????? ???? ???????????????? ? ??????????????? {�, �, ? , 礜 }, ??? 礿 ????????????? j-? ?????????????
??????.
????????? ??????? ? ?????? ?????? ??????? ??????????????? ?????? ??????????? ? ??????? ????????
i = 0, N ? ?????? ??? ?????????? ???????????? ???????? j = 0, N ? i. ??????????? ?????????? ??????? ? ????
?????????? ???????????? ????? Pi,j (t). ?? ????? ?????
K = (N + 1)(N + 2)?2. ?? ???. 1 ???????????? ?????????
????????? ????? ??????????? ???????.
??? ??????????? ????????????? ????????????
?????????? ??????? ???????????? ? ?????????? (i, j)
?????????? ?????? ???????????? Pi,j (t) ??????? ???,
?????? ?? ??????? ???????? ????????? ???????:
P?i,j (t) = u(i) ??i+j Pi?1,j (t) ? 礿+1 Pi,j (t)? +
+ u(j)礿 Pi+1,j?1 (t) ? u(N ? i ? j)?i+j+1 Pi,j (t).
????? u(t) ? ??????? ?????????, ???????? ???
1, ?? > 0,
??(??) = ?
0, ?? ? 0.
(1)
(2)
????????????? ??????? ??? (1) ???????? ????????
?????????? ???????? ?????????. ??? ????? ???? ???????????? ? ????????? ?????:
x? (t) = Ax(t),
(3)
??? x(t) ? ?????? ??????????? ??????? ??????????? K, ?
A ? ?????????? ???????.
???????????????? ?????????? ?? ??????????. 2015. ?2
32
Intellectual Technologies on Transport. 2015. ?2
???. 1. ????????? ????????? ????? ???????????
????? ???????? ????? ????????? ??????? ???, ??????????? ???, ??????????? ???????? ????????? ??????? ? ?? ?????????? ?????. ??, ??? ?????????? ? ??????
??? ????? ?????????????, ????? ??? ????????? ???????
????????????, ????????? ????????? ??????, ??????? ?
??????? ? ?.?., ????? ????????? ??????? ???????????
???????????. ??? ??????? ???????? ????? ?????????
??????? ???, ??????????? ????????????? ??????? ????????????, ??????????? ? [7].
??? ????? N (N=2) ??????? ????????????. ?? ???
????????? ? ???????? ? ??????????????? {??1 , ? , ???? }, ????????????? ????????????
�1
� = [�? , ??] = [�
??,1
�2
�?,2 ],
(4)
??? s=1,K ? ????? ???????; j = 1, 2 ? ????? ??????.
????????? ??????? ? ?????? ?????? ??????? ??????????????? ?????? ??????????? ???????? n (n=0, K),
?????? ??????????? ? ??? ???????? m (m=0, K-n) ? ???????? ????????? ??????? ????? = ??1 , ??2 = ????? ??, ??? ???? = 0,1.
(m, n, l)-??? ????????? ????????? ???:
??????,??,?? (??)
????
??=??
= ??(??) ? ????=1 ???? (?????1,??,???? (??) +
??(?? ? ??)?????1,??,?? ? (??) ? ????+?? + ??(??) ?
??=??
??=??+???2
{???=1 ??????? ???=0
????+1,???1,???? (??) ? �?,?? + ???? ?
????+1,???? (??) ? �?+???1,?? + ???? ? ????+1,???1,???? (??) ? �?+??,?? +
??=??+???1
?????? ? ???=0
????+1,???1,???? (??) ? �?,?? + ???? (??1 ) ?
???????????????? ?????????? ?? ??????????. 2015. ?2
33
Intellectual Technologies on Transport. 2015. ?2
??=??+??
???? (??2 ) ? ???=1
????+1,???1,???? (??) ? �?,?? ]? ? ???(?? ? ?? ?
??????.??????, ?????? = ??;
????????, ?????? = ????????,
??=??
