2. PRELIMINARIES AND NOTATIONS

Consider a reliability system consisting of independent components. At a specific instance, each component is either operational (working, up, on) or failed (down, off). The state of the system (operational/failed) is completely determined by the state of its elements, this dependence being reflected at the value of the so-called structure function. To set up our mathematical notation, denote by I the set of components of the structure and introduce, for each component α ∈ I, a binary random variable X_a indicating whether the component is up (X_a = 1) or down (X_a = 0). Then the state of the system will be described by the value of the structure function φ = φ(X_a,α ∈ I) (φ = 1 if the system is up and φ = 0 if the system is down). An alternative way to describe a coherent reliability structure is by providing system's minimal cut sets (C ⊆ I is a cut set if the failure of all components α ∈ C results in the system's breakdown).

As stated in Section 1, the present article focuses on a wide class of reliability structures whose failure can be efficiently described by absorbing states of finite Markov chains. The formal definition of these structures is as follows (see also Boutsikas and Koutras [2]).

Definition 2.1: A sequence of coherent structures with set of components J_n and respective family of minimal cut sets C_n,n = 0,1,2,…, will be called a Markov chain imbeddable sequence if the following hold:

The members of a Markov chain imbeddable sequence will be called Markov chain imbeddable systems (MISs).

The above definition is in compliance with the one given by Koutras [8] and can accommodate the structures studied by Chao and Fu [3,4] and Fu and Lou [7].

It is clear, by condition (a) of Definition 2.1 that state s_N is an absorbing state for the Markov chain {Y_n,n = 0,1,…}.

Let us next denote by

the transition probability matrix of the Markov chain and by

the vector of initial probabilities. Then the reliability function of the nth MIS can be expressed as (cf. Koutras [8])

In the sequel, we will confine ourselves to the special case in which the transition probabilities P(Y_n = s_j|Y_n−1 = s_i) do not depend on n; that is, P(Y_n = s_j|Y_n−1 = s_i) = p_ij for all i,j = 0,1,…, n and Λ_n = Λ₁ for all n = 1,2,… (homogeneous Markov chain imbeddable sequence or simply homogeneous MIS). This is usually the case when the components of the system are identical and work independently (i.i.d. system). Denoting by p the common survival probability of the components, the system's reliability h_n will be a function of p; that is,

Since state s_N is an absorbing state, the transition probability matrices Λ_n = Λ₁, n = 1,2,… can be decomposed as

where Λ is an N × N square matrix and

are column vectors. The nth power of Λ_n = Λ₁ now reads (see, e.g., Taylor and Karlin [11])

and substituting in formula (2.1), we arrive at the alternative expression

where

^N consists of the first N entries of π₀ and

. An assumption that will be taken for granted hereafter without explicitly mentioning it is that h_n(p) is not constant for n = 1,2,….

3. THE IFR PROPERTY PRESERVATION FOR MARKOV CHAIN IMBEDDABLE SYSTEMS

In this section we assume that the structure consists of i.i.d. components. For computational convenience, let us assume that J_i = {1,2,…, i} and denote by I = J_n = {1,2,…, n} the set of nth system's components. If at a given instance t each component has survival probability p = p(t), then the reliability of the nth system can be expressed as

The common failure rate r(t) of the n components is given by

whereas the system's failure rate is given by

Esary and Proschan [5] established a very interesting formula connecting r_S(t) and r(t), which proved to be instrumental in the development of our results. More specifically, they proved that

where

Expression (3.1) induces a simple sufficient condition on system structure that will preserve a monotone failure rate when the system is constructed out of independent alike components. More specifically, it is straightforward to verify the following:

On the basis of the above-mentioned points, the behavior of g as a function of p can determine the IFR property preservation from the component level to the system level. In the sequel we will provide sufficient conditions guaranteeing the monotonicity of g when the system under investigation belongs to the MIS family.

Note first that the transition probabilities

take on values from the set {0,p,1 − p}; that is, the nonvanishing entries of the matrix

are either p or 1 − p. Therefore, the entrywise derivative of Λ(p)

will contain only three types of element (0, 1, or −1) and does not depend on p.

