379Dependent Stochastic Processes and Time Series
as the sum of all the holding time densities corresponding to transitions out of state j. The mean
waiting time in role j is then the mean of W
j
(T), and denoted as M
j
,
j
i
ij
ij
M
p
j
∑
1
4
12 4µ ,, ,…
(11.20)
or the sum of all products of the probability p
ij
and the mean k
ij
/d
ij
of the h
ij
(t) density. In turn, the
mean time between transitions M is a weighted sum of the mean waiting times in each state
M
j
j
j
=
=
∑
1
4
*
(11.21)
where the weights used in the average are the stationary proportions of the embedded chain as given
in the previous section. The steady-state occupancy probabilities, and therefore the stationary pro-
portions X
**
i
for all the roles can be calculated as
jjj
XXMM
j
** *
,, ,
−1
12 4…