RELIABILITY AND AVAILABILITY ANALYSIS OF TWO DISSIMILAR UNITS BY USING LAPLACE TRANSFORMS Khaled Moh.El-said 1, Amany S. Mohamed 2, Nareman.

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1 Vol.4, No.4, pp.1-10, August 016 RELIABILITY AND AVAILABILITY ANALYSIS OF TWO DISSIMILAR UNITS BY USING LAPLACE TRANSFORMS Khaled Moh.El-said 1, Aany S. Mohaed, Narean.Mostafa 1 Departent of Math, Faculty of science, Helwan University, P.O.Box 11795,Cairo, Egypt Departent of Math Faculty of Science, Helwan University, Cairo, Egypt Deterent of Basic Science, Institute of Engineering, Canadian International College (CIC,Giza,Egypt. ABSTRACT: The paper studies the reliability and availability of two dissiilar units. In order to calculate reliability, a state dependent syste can be converted into the syste of first order ordinary differential equations based on Laplace Transfor technique. The syste of ordinary differential equations is solved using Inverse Laplace depend on Coplex Conjugate roots. Let failure rate and repair rate of each unit are taken as an exponential distribution. Availability, reliability and the ean tie to failure are derived. We analysis graphically to observe the effect of various systes Paraeters on the availability syste and ean tie to failure. KEYWORDS: Reliability, Availability, State Dependent Syste, Mttf, Laplace Transfor INTRODUCTION Studying the reliability of achine repair proble is very iportant in our life because it is widely used in the industrial syste and anufacturing syste.any syste becoes unreliable due to any reasons. The units of our syste have three states up and one down. However, in any cases, the units of the syste can have a finite nuber of states. Most reliability systes assue that the up and down ties of the coponents are exponential distribution. This assuption leads to a Monrovian odel with constant transition rates. The analysis in our cases is relatively siple and the nuerical results can obtain easily. In [1] evaluate the reliability and sensitivity analysis of a repairable -syste with iperfect coverage under service pressure condition.in [] have studied reliability based easures for a retrial syste with ixed standby coponents. In [] Reliability analysis of a war standby repairable syste with priority in use.in [4] Coparison of reliability and the availability between four systes with warstandby coponents and standby switching failures.in [5] reliability and availability of a war standby with coon cause failure and huan- error.in [6] reliability and availability analysis of n-unit outdoor power syste subject to an adjustable- repair facility. In [7] Reliability and availability analysis of a standby repairable syste with degradation facility, in [8] studies on reliability and availability of a repairable syste with ultiple degradations. In [9] Reliability and sensitivity analysis of the K-out-of-N: G war standby parallel repairable syste with replaceent at coon-cause failure using Markova odel, in [10] Reliability analysis of a two unit syste with coon cause shock failures. The ain object is to study syste with two dissiilar units where to develop the explicit expressions, for availability function, reliability function and ean tie to failure by Laplace transfor techniques then we show nuerical results to analyze the effects of the various syste paraeters on the syste reliability and syste availability. 1

2 Vol.4, No.4, pp.1-10, August 016 The objective of this paper is suarized as follows: In Section 1 we show the atheatical preliinaries and notation. Section shows the cubic equations roots and their cases, availability, reliability and ean tie to failure for every case. In section presents syste behavior through graphs. Finally, in section 4 we outline the ain conclusions. Syste Description and Assuptions The syste is analyzed under following practical assuptions: The syste consists of single unit having two dissiilar coponents, say A and B Initially, both the units are working. The syste fails copletely if during the repair of the failed unit, the another unit is also fails The failure of coponent changes the lifetie paraeter of the other After repair, the unit becoes as good as new. Table 1. Transition sates So S1 S S So λ 1 λ S1 µ 1 λ S µ λ 1 S µ µ 1 Considering these sybols, the syste ay be in one of the following states Up state: So = (A N,B N, S1 = (A F,B N, S= (A N,B F, Down state: S = (A F,B F A unit can be in one of the following states: A N B N A F B F First unit operative and in noral ode Second unit operative and in noral ode First unit failed and under repair Second unit failed and under repair Notations and States of the Syste λ 1 Failure rate fro A N to A F λ Transition rate fro B N tob F µ 1 Repair rate fro A F to A N µ Repair rate fro B F to B N p i (t Probability for i=0, 1,, p i (s Laplace transfor of p i (t

