Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 37 docx

Likewise, inherent availability becomes a useful term to describe com-bined reliability and maintainability characteristics or to define the one in terms of the other during the early co

e) Sizing Maximum or Design Capacity

The effective capacities of multiple system operations or processes within the same engineering design installation are usually different A bottleneck is a process that has the lowest effective capacity of any process in the designed installation and,

thus, limits total output Expansion of maximum or design capacity occurs only when bottleneck capacity is increased However, flexible flow processes may have

floating bottlenecks due to widely varying workloads on different processes at

dif-ferent times

The theory of constraints (TOC) in designing for availability focuses on design

alternatives that impede maximum capacity (i.e bottlenecks), with the objective

of maximising total product or materials process flow (Goldratt 1990) Also, the

focus on bottlenecks is the means to increasing throughput and, consequently, the

mass-flow rate of product and materials The performance of the overall process

design is a function of minimum bottleneck operations or processes TOC provides

the ability to descriptively characterise the functional relationships responsible for

a typical complex process environment Basically, through the application of system dynamics (SD) models, which are developed from TOC logic diagrams, insights into

the dynamics of design alternatives that impede maximum capacity are obtained

The application of TOC in designing for availability involves the following steps:

• Identification of system bottleneck(s).

• Exploitation of the bottleneck(s)

(i.e maximising throughput).

• Elevating the bottleneck(s)

(i.e considering increasing capacity at the bottleneck(s))

Criteria for sizing design capacity Besides increasing the capacity of system

bot-tlenecks in order to expand design capacity, further criteria for sizing design capacity

are concerned with predicted process utilisation rates that are close to 100%, indicat-ing the need to increase capacity because of the probability of declinindicat-ing productivity over time (i.e diminished output against constant input) Process utilisation tends to

be higher in capital-intensive processes, where prediction of utilisation between 90 and 100% is not uncommon In such cases, occurrences of bottlenecks in the total

process are inevitable, resulting in the essential application of TOC in designing for

availability

A further consideration is economy of scale In designing for availability, this

im-plies not only increasing a design’s size or capacity but at the same time attempting

to decrease the average unit cost through various options, such as:

• Spreading fixed costs:

As the system utilisation rate increases, the average unit cost is reduced

• Reducing manufacturing/construction costs:

Doubling facility size usually does not double costs

• Reducing material costs:

Higher volumes allow for bulk acquisition and handling

• Exploiting process advantages:

High volume may justify investment in more efficient technology

• Increasing inherent availabilities:

Determining initial system operational characteristics

In contrast to consideration of economy of scale is the need also to consider disec-onomies of scale, whereby excessive size can bring about complexity and

inefficien-cies that, in turn, can raise the average unit cost, and result in a non-linear growth of overhead

4.2.1.3 Inherent Availability (Ai ) Modelling with Uncertainty

Under initial conditions of uncertainty, it is feasible to define system availability only in terms of operable time and corrective maintenance Availability defined in this manner is termed inherent availability (Ai) Under such idealised conditions,

standby and delay times associated with scheduled or preventive maintenance, as

well as administrative and logistics downtime are ignored Inherent availability is thus useful in determining initial system operational characteristics under specified conditions, such as testing in a contractor’s facility, or any other controlled test

en-vironment Likewise, inherent availability becomes a useful term to describe com-bined reliability and maintainability characteristics or to define the one in terms of

the other during the early conceptual phase of the engineering design process when, generally, these terms cannot be defined individually and are rather related to system performance

Since such a definition of availability is easily measured, it is frequently used as

a contract-specified requirement Inherent availability is primarily the concern of the design engineer during the establishment of functional interface with the contrac-tor and manufacturers in the early phases of the engineering design process Inher-ent availability looks at availability from a design perspective; thus, reliability and maintainability are considered complementary measures in the inherent availability equation Inherent availability is in effect a model of reliability and maintainability measures The inherent availability equation is given as (Eq 4.46), (DoD 3235.1-H 1982):

where:

MTBF is the mean time between failure

MTTR is the mean time to repair

Ai is the largest availability value that can be achieved because only the times re-lated to operational disruptions due to breakdowns and their repair are considered, whereas downtime associated with planned maintenance as well as administrative and logistics downtime are ignored

If the expected design reliability measure of mean time between failures (or, more particularly, mean time to breakdowns) is very large compared to the related

mean time to repair (or mean time to replace), then the inherent availability is high Similarly, if the design maintainability measure of mean time to repair (MTTR) is

a minimum, the inherent availability Aiwill be a maximum

It is obvious from the inherent availability equation that if design reliability de-creases (i.e MTBF becomes smaller), then better design maintainability (i.e shorter MTTR) is needed to achieve the same inherent availability Conversely, as engineer-ing design reliability increases, design maintainability is not so important in beengineer-ing able to achieve the same inherent availability

