Reliability modelling of complex systems - Part 2 Текст научной статьи по специальности «Математика»

Applying mathematical induction it is possible to prove that the reliability function of the homogeneous and regular two-state hierarchical parallel-series system of order r is given by

n (t) = [1 - [1 - n (t)]'" ]for k = 2,3,..., r

and

Ri,k"(t) = [1 -[F(t)]'" ]kn, t e(-»,»),

where kn and ln are defined in Definition 12.

Corollary 4. If components of the homogeneous and regular two-state hierarchical parallel-series system of order r have an exponential reliability function

R(t) = exp[-1t] for t > 0, l > 0, then its reliability function is given by

Rk,kn,'n (t) = [1 - [1 - Rk-i,kn,'n (t) ]ln ]kn for k = 2,3,...,r and

RUknn (t) = [1 -[1 - exp[-1t]]ln ]kn for t > 0.

Theorem 5. If

(i) — (t) = exp[-V (t)], t e (-¥, ¥), is a non-degenerate reliability function,

(ii) lim lrn-1 kjn = 0 for r > 1,

1 1

(iii) lim kl: +..+1 [F(ant + bn )]ln = V (t) for t e Cv,

r > 1, t e (-¥, ¥),

then

lim R . , (at + b ) = W (t) for t e C-, r > 1,

n®¥ r,kn ,ln y n n' W W ' '

t e (-¥, ¥).

Proposition 5. If components of the homogeneous and regular two-state hierarchical parallel-series system of order r have an exponential reliability function

R(t) = exp[-1t] for t > 0, l > 0,

lim lr-1k ln = 0 for r > 1, lim l = l, l e

n n ' n ' '

and

l

i i —+...+— ln lr

b = 0,

then

Â2(0 = exp[-/lr ] for / > 0 is its limit reliability function.

(11)

Example 3. We consider a hierarchical regular parallel-series homogeneous system of order r = 2 such that k n = 200, ln = 3, whose components have identical exponential reliability functions with the failure rate l = 0.01.

Its exact reliability function, according to Corollary 4, is given by

= [1 -[1 -[ 1 -[1 -exp[-0.01i]]3]200 ]T0

for t > 0.

Next applying Proposition 5 with normalising constants

1

1

= 9.4912, b = 0,

a" 0.01 ' 2001/3+1'9 ' " ' we conclude that

) = exp[-t9] for t > 0

is the system limit reliability function, and from (7), the following approximate formula is valid

R2 200,3 (t) @ C, (0.1054t) = exp[-(0.1054t)9 ] for t > 0. 6. Conclusion

Generalizations of the results on limit reliability functions of two-state homogeneous systems for these and other systems in case they are non-homogeneous, are mostly given in [8] and [9]. These results allow us to evaluate reliability characteristics of homogeneous and non-homogeneous series-parallel and parallel-series systems with regular reliability structures, i.e. systems composed of subsystems having the same numbers of components. However, this fact does not restrict the completeness of the performed analysis, since by conventional joining of a suitable number of components which do not fail, in series sub-systems of the non-regular series-parallel systems, leads us to the regular non-homogeneous series-parallel systems.

an =

n

Similarly, conventional joining of a suitable number of failed components in parallel subsystems of the non-regular parallel-series systems we get the regular non-homogeneous parallel-series systems. Thus the problem has been analyzed exhaustively. The results concerned with the asymptotic approach to system reliability analysis, in a natural way, have led to investigation of the speed of convergence of the system reliability function sequences to their limit reliability functions ([9]). These results have also initiated the investigations of limit reliability functions of "m out of «"-series, series-"m out of n" systems, the investigations on the problems of the system reliability improvement and on the reliability of systems with varying in time their structures and their components reliability described briefly in [9] and presented in Part 2 ([10]) of this paper.

More general and practically important complex systems composed of multi-state and degrading in time components are considered in wide literature, for instance in [14]. An especially important role they play in the evaluation of technical systems reliability and safety and their operating process effectiveness is described in [9] for large multi-state systems with degrading components. The most important results regarding generalizations of the results on limit reliability functions of two-state systems dependent on transferring them to series, parallel, "m out of n", series-parallel and parallel-series multi-state systems with degrading components are given in [9]. Some practical applications of the asymptotic approach to the reliability evaluation of various technical systems are contained in [9] as well.

