a universal generating function approach for the analysis

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A universal generating function approach for the analysisof multi-state systems with dependent elements

Gregory Levitin*

Planning, Development and Technology Division, Department of Reliability, The Israel Electric Corporation Ltd.,

Bait Amir, P.O. Box 10, Haifa 31000, Israel

Received 21 September 2003; accepted 1 December 2003

Abstract

The paper extends the universal generating function technique used for the analysis of multi-state systems to the case when the

performance distributions of some elements depend on states of another element or group of elements.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Multi-state system; Dependent elements; Universal generating function

1. Introduction

All technical systems are designed to perform their

intended tasks in a given environment. Some systems can

perform their tasks with various distinguished levels of

efficiency usually referred to as performance rates. A system

that can have a finite number of performance rates is called a

multi-state system (MSS). Usually, MSS is composed of

elements that in their turn can be multi-state. An element is

an entity in a system, which is not further sub-divided. This

does not imply that an element cannot be made of parts, but

only means that, in a given reliability study, it is regarded as

a self-contained unit and is not analyzed in terms of the

reliability performances of its constituents.

Actually, a binary system is the simplest case of a MSShaving two distinguished states (perfect functioning and

complete failure).

There are many different situations in which a system

should be considered to be a MSS

† Any system consisting of different units that have a

cumulative effect on the entire system performance has

to be considered as a MSS. Indeed, the performance rate

of such a system depends on the availability of its units,

as the different numbers of the available units can

provide different levels of the task performance.

† The performance rate of elements composing a system

can also vary as a result of their deterioration (fatigue,partial failures) or because of variable ambient con-

ditions. Element failures can lead to the degradation of

the entire MSS performance. The performance rates of

the elements can range from perfect functioning up to

complete failure. The failures that lead to the decrease in

the element performance are called partial failures. After

partial failure, elements continue to operate at reduced

performance rates, and after complete failure the

elements are totally unable to perform their tasks.

The examples of MSS are power systems or computer

systems where the component performance is character-

ized by generating capacity or data processing speed,respectively. For the MSS, the outage effect will be

essentially different for units with different performance

rate. Therefore, the reliability analysis of MSS is much

more complex in comparison with binary-state systems.

In real-world problems of MSS reliability analysis, the

great number of system states that need to be evaluated

makes it difficult to use traditional binary reliability

techniques.

The methods of MSS reliability assessment are based on

four different approaches: the Monte-Carlo simulation

technique, the extension of the Boolean models to the

multi-valued case, the stochastic process (mainly Markov

and semi-Markov) approach and the universal generating

function approach.

0951-8320/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ress.2003.12.002

Reliability Engineering and System Safety 84 (2004) 285–292www.elsevier.com/locate/ress

* Corresponding author. Tel.: þ972-4-8183726; fax: þ972-4-8183790.

E-mail address: [email protected] (G. Levitin).

http://www.elsevier.com/locate/ress

http://www.elsevier.com/locate/ress



Even though, almost every real world MSS can be

represented by the Monte-Carlo simulation for the

reliability assessment, the main disadvantages of this

approach are the time and expenses involved in the

development and execution of the model.The approach based on the extension of Boolean

models is historically the first method that was developed

and applied for the MSS reliability evaluation. It is based

on the natural expansion of the Boolean methods (that

were well-established for the binary-state system

reliability analysis) to the MSS. The main difficulties in

the MSS reliability analysis are the ‘dimension damna-

tion’ since each system element can have many different

states (not only two states as existed in the binary-state

system). This makes the Boolean approach overworked

and time consuming.

The stochastic process methods that are widely used forthe MSS reliability analysis are more universal. In fact, this

approach was successfully used for the reliability assess-

ment of multi-state power systems and some types of

communication systems even before MSS was theoretically

defined. The stochastic process method can be applied only

to relatively small MSS because the number of system states

increases dramatically with the increase in the number of

system elements.

