Research Article Stochastic Analysis of a Two-Unit Cold Standby System Wherein Both Units May Become Operative Depending upon the Demand

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1 Journal of Quality and Reliability Engineering, Article ID , 13 pages Research Article Stochastic Analysis of a Two-Unit Cold Standby System Wherein Both Units May Become Operative Depending upon the Demand Reetu Malhotra 1 and Gulshan Taneja 2 1 DepartmentofAppliedSciences,ChitkaraUniversity,Rajpura,Punjab1441,India 2 Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana 1241, India Correspondence should be addressed to Reetu Malhotra; reetu.malhotra@chitkarauniversity.edu.in Received 16 June 214; Revised 13 October 214; Accepted 28 October 214; Published 1 December 214 Academic Editor: Yi-Hung Chen Copyright 214 R. Malhotra and G. Taneja. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The present paper analyzes a two-unit cold standby system in both units may become operative depending upon the demand. Initially, one of the units is operative while the other is kept as cold standby. If the operative unit fails or the demand increases to the extent that one operative unit is not capable of meeting the demand, the standby unit becomes operative instantaneously. Thus, both units may become operative simultaneously to meet the increased demand. Availability in three types of upstates is as follows: (i when the demand is less than or equal to production manufactured by one unit; (ii when the demand is greater than whatever produced by one unit but less than or equal to production made by two units; and (iii when the demand is greater than the produces by two units. Other measures of the system effectiveness have also been obtained in general case as well as for a particular case. Techniques of semi-markov processes and regenerative processes have been used to obtain various measures of the system effectiveness. 1. Introduction In the literature of reliability, extensive studies have been madeondifferenttypesofone-unitortwo-unitstandby redundant systems owing to their frequent use in modern business and industrial systems. There are two major types of redundancy parallel and standby. In parallel redundancy, the redundant units form part of the system from the start, as in a standby system the redundant units do not form part of the system from the start (until they are needed. Standby units can be classified as cold, warm, or hot. Acold standby is completely inactive and since it is not hooked up, it cannot fail until it is replacing the primary unit. A warm standby hasadiminishedloadbecauseitisonlypartially energized. A hot standby is fully active in the system (although redundant. A lot of work has been done on reliability and cost analysis of various systems by various researchers including [1 17] who have analyzed these systems by considering variousconceptsliketheerlangianrepairtime,operatingand rest periods, hardware/software faults, congestion of calls, availability, two types of repair facility, human failure, regenerative point technique, priority repair discipline, instruction, accident, patience time, chances of nonavailability of expert repairman, one big unit and two small identical units with priority for operation/repair to big unit, patience time, partial failures, and optimized maintenance of the diesel system in locomotives. In all such studies, the demand was fixed. There may be situations demand may vary and hence it affects the operability of the units of a system. The concept of variation in demand has been studied by [18, 19]. This concept of variation in demand was considered for single unit systems, the system either is in working state on some demand orisputtoshutdownmodeonnodemand.however,the demand may be much more than whatever produced by a single unit system and hence there is need of having one additional unit to meet the demand. Study of the concept of variation in demand for the two-unit system thus becomes more important. Keeping the above observations in view, we, in the present paper, develop a reliability model for a two-unit standby system working in a cable manufacturing plant in cold

