IEEE TRANSACTIONS ON RELIABILITY, VOL. R-35, NO. 1, 1986 APRIL

New Models for Maintenance of Series and Parallel Systems

N. M. Khan, Member IEEEGovernment College of Engineering, Ujjain

Ashok Gupta, Student Member IEEEGovernment College of Engineering, Ujjain

Key Words-Preventive maintenance, Pending failure state,Equivalent model technique, Successive reduction technique.

Reader Aids-Purpose: To report derivationsSpecial math needed for derivations: Markov process, Frequency and

duration approachSpecial math needed to use results: SameResults useful to: System and maintenance analysts and theoreticians

Summary & Conclusions-This paper develops new models formaintenance of multi-component series and parallel systems with inspec-tion, preventive maintenance (pm), and repair; and subject to deteriorationwith use and stochastic failure. Use of 3-state models makes it possible toconsider wear and pm of components. The 3-state model for a component isreduced to an equivalent 2-state model. By combining 2-state models, multi-component systems are analysed. The error introduced by using equivalentmodels for the system is below 0.7% for a typical example. A case study ofan electrical power system illustrates the application of the technique.Details are available in a separately available Supplement.

1. INTRODUCTION

Although modern engineering systems have manycomponents, only a little work has been reported in theliterature on such systems. This is because of the complex-ities arising in the analysis of complex systems due to thelarge number of system states. Moreover, usually both thenormal operating and wearout periods of componentshave not been considered. Authors in [1] considered thisaspect and presented 3-state models for 1-componentsystems. In the present work the 3-state model for a com-ponent is reduced to an equivalent 2-state model. The term'failure rate of component with pm and repair' in theequivalent model is very important because by this termthe effect of wear and pm of the component on its failurerate can be considered independently of its repair. Theequivalent failure rate and mean outage duration of a2-component system are obtained in terms of parametersof the equivalent models for the components. By combin-ing a third component with the equivalent model of twocomponents the parameters for a 3-component system areobtained. Successive application of this technique makespossible the analysis of multi-component systems.

2. NOTATION

gi, pi, fi states of a component i: good, pending failure(but operating), and failed

ai transition rate of component i from state gi to pif3i transition rate of component i from state pi to fiTi mean inspection interval for component i, defined

as the mean value of the time interval between twosuccessive visits to component i for pm

Ai repair rate of component iui, di states of component i for equivalent model:

operating and non-operatingfailure rate of component i with pm and repair(obtained by equivalent model), defined asreciprocal of mean time to go from state ui and di

rf mean outage duration of component i (obtainedby equivalent model), defined as mean time to gofrom state di to uillrA repair rate of component i (obtained byequivalent model)

The subscript is omitted for a (single) component (system)1, 2, 3, ... states of systemn number of components in a systemup, dn states of system for equivalent model: operating

(up) and non-operating (down)The subscript es implies an equivalent parameter for a"series" system with pm and repairThe subscript ep implies an equivalent parameter for a"parallel" system with pm and repairEMT equivalent model techniqueSRT successive reduction technique

Other, standard notation is given in "Information forReaders & Authors" at rear of each issue.

3. SYSTEM MODELS AND PERFORMANCEPARAMETERS

3.1 Assumptions

1. Any component at any instant of time is in one ofthe states g, p, f. In statep there is an indication of pendingfailure and increase in failure rate. A component operatesin states g and p, and it does not operate in state f. Allstates are observed without error. Transition rates a and eare constant, and A > a.

2. Any time a technician visits a component he iscapable of detecting its state. If the component is inpending-failure state, it is preventively maintained andrestored to state g instantaneously and no damage is doneto anything. The inspection interval is a random variableexponentially distributed. Any pending failure notrestored by a technician eventually results in componentfailure. Then the component is repaired and restored tostate g and no damage is done to anything. The repair rateis constant.

0018-9529/86/0400-0068$01 .00 1986 IEEE

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KHAN/GUPTA: NEW MODELS FOR MAINTENANCE OF SERIES AND PARALLEL SYSTEMS

3. The system is maintained by pm and repair of itscomponents. For each component, separate and indepen-dent pm and repair facilities are available. There are nowaiting lines.

4. The failure rates of components are very small incomparison to their repair rates.

3.2 Equivalent Model Technique (EM7)

For a 1-component system, a basic 3-state model andits equivalent 2-state model are shown in figures la and lbrespectively. The equivalent model is formed by combiningthe operating states of 3-state model. The parameters ofthe equivalent model as derived in the Supplement [2] are:

XI =c/( + +1/T) (1)

r' = 1/1A' = l/x (2)

This technique of obtaining equivalent model is named asEMT.

a)3-STATE M1ODEL

o(+p +I,'t

A >\~

( b)E6UIVVALENT 2-STATE MODEL

Fig. 1. State transition diagrams: for 1-component system

3.3 1-out-of-2:F, "Series" Systems

The exact Markov model of a 1-out-of-2:F, "series"system has 8 states. By using equivalent models of figure lbfor the components, the state transition diagram shown infigure 2a is obtained, which has only 3 states. In figure 2a -

Xi = ioi3/(ti + fi + 1/-r)

ti/ = l/rf = i

for i = 1, 2

,'J

(3)

(4)

Xes - )1 + 2 (5)

res = 1//Les = (X4r' + X2r2)/(X + X') (6)

Detailed derivations are given in the Supplement [2].