The computational burden is the crucial factor when one

solves optimization problems where the reliability measures

have to be evaluated for a great number of possible solutions

along the search process. This makes using the three above-

mentioned methods in reliability optimization problematic.On the contrary, the universal generating function (UGF)

technique is fast enough. This technique allows one to find

the entire MSS steady-state performance distribution (PD)

based on the steady-state PD of its elements by using a fast

algebraic procedure. An analyst can use the same recursive

procedures for MSS with a different physical nature of performance and different types of element interaction.

The UGF (u-function) technique was introduced in Ref.

[1] and proved to be very effective for the reliability

evaluation of different types of MSS [2]. The u-function

extends the widely known ordinary moment generating

function. The essential difference between the ordinary and

UGF is that the latter allows one to evaluate probabilistic

distributions of overall performance for wide range of

systems characterized by different topology, different nature

of interaction among system elements and different physical

nature of elements’ performance measures. This can be

done by introducing different composition operators overthe UGF.

One of the main assumptions when using the UGF

technique was that the system elements are mutually

statistically independent. This assumption is not true in

many technical systems. This paper suggests an extension of

the UGF approaches to the cases when the PD of some

system elements are influenced by the states of other

elements or subsystems. The suggested approach is valid

only in the case of unilateral dependency between the

elements.

It is assumed that functioning of each system element j is

characterized by random discrete performance G j: The

performance of the entire system is an unambiguousfunction of the performances of its individual elements.

Nomenclature

MSS multi-state systemUGF (u-function) universal generating function

PD performance distribution

G j random performance rate of MSS

element j

g jk performance rate of MSS element j at

state k

gi set of possible performance rates of

element i

gmi mth subset of gi

p jk unconditional probability that MSS

element j is in state k

p jk lm conditional probability that element j

is in state k when the influencingelement i has the performance belong-

ing to the set gmi

~p jc vector of conditional probabilities that

dependent element j is in state c

K i number of different states of indepen-

dent element i

C j number of different states of dependent

element j

w minimal allowable level of MSS per-formance (demand)

uið zÞ u-function representing performance

distribution of ith independent

element

~u jð zÞ u-function representing performance

distribution of jth dependent element

d ðu jð zÞ; wÞ operator over MSS u-function which

determines probability Pr{G j $ w}

^par

;^ser

composition operators over u-func-

tions of mutually independent

elements connected in parallel and

in series

^)

par;^)

ser;^) composition operators over u-func-

tions of influencing and dependent

elements connected in parallel, in

series and not connected (in the

reliability logic diagram sense)

G. Levitin / Reliability Engineering and System Safety 84 (2004) 285–292286



The system fails if it cannot meet the demand w (i.e. its

performance is lower than w).

2. Extension of the UGF technique to the case of elements

with unilateral dependency

2.1. u-Functions of individual independent elements

and their compositions

The u-function of an independent discrete random

variable X is defined as a polynomial

uð zÞ ¼XK

k ¼1

qk z xk ; ð1Þ

where the variable X has K possible values and qk

is the

probability that X is equal to xk : To evaluate the probability

that the random variable X is not less than the value w; the

coefficients of polynomial uð zÞ should be summed for every

term with x j $ w

Pr{ X $ w} ¼X

xk $w

qk : ð2Þ

This can be done using the following d operator over uð zÞ

d ðuð zÞ; wÞ ¼ d XK

k ¼1

qk z xk ; w

!¼

XK

k ¼1

d ðqk z xk ; wÞ; ð3Þ

where for the individual term qk z xk

d ðqk z xk ; wÞ ¼

qk ; xk $ w

0; xk , w

(: ð4Þ

In our case, the polynomial u jð zÞ can define PD of element j

(probability mass function of random value G j), i.e. it

represents all of the possible states of the element by

relating the probabilities of each state to the performance of

the element in that state. Note that the PD of the basic

element j defined by the ordered sets g j ¼ {g jk ; 1 # k #

K j} and p j ¼ { p jk ; 1 # k # K j} can now be represented as

u jð zÞ ¼XK j

k ¼1 p jk z

g jk :ð5Þ

To obtain the u-function of a subsystem containing two

elements, composition operators are introduced. Theseoperators determine the u-function for two elements

