Análise de sistemas GIB/M/s/c via ordenação estocástica 1 Introdução

GI B /M/s/c
GI B /M/s/c
GI B /M/s/c
GI B /M/s/c
GI B /M/s/c
GI B /M/s/c
GI/M/s/c
GI B /M/s/c

GI B /M/s/c
¯π
π (N) ¯π (N) π (N) ≤st ¯π ≤st ¯π (N)
≤st
p
p
π (N) ¯π (N)
GI B /M/s/c
GI/M/s/c
λ
A
{bn, n ∈ IN}
¯ b
p q p
q p ≤st q j≥k pj ≤j≥k qj k
X Y
p q X ≤st Y p ≤st q
C D
C D C ≤K D
j≥k cij ≤j≥k dmj i ≤ m k
C C ≤K C

s
µ
ρ = λ ¯ b/sµ
c = ∞
c < ∞
n i
min{n, c + s − i}
X(t) GI B /M/s/c t, t ≥ 0
X = {X(t), t ≥ 0}
X
{( ¯ Xk, Sk), k ≥ 0} Sk k
¯ Xk = X(S −
k )
k
{X(t), t ≥ 0}
S = {0, 1, 2, . . . , c ⋆ } c ⋆ = s + c {µi = µ min(s, i), i =
1, 2, . . . , c ⋆ }
{X(Sk + t), 0 ≤ t

GI B /M/s/c
A
¯ Xk
¯ X
+∞ +∞
+∞
+∞
¯pij = bn · γδc⋆ (i+n)j(t) A(t) = bn ·
0 n=1
n=1 0
γ δc ⋆ (i+n)j(t) A(t)
i, j ∈ S
¯pij
pij = +∞
γij(t) A(t).
0
[S1, S2)
sµ Y
pij =
+∞
0
γij(t) A(t) =
+∞
m=0
αA(m, sµ) · [ ˆ P m ]ij
ˆ P
sµ Y
αA(m, sµ) αm
m A
sµ
¯P =
αA(m, sµ) =
+∞
+∞
bn
n=1 m=0
+∞
0
αm · ∆n( ˆ P m ) =
−sµt (sµt)m
e A(t).
m!
+∞
+∞
bn · ∆n
n=1
m=0
αm · ˆ P m
[∆n(D)]ij = [D] δc ⋆ (i+n)j D = [dij]i,j∈S
∆n n ∈ IN
s = 1 c < +∞
ˆ P
s > 1
¯ P
ˆ P
¯ P

M (N) =
¯M (N) =
∞
n=1
∞
n=1
bn · ∆n
bn · ∆n
N
m=0
N
m=0
αm ˆ P m +
αm ˆ P m +
+∞
m=N+1
+∞
m=N+1
αmZ
αm ˆ P N
N ∈ IN0 Z = (zij) = (δj0) δ
δab a = b
Y (N) ¯ Y (N) S
M (N) ¯ M (N)
M (N) ¯ M (N)
M (N) ≤K ¯ P ≤K ¯ M (N)
N ∈ IN0 M (N) ¯ M (N) ¯ P N
n0 = min{n ∈ IN : bn > 0} ¯ X Y (N)
N ≥ δ c ⋆ −1(n0) ¯ Y (N) N ≥ δ c ⋆(n0 + 1)
¯ X Y (N) N ≥ δ c ⋆ −1(n0) c < ∞ ρ =
λ ¯ b/sµ < 1 ¯ Y (N) N ≥ δ c ⋆(n0 + 1)
c < ∞ N c = ∞ ρ < 1
¯ Y (N) π (N) (¯π ¯π (N) )
Y (N) ( ¯ X ¯ Y (N) )
π (N) ≤st ¯π ≤st ¯π (N) .
N ∈ IN0
{αn, n ∈ IN0}
B (N) =
N
αm ˆ P m +
m=0
+∞
m=N+1
αmZ ¯ B (N) =
N
αm ˆ P m +
m=0
+∞
m=N+1
αm ˆ P N
ˆ P n n ∈ IN0 Z ∆n n ∈ IN
M (N) =
∞
n=1
bn · ∆n(B (N) ) ¯ M (N) =
∞
n=1
bn · ∆n( ¯ B (N) )