An important design integrity principle is thus obtained:

Trade-offs can be made between reliability and maintainability to achieve the same availability in the engineering design process.

a) The Exponential Function for Inherent Availability

Ifλ is designated the failure rate (1 /MTBF) and μ is designated the repair rate

(1/MTTR), and both rates are exponential, then the probability density function (p.d.f.) of a failure at time x is

f (x) =λe−λx (4.47)

The probability density function that a subsequent repair will be completed at time t, the end of the availability cycle, t > x, is

f (t − x) =μe−μ(t−x) (4.48) The availability cycle can be construed to have two consecutive periods; the first period is when operation is terminated by a failure, and the second period is when

downtime ends with a completed repair Inherent availability is the ratio of the aver-age time for the first period, to the averaver-age time for the cycle, which includes oper-ation and downtime The probability density function of a failure before t, followed

by a repair completed at t, is the convolution (accumulated product) of Eqs (4.47)

and (4.48)

f (t) =

t



0

f (t) = λ μe−μt

μ−λ e−λt − e −μt

 withμ>λ The average period of an availability cycle E (t) is

E (t) =

t



0

E (t) = λ+μ

λ μ .

The average period of an availability cycle E (t) is expressed in terms of mean time

between failure (MTBF) and mean time to repair (MTTR):

E (t) = λ1 +

1 μ

E (t) = MTBF + MTTR Thus, inherent availability Aiis the fraction of the availability cycle

(MTBF + MTTR)=

1/λ

1/λ + 1/μ=

μ

b) Confidence Determination of Inherent Availability Predictions

Equation (4.51) indicates that if both the MTBF and MTTR distributions are

expo-nential, then the inherent availability Ai is a function of the failure rateλ and the repair rateμ Since bothλ andμcan readily be used for Bayesian prior and poste-rior analysis, random values can be generated in repeated trials in order to simulate

a value for Ai The percentage values of the resulting distribution on Aiare the con-fidence limits of the inherent availability prediction

In predicting the value of Ai, the ratio of the mean operating period (MTBF) to that of the availability cycle (MTBF+ MTTR) can be established by known or es-timated distributions for these values However, establishing confidence levels on

different values of Ai(i.e quantitative assessment of Ai) can be done only by using known failure and repair data to establish distributions on MTBF and MTTR param-eters For example, if both the time between failures and time to repair are exponen-tial, then the values for MTBF and MTTR can be determined from Bayesian prior distributions, which are functions of the prior data Beyond such relatively simple

analysis, establishing confidence levels on different values of Aiis very difficult

Thus, predictions of Ai are feasible under initial conditions of uncertainty, as with conceptual design, if it is possible to define system availability with respect to

estimates of operable time and downtime due to corrective maintenance Standby

and delay times associated with scheduled or preventive maintenance, as well as administrative and logistics downtime are ignored A major problem arises, though,

when these estimates cannot be based on obtained data, and predicting the value of

Aicannot be quantitative However, as indicated in Sect 3.3.3.3 on reliability eval-uation, a statistically acceptable qualitative methodology to determine the integrity

of engineering design in the situation where data are not available or not

mean-ingful is included in the concept of information integration technology (IIT) The concept of IIT includes a combination of methods and tools for collecting,

organ-ising and analysing diverse information, and for utilorgan-ising that information to guide optimal decision-making, based on Bayesian prior and posterior analysis (Booker

et al 2000)

4.2.1.4 Preliminary Maintainability Modelling

Probability theory and statistics have an important role in designing for main-tainability, as much as they have in engineering design integrity methodology as

a whole Various probability distributions may be used to quantify repair time data, and even uncertainty of repair times Where repair time data are not available,

in-cluding any data representing failure rates or expected time to failure, qualitative methods involving possibility theory need to be used, similar to the prediction of

reliability considered in the previous section However, in the case of data being available, even censored data, repair time distributions may be identified and the corresponding maintainability function may be obtained The maintainability

func-tion is used to predict the probability that a repair, beginning at time t= 0, will be

accomplished in a time t The maintainability function M (t), for any distribution, is

expressed by the following relationship (Dhillon 1999b):

M (t) =

t



0

fr(t) is the probability density function of the maintenance (repair) time.

This maintainability function may be represented by various distribution functions, depending upon the statistical characteristics of the data and the function

param-eters The exponential distribution is particularly useful in presenting maintenance times that are random in duration.

The exponential distribution probability density function is defined by the fol-lowing relationship

fr(t) = (1/MTTR)e −(t/MTTR) (4.53)

where:

t is the variable repair time, and MTTR is the mean time to repair.