connected in parallel and in series, respectively, using

simple algebraic operations over the individual u-functions

of basic elements. All the composition operators take

the form

uið zÞ^par

u jð zÞ ¼XK i

k ¼1

pik zgik ^

par

XK j

h¼1

p jh zg jh

¼XK i

k ¼1

XK j

h¼1

pik p jh zparðgik ;g jhÞ ð6Þ

uið zÞ^ser

u jð zÞ ¼

XK i

k ¼1

pik zgik ^

ser XK j

h¼1

p jh zg jh

¼XK i

k ¼1

XK j

h¼1

pik p jh zserðgik ;g jhÞ: ð7Þ

The obtained u-function relates the probability of each state

of a subsystem (equal to the product of the probabilities of

states of its independent elements) to the performance rate

of the subsystem in this state. The functions parð·Þ and serð·Þ

in composition operators expresses the entire performance

rate of the subsystem consisting of two elements connectedin parallel or in a series in terms of the individual

performance rates of the elements. The definition of the

functions parð·Þ and serð·Þ strictly depends on the physical

nature of the system performance measure and on the natureof the interaction among the elements.

Consider the flow transmission type of MSS in which the

performance measure is defined as productivity or capacity

(continuous materials or energy transmission systems,manufacturing systems, power supply systems). In MSS of

this type, the total performance rate of a pair of elements

connected in parallel is equal to the sum of the performance

rates of the individual elements. When the elements are

connected in series, the element with the lowest perform-

ance rate becomes the bottleneck of the subsystem. There-

fore, for a pair of elements connected in series the

performance rate of the subsystem is equal to the minimum

of the performance rates of the individual elements.Therefore, the composition operators ^

parand ^

serdefined for the parallel and series connections of a pair of

elements, respectively, in the flow transmission MSS take

the form

uið zÞ^par

u jð zÞ ¼ uið zÞ^þ

u jð zÞ ¼XK i

k ¼1

XK j

h¼1

pik p jh zgik þg jh ; ð8Þ

and

uið zÞ^ser

u jð zÞ ¼ uið zÞ^min

u jð zÞ ¼XK i

k ¼1 XK j

h¼1

pik p jh zmin{gik ;g jh}

: ð9Þ

In the task processing type of MSS the performance measure

is characterized by an operation time (processing speed).This category may include control systems, information or

data processing systems, manufacturing systems with

constrained operation time, etc. The operation of these

systems is associated with consecutive discrete actions

performed by the ordered line of elements. When the work

sharing between the parallel elements is allowed, the system

processing speed depends on the rules of the work sharing.

The most effective rule providing the minimal possible time

of work completion, shares the work among the elements in

proportion to their processing speed. In this case, the

processing speed of the parallel system is equal to the sumof the processing speeds of all of the elements. The total

G. Levitin / Reliability Engineering and System Safety 84 (2004) 285–292 287



operation time of two elements connected in series is equal

to the sum of the operation times of these elements. When

one measures the element (system) performance in terms of

processing speed (reciprocal to the operation time), the total

processing speed of the subsystem consisting of two

elements with processing speeds Gi and G j is equal to

ðG21i þ G

21 j Þ

21¼ GiG j = ðGi þ G jÞ: ð10Þ

Therefore, the composition operators ^par

and ^ser

defined for

the parallel and series connections of a pair of elements,

respectively, in the task processing MSS take the form

uið zÞ^par

u jð zÞ ¼ uið zÞ^þ

u jð zÞ ¼XK i

k ¼1

XK j

h¼1

pik p jh zgik þg jh ; ð11Þ

and

uið zÞ^ser

u jð zÞ ¼ uið zÞ^£

u jð zÞ ¼XK i

k ¼1

XK j

h¼1

pik p jh zgik g jh = ðgik þg jhÞ:

ð12Þ

By changing the functions parð·Þ and serð·Þ one can obtain

the composition operators for various types of MSS. Some

additional operators for special types of systems are

presented in Ref. [2].

