Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance
We determine a stationary measure for a process defined by a differential equation with phase space on the segment $[V_0 , V_1]$ and constant values of a vector field that depend on a controlling semi-Markov process with finite set of states.
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| Datum: | 2006 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509556608598016 |
|---|---|
| author | Pogorui, A. О. Погоруй, А. О. |
| author_facet | Pogorui, A. О. Погоруй, А. О. |
| author_sort | Pogorui, A. О. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:55:07Z |
| description | We determine a stationary measure for a process defined by a differential equation with phase space on the segment $[V_0 , V_1]$ and constant values of a vector field that depend on a controlling semi-Markov process with finite set of states. |
| first_indexed | 2026-03-24T02:42:59Z |
| format | Article |
| fulltext |
UDK 519.21
A. O. Pohoruj (Ûytomyr. ped. un-t)
STACIONARNYJ ROZPODIL PROCESU VYPADKOVO}
NAPIVMARKOVS|KO} EVOLGCI} Z ZATRYMUGÇYMY
EKRANAMY U VYPADKU BALANSU
A stationary measure is obtained for a process given by a differential equation with phase space on the
interval [
V0
, V1
] and stable values of a vector field that depend on the controlling semi-Markov
process with a finite set of states.
Znajdeno stacionarnu miru dlq procesu, wo opysu[t\sq dyferencial\nym rivnqnnqm iz fazovym
prostorom na vidrizku [
V0
, V1
] ta stalymy znaçennqmy vektornoho polq, qki zaleΩat\ vid ke-
rugçoho napivmarkovs\koho procesu zi skinçennog mnoΩynog staniv.
U zadaçax nadijnosti pry obçyslenni stacionarnyx pokaznykiv efektyvnosti ta
nadijnosti system vynyka[ problema znaxodΩennq stacionarnyx rozpodiliv pro-
cesiv, wo modelggt\ ci systemy [1 – 3] (rozd. 3, 4). Dlq doslidΩennq bahato-
faznyx system iz nakopyçuvaçamy v qkosti modelggçyx vykorystovugt\sq sto-
xastyçni procesy perenosu z zatrymugçymy ekranamy v markovs\komu çy napiv-
markovs\komu seredovywi [2, 3]. U vypadku napivmarkovs\koho kerugçoho pro-
cesu doslidΩu[t\sq stacionarnyj rozpodil vidpovidnoho trykomponentnoho
markovs\koho procesu, perßa komponenta qkoho [ çasovog (ças, projdenyj ke-
rugçym procesom z momentu ostann\o] zminy stanu), druha vidpovida[ stanu ke-
rugçoho procesu i tretq [ prostorovog, wo opysu[ napovnenist\ nakopyçuvaçiv.
ZnaxodΩennq stacionarnoho rozpodilu v napivmarkovs\komu vypadku [ netryvi-
al\nog zadaçeg navit\ dlq najprostißoho vypadku al\ternuval\noho kerugço-
ho procesu [3].
U danij roboti rezul\tat, otrymanyj u [3] dlq odnofazno] systemy, uzahal\-
ng[t\sq na dovil\nyj skinçennyj fazovyj prostir kerugçoho procesu u vypadku
balansu.
Rozhlqnemo rivnqnnq
d t
dt
ν( )
= C ( κ ( t ), ν ( t ) ), (1)
de κ ( t ) — napivmarkovs\kyj proces iz fazovym prostorom G = X ∪ Y , X = { x1 ,
x2 , … , xn }, Y = { y1 , y2 , … , ym }, matryceg perexidnyx imovirnostej vkladenoho v
κ ( t ) lancgha Markova kl , l ∈ N, P = p Gαβ α β, , ∈{ }, pαβ = P κ α κl l+{ = /1 =
= β} i çasom τα perebuvannq u stani α ∈ G, wo ma[ zahal\nu funkcig roz-
podilu Fα ( t ).
Vidomo [5] (hl. 3), [6], wo rivnqnnq (1) opysu[ stoxastyçnyj proces perenosu
v napivmarkovs\komu seredovywi.
