On asymptotic approach to reliability improvement of multi-state systems with components quantitative and qualitative redundancy: „m out of n” systems Текст научной статьи по специальности «Клиническая медицина»

Kwiatuszewska-Sarnecka Bozena

Gdynia Maritime University, Poland

On asymptotic approach to reliability improvement of multi-state systems with components quantitative and qualitative redundancy: „m out of n" systems

Keywords

reliability improvement, limit reliability functions Abstract

The paper is composed of two parts, in this part the multi-state homogeneous „m out of n" systems with reserve components are defined and their multi-state limit reliability functions are determined. In order to improve of the reliability of these systems the following methods are used: (i) a warm duplication of components, (ii) a cold duplication of components, (iii) a mixed duplication of components, (iv) improving the reliability of components by reducing their failure rate. Next, the effects of the systems' reliability different improvements are compared.

1. Introduction

Presented paper is continuation of a work about reliability improvement of large system. In the first part of this work are defined the component's and system's multi-state reliability functions and next the asymptotic approach are brought forward. There are presented results concerned with improvement of large series and parallel systems, their multi-state limit reliability functions in case when the systems have reserve components and in case when the reliability of components is improved by reducing their failure rate. As the main result are found the forms of reducing their failure rate factor for both kinds of large systems.

2. Reliability improvement of a multi-state „m out of n" system

Definition 2.1. A multi-state system is called an „m out of n" system if its lifetime in the state subset {u,u+1,...,z} is given by

T(u) = T(n-m+i)(u), m = 1,2,...,n, u = 1,2,...,z,

where T(n-m+1) (u) is the m th maximal order statistics in

the sequence of the component lifetimes T (u), T»,..., Tn (u).

Figure 1. The scheme of a homogeneous „m out of n" system

The above definition means that the multi-state „m out of n" system is in the state subset {u,u+1,...,z} if and only if at least m out of n its components is in this state subset and it is a multi-state parallel system if m = 1 and it is a multi-state series system if m = n.

Definition 2.2. A multi-state „m out of n" system is called homogeneous if its component lifetimes Ti(u) in the state subsets have an identical distribution function

Fi(t,u) = F(t,u), u = 1,2,...,z, t e(-<x>,<x>), i = 1,2,...,n.

The reliability function of the homogeneous multi-state „m out of n" system is given either by

R nm)(t ,•) = [1R nm)(t,1),...R nm)(t,z)],

where

R(„m)(t, u) = 1 - "¿X" )[R(t, u)]' [F(t, u)]n-i ,

i=0

t e(-w,w), u = 1,2,...,z, or by

R(m)(t,) = [1, Rf )(t,i),..., R«(t, z)], where

R(m)(t,u) = £ (n)[F(t,u)]' [R(t,u)]n-i , t e(-¥,¥)

i=0

m = n -m, u = 1,2,...,z.

Lemma 2.1. case 1: If

(i) IM d)(m)(t, u)= 1 - exp[-V(t, u)],

i=o i!

u = 1,2,...,z, is non-degenerate reliability function,

(ii) IR<xi": (t,u) is the reliability function of non-degenerate multi-state „m out of n" system with a hot reserve of its components defined by (16),

(iii) an(u) > 0, bn (u)e (-¥,¥), u = 1,2,...,z,

(iv) m = constant (m / n ® 0, as n ® ¥), then

Definition 2.3. A multi-state system is called an „m out of n" system with a hot reserve of its components if its lifetime J(1)(u) in the state subset {u,u+1,...,z} is given by

(u) = T(n-m+1)(u), m = 1,2,...,n, u = 1,2,...,z,

where T(n-m+1) (u) is the m-th maximal order statistics in the sequence of the component lifetimes

Ti(u) = max{r.(u)}, i = 1,2,..,n, u = 1,2,...,z,

1<j<2 J

where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components.

The reliability function of the homogeneous multi-state „m out of n" system with a hot reserve of its components is given either by

IR(1)nm) (t,• ) = [1, IR(1,z),..., IR(1)nm; (t,z) ], where

,(1)( m\

?(1)(m),

IR(1)nm)(t,u) = 1 - z(n)[1 -(F(t,u))2]''[F(t,u)]2(n-i) , (1)

i=0

t e(-¥,¥), u = 1,2,...,z, or by

IR(1)nm)(t,-) = [1, IR(1)n™)(t,1),..., IR(l)f\t,z)], where

IR (1)nm }(t, u)

= £ (r )[(F (t, u))]2' [1 - (F (t, u))2]

i=0

m = n - m, t e(-¥,¥), u = 1,2,...,z.

