Operational queueing system in the scheme of diffusion approximation with evolution averaging
We consider an operational queuing system of the type $[SM | M | \infty]^N$ in the scheme of diffusion approximation. The queueing system is described by a semi-Markov random evolution.
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| Date: | 2006 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3488 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509588652032000 |
|---|---|
| author | Mamonova, G. V. Мамонова, Г. В. |
| author_facet | Mamonova, G. V. Мамонова, Г. В. |
| author_sort | Mamonova, G. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:55:42Z |
| description | We consider an operational queuing system of the type $[SM | M | \infty]^N$ in the scheme of diffusion approximation. The queueing system is described by a semi-Markov random evolution. |
| first_indexed | 2026-03-24T02:43:30Z |
| format | Article |
| fulltext |
UDK 519.21
H.�V.�Mamonova (Nac. akad. DPS Ukra]ny, Irpin\)
EKSPLUATACIJNA SYSTEMA OBSLUHOVUVANNQ
U SXEMI DYFUZIJNO} APROKSYMACI}
Z EVOLGCIJNYM USEREDNENNQM
The operational queueing system of [ SM | M | ∞ ]N
type is considered in the scheme of diffusion
approximation. The queueing system is described by a semi-Markov random evolution..
Rozhlqda[t\sq ekspluatacijna systema obsluhovuvannq typu [ SM | M | ∞ ]N
u sxemi dyfuzijno]
aproksymaci]. Systema obsluhovuvannq opysu[t\sq napivmarkovs\kog vypadkovog evolgci[g.
1. Postanovka zadaçi. Budemo rozhlqdaty ekspluatacijnu systemu obsluhovu-
vannq (ESO) typu [ SM | M | ∞ ]N, wo sklada[t\sq z N vuzliv obrobky informa-
ci]. Ças obrobky vymoh, wo nadxodqt\ do systemy, ma[ pokaznykovyj rozpodil z
intensyvnostqmy µ = ( µk , k = 1, N ) . Napivmarkovs\kyj potik vymoh nadxodyt\
do koΩnoho vuzla za zakonom, wo zada[t\sq napivmarkovs\kym qdrom
Q ( t ) = [ Qkr ( t ) ; k , r ∈ E ] ; Qkr ( t ) = pkr Gk ( t ) , k , r ∈ E ,
(1)
pkr = P ( κn +1 = r | κn = k ) , k , r ∈ E ; Gk ( t ) = P ( θn +1 ≤ t | κn = k ) = : P ( θk ≤ t ) .
Intensyvnosti momentiv vidnovlennq zadovol\nqgt\ umovu
U1) gk : = E θk = G t dtk ( )
0
∞
∫ , g k
2
( ) : = Eθk
2 = t G t dtk ( )
0
∞
∫ = 1
2
2Eθk < ∞ , λk : =
: = 1
gk
, k = 1, N .
Proces nadxodΩennq vymoh do ESO vyznaça[t\sq procesom markovs\koho
vidnovlennq (PMV) κn , θn , n ≥ 0, u fazovomu prostori E = { 1, 2, … , N } . Pere-
xidni jmovirnosti6PMV vyznaçagt\sq napivmarkovs\kog matryceg6(1) [1]. OtΩe,
vymohy nadxodqt\ do ESO u momenty markovs\koho vidnovlennq
τn = θm
m
n
=
∑
1
, n ≥ 1, τ0 = 0. (2)
Nomer vuzla, do qkoho nadxodyt\ vymoha v moment τn , vyznaça[t\sq znaçennqm
vkladenoho lancgha Markova (VLM) κn .
Uvedemo porodΩugçu matrycg suprovodΩugçoho markovs\koho procesu
κ0( t ) , t ≥ 0:
Q : = Λ ⋅ [ P – I ] = [ qkr , k , r = 1, N ] , Λ = λd : = [ λk δkr , k , r = 1, N ] ,
P : = [ pkr , k , r = 1, N ] .
Matrycq marßrutyzaci] P0 : = [ pkr
0 ; k , r = 1, N ] vyznaça[ rux vymoh u me-
reΩi.
Vykonu[t\sq umova vidkryto] mereΩi
U2) max :
k k kr
r
N
p p0
0 0
1
1= −
=
∑ > 0
ta nerozkladnosti matryci marßrutyzaci] P0 . Tut pk0
0 — jmovirnist\, z qkog
vymoha zalyßa[ mereΩu.
