by: Mohammad El-Moniem Soleha

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1 Reliability and Availability Characteristics of A Two-Dissimilar-Unit Cold Standby System With three Modes by Using Linear First Order Differential Equations by: Mohammad El-Moniem Soleha ABSTRACT : This paper introduces a statistical analysis of a twodissimilar-unit cold standby redundant system, by which its reliability and availability characteristics are evaluated. This proposed system has been investigated under the assumption that each unit works in three different modes: normal, partial failure and total failure. The failure and repair times are assumed to have exponential distributions. Linear first order diferentialequations areused forthestatisticalanalysisofthesystem. By adopting this mathematically tractable technique, the mean time to system failure MTSF andthesteady-state availability A( ) of the system have been derived. A special case of the proposed general system is given in which two similar units are considered. Keywords: Reliability, Availability, Linear first order differential equations, Mean time to system failure, Steady-state availability,failure and repair times, Exponential distribition. Author: Mohammad El-Moniem Soleha, Department of Statistics, Mathematics & Insurance, Ain Shams University, Cairo, Egypt. drsoleha@yahoo.com. Editor: Xie, Min mxie@nus.edu.sg

2 . INTRODUCTION: Statistical investigation of two-unit redundant systems has been receiving considerable attention during the last four decades. Several authorshavestudied such systemsasuming only two statesof operations,namelyoperativeandfailed[5],[].also,the similarity of units is considered as a main assumption for the statistical analysis such systems [4]. These systems, have been formulated to assess various reliability measures (viz. MTSF, expected number of repairs etc.), by using the theory of semi-markov process, Markov renewal process and regenerative processes [4], [6]. Some researches [], [3] have implemented varioustypesof costequation in theiranalysisto determinetheconditionsformaximizing theprofitability ofthesystem. Thispaperintroducesaprobabilisticanalysisofa generalized situation inwhichasystem of two-dissimilar-unit isasumedtobe workingwith three diferentmodes.anevaluationofthemeantimeto system failure (MTSF) and its steady-state availability is carried out using the straightforward linear differential equations method. This paper is outlined as follows. The next section presents the proposed system, by putting up a simple set of assumptions, upon which the states of the system are determined. A graphical representation of thesestatesisgivenbythe TransitionDiagram (figure),onwhichthe transitional parameters ( i, i, i and i, i = = ) are pointed out. In section 3, the mean time to system failure (MTSF) is calculated for the system. Matrix differential calculus approach [8] of linear first order is usedtosolvethesystem s diferential equationsto get a closed form of (MTSF). In section 4, availability analysis is conducted, with emphasis

3 on the steady-state availability of the proposed system. Again, the linear first order differential equation is employed to evaluate such steady-state case of the system. Section 5, reports a special case of the results of analyzing the reliability and availability of the proposed system. This clarifiesthe superiority ofproposedsystem overthepreviousones.the last section introduces some comparative and suggestive remarks concerning the simplicity and straightforwardness of our analysis of the proposed system using linear first order differential equations. Thus, this sectionprovidesa rationale forthesuperiorityofouranalysisoverthe tradition Markovian-based analysis.. CONSTRUCTION OF THE SYSTEM In this section, we start by giving some Notations or (terminologies), and putting up some reasonable assumptions to arrive at the possible states of the proposed system... Main Assumptions We start by giving the following set assumptions: The first assumption: the system works with one unit that is operative andtheotheriskeptasacoldstandby,i.e.itdoesnotfail while standing by, unless it goes into operation. The second assumption: The standby is switched to operative state in negligible time. The third assumption: The operative units have three modes of operation. These are: normal, O-mode which means the functioning of the units with full capacity, partial failure, P-mode which means the functioning of the units with reduced capacity at specified level and the 3

4 total failure, T-mode which means the capacity goes below a specified level. The fourth assumption: Aunit,inthenormalmode, must pas throughthepartialfailuremodeto getinto thetotalfailuremode.inthe partialfailuremode,aunitisworkingas beingrepaired. The fifth assumption: A unit, which is replaced or repaired in total failure mode. This assumed to go directly to the normal mode without passing through the partial failure mode. The sixth assumption: The system is down when all units are nonoperative. The seventh assumption: Forconstructing the probabilistic structure ofthesystem,isitasumedthatthefailureandrepairtimes have exponential distributions... Notations of the System I : the failure rate of the i th unit from O i = mode to P i - mode, i =,. i : the failure rate of the i th unit from P i mode to T i mode, i =,. i : the repair rate of the i th unit from P i mode to O i mode, i =,. i : the repair rate of the i th unit from T i mode to O i mode, i =,. Also, the following terminologies will be used: O i : the i th unit is operative, i =,. S i : the i th unit is in a cold standby mode, i =,. S i : Denotes the states of the system, i = 0, 8. P i : the i th unit is in a partial failure mode, i =,. T i : the i th unit is a total failure mode, i =,. 4

