Characterization and preservations of the variance inactivity time ordering and the increasing variance inactivity time class

Journal of Advanced Research (2012) 3, 29–34

Cairo University

Journal of Advanced Research

ORIGINAL ARTICLE

Characterization and preservations of the variance
inactivity time ordering and the increasing variance
inactivity time class
Mervat Mahdy

*

Department of Statistics, Mathematics and Insurance, College of Commerce, Benha University, Egypt
Received 25 December 2010; revised 2 February 2011; accepted 5 March 2011
Available online 16 April 2011

KEYWORDS
Conditional variance;
Increasing variance inactivity
time;
Mixing;
Convolution;
Formation of coherent
system;
Erlang distribution

Abstract If the random variable X denotes the lifetime of a unit, then the random variable
XðtÞ ¼ ½t À XjX 6 tŠ for a fixed t > 0 is known as the inactivity time. In this paper, based on the

random variable X(t), a new class of life distributions, namely increasing variance inactivity time
(IVIT) and the concept of inactivity coefficient of variation (ICV), are introduced. The closure
properties of the IVIT class under some reliability operations, such as mixing, convolution and formation of coherent systems, are obtained.
ª 2011 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

Introduction
Let the random variable X denote the lifetime (X > 0, with
probability one) of a unit, having an absolutely continuous distribution function F, survival function F ¼ 1 À F and density
function f. Let the random variable XðtÞ ¼ ½t À XjX 6 tŠ denote
* Tel.: +20 120682460/+20 225077099; fax: +20 13 323 0860.
E-mail addresses: , mervat_em@
yahoo.com
2090-1232 ª 2011 Cairo University. Production and hosting by
Elsevier B.V. All rights reserved.
Peer review under responsibility of Cairo University.
doi:10.1016/j.jare.2011.03.001

Production and hosting by Elsevier

the time elapsed after failure till time t, given that the unit has
already failed at time t, for t > 0. The random variable X(t) is
known as the inactivity time of a unit at time t. Recently, the
random variable X(t) has received considerable attention in the
literature, see Ahmad and Kayid [1], Li and Xu [2], Lai and
Xie [3], Mahdy [4], and Nair and Sudheesh [5].
In the literature, the function r~F ðxÞ ¼ fðxÞ=FðxÞ is known as
the reversed (or retro or backward) hazard rate function (cf.
Shaked and Shanthikumar [6]). In the analysis of left-censored
data, the reversed hazard rate function plays the same role as
that of the hazard rate function in the analysis of right-censored data (cf. Anderson et al. [7]). The reversed hazard rate

ordering is related to the random variable X(t). Ahmad and
Kayid [1] characterized the decreasing reversed hazard rate
(DRHR) based on variability ordering of the inactivity time
of k-out-of-n system given that the time of the (n À k + 1)th
failure occurs at or sometimes before time t P 0.
In this paper, we focus our attention on nonparametric
classes of life distributions defined in terms of the variance


30

M. Mahdy

of X(t).These classes are the increasing variance inactivity time
(IVIT) and inactivity coefficient of variation time (ICV). Section ‘preliminaries’ contains definitions, notation and basic
properties used through the paper. In this section, we study
some properties of the IVIT class and the ICV class. The main
results and their proofs are provided in Section ‘preservation
properties’, where we establish closure properties of the classes
under relevant reliability operations such as mixing, convolution and formation of coherent systems; we show, for example,
that the class IVIT is closed under convolution, mixing and the
formation of coherent systems. The variance inactivity time of
parallel systems is provided in Section ‘variance inactivity time
of parallel systems’.

d 2
½r ðtފ ¼ ~
rðtÞ½m2F ðtÞ À r2F ðtފ:
dt F
The following definition is essential to our work:

Definition 2.1. A random variable X having distribution
function F has increasing variance inactivity time life, which
we denote as IVIT, if
1
FðtÞ

