Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 283

and let

p ij =

τ→∞ H ij (τ)

lim (A.77)

In this stochastic process, sojourn times in states can be arbitrarily dis-
tributed; moreover their PDF may depend on the next state as well as on
the present one.

This class of stochastic processes is called semi-Markov

processes (SMPs).
The PDF of the time spent in state i given that the next state is going to
be j is given by

(

H ij (τ)/p ij p ij > 0

G ij (τ) = (A.78)

0 p ij = 0

The stochastic sequence { Y n ,n ≥ 0 } is a DTMC and is called the EMC of
the SMP { X(t),t ≥ 0 } .
The transition probabilities of the EMC are defined as in ( A.46) . It can be
shown that

r ij = p ij (A.79)

with p ij given by ( A.77) .
Note that in this case it is not necessary to exclude the possibility of a
transition from state i to itself, hence not necessarily r ii = 0.
The classification of the states of the process { X(t),t ≥ 0 } can be done by
observing whether the DTMC { Y n ,n ≥ 0 } comprises either transient or re-
current states. Also the irreducibility question can be answered by observing
the EMC. The periodicity of the process { X(t),t ≥ 0 } is, instead, to be ex-
amined independently from the periodicity of the EMC. A state i of the
SMP is said to be periodic if the process can return to it only at integer
multiples of a given time δ. The maximum such δ > 0 is the period of the
state. Note that a state may be aperiodic in the SMP and periodic in the
EMC, and viceversa.
Assume, for simplicity, that all states of the SMP { X(t),t ≥ 0 } are recurrent
aperiodic. The average sojourn time in state i can be found as

2 3

Z

X

∞

4

1 −

5

E [SJ i ] = H jk (t) dt (A.80)

0

k∈S

Define the quantities η (Y j as being the stationary probabilities of the EMC
{ Y n ,n ≥ 0 } , defined as in ( A.51) and ( A.52) . The limiting probabilities of
the SMP { X(t),t ≥ 0 } , defined as in (A.50) can be found as:

η (Y j E [SJ j ]

η (X)

= (A.81)

P j

k∈S η (Y k E [SJ k ]