??) ? ????+??+1 + ??(??) ???=1 ??(????) ? �???,?? ? ? ????,??,?? (??), (5)
??? N=2;
???? = ??1 ??? + ?? ? ???? ? ? ??(?? + ?? ? ???? ? 1);
?????? = ??1 ??? + ?? ? ???? ? ? ???? (?? + ?? ? ???? ? 1);
???? = ????(?? ? 2) ? ??(???? ? ????) + ????(?? ? 1) ? ????(????) ? ????;
???? = ??1 ????? ? ? ??(???? ) ? ??(?? + ?? ? ???? );
1, ???? ?? > 0;
??(??) = ?
0, ???? ?? ? 0.
1, ???? ?? = 0;
???? (??) = ?
0, ?????? ? 0.
[?? ? ??, ??2 ], ???? ?? ? 1;
???? = ? 1
[??1 , ??2 ? ??], ???? ?? = 1.
[?? ? (?? + ??) ? ??, ??2 ], ???? ?? ? 1;
/
???? = ? 1
[??1 , ??2 ? (?? + ??) ? ??], ???? ?? = 1.
[?? ? 1, ??2 ], ???? ?? ? 1;
//
???? = ? 1
[??1 , ??2 ? 1], ???? ?? = 1.
///
????
[?? + ?? + ??, ??2 ], ???? ?? ? 1;
=? 1
[??1 , ??2 + ?? + ??], ???? ?? = 1.
////
????
[?? + ??, ??2 ], ???? ?? ? 1;
=? 1
[??1 , ??2 + ??], ???? ?? = 1.
??????? ????? ????????? ? ?? ????????? ? ??? ??????
????? ??????. ??? ???? ?????????? ???? ????? ???????
???????, ????? ?????? ????????? ??? ??????? ?? ?????????????? ?????????. ????????, ??????? ??????, ????????? ? [5]. ??? ??????????? ??????? ???????????? ????????? ????????????? ????? ??????? ???? ??????? ??????? ??????????? ????? ????????? ??????? ? ??? ????????????? ?????? ????????? ??????? ???.
???? ??????
???? ?????? ??????????? ? ???, ??? ?????????? ??????? ??????? ???????? ?? ?????? ????????? ? ?????? ?
?????? ????????? ????????? ?????????. ??? ????????? ??
?????? ????????? ? ?????? ?????????? ??????????? ?????
?????? ?? ????????????, ???? ????????? ??????, ??????? ?
???????? ????????? ??????????????. ????????? ?????????? ????? ???? ????????? ???????, ? ??????? n=0 ?
m=0. ?? ????? ? ????? ???????? ????????? ????????? ?
??? ????? ? ?????? ?????????(Num). ??? ?????????? ????????? Num=1. ????? ????? ?????????? ???? ?? ????????? ????????? ?? ???????? ????????, ????????? ???? ??
????????????? ?????? ????????? ?? ????? ? ???? ???, ??
????????? ??? ? ?????? ????????? (xT). ????? ??????????
????????? ? ??????? ? ?? ????????? ???????:
? ??,
(6)
??? xT.len ? ????? ?????? ?????????, ??????????????? ??
?????? ??????, ? ? ????????????? ???????? ?? ?????????
????????? ? ?????. ?? ????? ???? ????????????? ??????????? ????? ??????, ???? ????????????? ????????? ????????? ??????. ???? ?????????? ????????? ??? ???? ????????????? ? ????????? ? ?????? ????????? ??? ???????
Num1, ?? ??????? (6) ????????? ???:
?????? = ???? ????? ? ? ??(???? ) ? ???? (?? + ?? ? ???? );
1, ???? ?? ? 1;
?? = ?
2, ???? ?? = 1;
??????
????????,
????????1, ?????? = ??;
?????? = ????????, ?????? ? ??.