The proposed method offers enough simplified formulae to allow significant simplifying of large systems' reliability evaluating and optimizing calculations.

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] Chernoff, H. & Teicher, H. (1965). Limit distributions of the minimax of independent identically distributed random variables. Proc. Americ. Math. Soc. 116, 474-491.

[3] Cichocki, A. (2003). Wyznaczanie granicznych funkcji niezawodnosci systemow hierarchicznych przy standaryzacji pot^gowej. PhD Thesis. Maritime University, Gdynia - Systems Research Institute, Polish Academy of Sciences, Warszawa.

[4] Fisher, R. A. & Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest and smallest member of a sample. Proc. Cambr. Phil. Soc. 24, 180-190.

[5] Frechet, M. (1927). Sur la loi de probabilité de l'écart maximum. Ann. de la Soc. Polonaise de Math. 6, 93-116.

[6] Gniedenko, B. W. (1943). Sur la distribution limite du terme maximum d'une serie aleatoire. Ann. of Math. 44, 432-453.

[7] Gumbel, E. J. (1962). Statistics of Extremes. New York.

[8] Kolowrocki, K. (1993). On a Class of Limit Reliability Functions for Series-Parallel and Parallel-Series Systems. Monograph. Maritime University Press, Gdynia.

[9] Kolowrocki, K. (2004). Reliability of Large Systems. Elsevier, Amsterdam - Boston -Heidelberg - London - New York - Oxford - Paris -San Diego - San Francisco - Singapore - Sydney -Tokyo.

[10] Kolowrocki, K. (2007). Reliability of complex systems - Part 2. Proc. Summer Safety and Reliability Seminars - SSARS 2007, Sopot.

[11] Kurowicka, D. (2001). Techniques in Representing High Dimensional Distributions. PhD Thesis. Maritime University, Gdynia - Delft University.

[12] Von Mises, R. (1936). La distribution de la plus grande de n valeurs. Revue Mathematique de l'Union Interbalkanique 1, 141-160.

[13] Smirnow, N. W. (1949). Predielnyje Zakony Raspredielenija dla Czlienow Wariacjonnogo Riada. Trudy Matem. Inst. im. W. A. Stieklowa.

[14] Xue, J. & Yang, K. (1995). Dynamic reliability analysis of coherent multi-state systems. IEEE Transactions on Reliability 4, 44, 683-688.

Kolowrocki Krzysztof

Maritime University, Gdynia, Poland

Reliability modelling of complex systems - Part 2

Keywords

reliability, large system, asymptotic approach, limit reliability function Abstract

Series-"m out of n" systems and "m out of n"-series systems are defined and exemplary theorems on their limit reliability functions are presented and applied to the reliability evaluation of a piping transportation system and a rope elevator. Applications of the asymptotic approach in large series systems reliability improvement are also presented. The paper is completed by showing the possibility of applying the asymptotic approach to the reliability analysis of large systems placed in their variable operation processes. In this scope, the asymptotic approach to reliability evaluation for a port grain transportation system related to its operation process is performed.

1. Reliability of large series-"m out of n" systems

Definition 1. A two-state system is called a series-"m out of kn" system if its lifetime T is given by

T = T.. , m = 1,2,...,k№

(kn-m+1) ' ' ' ' ■

where T,, „ is the mth maximal order statistic in

(kn-m+1)

the set of random variables

Ti = min{Tij}, i = 1,2,...,kn.

R ( m )

kn ,l1,l2,.,lkn

(t )

1

S

r1,r2,..., rkn =0 1

q +r2 +...+i"k £m

k l- l ■

Kn 'i 'i ,

n[1 -n R-(t)]ri[n R-(t)]1

<=i -=1 -=1

for t e (-¥,¥), where m = kn - m.

Definition 2. The series-"m out of kn" system is called regular if

1£ j£li

I1 = I2 = ■ ■ ■ =lk = ln , lnî N.

The above definition means that the series-"m out of kn" system is composed of kn series subsystems and it is not failed if and only if at least m out of its kn series subsystems are not failed. The series-"m out of kn" system is a series-parallel for m = 1 and it becomes a series system for m = kn. The reliability function of the two-state series-"m out of kn" system is given either by

Definition.3. The series-"m out of kn" system is called homogeneous if its component lifetimes Tj have an identical distribution function

F(t) = P(Tj £ t), t e (-¥,¥), i = 1,2,...,kn, j = 1,2,...,li,

i.e. if its components Ej have the same reliability function

R (m)

kn ,l1 ,l2,...,lkn

(t) = 1

R(t) = 1 - F(t), t Î (-¥,¥).

