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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)

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

η
(Y
j
E [SJ
j
]

η
(X)

=

(A.81)

P

j

k∈S
η
(Y
k
E [SJ
k
]

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