(7)
???? ??????????????? ????????? ?? ???? ???????? ?????, ?? ????? ???????? ?????????? ???????????? ?????????, ? ??????? ??????? ????? ??????? ?? ?????? ??? ???????????. ??? ???? ? ???????? ?????????? Num=xT.len.
???? ??? ?????? ???? ????????? ????????????? ??? ?????????, ? ??????? ????? ??????? ???????, ?? ???? ???? ???????? ???????????. ???????? ?????????? ??????, ?????
?????????? ??? ????? ????????. ????-????? ?????????
???????????? ?? ???. 2.
???????? ??? ????????????? ???????
???????????????? ????????? ???????? ?????????
??????? ? ?? ??????? ?????? ????????????? ???????,
????????? ????.
??? ??????????? ??????????? ?????? ??????, ??????? ????? ??????? ??????????? ? ?????? ??????, ? ?????? ??? ?????? ??? ??? ???????????? ? ????? ???, ????? =
0, ???? j-? ????? ? ?????? ?????? ????????, ????? = ??, ???? j? ????? ? ?????? ?????? ???????????? i-? ??????.
??? ??? ????????? ??????? ??????????????? ?????????? (n,m,???), ?? ? ?????? ??????? ???????? ????? ??????????? ?????????? ?? ????????? ????????:
1) ?? ????????? ????,??,???, n+m <K, ??????? ?????????
? ????????? ????1,??1,??1
???? , ??? n1=n,m1=m+1, ? ?????????????? ????+??+1 . ??? ???? ????? ???????????
???? ?? ???????:
?? + 1, ???12 = 0
,
a. ???? ???1 = 0, ?? ???11 = ?
?? + 2, ???12 > 0
???12 = ???2 , ???12 = ???2 .
b. ???? ???2 = 0 ? ???1 ? 0, ??
???11 = ???1 ,
?? + 1, ???11 = 0
???12 = ?
.
?? + 2, ???11 > 0
2) ?? ????????? ????,??,???, ???1 ? 0, ??????? ????????? ?
??????
????????? ??
???? , n1=n+1,m1=m-1, ??11 = 0,
???????????????? ?????????? ?? ??????????. 2015. ?2
??1,??1,??1
??????2 = ??????2 ? ?????????????? �? ,1 .
??1
1
34
Intellectual Technologies on Transport. 2015. ?2
?????
?????
??
(??????? ?????????)
S ? ????????? ???????
xT ? ?????? ???? ?????????
Num ? ????? ????????? S
???????????? ??? ???????
????????? ?????? ??????????
???
?????????????
????? ?????????
Snew ??
??????????
???????
??????
????????? ?
??????? ?
??
Snew ???? ? xT?
???
???????? Snew ?
xT
?????
S=Snew
xT=xT
Num=xT.len
??????????? ?????
???. 2. ????-????? ?????????
3) ?? ????????? ????,??,???, ???2 ? 0, ??????? ????? ??????? ? ????????? ????1,??1,??1
???? , n1=n+1,m1=m-1,
??????
???
??????
??11 = ??1 , ??12 = 0 ? ?????????????? �? ,2 .
2
? ???????? ?????????? ????????? ???? ??????? ????????? ????,??,???, n=0, m=0, ??? = [0,0].
???? ????? ?????????, ???????????? ????????????
????? ??? ????????????? ??????? ? ?????????????? ??????????? ??????, ????????? ?? ?????? ??????, ? ?????-
????????? ????????????, ????????? ?? ?????? ?????? ?
?????? ?????? ????????????, ???????????? ?? ??????? 3.
??? ???? ? ????????? ???????????? ??????????????
??????? isExist, ??????? ?????????? ????? ????????? S ?
?????? ????????? xT ??? 0, ???? ????????? ? ?????? ???.
?? ??????????? ???????? S ? ?????????, ????? ????????
????, xT ? ?????? ????????? ? Num1 ? ????? ? ??????
?????????, ??????? ????????? ?????. ???????? ??????
