Limit theorems for systems of the type M θ/G/1/b with resume level of input stream
A finite capacity queueing system of the type M θ/G/1/b is considered in which the input flow is regulated by some threshold level. Asymptotic properties of the first busy period and the number of customers served for this period are studied.
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| Дата: | 2007 |
|---|---|
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| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3353 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509428267089920 |
|---|---|
| author | Bratiychuk, A. M. Братійчук, А. М. |
| author_facet | Bratiychuk, A. M. Братійчук, А. М. |
| author_sort | Bratiychuk, A. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:52:14Z |
| description | A finite capacity queueing system of the type M θ/G/1/b is considered in which the input flow is regulated by some threshold level.
Asymptotic properties of the first busy period and the number of customers served for this period are studied.
|
| first_indexed | 2026-03-24T02:40:57Z |
| format | Article |
| fulltext |
UDK 519.21
A. M. Bratijçuk (Ky]v. nac. un-t im. T. Íevçenka)
HRANYÇNI TEOREMY DLQ SYSTEM TYPU M G bθθ// // //1
Z VIDNOVLGGÇYM RIVNEM VXIDNOHO POTOKU
A finite capacity queueing system of the type M G bθ / / /1 is considered in which the input flow is
regulated by some threshold level. Asymptotic properties of the first busy period and the number of
customers served for this period are studied.
Rassmatryvaetsq systema obsluΩyvanyq typa M G bθ / / /1 , v kotoroj vxodqwyj potok rehuly-
ruetsq s pomow\g nekotoroho porohovoho urovnq. Yssledugtsq asymptotyçeskye svojstva per-
voho peryoda zanqtosty y çysla trebovanyj, obsluΩenn¥x za πtot peryod.
1. Vstup. Rozhlqnemo systemu typu M G bθ / / /1 , qku formal\no moΩna opy-
saty takym çynom. Nexaj zadano poslidovnosti vypadkovyx velyçyn { }τn , { }θn ,
{ }βn , n ≥ 1, qki reprezentugt\ ças miΩ nadxodΩennqm hrup vymoh do systemy,
rozmir hrup ta ças obsluhovuvannq zamovlennq vidpovidno. V moment çasu
τii
n
=∑ 1
do systemy nadxodyt\ n-ta hrupa zamovlen\, qka mistyt\ θn vymoh.
Velyçyna βn — ças obsluhovuvannq n-] vymohy i dyscyplina obsluhovuvannq
ma[ typ FCFS (perßyj pryjßov — perßyj obsluΩyvsq). Vsi navedeni vywe vy-
padkovi velyçyny [ nezaleΩnymy z odnakovym rozpodilom dlq koΩno] posli-
dovnosti i P{ }τn x> = e x−λ , x > 0. Kil\kist\ vymoh, qki moΩut\ znaxodytys\ u
systemi, obmeΩena deqkym natural\nym çyslom b < ∞ . OtΩe, qkwo çerhova
hrupa vymoh ob’[mu θn nadxodyt\ do systemy, de vΩe [ k vymoh, to lyße
min { , }b k n− θ z nyx pry[dnugt\sq do çerhy, a reßta vtraça[t\sq. Proces nad-
xodΩennq vymoh ma[ osoblyvist\, qka polqha[ v tomu, wo qkwo kil\kist\ vymoh
u systemi dosqhla rivnq b, to ]x nadxodΩennq bloku[t\sq i vidnovlg[t\sq zno-
vu lyße todi, koly ]x zahal\na kil\kist\ u systemi zmenßyt\sq do deqkoho rivnq
a b∈ −[ , ]1 1 . Neobxidnist\ rozhlqdu takyx system moΩna poqsnyty tym faktom,
wo u vypadku obmeΩen\ na dovΩynu çerhy deqki vymohy budut\ vtraçatys\, i
baΩano kil\kist\ vtraçenyx vymoh skorotyty, korystugçys\ takym pravylom:
krawe zazdalehid\ poperedyty pro perepovnennq systemy, niΩ vtratyty vymohu.