Prypuska[mo vykonannq umovy
U1
) isnugt\ wil\nist\ fα ( t ) =
dF t
dt
α ( )
ta momenty
mα =
0
∞
∫ t f t dtα ( ) , mα
( )2 =
0
2
∞
∫ t f t dtα ( ) ∀ α ∈ G.
Nexaj V0 , V1 , ai , bj ∈ R, V0 < V1 , ai > 0, bj > 0, i = 1, n, j = 1, m , i funkciq z
pravo] çastyny (1) zadovol\nq[ umovy:
dlq xi ∈ X, i = 1, n, C ( xi , ν ) =
b V V
V
i , ,
, ,
0 1
10
≤ <
=
ν
ν
© A. O. POHORUJ, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3 381
382 A. O. POHORUJ
dlq yj ∈ Y, j = 1, m , C ( yj , ν ) =
− < ≤
=
a V V
V
j , ,
, .
0 1
00
ν
ν
Vvedemo na fazovomu prostori Z = [ 0, ∞ ) × G × [ V0 , V1 ] trykomponentnyj
proces
ξ ( t ) = ( τ ( t ), κ ( t ), ν ( t ) ), de τ ( t ) = t – sup : ( ) ( )u t t u≤ ≠{ }κ κ .
Naßa meta polqha[ v znaxodΩenni stacionarnoho rozpodilu procesu ξ ( t ).
Proces ξ ( t ) — markovs\kyj i joho infinitezymal\nyj operator ma[ vyhlqd
[3, 4]
A ϕ ( τ, α, ν ) =
∂
∂τ
ϕ τ α ν( , , ) +
+ rα τ ϕ α ν ϕ τ α ν( ) ( , , ) ( , , )P 0 −[ ] + C( , ) ( , , )α ν ∂
∂ν
ϕ τ α ν
z hranyçnymy umovamy ′ϕ ττ( , , )x V0 = ′ϕ ττ( , , )y V1 = 0, x ∈ X, y ∈ Y, de
rα τ( ) =
f
F
α
α
τ
τ
( )
( )1 −
, Pϕ α ν( , , )0 =
β
αβ ϕ β ν
∈
∑
G
p ( , , )0 .
Qkwo proces ξ ( t ) ma[ stacionarnu miru ρ ( ⋅ ), to dlq bud\-qko] funkci]
ϕ ( ⋅ ) z oblasti vyznaçennq operatora A
Z
A z dz∫ ϕ ρ( ) ( ) = 0. (2)
Nadali prypuska[mo vykonannq umovy:
U2
) isnu[ wil\nist\ ρ ( τ, α, ν ) stacionarnoho rozpodilu procesu ξ ( t ), pry-
çomu isnugt\ skriz\
∂ρ τ α ν
∂τ
( , , )
,
∂ρ τ α ν
∂ν
( , , )
i lim ( , , )τ τ α ν→+∞ .
Analiz vlastyvostej procesu ξ ( t ) pokazu[, wo v toçkax ( τ, x, V1 ), x ∈ X, ta
( τ, x, V0 ), y ∈ Y, fazovoho prostoru Z znaxodqt\sq synhulqrni komponenty
stacionarno] miry. Budemo poznaçaty ]x çerez ρ [ τ, x, V0 ], ρ [ τ, y, V1 ].
