The General Compound Renewal Damage Process and The Associated Failure Distribution

Consider a renewal process, with renewal epochs 0 < ti <t2 < ... < Tn < Let Ti = Ti — to (i = 1, 2,...,to = 0) be the interarrival times. Ti, T2,... is a sequence of independent identically distributed (i.i.d.) random variables, having a common distribution F. We assume in this paper that F is absolutely continuous, with density function f, and F(0) = 0. Let N(t) denote the number of arrivals in the time interval (0, t], with N(0) = 0, i.e.,

It is well known [K97] that

P {N (t)= n} = F (n)(t) — F (n+1)(t), n = 0,1,... (2)

where F(0)(t) = 1, all t > 0, F(1)(t) = F(t), and for n > 2

That is, F (n) is the n-fold convolution of F. Let f(n) denote the corresponding n-fold convolution of the density f.

We model the cumulative damage (C.D.) process by the compound renewal process

where Yo = 0, Yi,Y2,... are i.i.d. positive random variables having a common absolutely continuous distribution G, with density g, and {Yn,n > 1} is independent of {N(t),t > 0}. The distribution function of Y(t), at time t, is

where G(n) is the n-fold convolution of G.

Notice that D has a jump point (atom) at y = 0, and D(0; t) = 1 — F(t) = F(t). The density of Y(t) on (0, to) is d(y; t) = J2(F(n)(t) — F(n)(t))g(n)(t), (6)

n=i where g(n) is the n-fold convolution of g. Let ¡3, 0 < f3 < to, be a threshold value such that the system fails as soon as Y(t) > 3. Thus, we define the stopping time

Since Y(t) | to a.s. as t ^ to, P{T(f) < to} = 1 for all 0 < f < to. Moreover,

This is the reliability function of the system, which is obviously decreasing function of t, with lim D(f; t) = 0. The density function of T (f) as obtained t—>tt from (5), is given by tt

Pt (t; f) = f (t)G(f) + E f(n) (t)(G(n-i) (f) - G(n)(f)). (9)

{N*(t),t > 0} is the renewal process associated with {Yo,Yi,Y2,...}. Accordingly, the density of T(f) can be written as

Theorem 1 If the interarrival time Ti has a moment of order m, (m > 1) then tt

n=i where m

Proof. According to (11), ptt

E{T (m)(f)} = tmpT (t; f)dt o tt / f tt \ = E{J0 tmf(n)(t)dtjP{N*(f)= n - 1}.

Moreover, rtt


E{T2 (¡3)} = nP {N *(3) = n - 1} + lAYl n(n - 1)P {N *(3) = n - 1}

>From (14)-(15) we obtain the following formula for the variance of T(3), namely

In a similar fashion one can obtain formulae for higher moments of T(¡3).

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