Stochastic Comparisons of Lifetimes of used Standby Systems

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Stochastic Comparisons of Lifetimes of used Standby Systems

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Submitted:

03 June 2023

Posted:

05 June 2023

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Abstract

In this paper, we first establish upper stochastic bounds on the lifetime of a used cold standby system with arbitrary age, using the likelihood ratio order and the usual stochastic order. Then, stochastic comparisons are made between the lifetime of a used cold standby system with age t and the lifetime of a cold standby system consisting of used components with age t using the likelihood ratio order and the usual stochastic order. We use illustrative examples to explore the results presented.

Keywords:

Subject: Engineering - Industrial and Manufacturing Engineering

1. Introduction and Preliminaries

Cold standby systems are backup systems that are not ready for use until they are needed. In other words, they are kept on standby but not actively operated. This type of backup system is commonly used in situations where the primary system has a long lifetime and is unlikely to fail frequently (see, e.g., Kumar and Agarwal [17], Kou and Zuo [18], Yang [31], and Peng et al. [22]). Reliability analysis of cold standby systems involves evaluating the probability of failure of the primary system and the time required to switch to the backup system. The reliability of the primary system is determined by analyzing its failure rate, while the reliability of the backup system is determined by analyzing its startup time and the probability of failure during startup. In the context of replacement strategies and related optimization problems, cold standby systems have been used repeatedly in the literature (see, e.g., Coit [7], Yu et al. [32], Jia and Wu [13], Xing et al. [30], and Ram et al. [24]).

Several methods can be used to analyze the reliability of cold standby systems, including fault tree analysis, reliability block diagrams, and Markov models. These methods allow engineers to identify potential failure modes, estimate the probability of system failure, and evaluate the effectiveness of backup systems. The study of the reliability of complex systems using cold standby systems has been conducted by many researchers for engineering problems (See, for example, Azaron et al. [3], Wang et al. [28], Wang et al. [29], and Behboudi et al. [5]).

Stochastic comparisons between the random lifetimes of various complex systems have been a subject of increasing interest among engineers and system designers. This enables them to have, for example, an optimization problem to solve and, consequently, a plan to prepare a product with greater reliability. The theory of stochastic orderings in applied probability has been recently utilized to compare the lifetime of coherent systems equipped by cold-standby units from a stochastic point of view (cf. Boland and El-Neweihi [6], Li et al. [19], Eryilmaz [8], Eryilmaz [9], Roy and Gupta [26] and Roy and Gupta [25]).

However, stochastic comparisons of lifetimes of general cold-standby systems with an arbitrary age, have not been considered in the literature thus far. As will be clarified in the sequel, the lifetime of a component in a system with additional

(n - 1)

cold-standby spares is the sum of random variables, thus, in this regard, a few studies may be found in Zhao and Balakrishnan [33], Amiri et al. [2], Khaledi and Amiri [16] and Amiripour et al. [1] among others.

In the current study, we develop some stochastic ordering results using the well-known usual stochastic order and the likelihood ratio order, involving the lifetime of a used cold-standby system with an arbitrary age, and provide some bounds for the rf of the lifetime of such a system. The rf of the lifetime of a cold-standby system with used components of the age t, is used to provide upper bound and lower bound for the rf of the used cold-standby system with age t.

In the sequel we will use some notations. Let

X ̠ = (X_{1}, \dots, X_{n})

be a random vector and

x ̠ = (x_{1}, \dots, x_{n})

be a vector of observations as a realization of

X ̠

. Denote

S_{m, X ̠} = \sum_{i = 1}^{m} X_{i}

and

S_{m, x ̠} = \sum_{i = 1}^{m} x_{i}

with

m = 1, 2, \dots, n

. Consider the cold standby system consisting of n components. Initially, one component starts working and the other

n - 1

components are in cold standby mode. When the working component fails, the components in standby mode are replaced one by one until all components have failed and the cold standby system fails. The cold standby system means that the components do not fail or degrade in standby mode and that the standby period does not affect the life of the components in future use. When the failed component is replaced by the standby component, the switch is absolutely reliable and transmission is instantaneous. Let

X_{1}, \dots, X_{n}

be the lifetimes of the n components with cumulative distribution functions

F_{X_{1}}, \dots, F_{X_{n}}

and corresponding reliability functions

{\bar{F}}_{X_{1}}, \dots, {\bar{F}}_{X_{n}}

. We also assume

X_{1}, \dots, X_{n}

are independent. Then, the lifetime of the cold standby system is

S_{n, X ̠} = X_{1} + X_{2} + \dots + X_{n} .

(1)

The rf or the survival function (SF) of the lifetime of the cold standby system given in (1), is

{\bar{F}}_{S_{n, X ̠}} (t) = P (X_{1} + X_{2} + \dots + X_{n} > t) = {\bar{F}}_{X_{1}} (t) * {\bar{F}}_{X_{2}} (t) * \dots * {\bar{F}}_{X_{n}} (t),

(2)

where * represents the convolution operator. It is known that when

X_{i}

and

X_{j}

, for

i \neq j

are independent, then

{\bar{F}}_{X_{i}} (t) * {\bar{F}}_{X_{j}} (t) = \int_{- \infty}^{+ \infty} {\bar{F}}_{X_{i}} (t - x) f_{X_{j}} (x) d x,

where

f_{X_{j}}

is the probability density function (pdf) of

X_{j}

, which is the rf of the convolution of

X_{i}

and

X_{j}

, i.e. the rf of

X_{i} + X_{j}

. Thus, from (2), we can write

{\bar{F}}_{S_{n, X ̠}} (t) = {\bar{F}}_{S_{n - 1, X ̠}} (t) * {\bar{F}}_{X_{n}} (t) = \int_{- \infty}^{+ \infty} {\bar{F}}_{S_{n - 1, X ̠}} (t - x) f_{X_{n}} (x) d x .

(3)

We will need some other preliminaries in the continuing part of the paper. Suppose that X is the lifetime of a fresh item as its life span. We may need to recognize the distribution of the lifetime of that item at the age t. The random variable

X_{t} = [X - t | X > t]

, which is called the residual lifetime of an item with original life length X at the time t provided that the item is already alive at this time, is ordinarily utilized to represent and model the lifetime of a used component or item. Let X have pdf

f_{X}

(whenever it exists) and rf

{\bar{F}}_{X}

. Then,

X_{t}

has pdf

f_{X_{t}} (x) = \frac{f_{X} (t + x)}{{\bar{F}}_{X} (t)} d x, x \geq 0,

and it has rf

{\bar{F}}_{X_{t}} (x) = \frac{{\bar{F}}_{X} (t + x)}{{\bar{F}}_{X} (t)} : x \geq 0,

which are valid as

t \in {t \geq 0 : {\bar{F}}_{X} (t) > 0} .

The mean residual lifetime (MRL) function of X is given by

m_{X} (t) = E [X - t | X > t] = \int_{t}^{+ \infty} \frac{{\bar{F}}_{X} (x)}{{\bar{F}}_{X} (t)}, t \in {t \geq 0 : {\bar{F}}_{X} (t) > 0} .

Stochastic ordering of distributions have been a useful tool for statisticians in the context of testing statistical hypothesises. Two well-know stochastic order will be used throughout this paper. The following definition can be found in Shaked and Shanthikumar [27].

Definition 1.