2 2 Journal of Quality and Reliability Engineering standby may become operative depending on demand. Information of such a system was gathered on visiting a cable manufacturing plant in H.P., India. It was observed that there were two units in the plant which were used to manufacture polyvinyl chloride (PVC wires as per the demand in the market. If the demand is less than whatever produced by one unit, only one unit is kept operative, as if demand is greater than the production from a single unit, both units are put to operative mode. Various measures of the system effectiveness are analyzed by making use of semi-markov processes and regenerative point techniques. The rest of the paper is organized as follows. In Section 2, we develop the proposed semi-markov model and presented its description. In Section 3, we find the steady-state probabilities and mean sojourn times. Section 4 deals with the computation of steady-state measures, such as mean time to system failure (MTSF, availability in three types of upstates (i.e., when demand is less than or equal to production made by one unit; when demand is greater than production made by one unit, but less than or equal to production made by two units; when demand is greater than production made by two units, busy period of the repairman for repair, and expected number of visits by the repairman. In Section 5, cost-benefit hasbeenobtainedasageneralcase.onthebasisofthe data/information estimated from the cable manufacturing plant, a particular case has been discussed in Section 6 and various graphs have been plotted in Section 7. Final conclusions along with some future directions are presented in Section Model Description and Assumptions Figure 1 depicts the state transition model which we developed for a two-unit standby system working in a cable manufacturing plant in cold standby may become operative depending on demand. If one or both units are in working mode, then the system is in operative state. When both units are not working, that is, one is under repair and the other is waiting to be repaired, the system will stay in the failed state. Variousassumptionsforthemodelareasfollows. (1 The units are similar and statistically independent. (2 Demand cannot be decreased further, when at the most one unit is operative. (3 Each unit is new after repair. (4 If a unit is failed, standby unit takes over automatically. (5 The system becomes inoperable when both units fail in a two-unit system. (6 All the random variables are independent. (7 Switching is perfect and instantaneous. (8 Failure and repair time follow exponential and general time distribution, respectively. 3. Transition Probabilities and Mean Sojourn Times The transition diagram showing the various states of the system is shown as in Figure 1.Theepochsofentryintostates S, S 1, S 2, S 5, S 6,andS 7 are regeneration points and thus are regenerative states. States S 3, S 9,andS 1 are failed states. The transition probabilities are q 1 (t =γ 11 e (+γ 11t, q 2 (t =e (+γ 11t, q 1 (t =γ 12 e (+γ 12+γ 21 t, q 15 (t =γ 21 e (+γ 12+γ 21 t, q 16 (t =e (+γ 12+γ 21 t, q 2 (t =e (+γ 11t g (t, q 4 21 (t = (γ 11e (+γ 11t De (+γ 21t g(t, q 3 22 (t = (e (+γ 11t De γ 11t g(t, q 23 (t =e (+γ11t G (t, q (4,8 2,5 (t = (γ 11 e (+γ11t Dγ 21 e (+γ21t De t g(t, q (3,9 2,6 (t = (e (+γ 11t Dγ 11 e γ 11t De γ 21t g(t, q (4,9 2,6 (t = (γ 11 e (+γ 11t De (+γ 21t De γ 21t g(t, q (3,9,1 2,7 (t = (e (+γ 11t Dγ 11 e γ 11t Dγ 21 e γ 21t D1 g (t, q (4,8,1 2,7 (t = (γ 11 e (+γ 11t Dγ 21 e (+γ 21t De t D1 g (t, q (4,9,1 2,7 (t = (γ 11 e (+γ 11t De (+γ 21t Dγ 21 e γ 21t 1 g (t, q 4 29 (t = (γ 11e (+γ 11t De (+γ 21t G (t, q (4,8 2,1 (t = (γ 11e (+γ 11t Dγ 21 e (+γ 21t De t G (t, q 51 (t =γ 22 e (+γ 22t, q 57 (t =e (+γ 22t, q 61 (t =e (+γ 21t g (t, q 8 65 (t = (γ 21e (+γ 21t De t g(t, q 9 66 (t = (e (+γ 21t De γ 21t g(t, q (8,1 6,7 (t = (γ 21 e (+γ 21t De t D1 g (t, q (9,1 6,7 (t = (e (+γ 21t Dγ 21 e γ 21t D1 g (t, q 69 (t = (e (+γ 21t G (t,

3 Journal of Quality and Reliability Engineering 3 g(t γ 11 S 4 F R,Op p 1 <d p 2 S 9 F R,F W p 1 <d p 2 γ γ S S 1 S 5 Op, C Op, Op S Op, Op d p γ 12 p 1 <d p 2 1 d<p2 γ 22 g(t g(t g(t g(t S S 6 S 7 2 F F r,op r,op F r,op p 1 <d p 2 d>p2 d p 1 γ 21 g(t g(t g(t S 8 γ 11 S 3 F R,Op S 1 F F R,F d>p R,F W W 2 d>p d p 2 1 γ 21 γ 21 Regeneration point Operative state Failed state Figure 1: State transition diagram. q 8 6,1 (t = (γ 21e (+γ 21t De t G (t, q 75 (t =e t g (t, q 1 77 (t = (e t D1 g (t, q 7,1 (t =e t G (t. Clearly,forthemodeltobeaccurate,itisimportanttoestimate accurately the model parameters (i.e., mean sojourn times and the steady-state transition probabilities. The nonzero elements p ij = lim s q ij (s are obtained as γ p 1 = 11 ( + γ 11, p 2 = ( + γ 11, γ p 1 = 12 ( + γ 12 +γ 21, γ p 15 = 21 ( + γ 12 +γ 21, p 16 = ( + γ 12 +γ 21, p 2 =g (+γ 11, p 4 21 = γ 11 (γ 21 γ 11 (g (+γ 11 g (+γ 21, (1 p 3 22 =( g (+γ 11+g (γ 11, p 23 = ( + γ 11 (1 g (+γ 11, p (4,8 2,5 =γ 11 γ 21 ( g (+γ 11 (γ 11 γ 21 γ 11 g (+γ 21 (γ 11 γ 21 γ 21 + g ( (γ 11 γ 21, p (3,9 2,6 =γ 11 ( g (+γ 11 (+γ 11 γ 21 g (γ 21 ( + γ 11 γ 21 (γ 21 γ 11 p (4,9 + g (γ 11 (γ 21 γ 11, g ( + γ 2,6 =γ 11 ( 11 ( + γ 11 γ 21 (γ 21 γ 11 + g (+γ 21 (γ 21 γ 11 p (3,9,1 2,7 =γ 11 γ 21 g (γ + 21 (+γ 11 γ 21, g ( + γ ( 11 (+γ 11 (+γ 11 γ 21 g (γ 11 γ 11 (γ 21 γ 11 g (γ + 21 γ 21 (γ 21 γ 11 (+γ 11 γ 21