© H.6V.6MAMONOVA, 2006
708 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, #5
EKSPLUATACIJNA SYSTEMA OBSLUHOVUVANNQ U SXEMI DYFUZIJNO} … 709
Osnovnyj rezul\tat roboty [2] — teorema userednennq — da[ moΩlyvist\
otrymaty sprowenu model\ stoxastyçno] evolgci] ESO. Ale dlq povnoty ana-
lizu neobxidno doslidyty fluktuaci] procesu ESO navkolo useredneno] deter-
minovano] systemy ρ ( t ) , t ≥ 0. Qk vidomo, fluktuaci] opysugt\sq dyfuzijnym
procesom. U roboti [3] fluktuaci] procesu ESO doslidΩeno vidnosno toçky
rivnovahy ρ useredneno] systemy, wo vyznaça[t\sq rozv’qzkom rivnqnnq C ( ρ ) =
= 0. U6danij roboti dyfuzijna aproksymaciq budu[t\sq dlq fluktuaci] vidnosno
useredneno] evolgcijno] systemy v ( t ) , t ≥ 0, wo vyznaça[t\sq rozv’qzkom evo-
lgcijnoho rivnqnnq d v ( t ) / dt = C ( v ( t ) ) . Pry rozv’qzanni problemy userednennq
vypadkova evolgciq mala neperervnu ta strybkovu komponenty. Analohiçnyj
pidxid zberiha[t\sq i pry dyfuzijnij aproksymaci].
2. Proces obsluhovuvannq v ESO. Qk i v roboti [3], proces obsluhovuvan-
nq ρε( t ) = ( ρε
k t( ), k ∈ E ) vyznaça[ kil\kist\ vymoh u koΩnomu vuzli k ∈ E v
moment çasu t > 0.
Vvedemo neobxidni poznaçennq ta umovy:
1) funkci] ßvydkostej useredneno] systemy C ( v ) = ( ck ( v ) , k ∈ E ) , v ∈ RN,
vyznaçagt\sq spivvidnoßennqmy [2]
C ( v ) = γ ( v ) + λ , γ ( u ) = u∗
A , A = : µd
[ P0 – I ] ,
de λ : = ( λk , k ∈ E ) — vektor-stovpec\, µd = [ µk δkr , k , r = 1, N ] , γ ( u ) = ( γl ( u ) ,
l = 1, N ) , γl ( u ) = u pr r rl rl
r
N
µ δ( )−
=
∑
1
, funkciq v ( t ) , t ≥ 0, zadovol\nq[ rivnqnnq
U3)
d t
dt
v( ) = C ( v ( t ) ) ;
2) krytyçne zavantaΩennq ESO zada[t\sq u sxemi serij iz malym parametrom
seri] ε > 0, ε → 0 :
U4) µε
k =
µ
ε
k , k ∈ E ;
3) poçatkove navantaΩennq systemy v teoremi userednennq zadovol\nq[
umovu
U5) ε ρε( 0 ) ⇒ ρ , ε → 0 ;
4) normovanyj ta centrovanyj proces obsluhovuvannq zada[t\sq u sxemi
serij:
U6) ζε( t ) = ερ
ε
ε t
2
– v( )t
ε
, t ≥ 0,
d t
dt
v( )
= C ( v ( t ) ) , C ( v ) = γ ( v ) + λ .
V umovax U1 – U5 ma[ misce slabka zbiΩnist\ (dyv. [2], vysnovok 2)
ερ
ε
ε t
⇒ v ( t ) , ε → 0 . (3)
3. Dyfuzijna aproksymaciq procesu obsluhovuvannq vymoh v ESO typu
[[[[ SM |||| M |||| ∞∞∞∞ ]]]]N. Normovanyj ta centrovanyj proces obsluhovuvannq vymoh v ESO
typu [ SM | M | ∞ ]N zada[t\sq u vyhlqdi
ζε( t ) = ερ
ε
ε t
2
– v( )t
ε
, t ≥ 0. (4)
Teorema (dyfuzijna aproksymaciq). V umovax U 1 – U6 ma[ misce slabka
zbiΩnist\ procesiv
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
710 H.6V.6MAMONOVA
ζε( t ) ⇒ ζ0( t ) , ε → 0 . (5)
Hranyçnyj proces ζ 0( t ) [ dyfuzijnym procesom Ornßtejna – Ulenbeka typu,
wo vyznaça[t\sq heneratorom
L0
ϕ ( u ) = – u∗
A ϕ′ ( u ) + 1
2
Tr B t u( )( ) ( )v ′′ϕ . (6)
Pozytyvno oznaçena matrycq dyspersij vyznaça[t\sq spivvidnoßennqm
B ( v ) = hd – [ vd
A + A ′ vd
] + C
d
( v ) ,
hd
= [ hk δkr ; k , r ∈ E ] , hk : =
[ ]( )g g
g
k k
k
2 2
3
2−
= λk kh3 , hk : = g gk k
( )2 22− , (7)
A = : µd
[ P0 – I ] , C
d
( v ) = vd + Λ .
