Convergence of an impulsive storage process with jump switchings
We investigate an impulsive storage process switched by a jump process. The switching process is, in turn, averaged. We prove the weak convergence of the storage process in the scheme of series where a small parameter ε tends to zero.
Збережено в:
| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3243 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509293212598272 |
|---|---|
| author | Samoilenko, I. V. Самойленко, І. В. |
| author_facet | Samoilenko, I. V. Самойленко, І. В. |
| author_sort | Samoilenko, I. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:57Z |
| description | We investigate an impulsive storage process switched by a jump process. The switching process is, in turn, averaged. We prove the weak convergence of the storage process in the scheme of series where a small parameter ε tends to zero. |
| first_indexed | 2026-03-24T02:38:48Z |
| format | Article |
| fulltext |
UDK 519.21
I. V. Samojlenko (In-t matematyky NAN Ukra]ny, Ky]v)
ZBIÛNIST| IMPUL|SNOHO PROCESU NAKOPYÇENNQ
ZI STRYBKOVYMY PEREMYKANNQMY
We study impulsive storage process switching by a jump process. The switching process itself is an
averaging process. Weak convergence of the storage process in a series scheme when a small parameter
ε tends to zero is proved.
Yssledovan ympul\sn¥j process nakoplenyq, kotor¥j pereklgçaetsq s pomow\g skaçkoobraz-
noho processa. Pereklgçagwyj process, v svog oçered\, usrednqetsq. Dokazana slabaq sxo-
dymost\ processa nakoplenyq v sxeme seryj, kohda mal¥j parametr ε stremytsq k nulg.
U roboti [1] rozhlqda[t\sq zbiΩnist\ procesiv nakopyçennq z peremykannqmy v
sxemi serij, qki budugt\sq za dopomohog sum umovno nezaleΩnyx vypadkovyx ve-
lyçyn abo procesiv z umovno nezaleΩnymy pryrostamy na tra[ktoriqx procesiv,
wo peremykagt\sq. Vyvçeno takoΩ deqki zastosuvannq do analizu procesiv na-
kopyçennq v modelqx system obsluhovuvannq.
Bil\ß detal\no, v [1] rozhlqnuto poslidovnist\ procesiv
S tn( ) =
k
nt
nk nk kS x
=
∑ ( )
0
ν
γ
( )
; ,
de Snk +1 = Snk + ξnk kx( , Snk ), ξnk , γ nk — nezaleΩni sim’] nezaleΩnyx u sukup-
nosti vypadkovyx velyçyn, xk — markovs\kyj proces, ν( )t = min k{ : k ≥ 0,
t tk + ≥ }1 , t ≥ 0, — zahal\na kil\kist\ toçok peremykannq na promiΩku 0, t[ ].
Takym çynom, procesy S tn( ) utvorggt\ procesy nakopyçennq z peremykan-
nqm na rekurentnyx procesax napivmarkovs\koho typu. Bulo vyvçeno zbiΩnist\
za parametrom n. Vykorystovugçy metody robit [2, 3] , zokrema metod xarakte-
rystyçnyx funkcij, dovedeno zbiΩnist\ S tn( ) do neodnoridnoho procesu z neza-
leΩnymy pryrostamy.
My proponu[mo rozhlqnuty analohiçnu zadaçu u dewo sprowenomu varianti v
terminax maloho parametra serij, wo prqmu[ do nulq, ta zastosuvaty dlq dove-
dennq zbiΩnosti metod, zaproponovanyj v roboti [4] (dyv. takoΩ [5]).
Dotrymugçys\ roboty [4], poznaçymo çerez x t( ), t ≥ 0, markovs\kyj proces
strybkiv u standartnomu prostori staniv ( , )E E . Nexaj cej proces vyznaça[t\sq
heneratorom
Q xϕ( ) = q x P x dy y x
E
( ) ( , ) ( ) – ( )∫ [ ]ϕ ϕ .
