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Figure A.2: The negative exponential pdf and its memoryless characteristics
An important implication of the Markov property is that the distribution
of the sojourn time in any state must be memoryless. Indeed, if the future
evolution depends on the present state only, it cannot depend on the amount
of time already spent by the process in the state. This, coupled with the
observation that for a continuous random variable X the only pdf that
satisfies the memoryless property
P
{
X
≥
t + τ
|
X
≥
t
}
= P
{
X
≥
τ
}
(A.5)
is the negative exponential
f
X
(x) = a e
−ax
x
≥
0
(A.6)
leads to the conclusion that sojourn times in CTMC states must be expo-
nentially distributed random variables.
Similarly, in the case of DTMCs, sojourn times in states must be geometri-
cally distributed random variables
P
{
SJ = i
}
= p
i−1
(1
−
p)
i = 1, 2, 3,
···
(A.7)
The importance of the memoryless distribution of the times spent in states
can be better understood by noting that, in order to check whether a stochas-
tic process satisfies the Markov property, it su
ces to check whether the
distributions of sojourn times are memoryless, and whether the probabili-
ties of going from one state to another only depend on the state the process
is leaving and on the destination state.
The memoryless property of the exponential pdf can be visualized by ob-
serving its shape, depicted in Fig.
A.2.
If we refer this distribution to a
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