Improving the Reliability of a Series-Parallel System Using Modified Weibull Distribution

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1 Internationa Mathematica Forum, Vo. 12, 217, no. 6, HIKARI Ltd, Improving the Reiabiity of a Series-Parae System Using Modified Weibu Distribution Abdefattah Mustafa Department of Mathematics, Facuty of Science Mansoura University, Mansoura, Egypt Copyright c 217 Abdefattah Mustafa. This artice is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. Abstract System reiabiity is a main criterion to design systems. Among various theories, which are appied to improve system reiabiity. The reiabiity equivaence factors theory is known as a powerfu too as improvement methods wi be appied to enhance system reiabiity. In this artice, a series-parae system is improved by using the reiabiity equivaence factors. A components have modified Weibu ifetime distribution. Three different methods; reduction, hot and cod are used to improve the given system. The reiabiity function and mean time to faiure for each method are obtained. The reiabiity equivaence factors and δ-fracties are introduced. Numerica exampes are studied. Mathematics Subject Cassification: 6E5, 62N5, 9B25 Keywords: Modified Weibu distribution, improving methods, reiabiity equivaence 1 Introduction The reiabiity equivaence is discussed in [13]. Various systems improved by appying such concept. Some authors improved the constant faiure rate systems such as series, parae, series-parae (parae-series) and bridge structure systems, [9, 1, 11, 14]. Aso, there are some systems have non-constant faiure rate are improved in [1, 3, 4, 5, 6, 7, 8, 15, 16].

2 258 Abdefattah Mustafa The reiabiity equivaence factors (REF) of a parae system with n independent and identica components with Gamma ife time distribution introduced in [16]. A series system with non-constant faiure rates studied in [7]. A simpe series-parae system with inear-exponentia distribution improved in [4]. The bridge structure system with modified Webiu distribution discussed in [5]. We focus on the equivaence anaysis of different designs of a series-parae system with independent and identica components with modified Weibu distribution. The random variabe T has modified Weibu distribution (MWD) with parameters α, β and µ if its cumuative distribution function is given by F (t; α, β, µ) = 1 exp{ (αt + βt µ )}, t, (1) where α, β, µ >. The reiabiity function for MWD(α, β, µ) can be obtained as R(t; α, β, µ) = 1 F (t) = exp{ (αt + βt µ )}, t. (2) The faiure rate function is h(t) = f(t) R(t) = α + βµµ 1, t. (3) Figures 1 and 2 show the cumuative distribution, reiabiity and hazard rate function respectivey, for α =.2, β =.3 and different vaues of shape parameter µ. Figure 1: The F (t) and R(t) for α =.2, β =.3 and different vaues of µ. The MWD(α, β, µ) has a bathtub faiure rate. 1. If the shape parameter µ = 1, then h(t) = α, is constant faiure rate. 2. If the shape parameter µ > 1, then h(t), is increasing (IFR). 3. If the shape parameter µ < 1, then h(t), is decreasing (DFR).

3 Improving the reiabiity of a series-parae system 259 Figure 2: The h(t) for α =.2, β =.3 and different vaues of µ. This paper can be organized as foows. In Section 2, we discuss the seriesparae system and derive its reiabiity function (RF) and mean time to faiure (MTTF). Section 3 introduces the improved systems. The δ-fracties are obtained in Section 4. The REFs of the series-parae system are derived in Section 5. Some numerica resuts are given in Sections 6. Finay, the concusion is introduced in Section 7. 2 Series-parae System The structure of the series-parae system is iustrated in Figure 3. The origina system consists of n subsystems, and each subsystem i has m i components, for i = 1, 2,, n. Figure 3: The Series-parae structure. Let R s (t) be the reiabiity function of the series-parae system, R i (t) be the reiabiity function of the system i. The function R s (t) can be derived as foows n n R S (t) = R i = {1 [ 1 e ] } (αt+βtµ ) m i. (4) The MTTF can be given by, (see [2]). MTTF = R s (t)dt. (5) Equation (5) can be computed numericay by using Mathematica Program system.