??????? ?? ??????????, ?.?. ??? ??????????? ????????
?????? ??????? ? ??????.
.
???????????????? ?????????? ?? ??????????. 2015. ?2
35
Intellectual Technologies on Transport. 2015. ?2
(??????? ?????????)
S-????????? ???????
xT-?????? ???? ?????????
Num-????? ????????? S ? ?????? xT
?????
S.n+S.m<K
??
Snew.n=S.n
Snew.m=S.m+1
???
S.l[2]=0
Snew.l[1]=max(S.l[1],S.l[2])+1
Snew.l[2]=S.l[2]
??
???
Snew.l[1]=S.l[1]
Snew.l[2]=S.l[2]
??
S.l[1]=0
Snew.l[1]=S.l[1]
Snew.l[2]=max(S.l[1],S.l[2])+1
???
isExist
Num1=0
xT=xT
S=Snew
??
Num1=0
Num1=xT.len+1
xT.add(Snew)
A[Num][Num]-=?[n+m+1]
A[Num1][Num]+=?[n+m+1]
???
A[Num][Num]-=?[n+m+1]
A[Num1][Num]+=?[n+m+1]
?????
S=Snew
Num=Num1
xT=xT
S.l[1]=0
???
Snew.n=S.n+1
Snew.m=S.m-1
Snew.l[1]=0
Snew.l[2]=S.l[2]
1
??
2
???????????????? ?????????? ?? ??????????. 2015. ?2
36
Intellectual Technologies on Transport. 2015. ?2
2
1
isExist
Num1=0
xT=xT
S=Snew
??
???
Num1=0
Num1=xT.len+1
xT.add(Snew)
A[Num][Num]-=?[S.l[1]]
A[Num1][Num]+=?[S.l[1]]
A[Num][Num]-=?[S.l[1]]
A[Num1][Num]+=?[S.l[1]]
?????
S=Snew
Num=Num1
xT=xT
S.l[2]=0
???
Snew.n=S.n+1
Snew.m=S.m-1
Snew.l[1]=S.l[1]
Snew.l[2]=0
isExist
Num1=0
xT=xT
S=Snew
??
???
Num1=0
Num1=xT.len+1
xT.add(Snew)
A[Num][Num]-=?[S.l[2]]
A[Num1][Num]+=?[S.l[2]]
??
A[Num][Num]-=?[S.l[2]]
A[Num1][Num]+=?[S.l[2]]
?????
S=Snew
Num=Num1
xT=xT
?????
???. 3. ???????? ??? ????????????? ??????
???????????????? ?????????? ?? ??????????. 2015. ?2
37
Intellectual Technologies on Transport. 2015. ?2
??? ????? ?? ??????? ???? ???????????? ????????
??????????? ???????? ??????? ??????????? ?????????
??? ????????? ??????? ????????????? ? ??? ???????
???????????????? ?????????. ?????????? ?????? ?????? ???? ????????? ?????? ???????, ??? ??????? ????????????? ? ????????????? ? ??????? ?????? ????????? ??????? ???.
????????? ??????? ???????????? ?????????? ????????? ????????? ??????? ? ??? ??????? ??????, ????????? ?
[7], ??? ??? ????? ????????? ??? ??? ??? ? ?? ???? ????????.
???????? ??? ??????? ??????
?????? ??????????????? ?????????? ???????????:
1. ? ? ?????????? ????? ??????, ?????? ??? ??????
????????? ????????????? ????????? ? i-?? ????
???????
??).
????????? ????, ??? ? (i+1)-?? ???? i=(1,
??????
2. ???? (?? = 1, ??) ? ?????????? ?????? i-?? ????,
???????????
?
???
??
????????
?????????????.
?
???????
???????
?????????????
3. ??????
??????????? ?????? i-?? ???? ? ??????? j.
4. ??? ????? ?????? ????????????.
1, ??, ?? = ?????
1, ??)
?
???????
???????
5. �??? (?? = ??????
