Improving The Performance of the Series Parallel System ... · 1038 Abdelfattah Mustafa, Beih S. El-Desouky and Ahmed Taha Keywords: Reliability equivalence, Series-parallel system,

International Mathematical Forum, Vol. 11, 2016, no. 21, 1037 - 1052HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/imf.2016.67107

Improving The Performance of the Series Parallel

System with Linear Exponential Distribution

Abdelfattah Mustafa1

Department of Mathematics, Faculty of ScienceMansoura University, Mansoura 35516, Egypt

Beih S. El-Desouky


Ahmed Taha


Copyright c© 2016 Abdelfattah Mustafa, Beih S. El-Desouky and Ahmed Taha. This

article is distributed under the Creative Commons Attribution License, which permits un-

restricted use, distribution, and reproduction in any medium, provided the original work is

properly cited.

Abstract

In this paper we study a series-parallel system consists of n blocksconnected in series. Each block consists of mi components connectedin parallel such that 1 ≤ i ≤ n. All components are independent andnon-identically distributed and have a failure rate as a function of time.All components are distributed as linear exponential distribution. Wediscuss four methods to improve the performance of the original systemreliability. We derive the equivalence factors of the original system to beas an improved system according to these methods. We discuss specialcases of this system. Finally, numerical results are presented.

Mathematics Subject Classification: 60E05, 62N05, 90B25

1Corresponding author

1038 Abdelfattah Mustafa, Beih S. El-Desouky and Ahmed Taha

Keywords: Reliability equivalence, Series-parallel system, Improving meth-ods, Reliability engineering

1 Introduction

In our life, series-parallel structure systems are found in many applications andthen we shall study and improve their performance. Many papers studied re-liability and equivalence factor for simple and complex systems with constantfailure rate and changeable failure rate as a function in time.

Alghamdi and Percy [13], improved a system of a series parallel systemdistributed as exponentiated weibull life time distribution. Sarhan [10], im-proved another system of components connected in series and each componentwas distributed as exponential distribution. Rade [7, 8, 9] Sarhan and Mustafa[11], Mustafa and El-Bassoiuny [3] and Mustafa and El-Faheem [5] improvedother various systems by applying the concept of reliability equivalence factor.Mustafa [2], applied the reliability equivalence factor techniques to a systemconsists of n independent and non-identical components connected in serieswith mixed constant failure rates. Mustafa and El-Faheem, [4], generalizedreliability equivalence factor technique and applied it to a system consists ofm independent and non-identical lifetimes distributions with mixed failure anddelayed lifetimes rates. Xia and Zhang [12] applied the reliability equivalencefactor of a parallel system with n independent and identical components dis-tributed as Gamma life time distribution. Mustafa [2] derived the reliabilityequivalence factor for a series system with non-constant failure rates. Ezzatiand Rasouli [1] studied the Radar system with linear-exponential distributionwith non-identical components.

In this paper, we study the series-parallel system with failure rate as afunction of time such that each component is distributed as linear exponentialdistribution and improve it according to four different methods:

1. Reduction method. In this method it is assumed that the componentcan be improved by reducing its failure rate of the systems componentsby the factor ρ, 0 < ρ < 1.

2. Hot duplication method. This method assumed that the component isduplicated by another standby one connected on parallel.

3. Cold duplication method. It is assumed in this method that the compo-nent is duplicated by another standby one with a perfect switch.

Improving the performance of the series-parallel system 1039

4. Cold duplication with imperfect switch method. It is assumed in thismethod that the component is duplicated by another standby one withan imperfect switch.

Reliability equivalence factor is a factor by which a characteristic systemdesign of components has to be multiplied to equal a characteristic of this de-sign and a different design.

In Section 2, the original system is introduced and its reliability functionis calculated. In Section 3, the original system is improved according to re-duction, hot, cold and cold with imperfect switch methods. In Section 4, theequivalence factor is calculated for the original system and improved systems.The δ-fractiles for the original system and improved systems are introduced insection 5. Finally, numerical example and numerical results are presented insection 6.