Dlq c\oho i vvodyt\sq vidnovlggçyj riven\. Naskil\ky vidomo avtoru, taku
ideg bulo vperße zaproponovano v [1] (u vypadku, koly vymohy nadxodqt\ po od-
nij), a v [2] navedeno formuly dlq erhodyçnoho rozpodilu dovΩyny çerhy v ta-
kij systemi. V [3] zaproponovano inßyj pidxid, qkyj bazu[t\sq na metodi poten-
cialu Korolgka [4] i faktyçno ne rozrizng[ systemy z indyvidual\nym çy hru-
povym vplyvom vymoh.
Navedemo poznaçennq, qki budemo vykorystovuvaty u podal\ßomu. Pozna-
çymo
P{ }βi x≤ = F ( x ) , P{ }θi k= = ak
, x ≥ 0, k = 1, 2, … ,
f ( s ) =
0
∞
−∫ e dF xsx ( ), a ( z ) = z ak
k
k=
∞
∑
1
, Re s ≥ 0, z ≤ 1,
mi =
0
∞
∫ x dF xi ( ), αi = k ai
k
k=
∞
∑
1
, i ≥ 1.
Nexaj ξ ( t ) , t ≥ 0, poznaça[ çyslo vymoh u systemi v moment çasu t ; symvol
Mn — umovne matematyçne spodivannq pry umovi, wo ξ ( 0 ) = n ≥ 0; F xk∗
( ) ta
© A. M. BRATIJÇUK, 2007
884 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
HRANYÇNI TEOREMY DLQ SYSTEM TYPU M G bθ / / /1 … 885
ai
k∗
— k-kratni zhortky rozpodiliv F ( x ) ta { ak } samyx iz sobog. Poznaçymo
takoΩ çerez τ = τb = inf ; ( ){ }t t≥ =0 0ξ perßyj period zajnqtosti, çerez
N ( τ ) = N ( τb ) kil\kist\ vymoh, qki bulo obsluΩeno za period zajnqtosti.
Nexaj
pl ( s ) ≡ f s a e
x
i
dF xl
i
i
l
s x
i
−
+
=
+ ∞
− +∗
∑ ∫1
1
0
1
0
( )
( )
!
( )( )λ λ
, k = – 1, 0, 1, … .
Oskil\ky pi ( s ) > 0, s > 0, i
i=−
∞∑ 1
pi ( s ) = 1, to çysla pi ( s ) , i ≥ – 1, moΩe-
mo interpretuvaty qk rozpodil skaçkiv deqkoho blukannq, neperervnoho znyzu, i
qkwo a ( s, z ) = s + λ ( 1 – α ( z )) , to nevaΩko perekonatys\, wo
z p si
i
i
( )
=−
∞
∑
1
=
f a s z
z f s
( ( , ))
( )
, z ≤ 1.
Oznaçymo poslidovnist\ Rk ( s, θ ) , s > 0, θ ≤ 1, takym çynom:
z R sk
k
k
( , )θ
=
∞
∑
1
= z
f a s z zθ ( ( , )) −
, z ≤ ν−( )s , (1)
de ν−( )s — [dynyj korin\ rivnqnnq
θ f a s z z( ( , )) − = 0, 0 < z < 1, (2)
na intervali [ 0, 1] . Poznaçymo takoΩ
Q sn( , )θ = R s d sn k k
k
n
− −
=
−
∑ 1
0
2
( , ) ( , )θ θ ,
d sk ( , )θ = 1
1
1
−
=−
−
∑θ f s p si
i
k
( ) ( ), p sn( ) = p si
i n
( )
=
∞
∑ . (3)
2. Perßyj period zajnqtosti ta çyslo vymoh, obsluΩenyx na c\omu
periodi. Poznaçymo
φ ( n ) ≡ φ ( n, s, θ ) = M en
s N− τ τθ ( ), Re s ≥ 0, θ ≤ 1, 0 ≤ n ≤ b.
