On SyStem Reliability Of incReaSing multi-State lineaR k-within-( m , s )-Of-( m , n ) : f lattice SyStem

Notations m,s,n,k,pj system parameters. N = n – s + 1. H highest state for the system and components. kj minimum number of components that must be in state less than j in the sub matrix of the size m×s, j = 1 , 2, ..., H. kj minimum number of components that must be in state greater than or equal j in the sub matrix of the size m×s, j = 1 , 2, ..., H; kj = (m×s)-kj+1. k a vector of kj-s. kj a vector of kj-s. δ j number of components that are in state less than j inside


Notations
m,s,n,k,p j system parameters.N = n -s + 1.

H
highest state for the system and components.k j minimum number of components that must be in state less than j in the sub matrix of the size m×s, j = 1 , 2, …, H. k j G minimum number of components that must be in state greater than or equal j in the sub matrix of the size m×s, j = 1 , 2, …, H; k j G = (m×s)-k j +1.k a vector of k j -s.k j G a vector of k j G -s.
δ j number of components that are in state less than j inside the sub-matrix of the size m×s.x i,j the state of the component, which are located in the row i and the column j, , {0,1,..., } ij x H ∈ .
the structure function of the system, ( ) {0,1, 2,..., } x H φ ∈ .j p probability that state of the component is j, probability that state of the component is greater than or equal j, probability that state of the component is less than j, Q j = 1 -P j .
, i j

A
an event that at least l k components in state less than l, jlH ≤≤ , of the sub-matrix with the size m s × , that begin with the component (1, i) and end with the component (m, i + s -1).
µ denote the random number of events among , , Pr( ) probability that state of the system is greater than or equal j. j F probability that state of the system is less than j, the lower integer part for z.

Introduction
A binary L(k,m,s,n:F) is a two dimensional grid.Its components have only the state 1 (operating) or state 0 (failed), and arranged in m rows and n columns.This system fails if at least one (m,s) sub-matrix of its components contains k or more failed components.Many papers studied its reliability, such as [13-16, 21, 23].In the last few years, many systems generalized to MS systems, because the MS models give more limberness for modelling the equipment conditions.Such as, MS consecutive k-out-of-n:F system [9,22,24], MS k-out-of-n:F system [1,10,20], MS consecutive k-out-of-r-from-n: F system [8,19] and MS L(k,m,s,n:F) [7].In this paper, we study MS L(k,m,s,n:F).This system is a model for many applications.The system definition and illustration examples of modelling the system are given in section 2. The Boole-Bonferroni bounds are generalized in section 3, that will used for evaluation the proposed bounds.In section 4, the proposed bounds and an illustration example are given.The numerical results are presented in section 5.

The MS L(k,m,s,n:F)
The MS L(k,m,s,n:F) contains m×n components, that are ordered as a matrix of the degree m×n.The possible states of MS L(k,m,s,n:F) and its components are: 0, 1, …,H.The state of MS L(k,m,s,n:F) is less than j whenever there is at least one sub-matrix of the size m×s which contains k l or more components that are in state less than l for all j ≤ l ≤ H.In other words, ( ) < if at least one sub-matrix of the size m×s is in state less than j.The state of a sub-matrix of the size m×s is less than j if all the following inequalities are satisfied: , , categorize the MS L(k,m,s,n:F) to three cases: the system is called a decreasing MS L(k,m,s,n:F).The exact reliability of decreasing MS L(k,m,s,n:F) evaluated in Ref. [7].
the system is called an increasing MS L(k,m,s,n:F).In this case, that is more difficulty, new lower and upper bounds are proposed.
Case3: When k 1= k 2= … = k H , the system is called a constant MS L(k,m,s,n:F).This system is a special case of the increasing MS L(k,m,s,n:F) and decreasing MS L(k,m,s,n:F).
As with the binary system, the MS L(k,m,s,n:F) and the MS L( G k ,m,s,n:G) are considered as mirror images of each other.Further, the decreasing MS L(k,m,s,n:F) is an increasing MS L( G k ,m,s,n:G).The following examples illustrate this system.

The state of
For state 3: < , if at least one sub-matrix of the degree 2×2 is in state less than 3.For example, when x = ( ) δ= and 3 3 δ=.

Example 3: A Surveillance Cameras System
Given, a surveillance cameras system consists of 20 cameras that arranged in 4 rows and 5 columns.This system has 4 different surveillance levels: Good surveillance (state 3).

Each camera also has 4 different surveillance levels:
Good surveillance, in the first time (state 3).

•
Non surveillance, the camera not works (state 0).• Then: The system state is less than 1, if at least one sub-matrix of the • degree 4×3 contains at least 6 components in state less than 1.
The system state is less than 2, if at least one sub-matrix of the • degree 4×3 contains at least 4 components in state less than 2.
The system state is less than 3, if at least one sub-matrix of the • degree 4×3 contains at least 2 components in state less than 3.