Z

0

Let X be a random variable with distribution
R 1 function F(t),
survival function F ¼ 1 À F, mean life l ¼ 0 FðuÞdu and variance r2 = Var(X). So the mean inactivity time (MIT), mF(t),
and variance inactivity time, r2F ðtÞ, respectively, can be defined
as follows:
mF ðtÞ ¼ Eðt À XjX 6 tÞ;
Rt
FðuÞdu
; 0 6 X 6 t; t P 0;
¼ 0
FðtÞ

ð2:1Þ

and
¼ Var½t À XjX 6 tŠ:

ð2:2Þ

The following definitions extend the increasing mean inactivity

time, IMIT, and IVIT classes into the orderings between
variables.
Let X and Y be two non-negative and absolutely continuous random variables, having distribution functions F and G,
reversed hazard rate functions ~
rF and r~G , the mean inactivity
time functions mF(t) and mG(t), and the variance inactivity
time functions r2F ðtÞ and r2G ðtÞ, respectively. The mean and
the variance inactivity time orderings can be defined as follows:
Definition 2.2. X is said to be smaller than or equal to Y in
mean inactivity time ordering (X 6mit Y) if
Rt

Note that
r2F ðtÞ

uðyÞdy 6 m2F ðtÞ;
0

Ry
where uðyÞ ¼ 0 FðxÞdx.Equivalently, X 2 IVIT if, and only if,
Z t
uðyÞdy 6 u2 ðtÞ:
FðtÞ

Preliminaries

r2F ðtÞ

t

0
2

2

¼ E½ðt À XÞ jX 6 tŠ À ½mF ðtފ :

FðuÞdu
P
FðtÞ

Rt
0

GðuÞdu
; for all t P 0:
GðtÞ

Clearly
r2F ðtÞ ¼ 2tmF ðtÞ À m2F ðtÞ À
Consider E½U2 jtŠ ¼
one has
r2F ðtÞ þ m2F ðtÞ ¼

R1

2
FðtÞ

2

FðtÞ

Z

t

xFðxÞdx:
0

0

u2 dF½ujtŠdt, using integration by parts

Z

t

Z

y

Definition 2.3. X is said to be smaller than or equal to Y in
variance inactivity time ordering (X 6vit Y) if

FðxÞdxdy;
0

0

so, Eq. (2.2) is equivalent to

Z tZ y
2
r2F ðtÞ ¼
FðxÞdxdy À m2F ðtÞ:
FðtÞ 0 0
Ry
Let uðyÞ ¼ 0 FðxÞdx, then
Z t
2
r2F ðtÞ þ m2F ðtÞ ¼
uðyÞdy;
FðtÞ 0

Rt Rx
0

uðtÞ
;
FðtÞ

Rt Rx
ð2:3Þ

Differentiating (2.2) with respect to t, we have
Rt
2½uðtÞFðtÞ À fðtÞ 0 uðyÞdyŠ
d 2
½rF ðtފ ¼
À 2mF ðtÞmnF ðtÞ:
dt

F2 ðtÞ
and using (2.3)–(2.5) in (2.6), we obtain that

R 0t R 0x
0

ð2:4Þ

also from (2.4), we get
m0F ðtÞ ¼ 1 À r~ðtÞmF ðtÞ:

0

FðuÞdudx
P
FðtÞ

Rt Rx
0

0

GðuÞdudx
; for all t P 0:
GðtÞ

It can be written as

and from (2.1), we get
mF ðtÞ ¼

It can be written as
Rt
FðuÞdu
is increasing in t P 0:
R 0t
GðuÞdu
0

ð2:5Þ

ð2:6Þ

0

FðuÞdudx
GðuÞdudx

is increasing in t P 0:

In probability theory and statistics, the coefficient of variation
(CV) is a normalized measure of dispersion of a probability
distribution. It is defined as the ratio of the standard deviation
r to the mean l:
r
CV ¼ :
l
This is only defined for a non-zero mean, and it is most useful
for variables that are always positive. It is also known as unitized risk, also it is used in some applied probability fields such
as renewal theory, queueing theory, and reliability theory.

