Template:The effect of beta on Weibull failure rate

The Effect of β on the Weibull Failure Rate

The value of β has a marked effect on the failure rate of the Weibull distribution and inferences can be drawn about a population's failure characteristics just by considering whether the value of β is less than, equal to, or greater than one.

As indicated by above Figure, populations with β < 1 exhibit a failure rate that decreases with time, populations with β = 1 have a constant failure rate (consistent with the exponential distribution) and populations with β > 1 have a failure rate that increases with time. All three life stages of the bathtub curve can be modeled with the Weibull distribution and varying values of β. The Weibull failure rate for 0 < β < 1 is unbounded at (or γ). The failure rate, λ(t), decreases thereafter monotonically and is convex, approaching the value of zero as t→∞ or λ(∞) = 0. This behavior makes it suitable for representing the failure rate of units exhibiting early-type failures, for which the failure rate decreases with age. When encountering such behavior in a manufactured product, it may be indicative of problems in the production process, inadequate burn-in, substandard parts and components, or problems with packaging and shipping. For β = 1, λ(t) yields a constant value of [math]\displaystyle{ { \frac{1}{\eta }} }[/math] or:

[math]\displaystyle{ \lambda (t)=\lambda ={\frac{1}{\eta }} }[/math]

This makes it suitable for representing the failure rate of chance-type failures and the useful life period failure rate of units.

For β > 1, λ(t) increases as increases and becomes suitable for representing the failure rate of units exhibiting wear-out type failures. For 1 < β < 2, the λ(t) curve is concave, consequently the failure rate increases at a decreasing rate as increases.

For β = 2 there emerges a straight line relationship between λ(t) and , starting at a value of λ(t) = 0 at t = γ, and increasing thereafter with a slope of [math]\displaystyle{ { \frac{2}{\eta ^{2}}} }[/math]. Consequently, the failure rate increases at a constant rate as increases. Furthermore, if η = 1 the slope becomes equal to 2, and when γ = 0, λ(t) becomes a straight line which passes through the origin with a slope of 2. Note that at β = 2, the Weibull distribution equations reduce to that of the Rayleigh distribution.

When β > 2, the λ(t) curve is convex, with its slope increasing as increases. Consequently, the failure rate increases at an increasing rate as increases indicating wear-out life.