GI B /M/s/c
∆n(B (N) ) ∆n( ¯ B (N) ) n ∈ IN
M (N) − ¯ P =
¯M (N) − ¯ P =
∞
m=N+1
∞
αm
Z −
αm
m=N+1 n=1
∞
bn · ∆n( ˆ P m
)
n=1
∞
bn · ∆n( ˆ P N − ˆ P m ).
∞
n=1 bn · ∆n( ˆ P m )
m ∈ IN M (N) ¯ M (N) ¯ P N
∞ m=N+1 αm N
ˆ P
ˆP n2 ≤K ˆ P n1 0 ≤ n1 ≤ n2 Z ≤K C
C S Z ≤ ˆ P n
n ∈ IN0 C D
S C ≤K D ∆n(C) ≤K ∆n(D) n ∈ IN
B (N) ≤K B ≤K ¯ B (N) ,
B = ∞ m=0 αm ˆ P m
M (N) =
∞
bn·∆n(B
n=1
(N) ) ≤K
n=1
∞
bn·∆n(B) = ¯ P ≤K
∞
n=1
bn·∆n( ¯ B (N) ) = ¯ M (N)
¯ X Y (N) N ≥ δc⋆−1(n0) ¯ Y (N) N ≥ δc⋆(n0 + 1)
bn0
Q = [qij]i,j∈S
qij > 0 i ≤ j ≤ δc⋆(i + n0) qi0 > 0 qi,i−1 > 0 i > 0
i ∈ S c < ∞
c = ∞ ρ < 1 ¯ X
c = ∞ ρ = λ¯b/sµ < 1
n≥1 nαn = sµ/λ = ¯b/ρ i ≥ s
lim sup
i
Xk+1
¯ − i| ¯ Xk = i ≤ lim sup
i
∞
k=1
i−s+1
k bk −
n=1
nαn
= ¯ b(1 − ρ −1 ) < 0.

¯ Xk+1 − i| ¯ Xk = i ≤ ¯ b i ∈ S
¯ X
c = ∞ ρ < 1 N ≥ δ c ⋆ −1 (n0)
Y (N) M (N) ≤K ¯ P
Y (N)
c = ∞ ρ = λ ¯ b/sµ < 1 ɛ > 0 λ ¯ b(1 + ɛ) < sµ
ɛ ⋆ = ¯ b − sµ/λ(1 + ɛ) < 0 N(ɛ) > ¯ b
N
n=0 nαn ≥ sµ/λ(1 + ɛ) = ¯ b − ɛ ⋆ N ≥ N(ɛ)
N ≥ N(ɛ) i ≥ s − 1 + N
Y ¯ (N)
k+1 − i| ¯ Y (N)
k = i =
∞
k=1
= ¯ b −
bk
N
(k − n)αn +
n=0
N
nαn − N
n=0
Y ¯ (N)
k+1 − i| ¯ Y (N)
k
∞
n=N+1
∞
n=N+1
αn ≤ ¯ b −
(k − N)αn
N
nαn ≤ ɛ ⋆ < 0.
n=0
= i ≤ ¯b i ∈ S
¯ Y (N) N ≥ N(ɛ)
¯ Y (N) ¯ X Y (N)
π (N) ¯π ¯π (N)
Y (N) ¯ X ¯ Y (N)
a(M (N) ) n ≤st a ¯ P n ≤st a( ¯ M (N) ) n
n, N ∈ IN a
S n
¯π
π (N) π (N)
W1 W2
IN0
P1 P2 P1 ≤K P2 W1
W2

GI B /M/s/c
¯αA(m, β)
A β
+∞
¯αA(m, β) =
0
−βt (βt)m
e
m!
Ā(t)t , m ∈ IN0, β > 0
Ā
Ā(t) = 1 − A(t)
GIB /M/s/c
c < ∞ ρ < 1
pk = lim (X(t) = k) = λ ¯πj · ηjk
t→+∞
c ⋆
j=0
k ∈ S ηjk
X k
j
S2
ηjk = 1 {X(t)=k} |X(S
S1
−
1 ) = j =
+∞
ψlk =
m=0
c ⋆ −1
n= ¯ δk(j+1)
¯αA(m, sµ) · [ ˆ P m ]lk
j, k, l ∈ S k ∈ {1, 2, ..., c ⋆ }
pk =
j=0
bn−jψnk + ¯ Bc ⋆ −j−1ψc ⋆ k
k−1
λ
¯πj
µ min(k, s)
¯ Bk−j−1.
X
S
{( ¯ Xk, Sk), k ≥ 0}
1/λ