1 kn li li

S n[n R-(t)]ri[1 -n Ri-(t)]1

,..., rkn =0 i=1 j=1 j=1

From the above definitions it follows that the reliability function of the homogeneous and regular series-"m out of kn" system is given either by

for t e (-œ,œ) or by

1 +r2 +...+rkn £m-1

m (t) =1 - ? (kn )[Rln (t)] [1 - Rln (t)]'

: 1 - )exp[-ita ][1 - exp[-ta]]k-i for t > 0

for t e (-¥,¥) or by

Rk-ml (t) = 2 (kn )[1 -Rln (t)]! [Rln (t)]kn-i

is its limit reliability function, i.e., for t > 0, we have

Rkmi (t) @*9(2)( )

a„

for t e (-¥,¥), m = kn - m, where kn is the number of series subsystems in the "m out of kn" system and ln is the number of components of the series subsystems.

Corollary 1. If components of the homogeneous and regular two-state series-"m out of kn" system have Weibull reliability function

R(t) = exp[-pta ] for t > 0, a > 0, b > 0, then its reliability function is given either by R« (t)

kn ,ln v '

m-1 /, \ a i

= 1 - 2((n )[exp[-ilnbta ]][1 - exp[-ln bt ]]kn-i

i=0

for t > 0 or by

1 - )exp[ -iblnta ][1 - exp[-blnta ]]k-i.

(3)

Rkmi (t)

= 2 ((n )[1 - exp[-lnbta ]];[exp[-(kn - i)lnbta ]] (2)

i=0

for t > 0, m = kn - m.

Proposition 1. If components of the two-state homogeneous and regular series-"m out of kn" system have Weibull reliability function

R(t) = exp[-bta ] for t > 0, a > 0, b > 0,

and

Example 1. The piping transportation system is set up to receive from ships, store and send by carriages or cars oil products such as petrol, driving oil and fuel oil. Three terminal parts A, B and C fulfil these purposes. They are linked by the piping transportation systems. The unloading of tankers is performed at the pier. The pier is connected to terminal part A through the transportation subsystem S\ built of two piping lines. In part A there is a supporting station fortifying tankers' pumps and making possible further transport of oil by means of subsystem S2 to terminal part B. Subsystem S2 is built of two piping lines. Terminal part B is connected to terminal part C by subsystem S3. Subsystem S3 is built of three piping lines. Terminal part C is set up for loading the rail cisterns with oil products and for the wagon carrying these to the railway station. We will analyse the reliability of the subsystem S3 only. This subsystem consists of kn = 3 identical piping lines, each composed of ln = 360 steel pipe segments. In each of lines there are pipe segments with Weibull reliability function

R(t) = exp[-0.0000000008t4] for t > 0.

We suppose that the system is good if at least 2 of its piping lines are not failed. Thus, according to Definitions 2-3, it may be considered as a homogeneous and regular series-"2 out of 3" system, and according to Proposition 1, assuming

1

1

(bln )v a (0.000000288)1

b = 0,

lim kn = k, k > 0, 0 < m < k, lim l = ¥,

and using (3), its reliability function is given by

a = (bl ) a , b = 0,

n Vr nJ ? n '

then * f(t)

R®(t) 9(2)(-)

1 f 3^

= 2

i=0

V i 0

exp[-i • 0.000000288t4] [1 - exp[-0.000000288t4]]3-i' for t > 0.

a =

n

a

n

2. Reliability of large "m out of «"-series systems

Definition 4. A two-state system is called an " mt out of li "-series system if its lifetime T is given by

T = min T(l +1), m = 1,2,...,I,

1<1<kn K1 1 '

where T., ,, is the mith maximal order statistic in

(li-mt +1) !

the set of random variables

T1^ Ti2, ..., Tih , 1 = l2^.^ kn .

Definition 6. The " mt out of l. "-series system is called regular if

l1 = l2 = . . . =lkn = ln

and

m1 = m2 = . . . = mk = m, where ln, me N, m £ ln.

