Kwiatuszewska-Sarnecka B. Reliability improvement of large series-parallel systems.

RELIABILITY IMPROVEMENT OF LARGE

SERIES-PARALLEL SYSTEMS

Kwiatuszewska-Sarnecka B.

Maritime University Gdynia , Departament of Mathematics, Gdynia, Poland

Abstract: The following excess methods of improving reliability of two-state series-parallel

systems are presented: hot reserve, cold reserve and mixed (hot and cold) reserve of single components. There is also introduced the method of system reliability improving by replacing their components by more reliable ones. New theorems on two-state limit reliability functions of homogeneous and non-homogeneous series-parallel large systems composed of components with improved reliability are presented and applied to comparison the effects of these systems different reliability improving methods.

1. Introduction

One of basic reliability structures are series-parallel systems. To define the series- parallel system, we assume that Eij, i = 1,2,...,kn, j = 1,2,...,li, kn, l1, l2,...,lk_n,

n

 N, are

components of a system, Tij, i = 1,2,...,kn, j = 1,2,...,li, kn, l1, l2,...,lkn,

n

N, are

independent random variables representing the lifetimes of components Eij , T is a random variable representing the lifetime of the system

Definition 1. A function

Rij(t) = P(Tij(u) > t) for t(,),i = 1,2,...,kn, j = 1,2,...,li, is called the reliability function of the component Eij .

Definition 2. A function

Rknln (t) = P(T( > t) for t  (-,), is called reliability function of a system.

Definition 3. A system is called series-parallel if its lifetime T is given by

T =

max

}}{min{

11



ljki

_in

T

ij

.

Definition 4. A series-parallel system is called regular if l1 = l2 = . . . =

l

k_n = ln, ln  N. Definition 5. A regular series-parallel system is called homogeneous if it consists of components with identical reliability function

R(t) =

1



F(t) for t(,).

E

₁₁

E

21

E

_kn1

E

12

E

22

E

_kn2

. .

E

_knln

E

_1ln

E

_2ln

. .

. .

p

11

p

12

p

1e₁

p

21

p

22

p

2e₂

p

a1

p

a2

p

ae_a

q

1

q

2

q

a

. . . .

Fig. 1. The scheme of a regular series-parallel system

The reliability function of the homogeneous regular series-parallel system is given by Rk_nl,_n(t) = 1[1[R(t)]ln]kn for t(,). (1)

2. Large two-state series-parallel system

In the asymptotic approach to multi-state system reliability analysis we are interested in the limit distributions of a standardized random variable (T bn)/an, where T is the

lifetime of the regular series-parallel system and an > 0, bnare some suitably chosen numbers, called normalizing constants. Since

) / )

((T b a t

P  n n  = P(T > ant + bn) = Rk ,n ln (ant + bn), then we assume the following definition.

Definition 6.

A function (t), t(,), _{is called the limit reliability function of the regular} series-parallel system with reliability function Rkn,ln(t)if there exist normalizing constants an>0, bn such that

 

nlim Rk ,n ln (ant + bn) = (t)

for t  C, where C is the set of continuity points of (t).

The knowledge of the system limit reliability function allows us, for sufficiently large n and t(,), to apply the following approximate formula

Rk ,nln (t)  ( n n

a

b

t



). (2) Definition 7.

A regular homogeneous series-parallel system is called exponential system if its components have exponential reliability function

1 ) (t 

R _{for t < 0,} R(t)exp[t] for t  0, 0.

Proposition 1. If components of the homogeneous regular series-parallel system have exponential reliability functions

and kn k, ln





, , 1 n n l a   bn 0, then ) t (  _{1[1exp[t]]}k for t  0,

is its limit reliability function.

From Proposition 1 and after applying (2), we get the following approximate formulas for this system reliability function

Rk ,nln (t) k nt l ]] exp[ 1 [ 1    for t  0. (3)

3. Reliability improvement of series-parallel system

There are many possibilities to improve reliability of a system. The reliability improvement by using hot, cold and mixed reserve of system’s components will be presented. First the systems with hot, cold, mixed reserve of its components will be defined. Next the exact reliability function of these types of series-parallel systems will be found. We assume that the basic and reserve components have the same reliability function.

Definition 8. A system is called series-parallel system with hot reserve of its components if its lifetime _T(1) _{is given by}

) 1 ( T

{max

}},max{min

211

1



i

_in

kljk



T

ijk



where Tij1 are lifetimes of its basic components and Tij2 of its reserve components.

Fig. 2. The schema of component with hot reserve

E₁₁₁

The reliability function of the homogeneous regular series-parallel system with hot reserve is given by ) ( ) 1 ( _t n nl k R = 1[1[1(F(t))2]ln]kn_{, for} t(,) (4) Definition 9. A system is called series-parallel system with cold reserve of its components if its lifetime _T(2) _{is given by}

) 2 ( T

min{max

}},{

2

11

1







 k

ljki

_in

T

ijk

where Tij1 are lifetimes of its basic components and Tij2 are lifetimes of its reserve

components.

