A more efficient imbedding technique

In this section we develop an alternative imbedding technique, which improves the computational efficiency of the approach proposed in Section 2.1, to compute the exact reliability of the systems. Before introducing this method, we first note that some states in the state space constructed in the previous section indeed provide the same information and hence they can be combined into one state; that is, the state space can be reduced. To see this, we notice that each state in the state space Ω^R obtained in Example 2.2 contains two 2 × 1 windows in different locations (alti-tudes). These two windows can be obtained by moving a 2 × 1 window vertically in the state. From this observation, we see that the total number of failed components in each window in the states [0 1 0]^> and [1 0 1]^> is one. In other words, these two states provide the same information about the total number of failed compo-nents in each window and thus they can be combined into one state. Although the state space can be reduced in this way, however, we may loose information at each component. Therefore the proposed method in this section does not work for the Markov dependent case (i.e. assumption (iii)) except for r = m = 1.

To introduce our procedure, we still follow the two steps discussed in Section 2.1. Let k_j, 0 ≤ k_j ≤ r, denote the number of failed components in the jth column

(circle) of a r × s window for j = 1, 2, . . . , s.

Step 1. Construction of a compound pattern based on a 2-dimensional scan statistic

Based on the scan statistic S_m,n^R [r, s], we define the set G^R_(k,r×s,m) by

G^R_(k,r×s,m) = {k₁k₂· · · k_t: min

t

Xt

j=1

k_j ≥ k, 1 ≤ k₁ ≤ r and 1 ≤ t ≤ s} (2.18)

= {Λ^R₁, Λ^R₂, . . . , Λ^R_L1}, (2.19) where L1 = card(G^R_(k,r×s,m)). From (2.18), we see that each simple pattern Λ^R_l represents the number of failed components within a r × s subregion that is greater than or equal to k. Further, we define a compound pattern Λ^R_(k,r×s,m) by

Λ^R_(k,r×s,m) =

L1

[

l=1

Λ^R_l .

From this construction, we see that a R-(k, r × s, m × n):F system works if and only if none of the simple patterns Λ^R_l , l = 1, . . . , L₁, is detected in the system; that is, the compound pattern Λ^R_(k,r×s,m) does not occur in the system. Similarly, for a C-(k, r × s, m × n):F system we define the set G^C_(k,r×s,m) based on the scan statistic S_m,n^C [r, s] by

G^C_(k,r×s,m) = {k₁k₂· · · k_t: min

t

Xt

j=1

k_j ≥ k, 1 ≤ k₁ ≤ r and 1 ≤ t ≤ s} (2.20)

= {Λ^C₁, Λ^C₂, . . . , Λ^C_L2}, (2.21) where L₂ = card(G^C_(k,r×s,m)) and define a compound pattern Λ^C_(k,r×s,m) by

Λ^C_(k,r×s,m)=

L2

[

l=1

Λ^C_l .

In fact, G^R_(k,r×s,m) is equivalent to G^C_(k,r×s,m) since we do not record the locations of the failed components. Therefore, we use Λ₁, Λ₂, . . . , Λ_L to denote all the simple patterns in G^R_(k,r×s,m) (G^C_(k,r×s,m)) where L = L₁ = L₂, and the compound pattern generated by these simple patterns is Λ_(k,r×s,m) =^SL_l=1Λ_l. The following example is given to illustrate the construction of the compound pattern Λ^R_(k,r×s,m) (Λ^C_(k,r×s,m)).

Example 2.4 Suppose that (k, r, s, m, n) = (4, 3, 3, 3, n). According to (2.18) and (2.20), we obtain the collection G^R_(4,3×3,3) (G^C_(4,3×3,3)) as

G^R_(4,3×3,3) = {103, 112, 113, 121, 122, 123, 13, 202, 203, 211, 212, 213, 22, 23, 301, 302, 303, 31, 32, 33}

= {Λ1, . . . , Λ20} = G^C_(4,3×3,3),

and the compound pattern Λ_(4,3×3,3) is given by

Λ_(4,3×3,3)=

[20

l=1

Λ_l.

