Reliability Analysis of Multi-state Systems with Heterogeneous

Multi-state ElementsDmitrij Birjukov

National Taras Shevchenko University of Kyiv Faculty of Cybernetics

birjukov@unicyb.kiev.ua

Multi-state system example and definition

Example. Let’s consider communication network that provide message passing. The information security achieved via information encoding.

In the case of encoding unit(subsystem) failure, the network will still perform message passing, but with the lower level of security. So this system has at least 3 states.

Definition. Systems that characterized by different levels of performance are known as multi-state systems (MSS).

Another examples of MSS are power systems and computer systems, where the elements’ performance is characterized by generating capacity and data processing speed respectively.

Most of reliability analysis and optimization models assume that system consists of binary-state components, where the states are functioning and failure. Binary-state reliability models don't allow to describe adequately: MSS operation processes, remaining resources, system state evolution, reasons of system failures and mechanisms of their prevention.

Report overview The aim of this work is to provide decision making framework for

reliable MSS design.

This report represents the following results:• reliability model of multi-state system with heterogeneous multi-

state elements; •redundancy optimization problem for multi-state elements (where

redundant elements are identical or non-identical).

Multi-state systems reliability models overview

MSS reliability assessment based on (1) extension of the Boolean models to the multi-valued case, (2) stochastic processes (mainly Markov and semi-Markov) approach, (3) universal generating function approach.

The main difficulty in the MSS reliability analysis is the “dimension damnation” since each system element can have many different states (not only two states as the binary-state system).

This makes the known approaches overworked and time consuming, because the number of system states increases dramatically with the increase in the number of system elements.

However, mentioned method were applied to simple models, such as the series, parallel systems (so called simple-structured systems models). In practice, technical systems are more complex, they include huge number of elements, that perform different tasks.

Multi-state systems reliability models overview

Main restrictions of known multi-state reliability models are the following:

•state sets of elements consist states with same meaning (therefore each element have same number of identical states);

•state sets of elements are ordered (partially-ordered); •system consists of elements with same functionality.

The reliability evaluation is strongly required on the stage of system engineering design. To evaluate MSS reliability in the case of complex technical systems special mathematical approach and software are needed.

Many technical systems unlike of known MSS reliability models consist of heterogeneous multi-state elements which perform non-identical functions and therefore have non-identical states. This type of MSS is described as MSS with heterogeneous elements (MSSHE) model. It should be underlined that MSSHE model is more general than known MSS models.

We consider complex-structured MSSHE, that consists of heterogeneous multi-state elements = {1,2,..., }i I n , which perform non-identical functions during operation period *

0T = [ , ]t t . Let ( ) = {1,..., }y t S m be the state of the system at the moment Tt . System state ( )y t is characterized by probabilities

( ) = { ( ) = }sp t P y t s , Tt , s S , that are incompatible, ( ) = 1, T.s

s S

p t t

Multi-state system with heterogeneous elements (1)

State of the element depends on the process of degradation, which is modeled by random process ( ) = {1,2,..., , T}i i it k t .

We assume that: 1. random processes ( ) = {1,2,..., , T}i i it k t , i I , are

stochastically independent, 2. probabilities ( ) ( ) =s

i t { ( ) = }iP t s , Tt , is are given.

Multi-state system with heterogeneous elements (2)

If degradation of the elements is modeled by Markov processes, than ( ) ( )s

i t , Tt , is , can be found as a solution of the system of differential equations:

( )( , ) ( ) ( ) ( , )

/ /

( ) = ( ) ( ) , , T,s

j s j s s jii i i i i

j S s j S s

d t t t s tdt

(1)

under given initial conditions ( ) ( )

0( ) = ,s si i it s , (2)

where ( , )s ji - constant intensity of transition from state s to state j ,

, is j , that is given. The exact solution of (1)-(2) can be obtained in the form

( ) ( ) = exp , , ,si q q i

q

t t s i I (3)

where q and q are coefficients. It should be noted that it is easy to perform algebraic operations over solution provided in the form (3).

Multi-state system with heterogeneous elements (3)

Complex system can be modeled by an oriented acyclic graph, in which nodes are correspondent to system elements and arcs are correspondent to functional links.

Consider the particular element i I . Lets’ denote 0

iI - the set of elements' indexes that are incident for element i I . Fig.1 illustrates part of graph that is corresponding for element i I .

Fig.1 Structured interaction between elements in MSS. Elements of MSS functionally interact through their input and

output. Taking into account functional specifics of the system and its elements the state function (SF) of the element i can be offered.

SF defines ( )iy t as a function of degradation process ( )i t and states ( )ly t of the elements 0

il I I :

0( ) = ( ), ( ),i i i l iy t t y t l I .

( )i t 0( ),l iy t l I ( ) ( )i iy t

Multi-state system with heterogeneous elements (4)

To define SF one of the following ways can be used: multi-state fault tree, analytical expression, tabular form. Then, using SF it is possible to define:

( ) ( ) = { ( ) = }, T, , .si i ip t P s t s S i I

The state function (SF) of system defines ( )y t as a function of elements’ states:

1( ) = ( ),..., ( )ny t y t y t . It is possible to define sp t by

1= ( ),..., ( ) = , T, .s np t P y t y t s t s S Thus state of the system depends on its structure, states and

functional interaction of its elements that is reflected in SF of elements.

Multi-state system with heterogeneous elements (5)

To evaluate sp t , T,t s S , recursive procedure should be started with sink node. The source node reliability is considered to be given. procedure recursive ( node i ) { for each node 0

il I do if ! is_evaluated(node l ) then recursive( node l ) ; evaluate _ reliability( node l ); } Logical function is_evaluated(node l ) returns true if

( ) ( ), T,sl lp t t s S are evaluated.