????????????? ???????????? ??????? ? ???????
j ?????? i-?? ????.
1, ??, ?? = ?????
1, ??) ? ??????? ???????
6. ?????? (?? = ??????
??????????? ???????. ?????? = 1 ?????? ???? I
???????? ????? ? ??????? j, ?????? = 0 ?
??????????????? ??????.
??????
7. ???? = {???? }(?? = 1,
??) ? ??????, ????????????
?????????? ?????? i-?? ????, ??????????? ?
???.
??????
??? = {???? }(?? = 1,
??) ? ??????, ????????????
8. ??
??????????
??????
???????
????,
???
?????????? ???????????? ? ?????????? ???.
?????
9. ??? = {???? }(?? = 1,
??) ? ?????? ????????? ???????
????????????, ???? = 0, ???? j-? ????? ????????,
???? = ??, ???? j-? ????? ??????????? ?????? ???? i.
10. ?????? ? ????? ??????? ? i-?? ?????? ????????????,
?? = ?????
1, ??.
????????
11. ????? = ??????? ?, ?? = 1,
?????? ? ?????? ??????? ? i-??
?????? ????????????, ??? ???? j-? ?????????? i?? ??????? ????? ?????? ???? ??????, ??????? ?
??????? ? i-?? ?????? ???????????? ?? j-??
?????.
??????
12. ?????
???? = {?????? }, ?? = 1,
?? ? i-? ?????????? ???????
?????????? ????????? ?????? i-?? ????.
????????? ??? ? ?????? ?????? ??????? ??????????????? ????????????? ??????? ????????:
??????, ??
??)
???, ???, ????
(????, ??
(8)
???????? ????? ?????????? ?????????????? ??????? isExist. ????? ????, ??? ??????????? ??????? ????????????? ????? ????? ?????? (???????? ????????????)
???? ??????? ??????? ????????? ?????? ?????? ????????????. ???????? ?????? ?????? ???????????? ??????????
? ?????????????? ??????? choiceChannel. ??? ???????
????? 3 ?????????: ?? ? ??????? ??????????? ???????; S ?
??????? ????????? ???????; zType ? ????? ???? ??????;
channel ? ??????????, ? ??????? ????????? ????????? ?????? ???????.
???????? ????? ??????????? ??????? ??????????
?? ????????? ????????:
1. ?? ????????? ??????,??
??????,??
?? ? ??????? ???? + ???? < ??????
????,???,????
???????
?????
???????
?
?????????
?
??????????????
??
????1
????,????1
?????????,??1
??????
??,???? +???? +1 . ???
??????,??1
???????,??1
???? ????? ?????? (j), ?? ??????? ??????? ??????,
????????????
?
???????
???????
choiceChannel, ???? ???? > 0, ?? ????1?? = ?????? + 1, ? i
??????????? ? ????? ????? ????? ????????, ???????? ??????? ?????????. ???? ???? = 0, ?? ??1?? =
??, ?????? ???? ? ??????? ?? ???????? ??? ?????????.
? ??1?? = ???? + 1. ?????? ??
??? ???????? ??? ?????????.
2. ?? ????????? ??????,??
??????,??
?? ? ??????? ???? > 0 ???????
????,???,????
????? ??????? ? ????????? ????1
????,????1
?????????,??1
?????? ? ????????,??1
???????,??1
???????????? �??? ,?? , ??? ??1?? = ???? ? 1, ??1?? = ???? +
0, ?????? = 0;
1, ??1?? = ?
????,1 , ?????? > 0,
??? ???? ?????????? ??????? ????,1
????? ????, ? ????????? ???????????? ??????? ???????? ??????? ????????? isExist, ????? ?? ??? ? ??? ????????????? ???????, ????????? ????. ? ???????? ?????????? ????????? ??????? ?????????, ? ??????? 0 ??????????? ?????? ? 0 ???????????. ????-????? ?????????
