Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 282

where the p 0 IJ (t) are the elements of the transition probability matrix P 0 (t)
of the process { Y (t),t ≥ 0 } .
If we are only interested in the analysis of the steady-state probabilities of
macrostates defined as
η 0 I

X

t→∞ P { X(t) ∈ A I } =

= lim η i (A.69)

i∈A I

we can, as in the discrete-time case, write the following equation for the
process { X(t),t ≥ 0 }

X

η i q ij = 0 (A.70)

i∈S

for all j ∈ S, and

X

η i = 1 (A.71)

i∈S

Summing (A.70) over all states j ∈ A J we obtain
X

η 0 I q 0 IJ = 0 (A.72)

A I ∈S 0

for all A J ∈ S 0 , and

X

η 0 I

= 1 (A.73)

A I ∈S 0

where

X X

q 0 IJ = q ij ν i|I (A.74)

i∈A I j∈A J

and ν i|I is as in (A.65) . Also in this case equations ( A.69) - ( A.74) hold,
regardless of the fulfillment of the lumpability condition, so that we can
directly evaluate the steady-state distribution of macrostates from ( A.72)
provided that we can evaluate the q 0 IJ from ( A.74) without solving for the
steady-state distribution of the CTMC { X(t),t ≥ 0 } .
If the CTMC { X(t),t ≥ 0 } is lumpable with respect to the partition S 0 , then
( A.67) holds, and ( A.72) reduces to
X X

η 0 I i ∈ A I

q ij = 0 (A.75)

A I ∈S 0 j∈A J

A.6 Semi-Markov Processes

Consider the stochastic process sample function shown in Fig. A.4. Assume
that the process { X(t),t ≥ 0 } is a finite-state continuous-time homogeneous
process which changes state at instants θ n , n = 0, 1, ··· and assumes the
value Y n in the interval [θ n ,θ n+1 ), and such that
P { Y n+1 = j,θ n+1 − θ n ≤ τ | Y 0 = k, ··· ,Y n = i,θ 0 = t 0 , ··· θ n = t n }

= P { Y n+1 = j,θ n+1 − θ n ≤ τ | Y n = i } = H ij (τ)

(A.76)