The reliability function of the two-state homogeneous and regular „ m out of ln "-series system is given either by

The above definition means that the " mi out of li "series system is composed of kn subsystems that are " mt out of li " systems and it is not failed if all its " mi out of li" subsystems are not failed. The " mi out of li "-series system is a parallel-series system if m1 = m2= . . . =mkn = 1 and it becomes a series system if mi = li for all i = 1,2, ... ,kn.

The reliability function of the two-state " mt out of L "series system is given either by

R(m1,m2,...,mkn ) ,,)

Rkn ,li,l2,...,lkn (t)

kn 1

= n Ü - Z

1 =1 r1,r2,...,rli =0 1 =1 r1+r2 +...+rl< <mi-1

li r li 1 r [n Rj (t)]ri [1 -n Rj (t)]1-ri ]

i=1

for t Î (-¥,¥) or by

R(m1,m2, ,mkn )(t) kn,l1,l2, ,lkn ( )

kn 1

= n [ Z .

1 =1 Î1 ,r2ri =0 1 =1 r1+r2 +...+ÎT <mi

li h [1 -n Rj (t)]ri [ n Rj (t )]1-ri ]

1=1

for t e (-¥,¥), where = lt - , i = 1,2,...,kn.

Definition 5. The two-state " m out of I. "-series system is called homogeneous if its component lifetimes Tij have an identical distribution function

F(t) = P(T^- £ t), t e (-¥,¥), i = 1,2,...,kn,j=1,2,...,li ,

i.e. if its components Eij have the same reliability function

^(t) = 1 - F(t), t e (-¥,¥).

Rf) (t) = [1 - "Z1 (ln )[R(t)]1 [1 -R(t)]ln-1 ]kn

1 =0

for t Î (-¥,¥) or by

R™] (t) = [ Z (ln )[1 -R(t)]1 [R(t)]ln-1 ]kn

1=0

for t e (-¥,¥), m = ln - m where kn is the number of "m out of ln" subsystems linked in series and ln is the number of components in the "m out of ln" subsystems.

Corollary 2. If the components of the two-state homogeneous and regular "m out of ln"-series system have Weibull reliability function

R(t) = exp[-pta ] for t > 0, a > 0, b > 0, then its reliability function is given either by

Rkrl (t )

= [1 - I ( )exp[-1ßta ][1 - exp[-ßta ]]'--1 ]kn (4)

1=0

for t > 0 or by

Rm (t) = [ Z (ln )[1 -exp[-ßta ]]1 exp[-(ln -1)ßta ]]k

1=0

v^n kn

for t > 0, m = l - m.

' n

Proposition 2■ If components of the two-state homogeneous and regular "m out of ln"-series system have Weibull reliability function

R(t) = exp[-pta ] for t > 0, a > 0, b > 0,

and

lim kn = k, k > 0, 0 < m < k, lim l = ¥,

R (5) (t) =

10,22 v'/

, bn = [^a

a log n b

then

(5)

[1 - S (?) exp[-/0.05t2][1 - exp[-0.05t2]]22- ]10

i=0

for t > 0.

Next, applying Proposition 2 with

[Â 3(0) (t)]k = [1 - exp[-exp[-t ]]S1expi_itl]k

i=0 i!

for t e (-¥, ¥), is its limit reliability function, i.e.

R& (t) @ [Â 3(0)(^-A)]k

t - b m-1

= [1 - exp[-exp[--^ ]] S

t - b exp[ -i-n- ]

-]k (6)

for t e (-¥, ¥), where an and bn are defined by (5).

Example 2. Let us consider the ship-rope transportation system (elevator). The elevator is used to dock and undock ships coming in to shipyards for repairs. The elevator is composed of a steel platform-carriage placed in its syncline (hutch). The platform is moved vertically with 10 rope hoisting winches fed by separate electric motors. During ship docking the platform, with the ship settled in special supporting carriages on the platform, is raised to the wharf level (upper position). During undocking, the operation is reversed. While the ship is moving into or out of the syncline and while stopped in the upper position the platform is held on hooks and the loads in the ropes are relieved.

In our further analysis we will discuss the reliability of the rope system only. The system under consideration is in order if all its ropes do not fail. Thus we may assume that it is a series system composed of 10 components (ropes). Each of the ropes is composed of 22 strands. Thus, considering the strands as basic components of the system and assuming that each of the ropes is not failed if at least m = 5 out of its strands are not failed, according to Definitions 5-6, we conclude that the rope elevator is the two-state homogeneous and regular „5 out of 22"-series system. It is composed of kn = 10 series-linked "5 out of 22" subsystems (ropes) with ln = 22 components (strands). Assuming additionally that strands have Weibull reliability functions with parameters a= 2, b= 0.05, i.e.