2 The Original System

Suppose a system consists of n blocks of components connected in series. Blocki consists of mi components connected in parallel and all components are in-dependent and non-identically distributed, see Figure 1.

Figure 1: Series-parallel system Structure.

Suppose that all components are distributed as linear exponential distri-bution. Let Rij(t) denote the reliability function of component j in block i(j = 1, · · · ,mi, i = 1, · · · , n), Ri(t) denote the reliability function of the blocki and R(t) denote the reliability function of the whole system. It is obviousthat:

R(t) =n∏i=1

Ri(t),

where


Ri(t) = 1−mi∏j=1

[1−Rij(t)] ,

then

R(t) =n∏i=1

1−mi∏j=1

[1− exp{−(aijt+

1

2bijt

2)}] . (1)

The mean time to failure (MTTF), is given as

MTTF =∫ ∞

0R(t)dt. (2)

3 Improved Methods

In this section, we improve the performance of the series-parallel system byimproving some systems components according to four methods:

1. Reduction method. In this method we improve the system performanceby reducing the failure rate of the components belong to the set A of thesystems components by the factor ρ, 0 < ρ < 1.

2. Hot duplication method. In this method we improve the system perfor-mance by adding hot component to some components belong to the setB of the systems component.

3. Cold duplication method. In this method we improve the system per-formance by adding cold component connected by a perfect switch withthe same component belong to the set B of the systems components.

4. Cold duplication with imperfect switch method. In this method we im-prove the system performance by adding cold component connected byan imperfect switch with the same component belong to the set B of thesystems components.

3.1 Reduction method

In this method, the system is improved by improving the set A of the systemcomponents, such that |A| = r, 0 < r ≤ N , N =

∑ni=1mi. We reduced the

failure rates of the components belong to the set A by the factor ρ, 0 < ρ < 1.In each subsystem we reduce the failure rate for some of its components saythe set Ai, where |Ai| = ri, 0 < ri ≤ mi and

⋃ni=1Ai = A,

∑ni=1 ri = r.

Let Rij,ρ(t) be the reliability function of the component j in subsystem i,that improved by reduction method. Let RAi,ρ(t) be the reliability function of


the subsystem i, when the set of its components, Ai are improved by reductionmethod.

Rij,ρ(t) = exp{−ρ(aijt+1

2bijt

2)},

and

RAi,ρ(t) = 1−∏j∈Ai

[1−Rij(t)]∏j∈Ai

[1−Rij,ρ(t)]

= 1−∏j∈Ai

[1− e−(aijt+

12bijt

2)] ∏j∈Ai

[1− e−ρ(aijt+

12bijt

2)].

Then the reliability function of the improved system using the reductionmethod, R

(A1,A2,···,An)A,ρ (t) is given by

R(A1,A2,···,An)A,ρ (t) =

n∏i=1

1−∏j∈Ai

[1− e−(aijt+

12bijt

2)] ∏j∈Ai

[1− e−ρ(aijt+

12bijt

2)] .

(3)The MTTF, is

MTTF(A1,A2,···,An)A,ρ =

∫ ∞0

R(A1,A2,···,An)A,ρ (t)dt. (4)

3.2 Hot duplication method

In this method, the system is improved by improving the set B of the systemcomponents, such that |B| = h, 0 < h ≤ N , N =

∑ni=1mi. In each sub-

system we improve some of its components say the set Bi, where |Bi| = hi,0 < hi ≤ mi, and

⋃ni=1Bi = B,

∑ni=1 hi = h.

Let RHij (t) be the reliability function of the component j in subsystem i,

that improved by hot duplication method. RHi (t) be the reliability function

of the subsystem i, when the set of its components, Bi are improved by hotduplication method.

RHij (t) = 1− [1−Rij(t)]

2 = 1−[1− e−(aijt+

12bijt

2)]2

andRHi (t) = 1−

∏j∈Bi

[1− e−(aijt+

12bijt

2)] ∏j∈Bi

[1− e−(aijt+

12bijt

2)]2.

