Kołowrocki K. Reliability, availability and risk evaluation of complex systems in their operation processes.

RELIABILITY, AVAILABILITY AND RISK

EVALUATION OF COMPLEX SYSTEMS IN

THEIR OPERATION PROCESSES

Kołowrocki K.

Maritime University, Gdynia, Poland

Abstract: The paper proposes an approach to the solution of the practically very important prob -lem of linking multi-state systems’ reliability and their operation processes. To involve the inter-actions between the systems’ operation processes and their reliability structures that are changing in time a semi-markov model of the system operation processes is applied. Application of the pro-posed methods is illustrated in the reliability, risk and availability evaluation of the port grain transportation system.

1. Introduction

Most real transportation systems are very complex and it is difficult to analyze their reli ability and availability. Large numbers of components and subsystems and their operat -ing complexity cause that the evaluation and optimization of their reliability and avail-ability is complicated. One of the important techniques that simplify the reliavail-ability and availability evaluation of large systems is the asymptotic approach [3]. The paper pro poses the usage of the system limit reliability functions in their reliability and availabil -ity evaluation and illustrates their practical application.

2. System operation process

We assume that the system during its operation process is taking v, N, _different

operation states. Thus, we can define the system operation process Z(t), ,

,

0 

 

t with discrete states from the set of states Z {z₁, z₂,..., z}. If the system operation process Z(t) is semi-Markov ([2]) with its conditional sojourn time

kl

 at the state z when its next state is k z l, k,l 1,2,...,v, k l, then it

– the vector of probabilities of the initial states [pk(0)]1x , where ) ) 0 ( ( ) 0 ( _k k P Z z p   for k 1,2,...,v,

– the matrix of probabilities of its transitions between states [pkl]x , where pkk 0

for k 1,2,...,v,

– the matrix of conditional distribution functions of the sojourn times kl ,

 x

)] (

[H_kl t , where H_kl(t)P(_kl t) for k,l 1,2,...,v, k l, and

0 )

(t 

H for k 1,2,...,v.

Then, the sojourn time kl mean values are given by:

E[kl]





 0

),

(t

tdH

kl k,l 1,2,...,v, k  l. (1)

The unconditional distribution functions of the sojourn times k of the process Z(t)

at the states zk, k 1,2,...,v, are given by: H_k(t) =



 v l 1

p

kl

H

kl

(

t

),

. ,..., 2 , 1 v k  (2)

The mean values E[k ] of the unconditional sojourn times k are given by

E[k] =



 v l 1

p

kl

E

[

kl

]



, k 1,2,...,v, (3)

where E[kl ] are defined by (1).

Limit values of the transient probabilities at the states pk(t) = P(Z(t) = z ) arek

given by p = k

lim

p

k

(

t

)

t =

,

]

[

]

[

1



 v l l l k k

E

E









, ,..., 2 , 1 v k  (4)









 





v

l

l

kl

k

k

p

1

.1

]

][

[

]

[







(5)

3. Application

As an example we will analyse the reliability of the port grain elevator in its operation process ([3]). This system is composed of four multi-state non-homogeneous series-parallel transportation subsystems and it is the basic structure in the Baltic Grain Terminal of the Port of Gdynia assigned to handle the 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.

Taking into account the quality of work of the considered transportation system, we dis-tinguish the following three reliability states (

z



2

) of its components ([3], [4]): state 2 – the state ensuring the largest quality of the conveyor work, state 1 – the state ensur-ing less quality of the conveyor work caused by throwensur-ing grain off the belt, state 0 – the state involving failure of the conveyor.

Subsystem S1 consists of 2 identical belt conveyors of the first type, each composed of 129 components. In each conveyor there are:

- one ribbon belt with reliability functions

R(1,1)_{(t,1) = exp[-0.1262t], R}(1,1)_{(t,2) = exp[-0.1674t],} - two drums with reliability functions

R(1,2)_{(t,1) = exp[-0.044t],R}(1,2)_{(t,2) = exp[-0.048t],} - 117 channelled rollers with reliability functions R(1,3)_{(t,1) = exp[-0.0798t], R}(1,3)_{(t,2) = exp[-0.0978t],}

- nine supporting rollers with reliability functions R(1,4)_{(t,1) = exp[-0.0714t], R}(1,4)_{(t,2) = exp[-0.0798t].}

Subsystem S2 consists of 3 identical bucket elevators, each composed of 743 components. In each elevator there are:

- one belt with reliability functions

R(1,1)_{(t,1) = exp[-0.178t], R}(1,1)_{(t,2) = exp[-0.182t],} - two drums with reliability functions