R(t) = exp[-0.05t2] for t > 0, from (4), we conclude that the elevator reliability function is given by

@ 1.2718, bn = @ 7.8626,

2 log 22 n 0.05

and (6) we get the following approximate formula for the elevator reliability function

Ri(o,22(t) @ [Â 3 (0.7863t - 6.1821)]

= [1 - exp[- exp[-0.7863t + 6.1821]]

■S

i=0

exp[-0.7863it + 6.1821i]

]10, t Î (-¥, ¥).

3. Asymptotic approach to systems reliability improvement

We consider the homogeneous series system illustrated

in Figure 1.

E11 E21

E,

n1

Figure 1. The scheme of a series system

It is composed of n components En, i = 1,2,...,n, having lifetimes Ti1, i = 1,2,..., n, and exponential reliability functions

R(t) = exp[-1t] for t > 0, l > 0.

Its lifetime and its reliability function respectively are given by

T(0) = min{T;1},

1<!<n

Rn (t) = [R(t)]n = exp[-1nt], t > 0.

In order to improve of the reliability of this series system the following exemplary methods can be used:

- replacing the system components by the improved components with reduced failure rates by a factor p, 0 < p < 1,

- a warm duplication (a single reservation) of system

an =

a

u

an i=0 i!

components,

- a cold duplication of system components,

- a mixed duplication of system components,

- a hot system duplication,

- a cold system duplication.

It is supposed here that the reserve components are identical to the basic ones.

The results of these methods of system reliability improvement are briefly presented below, giving the system schemes, lifetimes and reliability functions.

Case 1. Replacing the system components by the improved components En i = 1,2,...,n, with reduced failure rates by a factor p, 0 < p < 1, having lifetimes Tl, i = 1,2,..., n, and exponential reliability functions

E'n E'21

E\

n1

R(pt) = exp[-pit] for t > 0, 1> 0.

Figure 2. The scheme of a series system with improved components

T(1) = m in{Tn},

1<i<n

Rn\t) = [R(pt)]n = exp[-p1nt], t > 0. Case 2. A hot reservation of the system components

E

11

Ei

E

21

E2

. En-

-<n-11

E

n

E

n1

En

Figure 3. The scheme of a series system with components having hot reservation

T(2) = min{max{Ti(.}},

1<i<n 1< j<2

Rn2)(t) = [1 - [F(t)]2]n = [1 - [1 - exp[-1t]]2]n, t > 0. Case 3. A cold reservation of the system components

E

11

E

12

E2

— . -- E,

E

22

En

E n-

n-12

E

n2

Figure 4. The scheme of a series system with components having cold reservation

T(3) = m in{ S T. },

1<i<n .=1

R(n3) (t) = [1 - [F(t)] * [F(t)]]n = [1 + 1t]n exp[-nit], t > 0. Case 4. A mixed reservation of the system components

Figure 5. The scheme of a series system with components having mixed reservation

T(4) = min{ min{ST.}, min {max{Tiy}}},

1<i<m j=\ m+1<i<n 1<j<2

R n4) (t) = [1 - [F(t)] * [F(t)]]m [1 - R 2(t)]n-m

= [1 + 1t]m exp[-1nt][2 - exp[-1t]]n-m, t > 0. Case 5. A hot system reservation

E11 E21

E

n1

E12 E22

E

n2

Figure 6. The scheme of a series system with hot reservation

T(5) = max{min {T..}},

1<j<2 1<j<n J

R(n5)(t) = 1 - [1 - [R(t)]n ]2 = 1 - [1 - exp[-n1t]]2, t > 0.

Case 6. A cold system reservation

E11 E21

E

12

E

22

En1

En2

Figure 7. The scheme of a series system with cold reservation

T(6) = Z minT },

j=1

1<i<n

R n6)(t) = 1 - [1 - [ R(t)]n ] * [1 - [R(t)]n ] = [1 + n1t]exp[-n1t], t > 0.

