1. Introduction
In recent years, researchers in reliability theory have shown intensified interest in the study of stochastic and reliability properties of technical systems. The 
-out-of-
system structure is a very popular type of redundancy in technical systems. A 
-out-of-
system is a system consisting of 
components (usually the same) and functions if and only if at least 
out of 
components are operating
. Hence, such system fails if 
or more of its components fail. Let 
denote the component lifetimes of the system and assume that 
represent the ordered lifetimes of the components. Then it is easy to argue that the lifetime of the system is 
where 
denotes the
, the order statistics corresponding to 
's,
. Under the assumption that 
's are continuous random variables, several authors have studied the residual lifetime and the mean residual lifetime (MRL) of the system under different conditions. Assuming that at time 
at least 
components of the system are working, the residual lifetime of the system can be defined as follows:
(1)
Among the researchers who investigated the reliability and aging properties of the conditional random variable
, under various conditions and for different values of 
and
, we can refer to Bairamov et al. [1], Asadi and Bairamov [2,3], Asadi and Goliforushani [4], Li and Zhao [5] and Zhang and Yang [6]. The extension to coherent systems has also been considered by several authors; see, among others, Li and Zhang [7], Navarro et al. [8], Zhang [9,10], Zhang and Li [11], Asadi and Kelkin Nama [12], and references therein.
Recently Mi [13] considered the situation in which the components of the system had discrete lifetimes and investigated some of aging properties of the system. The aim of the present paper is to study the MRL of 
-out-of-
system under discrete setting. For this purpose, we assume that 
are nonnegative integer valued random variables denoting the lifetimes of the components of an 
-out-of-
system. Furthermore, we assume that
, 
are independent and have a common probability mass function
and survival function
The hazard rate of the components, denoted by 
and
, is defined as follows:
One can easily show that the survival and probability mass functions can be recovered from the hazard rate, respectively, as follows:
The MRL function of the components, denoted by
, plays an important role in reliability engineering and survival analysis. Assuming each component of the system has survived up to times
, the MRL function 
of each component is defined as
It is not difficult to show that the survival function 
can be represented in terms of 
as below:
The reset of the paper is organized as follows :
We first assume that at time 
all components of the system are working and obtaining the functional form of the mean of
. This is in fact the MRL of the system, denoted by
, under the condition that all components of the system are operating at time
. It is shown that when the components of the system have geometric distribution, 
is free of time. Then, we prove that if the components of the system have increased failure rate, 
is a decreasing function of
. It is also shown that when the components of two independents are ordered in terms of hazard rate ordering, under the condition that all components of the two systems are alive, their corresponding MRLs are also ordered. The results are then extended to the case where at least 
components of the system are operating. In this case, we obtain the functional form of the MRL of the system, denoted by
. It is shown that 
can be represented as the mixture of
, where the mixing function is
We prove that in the case where the components of the system have increased hazard rate, then 
is decreasing in time. However, it is shown, using a counter example, that when the components of the system have decreased hazard rate, it is not necessarily true in general that 
is increasing in time.
The function
, mentioned above, has its own interesting interpretation. It shows the probability that there are exactly 
failed components in the system, 
, given that at least 
components are working at time
. Several properties of 
are also investigated.
2. The Mean Residual Life Function of System at the Component Level
In this section, we consider a 
-out-of-
system and assume that the components of the system have independent discrete lifetimes 
with common probability mass function 
and survival function
, where
. Let also 
be the order statistics corresponding to 
's. In what follows, first, we assume that at time
, all the components of the system are working, i.e.
. The residual lifetime of the system, under the condition that all components of the system are working at time
, is 
(see Asadi and Bairamoglu [3]).
Using the standard techniques, one can easily show that
(2)
Hence the MRL function of the system, denoted by
, can be obtained as follows
(3)
(4)
where
denotes the MRL function of a series system consisting of 
components,
.
Example 2.1 Let the components of the system have geometric distribution with probability mass function
and survival function
We have
Note that the MRL of a system having independent geometric components does not depend on
.
The distribution function of the order statistics 
can be represented in terms of incomplete beta function as follows (see David and Nagaraga [14]):
where
Hence the MRL function of the system can be represented as
(5)
This representation is useful to prove the following two theorems.
Theorem 2.2 If the components of the 
-out-of-
system have an increasing (decreasing) hazard rate, then 
is decreasing (increasing) in
.
Proof:
If 
denotes the hazard rate of the components, then 
is increasing (decreasing) if and only if for non-negative integer valued 
is decreasing (increasing) in
. Now the result follows easily by representation (5).
The following example gives an application of this theorem.
Example 2.3 Let the components of the system have discrete Weibull distribution with survival function
Then the MRL 
of the system is decreasing for 
and increasing for
.
