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APPLIED STOCHASTIC MODELS AND DATA ANALYSIS,VOL.11,167-180 (1995)
A DISCRETE-TIME MODEL OF FAILURE AND REPAIR
JOS6 M.ROCHA-MAR=
Escuela Superior de Ffsica y Matenutticas,Instituto PolitCcnicoNacional, Edificio 9. Unidad ‘Adolfo Mpez Mateos’
Zacatenco, 07300 MCxico, D.F.. MCxico
.
AND
MOSHE SHAKED
Department of Mathematics, University of Arizona, Tucson,Arizona 85721, U.S.A.
SUMMARY
In this paper the authors introduce and study a model of failures and repairs of units with discrete
lifetimes. They suppose that a unit has a sequence of tasks to perform and that its lifetime is measured by
the number of tasks performed before its final, fatal failure. Upon a failure the unit may be repaired (with
some probability) and then it may attempt again to perform the current task. The unit dies when (with
some probability) a repair cannot be completed. We derive some stochastic comparisons of pairs of such
models. The stochastic comparisons are then applied for obtaining results regarding the inheritance of
several aging properties by the repaired unit. Various examples illustrate the applicability of the model.
Some variants of the model of this paper can be viewed as discrete analogues of the notion of imperfect
repair.
1. INTRODUCTION AND SUMMARY
In this paper we introduce and study a model of failures and repairs of a unit that has a
sequence of tasks to perform, numbered 0, 1,2, ... We assume that task m is executed
successfully with some probability that we denote by 1 - A(m). With probability A(m) the unit
fails during the execution of task m. Upon failure of the unit, an attempt is made to repair it. If
the repair is successful then the unit restarts the execution of task m (the execution of task m
now, again, may or may not be successful, with probabilities 1 - A(m) and A(m), respectively,
as above). If the repair is not successful then the failure is considered to be fatal and the
lifetime of the item, Mu,is defined then to be Mu= m. We assume that the probability of a
successful repair depends on the task m and on the number of times j that the repair has already
been successful while executing task m. Thus we denote by 1 - q(rn,j) the probability of a
successful repair at the jth trial, while executing task m. The probability of an unsuccessful
repair at the jth trial, while executing task m,is then q ( m ,j).This stochastic model is illustrated
in Figure 1. Some settings, in which such a model arises naturally, are described below.
(i) Space exploration. A probe (such as Pioneer 10 or Voyager 1 and 2) is sent on a journey
in the solar system (and perhaps also beyond it). The probe has a predetermined list of ordered
objectives to perform during its travel. The list of objectives may consist of a number of ‘close
up’ photographs of some planets, of measurements of the effect of the Sun in various regions
CCC 8755-OO24/95 /020 167- 14
0 1995 by John Wiley & Sons, Ltd.
Received 27 May I994
Revised I6 February 1995
168
J. M. ROCHA-MARTINEZAND M. SHAKED
Task m
Set
j = j + ~.
Unit
attempts
to execute
taskm
successfully
completes
execution
of task m
Unit
proceeds
to execute
taskrn+l
during execution
of task m
Figure 1. Illustration of the stochastic model
of the solar system, etc. The order of the objectives in the list depends, of course, on the
particular route of the probe. The probe may have some failures during its mission, but, with
some probability, it is possible to repair the probe by radio communications. However, it is
conceivable to expect that after a while the probe will experience a fatal failure which will be
beyond repair. The probe is useless afterwards. Here M uis the number of the last objective to be
executed.
(ii) A server in a repairfaciliry. A repair facility has a number of items to repair. In order to
do it, a server (or a special tool) is brought to the facility. The repair tasks are ordered according
to their importance (or according to their difficulty) and the server is used to repair the items in
the list according to their order. The server wears out as it performs its objectives. Upon a
failure, the server may sometimes be repaired (with some probability), but after a while a fatal
failure, beyond repair, occurs. The server then become useless. Here M uis the number of the
last repair objective that has been performed.
(ii) Educafional foys. A mini electronic lab (or a construction set) consists of some components,
one of which is fragile (or relatively weak). A booklet describes a list of experiments (or projects)
that can be performed using the items in the set. A child performs the experiments in order. The
fragde item is used in many of the experiments, and it breaks down every once in a while.