Rn2)(t) @ K {2\l4nt) = exp[-12nt2] for t > 0

and

T(2) = E[T(2)] @ r(-)

3 1

2 l4n

The difficulty arises when selecting the right method of improvement of reliability for a large system. This problem may be simplified and approximately solved by the application of the asymptotic approach. Comparisons of the limit reliability functions of the systems with different types of reserve and such systems with improved components allow us to find the value of the components' decreasing failure rate factor p, which gives rise to an equivalent effect on the system reliability improvement. Similar results are obtained under comparison of the system lifetime mean values. As an example we will present the asymptotic approach to the above methods of improving reliability for homogeneous two-state series systems.

Proposition 3.Case 1. If

Case 3. If

an = V2/ l4n, bn = 0, then

K(3)(0 = exp[-t2] for t > 0,

is the limit reliability function of the homogeneous exponential series system with cold reservation of its components, i.e.

and

an = 1/lpn, bn = 0, then

K (1)(t) = exp[-t] for t > 0,

is the limit reliability function of the homogeneous exponential series system with reduced failure rates of its components, i.e.

R (n1)(t) = K(1)( Ipnt) = exp[-1pnt] for t > 0

and

T(1) = E[T(1)] = ■

1

Ipn

Case 2. If

T(3) = E[T(3)] l I-

2 l V n

3 1 2

Case 4. If

an , bn = 0,

l V 2n - m

then

K(4)(t) = exp[-t2] for t > 0,

is the limit reliability function of the homogeneous exponential series system with mixed reservation of its components, i.e.

R n4)(t) @ K (4)(^ ^^)

= 1/iV«, bn = 0,

then

K(2)(t) = exp[-t2] for t > 0,

is the limit reliability function of the homogeneous exponential series system with hot reservation of its components, i.e.

r 2n - m „ 2 2, „ = exp[---—l212] for t > 0

and

T(4) = E[T(4)] @ T(-)-

3 1 2

2 l V 2n - m

Case 5. If

a

n

p = p (t ) = 1 - log[2 - exp[-1nt]] for i = 5,

an = —, bn = 0,

In

then

K(5)(t) =1 - [1 - exp[-t]]2 for t > 0,

is the limit reliability function of the homogeneous exponential series system with hot reservation, i.e.

R („5)(t) = K (5)(1nt)= 1 - [1 - exp[-Int]]2 for t > 0

and

p = p (t ) = 1--log[1 + 1nt ] for i = 6,

Int

while comparison of the system lifetimes

7%) = T(1)(t), i = 2,3,...,6

results respectively in the following values of the factor p :

P =-

1

o

Y(2)4n

for i = 2,

T(5) = E[T(5)] = -

21n

Case 6. If

1

an = —, bn = 0,

In

then

Â(6)(t) = [1 +1] exp[-t] for t > 0,

P =-

1

o

r(2)V2n

for i = 3,

p =-

for i = 4,

2 V 2n - m

p = — for i = 5, 3

is the limit reliability function of the homogeneous exponential series system with cold reservation, i.e.

R (n6)(t) = Â (6) (Int) = [1 + 1nt]exp[-1nt] for t > 0

and

T(6) = E[T ] = " In

2

p = — for i = 6. 2

Example 3. We consider a simplified bus service company composed of 81 communication lines. We suppose that there is one bus operating on each communication line and that all buses are of the same type with the exponential reliability function

R(t) = exp[-It] for t > 0, l > 0.

Corollary 3. Comparison of the system reliability functions

K(i)(t) = K(1)(t), i = 2,3,...,6,

results respectively in the following values of the factor p :

p = p (t) = It for i = 2,

p = p (t) = ~1t for i = 3,

. . 2n - m

p = p (t ) =-It for i = 4,

2n

Additionally we assume that this communication system is working when all its buses are not failed, i.e. it is failed when any of the buses are failed. The failure rate of the buses evaluated on statistical data coming from the operational process of bus service company transportation system is assumed to be equal to 0.0049 h-1.

Under these assumptions the considered transportation system is a homogeneous series system made up of components with a reliability function

R(t) = exp[-0.0049t] for t > 0.

Here we will use four sensible methods from those considered for system reliability improvement.

3

1

Namely, we apply the four previously considered cases.

Case 1. Replacing the system components by the improved components with reduced failure rates by a factor p.

Applying Proposition 3 with normalising constants

1

1

b81 = 0, 0.0049 • 81p 0.397p 81

we conclude that

^(1)(0 = exp[-t] for t > 0,

is the limit reliability function of the system, i.e.