4 26 Abdefattah Mustafa 3 The Improved Systems In this section, three different methods can be used to improve the seriesparae system. We wi compare the reiabiity of the origina system and that for the improved systems. 3.1 The reduction method In this method, the system can be improved by reducing the faiure rates of A of the system components by the factor ρ, < ρ < 1. We assume that the A = r components can be distributed into n subsystems. Such that r = n r i, r i denotes the number of components from the subsystem i whose faiure rates are reduced. Where r i m i, r N and N = n m i. Let R (r 1,r 2,,r n) r,ρ (t) denote the RF of the system improved by reducing the faiure rates of the set A components. The function R (r 1,r 2,,r n) r,ρ (t) can be obtained as foows. R (r 1,r 2,,r n) r,ρ (t) = n {1 [ 1 e ρ(αt+βtµ ) ] r i [ 1 e (αt+βt µ ) ] m i r i }. (6) The MTTF to the reduction system can be evauated by sove the foowing integra numericay MTTF r,ρ = 3.2 The hot dupication method R (r 1,r 2,,r n) r,ρ (t)(t)dt. (7) Let us assume that the system improved by improving the components of the set B by a hot redundant identica standby component. The set B consists of components ( N), can be distributed into the n subsystems of the system such that = n. We wi improve i components from the subsystem i by hot dupication method, that is i m i and N. The function R H( 1, 2,, n) (t) denotes for the RF of the improved system by hot dupication method. We can derive the function, R H( 1, 2,, n) (t) as foows R H( 1, 2,, n) (t) = n {1 [ 1 e (αt+βtµ ) ] m i + i }. (8) The hot dupication mean time to faiure can be computed by MTTF H = R H( 1, 2,, n) (t)dt. (9)

5 Improving the reiabiity of a series-parae system The cod dupication method In this method it is assumed that each component of the set B is connected with an identica component via a perfect switch. Let R C( 1, 2,, n) (t) denote the RF of the system improved by improving = B components according to the cod dupication method. The function R C( 1, 2,, n) (t) is given as foows R C( 1, 2,, n) (t) = n {1 [ 1 R Cij(t) ] i [1 Rij (t)] m i i }, (1) where Rij(t) C denotes the RF of the improved component j in subsystem i according to cod dupication method. R ij (t) = e (αt+βtµ), t Rij(t) C = R 1 (t) + f 1 (x)r 2 (t x)dx (11) t = e (αt+βtµ) + e αt (α + βµx µ 1 )e β[xµ +(t x) µ] dx. If we setting µ = 2 in (11), we get Rij(t) C = e (αt+βt2) + e (αt+βt2) e βt2 2 1 e βt2 2 [ ] π (α + βt) 2β + 1 e βt2 2. The MTTF for the cod dupication system can be determined as foows MTTF C = R C B(t)dt. (12) Remark 1. The cod redundant system having a singe main component and one standby component via a perfect switch. The component in a standby mode is ready to take over system operation when the main or normay operating component fais. 1. When the system has identica components with constant faiure rate, the Poisson distribution is used for the standby system operation P x (t) = (λt)x e λt, x =, 1, x! where P x (t) be the probabiity that x faiure occurs in interva (, t). The reiabiity for this system can be derived by using R(t) = P (t) + P 1 (t) = (1 + λt)e λt.

6 262 Abdefattah Mustafa 2. When the components have non-constant faiure rates or non-identica, we haven t Poisson distribution, so in this case, there are many techniques can be used to evauate the system reiabiity, (see [2]). Consider the standby system with two non-identica components A and B. The events eading to the system success are either: The main component, A, does not fai for an interva of time to t, or the main component A fais at time x < t and standby component B does not fai in the interva x to t. Let R a (t) and R b (t) be the reiabiity associated with these two events respectivey, then R(t) = R a (t) + t f a (x)r b (t x)dx This technique is avaiabe aso when we have identica components. Many authors are used the technique in (1) for the non-constant faiure rate, see [1, 3, 4, 5, 15]. This mistake in evauation the system reiabiity for cod standby method. Technique in (2) used by [6, 9, 1, 12]. 4 The δ-fracties In this section, the important measure of the performance of system reiabiity is obtained. The δ-fracties of the origina and improved systems, L(δ), L D (δ), respectivey, can be found by soving the foowing equations with respect to L: ( ) ( ) L(δ) R = δ, R D,( 1, 2,, n) L(δ) = δ, (13) Λ Λ where Λ = n(α + β) and D = H, C. Substituting from (4)into the first equation in (13), L(δ) satisfied the foowing equation { n 1 [1 e (α L Λ +β( L Λ )µ ) ] m i } n (δ) =, (14) From equation (8) and the second equation in (13), L = L H (δ) is obtained by sove the foowing noninear equation with respect to L { [ ] } n 1 1 e (α L Λ +β( L mi +h i Λ )µ ) n (δ) =. (15)