4 4 Journal of Quality and Reliability Engineering p (4,8,1 2,7 =γ 11 γ 21 p (4,9,1 2,7 =γ 11 γ 21 p 4 29 = γ 11 (γ 21 γ γ 11 γ 21 ( + γ 11, g (+γ ( 11 γ 11 ( + γ 11 (γ 21 γ 11 g (+γ 21 γ 21 ( + γ 21 (γ 21 γ 11 g ( 1 + γ 11 γ 21 (+γ 11 (+γ 21, g (+γ ( 11 ( + γ 11 (+γ 11 γ 21 (γ 21 γ 11 g (+γ 21 (+γ 21 (γ 21 γ 11 g (γ 21 γ 21 ( + γ 11 γ γ 21 ( + γ 11 (+γ 21, ( (1 g (+γ 11 ( + γ 11 p (4,8 2,1 =γ 11γ 21 (1 g (+γ 21, ( + γ 21 ( (1 g (+γ 11 γ 11 ( + γ 11 (γ 11 γ 21 (1 g (+γ 21 γ 21 ( + γ 21 (γ 11 γ 21 + (1 g (, (γ 11 γ 21 p 51 = p 57 = γ 22 ( + γ 22, ( + γ 22, p 61 =g (+γ 21, p 8 65 =g (+γ 21+g (, p 9 66 = g (+γ 21+g (γ 21, p (8,1 =γ 21 ( g (+γ 21 γ 21 ( + γ 21 g ( (γ (+γ 21, p (9,1 =γ 21 ( g (+γ 21 g (γ (+γ 21 (γ 21 γ 21 ( + γ 21, p 69 = ( + γ 21 (1 g (+γ 21, p 8 6,1 =γ 21 ( (1 g ( + γ 21 γ 21 ( + γ 21 p 75 =g (, + p 1 77 =1 g (, p 7,1 =1 g (. (1 g (, (γ 21 By these transition probabilities, it can be verified that p 1 +p 2 =p 1 +p 15 +p 16 =p 2 +p p 23 +p (4,8 +p p(4,8 2,1 =1, p 2 +p p3 22 +p(4,8 +p (3,9 +p (4,9 +p (3,9,1 +p (4,8,1 +p (4,9,1 =1, p 51 +p 57 =p 61 +p p 69 +p 8 6,1 =p 61 +p p9 66 +p(8,1 +p (9,1 =1, p 75 +p 7,1 =p 75 +p 1 77 =1. The mean sojourn time (μ i instatei is μ 5 = μ = 1 ( + γ 11 ; μ 1 1 = ( + γ 11 +γ 21 ; μ 2 = ( + γ 11 (1 g (+γ 11, 1 ( + γ 22 ; μ 6 = ( + γ 21 (1 g (+γ 21 ; (2 (3 μ 7 = 1 (1 g (. (4 Analysis carried out in this paper depends only on the mean sojourn time and is independent of the actual sojourn time distributions for the semi-markov processes states. If we were to carry out a transient analysis of the semi-markov processes, this will no longer be true. The unconditional mean time taken by the system to transit for any state j when it is counted from epoch of entrance into state i is mathematically stated as m ij = tq ij (t dt = q ij (. (5

5 Journal of Quality and Reliability Engineering 5 Thus, +(q 69 (s +q 8 6,1 (s m 1 +m 2 =μ ; m 1 +m 15 +m 16 =μ 1, m 2 +m m 23 +m (4,8 +m m(4,8 2,1 =μ 7, m 2 +m m3 22 +m(4,8 +m (3,9 +m (4,9 +m (3,9,1 +m (4,8,1 +m (4,9,1 =k 1, m 51 +m 57 =μ 5, m 61 +m m 69 +m 8 6,1 =μ 7 =m 75 +m 7,1 m 61 +m m9 +m(8,1 +m (9,1 =k 1 =m 75 +m 1 77, (6 (q 1 (s q 16 (s (1 q 57 (s q 75 (s q 2 (s q 16 (s ( q 4 21 (s (1 q 57 (s q 75 (s q 51 (s q (4,8 (s q 7,1 (s (q 1 (s ( q 15 (s q 57 (s +q 2 (s q 16 (s q 57 (s q 8 65 (s k 1 = tg(tdt. 4. Measures of System Effectiveness 4.1.MeanTimetoSystemFailure.To determine the mean time to system failure (MTSF of the system, we regard the failed states as absorbing states. By probabilistic arguments, we obtain the following recursive relations for φ i (t, i =, 1, 2, 5, 6, 7 are given by φ (t =Q 1 (t φ 1 (t +Q 2 (t φ 2 (t, φ 1 (t =Q 1 (t φ (t +Q 15 (t φ 5 (t +Q 16 (t φ 6 (t, φ 2 (t =Q 2 (t φ (t +Q 4 21 (t φ 1 (t +Q 23 (t +Q (4,8 (t φ 5 (t +Q 4 29 (t +Q(4,8 2,1 (t, φ 5 (t =Q 51 (t φ 1 (t +Q 57 (t φ 7 (t, φ 6 (t =Q 61 (t φ 1 (t +Q 8 65 (t φ 5 (t +Q 69 (t +Q 8 6,1 (t, φ 7 (t =Q 75 (t φ 5 (t +Q 7,1 (t. Taking Laplace-Stieltjes Transform (L.S.T. of the relations given by (7 and solving them for Ls(φ (t,weobtain Ls (φ (t = N (s =q 2 (s (q 23 (s +q 9 24 (7 N (s D (s, (8 (s +q(4,8 2,1 (s (1 q 57 (s q 75 (s q 15 (s q 51 (s q 16 (s (q 51 (s q 8 65 (s +q 61 (s (1 q 57 (s q 75 (s ( q (4,8 (s q 57 (s q 15 (s q 4 21 (s q 57 (s q 16 (s q 57 (s D (s =(1 q 57 (s q 75 (s (q 4 21 (s q8 65 (s q 61 (s q (4,8 (s, (1 q 16 (s q 61 (s q 1 (s q 1 (s q 2 (s (q 1 (s q 4 21 (s +q 2 (s (1 q 16 (s q 61 (s q 51 (s ((1 q 2 (s q 2 (s (q 15 (s +q 16 (s q 8 65 (s +q 2 (s q 1 (s q (4,8 (s. Now, the reliability R(t of the system at time t is given as (9 R (t = Inverse Laplace transform of (1 Ls (φ (t. s (1 Now,themeantimetosystemfailure(MTSFwhenthe system starts from the state is MTSF = R (t dt = limr (s. s (11 Using L Hospital rule and putting the value of Ls(φ (t from (8,wehave MTSF = N D, (12