5 .3. States of the System The system may be in one of the following states: S 0 (O, S ), S (S, O ), S (P, S ), S 3 (S, P ), S 4 (T, O ), S 5 (O, T ), S 6 (T, P ), S 7 (P, T ), S 8 (T, T ) S 0 (O, S ) S (P, S ) S 4 (T, O ) S 6 (T, P ) S 8 (T, T ) S (S, O ) S 3 (S, P ) S 5 (O, T ) S 7 (P, T ) : Up state : Down state 5

6 : Regeneration point 3. RELIABILITY ASSESSMENT: In this section, the mean time to system failure (MTSF) for the proposed system will be evaluated using the above-mentioned set of assumptions and the linear first order differential equations. For Fig., let P i (t) be the probability that the system at time t, (t 0) in state S i. If we let P (t) denote the probability row vector at time t, the initial conditions for this problem are P(0) = [P 0 (0), P (0), P (0), P 3 (0), P 4 (0), P 5 (0), P 6 (0), P 7 (0), P 8 (0)] = [, 0, 0, 0, 0, 0, 0, 0, 0 ] (3.) By employing the method of linear first order differential equations, we obtain the following differential equations: / P (t) = - P 0 (t) + P (t) + P 4 (t), 0 / P (t) = - P (t) + P 3 (t) + P 5 (t), / P (t) = - P (t) + P 7 (t) + P 0 (t), / P (t) = - P 3 (t) + P 6 (t) + P (t), 3 / P (t) = - P 4 (t) + P (t) + P 8 (t), 4 / P (t) = - P 5 (t) + P 3 (t) + P 8 (t), 5 / P (t) = - P 6 (t) + P 4 (t), 6 / P (t) = - P 7 (t) + P 5 (t), 7 6

7 / P (t) = - P 8 (t) + P 6 (t) + P 7 (t). (3.) 8 This can be written in the matrix form as: P = Q P (3.3) where, ( + ) ( + ) Q = ( + ) ( + ) ( + ) ( + ) ( + ) To calculate the MTSF, we take the transpose of the matrix Q and delete the rows and columns for the absorbing state. The new matrix is called (A). The expected time to reach an absorbing state is calculated from MTSF = P(0) (-A - ) (3.4) where, ( + ) ( + ) A = ( + ) ( + ) ( + ) ( + ) 7

8 8

9 Then, after some mathematical manipulations [0], [4], we arrive at a closed form for the mean time to system failure, which is MTSF = a a a a 5 3 a 4 (3.5) where, a = ( + ) ( + ) ( + ) ( + ), a = ( + ) ( + ) ( + + ), a 3 = ( + ) [ ( + ) ( + ) + ( + ) ], a 4 = ( + ), a 5 = ( + + ), (3.6) 4. AVAILABILITY ANALYSIS OF THE SYSTEM: For the availability case of the states of the system represented by Fig., the initial conditions for this problem are the same as for the reliability case: P(0) = [ P 0 (0), P (0), P (0), P 3 (0), P 4 (0), P 5 (0), P 6 (0), P 7 (0), P 8 (0) ] = [, 0, 0, 0, 0, 0, 0, 0, 0 ] (4.) The differential equations (3.) can be expressed as:. P = Q P (4.) The steady-state availability can be obtained using the following procedure. In the steady-state situation, the derivatives of the state probabilities become zero. That allows us to calculate the steady-state availability of the system defined as: A ( ) = P 8 ( ) (4.3) Also we have that Q P ( ) = 0 (4.4) 9

10 Thus, to obtain P 8 ( ), we solve (4.4) under the following normalizing condition: 8 i 0 P ( ) = (4.5) i This is done by substituting (4.5) in any one of the redundant rows in (3.9) to yield P 0 ( ) P ( ) 0 -( + ) P ( ) 0 0 -( + ) P 3 ( ) ( + ) P 4 ( ) ( + ) 0 0 P 5 ( ) ( + ) 0 0 P 6 ( ) ( + ) 0 P 7 ( ) P 8 ( ) = (4.6) The solution of (4.6) provides the steady-state probabilities of the availability case. For Fig., the steady state availability of the proposed system A( ) is given by N A ( ) = - D. (4.7) 0

11 where, N = 5 D = (b + b ) + (b 4 + b 3 ) + Also, the values of b i (i=,.,5)aregivenby: b 5 b = ( + ) [ ( + ) + ( + ) + ( + ) ], b = ( + ) ( + + ) + ( + ) + ( + + ) + ( + ), b 3 = ( + ) ( + + ) + ( + ) + ( + + ) ), b 4 = ( + ) ( + ) + [ ( + ) + + ) ], b 5 = ( + ) ( ). (4.8) 5. A SPECIAL CASE: By a simple substitution ( i =,, i = ), i =,, in the set of equation [3.6], we can get the MTSF which is defined before as MTSF = a a a a 5 3 a 4 (5.)