ZauvaΩennq. U vypadku pokaznykovoho rozpodilu velyçyna hk = 0. Razom
z tym dlq majΩe monotonnoho klasu rozpodiliv [4] hk > 0. Isnugt\ takoΩ roz-
podily, dlq qkyx hk < 0.
Teorema dovodyt\sq za ti[g Ω sxemog, wo i v roboti [3]. Spoçatku dlq roz-
ßyrenoho procesu markovs\koho vidnovlennq (RPMV) vyznaça[t\sq kompensug-
çyj operator. Dali budu[t\sq asymptotyçnyj rozklad kompensugçoho operato-
ra za stepenqmy maloho parametra ε . Pislq c\oho vykorystovu[t\sq rozv’qzok
problemy synhulqrnoho zburennq, qkyj da[ vyraz dlq operatora hranyçno]
dyfuzi]. Zaverßu[t\sq dovedennq teoremy vykorystannqm slabko] zbiΩnosti
vypadkovyx evolgcij [5].
4. Evolgciq vymoh u mereΩi. Dlq test-funkci] ϕ ( u ) ∈ C2
( RN
) vyznaçymo
vektor ϕ′ ( u ) : = ( ′ϕk u( ) , k = 1, N ) z komponentamy ′ϕk : = ∂ ϕ ( u ) / ∂ uk i matrycg
ϕ″ ( u ) : = [ ′′ϕkr u( ); k , r = 1, N ] z elementamy ′′ϕkr u( ) = ∂2
ϕ ( u ) / ∂ uk ∂ ur . Vvedemo
vektorni operatory λ ϕ ( u ) : = λ∗
ϕ′ ( u ) = λ ϕk k
k
N
u′
=
∑ ( )
1
, γ ( v ) ϕ ( u ) = γ∗( v ) ϕ′ ( u ) , a
takoΩ matryçnyj operator Λ ϕ ( u ) : = λ ϕk kk
k
N
u′′
=
∑ ( )
1
.
TverdΩennq 1. Evolgciq vymoh u mereΩi E = { 1, 2, … , N } opysu[t\sq v
evklidovomu prostori RN markovs\kym procesom ηε( t ) , t ≥ 0, wo zada[t\sq
heneratorom na test-funkci] ϕ ( u ) ∈ C ( RN
) :
Γε
( v ) ϕ ( u ) =
γ ϕ ε ϕkr rk
k r
N
u e u( )[ ( ) ( )]
,
v + −
= =
∑
1 0
, (8)
de vektory strybkiv
erk : = er – ek , ek : = ( δkl , l = 1, N ) , e0 : = 0 , (9)
intensyvnosti strybkiv
γkr ( v ) : = vk µk krp0 , k = 1, N , r = 0, N , k ≠ r . (10)
Vidpovidno do postanovky zadaçi v p.61 intensyvnosti γk r ( v ) , wo vyznaça-
gt\sq formulog (10), zadagt\ intensyvnosti perexodu vymohy zi stanu k v stan
r ≥ 1, wo opysu[t\sq vektorom strybkiv ekr , zadanym formulog (9). U6vypadku
r = 0 vymoha zalyßa[ systemu z intensyvnistg γk0 ( v ) .