Theorem 2.4 Let 
and 
be two 
-out-of-
systems with independent components. Let the components of 
and 
have the probability mass function
, and
, survival functions
and
; and hazard rates 
and
, respectively. If, for
, 
, then
, where 
and 
denote the mean residual life of 
and
, respectively.
Proof: Note that, for
, 
if and only if
The required result is immediate now from (5).
Khorashadizadeh et al. [15] studied discrete variance residual life function for one component.
Using the fact that
one can easily prove the following lemma.
Lemma 2.5
(6)
(7)
Using this, the variance of the residual life function of 
-out-of-
system under the condition that all components are working can be derived in terms of
.
Theorem 2.6 If
, the variance residual life function 
and mean residual life function 
are related as
wang#title3_4:spProof:
We have
Using Lemma 2.5, we get the required result.
Now, we study the MRL of 
-out-of-
system under the condition that at least 
components of the system are working. That is, we concentrate on
, 
First note that
where
and 
is a binomial random variable with parameters
.
(8)
Equation (8) shows that 
is a convex combination of
,
. Note that 
is given by (2).
Example 2.7 Let 
denote the lifetimes of 
independent components which are connected in a 
-out-of-
system. Let 
be distributed as discrete Weibull 
with
and
Then
and
Hence, the MRL 
is given by (8). Figures 1 and 2 show the graphs of 
in example 2.7 when
, 
, 
for different values of 
and
.
Remark 2.8 Let us consider again the condition random variable 
for which the survival function is given by (2). The representation (2) shows that 
is in fact the
, the order statistics
form of a distribution with survival function
. Hence using the result of David and Nagaraje [8]one can write
Hence
and
(9)
This indicates the MRL 
can be expressed in terms of simpler MRL 
which is in fact the MRL of series systems.
The following theorem gives bounds for
.
Theorem 2.9 It is always true that
Proof: The proof is similar to the proof of Theorem 2.3 of [4] and hence is omitted.
The next theorem proves that when the parent distribution has increased hazard rate, 
increases in terms of time.
Theorem 2.10 If 
is increasing in
, then 
is decreasing in
.
Proof: In order to prove the result, we need to show that, for 
and 
fixed,
We have, from (8), after some algebra
But the first term in the above equality is positive by Theorem 2.2. Hence we just need to prove that the second term in the above equality is positive. Assume that 
and note that 
is an increasing function of
. Then
After some algebraic manipulations, one can show that the numerator of the expression is equal to
(10)
It can be easily shown that for
, 
(see, [2,3]). On the other hand, as 
is an increasing function of
, we have
. This implies that the expression in (10) is non-negative and hence the proof is complete.
Remark 2.11 As it was already mentioned for a system with decreasing failure rate components, 
is increasing in time. This result, however, is not generally true for MRL
. Figures 3 and 4 show the graphs of 
and 
in Example 2.7. As the graphs show that 
is a decreasing function of time, however, 
is an increasing function of 
for a period of time and then starts to decrease.
Remark 2.12 In the following, we show that 
has its own interesting interpretation. In fact, under the condition that the system is working at time
, 
shows the probability that there is exactly 
component failure in the system. The mentioned conditional probability can be written as
Figure 4. The failure rate of the system for the discrete weibull 
distribution.
where 
for 
such that 
shows the odds of the event that a component has a lifetime less than
. Also in the following, we study some properties of
.
Theorem 2.13 For 
is decreasing function of t and for
, it is increasing function of t. Also, for 
Proof: We have
(11)
which is obviously a decreasing function of 
(
is a increasing function) since 
and
. From (11), we easily conclude that 
and
.
In this case, it is easily seen that 
is an increasing function of
, 
and
.
Theorem 2.14 The survival function 
can be uniquely determined by 
and
, 
as follows:
(12)
wang#title3_4:spProof:
The result easily follows from the fact that for
,
which gives (12).
Consider the vector
. Obviously, 
is a probability vector. we can then prove the following theorem.
Theorem 2.15 For all
,
Proof: In order to prove the result, we need to show that for
,
or equivalently
This is equivalent to show that
or
(13)
But, as 
is increasing in
, the bracket in the summations, for
, is always negative. Hence the inequality in (13) is valid. This completes the proof of the theorem.
Theorem 2.16 Consider two 
-out-of-
systems. Assume that the components of the systems have independent lifetimes, with survival function 
and
, respectively and odds functions 
and
, respectively. If for all
, 
, then
.
Proof: Asadi & Berred [16] proved that 
for fixed 
and 
is an increasing function of
.
The assumption that 
implies
, then
which is equivalent to say that for all 
and all
,