3 Vol.4, No.4, pp.1-10, August 016 A (t: availability functions of the syste. R (t: reliability functions of the syste. MTTF: ean tie to syste failure. Laplace transfor of p i (t is defined as: p i (s = e St 0 p i (t It, i=0, 1,, Matheatical forulation of the odel According to syste configuration diagra in table.1, the difference differential equations for this stochastic process which is continuous in tie and discrete in space are given as follows. dp 0 (t dt dp 1 (t dt dp (t dt dp (t dt = - [λ 1 +λ ] P 0 (t + µ 1 P 1 (t + µ P (t (1 = - [µ 1 +λ ] P 1 (t + λ 1 P 0 (t + µ P (t ( = - [µ +λ 1 ] P (t + λ P 0 (t + µ 1 P (t ( = - [µ 1 +µ ] P (t + λ 1 P (t + λ P 1 (t (4 Initial conditions: 1 where i = 0 P i (0 = { 0 otherwise Taking Laplace transfor of equations (1 (4, we get [λ 1 + λ + s] P 0 (s - µ 1 P 1 (s - µ P (s = P 0 (0 (5 [µ 1 + λ + s] P 1 (s - λ 1 P 0 (s - µ P (s = P 1 (0 (6 [λ 1 + µ + s] P (s - λ P 0 (s - µ 1 P (s = P (0 (7 [µ 1 + µ + s] P (s - λ 1 P (s - λ P 1 (s = P (0 (8 Solving equations (5-8 by craer rule, we obtain: P 0 (s = s +As +BS+ s[s +a 1 s +a S+a ] Cubic equations roots have are cases First case (D > 0 [1 root is real and coplex] P 0 (s = s +As +BS+ s(s+ A 1 W(S+ A 1 + w 1 i v 1 (S+ A 1 + w 1 + i v 1 q = a a 1 9, r = 9a 1a a 1 7a 54

4 Vol.4, No.4, pp.1-10, August 016 D = q + r u = r + q + r, t = r q + r (u + t w 1 = (u t, w= (u + t, v 1 =, A 1 = a 1 By taking inverse Laplace transfor of equations, we get P 0 (t = ( A 1 w(a 1 +A1 w+ w 1 +v 1 + ( A 1+ w +A( A 1 + w +B( A 1 + w+ ( A 1 + w(9w e ( A 1+ Wt 1 +v 1 + { [PX+HT][cos ( v 1t] [( H X (TP]( sin ( v 1 t } e ( A 1 w 1 t X +T P = (- A 1 - A 1 w A 1 v 1 - w w 1 v 1 - A 1 w 1 + A (A 1 + A 1 w - v 1 + w 1 + B ( A 1 w 1 + H = ( 6A 1 v 1 w 1 - v 1 + w 1 v 1 + A 1 v 1 - AA 1 v - A v 1 w + Bv 1 X = v w + 6v 1 A 1 T = wv 1 A 1 + w 1 v 6v 1 a =µ 1 (µ +λ + λ 1 (µ +λ + µ 1 µ (λ 1 + λ + λ 1 λ (µ 1 + µ + λ (µ 1 + λ 1 + µ (µ 1 +λ 1 a = µ 1 (µ 1 + µ + λ + λ 1 + µ (µ +λ 1 + λ + λ 1 + λ 1 λ + λ a 1 =λ 1 + λ + µ 1 + µ = µ 1 µ + µ 1 µ + µ 1 µ λ 1 + µ 1 µ λ B=µ 1 µ + µ 1 λ 1 + µ 1 λ + µ λ 1 + µ λ + λ 1 λ + µ 1 + µ A=µ 1 + µ + λ 1 + λ Syste availability and reliability Availability analysis of the Syste We find that A(t= ( A 1 w(a 1 +A1 w+ w 1 +v 1 4

5 Vol.4, No.4, pp.1-10, August ( A 1+ w +A( A 1 + w +B( A 1 + w+ e ( A 1+ Wt ( A 1 + w(9w 1 +v 1 + { [PX+HT][cos ( v 1t] [( H X (TP]( sin ( v 1 t } e ( A 1 w 1 t X +T The steady state availability can be obtained fro the following relation A=li t A(t A= ( A 1 w(a 1 +A1 w+ w 1 +v 1 Reliability analysis of the Syste To obtain the reliability function for this odel, we assue that at least one of failed states is absorbing state and the transition rate fro this state equal to zero R(t = ( A 1 w(a 1 +A1 w+ w 1 +v 1 + ( A 1+ w +A( A 1 + w +B( A 1 + w+ e ( A 1+ Wt ( A 1 + w(9w 1 +v 1 + { [PX+HT][cos ( v 1t] [( H X (TP]( sin ( v 1 t } e ( A 1 w 1 t X +T As we know, we have two failed states this lead to three cases of reliability function. 1- Failed state [coponents A] is absorbing when. µ 1 =0 - Failed state [coponents B] is absorbing when. µ =0 - Failed states [coponents A& B] are absorbing when. µ 1 = µ = 0 The ean tie to failure The ean tie to syste failure MTTF can be obtained fro the following relation MTTF = 0 R(tdt=li R(tdt t 0 t MTTF =li R(tdt = li SL { R(tdt}= li t 0 s 0 0 MTTF=li s 0 R (S, R (S= L (R(t t t S R (S s 0 S As ention above, reliability function has three cases so we find MTTF has following cases: When µ 1 =0 we find MTTF= ( ( A 1+w +A( A 1 + w+b ([PX+HT]( A 1 w 1 + 6v 1 [H X TP ] ( A 1 + w(9w 1 +v 1 (X +T [( A 1 w 1 +v 1 ] 5