Napivmarkovs\ke qdro
Q x B t( , , ) = P x B e q x t( , ) – – ( )1( ), x E∈ , B∈E , t ≥ 0,
vyznaça[ asocijovanyj markovs\kyj proces vidnovlennq ( , )xk kτ , k ≥ 0, de xk ,
k ≥ 0, — vkladenyj lancgh Markova, wo zadanyj stoxastyçnym qdrom
P x B( , ) = P( )x B x xk k+ ∈ =1 ,
a τk , k ≥ 0, — toçkovyj proces momentiv strybkiv, qkyj vyznaça[t\sq funkci[g
rozpodilu çasu perebuvannq θk +1 = τk +1 – τk , k ≥ 0,
P( )θk kt x x+ ≤ =1 = 1 – e q x t– ( )
.
© I. V. SAMOJLENKO, 2008
1282 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
ZBIÛNIST| IMPUL|SNOHO PROCESU NAKOPYÇENNQ ZI STRYBKOVYMY … 1283
V rozdili 3 roboty [4] vyvça[t\sq proces
S tε( ) = s +
k
t
k kC S x
=
∑ ( )
1
ν ε
ε
( / )
; ,
de s d∈ ′R , Sk
ε = S k( )ετ , ν( )t = max k{ : τk t≤ } — liçyl\nyj proces strybkiv.
V [4] dovedeno, wo proces S tε( ) pry ε → 0 zbiha[t\sq do rozv’qzku rivnqnnq
d
dt
S tˆ( ) = ˆ ˆ( )C S t( ),
de
ˆ( )C s =
E
dx C s x∫ π( ) ( ; ) . (1)
U danij roboti my dewo uzahal\ng[mo ostanng zadaçu ta rozhlqda[mo im-
pul\snyj proces u prostori Rd
U tε( ) = u +
k
t
k k kS x
=
∑ ( )
1
ν ε
ε εα
( / )
; , (2)
de αε
k s x( ; ) , k ≥ 1, s
d∈ ′R , x E∈ , — sim’q vypadkovyx velyçyn iz znaçennqmy
vJRd
.
Osnovnog metog roboty [ dovedennq slabko] zbiΩnosti procesu (2).
ZauvaΩennq. Na vidminu vid roboty [1] , de vyvça[t\sq zbiΩnist\ za paramet-
rom n → ∞, my vvodymo normuvannq çasu malym parametrom ε → 0. Krim toho,
vidminnist\ vid roboty [1] polqha[ v deqkyx obmeΩennqx na vypadkovi velyçyny,
qki vxodqt\ v oznaçennq procesu (2). Zokrema, dali vvedemo umovy na αε
k .
Nexaj vykonugt\sq nastupni umovy:
C
1
. Prypustymo, wo x t( ), t ≥ 0, — rivnomirno erhodyçnyj proces zi stacio-
narnym rozpodilom π( )B , B∈E . Takym çynom, vkladenyj lancgh Markova
xk , k ≥ 0, takoΩ [ rivnomirno erhodyçnym ta ma[ stacionarnyj rozpodil ρ( )B ,
B∈E , i vykonugt\sq spivvidnoßennq
π( ) ( )dx q x = q dxρ( ) , q : =
E
dx q x∫ π( ) ( ).
C
2
. Sim’q vypadkovyx velyçyn αε
k s x( ; ) , k ≥ 1, s d∈ ′R , x E∈ , rozhlqda[t\-
sq v sxemi serij z malym parametrom ε > 0 ta vyznaça[t\sq funkci[g rozpodilu
Φε( , ; )s x z = P αε
k s x z( ; ) <{ } , s
d∈ ′R , z
d∈R , x E∈ .
C
3
. Sim’q vypadkovyx velyçyn αε
k s x( ; ) , k ≥ 1, s
d∈ ′R , x E∈ , [ rivnomirno
kvadratyçno intehrovnog:
sup sup ( , ; )
ε
ε
> ∈ >
∫
0
2
x E z c
z s x dzΦ → 0, c → ∞.