2 (n-i)

(2)

lim IR(1)(„m) (an (u)t + bn (u)) = M (1)(m) (t, u),

t e C, u = 1,2,...,z,

if and only if

lim n[1-F (an (u)t + bn (u))] = V(t,u), t e CV, u = 1,2,...z,

case 2: If

x2

(i) IM (1)(mJ(t,u) = 1 —— J e 2 dx,

u = 1,2,...,z, is non-degenerate reliability function,

(ii) IR {l)<n'

(t, u) is the reliability function of non-degenerate multi-state „m out of n" system with a hot reserve of its components defined by (16),

(iii) an (u) > 0, bn (u)e (-¥,¥), u = 1,2,...,z,

(iv) m / n , 0 <m< 1, as n ®¥ ,

then

lim IR(1)(nm) (an (u)t + bn (u)) = IM (t, u),

t e Cm, u = 1,2,...,z,

if and only if

(n +1)[1 - F2 (an (u)t + bn (u))] - m

lim-. —-= v (t, u),

m(n - m +1)

n +1 u = 1,2,...,z.

case 3: If

(i) IM }(t,u) = £ V^Lexp[-V(t,u)],

i=0 i!

(m),

m = n - m, u = 1,2,...,z, is non-degenerate reliability function,

JR (l)(m )

an(u) =

4n1 (u)

, bn (u) =0, u = 1,2,...,z,

(ii) JR(1)n (t, u) is the reliability function of non- then

(iii) (iv) then

degenerate multi-state „m out ofn" system with a hot reserve of its components defined by (17),

an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z, n - m = m = constant (m / n ® 1 as n ® <),

lim (1)im\an(u)t + bn(u)) = IÂ (1)(M}(t,u), t eC, u = 1,2,...,z,

if and only if

lim n[ F (an (u)t + bn (u)]2 = V (t, u),t ÎC- , u = 1,2,...,z.

Proposition 2.1. If components of the homogeneous multi-state „m out of n" system with a hot reserve of its components have multi-state exponential reliability functions and

case 1 m = constant,

an(u) = , bn(u) = —^log 2n , u = 1,2,...,z,

I (u) I (u)

then

JÂ (1)(m)(t,u) = 1- "l-'exp|-//| exp[-exp[-t]],

i=o i!

t e (-œ,œ), u = 1,2,...,z, case 2 m / n ® m ,0 < m < 1, n ® ¥,

an (u) =-^-, bn(u) =T^yl1 -m ,

1(u)Wn + 1 l (u)

u = 1,2,...,z,

then

x2

1 t

JÂ (1)(mV,u) = 1 - Je 2 dx, V2p -¥

t e (-œ,œ), u = 1,2,...,z, case 3 n - m = m = constant, ( m / n ® 1, n ® ¥ ),

IÂ(1)(m}(t,u) = 1, t <0,

-ri\(m) n-m t -,

IÂ (1)( )(t,u) = 2 — exp[-t2], t > 0, u = 1,2,...,z,

i=o i!

is its limit reliability function.

Proof:

case 1: Since for all fixed u, we have

an (u)t + bn (u) as n . Therefore

F(t, u) = lim n[1 - F (an (u)t + bn (u))]

= lim n[2exp[-l(u)(an (u)t + bn (u))]

- exp[-21 (u)(an (u)t + bn (u ))]] = lim2n exp[-1 (u)(an (u)t + bn (u))]

[1 -1 exp[-1 (u)(an (u)t + bn (u))]]

= lim exp[-t][2n exp[-l(u)bn (u)] - n exp[-t ] exp[-21 (u)bn (u)]]

= lim exp[-t][2n1 -n—!-exp[-t]]

n®¥ n n

= exp[-t], t e (-¥, ¥), u = l,2,...,z, which by case l in Lemma 2. l completes the proof. case 2: Since for all fixed u, we have

an (u)t + bn (u) as n , moreover

l - F 2(an (u)t + bn (u))

1

= 2 exp[-1 (u)(an (u)t + bn (u))]

= [1 - exp[-1 (u )(an (u)t + bn (u))]]

- exp[-21 (u)(an (u)t + bn (u))]

= 2[1 -1(u)(an (u)t + bn (u))

t i

= [1 - exp[ —]]2 , t > 0. Vn

Therefore

-■212(u)(an (u)t + bn (u))2]

- [1 - 21 (u)(an (u)t + bn (u))

V(t, u) = lim n[F(an (u)t + bn (u))]2 = 0 , t < 0,

u = 1,2,...,z, and

1 2 2 1

+ - 412 (u)(an (u)t + bn (u))2] + o(---)

2 (n +1)

V(t, u) = lim n[F(an (u)t + bn (u))]

= 1 -12(u)(an (u)t + bn (u))2 + o(-L_),

(n +1)

t 2

= lim n[1 - exp[——]]

n®¥ Vn

next

(n +1)[1 - F2(an (u)t + bn (u))]] - m

v (t, u) = lim-, ==-

m(n - m +1)

n+1

= lim

, Vm(i-m) , 1

(n +1)[--—=-1 + m- o(^=)] - m

Vn + 1 Vn + 1

m(n - m +1) n +1

t 1 1 1 = lim n[— + o(—)]2 = t2, t > 0,

n®¥ Vn Vn u = 1,2,...,z,

which by case 3 in Lemma 2.1 completes the proof.