R(1,2)_{(t,1) = exp[-0.0437t],R}(1,2)_{(t,2) = exp[-0.0479t],} - 740 buckets with reliability functions

R(1,3)_{(t,1) = exp[-0.1954t], R}(1,3)_{(t,2) = exp[-0.2764t].}

Subsystem S3 consists of 2 identical belt conveyors of the second type, each composed of 139 components. In each conveyor there are:

- one belt with reliability functions

R(1,1)_{(t,1) = exp[-0.126t], R}(1,1)_{(t,2) = exp[-0.167t],} - two drums with reliability functions

R(1,2)_{(t,1) = exp[-0.0437t], R}(1,2)_{(t,2) = exp[-0.048t],} - 117 channelled rollers with reliability functions R(1,3)_{(t,1) = exp[-0.0798t], R}(1,3)_{(t,2) = exp[-0.0978t],} - 19 supporting rollers with reliability functions R(1,4)_{(t,1) = exp[-0.0714t], R}(1,4)_{(t,2) = exp[-0.0798t].}

Subsystem S4 consists of three chain conveyors. Two of these are composed of 162 components and the remaining one is composed of 242 components. In two of them there are:

- two driving wheels with reliability functions

- 160 links with reliability functions

R(1,2)_{(t,1) = exp[-0.124t], R}(1,2)_{(t,2) = exp[-0.151t].} The third conveyer is composed of:

- two driving wheels with reliability functions

R(2,1)_{(t,1) = exp[-0.167t], R}(2,1)_{(t,2) = exp[-0.182t],} - 240 links with reliability functions

R(2,2)_{(t,1) = exp[-0.208t], R}(2,2)_{(t,2) = exp[-0.231t].}

Taking into account the operation process of the considered transportation system we distinguish the following as its operation states:

1

z – the system operation with the largest efficiency when all components of the sub-systems S1, S2, S and 3 S are used, 4

2

z – the system operation with less efficiency system when the first conveyor of sub-system S1, the first and second elevators of subsystem S2, the first conveyor of

subsystem S and the first and second conveyors of subsystem 3 S are used, 4 3

z – the system operation with least efficiency when only the first conveyor of sub-system S₁,the first elevator of subsystem S₂, the first conveyor of subsys-tem S and the first conveyor of subsystem 3 S are used.4

Moreover, we arbitrarily assume the following matrix of the conditional distribution functions of the system sojourn times kl, k,l 1,2,3,

_         - -0 1 1 1 0 1 1 1 0 )] ( [ 20 5 . 12 5 20 5 78 . 2 t t t t t t kl e e e e e e t H

and the probabilities of transitions between the states are given by

. 0 615 . 0 385 . 0 2 . 0 0 8 . 0 643 . 0 357 . 0 0 ] [            kl p

Further, according to (2), the unconditional distribution functions of the process Z(t) sojourn times k in the states zk, k 1,2,3, are given by

H₁(t)1-0.357exp[-2.78t]-0.643expexp[-5t],

H2(t)1-0.8exp[-20t]-0.2exp[-5t],

H₃(t)1-0.385exp[-12.5t]-0.615exp[-20t],

and their mean values, from (3), are

M₁ E[₁]0.3570.360.6430.20.25712, M2 E[2]0.80.050.20.20.08,

M₃ E[₃]0.3850.080.6150.050.06155. Since from the system of equations (5)

,

1

0

615

.

0

385

.

0

2

.

0

0

8

.

0

643

.

0

357

.

0

0

]

,

,

[

]

,

,

[

3 2 1 3 2 1 3 2 1





























































we get 1 0.3742, 2 0.321, 3 0.3048,

then the limit values of the transient probabilities p_k(t) at the operational states z ,k

according to (4), are given by

p1 0.684, p2 0.1826, p3 0.1334.

(6)

After considering changing in operation states the system reliability structures, the sys-tem unconditional reliability is given by

R(t,)= [1,R(t,1), R(t,2)], (7) where R(t,1)



0

.

684



_R(1)(_t,1) ₊

0

.

1826



_R(2)_{(t,1) +}

0

.

1334



), 1 , ( ) 3 ( _t R R(t,2)



0

.

684



_R(1)(_t,2) ₊

0

.

1826



_R(2)_{(t,2) +}

0

.

1334



), 2 , ( ) 3 ( _t R and _R(1)(_t,1), _R(2)_{(t,1), R}(3)_(t,1) and R(1)(t,2), R(2)(t,2), ) 2 , ( ) 3 ( _t

R _{are conditional reliability functions in particular system operation states}

determined in [3]. Hence, we get the mean values and standard deviations of the system unconditional lifetimes in the reliability state subsets given by

(1)0.6840.011970.18260.00860.13340.0054 0104782 . 0  , (1) 0.000048 0.0069321, (2)0.6840.00860.18260.00620.13340.0039 0075347 . 0  , (8) (2) 0.00002470.004973, (9)

and the mean values of the system unconditional lifetimes in particular reliability states are:

ˆ(1)  (1)-(2)0.0029435, ˆ(2)(2) 0.0075347.

