Page 12 - SUBHARADEV SEN AND AHMED

SEN AND AFIFY

,
, ,

,
, ,

(20)
respectively.
When the two components in each system are independently and identically
distributed (IID) as finite range distribution with parameters p and θ, the
above failure rates boil down to

,

2

, , respectively.

,

We now concentrate on investigating the nature of failure rate functions for
series and parallel systems, noting the fact that the failure rate function (20)
for cold redundant (standby) system is quite complicated in its mathematical
form.
For all θ (> 0) and p > 1, the failure rate function for series system is
monotonically increasing (IFR) for all t and α (see Figure 1(b)). But when
p < 1, the failure rate function for series system is initially decreasing (DFR)
and then increasing (IFR) depending on the values of t and α, for all θ (> 0)
(see Figure 1(a)).

In univariate case, as shown by Mukherjee and Islam ([4], 1983) and the
subsequent note in Lai and Mukherjee ([3], 1986), finite range distribution is
IFR when p > 1. Hence, for p > 1, the property of IFR is well preserved for
the distribution of dependent coherent series system. Figure 1 show some

plots of failure rate function of series system for different values of p, α and
t, fixing θ at 5.0.

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