The reliability function of the improved system by using the hot duplicationmethod is obtained as follows

RH,(B1,B2,···,Bn)B (t) =

n∏i=1

1−∏j∈Bi

[1− e−(aijt+

12bijt

2)] ∏j∈Bi

[1− e−(aijt+

12bijt

2)]2 .

(5)


The MTTF, is

MTTFH,(B1,B2,···,Bn)B =

∫ ∞0

RH,(B1,B2,···,Bn)B (t)dt. (6)

3.3 Cold duplication with perfect switch method

In this method, the system is improved by improving the set B of the systemcomponents, such that |B| = c, 0 < c ≤ N . In each subsystem we improvesome of its components say the set Bi, where |Bi| = ci, 0 < ci ≤ mi, and⋃ni=1Bi = B,

∑ni=1 ci = c. If any component of these components improved

according to this method fails, the switch will convert to the standby compo-nent with probability 1.

Let RCij(t) be the reliability function of the component j in subsystem i,

that improved by cold duplication method. RCi (t) be the reliability function

of the subsystem i, when the set of its components, Bi are improved by coldduplication method.

RCij(t) =

[1 +

∞∑`=0

∑k=0

(−1)`+kb`ij`!

(`

k

)(aij

2`− k + 1+

bijt

2`− k + 2

)t2`+1

]e−(aijt+

12bijt

2),

andRCi (t) = 1−

∏j∈Bi

[1−Rij(t)]∏j∈Bi

[1−RC

ij(t)].

The reliability function of the improved system by using the cold duplica-tion method is obtained as follows.

RC,(B1,B2,···,Bn)B (t) =

n∏i=1

{1−

∏j∈Bi

[1− e−(aijt+

12bijt

2)] ∏j∈Bi

[1−

([1 +

∞∑`=0

∑k=0

(−1)`+kb`ij`!

(`

k

)(aij

2`− k + 1+

bijt

2`− k + 2

)t2`+1

]

e−(aijt+12bijt

2))]}

. (7)

The MTTF, is

MTTFCB =

∫ ∞0

RC,(B1,B2,···,Bn)B (t)dt. (8)

3.4 Cold duplication with imperfect switch method

In this method, the system is improved by improving the set B of the systemcomponents, such that |B| = p, 0 < p ≤ N . In each subsystem we improve


some of its components say the set Bi, where |Bi| = pi, 0 < pi ≤ mi, and⋃ni=1Bi = B,

∑ni=1 pi = p and let B, |B| = n − p, refers to the other not

improved components. If any component of these components improved ac-cording to this method fails, the switch will convert to the standby componentwith probability less than 1.

Let RIij(t) be the reliability function of the component j in subsystem i, that

improved by imperfect duplication method. RIi (t) be the reliability function

of the subsystem i, when the set of its components, Bi are improved by coldduplication method with imperfect switch. In this method, the switches havelinear failure rate distribution with parameters αij and βij.

RIij(t) =

[1 +

∞∑`=0

∑k=0

(`

k

)(−1)`(αij − bijt)k(βij + 2bij)

`−k

2`−k`!

(aij

2`− k + 1+

bijt

2`− k + 2

)t2`−k+1

]e−(aijt+

12bijt

2),

andRIi (t) = 1−

∏j∈Bi

[1−Rij(t)]∏j∈Bi

[1−RI

ij(t)].

The reliability function of the improved system by using the cold duplica-tion method is obtained as follows.

RI,(B1,B2,···,Bn)B (t) =

n∏i=1

{1−

∏j∈Bi

[1− e−(aijt+

12bijt

2)] ∏j∈Bi

[1−

(1 +

∞∑`=0

∑k=0

(`

k

)(−1)`

`!