Let X and Y be two non-negative random variables with pdfs

f_{X}

and

f_{Y}

, and refs

{\bar{F}}_{X}

and

{\bar{F}}_{Y}

, respectively. Then it is said that X is smaller or equal than Y in the

(i) likelihood ratio order (denoted by

X \leq_{l r} Y

) whenever

\frac{f_{Y} (t)}{f_{X} (t)}

is non-decreasing in

t \geq 0

(ii) usual stochastic order (denoted by

X \leq_{s t} Y

) whenever

{\bar{F}}_{X} (t) \leq {\bar{F}}_{Y} (t),

for all

t \geq 0

The stochastic orders in Definition 1 are connected to each other as it has been proved that

X \leq_{l r} Y

implies

X \leq_{s t} Y

(see, e.g., Shaked and Shanthikumar [27]). Nonparametric classes of life distributions are usually based on the pattern of aging in some sense. For example by comparing

X_{t_{1}}

with

X_{t_{2}}

for

t_{1} \leq t_{2}

as two time points according to the likelihood ratio order, new patterns of aging are produced. The common parametric families of life distributions also feature monotone aging. In this context, the following definition is also applied in the paper.

Definition 2.

The random variable X with pdf

f_{X}

is said to have

(i) Increasing likelihood ratio property (denoted as

X \in I L R

) whenever

f_{X} (t)

is log-concave in

t \geq 0

(ii) Decreasing likelihood ratio property (denoted as

X \in D L R

) whenever

f_{X} (t)

is log-convex in

t \geq 0

For example the exponential distribution has both

I L R

and

D L R

properties. The gamma distribution with shape parameter

α

and scale parameter

λ

has

I L R

property if

α > 1

and it has

D L R

property if

α < 1

. For recognizing

I L R

and

D L R

properties in further well-known distributions we refer the readers to Bagnoli and Bergstrom [4]. The following definition is due to Karlin [14].

Definition 3.

Suppose that

h (x, y)

is a non-negative function for all

x \in X \subseteq R

and for all

y \in Y \subseteq R

. Then, it is said that

h (x, y)

is totally positive of order 2 (denoted as

T P_{2}

) in

(x, y) \in X \times Y

, whenever

h (x_{1}, y_{1}) h (x_{2}, y_{2}) \geq h (x_{1}, y_{2}) h (x_{2}, y_{1})

for all

x_{1} \leq x_{2} \in X

and for all

y_{1} \leq y_{2} \in Y .

According to Definition 3, by using the convention that

a / 0 = + \infty

for any

a > 0

, then

h (x, y)

T P_{2}

(x, y)

in the desired subset of

R^{2}

, i.e. over the set

X \times Y

if and only if,

\frac{h (x_{2}, y)}{h (x_{1}, y)}

is non-decreasing in

y \in Y

, or equivalently if,

\frac{h (x, y_{2})}{h (x, y_{1})}

is non-decreasing in

x \in X

2. Stochastic Bounds for the Lifetime of a Used Standby System

Let us consider independent and non-negative rvs

X_{1}, X_{2}, \dots, X_{n}

, and denote by

X_{1} (t), X_{2} (t), \dots,

X_{n} (t)

the corresponding conditional rvs, so that

X_{i} (t) = [X_{i} | X_{i} > t]

, for all

t \geq 0

for which

R_{X_{i}} (t) > 0

. Note that for the random vector

Z = (Z_{1}, Z_{2}, \dots, Z_{m})

, the conditional rv

[Z | Z \in A]

is a vector-valued rv whose joint distribution is equal with conditional join distribution of Ẕ give

Z \in A

, where A is a subset of

R^{m}

for which

P (Z \in A) > 0

. The rv

X_{i} (t)

has pdf

f_{X_{i} (t)} (x) = \frac{f_{X_{i}} (x) I [x > t]}{R_{X_{i}} (t)} .

(4)

Consider, now, the vector

X ̠ (t) = (X_{1}, X_{2}, \dots, X_{n} | X_{1} > t, X_{2} > t, \dots, X_{n} > t)

which, since

X_{i}

’s are independent, it has joint cdf

f_{X ̠ (t)} (x ̠) = \prod_{i = 1}^{n} \frac{f_{X_{i}} (x_{i}) I [x_{i} > t]}{R_{X_{i}} (t)} .

(5)

Note that the ith marginal distribution of

X ̠ (t)

is the distribution of

X_{i} (t)

provided that

X_{1}, X_{2}, \dots, X_{n}

are independent. We can find that for any function

ψ : R^{n} \mapsto R

E [ψ (X ̠ (t))] = \frac{E [ψ (X ̠) I [x ̠ > t ̠]]}{{\bar{F}}_{X ̠} (t ̠)}

(6)

where

t ̠ = (t, t, \dots, t)

is a vector with size n and

{\bar{F}}_{X ̠}

is the joint reliability function of

X ̠

. We will utilize

ψ (x ̠) = S_{n, x ̠} .

Note that

(S_{n, X ̠}) (t)

is equal in distribution with the conditional rv

[S_{n, X ̠} | S_{n, X ̠} > t]

, where

S_{n, X ̠} = \sum_{i = 1}^{n} X_{i}

. The following lemma will be used in the sequel.

Lemma 4.

Let

X ̠ = (X_{1}, X_{2}, \dots, X_{n})

be a random vector, having joint pdf

f_{X ̠} (x ̠)

. Consider

Y ̠ = (Y_{1}, Y_{2}, \dots, Y_{n})

as a random vector with joint pdf

f_{Y ̠} (y ̠) = \frac{I [S_{n, y ̠} > t] f_{X ̠} (y ̠)}{P (S_{n, X ̠} > t)},

(7)

Then,

(S_{n, X ̠}) (t) =_{s t} S_{n, Y ̠}

, where

=_{s t}

means equality in distribution.

Proof.

It is sufficient to show that

R_{(S_{n, X ̠}) (t)} (s) = R_{S_{n, Y ̠}} (s),

for all

s \geq 0

. We can write

\begin{matrix} R_{(S_{n, X ̠}) (t)} (s) & = P (S_{n, X ̠} > s) \\ = P (S_{n, X ̠} > s | S_{n, X ̠} > t) \\ = \{\begin{matrix} 1 & : s \leq t \\ \frac{R_{S_{n, X ̠}} (s)}{R_{S_{n, X ̠}} (t)} & : s > t . \end{matrix} \end{matrix}

On the other hand, from (24), we obtain

\begin{matrix} R_{S_{n, Y ̠}} (s) & = P (S_{n, Y ̠} > s) \\ = \frac{P (S_{n, X ̠} > t \lor s)}{P (S_{n, X ̠} > t)} \\ = \{\begin{matrix} 1 & : s \leq t \\ \frac{R_{S_{n, X ̠}} (s)}{R_{S_{n, X ̠}} (t)} & : s > t, \end{matrix} \end{matrix}

in which

t \lor s = max {t, s}

. Thus, we proved the desired identity and, hence, the result follows. □

The following result indicates that the lifetime of a used standby system of age t with n components is dominated in the sense of the usual stochastic order by the lifetime of a standby system composed of used components plus

(n - 1) t

. Denote by

X_{i, t} = [X_{i} - t | X_{i} > t]

which is valid for all

t \geq 0

for which

R_{X_{i}} (t) > 0

and note that

X_{i}

is the random lifetime of the ith component in the standby system, with

i = 1, 2, \dots, n

. Denote

X ̠_{t} = (X_{1} - t, X_{2} - t, \dots, X_{n} - t | X_{1} > t, X_{2} > t, \dots, X_{n} > t), t : P (X ̠ > t I) > 0,

where

I = {(1, 1, \dots, 1)}^{⊤}

is a vector with n components and

X ̠ = {(X_{1}, X_{2}, \dots, X_{n})}^{⊤}

. Notice that the marginal distribution of the ith random element in

X ̠_{t}

corresponds with the distribution of

X_{i, t}

i = 1, 2, \dots, n

, provided that

X_{1}, X_{2}, \dots, X_{n}

are independent.