6 6 Journal of Quality and Reliability Engineering N=μ [(1 p 57 p 75 (p 16 (p 69 +p 8 6,1 +p 1 +p 7,1 (p 15 p 57 +p 16 p 57 p 5 68 ] +μ 1 [p 1 (1 p 57 p 75 +p 2 p 4 21 (1 p 57p 75 +p 51 p 2 p 48 ] +μ 5 [((1 p 2 p 2 (p 15 +p 16 p p 2p 1 p (4,8 (p 23 +p p(4,8 2,1 p 2 (p 15 +p 16 p (p 69 +p 8 6,1 p 2p 16 p 48 ] +μ 7 [p 2 ((1 p 57 p 75 (1 p 16 p 61 D=(1 p 57 p 75 p 51 (p 15 +p 16 p p 1 p 16 (1 p 57 p 75 p 2 p 16 ( p 4 21 (1 p 57p 75 p 51 p 48 +p 57 (p 1 (p 15 +p 16 p p 2 (p 48 +p4 21 p 15 (1 p 16 p 61 p 1 p 1 +p 16 (p 4 21 p8 65 p 61p 48 ], p 2 (p 1 p p 2 (1 p 16 p 61 p 51 ((1 p 2 p 2 (p 15 +p 16 p p 2p 1 p 48. ( Availability Analysis When Demand Is Less Than or Equal to Production Made by One Unit (d p 1. Letting i (t i =, 1, 2, 5, 6, 7 as the probability that the system is in upstate when demand is not less than production made by one unit at instant t giventhatitenteredthestateiat t=and using the arguments of the theory of the regeneration process, the availability i (t isseentosatisfythefollowingrecursive relations: +q (3,9,1 (t D 7 (t +q(4,8,1 (t D 7 (t +q (4,9,1 (t DA p1 7 (t, 5 (t =q 51 (t D 1 (t +q 57 (t D 7 (t, 6 (t =q 61 (t D 1 (t +q8 65 (t DAp 1 5 (t +q 9 66 DAp 1 6 +q (9,1 (t D 7 (t, (t +q(8,1 (t D 7 (t 7 (t =q 75 (t D 5 (t +q1 77 (t DAp 1 7 (t, (14 M (t = e (+γ 11t and M 2 (t =e (+γ 11t G (t +(γ 11 e (+γ 11t De (+γ 21t G (t D (γ 11 e (+γ 11t Dγ 21 e (+γ 21t De t G (t. (15 Taking Laplace transforms of above equations (14 and solving them for A (s,weget (s = N 1 (s D 1 (s, (16 N 1 (s =(M (s (1 q 3 22 (s+m 2 (s q 2 (s [(1 q 9 66 (s ((1 q 1 77 (s q 57 (s q 75 (s q 15 (s q 51 (s (1 q 1 77 (s +(1 q 1 77 (s ( q 16 (s (q 51 (s q 8 65 (s +q 61 (s q 75 (s q 16 (s ((q (8,1 (s +q (9,1 (s (t =M +q 1 (t D 1 (t +q 2 (t D 2 (t, 1 (t =q 1 (t D (t +q 15 (t D 5 (t +q 16 (t D 6 (t, 2 (t =M 2 (t +q 2 (t D (t +q4 21 (t DAp 1 1 (t +q 3 22 (t DAp 1 2 (t +q(4,8 (t D 5 (t +q (3,9 (t D 6 (t +q(4,9 (t D 6 (t +q 61 (s q 57 (s], D 1 (s = ((1 q 3 22 (s q 2 (s q 2 (s [(1 q 9 66 (s (1 q 1 77 (s q 57 (s q 75 (s q 15 (s q 51 (s (1 q 1 77 (s (17

7 Journal of Quality and Reliability Engineering 7 +(1 q 1 77 (s ( q 16 (s (q 51 (s q 8 65 (s +q 16 (s q 75 (s q 16 (s (q (8,1 (s +q (9,1 (s +q 61 (s q 57 (s]+q 1 (s [(1 q 9 66 (s ((1 q 1 77 (s q 57 (s q 75 (s (q 1 (s (1 q 3 22 (s+q 2 (s q 4 21 (s +q 2 (s (1 q 1 77 (s ((q 51 (s q 8 65 (s +q 61 (s +(q (3,9 (s +q (4,9 (s +(1 q 9 66 (sq(4,8 (s q 51 (s q 51 (s q 75 (s (q (8,1 (s +q (9,1 (s (q (3,9 (s +q (4,9 (s q 51 (s q 75 (s (1 q 9 66 (s (q (3,9,1 (s +q (4,8,1 (s +q (4,9,1 (s +q 61 (s q 57 (s q 75 (s (q (3,9 (s +q (4,9 (s]. In steady-state, the availability (18 of the system is given by = lim s (sap 1 (s = N 1 ( D 1 ( = N 1 D 1, (19 N 1 =(μ (1 p μ 2p 2 [(1 p 9 66 ((1 p 1 77 p 57p 75 p 15 p 51 (1 p (1 p 1 77 ( p 16 (p 51 p p 61 p 75 p 16 (p (8,1 +p (9,1 +p 61 p 57 ], D 1 = μ ((1 p 3 22 (1 p9 66 (1 p1 77 p 57p 75 p 1 μ 1 [(1 p 9 66 ((1 p1 77 p 57p 75 (p 1 (1 p p 2p p 2 (1 p 1 77 ((p 51 p p 61(p (3,9 +p (4,9 +(1 p 9 66 p48 p 51 p 51 p 75 ((p (8,1 +p (9,1 (p (3,9 +p (4,9 (1 p 9 66 (p (3,9,1 +p (4,8,1 +p (4,9,1 +p 61 p 57 p 75 (p (3,9 +p (4,9 ] +μ 2 (p 1 p 51 p 75 (1 p μ 5 ((1 p 3 22 p 2p 2 ( p 61 p 75 (p (3,9 +p (4,9 +p 1 p 75 (1 p 9 66 (p 1 (1 p p 2p p 75 (1 p p 16p 61 +k 1 (p 2 ((p 51 p p 61(p (3,9 +p (4,9 +(1 p 9 66 p48 p 51 +p 1 (1 p 9 66 (p 1 (1 p p 2p (p 57 (1 p 9 66 p 15p 51 +p 16 (p 51 p p 61 ((1 p 3 22 p 2p 2 +p 51 p 75 ((p (8,1 +p (9,1 (p (3,9 +p (4,9 +p 16 p 75 ((1 p 3 22 p 2p 2. ( Availability Analysis When Demand Is Greater Than the Production Made by One Unit and Less Than or Equal to Production Made by Two Units (p 1 <d p 2. Letting A p 2 i (t i =, 1, 2, 5, 6, 7 as the probability that the system is inupstatewhendemandisgreaterthantheproductionmade by one unit and less than or equal to production made by two units at instant t given that it entered the state i at t=and proceeding in the similar fashion as in 5.2, in steady-state, the availability A p 2 is given by A p 2 = lim s (sap 2 (s =N 2, (21 D 1 N 2 =μ 1 [(1 p 9 66 ((1 p1 77 p 57p 75 (p 1 (1 p p 2p p 2 (1 p 1 77 ((p 51 p p 61(p (3,9 +p (4,9 +(1 p 9 66 p48 p 51

8 8 Journal of Quality and Reliability Engineering p 51 p 75 ((p (8,1 +p (9,1 (p (3,9 +p (4,9 (1 p 9 66 (p (3,9,1 +p (4,8,1 +p (4,9,1 +p 61 p 57 p 75 (p (3,9 +p (4,9 ] +μ 7 [p 1 (1 p 3 22 (p 16 (1 p 1 77 p 57p 75 μ 7 [ p 2 (p (3,9,1 +p (4,8,1 +p (4,9,1 ((1 p 9 66 (1 p 15p 51 p 16 (p 51 p p 61 (p (8,1 +p (9,1 (p 1 p 16 (1 p 3 22 p 2 ((1 p 15 p 51 (p (3,9 +p (4,9 p 2 (p 16 (( p 4 21 (1 p1 77 p 57p 75 +p 51 (p 48 (1 p1 77 p 75 (p (3,9,1 +p (4,8,1 and D 1 is already specified. +p 16 (p p 51p (4,8 ], (24 +p 2 (p (3,9 +p (4,9 +p (4,9, Busy Period Analysis of the Repairman. The total fraction of the time for which the system is under repair of the repairman, in steady-state, is given by ((1 p 1 77 (1 p 15p 51 p 57 p 75 ], (22 B = lim s (sb (s = N 4 D 1, ( and D 1 is already specified Availability Analysis When Demand Is Greater Than Production Made by Two Units (d > p 2. Letting A d i (t i =, 1, 2, 5, 6, 7 as the probability that the system is in upstate when demand is greater than the production made by two units at instant t giventhatitenteredthestatei at t = and proceeding in the similar fashion as in Section 4.2, the availability A d in steady-state is