7 Improving the reiabiity of a series-parae system 263 Simiary, substituting from (1) into (13), L = L C (δ) satisfies the foowing equation { n 1 [ 1 R C ij ( )] ci [ ( )] } mi c L L i 1 R ij n (δ) =. (16) Λ Λ As it seems, equations (14) to (16) can be soved numericay. 5 The Reiabiity Equivaence Factors The REFs of the series- parae system are derived in this section, say ρ D r, (δ). The set B ( B = ) of system components is improved according to one of the dupication methods (H and C) and A ( A = r) is the set of system components improved according to a reduction method. The ρ D r, (δ), is the soution of the foowing system of two equations,see [14] R D ( 1, 2,, n) (t) = δ, R (r 1,r 2,,r n) r,ρ (t) = δ, D = H and C. (17) Substituting from (6) and (8) into (17), the hot REF, say ρ = ρ H r, (δ) is the soution of the foowing equations: n {1 [ 1 e ] ρ(αt+βtµ ) r i [ ] } 1 e (αt+βt µ ) m i r i n {1 [ 1 e ] } (αt+βtµ ) m i +h i n (δ) =. (18) n (δ) = Simiary, from equations (8), (15) and (17), the cod REF, say ρ = ρ C r, (δ) is given by the foowing equations: n {1 [ 1 e ] ρ(αt+βtµ ) r i [ ] } 1 e (αt+βt µ ) m i r i n (δ) = n { 1 [ 1 Rij(t) ] C c i [1 Rij (t)] m i c i } n (δ) = (19) The systems (18) and (19) have no cosed form soutions. The ρ = ρ D r, (δ), where D = H and C, respectivey, can be obtained by using Mathematica Program System.

8 264 Abdefattah Mustafa 6 Numerica Resuts and Concusions In this section, we introduce a numerica exampe to expain how one can utiize the previousy obtained theoretica resuts. Consider the foowing assumptions: 1. The series-parae system has two subsystems, n = 2, connected in series. 2. The first subsystem has ony one component, m 1 = 1, the second subsystem has two components, m 2 = 2, connected in parae and N = n 1 + n 2 = The parameters α =.1, β =.2 and µ = In the hot and cod dupication method, we improve = B components of the system components. Such that 1 component from the first subsystem, 2 from the second subsystem. We wi use the notation B ( 1, 2 ). 5. In the reduction method, we wi use the notation A (r 1,r 2 ) r. We improve r = A components of the system components. Such that r 1 component from the first subsystem, r 2 from the second subsystem. For this exampe, we have found that the MTTF= for the origina system, and the MTTF D for the improved systems are presented in Tabe 1. Tabe 1: The MTTF D, D = H, C. B (1,) 1 B (,1) 1 B (,2) 2 B (1,1) 2 B (1,2) 3 MTTF H MTTF C Figures 4 6 show the reiabiity for the improved and origina systems, when B = 1, 2, 3. (a) B (1,) 1 (b) B (,1) 1 Figure 4: The RFs for the origina and improved system, B = 1.

9 Improving the reiabiity of a series-parae system 265 (a) B (,2) 2 (b) B (1,1) 2 Figure 5: The RFs for the origina and improved system, B = 2. Figure 6: The RFs for the origina and improved system, B = 3. Figure 7 shows a comparison of the reiabiity for the improved methods with the origina system for different sets of system components, D = H, C. Figure 7: The RFs for the origina and improved system. Figure 8 gives the reiabiity for the improved system by using the reduction method with the origina system when ρ =.5. Figure 8: The RFs for the origina and improved system by reduction method, ρ =.5.

