In this problem, we show how to calculate the residual life distribution Y(t) as a transient in t. Let μ(t) = dm(t)/dt, where m(t) = E [N(t)], and let the interarrival distribution have the density fX (x). Let Y(t) have the density fY(t)(y).

(a) Show that these densities are related by the integral equation

μ(t + y) = fY(t)(y) + u=0

μ(t + u)fX (y - u) du.

(b) Let Lμ,t(r) = (μ(t + y)e-rydy  and  let  LY(t)(r) and  LX (r)  be  the Laplace transforms of fY(t)(y) and fX (x) respectively. Find LY(t)(r) as a function of Lμ,t and LX .

(c) Consider the inter-renewal density fX (x) = (1/2)e-x + e-2x for x ≥ 0 (as in Example 5.6.1). Find Lμ,t(r) and LY(t)(r) for this example.

(d) Find fY(t)(y). Show that your answer reduces to that of (5.30) in the limit as t → ∞.

(e) Explain how to go about finding fY(t)(y) in general, assuming that fX has a rational Laplace transform.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.