given by A d = lim s (sad (s =N 3 D 1, (23 N 3 =p 1 (1 p 3 22 (p 15 (1 p p 16p 8 65 ( μ 5 p 16 p 75 (p (8,1 +p (9,1 +p 57 μ 7 p 2 (( p (4,8 (1 p 9 66 (p8 65 +p 15p 61 (p (3,9 +p (4,9 p 15 p 4 21 (1 p9 66 p 16 (p 4 21 p8 65 p 61p (4,8 ((1 p 1 77 μ 5 +p 57 μ 7 μ 5 p 2 [ (p (3,9,1 +p (4,8,1 +p (4,9,1 (p 75 (1 p 9 66 p 16p 61 p 75 p 75 (p (8,1 +p (9,1 ((p (3,9 +p (4,9 +p 16 p 4 21 ] N 4 =k 1 [p 2 ((1 p 9 66 ((1 p 1 77 p 57p 75 p 15 p 51 (1 p (1 p 1 77 ( p 16 (p 51 p p 61 p 75 p 16 ((p (8,1 +p (9,1 +p 61 p 57 +(p 1 (1 p 3 22 (p 16 (1 p 1 77 p 57p 75 p 2 (p 16 (( p 4 21 (1 p1 77 p 57p 75 +p 51 (p 48 (1 p1 77 p 75 (p (3,9,1 +p (4,8,1 +p 2 (p (3,9 +p (4,9 +p (4,9,1 ((1 p 1 77 (1 p 15p 51 p 1 (1 p 3 22 p 57 p 75 (p 15 (1 p p 16p 8 65 p 57 p 2 (( p (4,8 (1 p 9 66 (p p 15p 61 (p (3,9 +p (4,9

9 Journal of Quality and Reliability Engineering 9 and D 1 (s is already specified. p 15 p 4 21 (1 p9 66 p 16 (p 4 21 p8 65 p 61p (4,8 p 57 ( p 2 (p (3,9,1 +p (4,8,1 +p (4,9,1 ((1 p 9 66 (1 p 15p 51 p 16 (p 51 p p 61 (p (8,1 +p (9,1 (p 1 p 16 (1 p 3 22 p 2 (1 p 15 p 51 (p (3,9 +p (4,9 +p 16 (p p 51p (4,8 ], ( 4.6. Expected Number of Visits by the Repairman. The expected number of visits per unit time by the repairman, in steady-state, is given by N 5 =p 2 (1 p 3 22 V = lim s (sv (s = N 5 D 1, ( [(1 p 9 66 ((1 p1 77 p 57p 75 p 15 p 51 (1 p (1 p 1 77 ( p 16 (p 51 p p 61 p 75 p 16 ((p (8,1 +p (9,1 +p 61 p 57 ] p 16 [(1 p 9 66 ((1 p1 77 p 57p 75 (p 1 (1 p p 2p p 2 (1 p 1 77 ((p 51 p p 61 (p (3,9 +p (4,9 +(1 p 9 66 p48 p 51 p 51 p 75 ((p (8,1 +p (9,1 (p (3,9 +p (4,9 (1 p 9 66 (p (3,9,1 +p (4,8,1 +p (4,9,1 +p 61 p 57 p 75 (p (3,9 +p (4,9 ] p 57 p 1 (1 p 3 22 (p 15 (1 p p 16p 8 65 ( p 16 p 75 (p (8,1 +p (9,1 p 2 (( p (4,8 (1 p 9 66 (p p 15p 61 (p (3,9 +p (4,9 p 15 p 4 21 (1 p9 66 p 16 (p 4 21 p8 65 p 61p (4,8 (1 p 1 77 p 2 ( (p (3,9,1 +p (4,8,1 +p (4,9,1 and D 1 is already specified. (p 75 (1 p 9 66 p 16p 61 p 75 p 75 (p (8,1 +p (9,1 5. Cost-Benefit Analysis ((p (3,9 +p (4,9 +p 16 p 4 21, (28 The expected profit can be calculated by expected total revenue in (, t] minus expected total repair in (, t] minus expected cost of visits by repairman in (, t].hencethetotal profit in (, t] is given by Profit P (t =(C (t +C 1A p 2 (t +C 2A d (t (29 C 3 B (t C 4 V (t. In steady-state, the expected profit per unit time incurred to the system is given by P (t P= lim ( =lim (s 2 P (s, t t s (3 Profit P=(C +C 1A p 2 +C 2A d C 3B C 4, C is revenue per unit uptime when demand is less than or equal to production made by one unit (d p 1, C 1 is revenue per unit uptime when demand is greater than theproductionmadebyoneunitandlessthanorequalto production made by two units (p 1 <d p 2, C 2 is revenue per unit uptime when demand is greater than the production made by two units (d >p 2, C 3 is cost per unit uptime for engaging the repairman for repair, and C 4 is cost per visit of the repairman. 6. Particular Case For the particular case, let us take g(t = αe αt and the values estimated from the system, that is, γ 11 =.235/hr, γ 12 =.7/hr, γ 21 =.4213/hr, γ 22 =.353/hr, α=1/hr, =.3/hr, C = INR 7, C 1 = INR 1, C 2 = INR 2, C 3 = INR 1, andc 4 = INR 4 (all costs are in Indian rupees.

10 1 Journal of Quality and Reliability Engineering Various measures of the system effectiveness are given by the following: (i mean time to system failure (MTSF = hrs; (ii steady-state availability when demand is less than or equal to production made by one unit (d p 1 ( =.2366; (iii steady-state availability when demand is greater than theproductionmadebyoneunitandlessthanor equal to production made by two units (p 1 <d p 2 (A p 2 = ; (iv steady-state availability when demand is greater than the production made by two units (d >p 2 (A d = ; (v busy period of the repairman for repair (B =.4613; (viexpectednumberofvisitsbytherepairman(v =.157; (vii profit incurred to the system = INR Graphical Analysis For the numerical results, let us take g(t = αe αt, α is repair rate. Data/information was gathered on failure and repair times from cable manufacturing system. On the basis of the gathered information, the following values are estimated: γ 11 =.235/hr, γ 21 =.4213/hr, γ 12 =.7/hr, γ 22 =.353/hr. (31 Various graphs have been plotted for the MTSF, the availability, and the profit with respect to rates/revenue per unit uptime for different values of rates/costs. Some have been shownhereandtheothershavenotbeenshowntoavoid space. The following interpretations can be made from the graphs. (i MTSF gets decreased with increase in the values of failureratebutincreasedwithincreaseinthevalues of repair rate. The values of other parameters per hour are taken as