(αij − bijt)k(βij + 2bij)`−k

2`−k×(

aij2`− k + 1

+bijt

2`− k + 2

)t2`−k+1

)e−(aijt+

12bijt

2)]}.(9)

The MTTF is

MTTF IB =

∫ ∞0

RI,(B1,B2,···,Bn)B (t)dt. (10)

4 Reliability Equivalence Factor

The reliability equivalence factor, ρD(α), D = H,C and I, is the factor bywhich the failure rates of the original system components should be reducedto equal the failure rates of the improved system according to hot (cold withperfect switch, cold with imperfect switch) duplication method. Then, ρD(α)is the solution of two equations:


RD,(B1,B2,···,Bn)B (t) = β, R

(A1,A2,···,An)A,ρ (t) = β, D = H(C, I). (11)

This system of equations has no closed form of solution, so we must usesome Numerical Techniques to solve this system.

5 δ-Fractiles

In this section we introduced the δ-fractiles of the original system and im-proved systems. Let L(δ) be the δ-fractile of the original system and LDB(δ)denotes the δ-fractile of the improved system obtained by improving the set Bof components according to duplication methods.

The fractile L(δ) of the original system can be found by solving the followingequation with respect to L:

R(L

Λ

)= δ, (12)

where Λ =∑ni=1

∑mij=1(aij + bij).

Substituting from Eq. (1) into Eq. (12), we have

n∏i=1

{1−

mi∏j=1

[1− e−(aij

LΛ

+ 12bij

LΛ

2)]}

= δ. (13)

The fractile LDB(β) can be found by solving the following equation withrespect to L:

LD,(B1,B2,···,Bn)B

(L

Λ

)= δ. (14)

Substituting from Eq. (5), (7) and (9) into Eq. (14) we can obtainLD(δ), D = H,C and I of the hot (cold and cold with imperfect switch),δ-fractiles which have not closed solution in L and we have to use the numer-ical technique methods.

6 Numerical Results

In this section we study the radar system as an example for a series-parallelsystem which consists of two subsystems (i = 1, 2) such that the first subsystemconsists of one component (m1 = 1) and the second subsystem consists oftwo components connected on parallel (m2 = 2). The first and the secondsubsystems are connected on series (n = 2) as shown in Figure 2, see [1].


Figure 2: The radar system structure.

We study the Radar system, shown in Figure 2 under these assumptions:

1. The parameters for subsystem 1 are a11 = 0.09 and b11 = 0.1.

2. The parameters for subsystem 2 are a21 = 0.1, b21 = 0.19, a22 = 0.12 andb22 = 0.21.

3. The parameters for the switches are αij = 0.08, βij = 0.07, for i ≥ j =1, 2.

4. The set B of system components are improved according to one of theprevious duplication methods, where B = B1∪B2 to improve the systemreliability, Bi the set of system components that improved from subsys-tem i, i = 1, 2.

5. In the reduction method, we reduce the failure rates of the set A =A1 ∪ A2, by the same factor ρ, Ai the set of system components thatimproved by reduction method from subsystem i, i = 1, 2.

For this example, the mean time to failure for the Radar system (originalsystem) is 2.2103. Table 1, contains the MTTFD

B for the improved systems.

Table 1: The MTTFDB , B = B1 ∪B2.

B1 B2 D = H D = I D = C{1} Φ 2.6924 2.7870 2.9377Φ {1} 2.4396 2.6162 2.7660{1} {1} 3.0232 3.5035 4.2509Φ {1,2} 2.5528 2.7914 2.9401{1} {1,2} 3.1928 3.8414 4.7510

Figures 3–5 show the comparing the reliability for the original and improved systemsfor different sets of components

Figure 3: The reliability function, R(t), RDB (t), for |B| = 1.




Figures 6 and 7, show comparing the reliability for different sets of system componentsfor the improved methods D = H, I and C with the original system.

Figure 6: The reliability function, R(t), RDB (t) for |B| = 1, 2, 3, D = H and I.

Figure 7: The reliability function, R(t), RCB(t) for |B| = 1, 2, 3.


The δ-fractiles, L(δ), LDB (δ) and ρDA,B(δ), D = H,C, I are calculated using some numer-

ical techniques according to the previous theoretical formulae. In such calculations the levelδ is chosen to be 0.1, 0.2, · · · , 0.9.