Klgçovyj etap dovedennq teoremy pro dyfuzijnu aproksymacig zabezpeçu[
nastupna lema.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
EKSPLUATACIJNA SYSTEMA OBSLUHOVUVANNQ U SXEMI DYFUZIJNO} … 711
Lema 1. Ma[ misce asymptotyçne zobraΩennq heneratora (8)
Γε
( v ) ϕ ( u ) = ε Γ ( v ) ϕ ( u ) +
ε ϕ ϕ ε θ ϕγ
ε2
0
21
2
Γ( ) ( ) ( ) ( ) ( ) ( )u u B u u+
+v v , (11)
de, za oznaçennqm, operatory digt\ takym çynom:
Γ ( v ) ϕ ( u ) = γ∗( v ) ϕ′ ( u ) , (12)
B0 ( v ) ϕ ( u ) = C
d
( v ) ϕ ( u ) + A ( v ) ϕ ( u ) – Λ ϕ ( u ) , (13)
A ( v ) = – [ vd
A + A′ vd
] ϕ′′ ( u ), Λ ϕ ( u ) = λ ϕk kk
k
N
u′′ ( )
=
∑
1
,
C
d
( v ) ϕ ( u ) = Tr [ C
d
( v ) ϕ′′ ( u ) ]. (14)
Zalyßkovyj çlen
θ ϕγ
ε ( ) ( )v u → 0, ε → 0.
Vykorysta[mo linijni vlastyvosti napivhrupy operatoriv
Γε ( v ) ϕ ( u ) = Γ
ε
( v + ε u ) ϕ ( u ) = Γ
ε
( v ) ϕ ( u ) + ε Γ
ε
( u ) ϕ ( u ). (15)
Za formulamy (8) – (10)
Γ
ε
( v ) ϕ ( u ) = γ ϕ ε ϕ ε γ ϕ ϕkr rk
k
r
N
kr r k
k
r
N
u e u u u( ) ( + ) − ( ) = ( ) ′ ( ) − ′ ( )[ ] [ ]
=
=
=
=
∑ ∑v v
1
0
1
0
+
+
ε γ ϕ ε θ ϕε
2
1
0
2
2 kr r k r k
k
r
N
e e e e u u( ) ( − )( − ) ( ) + ( ) ( )[ ]
=
=
∑ v v* . (16)
Vvedemo poznaçennq er ek ϕ ( u ) = ′′ ( )ϕrk u , er
2ϕ ( u ) = ′′ ( )ϕrr u . Vykorystovugçy oçe-
vydnu totoΩnist\
γ γkr kr
N ( ) + ( )=∑ v v01
= vk µk , dlq perßo] sumy ma[mo
γ ϕ ϕ γ ϕ ϕkr r k
k
r
N
kr r k
k
r
N
u u( ) ′ ( ) − ′ ( ) = ( ) ′ − ′[ ] [ ]
=
=
=
=
∑ ∑v v
1
0
1
0
=
( )( ) ′ − ( ) ′
=
∑ γ ϕ γ ϕkr r kr k
k r
N
v v
, 1
–
– γ ϕk k
k
N
0
1
( ) ′
=
∑ v = γ ϕ ϕ γ ϕ γkr r
r
N
k
N
k kr
r
N
k
N
k k
k
N
( ) ′ − ′ ( ) − ′ ( )
== == =
∑∑ ∑∑ ∑v v v
11 11
0
1
=
=
γ ϕ ϕ γ γkr r
r
N
k
N
k kr k
r
N
k
N
( ) ′ − ′ ( ) + ( )
== ==
∑∑ ∑∑v v v
11
0
11
=
= γ ϕ µ ϕ γ ϕkr r k k k
r
N
k
N
u( ) ′ − ′
= ( ) ′( )
==
∑∑ v v v
11
* .
Dlq druhoho dodanka v (16) otrymu[mo
γ ϕkr r k r k
k
r
N
e e e e u( ) ( − )( − ) ( )[ ]
=
=
∑ v *
1
0
=
( )( ) ′′ + ( ) ′′
=
=
∑ γ ϕ γ ϕkr rr kr kk
k
r
N
v v
1
1
–
–
[ ]( ) ′′ + ( ) ′′ = ( ) ′′ + ( ) ′′
=
=
=
=
=
=
∑ ∑ ∑γ ϕ γ ϕ γ ϕ γ ϕkr kr kr rk
k
r
N
kr rr
k
r
N
kr kk
k
r
N
v v v v
1
1
1
1
1
1
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
712 H.6V.6MAMONOVA
+
γ ϕ γ γ ϕ γ ϕk kk
k
N
kr rk
k
r
N
rk kr rr
k
r
N
0
1 1
1
1
1
( ) ′′ − ( ) + ( ) ′′ = ( ) ′′
= =
=
=
=
∑ ∑ ∑( )v v v v +
+
v v v vk k kk
k
N
k k kk
k
N
kr rk
k
r
N
rkµ ϕ µ ϕ γ γ ϕ′′ ′′ − ( ) + ( ) ′′
= = =
=
∑ ∑ ∑ ( )
1 1 1
1
2∓ =
=
γ ϕ µ ϕ γ ϕ µ ϕkr rr k k kk
r
N
k
N
k
N
kr kr k k kr
r
N
( ) ′′ − ′′
−
( ) ′′ − ′′
== = =
∑∑ ∑ ∑( )v v v v
11 1 1
+
+
γ ϕ µ ϕrk rk k k kk
r
N
( ) ′′ − ′′
=
∑ v v
1
= γ
d
( v ) ϕ′′ ( u ) – [ vd
A + A′ vd
] ϕ′′ ( u ) =
= C
d
( v ) ϕ ( u ) – Λ ϕ ( u ) + A ( v ) ϕ ( u ) = B0 ( v ) ϕ ( u ).