Sometimes some ‘home remedies’ may be used in order to repair the failed fragde item. But after a
while a serious failure occurs which is beyond repair. The fragile item (and the whole set) is then
useless. Here M uis the number of the last experiment which has been executed.
(iv) Rescue missions. A vehicle is to be used for a sequence of complicated rescue missions
(say, beyond enemy lines). The missions are ordered according to some criterion. The vehicle
performs the missions in order. Sometimes, upon failure, the vehicle can be repaired. However,
after a while a fatal failure, beyond repair, occurs. Here M uis the number of the last mission to
be performed.
In order to obtain some technical insight into the stochastic model described above, we need
the following preliminaries. Let M be a discrete random variable that takes on values in
DISCRETE-TIME MODEL OF FAILURE/REPAIR
N
+
E
169
( 0 , l ,2, ...). The failure rate function of M is defined as
A(m) = p ( M = m IM
Pm
),
m EN+
(1.1)
provided P ( M 2 m ) > 0. Otherwise, for notational convenience, we define A(m) = 1. It is well
known (see, e.g., Shaked et aZ.’) that if M is a proper random variable then
OcA(m)cl,
mEN,
(1.2)
and
(1.3)
m=O
In fact, every function A ( - ) that satisfies (1.2) and (1.3) uniquely determines a proper discrete
random variable M whose failure rate function is A(.) (again, see, e.g., Shaked et ul.’).
If A(.) is such that (1.2) holds, but instead of (1.3) we have
(1.3’)
m=O
then A ( - ) still uniquely determines a discrete random variable M with failure rate function A ( - ) ,
but this random variable is such that lim,,,+ P ( M 2 m }>O. We will call such a random variable
improper and say that it can take on the value 00 with probability
P ( M = 00) = lim,+, P ( M 2 m ) . For notational convenience, we also set A(=) = 1. In both cases,
the survival function of M and its discrete density function can be expressed by means of L as
follows:
where l’lf:; x , is taken to be equal to 1 for every (I E M +,
In order to see the relationship between the random variable M (described above) and the
random variable M u (described in the first paragraph of this section) notice the following. If
the unit (described in the first paragraph of this section) were not repaired at all, then, upon
a failure to execute task m we would set its lifetime to be m , and assume that the unit is
worthless for the execution of any further tasks. It is seen then that the failure rate function
associated with the number of tasks executed by the unit would be A(.) described in (1.1).
From (1.4)-or (1.5)-it follows that in such a case the number of tasks executed by the
unit has the same distribution as M. In general, however, M u and M have different
distributions. The distribution of M udepends on the distribution of M (through A ( . ) ) , and it
depends also on the probabilities q ( m , j ) ’ s . Below we call M the original lifetime of the
unit, and we call M uthe lifetime resulting from M (by the incorporation of the possibility
of repairs).
For a fixed m , the q ( m , j ) ’ s (described in the first paragraph of this section) can be
interpreted as the failure rates of a discrete random variable R,, i.e. q ( m , j )= P ( R , = j l R , a j } ,
j = 0, 1,2, ... We will call R, the allowed number of repairs at task m. Physically, A(m)
indicates the level of difficulty that the unit encounters while attempting to execute task m, and
170
J. M.ROCHA-MARTINEZ AND M.SHAKED
q(m,j) indicates the level of deterioration of the unit after it had already failed to complete
task m for j consecutive times.
Remark
Note that if
A(.) = l
and
q(rn,j)=Oforalljaj,, forsomejoEN+
(1.6)
then the unit may stay forever inside a loop of failures, each followed by a successful repair,
while trying to perform task m. In order to avoid this situation, we will always exclude at least
one of the two conditions described in (1.6).
Formally, the lifetime Muis defined as the lifetime with the discrete hazard rate function AJ.)
given by
If, for all m E N+, R, is a proper random variable with support in N+,then it is clear from (1.7)
that
=A(m)@Rm(A(m)),
(1 -8)
+
where OR,(s)= E(sRm), s E [0,1],is the probability generating function of R,.