γ 11 =.235/hr, γ 21 =.4213/hr, γ 12 =.7/hr, γ 22 =.353/hr. (32 (ii The graph of availability ( when demand is less than or equal to whatever production given by a single unit (d p 1 versusfailurerate( fordifferent values of repair rate (α shows that it gets decreased monotonically with increase in the values of failure rate but increases with increase in the values of repair rate. Similar behavior has been shown for the availability (A p 2 when demand is greater than the production by one unit but less than or equal to production Profit (P (INR Profit (P versus failure rate ( for different values of repair rate (α α =.6/hr α = 1.1/hr α = 1.6/hr Failure rate ( (/hr Figure 2: Profit (P versus failure rate( for different values of repair rate (α. made by two units (p 1 < d p 2 andalsoforthe availability (A d when demand is greater than the production made by two units (d > p 2. The graphs revealed that the availabilities in three types of the upstates (, (Ap 2, (Ad are different in magnitude for the same values of parameters, α, γ 11, γ 12, γ 21, and γ 22. (iii Behavior of the profit with respect to failure rate/ revenue per unit uptime for different values of repair rate/cost of engaging the repairman has also been observed from the graphs. Some of these graphs are shown in Figures 2, 3, 4,and5.Thegraphsdepictthat theprofitgetsdecreasedwithincreaseinthevaluesof failurerateandalsowiththeincreaseincostofengaging/visiting the repairman but increases with increase in the values of repair rate/revenue per unit uptime. Cut-off points as to when the profit is positive or negative have been obtained from the graphs and are shown in Table Conclusion A two-unit cold standby system in cable manufacturing plant in both units may become operative depending upon the demand has been analyzed. A state transition model that describes the dynamic behavior of such a system is used as a basis for developing a stochastic model. In this paper, we have obtained several general probability distribution functions thatcanbeusedtodescribethesystembehavior.various measures of system effectiveness are estimated using semi- Markov processes and regenerative point techniques. These include the following: (i mean sojourn times; (ii steady-state probabilities; (iii mean time to system failure (MTSF; (iv availability in three types of upstates (i.e., when demand is less than or equal to production made by

11 Journal of Quality and Reliability Engineering 11 Table 1 Fixed parameter γ 11 =.235/hr, γ 12 =.7/hr, γ 21 =.4213/hr, γ 22 =.353/hr, C = INR 7, C 1 = INR 1, C 2 = INR 2, C 3 = INR 1, C 4 =INR4 γ 11 =.235/hr, γ 12 =.7/hr, γ 21 =.4213/hr, γ 22 =.353/hr, α=1/hr, =.3/hr, C 1 = INR 1, C 2 = INR 2, C 4 = INR 4 γ 11 =.235/hr, γ 12 =.7/hr, γ 21 =.4213/hr, γ 22 =.353/hr, α =1/hr, =.3/hr, C = INR 7, C 2 = INR 2, C 4 =INR 4 γ 11 =.235/hr, γ 12 =.7/hr, γ 21 =.4213/hr, γ 22 =.353/hr, α =1/hr, =.3/hr, C = INR 7, C 1 = INR 1, C 4 = INR 4 γ 11 =.235/hr, γ 12 =.7/hr, γ 21 =.4213/hr, γ 22 =.353/hr, α =1/hr, =.3/hr, C 1 = INR 1, C 2 = INR 2, C 3 =INR 1 γ 11 =.235/hr, γ 12 =.7/hr, γ 21 =.4213/hr, γ 22 =.353/hr, α = 1/hr, =.3/hr, C = INR 7, C 2 = INR 2, C 3 =INR1 γ 11 =.235/hr, γ 12 =.7/hr, γ 21 =.4213/hr, γ 22 =.353/hr, α =1/hr, =.3/hr, C = INR 7, C 1 = INR 1, C 3 = INR 1 Varied parameter Profit Positive Negative Zero (cut-off point α =.6/hr <.598 >.598 =.598 α = 1.1/hr, <.796 >.796 =.796 α = 1.6/hr <.943 >.943 =.943 C 3 =INR24 C > 395. C < 395. C =395. C 3 =INR C > 53. C < 58.2 C = 53. C 3 = INR C > 69. C < 69. C = 69. C 3 =INR18 C 1 > C 1 < C 1 = C 3 = INR 22 C 1 > C 1 < C 1 = C 3 = INR C 1 > C 1 < C 1 = C 3 =INR18 C 2 > C 2 < C 2 = C 3 = INR 22 C 2 > C 2 < C 2 = C 3 = INR C 2 > 5.82 C 2 < 5.82 C 2 =5.82 C 4 = INR 3 C > C < C = C 4 =INR31 C > 58.2 C < 58.2 C = 58.2 C 4 =INR32 C > C < C = C 4 = INR 3 C 1 > C 1 < C 1 =422.9 C 4 =INR32 C 1 > C 1 < C 1 = C 4 =INR34 C 1 > C 1 < C 1 =599.2 C 4 =INR18 C 2 > C 2 < C 2 = C 4 =INR24 C 2 > C 2 < C 2 = C 4 = INR 3 C 2 > C 2 < C 2 = Profit (P (INR Profit (P versus revenue per unit uptime (C for different values of cost (C Revenue per unit uptime (C (INR C 3 = INR 24 C 3 = INR C 3 = INR Figure 3: Profit (P versus revenue per unit uptime (C for different values of cost (C 3. one unit; when demand is greater than production made by one unit, but less than or equal to production made by two units; when demand is greater than production made by two units and other measures of the system effectiveness which have been obtained in general case as well as for a particular case; it can also be concluded that these steady-state availabilities have different values in magnitude for the same value