Table 2, represents the β-fractiles, of the original and improved systems, L(δ) and LDB (δ).

Table 2: The The L(δ) and LDB (δ), B = B1 ∪B2, D = H, I and C.

B1 = {1}, B1 = Φ, B1 = {1}, B1 = Φ, B1 = {1},B2 = Φ B2 = {1} B2 = {1} B2 = {1, 2} B2 = {1, 2}

δ L D = H0.1 3.0751 3.4041 3.3096 3.6450 3.4315 3.76520.2 2.5885 2.939 2.8296 3.1966 2.9596 3.33050.3 2.2501 2.6169 2.4910 2.8859 2.6223 3.02850.4 1.9696 2.3510 2.2050 2.6287 2.3330 2.77770.5 1.7150 2.1107 1.9393 2.3951 2.0587 2.54860.6 1.4683 1.8791 1.6742 2.1682 1.7782 2.32420.7 1.2142 1.6414 1.3909 1.9328 1.4700 2.08880.8 0.9325 1.3782 1.0631 1.6673 1.1073 1.81820.9 0.5814 1.0432 0.6425 1.3180 0.6525 1.4504

δ L D = I0.1 3.0751 3.5405 3.6366 4.3112 3.893 4.63090.2 2.5885 3.0491 3.0742 3.7397 3.3113 4.06190.3 2.2501 2.7101 2.6817 3.3515 2.8956 3.66960.4 1.9696 2.4311 2.3532 3.0346 2.5400 3.34540.5 1.7150 2.1796 2.0507 2.7502 2.206 3.05080.6 1.4683 1.9378 1.7521 2.4766 1.8709 2.76360.7 1.2142 1.6903 1.4380 2.1954 1.5159 2.46360.8 0.9325 1.4170 1.0836 1.8811 1.1208 2.12110.9 0.5814 1.0702 0.6459 1.4722 0.6536 1.6608

δ L D = C0.1 3.0751 3.7360 3.9071 5.2149 4.1891 5.61770.2 2.5885 3.2148 3.2867 4.5312 3.5431 4.97090.3 2.2501 2.8555 2.8519 4.0627 3.0745 4.51920.4 1.9696 2.5602 2.4875 3.6798 2.6717 4.14240.5 1.7150 2.2946 2.1526 3.3365 2.2951 3.79730.6 1.4683 2.0395 1.8239 3.0075 1.9231 3.45850.7 1.2142 1.7791 1.4817 2.6708 1.5397 3.10220.8 0.9325 1.4921 1.1032 2.2965 1.1275 2.69250.9 0.5814 1.1291 0.6498 1.8121 0.6542 2.1357

It seems from the results shown in Tables 1, 2 and Figures 2 to 6 that:

1. R(t) < RHB (t) < RI

B(t) < RCB(t) for all B.

2. MTTF < MTTFHB < MTTF I

B < MTTFCB , in all studied cases.

3. L(δ) < LHB (δ) < LI

B(δ) < LCB(δ) for all B.

4. The cold duplication method is the best method to improve system reliability and coldduplication with imperfect switch method is much better than the hot duplicationmethod.


Tables 3-5 show the reliability equivalence factors of the improved systems using each du-plication method and A,B.

Table 3: The ρHA,B(δ), when A = A1 ∪A2 and B = B1 ∪B2.