Note that it is possible that Muis an improper random variable even if M and the R,’s are
proper. That is, it is possible that C”,o A,(m) < m even if
A(m)= 00 and
q(rn,j)= 00
for all m E N + . For example, if A(m)=l/(m+l), m E N + , and, for m E N + , we have
q(rn,j)=l/(m+l), / ’ E N + , then @Rm(s)=l/(m+l-sm), and, by (1.8), we see that
A,(rn) = l/[(m + - m].It is easy to verify that C;=l A,(m) <-.
In Section 2, two results about stochastic comparisons of pairs of such models are stated and
proved. In Section 3, we obtain some closure results connecting the aging properties of M and
Mu.The aging notions that we consider are the discrete IFR (increasing failure rate), DFR
(decreasing failure rate), IFRA (increasing failure rate average), DFRA (decreasing failure rate
average), SSLSF (star shaped log survival function) and ASSLSF (anti star shaped log survival
function). The exact definitions of these aging notions will be given in Section 3; they can also
be found in the work of Shaked et al.’ and Ross et u I . , ~ where their properties and usefulness
are studied.
Section 4 consists of some illustrative examples. In Section 5, the relationship is studied
between the imperfect repair model of Shaked et al.’ and the general stochastic model studied
in this paper.
Throughout this paper ‘increasing’ means ‘non-decreasing’ and ‘decreasing’ means ‘nonincreasing’.
z;=o
xTs0
2. STOCHASTIC COMPARISONS
The distribution of Mudepends on the distributions of M and of the R,’s, or, equivalently, on
the univariate function A ( - ) and on the bivariate function q ( . , -). Thus these functions can be
called the parameters of Mu.Below we sometimes denote the dependence of Muon these
functions by writing M umore explicitly as Mu(A,4).
DISCRETE-TIME MODEL OF FAILURE/REPAIR
171
Before we can formally state the results of this section, we need the following definitions.
Let X and Y be two discrete random variables (proper or improper) with failure rates functions
p ( . ) and p(.), respectively. We say that X is smaller than Y in the sense ofthe hazard rate
ordering, denoted by X d h Y, if p(m)2 p ( m ) for all m E N+.We say that Xis smaller than Y in
the sense of usual stochastic ordering, denoted by XC, Y, if P ( X a m }c P( Y a m ) for all
m E N+.The following implication is well known:
Y*x*& Y
(2.1)
Below, we sometimes abuse notation and write p d h p or p S a p instead of writing X C, Y or
Xd,Y.
xdh
Counterexample 2.1
One may be tempted to believe that if A d, A‘ then M,(A,q ) caM,(A’, q). The following
counterexample shows that this is not true in general. Let the R,’s be identically distributed
such that P ( R , = l } = l , i.e. q ( m , 0 ) = 0 and q ( m , j ) = l f o r j a l . IfA,(.) denotes the failure
rate function of Mu(&q ) and A:(.) denotes the failure rate function of M,,(A’, q ) , then it
is seen from (1.7) that A,(m)= ( A ( r n ) ) 2 and A:(rn)= (A’(m))’ for r n E N,. Now, let
A(0)=0.6,1(1)=0.4, and A(rn)=l, rn22. Also, let A’(O)=0-2, A’(l)=0.7, and A’(rn)=l,
m 5 2. It is easy to see that A d,,A’. But p(M,(A, q ) 2 2 ) =(1- A,(O))(l - A,(l)) =
(1 - (A(0))2)(1-(A(l))’) = 0.5376>0.4896 = (1 - A:(O))(l- G(1)) = P(Mu(A’, q ) b 2). Therefore [email protected],q ) #aMJA’, 4).
However, we have the following result. Intuitively, it says that if two systems have the same
number of allowed repairs at each task rn, but the level of difficulty of each task for the first
system is higher than the corresponding level of difficulty for the second system, then the
number of tasks executed until a fatal failure in the first system is smaller in a stochastic sense
(in fact, in the hazard rate ordering) than the number of tasks executed until a fatal failure in
the second system.
Theorem 2.2
Suppose that the R,’s are proper random variables. Using the notation described above, the
following implication holds:
A < h X * M u ( A s 4)shMu(A’, 4)
In particular, from (2.1), we have that A ehA’*
Mu(&q ) d,M,,(A’, q).