of failure rate and repair rate; (v busy period of the repairman for repair; Profit (P (INR Profit (P versus revenue per unit uptime (C 1 for different values of cost (C Revenue per unit uptime (C 1 (INR C 4 = INR 3 C 4 = INR 32 C 4 = INR 34 Figure 4: Profit (P versus revenue per unit uptime (C 1 for different values of cost (C 4. (vi expected number of visits by the repairman; (vii profit incurred to the system; (viii various graphs which are plotted to provide a better understanding of the behaviour of the system, help to refine its stochastic description, and lead to better estimates of the model parameters; (ix graphs from which the behavior of mean time to system failure, different types of availability, and the

12 12 Journal of Quality and Reliability Engineering Profit (P (INR Profit (P versus revenue per unit uptime (C 2 for different values of cost (C Revenue per unit uptime (C 2 (INR C 3 = INR 18 C 3 = INR 22 C 3 = INR Figure 5: Profit (P versus revenue per unit uptime (C 2 for different values of cost (C 3. profit can be depicted with respect to failure rate/ repair rate/cost for engaging repairman/revenue per unituptime/costpervisitoftherepairmanthathave been plotted; (x various cut-off points for the profit of the system whichhelptodecideabouthowmuchshouldbethe maximum tolerable value of the failure rate of the system and how much the cost price of the product (cable wires manufactured by the system should be soldtogetatleastthatvalueofrevenueperunit uptime so as to get positive profit; (xi the lower limits for the revenue per unit uptime for positive profit that have been obtained which may be quite useful for the system manufacturers, engineers, and the system analysts; (xii the upper limit for the cost per visit as well as for engaging the repairman which has been obtained. The above conclusions have been drawn on the basis of a particular case and the data collected. However, our model can be used by anyone using such system and can draw the conclusions in the similar fashion by putting those values of parameters, which exist for his/her system, in the general expressions obtained by us for the model. Further, we are focusing on developing some more realistic models (e.g., hot standby system related to the given system. Cost-benefit analysis for the system will be carried out to increase the uptime of the system and to reduce the cost involved in system. Nomenclature Symbols for the Various States : Operativestateforthesystem :Failedstateforthesystem D: Symbol for demand p 1 : Symbol for production by single unit p 2 : Symbol for production by two units. Notations : Failure rate of the operative unit γ 11 : Rate of increase of demand when demand is greater than the production made by one unit and less than or equal to production made by two units (p 1 <d p 2 γ 12 : Rateofdecreaseofdemandsoasto become less than or equal to production made by one unit (d p 1 γ 21 : Rate of further increase of demand when demand is greater than the production made by two units (d > p 2 γ 22 : Rate of decrease of demand when demand is greater than the production made by one unit and less than or equal to production made by two units (p 1 <d p 2 i (t: Probability that the system is in upstate when demand is less than or equal to production made by one unit (d p 1 at instant t giventhatitenteredthestatei at t= A p 2 i (t: Probability that the system is in upstate when demand is greater than the production made by one unit and less than orequaltoproductionmadebytwounits (p 1 <d p 2 at instant t given that it entered the state i at t= A d i (t: Probability that the system is in upstate when demand is greater than production made by two units (d > p 2 at instant t giventhatitenteredthestatei at t= B i (t: Probability that the repairman is busy to repair the failed unit at instant t given that it entered the state i at t= M i (t: Probability that the system, initially up in the regenerative state i,isupattimet without passing through any other regenerative state m ij : Contributiontomeansojourntimein regenerative state i before transitingto regenerative state j withoutvisitingany other state μ i : Mean sojourn time in regenerative state i before transiting to any other state : Symbol for derivative : Symbol for Laplace transforms Ls: Symbol for Laplace Stieltjes transforms D: Symbol for Laplace convolution : Symbol for Stieltjes convolution q ij (t, Q ij (t: p.d.f. and c.d.f of the first passage time from a regenerative state i to a regenerative state j or to a failed state j without visiting any other regenerative state in (, t] g(t, G(t: p.d.f. and c.d.f of repair rate at instant t.