Corollary 2.1. The reliability function of exponential „m out of n" system with a hot reserve of its components is given by

case 1

-vW -m)t+o(1)

= lim --- = -t, t £ (-¥, ¥),

>(1 -m)

u = 1,2,...,z,

which by case 2 in Lemma 2.1 completes the proof. case 3: Since for all fixed u, we have

TD (1)(m) „ X - 1 m-1exp[-i(1 (u)t - log 2n)]

IR 'n (t, u) - 1 -:-

i=0

exp[-exp[-1 (u)t + log2n],

t £ (-¥,¥), u = 1,2,...,z.

case 2

(3)

t

an (u)t + bn (u) =-p< 0 for t < 0

1 (u)V n

and

an (u)t + bn (u) =

i (u)Vn

> 0 for t > 0,

then

F 2(an (u)t + bn (u)) = 0, t < 0 and

F 2(an (u )t + bn (u))

1 3 x IR^\t,u) -1 --= Je" 2 dx

V 2p

where

(4)

_ 21 (u)4n + \ 2^fn + ^yf\—m , ,

3 = --t--^-— , t £ (-¥, ¥), (5)

Vm

u = 1,2,...,z. case 3

IR(1)(nm}(t, u) = 1, t < 0,

2

2

2

t

IR(1)(nm)(t,u) @ Y [1 (u)4~ntf exp[-l2(u)nt2], (6)

i=o i!

t > 0, u = 1,2,...,z.

Definition 2.4. A multi-state system is called an „m out of n" system with a cold reserve of its components if its lifetime T2)(u) in the state subset {u,u+1,...,z} is given by

T-2) (u) = T(n-m+1)(uXm = 1,2,...,n, u = 1,2,...,z,

where T(n-m+1) (u) is the m-th maximal order statistics in the sequence of the component lifetimes

2

Ti(u) = 2Tij (u), i = 1,2,..,n, u = 1,2,...,z,

j=1

where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components.

The reliability function of the homogeneous multi-state „m out of n" system with a cold reserve of its components is given either by

IR(2)nm)(t,) = [1, IR^ (t,1),..., IR^ (t,z)], where

IR (2)(nm)(t, u)

m-1/ \

1 - s(r)[1 - F(t,u) * F(t,u)]

(7)

= •[F(t, u) * F(, u)]n-i t e (-<,<), u = 1,2,...,z,

or by

IR^(t,) = [1, IR(2)(„m)(t,1),...,IR^(t,z)], where IR(2) f)(t,u)

= 2 (n )[F(t, u) * F(t, u)] [1 - F(t, u) * F(t, u)]n-i, (8)

i=0

t e (-<,<), m = n - m, u = 1,2,...,z.

Lemma 2.2. case 1: If

(i) I« (2)(m) (t, u)= 1 - exp[-V(t, u)],

i=0 i!

u = 1,2,...,z, is non-degenerate reliability function,

(ii) IR(2)nm (t, u) is the reliability function of non-degenerate multi-state „m out of n" system with a cold reserve of its components defined by (24),

(iii) an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z,

(iv) m = constant (m / n ® 0, as n ), then

lim IR(2)nm)(an (u)t + bn (u)) = I« (1)(m)(t, u), t e Ci«,

if and only if

lim n [F(an (u)t + bn (u)) * F(a„ (u)t + bn (u))] ]

= V(t,u), t e CV, u = 1,2,...z, case 2: If

x2

(i) I« (2)(mJ(t, u) = 1 —— J e 2 dx,

u = 1,2,...,z, is non-degenerate reliability function,

(i) IR{2)(n') (t,u) is the reliability function of non-degenerate multi-state „m out of n" system with a cold reserve of its components defined by (24),

(iii) an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z,

(iv) m / n , 0 <m< 1, as n ,

then

lim IR(2) (nm) (an (u)t + bn (u)) = I« (2)(m} (t, u),

t e C

i«,

if and only if

lim (n +1)[1 - F(an (u)t + bn (u)) * F(an (u)t + bn (u))] - m

m(n - m +1) n +1

=v (t,u), u = 1,2,...,z. case 3: If

(i) I« (2)(m )(t, u) = 2 ^^ exp[-V (t, u)]

i=0 i!

(111) (1V)

then

m = n - m, u = 1,2,...,z, is non-degenerate reliability function,

IR(2)i ) (t, u) is the reliability function of non-degenerate multi-state „m out of n" system with a cold reserve of its components defined by (25),

an(u) > 0, bn (u)e (-¥,¥), u = 1,2,...,z, n - m = m = constant (m / n ® 1 as n ® ¥),

l1m Itf (2)(nm)(a„ (u)t + bn (u)) = IÂ (2)(m\t, u),

t eC

iâ

1f and only 1f

then

IÂ(2)(™}(t, u) = 1, t < 0,

-r*,\(m) n-m t 1 i

IÂ (2)( J(t,u) = 2 — exp[-t2] , t > 0, u = 1,2,...,z,

i=o i!

is its limit reliability function.