2.2. u-Functions of dependent elements

Consider a subsystem consisting of a pair of multi-state

elements i and j in which the PD of element j depends onstate of element i: Since, the states of the elements are

distinguished by the corresponding performance rates, we

can assume that the PD of element j is determined by the

performance rate of element i: Let gi be the set of possible

performance rates of element i: In general, this set can be

separated into M mutually disjoint subsets gmi ð1 # m # M Þ

[ M

m¼1

gmi ¼ gi; g

mi

\g

li ¼ 0; if m – l; ð13Þ

such that when element i has the performance rate gik [ gmi ;

the PD of element j is defined by the ordered sets

g jlm ¼ {g jclm; 1#

c#

C jlm} and q jlm ¼ { p jclm; 1#

c#

C jlm}; where

q jclm ¼ Pr{G j ¼ g jclmlGi ¼ gik [ gmi }: ð14Þ

If each performance rate of element i corresponds to

different PD of element j; we have M ¼ K i and gmi ¼ {gim}:

We can define the set of all the possible values of

performance rate of element j as g j ¼S

M m¼1 g jlm and redefine

the conditional PD of element j when element i has the

performance rate gik [ gmi using two ordered sets

g j ¼ {g jc; 1 # c # C j} and p jlm ¼ { p jclm; 1 # c # C j};

where

p jclm ¼0; g

jcÓ g

jlm

q jclm; g jc [ g jlm

(: ð15Þ

According to this definition

p jclm ¼ Pr{G j ¼ g jclGi ¼ gik [ gmi } ð16Þ

for any possible realization of G j and any possible

realization of Gi [ gmi :

Since, the sets gmi ð1 # m # M Þ are mutually disjoint, the

unconditional probability that G j ¼ g jc can be obtained as

p jc ¼X M

m¼1

Pr{G j ¼ g jclGi [ gmi }Pr{Gi [ g

mi }

¼X M

m¼1

p jclm

XK i

k

pik 1ð pik [ gmi Þ: ð17Þ

In the case when gmi ¼ {gim}

p jc ¼X M

m¼1 pim p jclm: ð18Þ

The unconditional probability of the combination Gi ¼ gik ;

G j ¼ g jc is equal to pik p jclmðk Þ; where mðk Þ is the number of

set gik belongs to: gik [ gmðk Þi :

Consider an example in which element 1 has the PDg1 ¼ {0; 1; 2; 3}; p1 ¼ {0:1; 0:2; 0:4; 0:3} and the PD of

element 2 depends on the performance rate of element 1

such that when G1 # 2ðG1 [ g11 ¼ {0; 1; 2}Þ element 2 has

the PD g2l1 ¼ {0; 10}; q2l1 ¼ {0:3; 0:7} while when G1 .

2ðG1 [ g21 ¼ {3}Þ element 2 has the PD g2l2 ¼ {0; 5};

q2l2 ¼ {0:1; 0:9}: The conditional PDs of element 2 can be

represented by the sets g2 ¼ {0; 5; 10} and p2l1 ¼{0:3; 0; 0:7}; p2l2 ¼ {0:1; 0:9; 0}:

The unconditional probabilities p2c are

p21 ¼ Pr{G2 ¼ 0} ¼ Pr{G2 ¼ 0lG1 [ g11}Pr{G1 [ g

11}

þ Pr{G2 ¼ 0lG1 [ g21}Pr{G1 [ g

21}

¼ p21l1ð p11 þ p12 þ p13Þ þ p21l2ð p14Þ

¼ 0:3ð0:1 þ 0:2 þ 0:4Þ þ 0:1ð0:3Þ

¼ 0:24;

p22 ¼ Pr{G2 ¼ 5}¼ Pr{G2 ¼ 5lG1[ g11}Pr{G1[ g

11}

þPr{G2 ¼ 5lG1[ g21}Pr{G1[ g2

1}

¼ p22l1ð p11þ p12þ p13Þþ p22l2ð p14Þ

¼ 0ð0:1þ0:2þ0:4Þþ0:9ð0:3Þ¼ 0:27;

p23 ¼ Pr{G2 ¼ 10}¼Pr{G2 ¼ 10lG1[ g11}Pr{G1[ g1

1}

þPr{G2 ¼ 10lG1[ g21}Pr{G1[ g

21}

¼ p23l1ð p11þ p12þ p13Þþ p23l2ð p14Þ

¼ 0:7ð0:1þ0:2þ0:4Þþ0ð0:3Þ ¼ 0:49:

The probability of the combination G1 ¼ 2; G2 ¼ 10 is

p13 p23lmð3Þ ¼ p13 p23l1 ¼ 0:4·0:7 ¼ 0:28:

The probability of the combination G1 ¼ 3; G2 ¼ 10 is

p14 p23lmð4Þ ¼ p14 p23l2 ¼ 0:3·0 ¼ 0:

Having the sets g j and p jl m 1 # m # M that define theconditional PDs of element j; we can represent them in




the form of the u-function with vector coefficients

~u jð zÞ ¼ XC j

c¼1

~p jc zg jc ; ð19Þ

where

~p jc ¼ { p jcl1; p jcl2; …; p jcl M }: ð20Þ

Since, each combination of the performance rates of the two

elements Gi ¼ gik ; G j ¼ g jc corresponds to the subsystem

performance rate v ðgik ; gicÞ and the probability of the

combination is pik p jclmðk Þ; we can obtain the u-function of the

subsystem as follows

uið zÞ^)

v ~u jð zÞ¼

XK i

k ¼1

pik zgik ^

)

v

XC j

c¼1

~p jc zg jc ¼

XK i

k ¼1

pik

XC j

c¼1

p jclmðk Þ zv ðgik ;g jcÞ:

ð21Þ

The function v ðgik ; g jcÞ should be substituted by parðgik ; g jcÞ

or serðgik ; g jcÞ in accordance with the type of connection

between the elements. If the elements are not connected in

the reliability block diagram sense (the performance of

element i does not affect directly the performance of the

subsystem, but affects the PD of element j) the last equation

takes the form

uið zÞ^)

~u jð zÞ ¼XK i

k ¼1

pik zgik ^

) XC j

c¼1

~p jc zg jc ¼

XK i

k ¼1

pik

XC j

c¼1

p jclmðk Þ zg jc :

ð22Þ

2.3. u-Functions of group of dependent elements

Consider a pair of mutually independent elements n and j

and assume that both these elements depend on the same

element i: For any state k of the element i ðgik [ gmðk Þi Þ; the

PDs of the elements n and j are defined by the pairs of

vectors gn; pnlmðk Þ and g j; p jlmðk Þ; where pnlmðk Þ ¼ { pnclmðk Þl1 #

c # C n}: Having these distributions one can obtain the

u-function corresponding to the conditional PD of the

subsystem consisting of elements n and j by applying

the operators (6) or (7)

XC n

c¼1

pnclmðk Þ zgnc^v

XC j

h¼1

p jhlmðk Þ zg jh ¼XC n

c¼1

XC j

h¼1

pnclmðk Þ p jhlmðk Þ zv ðgnc ;g jhÞ;