Nexaj vykonugt\sq umovy puassonivs\ko] aproksymaci]:
PA
1
. Aproksymaciq serednix:
a s xε( ; ) = Eαε
k s x( ; ) =
Rd
z s x dz∫ Φε( , ; ) = ε θεa s x s xa( ; ) ( ; )+[ ]
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1284 I. V. SAMOJLENKO
ta
c s xε( ; ) =
Rd
zz s x dz∫ ∗Φε( , ; ) = ε θεc s x s xc( ; ) ( ; )+[ ].
PA
2
. Puassonivs\ka aproksymaciq qdra intensyvnosti
Rd
g z s x dz∫ ( ) ( , ; )Φε = ε θεΦg gs x s x( , ) ( ; )+[ ]
dlq vsix g ∈ C
d
3 R( ) ta qdro Φg s x( , ) obmeΩene dlq vsix g ∈ C
d
3 R( ) , tobto
sup ( , )
x E
g s x
∈
Φ ≤ Φg < ∞.
Çleny, qkymy moΩna znextuvaty θ θ θε ε ε
a c g, ,( ), zadovol\nqgt\ umovy
sup ( ; )
x E
s x
∈
⋅θ
ε → 0, ε → 0.
Osnovnym rezul\tatom roboty [ nastupna teorema.
Teorema 1. Za umov C
1
– C
3
, PA
1
, PA
2
ma[ misce slabka zbiΩnist\ pary
U t S tε ε( ), ( )( ) fi ˆ ( ), ˆ( )U t S t( ) , ε → 0.
Hranyçnyj proces
ˆ( ), ˆ( )U t S t( ) , t ≥ 0, vyznaça[t\sq heneratorom
ˆ( ) ( , )A s u sϕ = ˆ( ) ( , )a s u su′ϕ + ˆ( ) ( , )C s u ss′ϕ +
Rd
u z s u s s dz∫ +[ ]ϕ ϕ( , ) – ( , ) ˆ ( ; )Φ ,
(3)
de userednenyj determinovanyj zsuv vyznaça[t\sq qk
ˆ( )a s =
E
dx a s x∫ π( ) ( ; ) , (4)
a userednene qdro intensyvnosti — qk
ˆ ( ; )Φ s dz =
E
dx s x dz∫ π( ) ( , ; )Φ . (5)
Tut qdro Φ( ,s x ; dz) vyznaça[t\sq z rivnosti
Φg s x( ; ) =
Rd
g z s x dz∫ ( ) ( , ; )Φ , g z C d( ) ( )∈ 3 R .
Dovedennq. Dlq dovedennq slabko] zbiΩnosti vykorysta[mo rezul\taty,
otrymani v rozdili 3 roboty [4].
Nexaj C d
0
2 R( × E) — prostir dijsnoznaçnyx dviçi neperervno dyferenci-
jovnyx funkcij po perßomu arhumentu, vyznaçenyx na Rd × E i takyx, wo do-
rivnggt\ nulg na neskinçennosti, a C dR( × E) — prostir dijsnoznaçnyx nepe-
rervno obmeΩenyx funkcij, vyznaçenyx na Rd × E.
Ma[ misce nastupna teorema.
Teorema 2 [4] (teorema 6.3). Nexaj sim’q markovs\kyx procesiv ξε( )t , t ≥ 0,
ε > 0, zadovol\nq[ nastupni umovy:
CD1. Isnu[ sim’q test-funkcij ϕε( , )u x u prostori C Ed
0
2 R ×( ) takyx,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
ZBIÛNIST| IMPUL|SNOHO PROCESU NAKOPYÇENNQ ZI STRYBKOVYMY … 1285
wo
lim ( , )
ε
εϕ
→0
u x = ϕ( )u ,
rivnomirno na u, x.