Zminggçy v (2) porqdok intehruvannq, otrymu[mo vyrazy dlq A
*
ρ = 0, de
A
*
— sprqΩenyj do A operator, a same, dlq nesynhulqrno] çastyny miry
C( , ) ( , , )α ν ∂
∂ν
ρ τ α ν + rα τ ρ τ α ν( ) ( , , ) +
∂
∂τ
ρ τ α ν( , , ) = 0, (3)
ρ ( ∞, α, ν ) = 0, α ∈ G,
β
β βατ ρ τ β ν τ
∈
∞
∑ ∫
G
r d p
0
( ) ( , , ) = ρ ( 0, α, ν ), α ∈ G, (4)
i dlq synhulqrnyx komponent
ρ [ ∞, x, V1 ] = 0, x ∈ X, ρ [ ∞, y, V0 ] = 0, y ∈ Y,
d
d
x V d
τ
ρ τ τ, , 1[ ] + rx ( )τ ρ [ τ, x, V1 ] – bx ρ ( τ, x, V1 – ) = 0, (5)
d
d
y V d
τ
ρ τ τ, , 0[ ] + ry ( τ ) ρ [ τ, y, V0 ] – ay ρ ( τ, y, V0 + ) = 0, (6)
de ρ ( τ, x, V1 – ) = lim ( , , )ν ρ τ ν↑V
x
1
, ρ ( τ, x, V0 + ) = lim ( , , )ν ρ τ ν↓V x
0
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
STACIONARNYJ ROZPODIL PROCESU VYPADKOVO} NAPIVMARKOVS|KO} … 383
Dali,
y Y
y yzr y V d p
∈
∞
∑ ∫ [ ]
0
0( ) , ,τ ρ τ τ = ρ [ 0, z, V0 ], z ∈ Y,
x X
x xzr x V d p
∈
∞
∑ ∫ [ ]
0
0( ) , ,τ ρ τ τ = ρ [ 0, z, V0 ], z ∈ X,
(7)
y Y
y yxr y V d p
∈
∞
∑ ∫ [ ]
0
0( ) , ,τ ρ τ τ = b x V dx
0
0
∞
∫ +( )ρ τ τ, , , x ∈ X,
x X
x xyr x V d p
∈
∞
∑ ∫ [ ]
0
1( ) , ,τ ρ τ τ = a y V dy
0
1
∞
∫ −( )ρ τ τ, , , y ∈ Y.
Rozv’qzugçy rivnqnnq (3), znaxodymo
ρ ( τ, x, ν ) = f b ex x
r s dsx( )
( )
ν τ
τ
− ∫− 0
, x ∈ X,
ρ ( τ, y, ν ) = f a ey y
r s dsx( )
( )
ν τ
τ
+ ∫− 0
, y ∈ Y.
Vypadok balansu xarakteryzu[t\sq nezaleΩnistg nesynhulqrno] çastyny staci-
onarnoho rozpodilu ρ ( ⋅ ) vid ν. U takomu vypadku funkci] f bx x( )ν τ− = cx ,
f ay y( )ν τ+ = cy — konstanty. Todi z uraxuvannqm toho, wo
e
r s ds− ∫ α
τ
( )
0 = 1 – Fα ( τ ),
rozv’qzok rivnqnnq (3) ma[ vyhlqd
ρ ( τ, α, ν ) = cα (1 – Fα ( τ ) ), α ∈ G. (8)
Oskil\ky vykonu[t\sq umova U1
, to
∂ρ τ α ν
∂τ
( , , )
= – cα fα ( τ ), lim ( , , )τ τ α ν→+∞ =
= 0. Vraxovugçy nezaleΩnist\ ρ ( τ, α, ν ) vid ν, perekonu[mos\, wo rozv’qzok
(8) povnistg vidpovida[ umovi U2
.
Pidstavlqgçy cg rivnist\ v (4), otrymu[mo
α
α αβ
∈
∑
G
c p = cβ . (9)
Dali prypuska[mo, wo vykonu[t\sq umova
U3
) vkladenyj u proces κ ( t ) lancgh Markova nezvidnyj, erhodyçnyj i ma[
stacionarnyj rozpodil ρα , α ∈ G.
Todi rozv’qzok (9) nabyra[ vyhlqdu
cα = c ρα , α ∈ G, (10)
zvidky
0
∞
∫ ρ τ α ν τ( , , )d = c F dρ τ τα α
0
1
∞
∫ −( )( ) = c ρα mα , α ∈ G.
Rozv’qzugçy (5), (6), z uraxuvannqm (8), (10) otrymu[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
384 A. O. POHORUJ
ρ [ τ, x, V1 ] = c F b
x V
x x x
x
ρ τ τ ρ
σρ
1
0 1−( ) + [ ]
( )
, ,
,
(11)
ρ [ τ, y, V0 ] = c F a
y V
y y y
y
ρ τ τ ρ
σρ
1
0 0−( ) + [ ]
( )
, ,
.
