Numerical Example

In this section we apply the previous results to a three-state discrete time semi-Markov process described in Figure 1.

Let us consider that the state space E = {1, 2, 3} is partitioned into the up-state set U = {1, 2} and the down-state set D = {3}. The system is defined by the initial distribution p := (1 0 0), by the transition probability matrix V of the embedded Markov chain (Jn; n G N)

and by the conditional distributions of the sojourn time

Wq2,b2

Fig. 1. A three-state discrete time semi-Markov system

Wq2,b2

Fig. 1. A three-state discrete time semi-Markov system

0 f 12(k) 0 f (k):= ( f21 (k) 0 f23(k) I ,k G N. f 31(k) 0 0

We consider the following distributions for the conditional sojourn time:

• fi2 is a geometric distribution defined by f 12(0) := 0, f 12 (k) := p(1 - p)k-1,k > 1, where we take p = 0.8.

• f21 := Wqi,bl ,f23 := Wq2,b2 and f31 := Wq3,ba are discrete time, first type

Weibull distributions, defined by

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