Page 6 - SUBHARADEV SEN AND AHMED

SEN AND AFIFY

Here µ 1 is an increasing function in p and α.

In particular, since | α | ≤ 1, for p = 1 (which corresponds to dependent series
system with identically uniformly distributed with parameter θ) we have,
3 11

10 30 0.
On the other hand, the MTF for parallel system, with pdf (10), becomes

1 , say. (13)

In this case, µ 2 is a decreasing function in α and is increasing function in p
for a given α. In particular, since | α | ≤ 1, for p = 1 we have,
19 7 0.
30 10
The MTF for standby redundant system with pdf (11) is obtained as
,

1 4 , , , say. (14)

We note that µ 3 is a decreasing function in α. In particular, since | α | ≤ 1, for
p = 1 we have,
3 11 0.
10 30
3.1. Comparison of MTF's
Now, we compare the mean times to system failure, as obtained through
BVFR situation, with those of corresponding independently and identically
distributed (IID) cases. For comparison, we consider the values of p as 0.25,
0.50, 0.75, 1.00, 1.50, 2.00, 2.50, 3.00 and the values of α as -1.00, -0.75, -
0.50, -0.25, 0.00, 0.25, 0.50, 0.75, 1.00. Only one value of θ is considered
owing to the theoretical fact that each of µ 1, µ 2 and µ 3 is an increasing
function of θ and mean times to failure shall be proportionately higher for
greater values of θ. Statistical software R (version 3.5.1) is utilized for
numerical calculations. The results are shown in Table 1 for θ = 3.0.
The following observations are noted from Table 1.
(I) When the two components in the series system are dependent in
positive sense, mean times to failure are greater than those for the IID
case as the values of α increase towards +1. A completely reversed
scenario is observed when the two components in the series system
are dependent in negative sense and α decreases towards 1.

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