- , . , . , (), (). . , , . . . , , , .
, , . , . . , .
. , . . , .. . . , . , . 2 . 1... s. , . 1, 2. , . , , , . , .. . , , , , .
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. , . , . . , , .. , , 2. , , - . , , .. , . , , . , .
, , , . , O1 - ON. , , . , . , . , . , . 2. FIFO. - LIFO. , . . , . . . , . . , , .. . , . .
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, (). F(t) = P(t<t), P(t<t) - , t < t. t £ 0 F(t)= 0. . . , . . . , , .. . . - .
t N(t). N(t) - , N(t) . , k, k . Pk(t)=P[N(t)=k]. tSkPk(t)= 1. i j Pij(t1, t2), t1 i, t2 j. t1 - t2 = Dt Dt 0: Pij(Dt) fij Dt, fij = lim|Dt0[Pij(Dt)/Dt] - . , . U t1 < U< t2:
Pij(t1, t2)=Sr=0N Pir(t1, U) Prj(U, t2), 0 £ i, j£ N,
N - . t1 = 0; U = t; t2 = t + t, t > 0, Pij( 0 ,t+t) = Sr=0N Pir( 0 ,t)Prj(t, t+t),
Pk(t).
, , . . , . .
. . .. , l: l= const. , k t P(k, t)=(lt)k -lt(k!)-1; l - , .. , . t
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M = Sk=0¥ k(lt)k -lt(k!)-1= lt.
t
D = Sk=0¥ k2P(k, t)-lt)2=lt.
, l1, l2 (l1+ l2). . , .
b(t)= m -mt, m - : m=-1, - . , , . 0, N. . Pk(t), N+ 1 . : , - , , . , . , , , - . , k - e ,
dPk(t)/dt = - Sj=0N fkj Pk(t)+ Si=0N fkj Pi(t).
:
dP0(t)/dt= -lP0(t)+mP1(t);
...
dPk(t)/dt= -(l+m)Pk(t)+lPk-1(t)+mPk+1(t); k= 1 ,..., N;
...
dPN(t)/dt= -mPN(t)+lPN-1(t).
Sk=0N Pk(t) = 1.
:
0 = -lP0(t)+mP1(t);
...
0 = -(l+m)Pk(t)+lPk-1(t)+mPk+1(t); k= 1 ,..., N;
...
0 = -mPN(t)+lPN-1(t).
:
P0(t)=( 1 -r)( 1 -r N+1)-1; r=l/m.
k , , ,
Pk(t)=( 1 -r) r k ( 1 -r N+1)-1; 0 £ k £ N.
k>N k - Pk = 0. , .. N- 1 . , , .. N , . , N ().
S . , 2 S . S, , , . S 0 ,..., S - , . , :
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0 = -lP0(t)+mP1(t);
...
0 = -(l+ km)Pk(t)+lPk-1(t)+ (k+ 1 )mPk+1(t); k= 1 ,..., S;
...
0 = - SmPS(t)+lPS-1(t).
Sk=0S Pk(t) = 1. , k - :
Pk = (rk(k!) -1)[Sj=1S rj(j!) -1]-1; k = 0 ,..., S.
. , , S, :
P = PS = (rS(S!) -1)[Sj=1S rj(j!) -1]-1.
Pk . P0=[Sj rj(j!) -1]-1.
, m S.
r=l/m, , . , r £ S. , .
S . S () L. . , . FIFO. : 0 ,..., S+L. , , , . :
0 = -lP0(t)+mP1(t);
...
0 = -(l+ km)Pk(t)+lPk-1(t)+ (k+ 1 )mPk+1(t); k= 1 ,..., S;
...
0 = - (l+Sm)PS(t)+lPS-1(t)+ SmPS+1(t);
...
0 = -(l+ Sm)PS+n(t)+lPS+n-1(t)+ SmPS+n+1(t); n= 1 ,..., L- 1;
...
0 = - PS+L(t)+lPS+L-1(t).
Sk=0S Pk + Sn=1L PS+n = 1.
, k - Pk= P0 rk(k!)-1; k= 0 ,..., S.
(S+n): PS+n=PS(r/S)n(k!)-1; n= 0 ,...,L.
0=[ 1 +Sk=1S rk(k!)-1+ rS(S!)-1Sn=1L (r/S)n]-1.
, L , . .
P = PS+L = PS(r/S)L= rS+LSL(S!)-1[ 1 +S k=1S rk(k!)-1+ rS(S!)-1Sn=1L (r/S)n]-1.
- . r/S < 1 , r/S . - , .
. , , , S+ 1 S+L. ` l = Sk=S+1S+L kPk. Pk,
` l = PS r [ 1 -(r/S)L(S+ 1 -r)]S-1( 1 -r/S)-1.
, ( ). , t, , , t.
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P[ > t] = Sr=0L(l/(Sm))rPS Sk=0r(Smt)ke-Smt (k!)-1.
S S+L, - 0 r S t.
`T= PS [ 1 -(r/S)L(S+ 1 -r)]S-1m-1( 1 -r/S) -2.
. :
`T = `T + [ 1 -(r/S)LPS]/m.
. S = 1,
Pk =rk( 1 -r)( 1 -rL+2)-1; k= 0 ,...,L- 1.
P = [ 1 -rL+1] [ 1 -rL+2]-1.
`T=[m( 1 -r)]-1-L rL[m( 1 -r)L]-1.
, , . , . .