Theorem 5.

Let

X_{1}, X_{2}, \dots, X_{n}

be independent rvs which are non-negative and

X_{1, t}, X_{2, t}, \dots, X_{n, t}

be also independent, for a fixed

t > 0

. Then:

{(S_{n, X ̠})}_{t} \leq_{s t} S_{n, X ̠_{t}} + (n - 1) t .

(8)

Proof.

For

n = 1

, the result is trivial. Let us assume

n = 2

. From Lemma 4, they can be founds rvs

Y_{1}

and

Y_{2}

with joint pdf

f_{Y ̠} (y ̠) = \frac{I [S_{2, y ̠} > t] f_{X ̠} (y ̠)}{R_{S_{2, x ̠}} (t)}, y_{i} \geq 0, i = 1, 2,

(9)

in which

y ̠ = (y_{1}, y_{2}),

such that

(S_{2, X ̠}) (t) =_{s t} S_{2, Y ̠}

. From (9), since

X_{1}

and

X_{2}

are independent,

Y_{1}

has pdf

f_{Y_{1}} (y_{1}) = \frac{f_{X_{1}} (y_{1}) R_{X_{2}} (t - y_{1})}{R_{S_{2, x ̠}} (t)} .

Then:

\begin{matrix} \frac{f_{Y_{1}} (y_{1})}{f_{X_{1} (t)} (y_{1})} & = \frac{R_{X_{1}} (t) R_{X_{2}} (t - y_{1})}{R_{S_{2, x ̠}} (t) I [y_{1} > t]} \\ = \{\begin{matrix} + \infty & : y_{1} \leq t \\ \frac{R_{X_{1}} (t)}{R_{S_{2, X ̠}} (t)} & : y_{1} > t . \end{matrix} \end{matrix}

It is clear that

\frac{f_{Y_{1}} (y_{1})}{f_{X_{1} (t)} (y_{1})}

is decreasing in

y_{1}

, thus,

Y_{1} \leq_{l r} X_{1} (t) .

Since

\leq_{l r}

implies

\leq_{s t}

, thus

Y_{1} \leq_{s t} X_{1} (t) .

(10)

By Equation (9), the conditional pdf of

Y_{2}

given

Y_{1} = y_{1}

is derived as:

f_{Y_{2} | Y_{1}} (y_{2} | y_{1}) = \frac{f_{X_{2}} (y_{2}) I [S_{2, y ̠} > t]}{R_{X_{2}} (t - y_{1})} .

For any

y_{1} \geq 0,

we show that

\frac{f_{Y_{2} | Y_{1}} (y_{2} | y_{1})}{f_{X_{2} (t)} (y_{2})}

is decreasing in

y_{2} \geq 0

, as

y_{1} + y_{2} > t

. We have

\frac{f_{Y_{2} | Y_{1}} (y_{2} | y_{1})}{f_{X_{2} (t)} (y_{2})} = \{\begin{matrix} + \infty & : y_{2} \leq t, y_{1} > t - y_{2} \\ \frac{R_{X_{2}} (t)}{R_{X_{2}} (t - y_{1})} & : y_{2} > t, y_{1} > t - y_{2}, \end{matrix}

which is decreasing in

y_{2} \geq 0 .

Therefore,

[Y_{2} | Y_{1} = y_{1}] \leq_{l r} X_{2} (t)

. Hence,

[Y_{2} | Y_{1} = y_{1}] \leq_{s t} X_{2} (t) .

(11)

From Equations (10) and (11), using Theorem 6.B.3 in Shaked and Shanthikumar [27], one gets:

[X_{1} + X_{2} | X_{1} + X_{2} > t] = (S_{2, X ̠}) (t) \leq_{s t} X_{1} (t) + X_{2} (t) .

It is easily verified that

S_{2, X ̠} (t) \leq_{s t} S_{2, X ̠ (t)},

holds, if and only if,

S_{2, X ̠} (t) - t \leq_{s t} S_{2, X ̠ (t)} - t

. So,

{(S_{2, X ̠})}_{t} \leq_{s t} S_{2, X ̠_{t}} + t,

i.e., (13) holds when

n = 2

. Finally, we prove the result by induction. Suppose that for

n = m \geq 2,

it holds that

{(S_{m, X ̠})}_{t} \leq_{s t} S_{m, X ̠_{t}} + (m - 1) t

. From preservation property of

\leq_{s t}

under a change in location of distributions, we have:

S_{m, X ̠} (t) \leq_{s t} S_{m, X ̠ (t)} .

(12)

From (12), since for

m = 2

the result was proved, thus an application of Theorem 1.A.3(b) in Shaked and Shanthikumar [27] yields:

\begin{matrix} S_{m + 1, X ̠} (t) & \leq_{s t} S_{m, X ̠} (t) + X_{m + 1} (t) \\ \leq_{s t} S_{m, X ̠ (t)} + X_{m + 1} (t) \\ =_{s t} S_{m + 1, X ̠ (t)} . \end{matrix}

Equivalently, one can write:

{(S_{m + 1, X ̠})}_{t} \leq_{s t} S_{m + 1, X ̠_{t}} + m t

. Hence, the proof is obtained. □

In the context of Theorem 5 one may realize that, if in (13),

t = 0

, then

\leq_{s t}

becomes

=_{s t}

. Using Theorem 5, an upper bound for the mean residual lifetime (MRL) function of a standby system can be provided. Since

\leq_{s t}

implies the expectation order, thus, from Theorem 5,

\begin{matrix} m_{S_{n, X ̠}} (t) & = E [{(S_{n, X ̠})}_{t}] \\ \leq E [S_{n, X ̠_{t}}] + (n - 1) t \\ = \sum_{i = 1}^{n} m_{X_{i}} (t) + (n - 1) t, \end{matrix}

where

m_{S_{n, X ̠}} (t) = E [\sum_{i = 1}^{n} X_{i} - t | \sum_{i = 1}^{n} X_{i} > t]

is the MRL function of a standby system with independent component lifetimes

X_{1}, X_{2}, \dots, X_{n}

, and

m_{X_{i}} (t) = E [X_{i, t}]

is the MRL function of

X_{i}

for

i = 1, 2, \dots, n

. The bound provided for the MRL function of a standby system is valuable because the MRL function of the system which depends on the distribution of convolution of n rvs, which has no closed form in many situation, does not have an explicit formula. The rv X has a gamma distribution with the shape parameter

α

and the scale parameter

λ

whenever X has density

f_{X} (x) = \frac{λ^{α} x^{α - 1} e^{- λ x}}{Γ (α)}

(denote it by

X_{i} \sim G (α, λ)

). Let us suppose that under consideration is a standby system with independent heterogenous exponential component lifetimes

X_{1}, X_{2}, \dots, X_{n}

, so that

X_{i}

has mean

\frac{1}{λ_{i}}

(

X_{i} \sim E x p (λ_{i})

), where

λ_{i} > 0, i = 1, 2, \dots, n

. It is known that

m_{X_{i}} (t) = \frac{1}{λ_{i}}

, however, the

S_{n, X ̠} = \sum_{i = 1}^{n} X_{i}

as the random lifetime of the standby system, has a gamma distribution (

S_{n, X ̠} \sim G (n, \sum_{i = 1}^{n} λ_{i})

) with an indefinite MRL function. However,

m_{S_{n, X ̠}} (t) \leq \sum_{i = 1}^{n} \frac{n}{λ_{i}} + (n - 1) t, f o r a l l t \geq 0 .

In the following example, the result of Theorem 5 is applied.