Pry c\omu my skorystalysq rivnistg γ d
( v ) = Cd
( v ) – Λ, wo vyznaça[ umovu
balansu.
Lemu 1 dovedeno.
5. Kompensugçyj operator. Vvedemo neobxidni dlq podal\ßoho analizu
poznaçennq: napivhrupu Γt
ε( )v , t ≥ 0:
Γ Γ Γt s
t
I dsε ε ε( ) − = ( ) ( )∫v v v
0
; (17)
napivhrupu Ct
ε( )v , t ≥ 0:
C I C C dst s
t
ε ε ε( ) − = ( ) ( )∫v v v
0
, C u C uε ϕ ε ϕ( ) ( ) = − ( ) ′( )−v v1 * ; (18)
napivhrupu Ct , t ≥ 0:
C t I C C dss
t
( ) − = ∫
0
, C Cϕ ϕ( ) = ( ) ′( )v v v* .
Rozhlqnemo RPMV ζε
n, νε
n , κn
, n ≥ 0. (19)
Lema 2. Na test-funkciqx ϕ( )u, v = ( )( ) =ϕk u k N, , ,v 1 , ϕk ∈ C3
( RN × RN
)
kompensugçyj operator ma[ vyhlqd
L u G D I uε
ε
εϕ ε ϕ( ) = ( ) − ( )− [ ], ,v v v2Λ P , (20)
de
Gε ( v ) = G dt C Ck t t t
( ) ( ) ( )
∞
∫ Γε
ε
ε
εv v2 2
0
, D u u el
εϕ ϕ ε( ) = ( + ), ,v v . (21)
Dovedennq lemy bazu[t\sq na obçyslenni umovnoho matematyçnoho spodivan-
nq [6]:
E[ ]
+
( ) = = =+ϕ ζ κ ζκ
ε ε ε ε
n n n n n nk u
1 1, , ,v v v .
Lema 3. Kompensugçyj operator (20) dopuska[ asymptotyçne zobraΩennq
na test-funkciqx ϕ( )u, v ∈ C2
( RN × RN
)
L u Q u Q u uL
ε εϕ ε ϕ ϕ θ ϕ( ) = ( ) + ( ) + ( )−, , , ,v v v v2
2P , (22)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
EKSPLUATACIJNA SYSTEMA OBSLUHOVUVANNQ U SXEMI DYFUZIJNO} … 713
de
Q : = Λ [ P – I ], (23)
Q2 =
γ( ) + ( ) + + ( )u A h Cd d1
2
v v , (24)
hd = [ hk
δkr
, 1 ≤ k, r ≤ N ], hk = λk k kg g3 2 22( − )( ) , k = 1, N . (25)
Zalyßkovyj çlen θε
L zadovol\nq[ umovu znextuvannq:
θ ϕεL u( ), v → 0, ε → 0, ϕ ( u, v ) ∈ C3
( RN
).
Vykorystovugçy poznaçennq (21) ta totoΩnist\
g P d – I = P – I + ( g – I ) P + P ( d – I ) + ( g – I ) P ( d – I ),
zapysu[mo kompensugçyj operator (11) u vyhlqdi
Lε = ε– 2
Q + ε– 2
[ Gε ( x ) – I ] Q0 + ε– 2
[ Dε – I ] Q0 + ε– 2
[ Gε ( x ) – I ] [ Dε – I ] Q0. (26)
Dlq asymptotyçnoho zobraΩennq neperervno] skladovo] vykorysta[mo to-
toΩnist\
a b c – 1 = ( a – 1 ) + ( b – 1 ) + ( c – 1 ) + ( a – 1 ) ( b – 1 ) +
+ ( a – 1 ) ( c – 1 ) + ( b – 1 ) ( c – 1 ) + ( a – 1 ) ( b – 1 ) ( c – 1 ).
