519.872

.. , ..
E-mail: irinka_asf@mail.ru; smoiseeva@mail.ru
() . , , .
:
, , , .
Key words:
Parallel service, multiple demands, stream of Markov restoration, the method of moments.
, , - , , .
[1-2].
( ) .. , .. [3-5], , , , . , [6] , - .
.
(MR (2)|M2|<ї) , . , -
^^.-.^) - [7].
2
. |2|<ї
, /1 . , , . , .
4 - - . -, {^(), , {(1),1(?),1)^}, , 1^) - / -
MR , k(t) - .

. . u\k (t) = k ,z (t) < z, h(t) = \
P(k,z, iu i2,t) = P ! . .
[ = \ , i2 () = i2 P(k,z.,i1,i2,t) At- :
P(k, z - At, i1, i2, t + At) =
= (P(k, z, ii, i2, t) -P(k, At, ix, i2, t)) x x(1 - illAt )(1 - i2 2At) +
+LP(v, A^ ii -1 i2 -1 t)PvkAk(z) + v
+P( k, z, i1 +1, i2, t)(i1 +1) 1At +
+P( k, z, i1, i2 +1, t )(i2 +1) p2 At + o( At),
[8]
dP(k, z, i1, i2, t) = dP(k, z, i1, i2, t) dP (k ,0, i1, i2, t)
dt dz dz
-P (k, z, i1, 2, t)i1^1 - P(k, z, i1, 2, t )i22 +
+P(k, z, i1 +1, 2, t )(i1 +1)1 +
+P(k, z, i1, i2 +1, t )(i2 +1)2 +
>P(v,0, i1 -1, i2 -1, t)
+l-
v
dz
-PvkA (z).
dH (k, z, u1, u2)
du1
dH (k, z, u1, u2)
d = YLij eiu22 P(k, z, h, i2),
k=o i2=o
= 'L^e^1 eju2i2 P(k, z, i1, i2),
i =0 ij =0
(,1,,2) [9] (, , 1, 2) ( ,0, 1, 2)
dz
dz
+j^1 - e -*)H (k,Z, u1,u 2)
+i2(1 - e
du1
H (k, z, u1, u 2)
du2
+e
j(u1+u2)
L
dH (v,0, , ) dz
PvA (z) = 0.
(2)
H (z,ubu2)=[H(1,z,ubu2),H(2,z,ubu2). f A1(z) ... 0 "
D( z) =
0 ... Ak (z) j
' P11 P1k f 1 . . 0"
P = ,Pk1 . pkk j , I = ,0 . 1 j
(2)|2|<ї
( z, ) + (0, ul, ) ]+2) (. _ 1, +

+ ]^1 - -ї) ^- 2) +
+iV,(1 - e-J")
du1
dH( z, u1, u2) du
= 0.
(3)
:
dP (k, z, i1, i2) dP(k ,0, i1, i2)
dz dz
-P (k, z, i1, 2)1^1 - P (k, z, i1,i2)i22 +
+P(k, z, 1 +1, 2)(1 +1)1 +
+ P(k, z, 1, 2 +1)(2 +1)2 +
+LdP(v-0,i-1,i'-P,A (z) = 0. (1)
V dz
, [8]
H (k, z, ux, u2) = LLe1'lhe1'2'2 P(k, z, 1, 2).
1 =0 2 = 0
(1), ,
(^,1,2) - , (,0,0)=(), {/1(/)},{/2(/)}, , (2)|2|,
() = 1] = (ї, ,0) ,
(2) = ]22( "> = (ї, 2,0)
) - - {(1),,(1)}.
(2) [10] , [8].

[8]:
(, , 1, 2)
du1
dH (k, z, u1, u2)
= irnm(k, z),
du1
= irnm(k, z),
:
v
u =0
u =0
u =0
u =0
( z, 1, 2)
1
= 7 (2) =
^(0)(-() -1) + ^ \) = 0,

^(1, 2), ^(2, z),..],

1(1)(0) = (0)

*()( - *())
( , 1, 2)
.
= 2 (2) =
= 0 2 = 0
2(>(1,2),\2, 2 ]
(3) 1,
2 ( 2, 1, 2) 2 (0, 1, 2)

{ +2)0(2) - } +
1
21
2
(1) = (0) =
'0)
2
* ( )( - * (^ ))- (* (0) - ) -, (0)
2

(0, , 2)
2
(-)
]' ( 2) -
(2,,2)
(2)|2|
(1)(= (1'= 1 -(0) = ^
-^( -1)

2 ( 2, , 2)
1 ) () = (1 () =
=.
V 2 V
12
, |2|<ї,
-(-1)
2( 2, 1, 2)
= 0.
(4)
12
1=0, 2=0, - (1)()={1(1)(1,),1(1)(2,),^}
1(1)( ) 1(1)(0)
21) () = 2(1) ( ) =
1 0) =
V.2 2 V2
2
(0)
' 2
2
-((2) - ) +

