Cost- benefit analysis of stochastic models of A 2-out-of-3 redundant system with inspection

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Cost- benefit analysis of stochastic models of A 2-out-of-3 redundant system with inspection

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Abstract

In this paper two stochastic models for a 2-out-of-3 redundant system are developed and cost-benefit analysis is carried out by using semi-Markov and regenerative processes. There are three identical units in each model and the system is considered in up-state if 2-out-of-3 units are functioning. A single repair facility is available which plays the dual role of inspection and repair. In model1, server repairs the unit without inspection while in model 2, the unit is inspected at its failure to see the feasibility of repair. If repair of unit is not feasible, it is replaced by new one in order to increase the reliability of the system. The failure time distribution follows negative exponential while that of inspection and repair are taken as arbitrary. The expressions for MTSF, steady state availability, busy period and expected number of visits of the server are derived. The profit function of each model is also estimated. A particular case is considered to derive the results in parametric form.

Keywords - 2-out-of-3 redundant system, inspection, regenerative process and cost-benefit analysis

Abstract
Introduction
Notation
Transition probabilities and mean sojourn times
Analysis for model 1 and 2
1. Mean Time to System Failure (MTSF)
2. Steady state availability
3. Busy period of the server
4. Expected number of visits by the server
Particular case
References

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[...] The possible transition states of both the system models are shown in the following table1 S0 Model I Model II S1 S2 N0, Fwr, Fur N N wi , FuI S3 S4 S5 N N Cs N N Cs N N Fur N N Fui N N Fur N N wi , FuR N FwI , Fur For model S0, For model II: S1, S2, S3} The possible transition states along with transition rates for model I and model II are shown in fig and fig respectively Transition Probabilities and Mean Sojourn Times Simple probabilistic considerations yield the following expressions for the non-zero elements, pij = Qij = q ij (t)dt as: For Model I p01=1, p10=g*(2λ), p12=1-g*(2λ), p 11.2 Clearly, p01=p10+p12=p10+p 11.2 For Model II p01 = p10 = bh* (2λ p12 = 1 h * (2λ p13 = ah *(2λ p30 = (2λ p34 = 1 (2λ p 11.2 = b[1 h * (2λ p 11.25 = a[1 h (2λ p 31.4 = p34 It can be easily verified that * p01 = 1 = p10 + p12 = p30 + p34 = p30 + p 31.4 = p10 + p13 + p 11.2 + p 11.25 The unconditional mean time taken by the system to transit to any regenerative state Sj when it (time) is counted from epoch of entrance into that state Si, is given by mij = td{Qij = ij ' 0 and the mean Sojourn time in the state Si is given by µi = = P(T > t)dt 0 where T denotes the time to system failure. [...]

[...] Reliab. 655-659 (1985). Singh, S.K., Profit evaluation of a two unit cold standby system with random appearance and disappearance time of service facility, Microelectron Reliab., 29,705-709 (1989) Nakagawa, T., A replacement policy maximizing [...]

[...] A particular case is considered to obtain the results for MTSF and profit of the models in parametric form Notation E No Set of regenerative states Unit in normal mode and operative Unit in normal mode but not working Unit in normal mode and cold standby Probability that repair is feasible / not feasible Constant failure rate of an operative unit. Probability that the system is up initially in state Si E is up at time t without visiting to any other regenerative state. [...]

[...] The expected up time of the system in is given by t U ) = Aio ) , Taking Laplace transform, we get U * ( = 1 * A io ( s ) s 2. Expected down time of the system in Di(t) = i=1,2 Taking Laplace transform, we get = 1 sA* ) s2 3. Expected duration of busy time of server in is given by t µib ) = Bi 0 )du , 1 0r µ *ib ) = B 0 ) , s 4. [...]

[...] Therefore it is imperative that any amplifier system used in satellite communication be equipped with some form of automatic backup or spare to make the system more reliable and available for use. Hence, stochastic models for 2-out-of-3 modular amplifier system are under taken for study. In view of the above, two stochastic models for 2-out-of-3 redundant system of identical units are proposed in this paper. Initially two units work in parallel and third unit kept as cold standby. There is a single server who visits the system immediately whenever needed. The system is considered in up-state if two units are operative. [...]

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