Oznaçymo vyznaçnyk
∆ ( , )s θ =
1 1
1
1
1
1
1
1
1
+ − −
− +
−
− − −
= +
−
−
−
− −
=
−
∑
∑
( ( )) ( , ) ( ) ( , )
( ( )) ( , ) ( ) ( , )
θ θ θ
θ θ θ
f s R s p s Q s
f s R s p s Q s
b a
i a b i
i a
b
b a
b a
i b i
i
b
b
. (4)
Teorema81. Dlq dovil\nyx 1 ≤ n ≤ b – 1 ta s > 0, θ ≤ 1 ma[ misce zob-
raΩennq
M en
s N− τ τθ ( ) =
( ( , )) ( ( )) ( , ) ( )
( , )
1 1 1
1
1
+ +
−
−
− − −
= +
−
∑Q s f s R s p s
s
b n
b a
i a b i
i a
b
θ θ θ
θ∆
–
–
( ( )) ( ( , )) ( , ) ( )
( , )
θ θ θ
θ
f s Q s R s p s
s
b a
b a i n b i
i n
b
−
− − − −
= +
−
+ ∑1 1
1
1
∆
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
886 A. M. BRATIJÇUK
Dovedennq. Vykorystovugçy formulu povno] jmovirnosti, nevaΩko otry-
maty rivnqnnq
ϕ θ ϕ( ) ( ) ( ) ( )n f s p s n ii
i
b n
− +
=−
− −
∑
1
2
= ( ( )) ( ) ( )θ ϕf s p s ab a
b n
−
− −1 , 1 ≤ n ≤ b – 1,
z hranyçnog umovog ϕ ( 0 ) = 1.
Vykorystovugçy rezul\taty z [3], ma[mo
φ ( n ) = ( ( , )) ( ) ( ( )) ( ) ( , ) ( )1 1 1
1
1
1
+ − −−
− −
− − −
= +
−
∑Q s b f s a R s p sb n
b a
i n b i
i n
b
θ φ θ φ θ . (5)
Poklavßy v (5) n = a, otryma[mo perße rivnqnnq dlq φ( )a ta φ( )b − 1 :
φ θ θ θ φ( ) ( ( )) ( , ) ( ) ( ( , )) ( )a f s R s p s Q s bb a
i a b i
i a
b
b a1 1 11
1
1
+
− + −−
− − −
= +
−
−∑ = 0.
Z hranyçno] umovy ϕ ( 0 ) = 1 oderΩu[mo druhe rivnqnnq
– φ θ θ θ φ( )( ( )) ( , ) ( ) ( ( , )) ( )a f s R s p s Q s bb a
i b i
i
b
b
−
− −
=
−
+ + −∑ 1
1
1
1 1 = 1.
OtΩe, dlq φ( )a ta φ( )b − 1 ma[mo systemu linijnyx rivnqn\
φ θ θ θ φ
φ θ θ θ
( ) ( ( )) ( , ) ( ) ( ( , )) ( ) ,
( )( ( )) ( , ) ( ) ( ( ,
a f s R s p s Q s b
a f s R s p s Q s
b a
i a b i
i a
b
b a
b a
i b i
i
b
b
1 1 1 0
1
1
1
1
1
1
1
+
− + − =
− + +
−
− − −
= +
−
−
−
− −
=
−
∑
∑ )))) ( ) .φ b − =1 1
Rozv’qzugçy cg systemu ta pidstavlqgçy rozv’qzky v (5), zaverßu[mo dovedennq
teoremy.
3. Hranyçni teoremy. Poznaçymo ρ = λ αm1 1. Nastupna umova naklada[
pevni obmeΩennq na rozpodily ai , i ≥ 1, ta F ( x ) .
A. Nexaj z0 = sup : ( ( ( )){ }z f a z> − < ∞0 1λ > 1 i qkwo ρ < 1, to
f a z z( ( ( ))λ 1 00 0− − − > 0.
