Let \((Y, Z)\) be the typical cycle of an alternating renewal process, where \(Y\) and \(Z\) have an Erlang distribution with joint parameter \(\lambda\) and parameters \(n=2\) and \(n=1\), respectively. For \(t \rightarrow \infty\), determine the probability that the system is in state 1 at time \(t\) and that it stays in this state over the entire interval \([t, t+x], x>0\) (process states as introduced in section 7.3.6).

Data from Section 7.3.6

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So far it has been assumed that renewals take only negligibly small amounts of time. In order to be able to model practical situations, in which this assumption is not ful- filled, the concept of a renewal process has to be generalized in the following way: The renewal time of the system after its ith failure is assumed to be a positive random variable Z;; i = 1, 2,.... Immediately after a renewal the system starts operating. In this way, a marked point process {(Yi, Z;); i = 1,2,...} is generated, where Y; as before denotes the lifetime of the system after the ith renewal.