Proof
The function SH [email protected],Js)
is increasing in s 2 0 for each m E N+.Thus, from (1.8) it is seen
that if A s h A’ then A,(m) 2 A:(rn) for all rn E N, where A,(-) denotes the failure rate function
of M,(A, q ) , and A’,(.) denotes the failure rate of Mu(A’, q). That is, M,(A, q ) shM,(A’, 4). 0
If two systems have the same level of difficulty for each task, but the first system has a
stochastically smaller number of allowed repairs, in each task, than the second system, then one
would expect the number of tasks executed until a fatal failure of the first system to be
stochastically smaller (in some sense) than the number of tasks executed until a fatal failure of
the second system. This is indeed the case, as shown in the next result. Below, q ( m , .) and
4(m, denote the hazard rate functions of two different sets of allowed numbers of repairs
(R,)
and R,)
respectively.
a)
172
J. M.ROCHA-MARTINEZ AND M.SHAKED
Theorem 2.3
Suppose that the R,’s are proper random variables. Using the notation described above, the
following implication holds:
q ( m , -)cst4 ( m , for all m a M,(A, q ) bhMu(A,4 )
Proof
For every m E N, the function m A r ( m )is non-negative and decreasing in r E N+.It follows
implies that ~ ( ( ~ ( n ) ) ~Pm ~) ( ( n ( n ) ) R - ) for all n E N+. Therefore
that R , s * R ,
A(m)@,(A(m)) 2 A(rn)@Rm(A(m))and so, by (1.8), one obtains A,(m) a & ( m ) , m E N+,
where A,(.) denotes the failure rate function of M,(A,q) and I,(.) denotes the failure rate
0
function of M,(A, 4).
3. AGING PROPERTIES
In this section we derive some results regarding the inheritance, by Mu,of several aging
properties of M. The properties that we study are defined as follows. Let A4 be a discrete
random variable that takes on values in N+and let A(.) be its hazard rate function. We say that
M is IFR (DFR) if A(m) d [a]A(m’) whenever m’ m 2 0. Also, M is said to be IFRA
(DFRA) if n2-l
A ( k ) S [a]A(m), m a 1, and it is said to be SSLSF (ASSLSF) if
(n:=;’(lA(k)))”” 2 [dl 1 - A(m), m P 1. Note that the IFRA (DFRA) condition corresponds
to the requirement that the arithmetic average of the quantities 1 - A(k), k = 1,2, ..., m, is
decreasing (increasing) in m. The SSLSF (ASSLSF) condition corresponds to the requirement
that the geometric average of these quantities is decreasing (increasing) in m. Ross et a1.’
showed that some discrete random variables of importance in reliability theory are SSLSF. It is
known, and also it is not hard to prove, that IFR*SSLSF+IFRA
and that
DFR DFR4 3 ASSLSF.
It is easy to see that, in general, if M belongs to one of these classes then Muneed not belong
to same class. However we have the following results.
c;=i1
Theorem 3.1
Assume that the R,’s are proper discrete random variables such that R , aSt[dsl,d s t J R , + , ,
mEN,.Then
M is IFR (DFR, DFRA)
Muis IFR (DFR, DFRA)
Proof of the IFR part
We assume that A(m) S A(m + l), m E N+. Denote X ( m ) = A(m + l), m E N+. Then
A(m) s A’(m), m E N+,i.e. A ahx. It follows from Theorem 2.2 that
M u ( A s 4 ) PhMu(A‘,
4)
(3.1)
where q ( m , .) is the hazard rate function of R,, m E N +.
We also assume that R , 2 , t R , + 1 , m E N , . Denote R , = R , + l , m E N , . Then R,P,,R,,
m E N,. Let 4(m, be the hazard rate function of R,, i.e. 4 ( m , = q ( m + 1, m E N,. Then
q(m, .) @(m,.), m E N+.Therefore, by Theorem 2.3,
7
)
a )
a),
DISCRETE-TIME MODEL OF FAILURE/RJPAIR
173
From (3.1) and (3.2) we obtain
9) .hMu(A',
g)
(3.3)
The hazard rate of M,(A, q ) at task m is A,(m), m E N,, as given in (1.7). Also, from (1.7)it is
seen that the hazard rate of Mu(A', 4) at task m is A,(m + l), m E N,. Therefore, from (3.3) we
obtain that A,(m) C A,(m + l), m E N+,i.e. M uis IFR.The proof for the DFR part is similar.