Todi
Gε – I = G I G I G I G I G I G I G IC C C C
ε
γ
ε ε ε
γ
ε ε
γ
ε− + − + − + ( − )( − ) + ( − )( − ) +
+ ( − )( − ) + ( − )( − )( − )G I G I G I G I G IC C C C
ε ε ε
γ
ε ε . (27)
Dlq perßoho dodanka, vykorystovugçy oznaçennq napivhrup, metod intehru-
vannq çastynamy ta lemu 2, otrymu[mo
ε ε γ γ θε
γ
γ
ε− −[ − ] = ( ) + ( ) + ( ) + ( ) + ( )2
0
1
0
2 21
2
G I Q u B gv v v v( ) Γ , (28)
de Q0 = Λ P.
Analohiçno dlq druhoho ta tret\oho dodankiv u rivnosti (27) ma[mo
ε ε θε
ε− −[ − ] = ( ) + ( ) + ( )2
0
1 2 2G I Q C gC
Cv v v( ) Γ , (29)
ε θε
ε− [ − ] = + ( )2
0G I Q CC
C v . (30)
Operator C vyznaça[t\sq takym çynom: C ϕ ( v ) = C ( v ) ϕ′ ( v ).
Asymptotyka çetvertoho dodanka u (27)
ε θε
γ
ε γ
ε− ( − )( − ) = − ( ) ( ) + ( )2
0
22G I G I Q g CC
C
( ) Γ v v v . (31)
Reßta dodankiv u (27) magt\ zalyßkovyj xarakter. Napryklad,
( − )( − ) = ( ) ( ) ( )
∞
∫ ∫ ∫G I G I C dt ds C ds CC
k s
t
s
t
ε
γ
ε
ε
ε
ε
0 0 0
2
Γ Γv v =
=
2 22 2
0
2
2ε ε θ θε ε
ε
ε
γ
ε
γ
εΓ Γ Γ( ) ( ) = ( ) − ( ) = ( )
∞
∫ [ ]v vC C t dt C C g I x xk t t k C C
( ) .
Dali obçyslg[mo dodanky v rivnosti (26):
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
714 H.6V.6MAMONOVA
ε ε λ θε ε− −[ − ] = + +2
0
1 1
2
D I Q dΛ , (32)
ε θε
ε ε− ( )( ) − [ − ] = − ( ) +2
0G x I D I Q CC d
cdv . (33)
Na pidstavi lemy 2 ostannij dodanok u (26) obçyslg[t\sq tak:
ε γ θε
γ ε
γ
ε− ( )( ) − [ − ] = ( ) +2
0G I D I Q d
dv v . (34)
Reßta çleniv magt\ zalyßkovyj xarakter.
Ob’[dnugçy asymptotyçni zobraΩennq (28) – (34), ma[mo
Q2 =
γ γ γ( ) + ( ) − + ( ) + ( ) − ( ) ( )[ ]u A g C C1
2
1
2
22 2 2v v v v vΛ ( ) +
+
C Cd+ ( ) − +v Λ Λ1
2
,
abo
Q2 =
γ( ) + ( ) + + ( )u A h Cd d1
2
v v ,
hd = [ hk
δkr
, 1 ≤ k, r ≤ N ], hk = λk k kg g3 2 22( − )( ) , k = 1, N .
Takym çynom, operator hranyçno] dyfuzi]
L u u u B u C uu u
0 1
2
ϕ γ ϕ ϕ ϕ( ) = ( ) ′ ( ) + ( ) ′′( ) + ( ) ′ ( ), , , ,*v v v v v vv ,
B ( v ) : = A ( v ) + C
d
( v ) + 2hd.
OtΩe, hranyçnyj dyfuzijnyj proces ζ0
( t ) vyznaça[t\sq heneratorom
L u u u B t ut
0 1
2
ϕ γ ϕ ϕ( ) = ( ) ′( ) + ( ) ′′( )( )* v , γ ( u ) = u A.