Proof:

case 1: Since for all fixed u, we have

an (u)t + bn (u) ®¥ as n ®¥, t e (-¥, ¥). Therefore

l1m n[F (an (u)t + bn (u) * F(an (u)t + bn (u)] = V (t, u), V(t, u) = 11m n[1 + 1 (u)a (u)t + bn (u))]

t eC v, u = 1,2,...,z.

Proposition 3.2. If components of the homogeneous multi-state „m out of n" system w1th a cold reserve of 1ts components have multi-state exponential rel1ab1l1ty funct1ons and

case 1 m = constant,

1 exp[1 (u)bn (u )] an(u) =-, -:-= n , u = 1,2,...,z,

1 (u) 1 (u)bn (u)

then

IÂ (2)(m)(t,u)= 1-mi:1exp[~it] exp[-exp[-t]],

i=o i!

t e (-¥,¥), u = 1,2,...,z,

case 2 m / n ® m 0 < m < 1, n ®¥,

exp[-1 (u)(an (u)t + bn (u))]

= lim n[--

n®¥ exp[1 (u)bn (u)]

, l(u)bn(u)

+-n-] exp[-t]

exp[1 (u)bn (u)]

= exp[-t], t e (-¥,¥), u = 1,2,...z,

which by case 1 in Lemma 2.2 completes the proof.

case 2: Since for all fixed u, we have

an (u)t + bn (u) =

^ > / \ yî-^ , ~

an(u) =-bn (u) = —-, u = 1,2,...,z,

1(u)V n +1 1 (u)

—71 -m >0

m

1 (u^Vn+T

1 (u)

m > 0 as n ® ¥

1 (u)

and

then

IÂ (2)(m)(t,u)=1 - Je" 2 Jx, t e (-¥,¥), u = 1,2,...,z,

case 3 n - m = m = constant ( m / n ® 1, n ® ¥ ),

an(u) =

V2

(u)

bn (u) =0, u = 1,2,...,z,

1 - F (an (u)t + bn (u )) * F (an (u)t + bn (u))

= [1 + 1 (u)(an (u)t + bn (u))]

exp[-1 (u)(an (u)t + bn (u ))]

= [1 + 1 (u)(an (u)t + bn (u))]

[1 -1 (u)(an (u )t + bn (u )) +

2

l 2 2 l - 12(u)(an(u)t + bn(u))2 -o(—)]

n +1

= l -1 l\u)(an (u)t + bn (u ))2

- o( ~t)] , t e (-¥, ¥). n +1

. n tV2x r tV^

= l - (l +)exp[--—]

Vn Vn

t2 l = —+ o(—), t > 0. n n

Therefore

Therefore v(t, u)

lim (n + l)[l - F(an (u)t + bn (u)) * F(an (u)t + bn (u))] - m m(n - m +1) n +1

= lim

(n + ) t + m - o(J-)] - m Vn + l_n +1

m(n - m +1)

n+l

-^/m (1 -m)t + o(l)

= lim---= -t, t e (-¥, ¥),

n®¥ Vm (1 -m)

which by case 2 in Lemma 2.2 completes the proof. case 3: Since for all fixed u, we have

v(t, u) = lim n[F(an (u)t + bn (u))

* F (an (u)t + bn (u))]

ri n tV2x r tV2in

= lim n[l - (1 + —) exp[--—]]

n®¥ yjn Vn

12 1

= lim n[— + o(—)] = t2, t > 0, u = 1,2,...z,

n®¥ n n

which by case 3 in Lemma 2.2 completes the proof.

Corollary 2.2. The reliability function of exponential „m out of n" system whit a cold reserve of its components is given by

case 1

re(2)(m)„ m^pH1 (u)(t - bn (u ))]

IR 'n (t,u) = 1 -

i=o i!

an (u)t + bn (u) =

V2t

1 (u)^

< 0 for t < 0

exp[- exp[-1 (u)t + 1 (u )bn (u)], (9)

t e (-¥,¥), u = 1,2,...,z.

and

case 2

an (u)t + bn (u) =

tV2

1 (u)^

> 0 for t > 0,

1 3 - _

IR^}(t,u) @ 1 --= Je" 2 dx V2p

(l0)

then

where

F(an (u)t + bn (u)) = 0 for t < 0 and

F(an (u)t + bn (u)) * F(an (u)t + bn (u)) = [1 - [1 + 1 (u)(an (u)t + bn (u))]

1 (u^Vn+I Vn+Wl-m , ,

3 = v ^-t--;=—— , t e (-¥, ¥). (11)

case 3

IR(2)(nm}(t, u) = 1, t < 0,

exp[-1 (u)(an (u )t + bn (u))]]

IR (2)(nm }(t, u)]

2

- Y [1 (u)^^ exp[-l2(u)nt2/2], (12)

i=0 i!

t > 0, u = 1,2,...,z.