Step 2. Building a waiting time framework

Before constructing the finite state spaces Ω^R and Ω^C, we introduce a initial state space Ω^initial defined by

Ω^initial = {∅} ∪ {β} ∪

[L

l=1

S(Λ_l) ∪ {α₁, . . . , α_L}, (2.22)

where (i) ^SL_l=1S(Λl) is a collection of all proper subpatterns of Λ1, . . . , ΛL, (ii) α₁, . . . , α_L are absorbing states corresponding to the patterns Λ₁, . . . , Λ_L, respec-tively, and (iii) β indicates the state satisfying 0 ≤ k₁ < max(1, k − r(s − 1)).

With the same reason stated in Section 2.1, we can lump all the absorbing states α₁, . . . , α_L as one absorbing state α. Note that all the proper subpatterns in (2.22) always have the form k₁k₂· · · k_b, 1 ≤ b ≤ s − 1, and they can also be generated from the following inequalities:

max (1, k − r(s − 1)) ≤ k1 < min (k, r + 1), (2.23) and

max (0, k −

j−1X

t=1

kt− r(s − j)) ≤ kj < min (k −

j−1X

t=1

kt, r + 1), j = 2, . . . , b. (2.24) We interpret how to establish these inequalities as follows. The right-hand side of inequality (2.23) is trivial: if k₁ ≥ k, then k₁ becomes an absorbing state. On the other hand, suppose there are k₁ failed components in the first column of a

r × s window, then totally there are at most k₁ + r(s − 1) failed components in this window. If k₁ + r(s − 1) < k, then no any simple pattern can be formed under this situation. This illustrates the left-hand side of inequality (2.23) and also illustrates the meaning of the state β. The illustration for (2.24) is similar. For a fixed j, if k₁ + · · · + k_j ≥ k, then k₁k₂· · · k_j forms an absorbing state. Further, if k1+ · · · + kj + r(s − j) < k, then no any simple pattern can be formed.

With the initial state space Ω^initial, we next show how to construct the state spaces Ω^R and Ω^C.

Case (I): r = m

For r = m, the state spaces Ω^R and Ω^C are all defined to be the initial state space Ω^initial. Then there exists a Markov chain {Y_u : u = 0, 1, . . .} defined on Ω^initial and the transition probability matrix M_(k,r×s,m) has the same form as (2.9).

Under assumption (i), the transition probabilities for the imbedded chain {Y_u} are specified below:

(1) P (Y_u = k₁|Y_u−1= ∅) = P (Y_u = k₁|Y_u−1= β) =^³_k^r

1

´p^(r−k1)q^k1.

(2) P (Yu = α|Yu−1 = ∅) = P (Yu = α|Yu−1 = β) = ^P_x∈F^³r_x^´p^(r−x)q^x, where F = {x : k ≤ x < r + 1}.

(3) For j = 1, 2, . . . , s − 1,

P (Y_u = β|Y_u−1= ∅) = P (Y_u = β|Y_u−1= β)

= 1 − ^X

x1∈F1

P (Y_u = x₁|Y_u−1= β) − P (Y_u = α|Y_u−1= β)

= 1 − ^X

x1∈F1

Ãr x₁

!

p^(r−x1)q^x1 − ^X

x∈F

Ãr x

!

p^(r−x)q^x,

where F₁ = {x₁ : max (1, k − r(s − 1)) ≤ x₁ < min (k, r + 1)}.

(4) For a = 1, 2 . . . , j and j = 1, . . . , s − 1, there exists (a, j) excluding (1, s − 1) such that ^Pj+1_t=1k_t < k and {k_ak_a+1· · · k_jk_j+1} ∈ Ω^initial is the longest

subpattern. Then

P (Y_u = k_ak_a+1· · · k_jk_j+1|Y_u−1= k₁k₂· · · k_j) =

à r k_j+1

!

p^(r−kj+1)q^kj+1.

(5) For j = 1, 2, . . . , s − 1,

P (Yu = α|Yu−1 = k1k2· · · kj) = ^X

x2∈F2

Ãr x₂

!

p^(r−x2)q^x2,

where F₂ = {x₂ :^Pj_t=1k_t+ x₂ ≥ k}.