Dyfuzijna matrycq zaleΩyt\ vid useredneno] evolgci] v ( t ), t ≥ 0, wo vyznaça-
[t\sq rozv’qzkom evolgcijnoho rivnqnnq
d t
dt
C t
v
v
( ) = ( )( ).
Zaverßu[t\sq dovedennq teoremy za sxemog, navedenog v roboti [6], z vyko-
rystannqm hranyçno] teoremy dlq napivmarkovs\kyx vypadkovyx evolgcij u
sxemi userednennq [5] (teoremy 4.1, 4.3).
Vyslovlgg wyru podqku akademiku NAN Ukra]ny Volodymyru Semenovyçu
Korolgku za postanovku zadaçi ta postijnu uvahu pry ]] rozv’qzanni.
1. Korolgk7V.7S., Turbyn7A.7F. Process¥ markovskoho vosstanovlenyq v zadaçax nadeΩnosty
system. – Kyev: Nauk. dumka, 1982. – 2366s.
2. Mamonova7H.7V. Ekspluatacijna systema obsluhovuvannq typu [ SM | M | ∞ ]N
u sxemi use-
rednennq // Tr. Yn-ta prykl. matematyky y mexanyky NAN Ukrayn¥. – 2005. – 10. –
S.6135 – 144.
3. Mamonova7H.7V. Ekspluatacijna systema obsluhovuvannq u sxemi dyfuzijno] aproksymaci] //
Visn. Ky]v. un-tu. Ser. fiz.-mat. nauky. – 2005. – #63. – S.6333 – 337.
4. Karamaki N. Continues exponentive martingales and BMO // Lect. Notes Math. – 1999. – # 1579.
5. Korolyuk V. S., Swishchuk A. V. Random evolution. – Dordrecht: Kluwer Acad. Publ., 1994.
6. Svyrydenko7M.7N. Ob uslovyqx sxodymosty semejstva polumarkovskyx processov k mar-
kovskomu processu. – M¥tywy, 1986. – 176s.
OderΩano 17.06.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 5
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| id | umjimathkievua-article-3488 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:43:30Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a4/57874ed411464a8c2c78fee51be632a4.pdf |
| spelling | umjimathkievua-article-34882020-03-18T19:55:42Z Operational queueing system in the scheme of diffusion approximation with evolution averaging Експлуатаційна система обслуговування у схемі дифузійної апроксимації з еволюційним усередненням Mamonova, G. V. Мамонова, Г. В. We consider an operational queuing system of the type $[SM | M | \infty]^N$ in the scheme of diffusion approximation. The queueing system is described by a semi-Markov random evolution. Розглядається експлуатаційна система обслуговування типу $[SM | M | \infty]^N$ у схемі дифузійної апроксимації. Система обслуговування описується напівмарковською випадковою еволюцією. Institute of Mathematics, NAS of Ukraine 2006-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3488 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 5 (2006); 708–714 Український математичний журнал; Том 58 № 5 (2006); 708–714 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3488/3713 https://umj.imath.kiev.ua/index.php/umj/article/view/3488/3714 Copyright (c) 2006 Mamonova G. V. |
| spellingShingle | Mamonova, G. V. Мамонова, Г. В. Operational queueing system in the scheme of diffusion approximation with evolution averaging |
| title | Operational queueing system in the scheme of diffusion approximation with evolution averaging |
| title_alt | Експлуатаційна система обслуговування у схемі дифузійної апроксимації з еволюційним усередненням |
| title_full | Operational queueing system in the scheme of diffusion approximation with evolution averaging |
| title_fullStr | Operational queueing system in the scheme of diffusion approximation with evolution averaging |
| title_full_unstemmed | Operational queueing system in the scheme of diffusion approximation with evolution averaging |
| title_short | Operational queueing system in the scheme of diffusion approximation with evolution averaging |
| title_sort | operational queueing system in the scheme of diffusion approximation with evolution averaging |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3488 |
| work_keys_str_mv | AT mamonovagv operationalqueueingsystemintheschemeofdiffusionapproximationwithevolutionaveraging AT mamonovagv operationalqueueingsystemintheschemeofdiffusionapproximationwithevolutionaveraging AT mamonovagv ekspluatacíjnasistemaobslugovuvannâushemídifuzíjnoíaproksimacíízevolûcíjnimuserednennâm AT mamonovagv ekspluatacíjnasistemaobslugovuvannâushemídifuzíjnoíaproksimacíízevolûcíjnimuserednennâm |