By substituting Eq (4.53) into Eq (4.52), the following relationship is obtained

M (t) =

t



0

(1/MTTR)e −(t/MTTR) dt (4.54)

M (t) = 1 − e −(t/MTTR)

M (t) = 1 − e −μt .

The fundamental parameter is the repair rate,μ, the reciprocal of MTTR, rather than the failure rate,λ, the reciprocal of MTBF The treatment of ‘time to an event’ is also reversed, in that the objective should be to makeμas high as possible, so that repairs are completed quickly, and to makeλ as low as possible, so that the time between failures as long as possible

In the maintainability relationship given in Eq (4.54), let t denote a specified

or required ‘standard’ time to repair Since t is specified, it is necessary only to

evaluateμ Furthermore, suppose that available data consist of estimates of repair

times t1,t2, ,tr The total estimated time, T , on repair status is then

T=∑r

i=1

Because the repair events are all independent, the joint probability, or likelihood L,

of the first r repair times, t1,t2, ,tris the product of their respective probabilities

L=∏r

i=1

From Eq (4.53) we get

L=μexp

−μ



r

∑

i=1

t i



The maximum-likelihood estimate, E, is a valueμthat maximises the natural

loga-rithm of L

E = rlnμ−μT

∂E

∂ μ =

r

μ− T

Setting the derivative to zero, the maximum-likelihood estimate ofμis

μ= r

The best estimate m  (t) of the maintainability function, M(t), with standard mainte-nance time t, is then obtained where m  (t) = M, in the case of 0 ≤ M < 1, may be

viewed as having a Bayesian prior or posterior distribution with parameters that are

valid statistics for r repair actions and T total repair time (Eq (4.60)) If these esti-mates cannot be based on obtained data, the methodology of information integra-tion technology (IIT) is applicable, in which Bayesian prior and posterior analysis is

utilised

m  (t) = 1 − e −μ

t = 1 − e −r t /T = M

4.2.2 Theoretical Overview of Availability and Maintainability Assessment in Preliminary Design

Availability and maintainability assessment attempts to estimate the expected us-age of equipment over a period of operational time subject to both planned and unplanned maintenance downtime or, alternatively, the expected utilisation over

a specified period of each individual item of equipment at the upper systems

lev-els of the systems breakdown structure System availability is an important mea-sure of repairable systems, since it considers both reliability and maintainability,

whereas availability and maintainability modelling takes into account both the

fail-ure and repair states of a system More specifically, availability and maintainability assessment takes into account not only the failure and repair states of a system but downtime due to preventive maintenance as well Availability and maintainability assessment in this context is considered during the preliminary or schematic de-sign phase of the engineering dede-sign process The most applicable methodology for

availability and maintainability assessment in the preliminary design phase includes basic concepts of mathematical modelling such as:

i Markov modelling for design availability and maintainability

ii Achieved availability modelling subject to maintenance

iii Maintainability assessment with maintenance modelling

iv Maintenance strategy and cost optimisation modelling.

4.2.2.1 Markov Modelling for Design Availability and Maintainability

Markov modelling is a powerful engineering design analysis tool, and it can be

used in most cases of designing for reliability and designing for maintainabil-ity The method is useful in modelling systems, especially large complex sys-tems, with dependent failure and repair modes Markov models are particularly

useful to model repairable systems with random failure occurrences (i.e constant

or independent failure rates) and random repair times (i.e constant or independent repair rates) The method becomes unreliable for systems with

time-dependent failure and repair rates

a) The Two-State Markov Model

Several initial assumptions are important when applying Markov modelling to en-gineering design analysis (Dhillon 1999b):

Up State 0

System operating

Down State 1

System failed

λ μ

Fig 4.7 Markov model state space diagram

• All events are independent of each other.

• The probability of transition from the system operating state to the system failed state (state 0 to state 1) is given byλΔt, whereΔt is a finite time interval, andλ

is the constant failure rate, or the transition rate

• The probability of transition from the system failed state to the system operating state (state 1 to state 0) is given byμΔt, whereΔt is a finite time interval, andμ

is the constant repair rate, or the transition rate

• The probability of more than one transition from one state to another inΔt is very

small

The transition states can be represented in the following diagram (Fig 4.7) From Fig 4.7, the following mathematical model can be derived (Dhillon 1999b):

P0(t +Δt ) = P0(t)(1 −λft ) + P1(t)μrt (4.61) and

P1(t +Δt ) = P1(t)(1 −μrt ) + P0(t)λft (4.62)

Status variables and probabilities The various status variables and probabilities

of these two equations need to be evaluated:

λf is the system constant failure rate,

μr is the system constant repair rate,

P0(t +Δt) is the probability that the system is in an operating state 0 at the time

t+Δt,

P1(t +Δt ) is the probability that the system is in a failed state 1 at the time t +Δt,

P0(t) is the probability that the system is in an operating state 0 at time t,

P1(t) is the probability that the system is in failed state 1 at time t,

(1 −λft) is the probability of no failure in time interval t when the system is in

state 0,

(1 −μrt) is the probability of no repair in time interval t when the system is in

state 1,

λft is the probability of system failure in time interval t,

μrt is the probability of accomplishing system repair in time interval t.