Mu(A,
Proof of the DFRA part
The result follows from the following chain of equalities and inequalities:
1
-
In-'
2
A,(k) = 1
k=O
E[(A(k)p'
'1
k=O
m- 1
P
1
E[(A(k)f"+']
k=O
3 E [ [ L Zk =AO ( k ) ] " - + ' /
= A&),
m P
1
where the first and the last equalities follow from (1.8), the first inequality follows from
RkGaR,, the second inequality follows from the fact that for each r € N + one has
m-' ZL;'(A(k))'+I P ( m - l C,";' A(k))"',
and the third inequality follows from the
0
assumption that M is DFRA.
The conclusion of Theorem 3.1 is still true even if the R,'s are not proper, provided that
R , a h [ C , , C,] R,,], m E N+.This can be proved using (1.7) and (2.1), the latter in the form of
inequalities of algebraic expressions involving hazard rates-see the work of Rocha-Martinez3
for details.
In general, even when the R,'s are identically distributed, if M is SSLSF, IFRA or ASSLSF,
respectively, then Mu need not be SSLSF, IFRA or ASSLSF, respectively. This is shown in
Counterexamples 3.3-3.5 below. However, if the probability generating functions of the R,'s
satisfy some logconcavity property (see (3.4)) then Muinherits the ASSLSF and the SSLSF
properties from M. Although condition (3.4) looks very technical and without an intuitive
meaning, it is satisfied by most of the commonly used distributions. This is shown in the
examples discussed in Section 4.
Theorem 3.2
Assume that the R,'s are proper discrete random variables such that R , , , c , , [ ~ ~ ] R , + ~ ,
m E N+.If the probability generating function a,,,
of R , has the property that
g,(x)
log[l - (1 - ex)@,(l
-ex)]
is concave (convex) on (--, 01
then
M is ASSLSF (SSLSF) =$ M uis ASSLSF (SSLSF)
(3.4)
174
J. M.ROCHA-MARTINEZAND M.SHAKED
Proof
We only give the proof for the ASSLSF case. The proof for the SSLSF case is similar. Let
A(.) be the hazard rate function of M and let A,(.) be the hazard rate function of Mu.Fix an
m E N, and denote a ( m ) = 1- (nkmom-'(l- 1(k)))llm. Note that
grn(lOg(1- Y)) =lOg(l- y @ m ( y ) ) ,
0cy < 1
Therefore
fi
(...
log
c
-!-c
m- 1
(1 - A&)
= 2-
log(1 - iZ(k)@,(A(k)))
k=O
m- 1
c
log(1 - I(k)@,(L(k)))
k=O
= gm(lOg(1 - d m ) ) )
s gmOog(1 - 4m)))
= lOg(1 - A(m)@m(A(m)))
Am)
= log(1 where the first and the last equalities follow from (1.8), the first inequality follows from
RksaR,, the second inequality follows from the concavity of g,, and the third inequality
follows from the increasingness of gm(.) and from the assumption that M is ASSLSF. Thus we
have shown that
A , ( k ) ) } ' l " ' ~1- A,(m), m E N, i.e. that Muis ASSLSF.
0
In the next three counterexamples, we show some instances in which M udoes not inherit the
aging properties of M.
(n:;'(l-
Counterexample 3.3: M uis not SSLSF
Let M have the hazard rate function given by A(0)= 0.6, A(1) = 0.9 and 1(m) = 0.8, m 2. It
is easy to verify that M is SSLSF. Let the R,'s be identically distributed with a common failure
rate function given by q(m, j ) = 1/2, j E N, m E N, (i.e. the R,'s are geometric random
variables). From (1.8) it is seen, after some computation, that
Therefore (1 - A,(O))(l
- Au(l)) = 8/77 # 1/9= (1 - A,(2))*. Thus, M uis not SSLSF.