Example 6.

Suppose that

X_{1} \sim G (2, 3)

and

X_{2} \sim G (1 / 2, 3)

are two independent random variables and assume that

X_{1, t}

and

X_{2, t}

are also independent. Note that

X_{1} \in I L R

and

X_{2} \in D L R

. Consider a two-units standby system. Using Theorem 5, an upper bound for the rf of the used standby system with age

t = 0.1

is derived. Specifically, it is shown that

{(S_{2, X ̠})}_{t} = {(X_{1} + X_{2})}_{t} \leq_{s t} X_{1, t} + X_{2, t} + t = S_{2, X ̠_{t}} + t .

Since it is trivial that for all

x \leq 0.1,

one has

P (S_{2, X ̠_{t}} + t > x) = 1

, thus for all

x \leq 0.1

P (S_{2, X ̠_{t}} + t > x) \geq P ({(S_{2, X ̠})}_{t} > x)

. Therefore, it is enough to show that

P (S_{2, X ̠_{t}} + t > x) \geq P ({(S_{2, X ̠})}_{t} > x),

for all

x > 0.1

. It is seen that for

t = 0.1

P ({(S_{2, X ̠})}_{t} > x) = \frac{{\bar{F}}_{X_{1} + X_{2}} (0.1 + x)}{{\bar{F}}_{X_{1} + X_{2}} (0.1)} = \frac{\int_{0.1 + x}^{+ \infty} u^{\frac{3}{2}} e x p (- 3 u) d u}{\int_{0.1}^{+ \infty} u^{\frac{3}{2}} e x p (- 3 u) d u} .

On the other hand, for all

x > 0.1

, we can get

P (S_{2, X ̠_{t}} + t > x) = \int_{0.1}^{x} {\bar{F}}_{X_{1, 0.1}} (x - x_{2}) f_{X_{2, 0.1}} (x_{2} - 0.1) d x_{2} + {\bar{F}}_{X_{2, 0.1}} (x - 0.1),

in which the rf of

X_{1, 0.1} = [X_{1} - 0.1 | X_{1} > 0.1]

is acquired as

{\bar{F}}_{X_{1, 0.1}} (x) = \frac{{\bar{F}}_{X_{1}} (0.1 + x)}{{\bar{F}}_{X_{1}} (0.1)} = \frac{(1 + 3 (0.1 + x)) \exp (- 3 (0.1 + x))}{1.3 \exp (- 0.3)},

and, similarly, the rf of

X_{2, 0.1} = [X_{2} - 0.1 | X_{2} > 0.1]

is obtained as

{\bar{F}}_{X_{2, 0.1}} (x) = \frac{{\bar{F}}_{X_{2}} (0.1 + x)}{{\bar{F}}_{X_{2}} (0.1)} = \frac{\int_{0.1 + x}^{+ \infty} u^{- \frac{1}{2}} \exp (- 3 u) d u}{\int_{0.1}^{+ \infty} u^{- \frac{1}{2}} \exp (- 3 u) d u},

and, consequently,

f_{X_{2, 0.1}} (x) = \frac{{(0.1 + x)}^{- \frac{1}{2}} \exp (- 3 (x + 0.1))}{\int_{0.1}^{+ \infty} u^{- \frac{1}{2}} \exp (- 3 u) d u} .

In Figure 1, the graph of SFs of

{(S_{2, X ̠})}_{t}

and

S_{2, X ̠_{t}} + t

is plotted, which makes it clear that for

0.1 < x < 5

P (S_{2, X ̠_{t}} + t > x) \geq P {(S_{2, X ̠})}_{t})

when

t = 0.1

In the sequel of this section, the result of Theorem 5 is strengthened to the case where the likelihood ratio order is used. However, in this case, the random lifetimes of the components need to to have log-concave density functions which means that the components lifetimes have to fulfill the ILR property. We first give the following technical lemmas. The proof of the following lemmas, being straightforward, are omitted.

Lemma 7.

Suppose that Y and Z are two rvs with pdfs

f_{Y}

and

f_{Z},

respectively. The following assertions hold:

(a) If Y has support

S_{Y} = (l_{Y}, + \infty)

and Z has support

S_{Z} = (l_{Z}, + \infty),

so that

l_{Y} \geq l_{Z}

then,

Y \geq_{l r} Z

if, and only if,

\frac{f_{Y} (s)}{f_{Z} (s)}

is non-decreasing in

s > l_{Y}

(b) If

Y \geq_{l r} Z

, then

Y - c \geq_{l r} Z - c,

for

c > 0

Lemma 8.

The following assertions hold true:

(a) For any non-negative rv W, for all

t \geq 0

for which

P (W > t) > 0

, it holds that

W_{t} =_{s t} W (t) - t

where

W_{t} = [W - t | W > t]

and

W (t) = [W | W > t] .

(b) For all

t \geq 0,

for which

P (X_{i} > t) > 0, i = 1, 2, \dots, n

it holds that

S_{n, X ̠ (t)} =_{s t} S_{n, X ̠_{t}} + n t

Theorem 9.

Let

X_{1}, X_{2}, \dots, X_{n}

be non-negative independent rvs which are all

I L R

, and suppose that

X_{1, t}, X_{2, t}, \dots, X_{n, t}

are independent, for a fixed

t > 0

. Then:

{(S_{n, X ̠})}_{t} \leq_{l r} S_{n, X ̠_{t}} + (n - 1) t .

(13)

Proof.

Fix

t > 0

. Firstly, we prove that

Y = S_{n, X ̠ (t)} \geq_{l r} (S_{n, X ̠}) (t) = Z,

in which

S_{n, X ̠ (t)} = \sum_{i = 1}^{n} X_{i} (t)

and

(S_{n, X ̠}) (t) = (\sum_{i = 1}^{n} X_{i}) (t)

with

X_{i} (t) = [X_{i} | X_{i} > t]

. Since

l_{Y} = n t

and

l_{Z} = t

, thus by Lemma 7 it is enough to show that

\frac{f_{Y} (s)}{f_{Z} (s)}

is non-decreasing in s for all

s > n t

. One has

\begin{matrix} f_{Z} (s) & = \frac{f_{S_{n, X ̠}} (s) I [s > t]}{R_{S_{n, X ̠}} (t)} \\ = \frac{f_{S_{n, X ̠}} (s)}{R_{S_{n, X ̠}} (t)}, s > n t . \end{matrix}

Denote

S_{n - 1, X ̠ (t)} = \sum_{i = 1}^{n - 1} X_{i} (t)

, and similarly,

S_{n - 1, X ̠} = \sum_{i = 1}^{n - 1} X_{i}

. Therefore, for all

s > n t :

\begin{matrix} \frac{f_{Y} (s)}{f_{Z} (s)} & = R_{S_{n, X ̠}} (t) \frac{\int_{0}^{+ \infty} f_{X_{n} (t)} (s - y) f_{S_{n - 1, X ̠ (t)}} (y) d y}{\int_{0}^{+ \infty} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y) d y} \\ = \frac{R_{S_{n, X ̠}} (t)}{R_{X_{n}} (t)} \frac{\int_{0}^{s - t} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠ (t)}} (y) d y}{\int_{0}^{s} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y) d y} \\ = \frac{R_{S_{n, X ̠}} (t)}{R_{X_{n}} (t)} \int_{0}^{s} \frac{f_{S_{n - 1, X ̠ (t)}} (y)}{f_{S_{n - 1, X ̠}} (y)} \frac{I [y < s - t] f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y)}{\int_{0}^{s} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y) d y} d y \\ = \frac{R_{S_{n, X ̠}} (t)}{R_{X_{n}} (t)} \int_{0}^{s} \frac{f_{S_{n - 1, X ̠ (t)}} (y)}{f_{S_{n - 1, X ̠}} (y)} \frac{I [y < s - t] f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y)}{\int_{0}^{s - t} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y) d y} d y P_{n} (t | s), \end{matrix}