In the limiting case, Eqs (4.61) and (4.62) are represented by

lim

Δt→0

P0(t +Δt ) − P0(t)

dt = P1(t)μr− P0(t)λf (4.63) lim

Δt→0

P1(t +Δt ) − P0(t)

dt = P0(t)λf− P1(t)μr (4.64)

In order to solve Eqs (4.63) and (4.64) at time t= 0, the values for the following

probabilities are: P0(0) = 0, and P1(0) = 0

Then

P0(t) = μr

λf+μr+ λf

λf+μr

e−(λf+μr)t (4.65)

and

P1(t) = λf

λf+μr+ λf

λf+μr

e−(λf+μr)t. (4.66)

Thus, at any point in time t, the system’s availability may be represented by the

following

and

P0(t) = μr

λf+μr+ λf

λf+μr

e−(λf+μr)t

where:

A (t) = the system’s availability at a specified time t.

For engineering design availability assessment, estimate of availability for the sys-tem would be a steady-state availability, As, where t →∞ Thus

As= lim

and Asis A(steady state).

Substituting Eq (4.67) into Eq (4.68) gives the steady-state availability for the system

Thus, As= A(steady state) is given by

As = lim

t→∞



μr/λf+μr+λf/λf+μr(e−(λf+μr)t) (4.69)

As = μr

λf+μr .

b) Multi-State Markov Models—Method of Supplementary Variables

The components of most systems are assumed to fail with constant failure rates

(i.e failure times are governed by exponential distributions) However, though re-pair times of components are often non-exponentially distributed, they usually have

general distributions (i.e repair rates of the components are arbitrary functions of time) Multi-component repairable systems with general failure and/or repair time distributions are difficult to analyse mathematically These systems are known as non-Markovian systems, as the stochastic process is non-Markovian However, with

the inclusion of the method of supplementary variables, the Markov process

ap-proach provides a sufficient level of analysis that can be used to model complex systems with constant failure rates and non-exponential repair times Inclusion of sufficient supplementary variables in the specification of the state of the system can make a process Markovian (Dhillon 1983)

To enable the system to be characterised as a Markov system, a mathematical model is constructed with concise definitions of the various states for the system, together with a set of supplementary variables that include the concept of efficiency (or, rather, reduced efficiency) in the state definition of the system Because the

state at time t is an exact description of the circumstances prevailing in the system

at that time, the behaviour of the system over the passage of timeΔt may be found

by determining the state probabilities of the system A complex system can thus

be characterised as a Markov system by employing a set of supplementary variables with which a part of the system’s history is included in the state definition of the

sys-tem With the inclusion of supplementary variables, the Markov model represents

a multi-state stochastic system with modes of normal operation and total failure,

as well as operation at several different levels of performance (i.e with reduced efficiency)

The system has thus three operation modes: ‘normal operation’, ‘operation with reduced efficiency’ and ‘non-operation’ The supplementary variable technique en-ables a dynamic model of the behaviour of the system to be set up in the form of

a set of differential-difference equations with variable coefficients, and respective boundary and initial conditions (Virtanen 1977)

As an illustration of the method of supplementary variables, consider the system transition diagram in Fig 4.8 (Dhillon 1983)

The diagram represents a complex system that operates partially when some of system’s components fail and, if a catastrophic failure occurs, the system in its en-tirety fails When the system is operating partially, a repair process is expected to

be initiated to restore the system to its fully operational state However, the system may have a catastrophic failure from the partially operating state Once the system fails completely, it is expected to be restored to its normal operating state

System

Operating

Normally

System Operating Partially

System Operating Failed

λ λ2 λ1

μp

μf (x)

Fig 4.8 Multi-state system transition

Markov modelling is a powerful engineering design analysis tool, and it can be

used in most cases of designing for... class="page_container" data-page="7">

4.2.2 Theoretical Overview of Availability and Maintainability Assessment in Preliminary Design< /b>

Availability and maintainability. .. assessment in this context is considered during the preliminary or schematic de-sign phase of the engineering dede-sign process The most applicable methodology for

availability and maintainability