Counterexample 3.4: M uis not IFRA
Let M have the hazard rate function given by A (0) = 0.1, A(1) = 0.9 and 1(m)= 0.5, rn 3 2. It
is easy to verify that M is IFRA.Let the R,'s be identically distributed with a common failure
DISCRETE-TIME MODEL OF FAIL,URE/REPAIR
175
,
From (1.8)
rate function given by q(rn,0)=1/2, rnEN,, and q ( m , j ) = l , j ~ l rnEN,.
it is seen that Au(m)=4A(rn)+f(A(m))2.Therefore f(A,(O)+ Au(l))=91/200+257/600=
(A,(O) + A,(l) + Au(2)).Thus, Muis not IFRA.
Counterexample 3.5: M uis not ASSLSF
Let M have the hazard rate function given by A(0)=0.03, A(l)=O-Ol and
A(m)= 1 - [ (1 - A(O))( 1 - A( l))] ' I 2 = 1- [ (0.97)(O.99)]'I2, m 3 2. It is easy to verify that M is
ASSLSF. Let the R,'s be identically distributed with a common failure rate function given by
4(m,0)=0-8, mEN,, q(rn,l)=O, rnEN,, and q ( m , j ) = l , j.2, m E N , . From (1.8) it is
seen that A,(rn)=0.8A(m)+0-2(A(rn))3, rnE N,. It is easy to verify that
1-A,(m)=(1-A(rn))[l+0~2A(rn)+O~2(A(rn))2], rnEN,. In order to show that M u is not
ASSLSF it suffices to show that
[1+ (0*2)(0*03)
+ (0*2)(0*03)2][1
+ (0*2)(0-01)+ (0*2)(0*01)2]
> [1+ 0*21(2)+0.2(A(2))2]2
(3.6)
where A(2)= 1 - [(0.97)(0.99)]''2. Upon computing these expressions we get that
LHS(3.6)= 1.0082124836> 140820174845625= [1.0040925]2>RHS(3.6). Thus, Muis not
ASSLSF.
Before we close this section we return to condition (3.4). This condition holds for all the
common discrete distributions that are studied in Section 4 below. So one may conjecture that it
always holds. In the next counterexample it is shown that the function
g,(m) = log [ l - (1 - ex)O,(l-ex) J, x 8 0, may sometimesbe neither concave nor convex.
Counterexample 3.6
Consider the R,'s of Counterexample 3.5. The probability generating function of the R,'s is
O(s)=0.8+O-2s2, Is1 8 1. Thus, g,(x)=log [l -0.8(1-ex)-0.2(1-ex)3], xC0. In order to
show that g, is neither concave nor convex, it suffices to show that
is not monotone on ( - 0 0 , 01. But
1
gk(log(,))
=
$P > gL(log(3)) = ;;< gL(0) = ;
and this proves that g, is neither concave nor convex.
4. EXAMPLES
Example 4.1: A geometric model
Suppose that q(m, j ) = q, j E N,, m E N, i.e. q E (0,l) is independent of j (and also of m).
In other words, suppose that the R,'s are each a geometric random variable with a common
176
J. M.ROCHA-MARTINEZAND M.SHAKED
probability generating function
By (1.8) the failure rate function of M uin this model is
It is easy to see that in this example M uis proper whenever M is proper. We note that in this
example M uis IFR (DFR)if, and only if, M is IFR (DFR). Theorem 3.2 applies to this example
because the function
is concave on (-=, 01. Finally, it is not hard to find instances in which M is SSLSF (IFRA) but
M uis not SSLSF (IFRA);actually, Counterexample 3.3 is one of them.
Example 4.2: A one-repair model
Suppose that the R,'s are Bernoulli random variables, i.e. P ( R , = O ) = q and
P ( R , = l ) = l - q for all rnEN,, where q E ( 0 , l ) is fixed. Then q ( m , O ) = q and q ( m , j ) = l ,
j a 1 , rn E N,
and the common probability generating function of the R,'s is
@(s) = q + ( 1 - q ) s , s E [0,1]. By (1.8) the failure rate function of M u in this example is
A,(m) = qA(m)+ (1 - q)(A(m)>',rn E N,.