Now, we can define the coefficient of variation of the random variable X(t) as follows:


Characterization of variance inactivity time
cF ðtÞ ¼

31

rF ðtÞ
:
mF ðtÞ

By using Bool’s rule (Mathews and Fink [8]), we can show
that

The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to
the Exponential and Gamma distributions. The distribution
is used in the field of stochastic processes. The probability density function of the Erlang distribution is
kk xkÀ1 expðÀkxÞ
fðx; k; kÞ ¼
;
ðk À 1Þ!

for x; k P 0:

kÀ1
X
ðkxÞn
expðÀkxÞ:
n!

n¼0

(ii) An approximation of the variance inactivity lifetime
function is given by:
P ðkÀiÞ iÀ2
t2 À 2t kÀ1
þ ki¼1 ðiÀ1Þ!
k Bi
k
r2F ðtÞ %
À m2F ðtÞ;
PkÀ1 ðktÞn
1 À n¼0 n! expðÀktÞ
where
 i
512 À1kt  t i 96 À1kt  t i 512 À3kt 3t
28
e 4
e 4
þ e 2
þ
þ eÀkt ti :
45
4
45
2
135
4
45

(iii) The inactivity coefficient of variation of the random variable X(t) is greater than 1.

Proof. When X has the Erlang distribution and using Eq.
(2.4), we can show that
Rt
0

1À
1À

PkÀ1 ðkxÞn

n¼0 n!
PkÀ1 ðktÞn
n¼0 n!

expðÀkxÞdx

0

;

expðÀktÞ
hP
i
k
iÀ2 iÀ1
kÀi
t À kÀ1
þ expðÀktÞ

t
i¼1 ðiÀ1Þ! k
k
¼
; t P 0;
P ðktÞn
1 À kÀ1
n¼0 n! expðÀktÞ

where
Z t
kÀ1
X
ðkxÞn
expðÀkxÞdx
1À
n!
0
n¼0
"
#
k
X
kÀ1
k À i iÀ2 iÀ1
þ expðÀktÞ
k t
:
¼tÀ
k

ði À 1Þ!
i¼1
Hence from Eq. (2.7) the result follows.

y

FðxÞdxdy % t2 À 2t
0

k
kÀ1 X
k À i iÀ2
þ
k Bi ;
k
ði À 1Þ!
i¼1

where
 i
512 À1kt  t i 96 À1kt  t i 512 À3kt 3t
28
þ e 2
þ
þ eÀkt ti :
e 4
e 4
45
4
45

2
135
4
45

From (2.2), (2.7), and (2.8), we get the complete proof of
(ii).
Also, by using (i) and (ii), we get the complete proof (iii).
A hyper-exponential distribution is a continuous distribution with the probability density function as follows:
fX ¼

Proposition 2.1. If X is a non-negative random variable having
the Erlang distribution with scale parameter k, and shape
parameter k. Then
(i) The mean inactivity lifetime function is given by:
hP
i
k ðkÀiÞ iÀ2 iÀ1
t À kÀ1
þ expðÀktÞ
t
i¼1 ðiÀ1Þ! k
k
;
ð2:8Þ
mF ðtÞ ¼
P ðktÞn
1 À kÀ1
n¼0 n! expðÀktÞ

mF ðtÞ ¼

Z

ð2:7Þ

Now, we can discuss the behavior of ICV in the Erlang
distribution.

Bi ¼

t

Bi ¼

where ‘‘exp’’ is the base of the natural logarithm and ‘‘!’’ is the
factorial function. The parameter k is called the shape parameter and the parameter k is called the rate (scale) parameter.
The cumulative distribution function of the Erlang distribution
can be expressed as
Fðx; k; kÞ ¼ 1 À

Z

n
X

fYi ðyÞpi ;

ð2:9Þ

i¼1

where Yi is an exponentially distributed random variable with
rate parameter ki , and pi is the probability that X will take on
the form of the exponential distribution with rate ki . Now, we
can study some of properties of hyper-exponential distributions in terms of the following propositions:
Proposition 2.2.
If X has a hyper-exponential distribution, then
(i) The mean inactivity time function is given by
o
Pn n
1
1
i¼1 pi t þ ki expðÀki tÞ À ki
Pn
;
mF ðtÞ ¼
i¼1 pi f1 À expðÀki tÞg

ð2:10Þ

(ii) The variance inactivity time function is given by
n
o
P
2
i tÞ
2 ni¼1 pi t2 À kti À expðÀk
þ k12
k2i

2
i
Pn
rF ðtÞ ¼
i¼1 pi f1 À expðÀki tÞg
o32
2Pn n
1
1
i¼1 pi t þ ki expðÀki tÞ À ki
5;
À 4 Pn
i¼1 pi f1 À expðÀki tÞg
(iii) The coefficient of variation of X(t) is less than 1.

