Explain how to go about finding in general

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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 f(x). Let Y(t) have the density fY(t)(y).

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

μ(y) = fY(t)(y) + u=0

μ(u)f(udu.

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

(c) Consider the inter-renewal density f(x) = (1/2)e-e-2for ≥ 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 → ∞.

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

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

Reference no: EM13909261

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