(6) For j = 1, 2, . . . , s − 1,

P (Y_u = β|Y_u−1 = k₁k₂· · · k_j) = 1− ^X

x3∈F3

Ãr x3

!

p^(r−x3)q^x3− ^X

x2∈F2

Ãr x2

!

p^(r−x2)q^x2, where F3 = {x3 : kaka+1· · · kjx3 forms a subpattern for a = 1, . . . , j}.

(7) P (Y_u = α|Y_u−1= α) = 1.

(8) Otherwise the transition probability is zero.

Finally, the exact reliability of a R-(k, r × s, m × n):F system and that of a C-(k, r × s, m × n):F system are obtained by applying Equation (2.14). We give the following example to illustrate the procedure.

Example 2.5 Given (k, r, s, m, n) = (8, 3, 3, 3, n), it is easy to generate the initial state space

Ω^inital = {∅} ∪ {β} ∪ {2, 3, 23, 32, 33} ∪ {α},

where β represents the two states 0 and 1. Then we can define a Markov chain {Y_u : u = 0, 1, . . .} associated with W [Λ(8,3×3,3)] on Ω^inital and the transition probability matrix M_(8,3×3,3) for {Y_u} under assumption (i) is given by

M_(8,3×3,3) =

















0 1 − 3pq²− q³ 3pq² q³ 0 0 0 0 0 1 − 3pq²− q³ 3pq² q³ 0 0 0 0 0 1 − 3pq²− q³ 3pq² 0 q³ 0 0 0 0 1 − 3pq²− q³ 0 0 0 3pq² q³ 0 0 1 − 3pq²− q³ 0 0 0 3pq² 0 q³ 0 1 − 3pq²− q³ 3pq² 0 0 0 0 q³ 0 1 − 3pq²− q³ 0 0 0 0 0 3pq²+ q³

0 0 0 0 0 0 0 1

















=

"

N_(8,3×3,3) C

0 I

#

.

The exact reliability of the system can be easily determined.

Case (II): r < m

When r < m, a r × s window is able to move vertically (m − r + 1) (or m) steps in a rectangular (or cylindrical) system, or we can think there are (m − r + 1) (or m) windows in the system. Our idea of constructing the state space Ω^R (Ω^C) is that we record the total number of failed components in each column (circle) vector of a r × s window, then the states in Ω^initial can be used to described the status of the window. Before constructing the state space Ω^R, we introduce the concept of a basis which plays an important role in the construction.

For a column vector λ^R = [λ₁· · · λ_m]^>, λ_i ∈ {0, 1}, i = 1, . . . , m, in a rec-tangular system, we let v^R = [v₁· · · v_m−r+1]^> be a vector so that v_t = ^Pt+r−1_i=t λ_i for t = 1, . . . , m − r + 1. Further, we let V (m − r + 1) be the collection of all vectors v^R associated with a column vector λ^R and we call this collection a ba-sis. Similarly, for a circle vector λ^C = [λ₁· · · λ_m]^>, λ_i ∈ {0, 1}, i = 1, . . . , m, and λ_i = λ_i−m for i = m + 1, . . . , m + r − 1, in a cylindrical system, we let v^C = [v1· · · vm]^> = [v^R> vm−r+2· · · vm]^> be a vector so that vt = ^Pt+r−1_i=t λi for t = 1, . . . , m, and let V (m) denote the collection of all vectors v^C associated with a circle vector λ^C. The following is an example of how we construct the bases V (m − r + 1) and V (m).

Example 2.6 Consider a R-(5, 2 × 3, 3 × n):F system and a C-(5, 2 × 3, 3 × n):F system. According to (2.22), (2.23) and (2.24), the initial state space is given by

Ω^initial= {∅} ∪ {0, 1, 2, 12, 21, 22} ∪ {α}.