(14)

where

\begin{matrix} P_{n} (t | s) & = P (X_{n} > t | S_{n, X ̠} = s) \\ = P (S_{n - 1, X ̠} \leq s - t | S_{n, X ̠} = s) \\ = \int_{0}^{s - t} f_{S_{n - 1, X ̠} | S_{n, X ̠}} (y | s) d y \\ = \frac{\int_{0}^{s - t} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y) d y}{\int_{0}^{s} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y) d y} . \end{matrix}

Thus, from Eq. (14), one can write

\frac{f_{Y} (s)}{f_{Z} (s)} = \frac{R_{S_{n, X ̠}} (t)}{R_{X_{n}} (t)} P_{n} (t | s) E [Φ (Y^{★} (s))],

(15)

where

Φ (y) = \frac{f_{S_{n - 1, X ̠ (t)}} (y)}{f_{S_{n - 1, X ̠}} (y)}

and

Y^{★} (s)

is an rv with pdf

f^{★} (y | s) = \frac{I [y < s - t] f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y)}{\int_{0}^{s - t} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y) d y} .

(16)

Since

X_{i} \in I L R,

for all

i = 1, 2, \dots, n

thus from Theorem 1.C.53 in Shaked and Shanthikumar [27], we deduce that

[X_{n} | S_{n, X ̠} = s_{1}] \leq_{l r} [X_{n} | S_{n, X ̠} = s_{2}],

for all

s_{1} \leq s_{2}

. Hence,

[X_{n} | S_{n, X ̠} = s_{1}] \leq_{s t} [X_{n} | S_{n, X ̠} = s_{2}],

for all

s_{1} \leq s_{2}

which validates that

P_{n} (t | s) is non-decreasing in s > n t,

(17)

for all

t \geq 0

. From 16, we can write

f^{★} (y | s) = ϕ_{1} (y) ϕ_{2} (s) ξ (y, s)

, in which

ϕ_{1} (y) = f_{S_{n - 1, X ̠}} (y) \geq 0, ϕ_{2} (s) = {[\int_{0}^{s - t} f_{X_{n}} (s - y) f_{S_{n - 1, X ̠}} (y) d y]}^{- 1} > 0

and

ξ (y, s) = I [y < s - t] f_{X_{n}} (s - y)

. Since

X_{n}

I L R

, thus

f_{X_{n}} (s - y)

T P_{2}

(y, s) \in (0, + \infty) \times (0, + \infty) .

It is also plain to see that

I [y < s - t]

is also

T P_{2}

(y, s) \in (0, + \infty) \times (n t, + \infty) .

Hence, as the product of two

T P_{2}

functions in a common domain is itself another

T P_{2}

function, thus

ξ (y, s)

is also

T P_{2}

(y, s) \in (0, + \infty) \times (n t, + \infty) .

From Lemma 3 in Kayid and Alshagrawi [15],

f^{★} (y | s)

T P_{2}

(y, s) \in (0, + \infty) \times (n t, + \infty) .

This reveals that

Y^{*} (s_{1}) \leq_{l r} Y^{*} (s_{2}),

for all

s_{1} \leq s_{2} \in (n t, + \infty),

and consequently,

Y^{*} (s_{1}) \leq_{s t} Y^{*} (s_{2}), f o r a l l s_{1} \leq s_{2} \in (n t, + \infty) .

(18)

On the other hand, since

X_{i} \in I L R

, by Theorem 2.10 in Izadkhah et al. [12], the rv

X_{i} (t) = [X_{i} | X_{i} > t]

having weighted distribution with respect to

X_{i}

with the weight function

w (x) = I [x > t]

which is log-concave in x, for all

t \geq 0

, is also

I L R

. Now, by Theorem 1.C.9 of Shaked and Shanthikumar [27], since

X_{i} (t) \geq_{l r} X_{i}

for all

i = 1, 2, \dots, n

, thus

S_{n, X ̠ (t)} \geq_{l r} S_{n, X ̠} .

That is

Φ (y) = \frac{f_{S_{n - 1, X ̠ (t)}} (y)}{f_{S_{n - 1, X ̠}} (y)} is non-decreasing in y > 0 .

(19)

So, from (18),

E [Φ (Y^{★} (s))]

is also non-decreasing in

s \in (n t, + \infty) .

In view of (17) and Eq. (15), it follows that

\frac{f_{Y} (s)}{f_{Z} (s)}

is non-decreasing in

s \in (n t, + \infty)

. Therefore, we demonstrate it that

S_{n, X ̠ (t)} \geq_{l r} (S_{n, X ̠}) (t)

which by Lemma 8(b) it further implies that

S_{n, X ̠ (t)} + n t \geq_{l r} (S_{n, X ̠}) (t)

. By applying Lemma 7, by choosing

c = t,

one gets

S_{n, X ̠ (t)} + (n - 1) t \geq_{l r} (S_{n, X ̠}) (t) - t,

which by Lemma 8(a) yields

S_{n, X ̠ (t)} + (n - 1) t \geq_{l r} {(S_{n, X ̠})}_{t} .

The proof is completed. □

3. Comparison of a Used Standby System with a Standby System Composed of Used Components

In this section, we make stochastic comparisons between lifetimes of a used standby system with age t and another standby system composed of used components each has age t. We establish that when the

(n - 1)

ones of the n component lifetime distribution have a general density function (with absolutely continuous distribution) and the nth component is exponentially distributed, then the standby system with used components is more reliable and has smaller risk than the used standby system.

Theorem 10.

Let

X_{1}, X_{2}, \dots, X_{n}

be independent non-negative rvs with pdfs

f_{X_{1}}, f_{X_{2}}, \dots, f_{X_{n}}

, respectively, so that

S_{n - 1, X ̠_{t}} + t \geq_{l r} S_{n - 1, X ̠},

(20)

for a fixed

t \geq 0

, in which

S_{n - 1, X ̠} = \sum_{i = 1}^{n - 1} X_{i}

and

S_{n - 1, X ̠_{t}} = \sum_{i = 1}^{n - 1} X_{i, t}

and that

X_{n}

follows exponential distribution with parameter

λ_{n}

. Suppose that

X_{1, t}, X_{2, t}, \dots, X_{n, t}

where

X_{i, t} = [X_{i} - t | X_{i} > t], i = 1, 2, \dots, n

are also independent rvs. Then:

{(S_{n, X ̠})}_{t} \leq_{l r} S_{n, X ̠_{t}}

(21)

where

S_{n, X ̠} = \sum_{i = 1}^{n} X_{i}

and

S_{n, X ̠_{t}} = \sum_{i = 1}^{n} X_{i, t} .

Proof.