Naslidkom umovy A [ te, wo rivnqnnq (2) ma[ na intervali [ 0, z0 ) lyße dva
koreni 0 < ν θ−( , )s < 1 < ν θ+( , )s < z0 dlq vsix 0 < s < δ, 1 – δ < θ < 1 z
deqkym dostatn\o malym δ > 0. NevaΩko zrozumity, wo
ν θ+( , )s
s→ →
→
0 1, θ
ν ρ
ρ
+ ∈ <
>
( , ], ,
, ,
1 1
1 1
0z
ν θ−( , )s
s→ →
→
0 1, θ
ν ρ
ρ
− < >
<
1 1
1 1
, ,
, ,
ν θ±( , )s
s→ →
→
0 1, θ
1, ρ = 1,
de ν± — koreni rivnqnnq f a z z( ( ( ))λ 1 0− − = , qki vidminni vidJJ1.
Dali nam bude potribna bil\ß dokladna xarakterystyka povedinky funkcij
ν θ±( , )s pry s → 0 , θ → 1 . Ci rezul\taty podamo u vyhlqdi okremo] lemy, do-
vedennq qko] lehko vyplyva[ z teoremy pro neqvu funkcig [5].
Lema881. Nexaj s → 0 i θ = θ ( s )
s→
→
0
1, de θ ( s ) — deqka funkciq,
qka [ rehulqrnog v okoli toçky s = 0. Todi:
1) qkwo ρ > 1, to
ν θ−( , )s = ν θ ν λ ν
λ ν λ ν−
− −
− −
+
′ + ′ −
+ ′ ′ −
+( ) ( ( ( )))
( )) ( ( ( )))
( )
0 1
1 1
f a
a f a
s o s ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
HRANYÇNI TEOREMY DLQ SYSTEM TYPU M G bθ / / /1 … 887
ν θ+( , )s = 1
0
1
1− − ′
−
+m
s o s
θ
ρ
( )
( ) ;
2) qkwo ρ < 1, to
ν θ+( , )s = ν θ ν λ ν
λ ν λ ν+
+ +
+ +
+
′ + ′ −
+ ′ ′ −
+( ) ( ( ( )))
( )) ( ( ( )))
( )
0 1
1 1
f a
a f a
s o s ,
ν θ−( , )s = 1
0
1
1− − ′
−
+m
s o s
θ
ρ
( )
( ) ;
3) qkwo ρ = 1, to
ν θ±( , ( ))s s = 1
2 01
2 1 1
2
2
± − ′
+
+( ( ))
( )
( )
m s
m m
o s
θ
λα λα
.
Poznaçymo
α θ±( , )s =
′ + ′ −
+ ′ ′
+ +
± ±
θ ν λ ν
λθ ν θ ν θ
( ) ( ( ( )))
( ( , )) ( ( , ( , )))
0 1
1
f a
a s f a s s
, β θ±( , )s =
θ ν θ
θ ν θ
f s s
f s s
( ) ( , )
( ) ( ( , ))
+
−
±
±1
.
Dali ε, δ budut\ poznaçaty dodatni mali çysla, qki moΩut\ buty riznymy v
riznyx spivvidnoßennqx.
Lema882. Dlq vsix s < δ ta 1 − <θ δ magt\ misce asymptotyçni spiv-
vidnoßennq
Q sn( , )θ = – 1 1 1
0+ + + −+ +
− +
− −
− + −α θ ν θ α θ ν θ ε( , ) ( , ) ( , ) ( , ) (( ) )s s s s o zn n n , (6)
R s pi n i
i
n
( , )θ −
=
∑
1
= β θ α θ ν θ β θ α θ ν θ ε+ + +
−
− − −
− −+ + −( , ) ( , ) ( , ) ( , ) ( , ) ( , ) (( ) )s s s s s s o zn n n
0 ,
(7)
pryçomu ocinky o( )⋅ [ rivnomirnymy po s ta θ.