There are eight possible outcomes for a column vector of length 3:





0 0 0



,





1 0 0



,





0 1 0



,





0 0 1



,





1 1 0



,





1 0 1



,





0 1 1



,





1 1 1



,

and each vector corresponds to a vector v^R mentioned above. So we get a basis

As stated in the beginning of this section, the two states [0 1 0]^> and [1 0 1]^>

provide the same information about the total number of of failed components in each window of the same altitude. Therefore, they can be combined into one state [1 1]^>. This illustrates the meaning of the basis V (2) here. Now, we can obtain a basis V (3) associated with a C-(5, 2 × 3, 3 × n):F system in a similar fashion as

Now we are ready to construct a state space Ω^R. We adopt Fu’s (1996) forward and backward principle and Aki and Hirano’s (2004) idea to develop an algorithm in our construction. Initially, let

Ω0 = V (m − r + 1).

where hi is the longest subpattern (counting forward) of the composition ωivi in Ω^initial. In (2.25), if some of the h_i’s forms the absorbing state α ∈ Ω^initial, then

and

Ω^R_i = Ω^R_i−1∪ Ω_i. (2.27)

The process will be stopped until we find an i^∗ such that Ω^R₁ ⊂ Ω^R₂ ⊂ · · · ⊂ Ω^R_i∗ = Ω^R_i∗+1 = · · ·. Following the process discussed above, a finite state space Ω^R = Ω^R_i∗

can be generated by iteration.

With the state space Ω^R, we can define a Markov chain {Y_u} on Ω^R and again the transition probability matrix has the same form as (2.9). To specify the transition probability, we let P (v^R) denote the probability of forming a vector v^R. The transition probability P (Yu = y_u|Yu−1 = y_u−1) can be determined as follows:

(1) For y_u−1 = [∅ · · · ∅]^> and v^R ∈ V (m − r + 1), we have P (Y_u = v^R|Y_u−1 = y_u−1) = P (v^R).

(2) For y_u−1= [h₁· · · h_m−r+1]^>, y_u = [h^∗₁· · · h^∗_m−r+1]^> ∈ Ω^R\ {[∅ · · · ∅]^>, α^R}, we have

P (Yu = y_u|Yu−1= y_u−1) = P (v^R),

where v^R = [v₁· · · v_r−m+1]^> is a vector so that h^∗_i is the longest subpattern of the composition h_iv_i in Ω^initial for i = 1, . . . , m − r + 1.

(3) For y_u−1 = [h₁· · · h_m−r+1]^>∈ Ω^R\ {α^R}, we have

P (Yu = α^R|Yu−1 = y_u−1) = ^X v^R∈F

P (v^R),

where F is a collection of all vectors v^R = [v₁v₂. . . v_m−r+1]^> such that some of the longest subpatterns of the composition h_iv_i’s form the absorbing state α ∈ Ω^initial.

(4) P (Y_u = α^R|Y_u−1= α^R) = 1.

(5) Otherwise the transition probability is zero.

An iterated process for a R-(k, r × s, m × n):F system can be developed in a similar way to generate the state space Ω^C. We do not give further discussions but provide an example to make our idea more transparent.

Example 2.7 Recall Example 2.6, we first note that if we use the method devel-oped in Section 2.1 to construct the state space, then the size for the resulted state space will be 65. Now, from (2.25)−(2.27) we have

Ω^R = {

We can see a significant reduction on the state space by using the current approach.

To determine the transition matrix, we need the probability P (v^R):

P ( The transition probability matrix M^R_(5,2×3,3) has the form

M^R_(5,2×3,3) =

where the essential matrix N^R_(5,2×3,3) is shown in Figure 2.3. Some transition prob-abilities are illustrated below:

P (Y_u =

The exact reliability of a R-(5, 2 × 3, 3 × n):F system can be obtained by apply-ing Equation (2.14). If assumption (ii) is considered, we only need to modify the probability P (v^R) and the transition probabilities, and apply the equation

P (S_3,n^R [2, 3] < 5) = P (W [Λ^R_(5,2×3,3)] ≥ n + 1) = ξ

Yn

t=1

N^R_(5,2×3,3)(t)1^>

to get the desired reliability. For the case of a cylindrical system, we can obtain a finite state space Ω^C as

Ω^C = {

Again, the size of the state space significantly decreases from 65 to 40 compared to the method developed in Section 2.1. Under assumption (i), we list the probability P (v^R) below:

Following the same procedure, one can easily obtain the transition matrix M^C_(5,2×3,3) and the reliability of a cylindrical system.

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