????????? ??????? ????????? ??? ??????? ?????? ????????? ?? ???. 4.
???????????????? ?????????? ?? ??????????. 2015. ?2
38
Intellectual Technologies on Transport. 2015. ?2
(??????? ?????????)
S-????????? ???????
xT-?????? ?????????
Num-????? ????????? S ?
?????? xT
?????
i=1
1
???
i<=M
??
n[i]+m[i]<KT[i]
Snew=S
Snew.n[i]=S.n[i]+1
choiceChannel
Channel=0
?=?
S=S
zType=i
S.l[channel]=0
??
???
Snew.dq[channel]=S.dq[channel]+1
K=1
K>S.dq[channel] ???
pr[k]>pr[S.q[channel][k]
Snew.l[channel]=i
??
Snew.q[channel][k]=I
(??? ???? ???
???????? ? ???????
S.q[channel] ?
??????? ????????
?????????? ?? 1
???
k=k+1
???
isExist
Num1=0
xT=xT
S=Snew
??
???
Num1=0
Num1=xT.len+1
xT.add(Snew)
A[Num][Num]-=?[i][S.n[i]+S.m[i]+1]
A[Num1][Num]+=?[i][S.n[i]+S.m[i]+1]
A[Num][Num]-=?[i][S.n[i]+S.m[i]+1]
A[Num1][Num]+=?[i][S.n[i]+S.m[i]+1]
?????
S=Snew
Num=Num1
xT=xT
i=i+1
???????????????? ?????????? ?? ??????????. 2015. ?2
39
Intellectual Technologies on Transport. 2015. ?2
1
i=1
???
i<L
?????
??
S.l[i]=0
???
Snew=S
Snew.n[S.l[i]]=S.n[S.l[i]]-1
Snew.m[S.l[i]]=S.m[S.l[i]]-1
???
S.dq[i]=0
Snew.l[i]=S.q[i][1]
Snew.dq[i]=S.dq[i]-1
(??? ???? ??????????
??????? Snew.q[i][1]
??
Snew.l[i]=0
isExist
Num1=0
xT=xT
S=Snew
??
Num1=0
Num1=xT.len+1
xT.add(Snew)
A[Num][Num]-=?[S.l[i],i]
A[Num1][Num]+=?[S.l[i],i]
??
???
A[Num][Num]-=?[S.l[i],i]
A[Num1][Num]+=?[S.l[i],i]
?????
S=Snew
Num=Num1
xT=xT
i=i+1
???. 4. ????-????? ????????? ????????? ??????? ????????? ??? ??????? ??????
???????????????? ?????????? ?? ??????????. 2015. ?2
40
Intellectual Technologies on Transport. 2015. ?2
?????? ???????? ??? ?????????? ? ???????????? ?
?????????, ????????? ? [6]. ????? ????, ???????????
???????????? ??????? ??????. ? ????. 1 ????????? ??????????? ???????????? ????????? ??????? ? ?????????
??????? ???????, ?????????? c ??????? ???? ???????.
??? ????: x-y, ???????????? ????????????? ???????? ??
????????? ? ? ????????? ?.
??????? 3
???????? ??????? ????? ????????????
??????? 1
????????? ??????????? ???????
??????
???????
1
2
3
4
5
????????????
??????
0.010411
0.224826
0.564251
0.776363
0.869929
?????????????
0.0105599357
0.225337103263
0.56480951
0.776569477849
0.87000275886
?????????? ????????? ???????
??? ??????? ? [5], ??? ??????? 3 ?????????? ????????????? ???????, ???? ?????? ???????? ??????????? ?????
??????? ??. ????????? ???????? ??? ????, ??? ? ??????
???????????? ???? ??????? ?? (??? ?????????????? ?????
??????? ?? ?????? ???????????), ?? ??????????? ?????
???????? ? ????? ???? ?? ?????????. ????? ???????,
????????????? ??????? ??????? ????? ????? ?????, ????