6 Vol.4, No.4, pp.1-10, August 016 a =λ 1 (µ +λ + λ 1 λ µ + λ λ 1 + µ λ 1 a = µ (µ +λ 1 + λ + λ 1 + λ 1 λ + λ a 1 =λ 1 + λ + µ = 0, B=µ λ 1 + µ λ + λ 1 λ + µ, A=µ + λ 1 + λ When µ =0 we find MTTF= ( ( A 1+ W +A( A 1 + W+B ([PX+HT]( A 1 w 1 + 6v 1 [H X TP ] ( A 1 + w(9w 1 +v 1 (X +T [( A 1 w 1 +v 1 ] a =µ 1 λ + λ 1 λ + λ 1 λ µ 1 + λ (µ 1 +λ 1 a = µ 1 (µ 1 + λ + λ 1 + λ 1 + λ 1 λ + λ a 1 =λ 1 + λ + µ 1 =0, B=µ 1 λ 1 + µ 1 λ + λ 1 λ + µ 1, A= µ 1 + λ 1 + λ When µ1= µ =0 we find MTTF= ( ( A 1+ W +A( A 1 + W+B ([PX+HT]( A 1 w 1 + 6v 1 [H X TP ] ( A 1 + w(9w 1 +v 1 (X +T [( A 1 w 1 +v 1 ] a =λ 1 λ + λ λ 1, a = λ 1 + λ 1 λ + λ, a 1 = λ 1 + λ = 0, B= λ 1 λ, A= λ 1 +λ Second case D < 0 [All roots are real and unequal] P 0 (s = s +As +BS+ s(s+ A 1 w 0 (S+ A 1 w (S+ A 1 v s 1 = q cos( θ - a 1, s = q cos( θ a 1 s = q cos( θ a 1, θ = cos 1 r q w 0 = q cos( θ, w = q cos( θ + 10 v = q cos( θ + 40, A 1= a 1 By taking inverse Laplace transfor 6

7 Vol.4, No.4, pp.1-10, August 016 P 0 (t = ( A 1 w 0 (A 1 A1 w A 1 v +w v + ( A 1+ w 0 +A( A 1 + w 0 +B( A 1 + w 0 + ( A 1 + w 0 (w 0 w 0 w w 0 v +w v e ( A 1+ w 0 t + ( A 1+ w +A( A 1 + w +B( A 1 + w + ( A 1 + w (w w 0 w w v +w 0 v e ( A 1+ w t + ( A 1+ v +A( A 1 + v +B( A 1 + v + ( A 1 + v (v w 0 v w v +w 0 w e ( A 1+ v t Syste availability and reliability Availability analysis of the Syste We find that A(t = ( A 1 w 0 (A 1 A1 w A 1 v +w v + ( A 1+ w 0 +A( A 1 + w 0 +B( A 1 + w 0 + ( A 1 + w 0 (w 0 w 0 w w 0 v +w v e ( A 1+ w 0 t + ( A 1+ w +A( A 1 + w +B( A 1 + w + ( A 1 + w (w w 0 w w v +w 0 v e ( A 1+ w t + ( A 1+ v +A( A 1 + v +B( A 1 + v + ( A 1 + v (v w 0 v w v +w 0 w e ( A 1+ v t The steady state availability can be obtained fro the following relation A=li t A(t A= ( A 1 w 0 (A 1 A1 w A 1 v +w v... The ean tie to failure When µ 1 =0 we find ( A 1 + w 0 +A( A 1 + w 0 +B MTTF= ( ( ( A 1 + w 0 (w 0 w 0 w w 0 v +w v ( ( A 1 + v +A( A 1 + v +B ( A 1 + v (v w 0 v w v +w 0 w a =λ 1 (µ +λ + λ 1 λ µ + λ λ 1 + µ λ 1 a = µ (µ +λ 1 + λ + λ 1 + λ 1 λ + λ a 1 =λ 1 + λ + µ ( A 1 + w +A( A 1 + w +B ( A 1 + w (w w 0 w w v +w 0 v = 0, A= µ + λ 1 + λ, B=µ λ 1 + µ λ + λ 1 λ + µ When µ =0 we find 7