In this example, as in Example 4.1, it is easy to see that M uis proper if M is proper and, by
Theorem 3.1, M uis IFR (DFR,DFRA)if M is IFR (DFR, DFRA). Theorem 3.2 also applies to
this example. This can be verified by noting that the function g (x) = x + log [1 + (1 - q ) (1 - ex)]
is concave on (-=, 01.
Example 4.3: A Poisson model
Suppose that R , has a Poisson distribution with mean a(rn), rn E N, i.e. that R , has the
generating function @R,(s)=e-~('")('-", s E [0,1]. By (1.8), the failure rates of M uthen are
A,(rn) = A(rn)e-~(m)('-'(m)),
rn E N,.
Since Poisson random variables are stochastically ordered according to the values of their
means, it follows from Theorem 3.1 that if q(m) is decreasing (increasing, increasing) then
M is IFR (DFR,DFRA)
M uis IFR (DFR, DFRA)
In this example it is also true that M uinherits the ASSLSF property from M.This follows from
Theorem 3.2 once it is shown that the function g,(x)=log [ l - ( 1 -ex)e-q"] is concave on
(-=, 01 for all q. In order to do this, differentiate g to obtain
g&) = l + t l - a e " ,
-e+- - l . 1
ex ex
xtO
(4.1)
177
DISCRETE-TIME MODEL OF FAILURE/REPAIR
The numerator in the RHS(4.1) is clearly a decreasing function of x. By expending e w xas a
power series in qe" one obtains
Therefore, the denominator in the RHS(4.1) is a non-negative increasing function of x. It thus
follows that g ; ( x ) is decreasing in x and therefore g, in concave on (-w, 01.
It is not hard to find examples in which M of this example is SSLSF (IFRA) and M uis not
SSLSF (IFRA) even when q(m) is a constant independent of m. Since the q(m)'s are
arbitrary, it is possible to have situations in which M is proper, but M u is not proper. For
example, let
1
A(m) = m+2
and
m+2
q(m)= log(m + 2),
m+l
rn E N,
Then, the A(m)'s satisfy (1.3), but
That is, the A,(rn)'s satisfy (1.3').
As a final remark, let us mention that the model introduced in this paper can be generalized to
the multivariate case. This will be done elsewhere.
5 . AN APPLICATION: DISCRETE IMPERFECT REPAIR
Imperfect repair models for absolutely continuous distributions have been studied by Cleroux et
~ l . Berg
, ~ and Cleroux,' Brown and Proschan,6 Block et al.,' Bhattacharjee' and Shaked and
Shanthikumar' among others. However, continuous time is not always the best scale on which to
measure lifetimes. In some applications, the number of tasks completed by a unit before its
failure may be more important than the actual calendar age of the unit at failure. For example,
this is the case when the random variables of interest are the number of seasons a TV show is
run before being cancelled, the number of trips made by a vehicle before its failure or the
number of rounds fired by a gun before its failure.
Motivated by the importance of discrete time in some applications, Shaked et al.' introduced
a discrete time model of imperfect repair. In the following example it is shown that their model
is an important special case of the general stochastic model of the present paper.
Example 5.1: A one-attempt model
Suppose that the unit attempts to execute each task, m say, only once. If successful, then the
item proceeds to the next task. If it fails, then it is repaired (the repair is unsuccessful with
some probability, qm say, and then M u is set to be m), but upon a successful repair the unit
proceeds directly to execute task m + 1 (rather than attempting to execute task m again).
Alternatively, one can describe this model as one in which a successful repair guarantees that
the respective task is done. This model was studied by Shaked et al.' in an attempt to understand
the meaning of imperfect repair in the discrete time setting (in fact they only considered the
case in which qmdoes not depend on m).
178
J. M.ROCHA-MARTINEZ AND M.SHAKED
This model can be recast in the framework of Section 1 as follows. Suppose that R, has
the failure rates q ( m , O ) = q , ( q m E(0,l)) and q ( m , j ) = O , j a 1, m E N + . That is, suppose
that R, is an improper random variable such that P ( R , = 0 ) = q, and P(R, = -} = 1 - q,,
m E N+.Figure 2 describes the above model as a special case of the general model which we
study in this paper. It is easily seen, from (1.7), that in this example A,(m) = q,(m), m E N,.
