STOCHASTIC ANALYSIS OF A TWO UNIT SYSTEM WITH VACATION FOR THE REPAIR FACILITY AFTER m REPAIRS

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1 139 ORiON, Vol. 16, No. 2, pp ISSN X STOCHASTIC ANALYSIS OF A TWO UNIT SYSTEM WITH VACATION FOR THE REPAIR FACILITY AFTER m REPAIRS V.S.S. YADAVALLI & M. BOTHA University of South Africa Department of Statistics P.O.Box 392, UNISA 0003 South Africa ( yadavvss@unisa.ac.za, mbotha@unisa.ac.za) ABSTRACT Stochastic analysis of a two unit system with vacation period for the repair facility after the completion of 6 repairs is studie All the underlying distributions are assumed to be non Markovian. The reliability and availability analysis for such a system is studie A numerical illustration is given. Keywords: Repair facility, vacation period, reliability, avalability 1. INTRODUCTION Two unit standby redundant repairable systems have attracted the attention of many applied probabilists and system analysts. In the literature available so far on such systems, it is clear that the repair facility is continuously available to attend to the repair of the failed units. However, it is reasonable to expect that a vacation might be needed for the repair facility after 6 repairs, before the next repair could be taken up. Such a vacation period certainly arises in many mechanical and electrical systems. This principle of a vacation period was first introduced by Subramanian and Sarma [2]. Recent developments in the modeling and analysis of highly reliable systems require the use of some more sophisticated models. According to Platis et al. [1] Electricité De France (EDF) used this for the homogeneous Markov approach to model an electrical substation in order to evaluate some measures of system performance. However, in this model, the underlying distributions are all nonmarkovian, in the sense that, all the underlying distributions in this model are arbitrarily distribute In this paper, the concept of a vacation period for the repair facility after the completion of 6 repairs, and the arbitrary distributions described above have been introduce The reliability

2 140 and availability measures have been obtained using the regeneration point technique and product densities. A numerical example illustrates the results in section SYSTEM DESCRIPTION (a) The system consists of two identical units: one unit is operating online and the other is kept in cold standby. Either unit performs the system functions satisfactorily. (b) Switch is perfect and switchover is instantaneous. (c) After the completion of 6 repairs, the repair facility is not available for a random time, denoted by vacation time, the duration of which is governed by an arbitrarily distributed r.v. with p.f. E. (d) The life time of a unit while operating online is an arbitrarily distributed random variable (r.v.) with probability density function (p.f.) se. (e) The repair time of a unit is arbitrarily distributed with p.f. }E. 3. NOTATION. & G Eventthatthe& th repair commences after completion of the vacation time; & ' c 2c c 6 ( G Event that the system enters the down state s E? E G? fold convolution of se sec8ec 7 8 E G p.f., c.f. and survivor function (s.f) of the lifetime of the unit }EcCEcCE 7 G p.f., c.f. and s.f. of a repair time EcTEcT 7 E G p.f., c.f. and s.f. of the vacation time s W Er G Laplace transform of se 4. AUXILIARY FUNCTIONS Let us define the following auxiliary functions, which will be used in the reliability and availability analysis. We know that the events., 2 are regenerative. Since. can occur in two ways, it may or may not be a regenerative event.

3 141 Functions s o E and E For ccc 6 c let s o n in E c nce(c 'fm. at 'fo E ' *4 and n in E c nm. at 'fo E ' *4 At the epoch of occurrence of an. event, repair commences for a unit, while the other unit is just switched online.. Using probabilistic arguments we can find that s o E ' se CE and E ' se CE n}e 8E Functions s6 o 2 E and s 2 E Let s o 2 in E c nce(c 'fm. 6 at 'fo 6 2 E ' *4 and 2 in E c nm. 6 at 'fo 6 2 E ' *4 Figures 1 and 2 (Flowdiagram and Classification of events respectively) explain the various possibilities in order to derive the expressions for s6 o 2 E and s 2 E c namely s o 6 2 E ' se d}e f T E o f se CE n }E se 2 E CE se 2 _ _ 2 _ 'f 2 ' ' 2

4 142 Considering all the 6 cases D to I infigure2,weget 6 2 E ' se d}e f T E o f se CE n se d}e f T E o f }E 8 E n f f }E _ }E _ se 2 se 2 _ 2 E C E _ n 2 se 2 8 E 2 _ 2 E } E _ n 2 }E 8 E f se de f CE o n }E 8 E f 8E de f }E o Functions s22e o and s Let s22e o 2?E c nce. 2 c 'fce(c 'fm. 2 at 'fo ' *4 and s 2 in E c nce. 2 c 'fm. 2 at 'fo ' *4 s22e o is the p.f. of the interval between two successive ( avoiding. 2 events, and s is the p.f. of the interval between two successive. 2 events. Using probabilistic arguments, and observing that the events. c. e c c. 6 successively occur following an. 2 event, we have s22e o ' 6 s o E f s6 o 2 E & s E 's 2 E f s E f f s & E