Pidstavlqgçy (11) v (7), znaxodymo
x X
x x x xzb m p
∈
∑ ρ +
x X
xzx V p
∈
∑ [ ]ρ 0 1, , = ρ [ 0, z, V1 ], z ∈ X,
y Y
y y y yza m p
∈
∑ ρ +
y Y
yzy V p
∈
∑ [ ]ρ 0 0, , = ρ [ 0, y, V0 ], z ∈ Y,
y Y
y y y yxa m p
∈
∑ ρ +
y Y
yxy V p
∈
∑ [ ]ρ 0 0, , = b mx x xρ , x ∈ X,
x X
x x x xyb m p
∈
∑ ρ +
x X
xyx V p
∈
∑ [ ]ρ 0 1, , = a my y yρ , y ∈ Y.
Poznaçymo PX = p x z Xxz , , ∈{ }, PY = p y z Xyz , , ∈{ }, GX = ( I – PX )
–
1
= g xxz{ , ,
z X∈ }, G Y = ( I – PY )
–
1
= g y z Yyz , , ∈{ } . Matryci GX , GY magt\ znaçennq
potencialiv [7] (hl. 1, § 6). Rozv’qzugçy dva perßyx rivnqnnq (7), ma[mo
ρ [ 0, z, V0 ] =
x X
x x x
k X
xk kzb m p g
∈ ∈
∑ ∑ρ , z ∈ X,
ρ [ 0, z, V1 ] =
y Y
y y y
k Y
yk kza m p g
∈ ∈
∑ ∑ρ , z ∈ Y.
Pidstavlqgçy ci formuly v dva ostannix rivnqnnq (7), otrymu[mo umovu
U4
) magt\ misce spivvidnoßennq
x X
x x x xyb m p
∈
∑ ρ +
x X
x x x
k X
xk
z X
kz zyb m p g p
∈ ∈ ∈
∑ ∑ ∑ρ = b my y yρ , y ∈ Y,
y Y
y y y yxa m p
∈
∑ ρ +
y Y
y y y
k Y
yk
z Y
kz zxa m p g p
∈ ∈ ∈
∑ ∑ ∑ρ = b mx x xρ , x ∈ X.
OtΩe, dovedeno taku teoremu.
Teorema. Qkwo vykonugt\sq umovy U1 – U4
, to stacionarnyj rozpodil
ρ ( ⋅ ) procesu ξ ( t ) ma[ vyhlqd: dlq wil\nostej ρ ( τ , x , ν ) = c Fx xρ τ1 −( )( ) ,
x ∈ X, ρ ( τ, y, ν ) = c Fy yρ τ1 −( )( ) , y ∈ Y, dlq atomiv
ρ [ τ, x, V0 ] = c F b
x V
cx x x
x
ρ τ τ ρ
ρ
1
0 0−( ) + [ ]
( )
, ,
,
ρ [ τ, y, V1 ] = c F a
y V
cy y y
y
ρ τ τ ρ
ρ
1
0 1−( ) + [ ]
( )
, ,
,
de
ρ [ 0, z, V0 ] =
x X
x x x
k X
xk kzb m p g
∈ ∈
∑ ∑ρ , z ∈ X,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
STACIONARNYJ ROZPODIL PROCESU VYPADKOVO} NAPIVMARKOVS|KO} … 385
ρ [ 0, z, V1 ] =
y Y
y y y
k Y
yk kza m p g
∈ ∈
∑ ∑ρ , z ∈ Y,
a c znaxodyt\sq z umovy
Z
dz∫ ρ( ) = 1.
Teper pokaΩemo, wo za dodatkovyx umov, navedenyx nyΩçe, znajdena stacio-
narna mira [ [dynog, tobto ne isnu[ inßoho stacionarnoho rozpodilu, dlq qkoho
umovu U2 ne vykonano i qkyj by zadovol\nqv rivnqnnq (2).
U danyj ças u teori] vypadkovyx procesiv dobre vidomyj metod kaplinha
(sklegvannq), qkyj ßyroko vykorystovu[t\sq pry doslidΩenni markovs\kyx ta
blyz\kyx do nyx procesiv [8, 9]. A same, ma[ misce taka lema.