CD2. Ma[ misce zbiΩnist\
lim ( , )
ε
ε εϕ
→0
L u x = Lϕ( )u
rivnomirno na u, x. Sim’q funkcij Lε εϕ , ε > 0, [ rivnomirno obmeΩenog, a
Lϕ( )u i Lε εϕ naleΩat\ do C
dR( × E) .
CD3. Kvadratyçni xarakterystyky martynhaliv, wo vidpovidagt\ markov-
s\komu procesu ξε( )t , t ≥ 0, ε > 0, magt\ vyhlqd µε
t
=
0
t
s ds∫ ζε( ) , de vypad-
kovi funkci] ζε , ε > 0, zadovol\nqgt\ umovu
sup ( )
0≤ ≤s T
sE ζε ≤ c < + ∞.
CD4. Poçatkovi znaçennq zbihagt\sq ta
sup ( )
ε
εζ
>0
0E ≤ C < + ∞.
Todi ma[ misce slabka zbiΩnist\
ξε( )t fi ξ( )t , ε → 0.
Rozhlqnemo trykomponentnyj markovs\kyj proces
U tε( ), S tε( ), x t( / )ε , t ≥ 0,
de tretq peremykagça komponenta vyznaça[t\sq na standartnomu prostori
( , )E E za dopomohog heneratora
Q xϕ( ) = q x P x dy y x
E
( ) ( , ) ( ) – ( )∫ [ ]ϕ ϕ .
Cej proces xarakteryzu[t\sq martynhalom
µεt = ϕ εε εU t S t x t( ), ( ), ( / )( ) –
0
t
U S x d∫ ( )Lε ε εϕ τ τ τ ε τ( ), ( ), ( / ) , (6)
de henerator Lε ma[ vyhlqd [4] (rozdil 3)
Lεϕ( , , )u s x = ε ϕε ε– ( , ) ( , ) ( , , )1Q s x s x u s x+ +[ ]A C . (7)
Tut
A
ε ϕ( , ) ( )s x u = ε ϕ ε ϕ– ( ( ; )) – ( )1 u a s x u+[ ],
A( , ) ( )s x uϕ = a s x u( ; ) ( )′ϕ ,
C
ε ϕ( , ) ( )s x s = ε ϕ ε ϕ– ( ( ; )) – ( )1 s C s x s+[ ],
C( , ) ( )s x sϕ = C s x s( ; ) ( )′ϕ .
Rozv’qzok zadaçi synhulqrnoho zburennq dlq Lε na test-funkciqx ϕε(u, s,
x) = ϕ( , )u s + εϕ1(u , s , x) navedeno v [4] (lema 7.3). Zhidno z ci[g lemog, hra-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1286 I. V. SAMOJLENKO
nyçnyj dvokomponentnyj proces
ˆ( )U t( ,
ˆ( )S t ) vyznaça[t\sq heneratorom
Lϕ( , )u s = ˆ( ) ( , )a s u su′ϕ + ˆ( ) ( , )C s u ss′ϕ +
Rd
u z s u s s dz∫ +[ ]ϕ ϕ( , ) – ( , ) ˆ ( ; )Φ ,
(8)
de
ˆ( )C s , ˆ( )a s i
ˆ ( ; )Φ s dz vyznaçeno v (1), (4) i (5) vidpovidno.
Teper moΩemo zastosuvaty teoremuJ2.
Z (7) ta (8) oçevydno, wo rozv’qzok zadaçi synhulqrnoho zburennq zadovol\-
nq[ umovy CD1, CD2.
Umova CD3 vymaha[, wob kvadratyçna xarakterystyka martynhala, wo vidpo-
vida[ trykomponentnomu markovs\komu procesu, bula vidnosno kompaktnog.
Analohiçni umovy dlq stoxastyçnyx system z markovs\kym peremykannqm vy-
vçagt\sq v rozdili 6.4.1 roboty [4]. Tam dovedeno (dyv., zokrema, naslidok 6.1),
wo procesy, vyznaçeni martynhalamy typu (6), [ vidnosno kompaktnymy.