Procedure evaluate_reliability applies logical-probabilistic methods to fault tree of given element.

SF of the sink node is correspondent to the SF of the system.

Multi-state system with heterogeneous elements (6)

Reliability is considered to be a property of system that is characterized by ability to perform functions that intended to be done during determined period of time under certain exploitation conditions. According to this consideration complex system reliability can be measured by a number of different indexes.

Reliability, efficiency and availability of MSS are characterized quantitatively by indexes that should take into account evolution of the system state, be connected to the level of functional tasks performance, be dependent on time and periods of system operation.

Suppose su , s S - the effect of system operation or consequences of failure (depending on the contents of the state) is correspondent to each state of the system. Then efficiency index of MSS can be determined as expected efficiency:

T

(T) = ( ) .U s ss S

E u p t dt

(1)

Multi-state system reliability and efficiency indexes (1)

If system reliability connected with the ability of the system to execute functional tasks with efficiency higher the given level w , then the index of expected availability is used. MSS availability ( , )AE T w is the probability that the MSS will be in the states with performance level higher than w during operational period T . MSS availability is the function of time period and required demand w . It may be defined as

, T

(T, ) = ( ) .A s ss S u ws

E w u p t dt

(1)

In practice the interval of system operating is often divided into subintervals hT , *{1,..., }h H h , with different fixed efficiency levels

hw . Then expected workability of the system is defined as: ( , )I

A A h hh H

E E T w

.

Multi-state system reliability and efficiency indexes (2)

In general requirements to system's effectiveness on operation interval may be defined by function ( )w t , t T , and deviation from these requirements means losses, then system reliability can be charaterized by expected efficiency deviation:

2

T

(T, ( )) = ( ( ) ) ( ) .D s ss S

E w t w t u p t dt

(1)

If among the states of the system there is a subset KS of failure states that lead to disastrous results, then safety of system operating is defined by conditions

( ) , T, ,Ks sp t t s S S

where s - is upper bound for probability of system state Ks S .

Multi-state System reliability and efficiency indexes (3)

Redundancy is widely used approach in complex technical system design. Including of redundant elements in system structure increases its reliability, but also increases system cost, weight and effect on other parameters.

Redundancy allocation of multi-state elements

Case of identical redundant elements Let 1 2= ( , ,..., )n be a vector, where i - number of units in

subsystem i . In each subsystem i I simultaneously used one main and ( 1i ) redundant elements. As far system structure is given (determined by subsystems functionality) and different versions of subsystems’ implementation differ only by quantities of redundant elements, system version is determined by vector 1 2= ( , ,..., )n .

During operation process system and its components are degrading. Degradation process of subsystem, which contains i identical elements with 1 2( ), ( ),..., ( )i

i i it t t stochastically independent random processes with identical ( ) ( ) = { ( ) = }s k

i it P t s , = 1,2,...,i im , {1,2,..., }ik , is described by random process

1 2( , ) = max{ ( ), ( ),..., ( )} , Tii i i i i it t t t t

where ( ) ( , ) = { ( , ) = }si i i it P t s are the following

1( ) ( ) ( )

=1 =1

( , ) = ( ) ( ) ,i is s

s k ki i i i

k k

t t t

T, ,it s i I .

Redundancy allocation of multi-state elements

Case of non-identical redundant elements In this case iq different types (versions) of redundant elements can

be allocated in subsystem i .

Let’s denote ** * (( ((1) (1) (1)1 2

1 1 2 2)) )= ( ,..., , ..., ,..., ,..., )qq q n

n n - vector, that defines number of different type redundant elements in subsystems. Each subsystem i I simultaneously use q

i redundant elements of type *= {1,2,..., }i iq Q q , that are connected in parallel (hot stand-by).

Each element of subsystem i is characterized by parameters ( , )i qlg ,

*= {1,2,..., }l L l (cost, energy supply, weight). Supplying of l -th type of resource is defined by ( )lg and constrained by value lb , l L .

Redundancy allocation of multi-state elements

During operational period elements are degrading. When subsystem consist non-identical degrading elements, its degradation can be described by stochastic process

(1)( ) 1(1) 11 12( ,..., , ) = max{ ( ), ( ),..., ( ),iqi

i i i i i it t t t ( 2)221 22( ), ( ),..., ( ),...,i

i i it t t

*( )* * *1 2( ), ( ),..., ( )} , Tqi

i i i iq q qi i i it t t t ,

where ( )qiq

i t are series of stochastically independent processes with equal ( ) ( ) =q s

i t { ( ) = }qkiP t s , = 1,2,...,i im ,

*

{1,2,..., },iqik iq Q .

Thus *

1( )( ) (1) ( ) ( )

=1 =1

( ,..., , ) = ( ) ( ) ,q qi i

i

i i

s sqs k k

i i i i ik kq Q q Q

t t t

T, ,it s i I . (1)

MSSHE redundancy optimization problem consists in finding the quantity of different type redundant elements to be allocated in subsystems, that provides minimum deviation of expected system effectiveness from its desirable level on the operation period, and satisfies system-level resource constraints. Mathematical model of described problem is the following:

2( ( ) ) ( , ) mins ss S

w t u p t dt

under constrains: ( )l lg b , l L ,

*[1, ]i i i , i - integer, i I ,

1 2( , ,..., )n ii I

,

where ( )w t - given efficiency requirements, su - system effectiveness in the state s S .

MSSHE redundancy optimization problem

ConclusionsWhen considering the system composed from heterogeneous

elements (of different intending) known multi-state system reliability models can't be implemented.

This paper presents a novel approach to multi-state complex-structured system reliability analysis.

It provides decision making framework for reliable MSS design.