Lema. Qkwo proces Markova ζ ( t ) z fazovym prostorom Ψ zadovol\nq[
umovu
C) isnugt\ T > 0, γ > 0 ta proces Markova
�
θ( )t = ( θ1 ( t ), θ2 ( t ) ) na fazo-
vomu prostori Ψ
2
taki, wo:
a) obydvi komponenty procesu
�
θ( )t magt\ odnakovyj rozpodil z procesom
ζ ( t );
b) dlq bud\-qkyx x, y P θ θ θ θ1 2 1 20 0( ) ( ) ( ) , ( )T T x y= = ={ }/ ≥ γ,
to proces ζ ( t ) ma[ ne bil\ße odnoho stacionarnoho rozpodilu.
PokaΩemo qk moΩna zastosuvaty metod kaplinha u danomu vypadku.
I. Prypustymo, wo κ ( t ) — markovs\kyj proces, tobto zamist\ trykompo-
nentnoho procesu ξ ( t ) = ( τ ( t ), κ ( t ), ν ( t ) ) moΩemo rozhlqdaty dvokomponentnyj
ξ( )t = ( κ ( t ), ν ( t ) ). Bez vtraty zahal\nosti prypuska[mo, wo pα β =
= P κ α κ βl l+ = ={ }/1 ≥ δ > 0 ∀ α, β ∈ G. Nexaj ′κ l i ′′κ l , l ≥ 1, — dva ne-
zaleΩnyx lancghy Markova z spil\nym fazovym prostorom G, matryceg pere-
xidnyx imovirnostej P = { pα β , α , β ∈ G } i odnoçasnymy strybkamy. Todi ne-
vaΩko perekonatys\, wo P ′ = ′′= ′ = ′′ ={ }/κ α κ α κ α κ β1 0 1 0 0 0, , ≥ δ
2
> 0 ∀ α0
, α,
β ∈ G, tobto pislq perßoho strybka procesy ′κ l i ′′κ l , l ≥ 1, sklegt\sq z
imovirnistg ne menßog za δ
2
> 0. Rozhlqnemo dva nezaleΩnyx markovs\kyx
procesy κ ′ ( t ), κ ′′ ( t ) z vkladenymy lancghamy ′κ l i ′′κ l vidpovidno i z inten-
syvnostqmy perebuvannq u stanax takymy Ω, qk u κ ( t ). Dali, do deqkoho momen-
tu T0 > 0 z imovirnistg ne menßog za δ1 = λ λ
0
0T se ds∫ −
vidbudet\sq strybok
procesu κ ( t ), de λ = minα αλ∈G ( λα — intensyvnist\ perebuvannq κ ( t ) v
α ∈ G ). OtΩe, do momentu t0 procesy κ ′ ( t ), κ ′′ ( t ) sklegt\sq z imovirnistg ne
menßog za δ1 δ
2
> 0.
Poznaçymo νmin = min ( , ),i j G i ja b∈ , T = 2 1 0V V−
νmin
. Z imovirnistg δ2 =
= λ λ
T
se ds
0
∞ −∫ pislq momentu T0 ne vidbudet\sq strybkiv i, otΩe, za odnakovyx
κ ′ ( t ), κ ′′ ( t ) çerez ças ne bil\ßyj za T sklegt\sq v atomi komponenty ν ′ ( t ),
ν ′′ ( t ), de ν ′ ( t ) zadovol\nq[ rivnqnnq (1), kerovane procesom κ ′ ( t ), a ν ′′ ( t ) —
procesom κ ′′ ( t ). OtΩe, z imovirnistg ne menßog za δ2 δ1 δ
2
> 0 do momentu T +
+ T0 sklegt\sq procesy
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
386 A. O. POHORUJ
′ξ ( )t = ( κ ′ ( t ), ν ′ ( t ) ) ta ′′ξ ( )t = ( κ ′′ ( t ), ν ′′ ( t ) ).
II. Rozhlqnemo vypadok, koly κ ( t ) — napivmarkovs\kyj proces. Rozhlqnemo
nezaleΩni procesy ξ ′ ( t ) = ( τ ′ ( t ), κ ′ ( t ), ν ′ ( t ) ), ξ ′′ ( t ) = ( τ ′′ ( t ), κ ′′ ( t ), ν ′′ ( t ) ), de
κ ′ ( t ), κ ′′ ( t ) — nezaleΩni napivmarkovs\ki procesy z vkladenymy lancghamy ′κ l
i ′′κ l vidpovidno i z intensyvnostqmy perebuvannq u stanax takymy Ω, qk u κ ( t );
ν ′ ( t ) zadovol\nq[ rivnqnnq (1), kerovane procesom κ ′ ( t ), a ν ′′ ( t ) — procesom
κ ′′ ( t ). Spoçatku pokaΩemo, wo za pevno] umovy moΩna skle]ty ( τ ′ ( t ), κ ′ ( t ) ) ta
( τ ′′ ( t ), κ ′′ ( t ) ).