ð23Þ

where the function v ðgnc; g jhÞ is substituted by parðgnc; g jhÞ

or serðgnc; g jhÞ in accordance with the type of connection

between the elements. Applying Eq. (23) for any subset

gmi ð1 # m # M Þ; we can obtain the u-function representing

all of the conditional PDs of the subsystem consisting

of elements n and j using the following operator over the

u-functions ~unð zÞ and ~u jð zÞ

~unð zÞ^v

~u jð zÞ¼XC n

c¼1

~pnc zgnc^

v XC j

h¼1

~p jh zg jh ¼X

C i

c¼1XC j

h¼1

~pnc ~p jh zv ðgnc ;g jhÞ;

ð24Þ

where

~pnc ~p jh ¼ { pncl1 p jhl1; pncl2 p jhl2; …; pncl M p jhl M }: ð25Þ

2.4. Algorithm for MSS reliability (availability) evaluation

Consecutively, applying the operators (8), (9), (21) and

(22) and replacing pairs of elements by auxiliary equivalent

elements, one can obtain the u-function representing the PD

of the entire system and evaluate the system reliability

(availability). The following recursive algorithm obtains the

system availability

1. Obtain the u-functions of all of the independent elements

using Eq. (5).2. Obtain the u-functions of all of the dependent elements

using Eqs. (19) and (20).

3. If the system contains a pair of mutually independent

elements connected in parallel or in a series, replace this

pair with an equivalent element with u-function obtained

by^par

or^ser

operator using Eqs. (6) or (7), respectively (if

both elements depend on the same external element, Eq.

(24) with the corresponding function parð·Þ or serð·Þ

should be applied instead of Eqs. (6) or (7)).

4. If the system contains a pair of dependent elements,

replace this pair with an equivalent element with u-

function obtained by ^)

par; ^

)

seror ^

)

operator (accord-

ing to the type of interaction between the elements) usingEqs. (21) or (22), respectively.

5. If the system contains more than one element, return to

step 3.

6. Obtain the system availability AðwÞ for the given demand

w by applying the operator d (3) over the u-function of

the remaining single equivalent element representing the

system PD.

3. Illustrative examples

3.1. Task processing system

Consider an information processing system consisting

of three independent computing blocks (Fig. 1). Each block consists of a high priority processing unit and a low

priority processing unit that share access to a database.

When the high priority unit operates with the database, the

low priority unit waits for the access. Therefore, the

processing speed of the low priority unit depends on

the load (processing speed) of the high priority one. The

processing speed distributions of the high priority units

(elements 1, 3 and 5) are presented in Table 1. The

conditional distributions of the processing speed of the low

priority units (elements 2, 4 and 6) are presented in Table 2.

The high and low priority units share their work inproportion to their processing speed.




Two first computing blocks also share the compu-

tational load in proportion to their processing speed. The

third block obtains the output of the first two blocks andstarts the processing when these blocks complete theirwork. The system fails if its processing speed is lower

than the demand w:

The system belongs to the task processing type. In order

to obtain the UGF representing the system PD, we first

define the u-functions u1ð zÞ; u3ð zÞ; u5ð zÞ in accordance with

Eq. (5) and the u-functions ~u2ð zÞ; ~u4ð zÞ; ~u6ð zÞ in accordance

with Eq. (19)

u1ð zÞ¼ 0:2 z50þ0:5 z

40þ0:1 z

30þ0:1 z

20þ0:05 z

10þ0:05 z

0;

u3ð zÞ¼ 0:2 z60þ0:7 z

20þ0:1 z

0;

u5ð zÞ¼ 0:7 z100þ0:2 z80þ0:1 z0;

~u2ð zÞ ¼ ð0:8;0:4;0Þ z30þð0:15;0:55;0:9Þ z15

þð0:05;0:05;0:1Þ z0;

~u4ð zÞ¼ ð0:8;0:6;0Þ z30þð0:15;0:35;0:95Þ z15

þð0:05;0:05;0:05Þ z0;