Proof. By the expression followed from (2.4) for the distribution function given in (2.9), (i) is satisfied. Also, by using (2.2)
and (2.10) we get the complete the proof of (ii). It is easy to
check that maximum value of the inactivity coefficient of variation of the random variable X(t) is less than one; this is the
complete proof of (iii). h
Preservation properties
This section will develop some preservation of VIT order and
IVIT.
Theorem 3.1. X is IVIT if and only if X6vit X þ Y for any Y
independent of X.


32

M. Mahdy

Proof. Necessity: If r2F ðtÞ is increasing in t P 0, then by Fubini’s theorem, we have for any t P 0,
Rt Ry Rx
2 0 0 0 Fðx À uÞdGðuÞdxdy
2
2
rðXþYÞ ðtÞ þ mðXþYÞ ðtÞ ¼
;
Rt
Fðt À uÞdGðuÞ
0
R t R tÀu R yÀu
2
FðxÞdxdydGðuÞ
;
¼ 0 0Rt 0
Fðt À uÞdGðuÞ
0
Rt
Fðt À uÞ½r2ðXÞ ðt À uÞ þ m2ðXÞ ðt À uފdGðuÞ
¼ 0
Rt
Fðt À uÞdGðuÞ
0
6 r2ðXÞ ðtÞ þ m2ðXÞ ðtÞ:
By Proposition 2.3 in Li and Xu [2] we get that

Z

t

Z

0

Gn ðtÞFn ðuÞdudx À

0

0

t

Z

x

Fn ðtÞGn ðuÞdudx P 0:
0

Since, for any t P 0,
"

n
X
kðuÞ ¼
½GnÀi ðtÞFnÀi ðuފ½FiÀ1 ðtÞGiÀ1 ðuފ

#À1
;

i¼1

is non-negative and decreasing in u P 0, by Theorem 3.1 of Li
and Xu [2] we have,
Z tZ x
½GðtÞFðuÞ À FðtÞGðuފdudx
0

m2ðXþYÞ ðtÞ 6 m2ðXÞ ðtÞ:

x

Z

0

¼

Z
0

Thus

t

Z

x

kðuÞ½Gn ðtÞFn ðuÞ À Fn ðtÞGn ðuފdudx P 0;

0

h

X 6vit X þ Y:

which states that X 6vit Y.

Theorem 3.2. Assume that / is strictly increasing and concave

Variance inactivity time of parallel systems

/ð0Þ ¼ 0:

We consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F. It is assumed that at time t the system
failed. Under these conditions, Asadi [9], Asadi and Bayramoglu [10] and Bairamov et al. [11] introduced MIT of the components of this system. Also, they mention some of its
properties such as recovered distribution function by application of MIT, and comparison between MITs of two parallel
systems.
On the basic of the structure of parallel systems, when a
component with lifetime Tr:n = r = 1, 2, . . ., n À 1 fails the
system is continuing to work until Tn:n fails. In fact, the system
can be considered as a black box in the sense that the exact
failure time of Tr:n is unknown. Motivated by this, we assume
that at time t the system is not working and in fact, it has failed
at time t or sometime before time t.
Let

If X 6vit Y then /ðXÞ 6vit uðYÞ.
Proof. Without loss of generality, assume that / is differentiable with derivative /n . Thus X 6vit Y implies that for any

t P 0,
Z

Z

/À1 ðtÞ

0

/À1 ðx:Þ




FðuÞ
GðuÞ
À
dudx P 0:
Fð/À1 ðtÞÞ Gð/À1 ðtÞÞ

0

Since /n ðtÞ is non-negative and decreasing, by Theorem 3.1 of
Li and Xu [2] it holds that
Z

