Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 275

Figure A.4: Possible sample function of a CTMC (or SMP) from which an
EMC can be derived.

and that the transition probabilities r ij can be obtained as a function of the
transition rates q ij . Indeed, in the case of an ergodic CTMC,

(

− q ij /q ii

i 6 = j

r ij = (A.49)

0 i = j

If the CTMC { X(t),t ≥ 0 } is ergodic, then the DTMC { Y n ,n ≥ 0 } is ir-
reducible recurrent (possibly periodic), and the steady-state distribution of
the process X(t) can be determined from the stationary distribution of the
sequence Y n . Let

η (X)

t→∞ P { X(t) = j } (A.50)
and let the quantities η (Y j be obtained by solving the system of linear equa-
tions that gives the stationary distribution for the DTMC { Y n ,n ≥ 0 }

= lim

j

X

η (Y ) η (Y i r ij = (A.51)

j

i∈S

X

η (Y ) = 1 (A.52)

i

i∈S

The steady-state probabilities for the CTMC X(t) can then be shown to be

η (Y j / − q jj

η (X) = (A.53)

P
i∈S

j

η (Y i / − q ii

Or, equivalently,

h
SJ (X)

i

E

(1/ − q jj )

j

η (X) =

h i

=

h
SJ (X)

i

(A.54)

P
i∈S

P
i∈S v ij E

j

η (Y i / − q ii η (Y )

j i