We prove that

\frac{f_{S_{n, X ̠_{t}}} (s)}{f_{{(S_{n, X ̠})}_{t}} (s)}

is non-decreasing in

s \geq 0

. Let us write

\begin{matrix} \frac{f_{S_{n, X ̠_{t}}} (s)}{f_{{(S_{n, X ̠})}_{t}} (s)} & = \frac{R_{S_{n, X ̠}} (t)}{R_{X_{n}} (t)} \frac{\int_{0}^{+ \infty} f_{X_{n, t}} (s - y) f_{S_{n - 1, X ̠_{t}}} (y) d y}{\int_{0}^{+ \infty} f_{X_{n}} (s + t - y) f_{S_{n - 1, X ̠}} (y) d y} \\ = \frac{R_{S_{n, X ̠}} (t)}{R_{X_{n}} (t)} \frac{\int_{0}^{+ \infty} I [y < s] f_{X_{n}} (s + t - y) f_{S_{n - 1, X ̠_{t}}} (y) d y}{\int_{0}^{+ \infty} f_{X_{n}} (s + t - y) f_{S_{n - 1, X ̠}} (y) d y} . \end{matrix}

Hence, it suffices to prove that

\frac{\int_{0}^{+ \infty} I [y < s] f_{X_{n}} (s + t - y) d F_{S_{n - 1, X ̠_{t}}} (y)}{\int_{0}^{+ \infty} f_{X_{n}} (s + t - y) d F_{S_{n - 1, X ̠}} (y)} is non-decreasing in s \geq 0 .

(22)

By changing the variable y into

y - t

in the numerator in (22) one has

\begin{matrix} \frac{\int_{0}^{+ \infty} I [y < s] f_{X_{n}} (s + t - y) d F_{S_{n - 1, X ̠_{t}}} (y)}{\int_{0}^{+ \infty} f_{X_{n}} (s + t - y) d F_{S_{n - 1, X ̠}} (y)} & = \frac{\int_{0}^{+ \infty} I [y < s + t] f_{X_{n}} (s + 2 t - y) d F_{S_{n - 1, X ̠_{t}}} (y - t)}{\int_{0}^{+ \infty} f_{X_{n}} (s + t - y) d F_{S_{n - 1, X ̠}} (y)} \\ = \int_{0}^{+ \infty} I [y > t] \frac{f_{X_{n}} (s + 2 t - y)}{f_{X_{n}} (s + t - y)} \frac{f_{S_{n - 1, X ̠_{t}}} (y - t)}{f_{S_{n - 1, X ̠}} (y)} d F^{★} (y | s) \\ = E [Φ (s, Y^{★} (s))], \end{matrix}

where

Φ (s, y) = I [y > t] \frac{f_{X_{n}} (s + 2 t - y)}{f_{X_{n}} (s + t - y)} \frac{f_{S_{n - 1, X ̠_{t}}} (y - t)}{f_{S_{n - 1, X ̠}} (y)}

, and

Y^{★} (s)

is an rv with support

[0, s + t]

having pdf

\begin{matrix} f^{★} (y | s) & = \frac{f_{X_{n}} (s + t - y) f_{S_{n - 1, X ̠}} (y)}{\int_{0}^{+ \infty} f_{X_{n}} (s + t - y) f_{S_{n - 1, X ̠}} (y) d y} \\ = \frac{\exp (λ_{n} y) f_{S_{n - 1, X ̠}} (y) I [y < s + t]}{\int_{0}^{+ \infty} \exp (λ_{n} y) f_{S_{n - 1, X ̠}} (y) I [y < s + t] d y} . \end{matrix}

Note that, since

X_{n} \sim E x p (λ_{n}),

thus for all

y \in [0, s + t]

we have:

\frac{f_{X_{n}} (s + 2 t - y)}{f_{X_{n}} (s + t - y)} = \exp (- λ_{n} t) \frac{I [y < s + 2 t]}{I [y < s + t]} = \exp (- λ_{n} t) .

Therefore, for all

y \in [0, s + t]

Φ (s, y) = \exp (- λ_{n} t) I [y > t] \frac{f_{S_{n - 1, X ̠_{t}}} (y - t)}{f_{S_{n - 1, X ̠}} (y)} is non-decreasing in y \geq 0 .

(23)

Notice that

f^{★} (y | s)

T P_{2}

(y, s) \in (0, + \infty) \times (0, + \infty)

, which means that

Y^{★} (s_{1}) \leq_{l r} Y^{★} (s_{2}),

for all

s_{1} \leq s_{2} \in [0, + \infty)

and, consequently,

Y^{★} (s_{1}) \leq_{s t} Y^{★} (s_{2}),

for all

s_{1} \leq s_{2} \in [0, + \infty)

. This is enough together with the non-parenthetical part of Lemma 2.2(i) in Misra and van der Meulen (2003) to obtain

E [Φ (s_{1}, Y^{★} (s_{1}))] \leq E [Φ (s_{2}, Y^{★} (s_{2}))],

for all

s_{1} \leq s_{2} \in [0, + \infty)

which fulfills (22) as a correct statement and, thus, the stochastic order relation given is (21) stands valid. □

The following example of a single unit system equipped with a cold-standby unit which has an exponential lifetime distribution, fulfills the result of Theorem 10.

Example 11.

Let we have a cold-standby system with size

n = 2

heterogenous components with lifetimes

X_{1}

and

X_{2}

so that

X_{1}

has an arbitrary lifetime distribution (

F_{X_{1}} (0^{-}) = 0

) and

X_{2}

has an exponential lifetime distribution with parameter

λ_{2}

and we, further, assume that

X_{1}

and

X_{2}

are independent. Then, since

S_{1, X ̠_{t}} + t =_{l r} X_{1, t} + t =_{l r} X_{1} (t) \geq_{l r} X_{1}

thus the likelihood ratio ordering in Equation (20) holds true for

n = 2

, and hence by Theorem 10, one concludes

X_{1, t} + X_{2, t} \geq_{l r} {(X_{1} + X_{2})}_{t}

, for all

t \geq 0

. Let us suppose that

X_{1}

has a gamma distribution with

α_{1} = 2

and

λ_{1} = 3

and

X_{2}

has exponential distribution with

λ_{2} = 3

as the age

t = 5

is chosen. By routine calculation, for all

x \geq 0,

one has

f_{X_{1, t} + X_{2, t}} (x) = \frac{27 x}{16} (5 + \frac{x}{2}) e^{- 3 x} a n d f_{{(X_{1} + X_{2})}_{t}} (x) = \frac{27}{257} {(5 + x)}^{2} e^{- 3 x} .

In Figure 2, we plot the graph of the function

L R

given by

L R (x) : = \frac{f_{X_{1, t} + X_{2, t}} (x)}{f_{{(X_{1} + X_{2})}_{t}} (x)} = \frac{257 x (10 + x)}{32 {(5 + x)}^{2}}

The function

L R (x)

is increasing with respect to

x \geq 0,

according to Theorem 10 as Figure 2 confirms it.

Now, we prove another result to compare the lifetime of a used standby system with the lifetime of another system composed of used components with respect to the usual stochastic order. The following lemma is essential to our development.

Lemma 12.

Let

X ̠ = (X_{1}, X_{2}, \dots, X_{n})

be a random vector with non-negative random components with joint pdf

f_{X ̠} (x ̠)

. Fix

l \in {1, 2, \dots, n}

. Let

Y ̠ = (Y_{1}, Y_{2}, \dots, Y_{n})

be another random vector with joint pdf

f_{Y ̠} (y ̠) = \frac{f_{X_{l}} (t + S_{n, y ̠})}{f_{X_{l}} (S_{n, y ̠}) c (t, l)} f_{X ̠} (y ̠),

(24)

where

c (t, l) = E (\frac{f_{X_{l}} (t + S_{n, X ̠})}{f_{X_{l}} (S_{n, X ̠})}) < + \infty

. Then,

S_{n, X ̠}^{l} =_{s t} S_{n, Y ̠}

, where

S_{n, X ̠}^{l}

is an rv with pdf

f_{S_{n, X ̠}^{l}} (s) = \frac{f_{X_{l}} (t + s)}{f_{X_{l}} (s) c (t, l)} f_{S_{n, X ̠}} (s)

Proof.