R V(t) = ^(1)(0.397 pt) = exp[-0.397pf]for t > 0

and

T(1) = E[T(1)] =

1

0.397 p

h.

Case 2. Improving the reliability of the system by a single hot reservation of its components. This means that each of 81 communication lines has at its disposal two identical buses it can use and its task is performed if at least one of the buses is not failed. Applying Proposition 3 with normalising constants

1 1 , ft

a8i =-;= =-, b81 = 0,

81 0.0049 -V8l 0.0441 81

we conclude that

 (2)(t) = exp[-t2], t > 0.

is the limit reliability function of the system, i.e.

R82)(t) (2)(0.0441t) @exp[-0.0019t2], t > 0, and

T(2) = E[T(2)] @T(-)

1

2 0.0049V8T

@ 20.10 h.

Case 4. Improving the reliability of the system by a single mixed reservation of its components. This means that each of 81 communication lines has at its disposal two identical buses. There are m = 50 communication lines with small traffic which are using one bus permanently and after its failure it is replaced by the second bus (a cold reservation) and n - m = 81 - 50 = 31 communication lines with large

traffic which are using two buses permanently (a hot reservation).

Applying Proposition 3 with normalising constants

1

2

1

0.0049 V 112 0.0367

K = 0,

we conclude that

^(4)(0 = exp[-t2] for t > 0, is the limit reliability function of the system, i.e.

R(n4)(t) @ ^(4) (0.0367t) = exp[-0.00135t2] for t > 0 and

T(4) = E[T(4)] @ 24.15 h.

2 0.004^112

Case 5. Improving the reliability of the system by a single hot reservation.

This means that the transportation system is composed of two independent companies, each of them operating on the same 81 communication lines and having at their disposal one identical bus for use on each line. Applying Proposition 3 with normalising constants

1

1

b81 = 0,

81 0.0049 • 81 0.397 ' 81 we conclude that

^(5)(0 =1 - [1 - exp[-t]]2 for t > 0, is the limit reliability function of the system, i.e. R (n5)(t) = ^(5)(0.397t)

= 1 - [1 - exp[-0.397t]]2 for t > 0

and

T(5) = E[T(5)] =

3

2 • 0.0049 • 81

@ 3.78 h.

Comparing the system reliability functions for considered cases of improvement, from Corollary 3, results in the following values of the factor p:

p = p (t) = 0.0049t for i = 2, p = 0.0340t for i = 4,

an =

p = p (t) = 1 - log[2 - exp[-0.397t]] for i = 5,

while comparison of the system lifetimes results respectively in:

p = 0.1254 for i = 2,

p= 0.1043 for i = 4,

p= 0.6667 for i = 5.

Methods of system reliability improvement presented here supply practitioners with simple mathematical tools, which can be used in everyday practice. The methods may be useful not only in the operation processes of real technical objects but also in designing new operation processes and especially in optimising these processes. Only the case of series systems made up of components having exponential reliability functions with single reservations of their components and subsystems is considered. It seems to be possible to extend these results to systems that have more complicated reliability structures, and made up of components with different from the exponential reliability functions.

4. Reliability of large systems in their operation processes

This section proposes an approach to the solution of the practically very important problem of linking systems' reliability and their operation processes. To connect the interactions between the systems' operation processes and their reliability structures that are changing in time a semi-markov model ([1]) of the system operation processes is applied. This approach gives a tool that is practically important and not difficult for everyday use for evaluating reliability of systems with changing reliability structures during their operation processes. Application of the proposed methods is illustrated here in the reliability evaluation of the port grain transportation system. We assume that the system during its operation process is taking different operation states. We denote by Z(t), t e< 0, ¥>, the system operation process that may assume v different operation states from the set

Z = {z^ Z2 , . . ., Zv }.

In practice a convenient assumption is that Z(t) is a semi-markov process ([1]) with its conditional sojourn

times 0 bl at the operation state zb when its next operation state is zl, b, l = 1,2,...,v, b * l. In this case this process may be described by:

- the vector of probabilities of the initial operation states [ Pb (0)]1xn ,

- the matrix of the probabilities of its transitions between the states [ pbl ]nxn ,

- the matrix of the conditional distribution functions [Hbl (t)]vxv of the sojourn times 0bl, b * l, where

Hbl(t) = P(0bl < t) for b, l = 1,2,...,v, b * l,

and

Hbb(t) = 0 for b = 1,2,...,v.