Generalization of Boole-Bonferroni Bounds
The technique of Boole-Bonferroni bounds was derived by Prékopa and Boros [2,17], and improved by many papers such as [2-5, 11, 12, 17].This technique depends on the solution of the linear programming problem according to the definition of the binomial moments.
Let µ denote the random number of the events among where Pr( ) The proof of the definition of the expected value in formulae (1) can be found by Prékopa [18].The value S i,j is called the ith binomial moment of µ.

If we take
0, 0,..., 0,..., 0 The solutions of these problems give us the best possible lower and upper bounds respectively on the value of These bounds are called Boole-Bonferroni bounds.In the following, we give the known explicit solutions of the linear programming problems for V = 2 (the second order) and V = 3 (the third order).

The Second Order of Boole-Bonferroni Bounds:
By putting V=2 in the aforesaid linear programming problems and calculation 1, j S and 2,j S , j = 1, 2, 3,…, H, then the lower bound ( 1) Where: And the upper bound of j F is: sciENcE aNd tEchNology 2 .

The Third Order of Boole-Bonferroni Bounds:
By putting V=3 in the aforesaid linear programming problems and calculation 1, j S , 2,j S and 3, j S , j = 1, 2, 3,…, H, then the lower bound of j F is: And the upper bound of j F is: (

System Reliability of the Increasing MS L(k,m,s,n:F)
Calculation j F in equation ( 12) is very difficult, so we will propose an approximation for lower and upper bounds of increasing MS L(k,m,s,n:F) using Boole-Bonferroni bounds.Calculation these bounds required the knowledge of 1 , j S , 2, j S and 3, j S , that will be suggested in the following sections.Further, we can have the lower bounds and upper bounds of j R as follows: Estimation j R by one value can be given by the following formula: The maximum error is:

3. Calculation the Binomial Moment 3, j S
The binomial moment 3, j S can be given by: Pr( ) The , , ,

Pr( ) a j bj cj
A A A , 1 a b c N ≤ < < ≤ , can be calculated through the following five cases:

Case 1: c-a ≤ s-1
In this case, all the events , A is m×(s + a -c) components.Then: .

e L e e e e M j e L e e e e L g m M j m x D e e e L d e x a j b j c j j j M L e e L x t d e L m x D m
, In this case the events , In this case the events ,

Pr( ) a j b j c j
A A A by the following formula: ,

) a j b j c j a j b j c j A A A
Such that, the ,,

Pr( ) aj bj
A A can be obtained by formulas ( 23) , A can be obtained by formula (18).
In this case the two events ,

Pr( ) a j b j c j
A A A by the following formula: ) A A can be obtained by formulas ( 23), A can be obtained by formula (18).
In this case, all the events ,

j b j c j
A A A by the following formula: )

j b j c j
A A A can be obtained by formula (18).

Numerical Results
The numerical calculations of increasing MS L(k,m,s,n:F) reliability are carried out using Visual Basic Program.The computer codes were written very carefully.The new bounds and computer codes examined by previously published numerical examples for some special cases of increasing MS L(k,m,s,n:F), as shown in tables 1-3.When • m=1, the increasing MS L(k,1,s,n:F) becomes the increasing MS consecutive-k-out-of-s-from-n: F system.An example of increasing MS consecutive-k-out-of-s-from-n: F system in Ref [19] is examined by our bounds and given in table 1.When • H=1 and m=1, the increasing MS L(k,1,s,n:F) becomes the binary consecutive-k-out-of-s-from-n: F system.An example of binary consecutive-k-out-of-s-from-n: F system in Ref [6] is examined by our bounds and given in table 2. When • m=1 and s = n, the increasing MS L(k,1,n,n:F) becomes the increasing MS k-out-of-n:F system.An example of MS kout-of-n:F system in Ref [10] is examined using formula (18) and given in table 3.
The bounds of the increasing MS L(k,m,s,n:F) reliability with H = 3 and variant values of p j , k j , m, s, n are given in tables 4-7.These bounds are evaluated using second and third orders of Boole-Bonferroni bounds.The comparison between the results of second and third orders of Boole-Bonferroni bounds explained in tables 2-7 and figures 1-4.This comparison shows that the third order Boole-Bonferroni bounds are the best.

Conclusions.
In this paper, we proposed new lower and upper bounds for increasing MS L(k,m,s,n:F) reliability with i.i.d components using second and third orders Boole-Bonferroni bounds.The new bounds are examined by previously published numerical examples for some special cases of increasing MS L(k,m,s,n:F).The comparison between the results of second and third orders of Boole-Bonferroni bounds shows that the third order Boole-Bonferroni bounds are the best.

A
have common components.The number of common components between the events ,

AA
is m×(s + a -b) components and between , is m×(s + b -c) components.But there are no any common components between the events ,

A
and so the two events ,a j A , , c j A are disjoint.The number of common components between the events , is m×(s + a -b) components.So, we can find the , , ,

A
is m×(s + b -c) components.So, we can find the , , , b 1 , b 2 …,b N as variables and compute 1, 2,