Definition 2.5. A multi-state series system is called an „m out of n" system with a mixed reserve of its components if its lifetime T^u) in the state subset {u,u+1,...,z} is given by

T(3) (u) = T(n-m+1)(u),m = 1,2,...,n, u = 1,2,...,z,

where T(^^ (u) is the m-th maximal order statistics in the sequence of the component lifetimes

2

Ti (u) = { max {max{Tij (u)}}, max { £ T. (u)}},

1<i<s\n 1<j<2 s1n+1<i<n j=1

i = 1,2,..,n., u = 1,2,...,z,

where Ti1(u) are lifetimes of components in the basic system and Ti2(u) are lifetimes of reserve components and si, s2, where si+ s2 = 1 are fractions of the components with hot and cold reserve, respectively.

The reliability function of the homogeneous multi-state „m out of n" system with a mixed reserve of its components is given either by

IR^(t,•) = [1, IRQrn" (t,1),..., IRQrn" (t,z)], where

(3)(m)

(3)(m)

IR

(3)(m)(t) = 1 - ^(n)[1 - (F(t,u))2F

i=0

[1 - F (t, u) * F (t, u)]s2i [F (t, u)]2(n-i )s1

[F(t,u)*F(t,u)](n-i>2 , t £ (-¥,¥), u = 1,2,...,Z,

or by

IR®?^,.) = [1, IR^i),...,IR(3)?V,z)], where

(13)

IR

(3) (m )/A _

(t) = £ (n ) [ F (t, u)]2s1i [ F (t, u) * F (t, u)]s2i

case 1: If

(i) IM (3)(m)(t,u) = 1 - exp[-V(t,u)],

i=0 i!

u = 1,2,...,z, is non-degenerate reliability function,

(ii) IR(3)(nm) (t, u) is the reliability function of non-degenerate multi-state „m out of n" system whit a mixed reserve of its components defined by (30),

(iii) an(u) > 0, bn (u)e (-¥,¥), u = 1,2,...,z,

(iv) m = constant (m / n ® 0, as n ®¥),

then

lim IR (3)(nm)(an (u)t + bn (u)) = IM (3)(m)(t, u),

n®¥

t e Cim, u = 1,2,...,z, if and only if

lim n[sJ1 - [ F (an (u)t + bn (u))]2]

+ s2 [1 - F(an (u)t + bn (u)) * F(a„ (u)t + bn (u))]]

=V(t,u), t e CV, u = 1,2,...z, case 2: If

x2

(i) IM (3) '(t,u)= 1 —— J e 2 dx, u =

v2p ¥

1,2,...,z, is non-degenerate reliability function,

(ii) IR(3)(nm:

(t, u) is the reliability function of non-degenerate multi-state „m out of n" system whit a mixed reserve of its components defined by (30),

(iii) an(u) > 0, bn (u)e (-¥,¥), u = 1,2,...,z,

(iv) m / n i m , 0 < m < 1 , przy n i ¥ , then

lim IR(3) (nm)(an (u)t + bn (u)) = IM (3)(m )(t, u),

n®¥

t e Cim, u = 1,2,...,z, if and only if

lim(n + 1)[st[1 - [F(an (u)t + bn (u))]2] m(n - m +1)

n+1

[1 -(F(t,u))2](n-i)s1[1 -F(t,u)*F(t,u)](n-i)s2 , (14) t £ (-¥,¥), m = n -m, u = 1,2,...,z.

Lemma 2.3.

s 2 [1 - F (an (u)t + bn (u)) * F (an (u)t + bn (u))]] - m m(n - m +1) n +1

=v (t,u), u = 1,2,...,z.

case 3: If (i)

IÂ(3)(m}(t,u) = Z [V(t:u)]i exp[-V(t,u)]

i=0 i!

m = n - m, u = 1,2,...,z, is non-degenerat reliability function,

(ii) IR ^\t, u) is the reliability function of non-degenerate multi-state „m out ofn" system with a mixed reserve of its components defined by (31),

an(u) > 0, bn (u)e (-<,<), u = 1,2,...,z, n - m = m = constant (m / n ® 1 as n ® <),

(iii) (iv) then

lim IR

(3)(m )

(an(u)t + bn(u)) = IÂ (3)(m}(t,u),

t eCIÂ, u = 1,2,...,z, if and only if

lim n[sx[F(an(u)t + bn(u))]2

+ s 2 [ F (an (u)t + bn (u )) * F (an (u)t + bn (u))]]

= V (t, u )

t eC-, u = 1,2,...,z.