13 Journal of Quality and Reliability Engineering 13 States of System C S : Op (d p 1 : Unitisincoldstandbystate Unitisinoperativestatewhen demand is less than or equal to production made by one unit Op (p 1 <d p 2 :Unitisinoperativestatewhen demand is greater than the production made by one unit and less than or equal to production made by two units Op (d > p 2 : F r : F R : F w : Conflict of Interests Unitisinoperativestatewhen demand is greater than production made by two units Failed unit under repair Repair of failed unit continuing from previous state Failed unit waiting for repair. The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] P. Chandrasekhar, R. Natarajan, and V. S. Yadavalli, A study on a two unit standby system with Erlangian repair time, Asia- Pacific Journal of Operational Research,vol.21,no.3,pp.1 7, 24. [2] A.Goyal,G.Taneja,andD.V.Singh, Reliabilityandprofitevaluation of a 2-unit cold standby system working in a sugar mill with operating and rest periods, Caledonian Journal of Engineering,vol.5,pp.1 5,29. [3] R. Kumar and S. Kapoor, Cost-benefit analysis of a reliability model for a base transceiver system considering hardware/software faults and congestion of calls, International Journal of Science and Technology,vol.4,no.6,pp.13 23,212. [4] R. Kumar and M. Kumar, Performance and cost-benefit analysis of a hardware software system considering hardware based software interaction failures and different types of recovery, International Journal of Computer Applications, vol.53,no.17, pp. 32,212. [5] B. B. Madan, K. Goševa-Popstojanova, K. Vaidyanathan, and K. S. Trivedi, A method for modeling and quantifying the security attributes of intrusion tolerant systems, Performance Evaluation,vol.56,no.1 4,pp.1 186,24. [6] H. Mine and H. Kawai, Repair priority effect on avail ability of two-unit system, IEEE Transactions on Reliability, vol.28,no. 4, pp. 3 3, [7] K. Murari and V. Goyal, Reliability system with two types of repair facilities, Microelectronics Reliability, vol. 23, no. 6, pp , [8] D. Pandey and M. Jacob, Cost analysis, availability and MTTF of a three state standby complex system under common cause and human failures, Microelectronics Reliability, vol. 35, no. 1, pp.91 95,1995. [9] B. Parashar and G. Taneja, Reliability and profit evaluation of a PLC hot standby system based on a master-slave concept and two types of repair facilities, IEEE Transactions on Reliability, vol. 56, no. 3, pp , 27. [1] S. M. Rizwan, V. Khurana, and G. Taneja, Reliability analysis of a hot standby industrial system, International Journal of Modelling and Simulation,vol.3,no.3,pp ,21. [11] B. S. Siwach, R. P. Singh, and G. Taneja, Reliability and profit evaluation of a two-unit cold standby system with instructions and accidents, Pure and Applied Mathematika Sciences,vol.53, no. 1-2, pp , 21. [12] S. K. Srinivasan and M. N. Gopalan, Probabilistic analysis of atwo-unitsystemwithawarmstandbyandasinglerepair facility, Operations Research,vol.21,pp ,1973. [13] G. Taneja, A. Goyal, and D. V. Singh, Reliability and cost-benefit analysis of a system comprising one big unit and two small identical units with priority for operation/repair to big unit, Mathematical Sciences,vol.5,no.3,pp ,211. [14] G. Taneja and V. Naveen, Comparative study of two reliability models with patience time and chances of non-availability of expert repairman, PureandAppliedMathematikaSciences,vol. 57, no. 1-2, pp , 23. [15] R. K. Tuteja, R. T. Arora, and G. Taneja, Analysis of a two-unit system with partial failures and three types of repairs, Reliability Engineering & System Safety, vol.33,no.2,pp , [16] K. S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Application, JohnWiley&Sons,New York, NY, USA, 2nd edition, 21. [17] Z.Zhag,W.Gao,Y.Zhou,andZ.Zhiqiang, Reliabilitymodeling and maintenance optimization of the Diesel system in Locomotives, Maintenance and Reliability, vol. 14, no. 4, pp , 212. [18] R. Malhotra and G. Taneja, Reliability and availability analysis of a single unit system with varying demand, Mathematical Journal of Interdisciplinary Sciences,vol.2,no.1,pp.77 88,213. [19] G. Taneja and R. Malhotra, Cost-benefit analysis of a single unit system with scheduled maintenance and variation in demand, Journal of Mathematics and Statistics,vol.9,no.3,pp , 213.

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