Under these assumptions, the lifetime 0bl mean values are given by

¥

Mbl = E[0bl ] = J tdHbl (t), b, l = 1,2,...,v, b * l. (7)

0

The unconditional distribution functions of the sojourn times 0 b of the process Z (t) at the states zb, b = 1,2,..., v, are given by

Hb (t) = ZPblHbl (t), b = 1,2,..., v.

l=1

The mean values E[0 b ] of the unconditional sojourn times 0 b are given by

Mb = E[0b] = ZPblMbl , b = 1,2,...,v, (8)

l=1

where Mbl are defined by (7).

Limit values of the transient probabilities at the states

pb (t) = P(Z(t) = Zb), t e< 0,¥), b = 1,2,..., v,

are given by ([1]) Pb = lim pb (t) =

p bMb Z p Ml

b = 1,2,..., v, (9)

l=1

where the probabilities pb of the vector [pb ]1xn satisfy the system of equations

[p b ] = [p b ][Pbl]

Z p l = 1.

l=1

We consider a series-parallel system and we assume that the changes of its operation process Z(t) states

have an influence on the system components Eij

reliability and on the system reliability structure as well. Thus, we denote ([13]) the conditional reliability function of the system component Etj while the

system is at the operational state zb, b = 1,2,...,v, by

[R a1) (t)](b) = P(T(b) > t / Z(t) = zb),

for t e <0,¥), b = 1,2,...,v, and the conditional reliability function of the non-homogeneous regular series-parallel system while the system is at the

operational state zb, b = 1,2,...,v, by

(b)

[(t)](b)= P(T(b) >t/Z(t) = Zb)

(b)

=1 -n [1 - [[ R(i )(t )](b)]ln ]

i=1

n Hi^n

(10)

for t e< 0, œ) and

[R®(t)](b) = n[[R(U)(t)](b}]1 i = 1,2, . ,a. (11)

j=1

The reliability function [R^1 )(t)](b) is the conditional probability that the component Eij lifetime Ti(b) in the is not less than t, while the process Z(t) is at the operation state zb. Similarly, the reliability function [R, , (t)](b) is the conditional probability that the series-parallel system lifetime T (b) is not less than t,

while the process Z(t) is at the operation state zb. In the case when the system operation time is large enough, the unconditional reliability function of the series-parallel system is given by

RknJn (t) = P(T >t) @ Z Pb [Rknn (t)]

b=1

(b)

(12)

for t > 0 and T is the unconditional lifetime of the series-parallel system.

The mean values and variances of the series-parallel system lifetimes are

m PbMb,

b=1

where

(13)

Mb = 1 [Rh„, (t)](b)dt,

and

D[T(b)] = 2 J t[R kl (t)](b) dt - [Mb ]2, (15)

for b = 1,2,...,v.

Example 5. We analyse the reliability of one of the subsystems of the port grain elevator. The considered system is composed of four two-state non-homogeneous series-parallel transportation subsystems assigned to handle and clearing of exported and imported grain. One of the basic elevator functions is loading railway trucks with grain. In loading the railway trucks with grain the following elevator transportation subsystems take part: S1 -horizontal conveyors of the first type, S2 - vertical bucket elevators, S3 - horizontal conveyors of the second type, S4 - worm conveyors. We will analyze the reliability of the subsystem S 4 only.

Taking into account experts opinion in the operation process, Z(t), t > 0 of the considered transportation subsystem we distinguish the following as its three operation states:

an operation state z1 - the system operation with the largest efficiency when all components of the subsystem S 4 are used,

an operation state z2 - the system operation with less efficiency system when the first and second conveyors of subsystem S4 are used,

an operation state z3 - the system operation with least efficiency when the first conveyor of subsystem S 4 is used.

On the basis of data coming from experts, the probabilities of transitions between the subsystem S4 operation states are given by

0 0.357 0.643"

[Pbl ] =

0.8 0.385

0

0.615

0.2 0

(14)

and their mean values, from (8), are Mx = E[ej = 0.357 • 0.36 + 0.643 • 0.2 @ 0.257, M2 = E[e2] = 0.8 • 0.05 + 0.2• 0.2 @ 0.08, M3 = E[e3] = 0.385 • 0.08 + 0.615 • 0.05 @ 0.062. Since from the system of equations

¥