~u6ð zÞ~u2ð zÞ ¼ ð0:8;0:5;0:3Þ z50þð0:15;0:4;0:6Þ z30

þð0:05;0:1;0:1Þ z0:

Then, we apply the following operators producing the u-

functions of auxiliary equivalent elements

u7ð zÞ ¼ u1ð zÞ^)

þ~u2ð zÞ; u8ð zÞ ¼ u3ð zÞ^

)

þ~u4ð zÞ;


þ~u6ð zÞ:

The obtained u-functions represent the PD of the three

computing blocks. The PD of the subsystem consisting of

two parallel blocks (element 10) is represented by

u10ð zÞ¼ u7ð zÞ^þ

u8ð zÞ:

The entire system can be represented as two elements with

u-functions u10ð zÞ and u9ð zÞ connected in series. Since, the

system belongs to the task processing type, its u-function is

obtained by the operator (12)

u11ð zÞ¼ u10ð zÞ^£

u9ð zÞ:

The system availability can now be obtained by applying the

operator d over u11ð zÞ : AðwÞ ¼ d ðu11ð zÞ; wÞ: The system

availability as a function of demand w is presented in

Fig. 4 (curve A).

3.2. Flow transmission system

A continuous production system (Fig. 2) consists of two

consecutive production blocks. Each block consists of main

production unit and auxiliary production unit that share some

preventive maintenance resources (cooling, lubrication,

Fig. 1. Information processing system. (A. Structure of computing block; B.

system logic diagram).

Table 2

Conditional performance distributions of system elements 2, 4 and 6

(examples 1 and 2)

Condition

Element 2

G2 : 30 15 0

0 # G1 , 15 p2ðG1Þ : 0.8 0.15 0.05

15 # G1 , 35 0.4 0.55 0.05

35 # G1 , 70 0 0.9 0.1

Element 4

G4 : 30 15 0

0 # G3 , 15 p4ðG3Þ : 0.8 0.15 0.05

15 # G3 , 35 0.6 0.35 0.05

35 # G3 , 70 0 0.95 0.05

Element 6

G6 : 50 30 00 # G5 , 30 p6ðG5Þ : 0.8 0.15 0.05

30 # G5 , 90 0.5 0.4 0.1

90 # G5 , 150 0.3 0.6 0.1

Table 1

Unconditional performance distributions of system elements 1, 3 and 5

(examples 1 and 2)

G1 50 40 30 20 10 0

p1 0.2 0.5 0.1 0.1 0.05 0.05

G3 60 20 0

p3 0.2 0.7 0.1

G5 100 80 0 p5 0.7 0.2 0.1 Fig. 2. Continuous production system. (A. Structure of production block; B.

system logic diagram).




etc.).When themain production unit is intensively loadedthe

lack of resources prevent the auxiliary unit to be intensively

loaded with high availability.

The productivity distributions of the main production

units (elements 1 and 3) are presented in Table 1. The

conditional distributions of productivity of the auxiliary

units (elements 2 and 4) are presented in Table 2. The

system fails if it does not meet the demand w:

The system belongs to the flow transmission type. In

order to obtain the UGF representing the system PD, we first

define the u-functions u1ð zÞ; u3ð zÞ in accordance with Eq. (5)

and the u-functions ~u2ð zÞ; ~u4ð zÞ in accordance with Eq. (19)

as in the previous example.

Then, we apply the following operators producing the u-

functions of auxiliary equivalent elements corresponding to

the production blocks


þ~u2ð zÞ; u6ð zÞ ¼ u3ð zÞ^

)

þ~u4ð zÞ:

The entire system can be represented as two elements with

u-functions u5ð zÞ and u6ð zÞ connected in series. Since, the

system belongs to the flow transmission type, its u-function

takes the form

u7ð zÞ ¼ u5ð zÞ ^min

u6ð zÞ:

The system availability is obtained as AðwÞ ¼ d ðu7ð zÞ; wÞ:

The system availability as a function of demand w is

presented in Fig. 4 (curve B).