Poznaçymo µ t, x ( d s ) = P τ τ κ∈ = ={ }/ds t x, i budemo prypuskaty isnuvannq
neperervno] wil\nosti pt, x ( s ) > 0, tobto µ t, x ( d s ) = pt, x ( s ) d s. Nexaj vykonu[t\-
sq umova: dlq deqkoho T0 > 0
inf min ( ), ( )
( , ), ( , )
,
, ,t x t x
t t T
T
t x t xp s p s ds
1 1 2 2
1 2 0
0
1 1 2 2
0≤
∫ ( ) = δ3 > 0,
(12)
inf ,
,
,
t x
t x Tµ 0 0[ ]( ) = δ4 > 0.
Z uraxuvannqm (12) iz kaplinhovo] lemy [8, 9] vyplyva[, wo ( τ ′ ( t ), κ ′ ( t ) ) ta
( τ ′′ ( t ), κ ′′ ( t ) ) moΩna skle]ty do momentu T0 z imovirnistg ne menßog za
δ3 δ4 δ
2
, a dali analohiçno markovs\komu vypadku çeka[mo do sklejky kompo-
nent ν ′ ( t ) i ν ′′ ( t ) v atomi. Qkwo max ( t1 , t2 ) > T0 , to z imovirnistg ne menßog
za
δ5 =
0
1 0 1 21 1 2
∞
∫ + + − +( ) +F t T t F t t f t t dtx x x( ) ( ) ( )
v moment strybka κ ′′ ( t ) ma[mo τ ′′ = 0, τ ′ ≤ T0 i moΩna z uraxuvannqm (12) sko-
rystatys\ kaplinhovog lemog.
Dlq praktyçnoho zastosuvannq, napryklad pry obçyslenni stacionarnoho ko-
efici[nta hotovnosti systemy, neobxidno znaty funkcig ρ α ν( , ) , wo dorivng[
çastci çasu, provedenoho dvokomponentnym procesom ζ ( t ) = ( κ ( t ), ν ( t ) ) u stani
( α, ν ) [3]. Cq funkciq znaxodyt\sq tak:
ρ α ν( , ) =
0
∞
∫ ρ τ α ν τ( , , )d = c ρα mα , α ∈ G,
ρ x V, 1[ ] =
0
1
∞
∫ [ ]ρ τ τ, ,x V d =
= c b m b m p g
z X
z z z
z X
z z z
k X
zk kx
1
2
2 2
∈ ∈ ∈
∑ ∑ ∑+
ρ ρ( )
, x ∈ X,
ρ y V, 0[ ] =
0
0
∞
∫ [ ]ρ τ τ, ,y V d =
= c b m a m p g
z Y
z z z
z Y
z z z
k Y
zk ky
1
2
2 2
∈ ∈ ∈
∑ ∑ ∑+
ρ ρ( )
, y ∈ Y,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
STACIONARNYJ ROZPODIL PROCESU VYPADKOVO} NAPIVMARKOVS|KO} … 387
de
c =
β
β β β
β
β β βρ ρ ρ
∈ ∈ ∈ ∈ ∈
∑ ∑ ∑ ∑ ∑+ +
G G x X z X
z z z
k X
zk kxb m b m b m p g
1
2
2 2 +
+
y Y z Y
z z z
k Y
zk kya m p g
∈ ∈ ∈
−
∑ ∑ ∑
ρ 2
1
.
Na zaverßennq vyslovlgg podqku O. M. Kulyku, qkyj u pryvatnij besidi naviv
meni sxemu dovedennq odynyçnosti stacionarnoho rozpodilu za dopomohog meto-
du kaplinha.
1. Korolgk V. S., Turbyn A. F. Process¥ markovskoho vosstanovlenyq v zadaçax nadeΩnosty
system. – Kyev: Nauk. dumka, 1982. – 234 s.