Fig. 3. The schema of component with cold reserve

The reliability function of the homogeneous regular series-parallel system with cold reserve is given by ) ( ) 2 ( _t n nl k R _{= 1}_[1_[1_F(t)_F(t)]ln_]kn _{, for} t(,), (5) Definition 10. A system is called series-parallel system with mixed reserve of its components if its lifetime _T(3) _{is given by}

E₁₁₁

}}}}{min},

max{min{min{{max

2

112

111

)3(

11







 k

klsjki

ijk

ljls

TT

ijk

T

nn

nn

,

where Tij1 are lifetimes of its basic components and Tij2 are lifetimes of its reserve

components, and s1, s2, where s1 s2 1are fractions of the components with hot and

cold reserve, respectively.

Fig. 4. The schema of component with mixed reserve

The reliability function of the homogeneous regular series-parallel system with mixed reserve is given by

E₁₁₁

E₁₁₂

E_1j1

) ( ) 3 ( _t n nl k R = 1[1[1(F(t))2]s1ln [1F(t)F(t)]s2ln]kn _for ), , (   t ₍₆₎

4. Reliability improvement of large series-parallel system

The asymptotic reliability functions of the systems with hot, cold and mixed reserve of its components will be found and next the system with components with better reliability will be defined and its limit reliability function will be presented.

Proposition 2. If components of the homogeneous regular series-parallel system with hot reserve of its components have exponential reliability functions

and kn k, ln





, , 1 n n l a   b_n _0, then ) (t  _{1[1exp[t ]]}2 kfor t  0,

is its limit reliability function.

From Proposition 2 and after applying (2), we get the following approximate formulas for this system reliability function

) ( ) 1 ( _t n nl k R k nt l ) ]] ( exp[ 1 [ 1    2  for t  0 (7)

Proposition 3. If components of the homogeneous regular series-parallel system with cold reserve of its components have exponential reliability functions

and kn k, ln





, , 2 n n l a   b_n 0, then ) (t  _{1[1exp[t ]]}2 kfor t  0,

From Proposition 3 and after applying (2), we get the following approximate formulas for this system reliability function

) ( ) 2 ( _t n nl k R lnt _]]k 2 ) ( exp[ 1 [ 1 2



    for t  0 (8)

Proposition 4. If components of the homogeneous regular series-parallel system with mixed reserve of its components have exponential reliability functions

and kn k, ln





, , ) 2 ( 2 2 1 n n l s s a    bn 0, then ) (t  _{1[1exp[t ]]}2 kfor t  0,

is its limit reliability function.

From Proposition 4 and after applying (2), we get the following approximate formulas for this system reliability function



)

(

) 3 (

_t

n nl k

R

s s lnt _]]k 2 ) ) 2 ( ( exp[ 1 [ 1 2 2 1   



for t  0, i = 1,2,...,a, (9) Definition 11. A exponential regular series-parallel system is called the system with reduced failure rates of its components by the factor



, 01_{, if their reliability}

function is of the form

1 ) (

~ _{t }

R for t < 0, R~(t)exp[t] for t  0.

Proposition 5. If components of the homogeneous regular series-parallel system have exponential reliability functions with reduced failure rates by the factor



and kn k, ln





, , 1 n n l a   b_n 0, then ) (t  _{1[1exp[t]]}k for t  0,

From Proposition 5 and after applying (2), we get the following approximate formulas for this system reliability function



)

(

) 4 (

_t

n nl k

R

k nt l ]] exp[ 1 [ 1   for t  0 (10)

5. Comparison of relibility improvement effects

The comparison of the system reliability improvement effects in the case of the reservation to the effects obtained by its components reliability improvement may be got by solving with respect to the factor



at the moment t  0, the following equation

 n n a b t ( ) 4 ( ) =  n n a b t ( ) ( ),  1,2,3, (11) which results in the following corollaries.

Corollary 1

The factor



decreasing components failure rate of the homogeneous exponential regular series-parallel system equivalent with its components reserve as a solution of the comparison (11) is given by

for κ1 (from (10) and (7)) t



  for t  0,

for κ2 (from (10) and (8))

2

t



  for t  0,

for κ3 (from (10) and (9))

2

)

2

)(

s

₁



s

₂

t







for t  0.

6. Conclusions

Presented methods of the system reliability improvement supply with the practitioners a simple mathematical tools which can be used in everyday practice. The methods may be

useful not only in the technical objects exploitation processes but also in their new processes designing, especially in their optimisation. The case of series-parallel systems composed of components having exponential reliability functions with double reserve of their components is considered only. It is possible to extend the results to the systems having other much complicated reliability structures and components with different from the exponential reliability function.

References

1. Kolowrocki K.: Reliability of Large Systems. Amsterdam Boston Heidelberg London

-New York - Oxford - Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo: Elsevier, 2004

2. Kolowrocki K., Kwiatuszewska-Sarnecka B.: Asymptotic approach to reliability improvement

of non-renewal systems. Chapter 6 and Chapter 34 (in Polish). Gdynia: Maritime University.

Project founded by the Polish Committee for Scientific Research, 2005

3. Kwiatuszewska-Sarnecka B.: Analysis of Reserve Efficiency in Series Systems. PhD Thesis, (in