8 Vol.4, No.4, pp.1-10, August 016 ( A 1 + w 0 +A( A 1 + w 0 +B MTTF = ( ( ( A 1 + w 0 (w 0 w 0 w w 0 v +w v ( ( A 1 + v +A( A 1 + v +B ( A 1 + v (v w 0 v w v +w 0 w ( A 1 + w +A( A 1 + w +B ( A 1 + w (w w 0 w w v +w 0 v a =µ 1 λ + λ 1 λ + λ 1 λ µ 1 + λ (µ 1 +λ 1 a = µ 1 (µ 1 + λ + λ 1 + λ 1 + λ 1 λ + λ a 1 =λ 1 + λ + µ 1 =0, B=µ 1 λ 1 + µ 1 λ + λ 1 λ + µ 1, A= µ 1 + λ 1 + λ When µ1= µ =0 we find ( A 1 + w 0 +A( A 1 + w 0 +B MTTF= ( ( ( A 1 + w 0 (w 0 w 0 w w 0 v +w v ( ( A 1 + v +A( A 1 + v +B ( A 1 + v (v w 0 v w v +w 0 w ( A 1 + w +A( A 1 + w +B ( A 1 + w (w w 0 w w v +w 0 v a =λ 1 λ + λ λ 1, a = λ 1 + λ 1 λ + λ, a 1 = λ 1 + λ = 0, B= λ 1 λ, A= λ 1 +λ The syste behavior through graphs For ore the concrete study of ean tie to syste failure and availability. we plot the steady -state availability and MTTF for the odels, against λ1 keeping other paraeters λ = 0., µ =0.7, µ 1 =0.5 λ 1 = 0.1, 0., 0., 0.4, 0.5 8

9 Vol.4, No.4, pp.1-10, August 016 Fig 1 The Steady state Availability w.r.t. Failure Rate λ1 Fig The ean tie failure w.r.t. Failure Rate λ1 9

10 Vol.4, No.4, pp.1-10, August 016 CONCLUSIONS We use coputer software, to plot syste availability and MTTF in fig. 1 and respectively. It is noted that A decrease as λ increases and also MTTF decrease as λ increases. REFERENCES [1] Kuo-Hsiung Wanga, Tseng-Chang Yenb, Jen-Ju Jian., Reliability and Sensitivity analysis of a repairable syste with iperfect coverage under service pressure condition, Journal of Manufacturing Systes ( [] Ching-Chang Kuo, Shey-Huei Sheu, Jau-Chuan Ke, Zhe George Zhang., Reliabilitybased easures for a retrial syste with ixed standby coponents, Applied Matheatical Modelling. ( X. [] Li Yuan, Xian-Yun Meng., Reliability analysis of a war standby repairable a syste with priority in use. Applied Matheatical Modeling 5 ( [4] Kuo-Hsiung Wang, Wen-Li Dong, Jyh-Bin Ke., Coparison of reliability and the availability between four systes with war standby coponents and standby switching failures. Applied Matheatics and Coputation 18 ( [5] B.S. DHILLON and N. YANG., reliability and availability of a war standby with coon-cause failure and huan error. Microelectronic. Reliable (4( [6] J. NATESAN, K. THULASIRAMAN and M. N. S. SWAMV., ] reliability and availability analysis of n-unit outdoor power syste subject to the adjustable repair facility. Microelectronic. Reliable. 4( 6 (1984 I [7] M.A. El-Dances, M.S. Shaa., Reliability and availability analysis of a standby a repairable syste with degradation facility, IJRRAS 16 ( (01. [8] M. A. El-Dances, M.S. Shaa., studies on reliability and availability of a repairable syste with ultiple degradations. IJRRAS 5 ( (015 [9 M. A. El-Dances and N. H. El-Sodany, Reliability and Sensitivity Analysis of the k-out of-n: swar Standby Parallel Repairable Syste with Replaceent at Coon-Cause Failure using Markov Model. RT.A 04(9 (015. [10] M. JAIN, Reliability analysis of a two unit syste with coon cause shock failures, Indian J. pure apply. Math. 9(1 (