In order to investigate the monotonicity of g(p) (cf. (3.2)), it is essential to develop a neat expression for

The first step toward this direction is achieved by the following proposition

Proposition 3.1: If there exists c ∈ R such that

then

Proof: We will first prove that under assumption (3.3), the following identity holds true for all n = 1,2,…:

Our assertion is apparent for n = 1. Assume next that the identity is true for n = k; that is,

Then applying the product rule of the matrix entrywise differentiation (cf. Lutkepohl [9])

for A(p) = Λ^k(p) and B(p) = Λ(p), we can write

Replacing dΛ^k(p)/dp by virtue of (3.5) and making use of (3.3), we get

Hence, (3.4) holds true for n = k + 1. Consequently, by the induction principle, we conclude that (3.4) is valid for all n = 1,2,…

The desired result is now easily obtained by a direct substitution of (3.4) in the expression

It is worth mentioning that in the case c = 1 (i.e., when ΛΛ′ = Λ′Λ), h_n′(p) takes on the more familiar form

A wide class of MISs in which the previous formula can be applied is systems whose transition matrices Λ are upper diagonal, with all elements in the same diagonal being equal. This is ascertained by the following proposition.

Proposition 3.2: Let A = [α_ij]_m×m and B = [b_ij]_m×m denote two upper triangular matrices with the elements in each diagonal being the same; that is, a_i,i+s−1 = a_s and b_i,i+s−1 = b_s for i = 1,2,…, m and s = 1,2,…, m − i + 1. Then matrices A and B commute (i.e., AB = BA).

Proof: The proof follows immediately by simple algebraic calculations. █

It is clear that if Λ is upper diagonal with all elements in the same diagonal equal, Λ′ possess the same property, and applying Proposition 3.2 for A = Λ and B = Λ′, we conclude that Λ and Λ′ commute.

Before stating our main result on the monotonicity of g(p) and the IFR property preservation under the formation of MISs, we need some additional notation. Let

be the set of states from which a one-step transition to the absorbing state is feasible and

the set of states from which the absorbing state is not accessible directly. Manifestly, the state space S is now partitioned into three subsets as follows:

Moreover, let us introduce the notation

for the probabilities that the Markov chain is visiting (at time n) the states of A and B, respectively.

We are now ready to state the main result of this section.

Proposition 3.3: Consider a Markov chain imbeddable sequence such that condition (3.3) holds true. If the quantity

is an increasing function of p, then the nth member of the family preserves the IFR property.

Proof: Let a = (a₀,a₁,…, a_N−1)′ be the vector indicator of A; that is,

(note that the absorbing state s_N has been excluded from the vector). It is straightforward to verify that Λ1 = 1 − qa = 1 − (1 − p)a and, consequently, Λ′1 = a. Hence, the reliability function h_n(p) can be expressed as

whereas h_n′(p), in view of Proposition 3.1, takes on the form

In view of the last two expressions, (3.2) yields

which can be equivalently written in the form

Observe next that 1 − a = b is the indicator vector of B and

Therefore,

and the conclusion stated in the proposition is now obvious. █

If we use the notation

we can express R_n^(B)(p) as

and the next corollary follows readily from Proposition 3.3.

Corollary 3.1: Consider a Markov chain imbeddable sequence such that condition (3.3) holds true. If the quantities

are increasing functions of p for all j ∈ B, then the nth member of the family preserves the IFR property.

4. AN APPLICATION TO THE k-OUT-OF-n SYSTEM

An illustrative example of application of the methodology presented in the previous section is offered by the k-out-of-n:F system. This structure fails if at least k components fail to work. It is clear that the lifetimes of the k-out-of-n:F systems coincide with the order statistics of the random variables describing the components' lifetimes. Therefore, proving the IFR closure property for this specific class of reliability systems is in fact equivalent to the well-known result that the order statistics of i.i.d. IFR random variables hold the IFR property as well.

An imbedding of the system is easily achieved by considering the state space S = {s₀,s₁,…, s_k} (i.e., N = k) and the Markov chain {Y_n,n ≥ 0} defined as follows:

• Y_n = s_i if exactly i of the components 1,2,…, n are down (0 ≤ i < k).

• Y_n = s_k if at least k of the components 1,2,…, n are down.

Fu and Lou [7] (see also Koutras [8]) stated that the transition matrix is given by

where p_n, n = 1,2,…, denotes the survival probability of the nth component, and used this setup to derive asymptotic results for large k-out-of-n:F systems. In the i.i.d. case (p_n = p for n = 1,2,…), it is obvious that

where

The first derivative of matrix Λ is given by

and, therefore (cf. remark following Proposition 3.2), matrices Λ and Λ′ commute.

As a consequence, we can exploit Proposition 3.3 to study the IFR property preservation of the family of k-out-of-n systems. It is evident that

whereas for the quantities R_n^(j)(p),j = 0,1,…, k − 1, we have

Hence,

and an obvious conclusion drawn from the last expression is that the quantity under investigation is an increasing function of p for all s_j ∈ B (i.e., for j = 0,1,…, k − 2). A direct application of Corollary 3.1 reveals that the k-out-of-n:F system preserves the IFR property. It is worth mentioning that in the special case k = 1, which corresponds to a series system, set B is empty and, therefore, the condition required for the application of Corollary 3.1 is trivially fulfilled.

