Let A(t) and Y(t) denote respectively the age and excess at t. Find:

(a) P{Y(t)>x| A(t) = s}.

(b) P{Y(t)>x|A(t + x/2) = s}.

(c) P{Y(t)>x|A(t + x) > s} for a Poisson process.

(d) P{Y(t) > x, A(t) > y}.

(e) If is called a renewal-type equation. In convolution notation the above states that g=h+g* F. Either iterate the above or use Laplace transforms to show that a re- newal-type equation has the solution where m(x) = 8(t) = h(t) + f' h(t x) dm(x), F(x). If h is directly Riemann integrable and F nonlattice with finite mean, one can then apply the key renewal theorem to obtain lim g(t) = Sh(t) dt SF(t) di Renewal-type equations for g(t) are obtained by conditioning on the time at which the process probabilistically starts over. Obtain a renewal- type equation for:

(a) P(t), the probability an alternating renewal process is on at time t;

(b) g(t) =E[A(t)], the expected age of a renewal process at r.

Apply the key renewal theorem to obtain the limiting values in

(a) and (b)