12 where, a = a = a 3 = a 4 = a 5 = Also, a similar argument (substitution) can be implemented in the calculation of the steady-state availability A ( ) represented by relation (4.7). For the availability calculations, recall the equation of the steady state availability (4.7), then using the following relations for N and D. N = a 5, (5.3) where a 5 is defined by the set of equations (5.), and D = (b + b ) + (b 4 + b 3 ) + 4 b 5. (5.4) Here b i (i =, 5) are defined by the set of equations (4.8), after substituting ( i =, i =, i =, i = ), i =,. The importance of this special case stems from the fact that its points outa generalization featureoftheproposed two-dissimilar-unit cold standby system. That is, by dropping the subscripts (i =, ) from the transitional parameters i, i, i and i, one can get the special-more restricted-caseoftwo similar-unit system,beingcommonly assumed by many authors [], [4] and [6].

13 6. FINAL COMPARATIVE REMARKS: Thispaperpresentsa generalized situation ofthestochastic behavior of a proposed two-dissimilar-unit cold standby system, which is statistically analyzed using linear differential equation method. From a comparative viewpoint, this research can be considered as a generalization ofsomeotherpreviousworks[],[4],whichpresenteda statisticalinvestigationofthereliabilityoftwo identical or similar unit cold standby systems.this advantage overthepreviousworksis demonstrated by the above-mentioned special case where the subscripts (i=,)are dropped forthetransitionalparameters( i, i, i, i ) of the general setingoftheproposedsystem.thismeansthat,adopting more general asumptions(twodisimilarunits)forconstructingthe proposed system providesa rationale ofitssuperiorityoverthese traditionalsystemsoftwo identical components. Secondly,from the simplicity perspective,thestraightforward method oflineardiferentialequations, overweigh thetraditional tedious Markov chain approach [3], [5]. This is because, the latter technique,requiresanelaborateworkofsetingup integralequations for the states of the system and solving then using Laplace transformations. Thirdly, an extension can be suggested by implementing the idea of cost-benefit analysis, introduced by [9], in our system and using the same deferential equation technique in calculating the profitability of the proposed system. Finally, the proposed system which is analyzed in this paper can be considered as a type of parallel systems which allows the system to be functioning (operative) even if one of the components fails. This can be consideredaspracticalymore applicable thanthecaseofanalyzing series systems [3]. 3

14 REFERENCES. Arora J.R. (976): Reliability of a -unit standby redundant system with constrained repair time, IEEE Trans. Reliab., Vol. 5, No.3, Barlow, R.E. and Proschan (965): Mathematical theory of Reliability, Wiley, New York. 3. El-Said Kh. M. and El-Sherbeny M.S. (005): Profit analysis of a two unit cold standby system with preventive maintenance and random change in units. J. Math. Stat., (), El-Said KH.M, Soleha A.B., Abd El-Moniem M. and El- Shanshory G.H. (994): Stochastic analysis of a twosimilar unit Standby redundant system with three modes, The annual conference, ISSR, Cairo University, Vol. 9, Part (II). 5. Emara S.A.A. and El-Said Kh.M. (99): Repairable system with different failure modes, Math. Congress proceeding of non-linear analysis, Tempa, Florida, U.S.A., Aug Gupta R. and Mittal M. (006): Stochastic Analysis of a compound Redundant System involving Human Failure, Journal of Mathematics and Statistics, Vol., (3), pp

15 7. Jussi K.V. (003): Uncertainties and quantification of common cause failure rates and probabilities for system analysis. Reliability Engineering & System Safety, 8 (), pp Magnus J.R. and Neudecker H. (988): Matrix Differential Calculus, with applications in statistics and Econometrics. Chichester: Wiley. 9. Malwane M.A. Ananda. (999): Estimation and testing of availability of a parallel system with exponential failure and repair times. Journal of Statistical Planning and Inference, 77, pp Polyanin A.D. and Zaitsev V.F. (003): Handbook of Exact Solutions for Ordinary Differential Equations ( nd edition), Chapman & Hall/CRC Press, Boaca Raton.. Rander M.C., Suresh K. and Ashok K. (994): Cost analysis of two dissimilar cold standby system with preventive maintenance and replacement of standby, Microelectron. Reliab. 34, 7-74, Thomee L. (003): Partial deferential equations with numericalmethods,springer,germany. 3. Wang K.H., Hsieh C.H. and Liou C.H. (006): Cost benefit analysis of series systems with cold standby components and repairable service station, Quality Technology Quantitative Management 3, Zwillinger D. (997): Handbook of Differential Equations (3 rd edition), Academic Pres, Boston. 5

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Reliability Analysis of a Two Dissimilar Unit Cold Standby System with Three Modes Using Kolmogorov Forward Equation Method

Reliability Analysis of a Two Dissimilar Unit Cold Standby System with Three Modes Using Kolmogorov Forward Equation Method Available online at http://www.ajol.info/index.php/njbas/index Nigerian Journal of Basic and Applied Science (September, 03), (3): 97-06 DOI: http://dx.doi.org/0.434/njbas.vi3.5 ISSN 0794-5698 Reliability