We show that

S_{n, X ̠}^{l}

and

S_{n, Y ̠}

have the same cdf. It is notable that for any function

β : R \mapsto R,

one has

E [β (S_{n, X ̠}^{l})] = \frac{1}{c (t, l)} E (β (S_{n, X ̠}) \frac{f_{X_{l}} (t + S_{n, X ̠})}{f_{X_{l}} (S_{n, X ̠})}) .

(25)

For all

s \geq 0

, one one hand we can derive

P (S_{n, Y ̠} \leq s) = \frac{F_{S_{n, X ̠}} (s)}{c (t, l)} E (\frac{f_{X_{l}} (t + S_{n, X ̠})}{f_{X_{l}} (S_{n, X ̠})} ∣ S_{n, X ̠} \leq s) .

On the other hand, for all

s \geq 0

, one has

\begin{matrix} P (S_{n, X ̠}^{l} \leq s) & = \frac{1}{c (t, l)} \int_{0}^{s} \frac{f_{X_{l}} (t + s^{'})}{f_{X_{l}} (s^{'})} f_{S_{n, X ̠}} (s^{'}) d s^{'} \\ = \frac{F_{S_{n, X ̠}} (s)}{c (t, l)} \int_{0}^{s} \frac{f_{X_{l}} (t + s^{'})}{f_{X_{l}} (s^{'})} \frac{f_{S_{n, X ̠}} (s^{'})}{F_{S_{n, X ̠}} (s)} d s^{'} \\ = \frac{F_{S_{n, X ̠}} (s)}{c (t, l)} E (\frac{f_{X_{l}} (t + S_{n, X ̠})}{f_{X_{l}} (S_{n, X ̠})} ∣ S_{n, X ̠} \leq s) . \end{matrix}

The proof of the result is complete. □

We introduce some notation before stating the result. Let

f_{Z} (t)

be the density of the rv Z which is differentiable. The function

η_{Z} (t) = - \frac{f_{Z}^{'} (t)}{f_{Z} (t)}

is well-known as the Glaser’s eta function which is very useful in the study of the shape of the hazard rate function and the mean residual life function (see, e.g., Glaser [10] and Gupta and Viles [11]).

Theorem 13.

Let

X_{1}, X_{2}, \dots, X_{n}

be independent rvs which are non-negative whose densities are differentiable and

X_{1, t}, X_{2, t}, \dots, X_{n, t}

be also independent, for a fixed

t > 0

. Then if there exists an

l \in {1, 2, \dots, n}

such that

(i): $η_{S_{n, X ̠}} (t + x) - η_{S_{n, X ̠}} (x) \geq η_{X_{l}} (t + x) - η_{X_{l}} (x),$ for all $x \geq 0;$
(ii): $η_{X_{l}} (t + x) - η_{X_{l}} (x)$ is increasing in $x \geq 0;$
(iii): $η_{X_{i}} (t + x) - η_{X_{i}} (x) \leq η_{X_{l}} (t + x) - η_{X_{l}} (x),$ for every $i = 1, 2, \dots n$ and for all $x \geq 0;$

we have

{(S_{n, X ̠})}_{t} \leq_{s t} S_{n, X ̠_{t}} .

(26)

Proof.

Under the assumption (i), it is found that

\frac{f_{S_{n, X ̠}} (t + x) f_{X_{l}} (x)}{f_{S_{n, X ̠}} (x) f_{X_{l}} (t + x)}

is decreasing in

x \geq 0 .

Thus from Theorem 3.2.(a) of Misra et al. [21] ,

{(S_{n, X ̠})}_{t} \leq_{l r} S_{n, X ̠}^{l}

, and, therefore,

{(S_{n, X ̠})}_{t} \leq_{s t} S_{n, X ̠}^{l}

. Suppose now that

n = 2

. Note that the assumption (ii) is equivalent to

\frac{f_{X_{l}} (t + x)}{f_{X_{l}} (x)}

being a log-concave function in

x \geq 0

. By Lemma 12 and the Equation (24), there exist non-negative rvs

Y_{1}

and

Y_{2}

with joint pdf

f_{(Y_{1}, Y_{2})} (y_{1}, y_{2}) = \frac{f_{X_{l}} (t + S_{2, y ̠})}{f_{X_{l}} (S_{2, y ̠}) c (t, l)} f_{(X_{1}, X_{2})} (y_{1}, y_{2}),

(27)

so that

S_{2, X ̠}^{l} =_{s t} Y_{1} + Y_{2} .

By 27, since

X_{1}

and

X_{2}

are independent, thus

Y_{1}

follow pdf

f_{Y_{1}} (y) = C_{0} f_{X_{1}} (y) E (\frac{f_{X_{l}} (t + y_{1} + X_{2})}{f_{X_{l}} (y_{1} + X_{2})}), y_{1} \geq 0;

where

C_{0}

is the normalizing constant. Suppose that

X_{i}^{l}

is an rv with pdf

f_{X_{i}^{l}} (x) = C_{1} \frac{f_{X_{l}} (t + x)}{f_{X_{l}} (x)} f_{X_{i}} (x)

for every

i = 1, 2

. We can then write

\frac{f_{Y_{1}} (y_{1})}{f_{X_{1}^{l}} (y_{1})} = C_{2} E (\frac{f_{X_{l}} (t + y_{1} + X_{2}) f_{X_{l}} (y_{1})}{f_{X_{l}} (y_{1} + X_{2}) f_{X_{l}} (t + y_{1})}),

in which

C_{2}

is the normalizing constant. This ratio is decreasing in

y_{\geq} 0

since

\frac{f_{X_{l}} (t + x)}{f_{X_{l}} (x)}

is a log-concave function in

x \geq 0

. This is equivalent to saying that

Y_{1} \leq_{l r} X_{1}^{l},

and therefore,

Y_{1} \leq_{s t} X_{1}^{l} .

(28)

Once again, in spirit of (27), given that

Y_{1} = y,

the conditional pdf of

Y_{2}

is derived as

f_{Y_{2} | Y_{1}} (y_{2} | y_{1}) = \frac{f_{X_{2}} (t + y_{1} + y_{2}) f_{X_{2}} (y_{2})}{f_{X_{2}} (y_{1} + y_{2}) E (\frac{f_{X_{2}} (t + y_{1} + X_{2})}{f_{X_{2}} (y_{1} + X_{2})})} .

For any fixed

y_{1} \geq 0,

the log-concavity of

\frac{f_{X_{2}} (t + x)}{f_{X_{2}} (x)}

x \geq 0

, implies that

\frac{f_{Y_{2} | Y_{1}} (y_{2} | y_{1})}{f_{X_{2}^{l}} (y_{2})}

is decreasing in

y_{2} \geq 0

; that is

[Y_{2} | Y_{1} = y_{1}] \leq_{l r} X_{2}^{l}

Thus,

[Y_{2} | Y_{1} = y_{1}] \leq_{s t} X_{2}^{l} .