Z

/À1 ðtÞ

0

/À1 ðx:Þ

/n ðtÞ



0


FðuÞ
GðuÞ
À
dudx
Fð/À1 ðtÞÞ Gð/À1 ðtÞÞ

P 0; for any t > 0:
Equivalently,
Z

Z

/À1 ðtÞ

0

ITr:n;ðtÞ ¼ ½t À Tr:n jTn:n 6 tŠ; t > 0; r ¼ 1; 2; . . . ; n:

/À1 ðxÞ

0

Z

/À1 ðtÞ

Z

n

/ ðtÞFðuÞ
dudx
Fð/À1 ðtÞÞ
/À1 ðxÞ

P
0

0

where ITr:n,(t) shows, in fact, the time that has passed from the
failure of the component with lifetime Tr:n in the system given
that the system has failed at or before time t. If we denote the
expectation of ITr:n,(t) by Mrn ðtÞ and variance of ITr:n,(t) by
Vrn ðtÞ, i.e.

/n ðtÞGðuÞ
dudx;
Gð/À1 ðtÞÞ

that is, for any t > 0,
Z

t

Z

0

x
0

Fð/À1 ðuÞÞ
dudx P
Fð/À1 ðtÞÞ

Z

t
0

Z
0

Mrn ðtÞ ¼ EðITr:n;ðtÞ Þ; t > 0; r ¼ 1; 2; . . . ; n;
x

Gð/À1 ðuÞÞ
dudx;

Gð/À1 ðtÞÞ

which shows for any t P 0 that /ðXÞ 6vit uðYÞ.

and
h

Theorem 3.3. Let X1, . . ., Xn and Y1, . . ., Yn be independent
and identically distributed (i.i.d) copies of X and Y, respectively. If maxfX1 ; . . . ; Xn g 6vit maxfY1 ; . . . ; Yn g then X 6vit Y.
Proof. maxfX1 ; . . . ; Xn g6vit maxfY1 ; . . . ; Yn g implies that
Rt Rx
0

0

Fn ðuÞdudx
P
Fn ðtÞ

that is

Rt Rx
0

0

Gn ðuÞdudx
for any t > 0;
Gn ðtÞ

Vrn ðtÞ ¼ VarðITr:n;ðtÞ Þ; t > 0; r ¼ 1; 2; . . . ; n:
Then Mrn ðtÞ measures the MIT from the failure of the component with lifetime Tr:n given that the system has a lifetime less
than or equal to t. Also, Vrn ðtÞ measures the VIT from the failure of the component with lifetime Tr:n given that the system
has a lifetime less than or equal to t.
Let a parallel system with n non-negative independent components having a common continuous distribution function F
with left extremity a = inf {t:F(t) > 0} and right extremity
b = sup {t:F(t) < 1}. In the following, we derive the distribution of ITr:n,(t). Let RðxjtÞ denote the reliability function of
ITr:n,(t), for x < t and x, t 2 (a, b). Then


Characterization of variance inactivity time
RðxjtÞ ¼ PrðITr:n;ðtÞ P xÞ;
 
Pn n i
F ðt À xÞðFðtÞ À Fðt À xÞÞnÀi
i¼r
i
;
ð4:1Þ
¼
Fn ðtÞ


n  X
X
n À i Fðt À xÞ iþj
n nÀi
¼
ðÀ1Þj ð
Þ

; for r ¼ 1; .. .; n:
FðtÞ
j
i j¼0
i¼r
Using the survival function given in (4.1), Asadi [9] obtains the
MIT of Tk:k as follows:
Mk ðtÞ ¼ E½t À Tk:k P xjTk:k 6 tŠ; k ¼ 1; 2; . . . ;
Rt k
F ðuÞdu
:
¼ 0 k
F ðtÞ
Now, let us define the second non-central moment of the parallel system lifetime, which is denoted by Sk(t) as follows:
Rt Rx
2 0 0 Fk ðuÞdudx
;
Sk ðtÞ ¼
Fk ðtÞ
Rt k
2 uF ðuÞdu
;
ð4:2Þ
¼ 2tMk ðtÞ À 0 k
F ðtÞ
¼ 2tMk ðtÞ À Uk ðtÞ;