Oskil\ky Uε( )0 = ˆ( )U 0 , Sε( )0 = ˆ( )S 0 , xε( )0 = x( )0 , umova CD4, oçevydno,
vykonu[t\sq. Takym çynom, vsi umovy teoremyJ2 vykonugt\sq, tobto ma[ misce
slabka zbiΩnist\ U tε( )( , S tε( )) fi ˆ ( )U t( ,
ˆ( )S t ).
Hranyçnyj markovs\kyj proces
ˆ( )U t( ,
ˆ( )S t ), t ≥ 0, zhidno z rezul\tatamy [4],
vyznaça[t\sq heneratorom (3).
TeoremuJ1 dovedeno.
1. Anisimov V. V. ZbiΩnist\ procesiv nakopyçennq z peremykannqmy // Teoriq jmovirnostej ta
mat. statystyka. – 2000. – # 63. – S. 3 – 12.
2. Billingsley P. Convergence of probability measures. – New York: J. Wiley and Sons, 1968. –
368 p.
3. Hryhelyonys B. Y. Ob otnosytel\noj kompaktnosty mnoΩestv veroqtnostn¥x mer v
D X( , ) ( )0 ∞ // Lyt. mat. sb. – 1973. – 13, # 4. – S. 83 – 96.
4. Koroliuk V. S., Limnios N. Stochastic systems in merging phase space. – World Sci. Publ., 2005. –
330 p.
5. Koroliuk V. S., Limnios N. Diffusion approximation with equilibrium of evolutionary systems
switched by semi-Markov processes // Ukr. mat. Ωurn. – 2005. – 57, # 9. – S. 1253 – 1260.
OderΩano 29.01.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
|
| id | umjimathkievua-article-3243 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:38:48Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/5d/7bd72c4daa0b7dc3dcc0676309bc8b5d.pdf |
| spelling | umjimathkievua-article-32432020-03-18T19:48:57Z Convergence of an impulsive storage process with jump switchings Збіжність імпульсного процесу накопичення зі стрибковими перемиканнями Samoilenko, I. V. Самойленко, І. В. We investigate an impulsive storage process switched by a jump process. The switching process is, in turn, averaged. We prove the weak convergence of the storage process in the scheme of series where a small parameter ε tends to zero. Исследован импульсный процесс накопления, который переключается с помощью скачкообразного процесса. Переключающий процесс, в свою очередь, усредняется. Доказана слабая сходимость процесса накопления в схеме серий, когда малый параметр ε стремится к нулю. Institute of Mathematics, NAS of Ukraine 2008-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3243 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 9 (2008); 1282–1286 Український математичний журнал; Том 60 № 9 (2008); 1282–1286 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3243/3232 https://umj.imath.kiev.ua/index.php/umj/article/view/3243/3233 Copyright (c) 2008 Samoilenko I. V. |
| spellingShingle | Samoilenko, I. V. Самойленко, І. В. Convergence of an impulsive storage process with jump switchings |
| title | Convergence of an impulsive storage process with jump switchings |
| title_alt | Збіжність імпульсного процесу накопичення зі стрибковими перемиканнями |
| title_full | Convergence of an impulsive storage process with jump switchings |
| title_fullStr | Convergence of an impulsive storage process with jump switchings |
| title_full_unstemmed | Convergence of an impulsive storage process with jump switchings |
| title_short | Convergence of an impulsive storage process with jump switchings |
| title_sort | convergence of an impulsive storage process with jump switchings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3243 |
| work_keys_str_mv | AT samoilenkoiv convergenceofanimpulsivestorageprocesswithjumpswitchings AT samojlenkoív convergenceofanimpulsivestorageprocesswithjumpswitchings AT samoilenkoiv zbížnístʹímpulʹsnogoprocesunakopičennâzístribkovimiperemikannâmi AT samojlenkoív zbížnístʹímpulʹsnogoprocesunakopičennâzístribkovimiperemikannâmi |