?????? ??????? ?? ? ???????????.
???? ????????????? ????????? ?? ??????????? ?????
???????????? ????????, ? ?????? ???? ????? ?? ??????????? ????? ??????????? ???????? ?? ?? ???? ?? ?????????
??????????? ?????? ?? ????? ????? ??????????? ?? ???????????, ? ??????? ? ?????? ??????????? ???.
????????, ? ??????? ?????? ? ????? ????????? ?????
?????? ????? ????? ?????? ?????????, ?? ????????? ????????? ??????, ??? ??? ??? ????????? ?? ????? ????????
???? ?? ?????.
?? ?????? ???????????? ????? ?????? ?? ??????????? ??????? ?? ? ?????? xT, ? ??????? ???????? ??? ????????? ???????. ??? ?????????? ????????????? ???????? ??????? ???, ??? ??? ??????? ????. ??? ????? ?????????? ??????????????? ????? ?????????? ??????????.
??? ????, ??? ???????????? ???? ????????? ? ????????
x1 ? ?2 ?????????? ???????? ??????? 2 ?????? ? 2 ??????? ? ??????? ? ? ???? ?? ????????. ?????? ????? ???????????? ???????? ? ????. 2 ? 3.
??????? 2
???????? ??????? ?? ????????????
1
2
?
x1
x2
x3
1
1-1
2-1
?
x1-1
x2-1
x3-1
2
1-2
2-2
?
x1-2
x2-2
x3-2
?
?
?
?
?
?
?
x1
1-x1
2-x1
?
x1-x1
x2-x1
x3-x1
x2
1-x2
2-x2
?
x1-x2
x2-x2
x3-x2
x3
1-x3
2-x3
?
x1-x3
x2-x3
x3-x3
? ????. 3 ???????? ???????? ??????? ????????? ?.
?????? ??????? ? ?????? ?????? ???????? ???????? ?????? ?????????. ? ??????? ??????? ?????????? ???????-
1
2
?
x2
x1
x3
1
1-1
2-1
?
x2-1
x1-1
x3-1
2
1-2
2-2
?
x2-2
x1-2
x3-2
?
?
?
?
?
?
?
x2
1-x2
2-x2
?
x2-x2
x1-x2
x3-x2
x1
1-x1
2-x1
?
x2-x1
x1-x1
x3-x1
x3
1-x3
2-x3
?
x2-x3
x1-x3
x3-x3
???????? ? ???????? ?????????? ????????? ????????????? ???????? ??????? ????????? ? ???????? ?1 ? ?2.
??????? ????????????? ?? ????? ????? ????????????
???????????? ? ????. 2. ??? ????? ?? ????. 2, ???????????
????? ??????????? ?? ????????, ??? ?????????? ???????????? ????????????? ?????? ??????????.
??????????
????? ?? ??????? ? ?????????????????? ?? ???? ?????? ???, ???????????? ?????? ????????? ????????????? ??? ??? ???????? ????????? ??????? ?????????????. ?????????? ????? ?????????? ??????? ????? ? ?????????? ????? ????????? ?, ??????, ?????? ?????????, ??
????????? ?????? ???? ?????? ??????????? ?? ????????.
????? ???????, ????????????? ???????????? ?????? ??????? ?????? ?????? ?????? ????????? ??????? ???.
??????????
1. Osogami T. Analysis of transient queues with semidefinite optimization. / T. Osogami, R. Raymond // Queueing
Systems. ? 2013. ?73. ? ?. 195-234.
2. Wolff R. W. Little?s law when the average waiting
time is infinite. / R. W. Wolff, Y. Yao // Queueing Systems. ?
2014. ?76. ? ?. 267-281.
3. Sudhesh R., Vijayashree K. V. Stationary and transient
analysis
of
M/M
/1
G-queues
// Int. J. of Mathematics in Operational Research, 2013.
Vol. 5, No 2, pp. 282-299.