33
RðxjtÞ ¼

iþj

 t
n  X
X
n nÀi
nÀi
e À ex
ðÀ1Þj
;
et À 1
i j¼0
j
i¼r

x < t; t > 0:
By Asadi [4], we get that:
Rt
2 uFiþj ðuÞdu
;
Siþj ðtÞ ¼ 2tMiþj ðtÞ À 0 iþj
F ðtÞ


iþj
Rt
iþj 1
P
2t ðÀ1Þk
ð1 À eÀkt Þ À 2 0 uFiþj ðuÞdu

k
k
k¼0
¼
:
ð1 À eÀt Þiþj
Since
Z

t

uFiþj ðuÞdu ¼

0



iþj
X
iþj 1
ðÀ1Þk
½1 À eÀkt ðtk þ 1ފ;
2
k
k
k¼0

so that
Piþj
Siþj ðtÞ ¼

where
2

Rt


iþj 1
ð1 À eÀkt Þ
k
k
À Uiþj ðtÞ;
ð1 À eÀt Þiþj

k
k¼0 ðÀ1Þ



whereas

uFk ðuÞdu
:
Fk ðtÞ

Also, by using (4.2), we get the VIT of Tk:k as follows:



P

k iþj
1
2 iþj
½1 À eÀkt ðtk þ 1ފ
k¼0 ðÀ1Þ
k2
k
:
Uiþj ðtÞ ¼
ð1 À eÀt Þiþj

Vk ðtÞ ¼ Var½t À Tk:k P xjTk:k 6 tŠ;

So, we can show that for r = 1, . . ., n,

Uk ðtÞ ¼

0

k ¼ 1; 2; . . . ;

¼ 2tMk ðtÞ À M2k ðtÞ À Uk ðtÞ:
Furthermore, Asadi [9] mentioned that MIT of Tr:n is
Z t
RðxjtÞdx;
Mrn ðtÞ ¼
0




n  X
X
n nÀi
nÀi
Miþj ðtÞ:
ðÀ1Þj
¼
i j¼0
j
i¼r
Now, we can obtain the VIT of Tr:n as follows:
Z t
Vrn ðtÞ ¼ 2 ðt À xÞRðxjtÞdx À ½Mrn ðtފ2 :
0



Piþj
k iþj 1




ðÀ1Þ
½1 À eÀkt Š
n
nÀi
k¼0
X
n X

k k
j nÀi
r
Vn ðtÞ ¼ 2t
ðÀ1Þ
i j¼0
j
ð1 À eÀt Þiþj
i¼r


n  X
X
n nÀi
nÀi
À2
ðÀ1Þj
i j¼0
j
i¼r


Piþj
i
þ
j
k
1
Àkt
ðtk þ 1ފ

2 ½1 À e
k¼0 ðÀ1Þ
k k
Â
iþj
Àt
ð1 À e Þ


2
32
P
iþj 1
 X

 iþj
ðÀ1Þk
½1 À eÀkt Š
n
nÀi
k¼0
k
X
6
7
n
nÀi
k
7:
À6

ðÀ1Þj
4
5
i j¼0
j
ð1 À eÀt Þiþj
i¼r

But note that
Z t
Urn ðtÞ ¼ 2
xRðxjtÞdx;
0



n  X
X
n nÀi
j nÀi
ðÀ1Þ
¼
Uiþj ðtÞ;
i j¼0
j
i¼r
so we can define the second non-central moment of Tr:n as
follows:

Acknowledgement

The author is grateful to Dr. Ibrahim Ahmad (Professor and
Head, Department of Statistics, Oklahoma State University,
USA) for reading preliminary versions of this paper and making many useful comments.

Srn ðtÞ ¼ 2tMrn ðtÞ À Urn ðtÞ:
Consequently
Vrn ðtÞ ¼ 2tMrn ðtÞ À Urn ðtÞ À ½Mrn ðtފ2 :
Example. Let T0i s, i ¼ 1; . . . ; n, n P 1, be an independent
exponential with mean 1. Then

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