Dovedennq. Dlq dostatn\o malyx z z oznaçennq funkci] Q sn( , )θ (for-
mula (3)) ta spivvidnoßennq (1) ma[mo
z Q sn
n
n
( , )θ
=
∞
∑
1
=
z f a s z
f a s z z z
( ( ( , )))
( ( ( , )) )( )
1
1
−
− −
θ
θ
. (8)
Oskil\ky dlq 0 < <s δ , 0 1< − <θ δ funkciq v pravij çastyni (8) ma[
prosti polgsy v oblasti z z< 0 v toçkax z = 1, z = ν θ±( , )s , to v cij oblasti
vykonu[t\sq rivnist\
z f a s z
f a s z z z
( ( ( , )))
( ( ( , )) )( )
1
1
−
− −
θ
θ
=
= –
1
1 0−
+
−
+
−
++ +
+
− −
− =
∞
∑z
s s
s z
s s
s z
z Q sn
n
n
α θ ν θ
ν θ
α θ ν θ
ν θ
θ( , ) ( , )
( , )
( , ) ( , )
( , )
( , )
�
, (9)
za vynqtkom toçok z = 1, z = ν θ±( , )s , pryçomu dlq funkci]
�
Q sn( , )θ spraved-
lyvog [ ocinka
�
Q sn( , )θ = o z n(( ) )0 − −ε , (10)
rivnomirna po s ta θ. OtΩe, z (8), (9) ma[mo
z Q sn
n
n
( , )θ
=
∞
∑
1
= –
1
1 0−
+
−
+
−
++ +
+
− −
− =
∞
∑z
s s
s z
s s
s z
z Q sn
n
n
α θ ν θ
ν θ
α θ ν θ
ν θ
θ( , ) ( , )
( , )
( , ) ( , )
( , )
( , )
�
,
zvidky z uraxuvannqm (10) otrymu[mo (6). Spivvidnoßennq (7) dovodyt\sq
analohiçno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
888 A. M. BRATIJÇUK
Teorema882. Nexaj a b= α , n b= β , de 0 ≤ α , β ≤1.
1. Qkwo ρ < 1, to
lim / ( )/
b
n
s b N bM e e
→∞
− −τ µ τ = e
sm− +
−
−1
1
1
µ
ρ
β( )
. (11)
2. Qkwo ρ = 1, to
lim / ( )/
b
n
s b N bM e e
→∞
− −τ µ τ2 2
=
sh q s sh q s
sh q s sh q s
{ ( , )( )} { ( , )( )}
{ ( , )} { ( , ) }
µ β µ α β
µ µ α
1 − − −
−
, (12)
de
q s( , )µ =
2 1
2 1 1
2
2
( )
( )
sm
m m
+
+
µ
λα λα
.
Dovedennq. Nasampered doslidymo asymptotyçni vlastyvosti vyznaçnyka
∆( , )s θ z (4). Oçevydno, wo
∆( , )s θ = 1
1
1
1
1
1
1
1
1+ +
− −
− +
−
− − −
= +
−
−
− −
=
−
∑
∑
Q s f s
R s p s Q s
R s p s Q s
b
b a
i a b i
i a
b
b a
i b i
i
b
b
( , ) ( ( ))
( , ) ( ) ( , )
( , ) ( ) ( , )
θ θ
θ θ
θ θ
. (13)
U podal\ßomu dlq zruçnosti ne budemo pysaty v poznaçennqx funkcij α θ±( , )s ,
β θ±( , )s , ν θ±( , )s arhumentiv s, θ. Oznaçymo nastupni vyznaçnyky:
∆0( , )s θ =
α β ν α β ν α ν α ν
α β ν α β ν α ν α ν
+ + +
+ −
− − −
+ −
+ +
+ −
− −
+ −
+ + +
−
− − −
−
+ +
−
− −
−
+ − +
− − +
a b a b a b a b
b b b b
1 1 1 1
1 1 1 1
,
∆1( , )s θ =
o z
o z
a b a b a b
b b b
(( ) )
(( ) )
0
1 1
0
1 1
− − −
− +
−
+ +
+ −
− −
+ −
−
+ +
−
− −
−
ε α ν α ν
ε α ν α ν
,
∆2( , )s θ =
α β ν α β ν ε
α β ν α β ν ε
+ + +
+ −
− − −
+ − −
+ + +
−
− − −
− −
+ −
− − −
a b a b a b
b b b
o z
o z
1 1
0
1 1
0
(( ) )
(( ) )
,
∆3( , )s θ =
o z o z
o z o z
a b a b
b b
(( ) ) (( ) )
(( ) ) (( ) )
0 0
0 0
− −
− −
− −
− −
ε ε
ε ε
.
