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Compound Poisson With Exponential Damage

The process {Y(t),t > 0} is a homogeneous compound Poisson process if F(t) = 1 - e-xt, t > 0.

In this case the damage distribution is

-xt A)n where p(n; At) = e xt-:— is the probability function of the Poisson distri-

bution with mean At. We consider the special case where Yi,Y2, ... have a common exponential distribution, i.e., G(y) = 1 - e-^v, y > 0. In this case G(n) (y) is the cdf of the Erlang distribution, and we have

where P(■; ¡iy) is the cdf of the Poisson distribution with mean ¡iy. According to (8), (17) and (18) we obtain that the reliability function R(t; A,Z) = P(T(3) > t), where Z = , is

Theorem 2 For a Compound Poisson damage process, with G(y) = 1 -e-^v, the reliability function is

Proof.

By changing order of summation we obtain

This implies (19).

>From the definition of T(ß), it is obvious that Z ^ R(t; X; Z) is an increasing function. This follows also from (19), by applying Karlin's Lemma [K57]. The density of the failure times is, in this special case, fT(t; X,Z) = Xj2p(n Z)p(n; Xt), t > 0. (20)

Indeed, and n=0

-P(j; Xt) = -Xp(j; Xt) d fT(t; X,Z) = -^R(t; X, Z).

The expected value and variance of T(fl) are, in this special case,

Indeed, in the present case = j and E{N* (fl)} = pfl = Z. Similarly V{N*(fl)} = Z and a2 = • Equation (22) follows immediately from (16). The hazard function corresponding to (19)-(20) is

As proven in

Zacks [Z04], h(0; A, Z) = Ae-Z and lim h(t; A, Z) = A. Moreover, t—

one can show that t ^ h(t; A, Z) is strictly increasing. In Figure 1 we present the hazard function (23).

Wilson [W00] discusses the double stochastic Poisson process, in which the intensity parameter A, of the Compound Poisson process is integrated with respect to some Lebesgue density. If we consider A a gamma random variable with shape parameter v and scale parameter 1/A*, we obtain that the reliability function is

Fig. 1. Hazard Function, A = 1, Z = 5.

X* + t where NB(j; v) is the c.d.f. of the negative-binomial distribution

Notice that in (24) ■ = t/(X*+t). Also, R*(t; X*, Z, v) is an increasing function of t, from ve-ß to v/X*.

The density function corresponding to (24) is f*(t; X*,Z, v) = Yp(j; Z) V+Xnb(j; t+tX*,'

where nb j;-— ,vj is the probability function corresponding to

The density function (26) is equivalent to

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