j, k ∈ S
ηjk =
=
=
=
+∞
S2
S1
bn ·
n=1
0
+∞
∞ s
bn ·
n=1 0 0
+∞
∞
bn ·
n=1 0
1 {X(t)=k} |X(S − 1
) = j
S2−S1
1 {Y (t)=k} |Y (0) = δc⋆(j + n)
(Y (t) = k |Y (0) = δ c ⋆(j + n))tA(s)
γ δc ⋆ (j+n)k(t) Ā(t)t.
[S1, S2)
sµ
+∞
+∞
ηjk = bn ·
=
=
n=1
c ⋆ −j−1
n= ¯ δk−j(1)
c ⋆ −1
n= ¯ δk(j+1)
m=0
¯αA(m, sµ) · [ ˆ P m ] δc ⋆ (j+n)k
bn · ψ δc ⋆ (j+n)k + ¯ Bc ⋆ −j−1 · ψc ⋆ k
bn−j · ψnk + ¯ Bc ⋆ −j−1 · ψc ⋆ k.
k ∈ {1, 2, ..., c ⋆ } Ak = {0, 1, ..., k − 1}
Ak Ak
X
k Ak pkµk µk
X
k
µk = lim
t→+∞
t
0 1 {X(s − )=k,X(s)=k−1} M(s)
t
0 1 {X(s)=k} s
M
k µ min(s, k)
µk = µ min(s, k)
Ak
k − j − 1 X

GI B /M/s/c
j j ∈ Ak Ak λ k−1
j=0 ¯πj ¯ Bk−j−1.
Ak
k ∈ {1, 2, ..., c ⋆ }
r S
qk(r) = λ
c ⋆
j=0
rj · ηjk
k ∈ S p = q(¯π) q
S
j ∈ S c ⋆
l=j ηil i i ∈ S
j ∈ S
c ⋆
+∞
∞ s
ηil = bn ·
l=j n=1 0 0
(Y (t) ≥ j |Y (0) = δ c ⋆(i + n))tA(s)
i ∈ S Y
n ∈ IN δ c ⋆(i+n)
i s
0 (Y (t) ≥ j |Y (0) = δ c ⋆(i + n))t
δ c ⋆(i + n) Y
j [0, s]
i c ⋆
l=j ηil i
.1 .∞ L1 L∞
a b S
a ≤st b ⇒ q(a) ≤st q(b)
q c⋆ < ∞
.⋆ = .1, .∞
j ∈ S
βij =
q(a) − q(b)⋆ ≤ (c ⋆ + 1) a − b⋆
c ⋆
l=j η0l
⋆
c
l=j ηil − c ⋆
l=j ηi−1 l
i = 0
1 ≤ i ≤ c ⋆
.

βij i, j ∈ S
c ⋆
l=j ηil i, i ∈ S
c ⋆
l=j
ηil =
i
βkj, i, j ∈ S.
k=0
g S k ∈ S
ql(g) = λ gn · ηnl = λ
c ⋆
l=k
c ⋆
c ⋆
l=k n=0
= λ
c ⋆
gn
n=0 i=0
c ⋆
n=0
n
c
βik = λ
⋆
gn
c ⋆
l=k
c ⋆
βik
i=0 n=i
ηnl
gn = λ
c ⋆
i=0
βik ¯ Gi−1.
a b S a ≤st b
Āi−1 ≤ ¯ Bi−1 i ∈ S c = q(a) d = q(b)
βik
i, k ∈ S
¯Ck−1 =
ql(a) = λ βikĀi−1
≤ λ βik ¯
Bi−1 = ql(b) = ¯ Dk−1
c ⋆
l=k
c ⋆
i=0
k ∈ S
c⋆ < ∞ j ∈ S
|qj(b) − qj(a)| =
λ
c ⋆
(bn − an) · ηnj
n=0
c ⋆
i=0
c ⋆
l=k
≤ λ |bn − an| · ηnj ≤ |bn − an|
ηnj ηnj ≤ c ⋆
k=0 ηnk = λ −1
n ∈ S q(b)−q(a)1 = c ⋆
c ⋆
n=0
c ⋆
n=0
j=0 |qj(b)−qj(a)| ≤ (c ⋆ +1) b−a1.
· ∞
|qj(b) − qj(a)| ≤ c ⋆
n=0 |bn − an| j ∈ S
c < ∞ ρ < 1 N
¯ Y (N)
p (N) = q(π (N) ) ¯p (N) = q(¯π (N) )
p (N) ≤st p ≤st ¯p (N) .

GI B /M/s/c
c < ∞
¯p (N) − p (N) ⋆ ≤ (c ⋆ + 1) ¯π (N) − π (N) ⋆
.⋆ = .1, .∞
GIB /M/s/c
¯π p
¯π p
/M/s/c