10 266 Abdefattah Mustafa Using Mathematica Program system, one can cacuate the δ-fracties L(δ), L D (δ) and the REFs ρ D r, (δ), D = H, C. Tabe 2 contains the δ-fracties of the origina and improved systems. Tabe 3 shows the REFs of the improved systems. Tabe 2: The δ-fracties, L(δ), L D (δ), D = H, C, B(1,2) and = B. B (1,) 1 B (,1) 1 B (,2) 2 B (1,1) 2 B (1,2) 3 δ L(δ) H C H C H C H C H C Tabe 3: The REF, ρ D r, (δ), D = H and C, r = A = r 1 + r 2 and = B. B (1,) 1 B (,1) 1 B (,2) 2 B (1,1) 2 B (1,2) 3 δ (r 1, r 2 ) H C H C H C H C H C.1 (1,) NE.421 NE (,1) NE.1364 NE (,2).5983 NE NE NE.311 NE (1,1) (1,2) (1,) NE NE.3361 NE (,1).2583 NE NE.286 NE (,2).4657 NE NA NE NE NE (1,1) NE.7751 NE (1,2) NE.8197 NE (1,) NE NE.2739 NE (,1).1194 NE NE NE NE NE (,2).364 NE NE NE NE NE (1,1) NE.7632 NE (1,2) NE.881 NE (1,) NE.6681 NE.3153 NE.222 NE (,1).277 NE NE NE NE NE (,2). NE NE NE NE NE (1,1) NE.7596 NE (1,2) NE.829 NE (1,) NE.6847 NE.2664 NE.1759 NE (,1) NE NE NE NE NE NE (,2) NE NE NE NE NE NE (1,1) NE NE NE (1,2) NE NE NE (1,) NE.7151 NE.2192 NE.1337 NE (,1) NE NE NE NE NE NE (,2) NE NE NE NE NE NE (1,1) NE NE NE NE (1,2) NE NE NE (1,) NE.7615 NE.1718 NE.943 NE (,1) NE NE NE NE NE NE (,2) NE NE NE NE NE NE (1,1) NE NE NE NE (1,2) NE NE NE NE (1,) NE.8252 NE.1225 NE.573 NE (,1) NE NE NE NE NE NE (,2) NE NE NE NE NE NE

11 Improving the reiabiity of a series-parae system 267 (1,1) NE NE NE NE (1,2) NE NE NE NE (1,) NE.956 NE.684 NE.236 NE (,1) NE NE.946 NE NE NE NE NE (,2) NE NE.337 NE.944 NE NE NE NE NE (1,1) NE NE NE NE (1,2) NE NE NE NE NE From Tabes 1, 2 and Figures 4 8, it seems that: 1. MT T F < MT T F H < MT T F C, for a. 2. L(δ) < L H (δ) < LC (δ), = 1, 2 and Improving B (1,) 1 according to the dupication methods produces a better design than what can be designed by improving B (,1) 1 by the same methods. 4. Improving B (1,1) 2 using the dupication methods one gets a better design than what can be designed by improving B (,2) Improving B (1,2) 3 by the dupication methods gives the best design. 6. The cod dupication method is much better than the hot dupication method. Based on the resuts presented in Tabes 2 and 3: 1. The L(.1) wi increase from 2.279/Λ to 2.589/Λ when the system improved by hot dupication of B (1,) 1, see Tabe 2. The same resut on L(.1) can occur by reducing the faiure rates of (i) one component from first subsystem, A (1,) 1 by the factor ρ H =.6564, (ii) one component from second subsystem, A (,1) 1, by the factor ρ H =.475, (iii) two components from second subsystem, A (,2) 2, by the factor ρ H =.5983, (iv) one component from first and second subsystem, A (1,1) 2, by the factor ρ H =.7713, (v) one component from first subsystem and two components from the second subsystem, A (1,2) 3, by the factor ρ H =.8179, see Tabe Cod dupication of B (1,) 1, wi increase L(.1) from 2.279/Λ to 3.16/Λ, see Tabe 2. The same effect on L(.1) can obtain by reducing the faiure rates of (i) one component from first subsystem, A (1,) 1 by the factor ρ C =.2532, (ii) one components from second subsystem, A (,1) 1, by the factor ρ C =.23, (iii) one component from first and second subsystem, A (1,1) 2, by the factor ρ C =.5353, (iv) one component from first subsystem and two components from the second subsystem, A (1,2) 2, by the factor ρ C =.61, see Tabe 3.

