
562 Large Scale and Big Data
Steady-state probabilities of states (N−1,h), (N−1,w), and (N−1,c) are given by
π
(N−1,h)
= λ /δ
h
π
(N−2,h)
= λ /δ
h
A
(N−2,h)
π
(0,0)
(18.12)
πλδπ
δ
δ
π
(,)(,)
)
()
Nw wNw
hh
w
P
−− −
=+
−
12 1
1
/
(18.13)
=+
−
λδ π
δ
δ
λδ π
wNw
hh
w
hNh
A
P
A
(,)(,) (,)(,
()
20
00
1
))
(18.14)
πλδπ
δ
δ
π
λδ
(,)(,) (,)
(
()
Nc cNc
ww
c
Nw
c
P
A
−− −
=+
−
=
12 1
1
/
/
NNc
ww
c
wNw
h
w
P
A
Ph
+
−
+
−
1
1
,) (,)
()
()
δ
δ
λδ
δ
δ
λδ
/
/
hhNh
A
(,)(,)−
200
π
(18.15)
Observe that steady-state probabilities of all states in the CTMC of Figure 18.2
can be expressed as a function of the steady-state probability of state (0,0). Using
normalization,
ππππ
(,)(,) (, )(,)
00
0
1
++
=
−
∑
ih iw ic
i
N
,
(18.16)
we can compute ...