Statistics and Probability Letters 83 (2013) 1364–1371
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Statistics and Probability Letters
journal homepage: www.elsevier.com/locate/stapro
Reliability analysis using ageing intensity function
Subarna Bhattacharjee
a
, Asok K. Nanda
b,∗
, Satya Kr. Misra
c
a
Department of Mathematics, Ravenshaw University, Cuttack-753003, Odisha, India
b
Department of Mathematics and Statistics, IISER Kolkata, Mohanpur Campus, Mohanpur-741252, West Bengal, India
c
School of Applied Sciences, KIIT University, Bhubaneswar-751024, Odisha, India
article info
Article history:
Received 24 June 2012
Received in revised form 17 December 2012
Accepted 12 January 2013
Available online 4 February 2013
MSC:
primary 60E15
secondary 62N05
60E05
Keywords:
Ageing phenomenon
Generalized Weibull model
Hazard rate
abstract
Ageing intensity (AI) function analyzes the ageing property of a system quantitatively. We
study the behavior of a few generalized Weibull models and some system properties in
terms of AI function. We establish that AI function provides a major and altogether a new
role in studying system’s ageing behavior from reliability perspective.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
An important phenomenon in reliability theory is ‘ageing’ which is an inherent property of a unit that may be a living
organism or a system of components. Failure rate, mean residual life, etc. are various measures of ageing. Jiang et al. (2003)
claim that the representation of ageing of a system by failure rate is qualitative. They developed a new notion, called ageing
intensity (AI), to quantitatively evaluate the ageing property of a system, denoted by L
X
(t ), of a random variable X , defined
as the ratio of the instantaneous failure rate r
X
(t ) to the failure rate average H
X
(t ) =
t
0
r
X
(u)du
/t , for t > 0, i.e.,
L
X
(t ) =
r
X
(t )
H
X
(t )
, where defined,
=
−tf
X
(t )
¯
F
X
(t ) ln
¯
F
X
(t )
,
where f
X
(·) and
¯
F
X
(·) are the probability density function and the survival function of the random variable X respectively.
Let the random variable X have support S
X
= (l
X
, u
X
), where u
X
may be infinity. We define
L
X
(t ) =
0, if t < l
X
r
X
(t )
H
X
(t )
, if l
X
≤ t ≤ u
X
∞, if t > u
X
.
∗
Corresponding author.
E-mail addresses: asoknanda@yahoo.com, asok@iiserkol.ac.in (A.K. Nanda).
0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
doi:10.1016/j.spl.2013.01.016