(2)|2|. 1 (4):
(2) - / (2) = 0
(5)
3(2, 1, 2) 3(0, 1, 2)
{ +2} (2) -} +
2
-- [8]
1() = ]~(1()(2), * () = |~ {2).
0 0 (5) --,
/1^) 1 *
() + ;------------( ()-) +
02
+ ?(0) - *() - () = 0
2
21
2 (0, 1, 2) 21 2 (0, 1, 2)
^2 ( 2) +
21
2 ^ ( 2) +
+ (0,1,2) -2/^1 (2) -2
-11(- ,--^1) (2 ^ 2) - ,, - 1 2(2 ^ ) ( J ) ~ - 2

2( 2, , )

V-)() = '1)(0)(*() - ) + 2
+∞|∞> ї.
2
(6)
-^(-^-')
^( -1)
---JV2(-Je- -1)
^2 3( 2, 1, 2) 3
3( 2, 1, 2)

() =
1
1(1)(0) 2 (0)
( () - ) +
2
*()
2 = 0. (7)

: 2 (, 2, 1, 2)
1
2 (, 2, 1, 2)
(6) =, :
2
= /2(2)(, 2).
=0
=0
=0
=0
=0
=0
:
2( , 1, 2)

= /^ ) =

=2 -1(2) (0) _

= ), ^(2, ),...],
^2 (0) =
2( , 1, 2)
2
= ] 2(2\) =
= ]2[2(2\1, ), 2 (2, ),...].
(7) 1=0, 2=0, :
2 /2)() + . /2)(0)
]

+ ]

(() -1) +

+2/ ^ ()+] ^ ( +

+ -1]21 () - ! / 2(2) () = 0
<+^ (0( ) -1)+


2 +, 0()+
^ 2∞
^
+ () - 22(2) () = 0,
(8)
-.

1() = "^~(1^>(), 2() = j~2dm1(2'(),
0 0

* () = | -( )
0
(8) :
/2)(0) *, 2() +-----------( () -1) +

2 + 11 *() +
)
+ 1() - '2 2() = 0,
(2 -)2() = ()(0)(*() -1) +

01(1)(0) (0)' *, . ...
21-------------------------1-| () + 11(),


2() =
1
21 -
/2^)

(*() -1) +
^1(1)(0) (0) .__*,
2----1----+-------| () + ^()


!) +
2 !>) +^(0) ^ *(2 *)
)
( I - *(2 ,)) ~1-, ,

\0) = |dAk () = () - (0) =1,
0
*(0) = I, 1(0) = 1(1)() =
= 1 5(0^_
= , 1 1
(2)|2|
^2) () = /2 () = 2(0) =
^/ї + 0)
1
21
-+11(0) =
{2 + + \ =
21 [
1
= 1 | 1(1)(0)
+ [ =
= _[(I_ (1))-1 ) +1].
1
, (2)|2|:

2(2)() = [(I - *(2))-\2) +1].
2
, 2\.

(4) 2,
3(,1,2) + 3(0,1,2) ^ zu1u2
2(0, 1, 2)
(] (+2> () -I) +
2 2( , 1, 2)
]1 ]]2 ( ) +
]2 ]]1 ( ) +
(0, 2) ]]2]]!() -
=0
=0
=0
=0
- ]u1 2(0, , )
12
-Ju1 13( z, Ui, U2)
2
-J2(-J)e
-J2(eJu -1)
-Ju2 2H(z,Ui,U2)
12 3( z, 1, 2) 12
0.
, :
( z, 1, 2)
12
= Jm12(z) = J[m(1,z),m(2, z),..].
z
z
-(PD( z) -1) +
, (0) + 5m^ + R(0) |x z z z
xPD(z) - (1 + 2 )12 (z) = 0
,
,
12 () = 12 () = 12 (0) =
= 1 {1(1)(0) 1(2) (0) (0)]
1 +2
z
z
z
E =
1 J^^E+mmE+A^
1 +2 A
z
z
, 1=0, 2=0, :
12( ) +12(0),
[( - *( +2))^ *( +2) +1].
+ 2
, . , . , , , . .
=0
0

1. .. // . - 1995. - 6. - . 163-165.
2. . // . - 1999. - 7. - . 177-181.
3. .. // . . - : -, 1989. - . 109. - .
4. .. // Math. Operationsforsch. und Statist. Ser. Optimization. - 1983. - V. 14. - 3. - P. 433-444.
5. .., ., ., .. // -
. - 2001. - . 37. - 4. -. 130-140.
6. .., .. // . -, 2009. - . 2. - . 262-268.
7. .., .. . - , - , 2006. - 204 .
8. .., .. .
- , - , 2004. - 228 .
9. .. . - .: , 1969. - 424 .
10. .., .. . - , - , 2006.
- 112 .
19.11.2012 .