2. Pohoruj A. A., Turbyn A. F. Ocenka stacyonarnoj πffektyvnosty proyzvodstvennoj lynyy
s dvumq nenadeΩn¥my ahrehatamy // Kybernetyka y systemn¥j analyz. – 2002. – # 6. –
S.S35 – 42.
3. Turbyn A. F., Pohoruj A. A. Rasçet stacyonarn¥x pokazatelej πffektyvnosty system up-
ravlenyq zapasamy s obratnoj svqz\g // Yntellektualyzacyq system obrabotky ynformacy-
onn¥x soobwenyj. – Kyev: Yn-t matematyky NAN Ukrayn¥, 1995. – S. 191 – 204.
4. Korlat A. N., Kuznecov V. N., Novykov M. M., Turbyn A. F. Polumarkovskye modely vossta-
navlyvaem¥x system y system massovoho obsluΩyvanyq. – Kyßynev: Ítyynca, 1991. – 276 s.
5. Korolgk V. S., Svywuk A. V. Polumarkovskye sluçajn¥e πvolgcyy. – Kyev: Nauk. dumka,
1992. – 256 s.
6. Korolgk V. S. Stoxastyçni modeli system. – Ky]v: Lybid\, 1993. – 134 s.
7. Korolgk V. S., Turbyn A. F. Matematyçeskye osnov¥ fazovoho ukrupnenyq sloΩn¥x
system. – Kyev: Nauk. dumka, 1978. – 220 s.
8. Veretennikov A. Yu. Coupling method for Markov chains under integral Doeblin type condition //
Theory Stochast. Processes. – 2002. – 8(24), # 3 – 4. – P. 383 – 391.
9. Lindvall T. Lectures on the coupling method. – New York: Dover Publ., 1992. – 257 p.
OderΩano 28.05.2003,
pislq doopracgvannq — 22.07.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 3
|
| id | umjimathkievua-article-3461 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:42:59Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f4/8937f34adbe9660b6c86168838257ef4.pdf |
| spelling | umjimathkievua-article-34612020-03-18T19:55:07Z Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance Стаціонарний розподіл процесу випадкової напівмарковської еволюції з затримуючими екранами у випадку балансу Pogorui, A. О. Погоруй, А. О. We determine a stationary measure for a process defined by a differential equation with phase space on the segment $[V_0 , V_1]$ and constant values of a vector field that depend on a controlling semi-Markov process with finite set of states. Знайдено стаціонарну міру для процесу, що описується диференціальним рівнянням із фазовим простором на відрізку $[V_0 , V_1]$ та сталими значеннями векторного поля, які залежать від керуючого напівмарковського процесу зі скінченною множиною станів. Institute of Mathematics, NAS of Ukraine 2006-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3461 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 3 (2006); 381–387 Український математичний журнал; Том 58 № 3 (2006); 381–387 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3461/3659 https://umj.imath.kiev.ua/index.php/umj/article/view/3461/3660 Copyright (c) 2006 Pogorui A. О. |
| spellingShingle | Pogorui, A. О. Погоруй, А. О. Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance |
| title | Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance |
| title_alt | Стаціонарний розподіл процесу випадкової напівмарковської еволюції з затримуючими екранами у випадку балансу |
| title_full | Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance |
| title_fullStr | Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance |
| title_full_unstemmed | Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance |
| title_short | Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance |
| title_sort | stationary distribution of a process of random semi-markov evolution with delaying screens in the case of balance |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3461 |
| work_keys_str_mv | AT pogoruiao stationarydistributionofaprocessofrandomsemimarkovevolutionwithdelayingscreensinthecaseofbalance AT pogorujao stationarydistributionofaprocessofrandomsemimarkovevolutionwithdelayingscreensinthecaseofbalance AT pogoruiao stacíonarnijrozpodílprocesuvipadkovoínapívmarkovsʹkoíevolûcíízzatrimuûčimiekranamiuvipadkubalansu AT pogorujao stacíonarnijrozpodílprocesuvipadkovoínapívmarkovsʹkoíevolûcíízzatrimuûčimiekranamiuvipadkubalansu |