If a critical reliability state of the system is r = 2, then its risk function takes the form ([3])

r(t)  1 - R(t,2).

Hence, the moment when the risk exceeds the critical level d = 0.05 is ([3]) t = r-1_{(d)  0.00106 years.}

Further, assuming that the grain elevator is repaired after its failure and that the time of the system renovation is not ignored and it has the mean value ₀ _0.005 and the standard deviation ₀ 0.005, from Proposition 3 ([1]), applying (8) and(9), we ob-tain the following results:

i) the distribution function of the time

_S

_N

₍

₂

₎

until the Nth renovation of this sys-tem, for sufficiently large N, has approximately normal distribution

), 0070498 . 0 , 0125347 . 0 ( N N N i.e.  ( ,2) ) ( t F N ), 0070498 . 0 0125347 . 0 ( ) ) 2 ( ( _(0,1) N N t F t S P N   _N -

ii) the expected value and the variance of the time

_S

_N

₍

₂

₎

up to the Nth renovation take respectively forms

E

[

S

N

(

2

)]



0

.

0125347

N

,



,

0000496

.

0

)]

2

(

[

S

N

D

N





iii) the distribution function of the time

_S

-_N

₍

₂

₎

up to the Nth exceeding the reliability critical state of this system takes form

F(N)(t,2) ( (2) ) (0,1)( ₀0_.0000496.0125347 ₀0_.005.005), - -  N N t F t S P _{N N}

iv) the expected value and the variance of the time -

₍

₂

₎

N

S

up to the Nth exceeding the reliability critical state of this system are respectively given by

E

[

S

-N

(

2

)]



0

.

0075347

N



(

N

-

1

)

0

.

005

,

D

[

S

-N

(

2

)]



0

.

0000247

N



(

N

-

1

)

0

.

000025

,

v) the distribution of the number _N_(t,2) renovations of the system up to the moment

, 0 ,t  t _{is of the form} ) ) 2 , ( (N t N P   (0,1)(_0.00704980.0125347_79.778535 ) t t N FN - ), 778535 . 79 0070498 . 0 ) 1 ( 0125347 . 0 ( ) 1 , 0 ( t t N F_N  --

vi) the expected value and the variance of the number _N _(t,2) of renovations of the system up to the moment t,t 0, take respectively forms

H (t,2) 79.778535t, D(t,2) 25.235654t,

vii) the distribution of the number _N- _(t,2) exceeding the reliability critical state of this system up to the moment t,t 0, is of the form

( ( ,2) ) (0,1)(_0.00704980.0125347_79.778535(0.005_0.005))  -  t t N F N t N P N _{0.0070498 79.778535( 0.005)}), 005 . 0 ) 1 ( 0125347 . 0 ( ) 1 , 0 (  - -t t N F_N

viii) the expected value and the variance of the number _N-_{( rt, )} exceeding the relia-bility critical state of this system up to the moment t,t 0, are respectively given by

H(t,2)79.778535(t0.005), D(t,2) 25.235654(t0.005), ix) the limiting availability coefficient of the system for sufficiently large

t

is given by K(t,2) _0.00753470.0075347_005 0.6011073,

x) the availability coefficient of the system in the time interval t,tt),t 0, is given by





 t

t

,

2

)

79

.

7785535

,

(t

K

R(t,2)dt.

4. Conclusions

The paper proposes an approach to the solution of practically very important problem of linking the systems’ reliability and their operation processes. To involve the interactions between the systems operation processes and their changing in time reliability structures a semi-markov model of the system operation processes and the system limit reliability functions are applied. This approach gives practically important and not difficult in ev -eryday usage tool for reliability and availability evaluation of the large systems with changing reliability structures during their operation processes. In further developing of the proposed methods it seem to be possible to obtain the results useful in the complex technical systems and their operation processes reliability and availability evaluation, improvement and maintenance optimisation. Application of the proposed method is il-lustrated in the reliability, risk and availability evaluation of the port grain transportation system. The reliability data concerned with the operation process and reliability func tions of the components of the port grain transportation system are not precise and as -sumed almost arbitrarily. They are coming from experts and are concerned with the mean lifetimes of the system components and with the conditional sojourn times of the system in the operation states under arbitrary assumption that their distributions are ex-ponential.

References

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