Formulário M008 Capítulo 7

Prévia do material em texto

Teoria de Filas – Formulário 
Para todas as filas: 
µ
λρ = 
µ
1
)( =stE 
0
( ) ( ) ( ) k
k
E q E w E S k P
∞
=
= + = ⋅ )()()( swq tEtEtE += 01U P= − 
 
Little para filas sem bloqueio (buffer infinito): 
λ
)(
)(
qE
tE q = λ
)(
)(
wE
tE w = λ
)(
)(
SE
tE s = ρ=)(SE 
 
Little para filas com bloqueio (buffer finito): 
)1(
)(
)(
B
q
P
qE
tE
−
=
λ
 
)1(
)(
)(
B
w
P
wE
tE
−
=
λ
 
)1(
)(
)(
B
s
P
SE
tE
−
=
λ
 ( ) (1 )BE S Pρ= − 
 
Fila M/M/1/∞/∞/∞/FIFO: 
)1(
0
ρ−=P 0PP
k
k ⋅= ρ ρ
ρ
−
=
1
)(qE 
2
( )
1
E w
ρ
ρ
=
−
 
λµ −
= 1)( qtE 
( )
N
P q N ρ≥ = ( )( ) e TqP t T
µ λ− − ⋅≥ = 
 
 
Fila M/M/1/J/J+1/∞/FIFO: 
0
k
k PP ⋅ρ= 0 2
1
1
J
P
ρ
ρ +
−=
−
 1B JP P += 
2
2
( 2)
( )
1 1
J
J
J
E q
ρ ρ
ρ ρ
+
+
+ ⋅= −
− −
 
 
 
Fila M/M/m/0/m/∞/ FIFO: 
 
0
k
k P
!k
P ⋅ρ= 0
0
1
!
km
k
P
k
ρ
=
=

 
0
!
m
B m
P P P
m
ρ= = ⋅ 
[ ] [ ] (1 )
1
[ ] [ ]
B
q S
E q E S P
E t E t
ρ
µ
= = −
= =
 
 
Fila M/M/m/∞/∞/∞/FIFO: 
mkP
k
P
k
k ≤⋅= ,
!
0
ρ
 mk,P
!mm
P 0mk
k
k ≥⋅
⋅
ρ=
−
 
0 1
0
1
!
! 1
k mm
k
P
k
m
m
ρ ρ
ρ
−
=
=
 
+ 
   ⋅ − 
 

 



















 −
⋅⋅=
2
0
1
!
)(
m
m
m
P
wE
m
ρ
ρ
ρ
 
Fila M/M/m/J/K/∞/FIFO: 
mkP
k
P
k
k ≤⋅= ,
!
0
ρ
 
0
,
!
k
k k m
P P m k J m
m m
ρ
−= ⋅ ≤ ≤ +⋅
 B J mP P += 
0
0 1
1
! !
k km J m
k m
k k m
P
k m m
ρ ρ+
−
= = +
=
+
⋅ 
 
0 0
1 1
( )
! !
k km J m
k m
k k m
k k
E q P P
k m m
ρ ρ+
−
= = +
⋅ ⋅= +
⋅ 
 
 
Fila M/G/1/∞/∞/∞/FIFO: 
2 2
( )
( )
2(1 )
s
E t
E w
λ
ρ
⋅=
−
 
2
( )
( )
2(1 )
s
w
E t
E t
λ
ρ
⋅=
−
 
Para atendimento exponencial (M/M/1): 
2
2 1
µ
σ =ts [ ]2 2
1
( )
s
E t
µ
= 2
2
2
( )
s
E t
µ
= 
ρ
ρ
ρ
λ
−
=
−
⋅=
1)1(2
)(
)(
222
stEwE 
ρ
ρ
ρ
ρ
−
=+
−
=
1
)(
1
)(
2
sEqE 
Para atendimento constante: 02 =tsσ [ ]2 2
1
( )
s
E t
µ
= 2
2
1
( )
s
E t
µ
= 
)1(2)1(2
)(
)(
222
ρ
ρ
ρ
λ
−
=
−
⋅= stEwE ρ
ρ
ρ
ρ
ρ +
−
=+
−
=
)1(2
)(
)1(2
)(
22
sEqE 
Para qualquer atendimento (incluindo os casos anteriores): 
 ( ) ( )
ss s T s
E t t f t= ⋅ 2 2( ) ( )ss s T sE t t f t= ⋅ 
 
Filas com prioridades: 
1
R
r
r
λ λ
=
= 
1
R
r
r
ρ ρ
=
= ( ) ( )1 rR rs srE t E tλλ== ( ) ( )
2 2
1 r
R
r
s sr
E t E t
λ
λ=
= 
( ){ } ( ) { }
( )( ) ( )( )
2
1
. .
2 1 1
sp
p
p p
E t
E w
λ λ
β β−
=
⋅ − ⋅ −
 
( ) ( )
( )
1
0
0
i
i k
k
β ρ
β
=
 =

 =

 ( ) rp
λ λ=
 ( ) rp
ρ ρ=
 
 
( ){ } { }
( )( ) ( )( )
2
1
.
2 1 1
p
s
w
p p
E t
E t
λ
β β−
=
⋅ − ⋅ −
 
( ) ( )
( )
1
0
0
i
i k
k
β ρ
β
=
 =

 =

 ( ) rp
λ λ=
 ( ) rp
ρ ρ=
 
{ } ( ){ }
1
P
p
p
E w E w
=
=
 
{ }
( ){ }
1
p
P
w w
p
E t E t
=
=
 
Equacionamento para o caso sem prioridades: 
)1(2
)(
)(
22
ρ
λ
−
⋅= stEwE 
)1(2
)(
)(
2
ρ
λ
−
⋅= sw
tE
tE 
Fila M/M/m/0/m/S/FIFO (População finita): 
 
0 0
!
. . . .
( )! !
K K
K
S S
P P P
K S K K
ρ ρ = =  − 
 
0
0
1
.
m
K
K
P
S
K
ρ
=
=
 
 
 

 
 
Fila M/M/m/J/K/S/FIFO (População finita): 
0 0
!
. . . . 
( )! !
K K
K
S S
P P P K m
K S K K
ρ ρ = = ≤  − 
 
 
( )0
1
.
. 1 
K J
K K m
K
SP
P S m K m K m J
mm
ρ
− =
 
= ⋅ ∏ − − + < ≤ + 
 
 
 
( )
0
1
0 1
1
. . 1
Km m J J
K
K m
i
K K m
P
S S
S m i
K m m
ρρ
+
− == = +
=
    
+ ⋅∏ − − +    
    
 
 
 
Teorema de Little para filas de população finita: 
( )e
0
m J
K
K
S K Pλ λ
+
=
= − ⋅ ⋅ { } { }( )e 1q B
E q
E t
Pλ
=
−
 { } { }( )e 1w B
E w
E t
Pλ
=
−
 { } ( )e 1 BPE S λ
µ
−
= 
 
Sistemas multidimensionais (filas com múltiplos serviços): 
, , ,..., 0,0,0,...,0 1 2
1
. , com , ,...,
L
L
i
K
L L
a b c K Ki
L L
S
P P i a i b i K
λ
µ=
  
= = = =  
  
∏ 
 
Redes de filas. Teorema de Jackson: 
1
M
i i ji jj
rλ γ λ
=
= + ( ) ( )1 2 3
1
, , ,...,
M
M i
i
p q q q q p q
=
= ∏