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Formula Sheet
Markov Chains
Chapman–Kolmogorov Equations:
p
(
n
)
ij
=
M
X
k
=0
p
(
m
)
ik
p
(
n

m
)
kj
for any
m < n
.
SteadyState Probabilities:
π
j
=
M
X
i
=0
π
i
p
ij
∀
j
= 0
,
1
,...,M
M
X
j
=0
π
j
= 1
Queueing Theory
M/M/
1
M/M/s
M/G/
1
P
n
=
C
n
P
0
=
C
n
ˆ
∞
X
n
=0
C
n
!

1
C
n
=
±
λ
μ
¶
n
=
ρ
n
(
λ/μ
)
n
n
!
,
for
n
= 1
,
2
,...,s
(
λ/μ
)
n
s
!
s
n

s
,
for
n
=
s
+ 1
,s
+ 2
,...
±
λ
μ
¶
n
=
ρ
n
P
0
=
1

ρ
"
s

1
X
n
=0
(
λ/μ
)
n
n
!
+
(
λ/μ
)
s
s
!
1
1

λ/
(
sμ
)
#

1
1

ρ
L
q
=
λ
2
μ
(
μ

λ
)
P
0
(
λ/μ
)
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This homework help was uploaded on 04/07/2008 for the course IE 220 taught by Professor Storer during the Spring '07 term at Lehigh University .
 Spring '07
 Storer

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