(29)

Now, taking into accounts the ordering relations in (28) and (29) and using Theorem 6.B.3 in Shaked and Shanthikumar [27], we get

S_{2, X ̠}^{l} \leq_{s t} S_{2, X ̠^{l}},

where

S_{n, X ̠^{l}} = \sum_{i = 1}^{n} X_{i}^{l},

for every

n = 1, 2, \dots

. We use the induction method and assume that

S_{n, X ̠}^{l} \leq_{s t} S_{n, X ̠^{l}}

(30)

holds true for

n = m \geq 2,

i.e.,

S_{m, X ̠}^{l} \leq_{s t} S_{m, X ̠^{l}}

. From known properties of the usual stochastic order

S_{m + 1, X ̠}^{l} \leq_{s t} S_{m, X ̠}^{l} + X_{m + 1}^{l} \leq_{s t} S_{m, X ̠^{l}} + X_{m + 1}^{l} =_{s t} S_{m + 1, X ̠^{l}},

which means that (30) stands valid for

n = m + 1

. Thus, we proved that

S_{n, X ̠}^{l} \leq_{s t} S_{n, X ̠^{l}}

. Since the assumption (iii) means that

f_{X_{l}} (t + x) f_{X_{i}} (x) / f_{X_{l}} (x) f_{X_{i}} (t + x)

is decreasing in

x \geq 0

, thus an application of Theorem 3.2.(a) of Misra et al. [21] implies that

X_{i}^{l} \leq_{l r} X_{i}^{t},

X_{i}^{l} \leq_{s t} X_{i}^{t}

for all

i = 1, 2, \dots, n

. Note that

X_{i}^{t} =_{s t} X_{i, t}, i = 1, 2, \dots, n

where

X_{i, t} =_{s t} [X_{i} - t | X_{i} > t]

is the residual lifetime after age

t \geq 0

. From Theorem 1.A.3.(b) in Shaked and Shanthikumar [27] it follows that

S_{n, X ̠^{l}} \leq_{s t} S_{n, X ̠_{t}}

. Hence, the result is proved. □

The next example clarifies that the property that a standby system composed of used components has a greater reliability than a used standby system is fulfilled in the context of

D L R

lifetime components distribution.

Example 14.

We consider components with heterogenous independent lifetime distributions. Suppose that

X_{i} \sim G (α_{i}, λ), i = 1, 2, \dots, n

. Note that as

X_{1}, X_{2}, \dots, X_{n}

are independent rvs, consequently,

S_{n, X ̠} \sim G (\sum_{i = 1}^{n} α_{i}, λ)

, thus

f_{S_{n, X ̠}} (x) = \frac{λ^{\sum_{i = 1}^{n} α_{i}} x^{\sum_{i = 1}^{n} α_{i} - 1} e^{- λ x}}{Γ (\sum_{i = 1}^{n} α_{i})}

. We assume that

α_{i} \leq 1

for all

i = 1, 2, \dots, n

and that

α_{l} = {max}_{1 \leq i \leq n} {α_{i}}

. Therefore,

η_{X_{i}} (x) = \frac{1 - α_{i}}{x} + λ, η_{S_{n, X ̠}} (x) = \frac{1 - \sum_{i = 1}^{n} α_{i}}{x} + λ .

For all

x \geq 0,

it is seen that

\begin{matrix} η_{S_{n, X ̠}} (t + x) - η_{S_{n, X ̠}} (x) & = \frac{t (\sum_{i = 1}^{n} α_{i} - 1)}{x (t + x)} \end{matrix}

(31)

\begin{matrix} \geq \frac{t (α_{l} - 1)}{x (t + x)} = η_{X_{l}} (t + x) - η_{X_{l}} (x) . \end{matrix}

(32)

Thus, the assumption (i) in Theorem 13 holds true. It is further observed that

η_{X_{l}} (t + x) - η_{X_{l}} (x) = \frac{(α_{l} - 1) t}{x (t + x)}

which is an increasing function in

x \geq 0

, that is, the assumption (ii) in Theorem 13 is satisfied. Since

α_{i} \leq α_{l}

, for all

i = 1, 2, \dots, n

, thus the assumption (iii) in Theorem 13 is also valid, and consequently,

{(\sum_{i = 1}^{n} X_{i})}_{t} \leq_{s t} \sum_{i = 1}^{n} X_{i, t}

4. Concluding Remarks

With this work we have achieved two goals. The first is to develop some stochastic upper bounds on the random lifetime of a cold standby system that is not fresh or new and has age t, having been in operation and still functioning by time t. Two well-known stochastic orders, namely the likelihood ratio order and a weaker stochastic order, the usual stochastic order, were applied to obtain the stochastic upper bound. The interesting point is that the reliability function of the lifetime of the used cold standby system with n units is always dominated (without any further assumptions) by the reliability function of the lifetime of a cold standby system with

(n - 1)

units consisting of used components with common age t, provided that the lifetimes of the used components are shifted t times, as is the case, for example, in the burn-in process. For example, may be the case in the burn-in process, where a product is put into use for a time interval of length t before being handed over to the customer, is a realistic situation. However, the domination of this stochastic upper bound over the lifetime of the cold standby system used in terms of the likelihood ratio order requires the further assumption that the components have the

I L R

property. The second objective was to find conditions under which the lifetime of a used cold standby system with an age of t is dominated by the lifetime of a cold standby unit with used components, each with an age of t, in terms of the likelihood ratio order and the usual stochastic order. In general, and as confirmed by our research as a whole, it is found that the use of cold standby units that were previously in use for an equally long period of time (e.g., t) is preferable to a used cold standby system with age t because it satisfies larger stochastic lifetimes. Therefore, a cold standby system with components

X_{1, t}, X_{2, t}, \dots, X_{n, t}

that has random lifetime

X_{1, t} + X_{2, t} + \dots + X_{n, t}

is more reliable than a used cold standby system with random lifetime

{(X_{1} + X_{2} + \dots + X_{n})}_{t}

in most situations. We hope that the research conducted in this study will be useful to engineers and system designers.

In the future study, we will use the hazard rate order and the reversed hazard rate order to determine new bounds on the reliability function and the cumulative distribution function, respectively, of a second-hand cold standby system. We will look for conditions under which the lifetime of a used cold standby system of age t is dominated by the lifetime of a cold standby unit with used components each of age t, in terms of new stochastic orders. Stochastic comparisons between the inactivity times of cold standby systems according to known standard stochastic orders are another problem that can be studied in future work. The importance of the loss caused by further inactivity of engineering systems may motivate us to conduct the above study to design an optimal cold standby unit with less stochastic inactivity.

Author Contributions

Conceptualization, L.A. and M.K.; methodology, M.K.; software, L.A.; validation, M.K.; formal analysis, M.K.; investigation, L.A.; resources, L.A. and M.K.; writing—original draft preparation, L.A.; writing—review and editing, M.K.; visualization, M.K.; supervision, M.K.; project administration, M.K.; funding acquisition, M.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research is funded by Researchers Supporting Project number (RSP2023R392), King Saud University, Riyadh, Saudi Arabia.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors acknowledge financial support from the Researchers Supporting Project number (RSP2023R392), King Saud University, Riyadh, Saudi Arabia.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Plot of the survival function of

S_{2, X ̠_{t}} + t

(solid line) and the survival function of

{(S_{2, X ̠})}_{t}

(dot-dashed line) for

t = 0.1

and for

x \in (0.1, 5) .

Figure 1. Plot of the survival function of

S_{2, X ̠_{t}} + t

(solid line) and the survival function of

{(S_{2, X ̠})}_{t}

(dot-dashed line) for

t = 0.1

and for

x \in (0.1, 5) .

Figure 2. Plot of the likelihood ratio

L R (x) : = \frac{f_{X_{1, t} + X_{2, t}} (x)}{f_{{(X_{1} + X_{2})}_{t}} (x)}

for

t = 5

Figure 2. Plot of the likelihood ratio

L R (x) : = \frac{f_{X_{1, t} + X_{2, t}} (x)}{f_{{(X_{1} + X_{2})}_{t}} (x)}

for

t = 5

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Stochastic Comparisons of Lifetimes of used Standby Systems

Abstract

1. Introduction and Preliminaries

2. Stochastic Bounds for the Lifetime of a Used Standby System

3. Comparison of a Used Standby System with a Standby System Composed of Used Components

4. Concluding Remarks

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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