Esary and Proschan [5] have provided a completely different proof of the IFR closure property for k-out-of-n:F systems. More specifically, they exploited the well-known relation between the tail probability of the binomial distribution and the incomplete Beta ratio to arrive at the following integral expression for the function g(p) defined in (3.2):

which reveals that g is a decreasing function of p (a convenient reference for this approach is Ross [10]).

Finally, we will illustrate how the general approach presented in Section 3 can be applied to the weighted k-out-of-n:F systems introduced by Wu and Chen [12]. These systems comprise an extension of the ordinary k-out-of-n:F system obtained by assigning a positive integer weight w_i to each of the system components. Then, the system's breakdown occurs if and only if the total weight of failed components becomes at least k.

In order to imbed the system to a Markov chain, we can again consider the state space S = {s₀,s₁,…, s_k} (i.e., N = k) and define {Y_n,n ≥ 0} as follows:

• Y_n = s_i if the total weight of the components 1,2,…, n that are down is exactly i (0 ≤ i < k).

• Y_n = s_k if the total weight of the components 1,2,…, n that are down is at least k.

In the case of i.i.d. components with survival probabilities p_n = p,n = 1,2,…, the transition probability matrix associated with the nth component takes on the form (see Koutras [8])

where

and the system's reliability can be computed with the aid of the formula

It should be stressed that, in view of Proposition 3.2., matrices M_i and M_j commute for every i ≠ j and, therefore, the multiplication of matrices M₁,M₂,…,M_n can be performed in any order. In the sequel we will investigate two special cases of weighted k-out-of-n systems. In the first, we will prove that the IFR property is preserved. However, the second case illustrates a case in which the property is not preserved.

Let us consider first the special case of a weighted k-out-of-2n:F system with k = 6 and component weights

Then, the reliability of the system can be written as h_n(p) = π′(M₁M₂)ⁿ1 = π′Λⁿ1, where Λ = M₁M₂ and M₁ and M₂ are the following k × k matrices:

Note that the minimal cut sets of this system are not all of the same size (as it happens for the ordinary k-out-of-n system). More specifically, we have now two types of cut set: cut sets of the form {2i − 1,2j − 1,2r − 1}, where i, j, and r are three different positive integers, and cut sets of the form {2i,2j}, where i and j are two different positive integers. The matrix Λ appearing in the expression h_n(p) = π′(M₁M₂)ⁿ1 = π′Λⁿ1 equals

and its derivative with respect to p reads

By Proposition 3.2, the condition required in Proposition 3.1 holds true (for c = 1) for matrices Λ and Λ′ and, therefore, we arrive at the expression

By using the notations Λ′1 = (0,2 − 2p,2 − 2p,1,2p,2p)′ = α and γ = (1,p²,p²,0,−p²,−p²)′ we can write Λ1 = pα + γ, and substituting the quantities h_n(p) = π′Λⁿ1 = π′Λⁿ⁻¹Λ1 and h_n′(p) = nπ′Λⁿ⁻¹Λ′1 in (3.2), we deduce

where

Should the quantity f (p) be an increasing function of p, the nth member of the family would preserve the IFR property. In the special case considered here, f (p) can be immediately computed as

which is easily checked to be an increasing function of p.

Let us next consider a weighted k-out-of-3:F system with k = 6 and component weights w₁ = 2, w₂ = 3, and w₃ = 4. Then the reliability of the system can be expressed as h_n(p) = π′Λ1, where Λ = M₁M₂M₃ and M₁ and M₂ are as in the previous example and M₃ is the following k × k matrix:

The matrix Λ = M₁M₂M₃ equals

and the reliability of the system takes on the form

If we assume that the structure consists of i.i.d. components with (common) survival probabilities p = p(t), then the reliability of the system can be expressed as

and its failure rate equals

On substituting R_n(t) = h_n(p(t)) = p(t)(−(p(t))² + p(t) + 1) we can easily verify that system's failure rate takes on the form

and the monotonicity of the last expression cannot be guaranteed by the monotonicity of (d/dt)[ln p(t)]. For example, if the lifetime distribution p(t) of each component is exponential with p(t) = e^−λt,t > 0,λ > 0, the system's failure rate equals

and it can be proved that this quantity is not monotone with respect to t. The last example answers (to the negative) the question of whether the class of weighted-k-out-of-n:F systems preserves the IFR property. However, specific subclasses, such us the one examined in the first example might preserve it.