Proposition 2.3. If components of the homogeneous multi-state „m out ofn" system with a mixed reserve of its components have multi-state exponential reliability functions and

case 1 m = constant,

an (u) =

1

exp[1 (u)bn (u)] 1(u) ' 2sj + s21(u)bn(u)

= n, u = 1,2,...,z,

then

IÂ (3)(m)(t,u) = 1- Z

i=0 i!

t e (-œ,œ), u = 1,2,...,z,

exp[-it]

exp[-exp[-t]],

case 2 m / n 0 <m< 1, n ,

Vm/2

an(u)=

bn(u)=

1 (u )^/(2sj + s 2)(n +1)

1 (u) y

2(1 -m )

2sj + s 2

u = 1,2,...,z,

then

1 t

I (3)(m}(t,u)= 1 - Je~Tdx, t e (-œ,œ), u = 1,2,...,z, case 3 n - m = m = constant ( m / n ® 1, n ®<x> ),

V2

an (u) =

1 (u )^/(2sj + s 2)n

, bn (u) = 0, u = 1,2,...,z,

then

IÂ(3)(m}(t, u) = 1, t < 0,

-r-,\(m) n-m t -,

IÂ (3)( J(t,u) = Z — exp[-t2], t > 0, u = 1,2,...,z,

i=0 i!

is its limit reliability function.

Proof:

case 1: Since for all fixed u, we have

an (u)t + bn (u) ® < as n ® < for t e (-<, <), and

1 -[F(an (u)t + bn (u))]2 = 2 exp[-1 (u)(an (u)t + bn (u))] - exp[-21 (u)(an (u)t + bn (u))], t e (-<, <), 1 - F(an (u)t + bn (u)) * F(an (u)t + bn (u)) = [1 + 1 (u)(an (u)t + bn (u))]

exp[-1 (u)(an (u)t + bn (u))], t e (-<, <). Therefore

1

2

m-1

V(t, u) = lim n[s 1 [1 - [F(an (u)t + bn (u))]2] + s2[1 - F (an (u)t + bn (u))

* F (an (u)t + bn (u))]]

= lim[ns1 [2 exp[-1 (u)(an (u)t + bn (u))] - exp[-21 (u)(an (u)t + bn (u))]]

+ ns2 [1 +1 (u)(an (u)t + bn (u))]

exp[-1 (u)(an (u)t + bn (u))]]

= lim[ns12 exp[-1(u)(an (u)t + bn (u))]

[1 -1/2 exp[-1 (u)(an (u)t + bn (u))]]

1 + 1 (u)a„ (u)t + ns21 (u)bn (u)[1 + ^ ]

1 (u)bn (u)

exp[-1 (u)(an (u )t + bn (u))]]

= lim exp[-t ][ns12 exp[-1 (u )bn (u)]

[1 - ^ exp[-t] exp[-1 (u )bn (u)]]

+ ns21 (u)bn (u)[1 + . . ' + f. J

1 (u)bn (u)

exp[-1 (u)bn (u)]]

= lim exp[-t ][ns12exp[-1(u )bn (u)]

- ns1 exp[-t] exp[-21 (u)bn (u)] + ns21 (u)bn (u) exp[-1 (u)bn (u)] + ns 2[1 +1 ] exp[-1 (u)bn (u)]] = lim exp[-t]

[1 --

n(2s1 + 1(u )bn (u )s2)'

exp[-t]

+-2-(1 +1)]

(2^1 +1(u)bn (u)s2)

= exp[-t], t e (-¥, ¥), u = 1,2,...,z,

which by case 1 in Lemma 2.3 completes the proof.

case 2: Since for all fixed u, we have

an(u)t + bn(u) ^^Jf^ > 0 as n ®¥;

1(u) y 2s1 + s2

t e (-¥, ¥). and

1 - [F(an (u)t + bn (u))]2 = 2 exp[-1 (u)(an (u)t + bn (u))] - exp[-21 (u)(an (u)t + bn (u))] = 2[1 -1(u)(an (u)t + bn (u"

1 12(u)(an (u)t + bn (u))2]

- [1 - 21 (u)(an (u)t + bn (u"

1 2 2 1

+ - 412 (u)(a„ (u)t + bn (u))2] + o(--)

2 n +1

= 1 -12(u)(an (u)t + bn (u))2 + o(--)

n +1

t e (-¥, ¥),

1 - F(an (u)t + bn (u)) * F(an (u)t + bn (u))

= [1 + 1(u)(an (u)t + bn (u))]

[1 -1 (u)(an (u)t + bn (u))

1 9 9 1

^12(u)(an (u)t + bn (u))2 - o(--)]

2 n +1

= 1 -112(u)(an (u)t + bn (u"2 - o(^)L

2

t e (-¥, ¥). Therefore

s1[1 - [ F (an (u)t + bn (u ))]2]

n+1

s

+ s 2[F (an (u)t + bn (u)) * F (an (u)t + bn (u))]