5 RELIABILITY ANALYSIS We observe that the. 2 events constitute a renewal process. For the sake of simplicity we assume that an. 2 event has occurred at 'f Let E ' d E(c 'fm. 2 at 'fo To derive an expression for the reliability E of the system, we consider the following mutually exclusive and exhaustive cases: no. 2 event occurs up to c or at least one. 2 event occurs in Efc o Accordingly we have E ' 632 [ 6 "[ kes o22e l E??' s o E f 7 8E n s o E f 7 8E nse CE f 7 8 E n 6 f [ 7 ' s o E f 7 8 E n s o E f 8 7 E nse CE f 8E 7 : 8 Mean time to system failure (MTSF) can be obtained using the relation MTSF ' U " E_ ' W Ef f 6

6 AVAILABILITY ANALYSIS This is defined as the probability that the system is able to operate within the tolerances at a given instant of time. In symbols the pointwise availability is: E ' d system is up at m. 2 at 'fo Using renewal theoretic arguments we get E ' 632 [ E f 7 8 E n 6 "[ l E??' 6 E f 7 8 E nse CE f 7 8 E n 8 E }E f 7 8E n f [ 7 E f 7 8E n E f 8 7 E nse CE f 8E n}e 8 7 E f 8E 7 : 8 6 The steady state availability " can be obtained using the relation " '*4 <" E '*4 r<f r W Er

7 145 Figure 1: Flowdiagram E m occurs at t = 0. At the epoch of this repair completion the vacation time commences and the online unit is operable. At the epoch of failure of this online unit in the failed state. The standby unit is switched online. When the vacation time is over and the standby unit is switched online. Repair of the first failed unit commences (E 1 ) and the vacation time is not complete The standby unit is switched online (while the failed unit is waiting for repair). When this repair is completed before t. The online unit fails in (t, t + ). E 2 occurs at t. this repair is completed before (t, t + ). The online unit fails before t. E 2 occurs at t. the vacation time is over repair of the original unit starts and is completed before t. Online unit fails in (t, t + ). E 2 occurs in (t, t + ) the vacation time is over repair of the original unit starts and is completed in (t, t + ). The online unit fails before t. E 2 occurs in (t, t + )

8 146 Figure 2: Classification of events D n n & n m Unit repair Vacation Online unit Repair Online completion. time is over. fails. Standby complete unit fails. Vacation time unit switched. 2 occurs. commences. online. E n n & & m 6th repair Vacation Online unit Online 1st repair completion. time is fails. Standby unit completion. Vacation over unit switched fails.. 2 occurs. time in re online. st pair facility. repair commences. F 2 e n n n n m 6th repair Online unit Vacation Repair Online completion. fails. Stand time is over. complete unit fails. Vacation time by switched st repair. 2 occurs. commences. online. commences. G online fails && & 2 n n n m 6th repair Online unit Vacation time 1st repair completion. fails. Standby is over. completion Vacation time switched online. st repair. 2 occurs. commences. commences. H & & n n m Online 6th repair comple Vacation time st repair Online unit tion. Vacation time is over. st is com unit fails. fails. commences. Online repair com plete. 2 occurs. unit is just online. mences. I onlinefails & % n & m Online 6th repair comple Vacation time st repair unit fails. tion. Vacation time is over. st re is comcommences. One pair commences. plete unit is swited online.

9 NUMERICAL ILLUSTRATION A numerical example of some results obtained for the model are given in this section. For this purpose we assume that se ' be 3b c b : f }E ' >e 3> c > : f E ' _ 2 e 3_ c _ : f The special case of 6 'is considere Table 1 presents the mean time to system failure (MTSF) for different parametric values when b ' 2f ACKNOWLEDGMENTS The authors are thankful to the referees for their comments. Thanks are due to Anette Klaassen for her professional handling of the typing. REFERENCES [1] A.PLATIS, N.LIMNIOS and M.Le DU, Dependability analysis of systems modelled by nonhomogeneous Markov chains, Reliability Engineering and Systems Safety, 61, (1998). [2] R.SUBRAMANIAN and Y.V.S.SARMA, A twounit standby system with dead time for the repair facility, IEEE Transactions on Reliability, R30, 498 (1981).

10 148 TABLE 1 Rate of Vacation Repair MTSF _ ' ffd SHb2 D ffh 2D 2b f 2.e f e bhb. f. es.fd fc 2f D2 f _ ' ffd SD f ffh 2eff f fddb f e S.D f. e S22 f2f esbs _ ' ffd S2f D ffh 2 De f 2HHbH f e.be f. Hfef f2f e.sh _ ' ffd Sfff 2f ffh 22Dfe f 2..fe f e bbs f. DS f2f H. f _ ' ffd DH2D 2D ffh 22f22 f 2SHef f e f.2 f. b2e f2f SS H