Lifetime modeling of physical systems and biological organisms involves the use of failure distribution functions. The shape of the hazard rate (HR) function associated with the distribution function characterizes the effect of age (and other factors) on the failure. Examples of failure (and survival) data indicate that the shapes are many and varied, due to many underlying reasons. In this paper we investigate a model where the HR function is a rational function, which is by definition the ratio of two polynomials. The different possible shapes (increasing, decreasing, bathtub, inverted bathtub, modified bathtub, roller-coaster, etc.) of the HR rational functions are identified, and necessary and sufficient conditions on the parameters to ensure their existence are derived.
Received on February 10, 2009, revised on June 5, 2009