4. Sudhesh R., Francis Raj L. Stationary and transient
solution of Markovian queues ? an alternate approach // Int.
J. of Mathematics in Operational Research, 2013. Vol. 5,
No 3, pp. 407-421.
5. ?????? ?.?. ???????? ?????????????? ???????
???????????? ????????? ?????????????? ?????? ????????????. / ?.?. ?????? // ???????? ?????????????? ???????????? ????? ?????????. ? 2011. ?4. ? ?. 90-97.
6. ?????? ?.?. ??????????? ??????????? ?????????? ????????-?????????????? ?????? ??????? ???????
?????????????? ?????? ????????????. / ?.?. ??????,
?.?. ??????, ?.?. ??????? // ????? ???????. ? 2015.
?1. ? ?. 218-232.
7.
?????? ?.?. ?????????? ???????????? ???????
?????????????? ?????? ????????????. / ?.?. ??????,
?.?. ???????. ? ?. ?????-?????????, 1999, 65 ?.
???????????????? ?????????? ?? ??????????. 2015. ?2
41
Intellectual Technologies on Transport. 2015. ?2
The Method of Construction of Systems of
Homogeneous Differential Equations for
Calculating the Probability-Time Characteristics
of non-Stationary Service Systems
Sergeev S.A.
Petersburg State Transport University
Saint-Petersburg, Russia
serega_svetl@mail.ru
Abstract. It is proposed the method of construction of systems
of differential equations based on the use of a recursive algorithm
for generating the coefficient matrix for the homogeneous system
of differential equations describing the model of non-stationary
service systems. The proposed approach simplifies the construction of the matrix coefficients. With it was first implemented such
complex models of non-stationary systems as network and Lchannel models. The article also provides a detailed algorithm for
the network model.
Keywords: solution of non-stationary systems, service,
algorithm for generating the coefficient matrix, network model
non-stationary system inspection completed.
REFERENCES
1. Osogami T. Analysis of transient queues with semidefinite optimization. / T. Osogami, R. Raymond // Queueing
Systems. 2013. ? 73. pp. 195-234.
2. Wolff R. W. Little?s law when the average waiting
time is infinite. / R. W. Wolff, Y. Yao // Queueing Systems. ?
2014. ? 76. pp. 267-281.
3. Sudhesh R., Vijayashree K. V. Stationary and transient
analysis
of
M/M/1
G-queues
// Int. J. of Mathematics in Operational Research, 2013.
Vol. 5, No 2, pp. 282?299.
4. Sudhesh R., Francis Raj L. Stationary and transient
solution of Markovian queues ? an alternate approach // Int.
J. of Mathematics in Operational Research, 2013. Vol. 5,
No 3, pp. 407-421.
5. Bubnov V.P. Algoritm analiticheskogo raschyota
veroyatnostej sostoyanij nestacionarnyh system obsluzhivaniya: Izvestiya Peterburgskogo universiteta putej soobshcheniya. [Algorithm of analytical calculation of non-stationary state
probabilities service systems: News from the St. Petersburg
University of communication.] / V.P. Bubnov. 2011. ? 4. 9097 p.
6. Bubnov V.P. Osobennosti programmnoj realizacii
chislenno analiticheskogo metoda raschyota modelej nestacionranyh sistem obsluzhivaniya: Trudy SPIIRAN. [Features
of the software implementation of numerical-analytical method of calculation models non-stationary service systems:
SPIIRAS Proceedings.] / V.P. Bubnov, A.S. Eremin, S.A.
Sergeev. 2015. ?1. pp. 218-232.
7.
Bubnov V.P. Razrabotka dinamicheskih modelej
nestacionarnyh system obsluzhivaniya. [Developing dynamic
modeling of non-stationary systems.] / V.P. Bubnov, V.I.
Safonov. ? Saint-Petersburg, 1999, 65 p.
???????????????? ?????????? ?? ??????????. 2015. ?2
42
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