Note that in this example we must have A(m)c 1 for all m E N, i.e. the support of M in N+
cannot have an upper bound. The intent of introducing a model of discrete imperfect repair,
which applies to discrete random variables with bounded support, was the original goal of
the present study.
It is easy to see that in this example Muis IFR (IFR4, DFR, DFRA) if M is IFR @I%,
DFRA) provided q, G [s, a,.] qm+', m E N, (see Shaked et al.' for a discussion also about
some NBU-new better than used-properties). Moreover, in this example M ualso inherits the
SSLSF aging property from M as the next proposition shows. Note that the statement of
Proposition 5.2 does not simply follow from Theorem 3.2 because in Theorem 3.2 the R,'s are
assumed to be proper, whereas here they are not.
Proposition 5.2
Assume in the model of Example 5.1 that qmd qm+l,m E N,. Then, if M is SSLSF then Mu
in SSLSF.
Proof
The proof is similar to the proof of Theorem 3.2. Fix an m E N and denote
+
(k=O
I
Define hm(x)=log ( 1 - qm(l- ex)), x E (-=, 0). Note that h, in an increasing convex function
and that
Rep.,,
I8
Repair 01
"nsuc<ar(ul-
( ~ . pI - P m )
(-1
E
M.N D= m)
atkrnpkd
Figure 2. An illustration of the one-attempt model
DISCREI'E-TIME MODEL OF FAILURE/REPB
179
Therefore
where the first inequality follows from qkrq,, the second inequality follows from the
convexity of h,, and the third inequality follows from the increasingness of h,(.) and from the
assumption that M is SSLSF. Thus we have shown that {l'lE;'(l- A,(k))}""'> 1 -A,(m),
m E N+, i.e. that Muis SSLSF.
Counterexample 5.3
Even in the case that q, = q, m E N+, in Example 5.1, if M is ASSLSF then Muneed not be
ASSLSF. This is shown in the following example. Let M have the failure rates A(O)=0.9,
A(l)=0.6, A(m)=0.8, m b 2 . Then M is ASSLSF. Let q=1/2, then A,,(m)=;A(m).
Thus(1- A,(o))(l- A,(l)) = 0.385 4 0.36 = (1 - A,(2))' and therefore Muis not ASSLSF.
ACKNOWLEDGEMENT
The authors thank two referees for helpful comments that improved the presentation of the
results of this paper. They are also grateful to Professor J. George Shanthikumar, with whom
they had some interesting discussions regarding the results of this paper.
The work reported in this paper was supported by NSF Grant DMS 9303891.
REFERENCES
1. M. Shaked, J. G. Shanthhmar and J. B. Valdez-Toms, 'Discrete hazard rate functions', Computers & Ops Res.,
22.391-402 (1995).
2. S. M. Ross, M. Shkhahani and G. Weiss, 'On the number of component failures in systems whose component
lives arc exchangeable',M a t h of Ops Res., 5,358-448 (1986).
3. J. M. Rocha-Ma~tinez,'Discrete imperfect repair', Doctoral dissertation, University of Arizona.
180
J. M. ROCHA-MARTINEZAND M. SHAKED
4. R. Cleroux, S. Dubuc and C. Tilquin, ‘The age replacement problem with minimal repair and random repair cost’,
OpsRes., 27,1158-1167 (1979).
5. M. Berg and R. Cleroux, ‘A marginal cost analysis for an age replacement policy with minimal repair’, INFOR,
20.258-263 11982).
6. M..Brown andF. goschan, ‘Imperfect repair’, J . Appl. Probab., 20,851-859 (1983).
7. H. W. Block, W. S. Borges and T. H. Savits, ‘Age-dependent minimal repair’, J . Appl. Probob., 22, 370-385
(1985).
8. M. C. Bhattacharjee, ‘New results for the Brown-Proschan model of imperfect repair, J. Srutist. Planning ond
lnjerence, 16,305-316 (1987).
9. M.Shaked and J. G. Shanthikumar, ‘Multivariate imperfect repair’, OpsRes., 34,437448 (1986).
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