Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 284

which is analogous to ( A.53) . The EMC technique thus allows the steady-
state analysis of a broader class of stochastic processes than was previously
considered, namely of SMPs.
The mean recurrence time of state j, defined as the average time elapsing
between two successive visits to state j, is

= E [SJ j ]
η (X)

M j (A.82)

j

Note that also this equation is an extension of (A.37) .
The class of SMPs comprises the class of MCs, but SMPs themselves are a
subset of a more general class of stochastic processes named semi-regenerative
processes. We shall not venture into the analysis of semi-regenerative pro-
cesses, but it is useful to derive one result that, obviously, is valid for all
SMPs and MCs.
Consider the SMP sample function shown in Fig. A.4. From (A.76) it fol-
lows that the time instants at which a jump occurs are such that the future
evolution of the process is independent of the past, given the present state,
even if the sojourn time in the present state is not an exponentially dis-
tributed random variable. This is su cient to recognize that the process is
a semi-regenerative stochastic process.
In this case the characteristics of the process can be determined from the
behaviour in the period comprised between two instants at which jumps
occur. This time period is called a cycle. The cycle duration is a random
variable CY whose average is

X

η (Y j E [SJ j ]

E [CY ] = (A.83)

j∈S

Indeed, the cycle duration equals the sojourn time in state j provided that
the state at the beginning of the cycle is j. Using the total probability

theorem we obtain (A.83) .
The limiting state probabilities of a semi-regenerative process can be found
as the ratio of the average time spent in a state in a cycle divided by the
average cycle duration. This provides another method for the derivation of
( A.81) .
The cycle analysis can also be performed in a slightly different manner.
Consider as initial cycle times the instants at which the process enters state
j. In this case the average cycle duration is obtained as

η (Y )

X X

k
η (Y )

E [CY j ] = v kj E [SJ k ] = E [SJ k ] (A.84)

k∈S k∈S

j

where v kj gives the average number of visits to state k between two successive
visits to state j, and is given by (A.22) .

The average time spent in state