Vykorystovugçy lemuJJ2, moΩemo perepysaty (13) u vyhlqdi
∆( , )s θ = 1 0 1 2 3+ + + + +( )−Q s f s s s s sb
b a( , ) ( ( )) ( , ) ( , ) ( , ) ( , )θ θ θ θ θ θ∆ ∆ ∆ ∆ .
Za dopomohog prostyx obçyslen\ znaxodymo
∆0( , )s θ = α α ν ν ν ν β β+ − − +
−
+ − + −− −( ) ( )( )1 b a a .
Nexaj teper ρ < 1. Poznaçymo ˆ /s s b= , ˆ exp{ }/θ µ= − b , s , µ ≥ 0. Z teo-
remyJJ1 ma[mo
M e eb
s b N b
β
τ µ τ− −/ ( )/ =
1 11
1
1
1
1
+( ) +
−
−
− − −
= +
−
∑Q s f s R s p s
s
b
b
i b b i
i b
b
( )
( )( ˆ, ˆ) (ˆ (ˆ)) (ˆ, ˆ) (ˆ)
(ˆ, ˆ)
β
α
α
α
θ θ θ
θ∆
–
–
(ˆ (ˆ)) (ˆ, ˆ) (ˆ, ˆ) (ˆ)
(ˆ, ˆ)
( )
( )θ θ θ
θ
α
α β
β
f s Q s R s p s
s
b
b i b b i
i b
b
1
1 1
1
1
1−
− − − −
= +
−
+( ) ∑
∆
. (14)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
HRANYÇNI TEOREMY DLQ SYSTEM TYPU M G bθ / / /1 … 889
Oskil\ky θ̂ = 1 − +µ
s
s o sˆ (ˆ) pry ŝ → 0, to v c\omu vypadku ′θ ( )0 = – µ / s, i z
lemyJJ1 otrymu[mo
ν θ−(ˆ, ˆ)s = 1
1
1 1 1− +
−
+− −sm
b o b
µ
ρ
( ),
a otΩe,
( )(ˆ, ˆ)ν θ−
−s b = 1
1
1 1− +
−
+
−
−
sm
b
o b
bµ
ρ( )
( )
b→∞
→ e
sm− +
−
1
1
µ
ρ . (15)
Oçevydno, wo
( )ˆ (ˆ)θ f s b a− =
= ( )( ( )) ( ( )) ( )1 11 1 1
1
1 1 1− + − +− − − − − −µ αs b o b m b o b b
b→∞
→ e sm− − +( )( )1 1α µ .
Z oznaçennq vyznaçnykiv ∆i s(ˆ, ˆ)θ vyplyva[, wo
∆0(ˆ, ˆ)s θ = O b( )ν+
− , ∆i s(ˆ, ˆ)θ = o z b( )( ) ( )
0
1− − −ε α , i = 1, 2, 3.
Z (6) ta (15) znaxodymo
1 + Q sb(ˆ, ˆ)θ
b→∞
→ 1
1
1
1
−
+
−
ρ
µ
ρe
sm
, 1 + −Q sb a(ˆ, ˆ)θ
b→∞
→ 1
1
1
1
1
−
+
−
−
ρ
µ
ρ
α
e
sm
( )
,
(16)
1 + −Q sb n(ˆ, ˆ)θ
b→∞
→ 1
1
1
1
1
−
+
−
−
ρ
µ
ρ
β
e
sm
( )
.