= sJ1 -12(u )(an (u)t + bn (u))2]

1 2 2 1

+ s2 [1 - -12 (u)(an (u)t + bn (u))2] + o(--)

= 1 -

2s1 + s 2 12(u)(an (u)t + bn (u))2 + o(—)

2

Vm (1 -m)

n +1

1 n +1

1

- t + m-o(--), t e (-¥,<),

Vn +1 n + 1

next

„ , r (n +1)[sJ1 - [F(an (u)t + bn (u))]2]

v (t, u) = lim-, ==-

m(n - m +1)

n +1

+ -

s2[1 - F(an (u)t + bn (u)) * F(an (u)t + bn (u))]] - m

m(n - m +1) n +1

= lim

, ixr Vm(1 -m) , 1

(n +1)[- ,- t + m- o(--)] - m

Vn + 1_n + 1

m(n - m +1)

n+1

and for t > 0 [F(an (u)t + bn (u))]2 = [1 - exp[-1 (u)(an (u)t + bn (u ))]]2, F(an (u)t + bn (u)) * F(an (u)t + bn (u)) = [1 - [1 + 1 (u)(an (u)t + bn (u))] exp[-1 (u)(an (u)t + bn (u))]]. Therefore V(t,u) = lim n[s1[F(an (u)t + bn (u))]2

+ s2 [F(an (u)t + bn (u)) * F(an (u)t + bn (u))]]

= lim [ns1 [1 - exp[-1 (u)(an (u)t + bn (u))]]2

+ ns2 [1 - [1 + 1(u)(an (u)t + bn (u))] exp[-1 (u)(an (u)t + bn (u))]]] = lim[ns1[1 (u)(an (u)t + bn (u ))]2

+ ns2 [12 (u)(an (u)t + bn (u))2

^m (1 -m )t + o(1)

= lim---= -t, t e (-¥, <),

n®< Vm (1 -m)

u = 1,2,... , z,

which by case 2 in Lemma 2.3 completes the proof. case 3: Since for all fixed u, we have

an (u)t + bn (u) =

V2t

1 (u)*J(2s1 + s 2)n

< 0, t < 0

and

an (u)t + bn (u) =

tV2

1 (u)*J(2s1 + s 2)n

> 0 , t > 0,

then

F(an (u)t + bn (u)) = 0, t < 0,

-112(u)(an (u)t + bn (u))2]]

2s + s

= lim n[-1-(u)an (u)t )2]

= lim n[—] = t , t > 0, t e (-<,<), u = 1,2,...,z,

n®< n

which by case 3 in Lemma 2.3 completes the proof.

Corollary 2.3. The reliability function of exponential „m out of n" system with a mixed reserve of its components is given by

case 1

IR(3)(nm) (t, u) @ 1 - m21exp[-i1 (u)(t - bn(u))]

i=0 i!

exp[- exp[-1 (u)t + 1 (u)bn (u)], (15)

2

t e (-¥, ¥), u = 1,2,...,z.

case 2

IR(3)(;}(t, u) @ 1 Je" 2 dx,

v2p ¥

(16)

case 3 n - m = m constant ( m / n ® 1, n ® ¥ ),

an(u) =

then

1

n1 (u) p (u)

bn (u) =0, u = 1,2,...,z,

where

J =

1 (u)4(2sx + s2 )(n +1)2 - 2Vn+iyr-m (17) t <- , (17)

t e (-¥,¥), u = 1,2,...,z.

IÂ(4)(m}(t, u) = 1, t < 0,

__ .. (m) n-m t

IÂ (4)( \t, u) = 2 - exp[-t], t > 0.

i=0 i!

1s 1ts l1m1t rel1ab1l1ty function.

case 3

IR(3)(nm}(t,u) = 1, t < 0,

Corollary 2.3. The reliability function of exponential „m out ofn" system with improved reliability functions of its components is given by

case 1

(3)(m) n-rn [1 (u)^(2sl + s2)nt /V2]2

IR n (t, u) = 2

i=0

IR(4)(m) (t u) @ 1 - '21exp[-i1 (u)p (u)t - log n)]

exp[-12(u)(2sj + s 2)nt2/2], (18) t > 0 u = 1,2,...,z.