3.4 I-out-of-n:F, "Series" System

To get the equivalent model of this system, first com-ponents 1 and 2 are combined and the equivalent failurerate (X)", say) and mean outage duration (r", say) of theircombination are obtained using (5) and (6). Then, thesingle component characterised by X", r", and component3 are combined and the equivalent failure rate and meanoutage duration of combination of first three componentsare obtained using (5) and (6) again. Proceeding in thisway, adding one component at a time, the parameters ob-tained for the n-component system are:

n

=e F, X!')\es I

i=l

n n

res = 1l/les = E X)ri/ E xi,i=l i=l

(7)

(8)

where, X'and r, for i = 1, n are given by (3) and (4) respec-tively. This technique of adding one component at a time isnamed SRT. Details of derivations of (7), (8) are given inthe Supplement [2].

3.5 1-out-of-n:G, "Parallel" System

Applying the SRT described in section 3.4 to a"parallel" system the following parameters are obtained inthe Supplement [2]:

Xep = Il Xir)I( E (1/rr)n

rep =I1/ep = I1/ E (I Iri)

(9)

(10)

where, X/ and rf, for i = 1, n, are given by (3) and (4)respectively.

tes

ye es

(b) Equivalent M od el

ACKNOWLEDGMENT

The authors are grateful to Dr. Ralph A. Evans,Editor, for many suggestions for improving this paper.

Fig. 2. State transition diagrams for 1-out-of-2:F series system

Combining non-operating states of figure 2a, theequivalent 2-state model of figure 2b is obtained for thesystem, where

REFERENCES

[1] A. Gupta, N. M. Khan, "Availability analysis of 3-state system",IEEE Trans. Reliability, vol R-34, 1985 Apr, pp 86-87.

[2] Supplement: NAPS document No. 04300-D, 15 pages in this supple-ment. For current ordering information, see "Information for

69

IEEE TRANSACTIONS ON RELIABILITY, VOL. R-35, NO. 1, 1986 APRIL

Readers & Authors" in a current issue. Order NAPS document No.04300, 95 pages. ASIS-NAPS; Microfiche Publications; POBox3513, Grand Central Station; New York, NY 10163 USA.

AUTHORS

Dr. Nasir M. Khan; Electrical Engineering Department; GovernmentCollege of Engineering; Ujjain - 456 009 MP, INDIA.

Nasir M. Khan (M'81): For biography, see vol R-32, 1983 Oct, p398.

Er. Ashok Gupta; Electrical Engineering Department; Government Col-lege of Engineering; Ujjain - 456 009 MP, INDIA.

Ashok Gupta (S'84): For biography, see vol R-30, 1981 June, p 162.

Manuscript TR83-166 received 1983 November 10; revised 1985 October30

EXTENDEDABSTRACT EXTENDEDABSTRACT EXTENDEDABSTRACT EXTENDEDABSTRACT EXTENDEDABSTRACT

Cost-Benefit Analysis of a 2-Unit Warm-StandbySystem with Inspection, Repair, and Post Repair

L. R. GoelMeerut University, Meerut

Rakesh GuptaMeerut University, Meerut

S. K. SinghUdai Pratap College, Varanasi

Key Words-Warm standby, Inspection, Repair, Post repair,Regenerative point.

1. INTRODUCTION

Many papers with two types of repair have widelybeen studied in literature. Goel et alii analysed a single unitmulti-component system with two types of repair (minorand overhaul) where the decision about the type of repairwas taken by inspection. In many systems post-repair ofthe repaired unit is essential. In view of this we study a

2-unit standby system with two types of repair and postrepair. Using regenerative point technique in Markovrenewal process, the mean time to system failure, mean

time to system recovery, mean-up-time of the system in (0,tf, mean busy period of the repairman in (0, t], and mean

number of visits by the repairman in (0, tJ are obtained.The complete derivations are given in a separatelyavailable supplement [1].

2. ASSUMPTIONS

Consider a 2-unit warm-standby system with twotypes of repair I and II. Each unit of the system has twomodes - normal and failed.

1. Whenever a unit fails, it requires inspection fordeciding the type of repair. After inspection, the unit goesfor repair of type I with known probabilityp and repair oftype II with probability 1 - p. Upon completion of repairthe unit undergoes post-repair.

2. A single service facility is available for inspection,repair, and post repair of the failed unit.

3. If during the inspection, repair, or post-repair of afailed unit, the other unit also fails, the later unit has towait for inspection until the first unit completes post-repair. After post repair the unit is as good as new.

4. Failure time distributions are negative exponentialwhile inspection, repair, and post repair time distributionsare arbitrary.

SUPPLEMENT

[1] NAPS document No. 04300-F; 19 pages in this Supplement. For cur-rent ordering information, see "Information for Readers &Authors" in a current issue. Order NAPS document No. 04300, 95pages. ASIS-NAPS; Microfiche Publications; POBox 3513, GrandCentral Station; New York, NY 10163 USA.

AUTHORS

L. R. Goel, Head; Department of Statistics; Institute of AdvancedStudies; Meerut University; Meerut - 250 005 INDIA.

Rakesh Gupta, Lecturer; Department of Statistics; Institute of AdvancedStudies; Meerut University; Meerut - 250 005 INDIA.

S. K. Singh, Lecturer; Department of Statistics; Udai Pratap College;Varanasi - 221 002 INDIA.

Manuscript TR85-010 received 1985 February 1; revised 1985 November4.

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