Z cyx spivvidnoßen\, a takoΩ z formuly (13) ma[mo
∆ (ˆ, ˆ)s θ
b→∞
→ 1
1
1
1
−
+
−
ρ
µ
ρe
sm
. (17)
Z (7) otrymu[mo
R s p si b b i
i b
b
− − −
= +
−
∑ β β
β
θ(ˆ, ˆ) (ˆ)1
1
1
b→∞
→
ρ
ρ
µ
ρ
β
1
1
1
1
−
+
−
−
e
sm
( )
. (18)
Teper (11) vyplyva[ z (14) ta (16) – (18). Dovedennq (12) [ analohiçnym.
1. Takagi H. Analysis of finite-capasity M G/ /1 queue with a resume level // Performance Evaluat.
– 1985. – 5, # 3. – P. 197 – 203.
2. Takagi H. Queueing analysis. – Netherlands: Elsevier Sci. Publ., 1993. – Vol. 2. – 630 p.
3. Bratijçuk A. M. Systema M G bθ / / /1 z vidnovlggçym rivnem vxidnoho potoku // Visn.
Ky]v. un-tu. Ser. fiz.-mat. nauky. – 2007. – # 1. – S.J12 – 18.
4. Korolgk V. S. Hranyçn¥e zadaçy dlq sloΩn¥x puassonovskyx processov. – Kyev: Nauk.
dumka, 1975. – 175 s.
5. Evhrafov M. A. Analytyçeskye funkcyy. – M.: Nauka, 1968. – 471 s.
OderΩano 02.04.2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
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| id | umjimathkievua-article-3353 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:40:57Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/08/059a379ce5985dadfa08530cd1abf408.pdf |
| spelling | umjimathkievua-article-33532020-03-18T19:52:14Z Limit theorems for systems of the type M θ/G/1/b with resume level of input stream Граничні теореми для систем типу M θ/G/1/b з відновлюючим рівнем вхідного потоку Bratiychuk, A. M. Братійчук, А. М. A finite capacity queueing system of the type M &theta;/G/1/b is considered in which the input flow is regulated by some threshold level. Asymptotic properties of the first busy period and the number of customers served for this period are studied. Рассматривается система обслуживания типа M &theta;/G/1/b, в которой входящии поток регулируется с помощью некоторого порогового уровня. Исследуются асимптотические свойства первого периода занятости и числа требовании, обслуженных за этот период. Institute of Mathematics, NAS of Ukraine 2007-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3353 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 7 (2007); 884-889 Український математичний журнал; Том 59 № 7 (2007); 884-889 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3353/3450 https://umj.imath.kiev.ua/index.php/umj/article/view/3353/3451 Copyright (c) 2007 Bratiychuk A. M. |
| spellingShingle | Bratiychuk, A. M. Братійчук, А. М. Limit theorems for systems of the type M θ/G/1/b with resume level of input stream |
| title | Limit theorems for systems of the type M θ/G/1/b with resume level of input stream |
| title_alt | Граничні теореми для систем типу M θ/G/1/b з відновлюючим рівнем вхідного потоку |
| title_full | Limit theorems for systems of the type M θ/G/1/b with resume level of input stream |
| title_fullStr | Limit theorems for systems of the type M θ/G/1/b with resume level of input stream |
| title_full_unstemmed | Limit theorems for systems of the type M θ/G/1/b with resume level of input stream |
| title_short | Limit theorems for systems of the type M θ/G/1/b with resume level of input stream |
| title_sort | limit theorems for systems of the type m θ/g/1/b with resume level of input stream |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3353 |
| work_keys_str_mv | AT bratiychukam limittheoremsforsystemsofthetypemthg1bwithresumelevelofinputstream AT bratíjčukam limittheoremsforsystemsofthetypemthg1bwithresumelevelofinputstream AT bratiychukam graničníteoremidlâsistemtipumthg1bzvídnovlûûčimrívnemvhídnogopotoku AT bratíjčukam graničníteoremidlâsistemtipumthg1bzvídnovlûûčimrívnemvhídnogopotoku |