Proposition 2.3. If components of the homogeneous

mult1-state „m out of n" system have 1mproved

component rel1ab1l1ty funct1ons 1.e. 1ts components

fa1lure rates 1(u) 1s reduced by a factor p(u), p(u)e<0,1>, u = 1,2,...,z, and

case 1 m = constant,

exp[-exp[-1 (u)p(u)t + logn], (19)

t e (-¥,¥), u = 1,2,...,z. case 2

1 3 - _

IR(4)(n \t, u) @ 1 —^ Je" 2 dx,

V 2^ -¥

where

(20)

an(u) = , * N , bn(u) = —logn—, u = 1,2,...,z,

1 (u ) p (u)

1 (u) p (u )

then

IÂ (4)(m) (t, u) = 1- mr1^xp[ziilexp[-exp[-t]],

i=0 i!

t e (-¥,¥), case 2 m / n ® m ,0 <m< 1, n ®¥,

an(u) =

1

1 (u)p (u)Vn+1 v m u = 1,2,...,z,

then

1 -m , bn(u) =log(1/ m)

1 (u) p (u)

IÂ (4)(m}(t,u) = 1 - Je" 2 dx , t e (-¥,¥),

J =

1 (u)p (uW(n+vn+ivm, „. x

t--,— log(1/ m ), (21)

m

m

t e (-¥,¥), u = 1,2,...,z.

case 3

IR(4)(nmV,u) = 1, t < 0,

IR (4)nm }(t, u )

@ ^ [1 (u)p (u)nt]i exp[-1 (u)p (u)nt]

i =0 i! t > 0 , u = 1,2,...,z.

3. Comparison of reliability improvement effects

(22)

2

i=0

2

2

The comparisons of the limit reliability functions of the systems with different kinds of reserve and such systems with improved components allow us to find the value of the components decreasing failure rate factor p(u), which warrants an equivalent effect of the system reliability improvement.

3.1. The "m out of n" system

The comparison of the system reliability improvement effects in the case of the reservation to the effects in the case its components reliability improvement may be obtained by solving with respect to the factor p (u) = p (t, u)the following equations

JÂ

(4)1»

((t - bn (u))/ an (u))

= JÂ )(m)

((t - bn (u))/ an (u)),

(23)

u = 1,2,...,z, k = 1,2,3.

The factors p (u) = p (t, u) decreasing components failure rates of the homogeneous exponential multistate „m out of n" system equivalent with the effects of hot, cold and mixed reserve of its components as a solution of the comparisons (23) are respectively given by

k = 1

ln 2

case 1 p (u) = p (t, u) = 1--, u = 1,2,...,z,

1 (u)t

case 2 p (u) = -

2V1 -m 2(1 - m ) + m log m

m

i (u) mt

u = 1,2,...,z,

case 3 p (u) = p (t, u) = 1(u)t, u = 1,2,...,z, k = 2

case 1 p(u) = 1 -

1(u)bn (u) - log n

l(u)t

log 1(u)bn (u)

= 1--, u = 1,2,...,z,

l(u)t

-> / \ V1 - m 1 -m + m log m , 0

case 2 p(u) =---, u = 1,2,...,z,

m i (u)mt

case 3 p(u) = p(t, u) = 1 (^ , u = 1,2,...,z,

k = 3

case 1 p (u) = 1 -

1 (u)bn (u) - log n 1 (u)t

log(2sj + s 21 (u)bn (u))

= 1--, u = 1,2,...,z,

1(u)t

case 2

p (u) =

^J2(2s1 + s 2 - m 2(1 - m) + m log m

u = 1,2,...,z,

i (u)mt

(2sj + s 2)1 (u)t

case 3 p (u) = p (t, u) = ^—1—^—u = 1,2,...,z.

4. Conclusion

Proposed in the paper application of the limit multistate reliability functions for reliability of large systems evaluation and improvement simplifies calculations. The methods may be useful not only in the technical objects operation processes but also in their new processes designing, especially in their optimization. The case of series, parallel (in part 1) and ,,m out of n" systems composed of components having exponential reliability functions with double reserve of their components is considered only. It seems to be possible to extend the results to the systems having other much complicated reliability structures and components with different from the exponential reliability function. Further, it seems to be reasonable to elaborate a computer programs supporting calculations and accelerating decision making, addressed to reliability practitioners.

References

[1] Barlow, R. E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Probability Models. Holt Rinehart and Winston, Inc., New York.

[2] Kolowrocki, K. (2004). Reliability of Large Systems. Elsevier.

M-

[3] Kw1atuszewska-Sarnecka, B. (2002). Analyse of Reserve Efficiency in Series Systems. PhD thes1s, Gdyn1a Maritime Un1vers1ty, (in Polish).

[4] Kolowrock1, K., Kw1atuszewska-Sarnecka, B. at al. (2003). Asymptotic Approach to Reliability Analysis and Optimisation of Complex Transportation Systems. Part 2, Project founded by Pol1sh Comm1ttee for Stientific Research, Gdyn1a: Mat1t1me Un1vers1ty, (in Polish).

[5] Kw1atuszewska-Sarnecka, B. (2006). Rel1ab1l1ty 1mprovement of large mult1-state ser1es-parallel systems. International Journal of Automation and Computing 2, 157-164.

[6] Xue, J. (1985). On multi-state system analys1s, IEEE Transactions on Reliability, 34, 329-337.