3.3. System with indirect influence of part of elements

on its performance

A chemical reactor consists of six heating elements and

two identical mixers (Fig. 3). Two heating elements have

heating power 8 and availability 0.9, four heating elements

have heating power 5 and availability 0.85. The heating

elements are powered by two independent power sources

with nominal power 25 and availability 0.95 each one. The

heating power of the elements cannot exceed the total power

of the available sources.

The productivity distribution of each one of the mixers

depends on the cumulative power of the heaters. The greater

the heating effect, the greater the productivity and

availability of the mixers. The mixers do not depend one

on another. The conditional distributions of productivity of

the mixers (element 4) are presented in Table 3. The total

productivity of the reactor is equal to the cumulated

productivity of the two mixers. The system fails if it does

not meet the demand w:

The heating subsystem is the flow transmission typeseries-parallel system. In order to obtain the UGF

representing the subsystem PD, we first define the u-functions u1ð zÞ; u2ð zÞ; u3ð zÞ in accordance with Eq. (5) as

u1ð zÞ ¼ 0:95 z25þ 0:05 z

0; u2ð zÞ ¼ 0:9 z

8þ 0:1 z

0;

u3ð zÞ ¼ 0:85 z5þ 0:15 z

0;

and then obtain the u-function representing the PD of the

subsystem by consecutively applying the composition

operators.

The u-function of power supply system is

u5ð zÞ ¼ u1ð zÞ^þ

u1ð zÞ:

The u-function of heaters is obtained as follows


u2ð zÞ; u7ð zÞ ¼ u6ð zÞ^þ

u3ð zÞ;



u3ð zÞ;


u3ð zÞ:

Observe that this u-function can be obtained in a simpler

way by defining an auxiliary element with u-function u7ð zÞ

Fig. 3. Chemical reactor. (A. Structure of reactor; B. system logic diagram).

Table 3

Conditional performance distributions of the mixers (example 3)

Condition G4 : 40 30 15 0

0 # Gh , 10 p4ðGhÞ : 0 0 0.2 0.8

10 # Gh , 20 0 0 0.8 0.2

20 # Gh , 25 0 0.2 0.6 0.2

25 # Gh , 30 0.3 0.4 0.2 0.1

30 # Gh , 40 0.7 0.1 0.1 0.1

Fig. 4. System availability as a function of demand w:




equivalent to u-function of two parallel elements 3



u3ð zÞ;



u8ð zÞ:

The u-function of the entire heating system (power sources

and heaters) is

uhð zÞ ¼ u5ð zÞ ^min

u10ð zÞ:

The mechanical system consists of two parallel mixers and

belongs to flow transmission type. Having the u-function

~u4ð zÞ of single mixer defined in accordance with Eq. (19) as

~u4ð zÞ¼ð0;0;0;0:3;0:7Þ z40þð0;0;0:2;0:4;0:1Þ z30

þð0:2;0:8;0:6;0:2;0:1Þ z15þð0:8;0:2;0:2;0:1;0:1Þ z0

;

we obtain the u-function representing the conditional PDs of

the system

~u11ð zÞ ¼ ~u4ð zÞ^þ

~u4ð zÞ:

Since, the heating system affects the reactor productivityonly by influencing the PD of the mixers, we apply the ^

)

operator

u12ð zÞ ¼ uhð zÞ^)

~u11ð zÞ:

The system availability can now be obtained as AðwÞ ¼

d ðu12ð zÞ; wÞ: The system availability as a function of demand

w is presented in Fig. 4 (curve C).

References

[1] Ushakov I. A universal generating function, Sov J Comput Syst Sci;

24:37–9.[2] Lisnianski A, Levitin G. Multi-state system reliability. Assessment,

optimization and applications. Singapore: World Scientific; 2003.


a universal generating function approach for the analysis

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