12 268 Abdefattah Mustafa 3. In the same manner, one can read the rest of resuts presented in Tabes 2 and The notation NE, means that there is no equivaence between the two improved systems obtained by reduction method and dupication methods. 7 Concusion In this artice, the REFs of series-parae system are derived. Three methods, reduction method, hot and cod dupication method are used to improve the system reiabiity. It is shown that the improved systems is higher reiabiity than the origina system in a studied cases. It s found that if reduction method is appied to improve system reiabiity using different items, a combination of one items from first subsystem and two component from second subsystem is the best option for reiabiity improvement. Aso, it s concuded that when a singe set of items from subsystem 2 is improved using the reduction method, the smaer reduction factor appied for reiabiity improvement. Moreover, the best method to improve the system reiabiity is the cod dupication method. Many specia cases can be derived from the modified Weibu distribution. References [1] Y. H. Abdekader, A. I. Shawky and M. I. A-Ohay, Reiabiity equivaence of independent non-identica parae and series systems, Life Science Journa, 9 (212), no. 3, [2] R. Biinton and R. Aan, Reiabiity Evauation of Engineering Systems: Concepts and Techniques, Penum Press, New York, [3] M. A. E-Damcese, Reiabiity equivaence factors of a series-parae system in Weibu distribution, Internationa Mathematica Forum, 4 (29), no. 19, [4] G. Ezzati and A. Rasoui, Evauating system reiabiity using inearexponentia distribution function, Internationa Journa of Advanced Statistics and Probabiity, 3 (215), no. 1, [5] A. H. Khan and T. R. Jan, Reiabiity evauation of engineering system using modified Weibu distribution, Research Journa of Mathematica and Statistica Sciences, 3 (215), no. 7, 1-8.

13 Improving the reiabiity of a series-parae system 269 [6] H. S. Migdadi and M. S. A-Batah, Testing reiabiity equivaence factors of a series-parae systems in Burr type X distribution, British Journa of Mathematics & Computer Science, 4 (214), no. 18, [7] A. Mustafa, Reiabiity equivaence factor of n- components series system with non-constant faiure rates, Internationa Journa of Reiabiity and Appications, 1 (29), no. 1, [8] A. Mustafa and A. H. E-Bassoiuny, Reiabiity equivaence of some systems with mixture inear increasing faiure rates, Pakistan Journa of Statistics, 25 (29), no. 2, [9] A. Mustafa and A. A. E-Faheem, Reiabiity equivaence factors of a genera parae system with mixture of ifetimes, Appied Mathematica Sciences, 6 (212), no. 76, [1] A. Mustafa and A. A. E-Faheem, Reiabiity equivaence factors of a system with mixture of n independent and non-identica ifetimes with deayed time, Journa of the Egyptian Mathematica Society, 22 (214), [11] A. Mustafa, B. S. E-Desouky and M. E-Dawoody, Reiabiity equivaence factors of non-identica components series system with mixture faiure rates, Internationa Journa of Reiabiity and Appications, 1 (29), no. 1, [12] A. Mustafa, B. S. E-Desouky and A. Taha, Improving the performance of the series-parae system with inear exponentia distribution, Internationa Mathematica Forum, 11 (216), no. 21, [13] L. Råde, Reiabiity surviva equivaence, Microeectronics Reiabiity, 33 (1993), [14] A. M. Sarhan and A. Mustafa, Reiabiity equivaence of a series system consists of n independent and non-identica components, Internationa Journa of Reiabiity and Appications, 7 (26), no. 2, [15] A. I. Shawky, Y. H. Abdekader and M. I. A-Ohay, Reiabiity equivaence factors in exponentiated exponentia distribution, Wufenia Journa, 2 (213), no. 3, [16] Y. Xia and G. Zhang, Reiabiity equivaence factors in gamma distribution, Appied Mathematics and Computation, 187 (27), Received: December 7, 216; Pubished: January 31, 217