δ B1 = {1}, B1 = Φ, B1 = {1}, B1 = Φ, B1 = {1},A1 A2 B2 = Φ B2 = {1} B2 = {1} B2 = {1, 2} B2 = {1, 2}

0.1 {1} Φ 0.572 0.684 0.315 0.540 0.201Φ {1} 0.598 0.688 0.420 0.574 0.351{1} {1} 0.785 0.838 0.671 0.770 0.624Φ {1, 2} 0.735 0.804 0.579 0.715 0.511{1} {1, 2} 0.837 0.88 0.742 0.825 0.701

0.2 {1} Φ 0.506 0.646 0.218 0.481 0.089Φ {1} 0.501 0.618 0.301 0.482 0.224{1} {1} 0.757 0.811 0.612 0.730 0.558Φ {1, 2} 0.669 0.761 0.481 0.653 0.396{1} {1, 2} 0.804 0.851 0.695 0.795 0.647

0.3 {1} Φ 0.456 0.624 0.148 0.449 0.008Φ {1} 0.413 0.560 0.198 0.407 0.118{1} {1} 0.706 0.791 0.565 0.703 0.507Φ {1, 2} 0.603 0.720 0.382 0.598 0.277{1} {1, 2} 0.775 0.843 0.656 0.772 0.603

0.4 {1} Φ 0.411 0.613 0.089 0.435 NAΦ {1} 0.320 0.504 0.096 0.339 0.017{1} {1} 0.671 0.776 0.521 0.683 0.461Φ {1, 2} 0.526 0.687 0.259 0.545 0.097{1} {1, 2} 0.746 0.830 0.618 0.755 0.562

0.5 {1} Φ 0.369 0.612 0.038 NA NAΦ {1} 0.212 0.446 NA 0.273 NA{1} {1} 0.634 0.765 0.478 0.671 0.417Φ {1, 2} 0.424 0.640 0.001 0.488 NA{1} {1, 2} 0.713 0.802 0.578 0.743 0.521

0.6 {1} Φ 0.326 NA NA NA NAΦ {1} 0.079 0.384 NA 0.206 NA{1} {1} 0.594 0.760 0.434 0.669 0.372Φ {1, 2} 0.251 0.601 NA 0.425 NA{1} {1, 2} 0.675 0.813 0.534 0.738 0.476

0.7 {1} Φ 0.281 NA NA NA NAΦ {1} NA 0.314 NA 0.137 NA{1} {1} 0.545 NA 0.384 NA 0.325Φ {1, 2} 0.001 0.544 NA 0.349 NA{1} {1, 2} 0.628 0.810 0.483 NA 0.426

0.8 {1} Φ 0.231 NA NA NA NAΦ {1} NA 0.229 NA 0.071 NA{1} {1} 0.481 NA 0.325 NA 0.271Φ {1, 2} NA 0.460 NA 0.253 NA{1} {1, 2} 0.560 NA 0.417 NA 0.363

0.9 {1} Φ 166 NA NA NA NAΦ {1} NA 0.122 NA 0.019 NA{1} {1} 0.382 NA 0.245 NA 0.202Φ {1, 2} NA 0.343 NA 0.133 NA{1} {1, 2} 0.447 NA 0.316 NA 0.273


Table 4: The ρIA,B(δ), when A = A1 ∪A2 and B = B1 ∪B2.


0.1 {1} Φ 0.421 0.323 NA 0.089 NAΦ {1} 0.490 0.425 0.131 0.288 NA{1} {1} 0.717 0.674 0.467 0.580 0.405Φ {1, 2} 0.643 0.596 0.253 0.443 0.122{1} {1, 2} 0.781 0.745 0.550 0.660 0.483

0.2 {1} Φ 0.377 NA NA NA NAΦ {1} 0.405 0.385 0.055 0.234 NA{1} {1} 0.682 0.669 0.434 0.565 0.365Φ {1, 2} 0.584 0.563 0.161 0.407 NA{1} {1, 2} 0.755 0.744 0.527 0.654 0.455

0.3 {1} Φ 0.342 NA NA NA NAΦ {1} 0.326 0.351 NA 0.192 NA{1} {1} 0.652 0.667 0.406 0.561 0.335Φ {1, 2} 0.522 0.532 NA 0.375 NA{1} {1, 2} 0.73 0.743 0.506 0.652 0.431

0.4 {1} Φ 0.311 NA NA NA NAΦ {1} 0.242 0.317 NA 0.155 NA{1} {1} 0.622 NA 0.379 NA 0.307Φ {1, 2} 0.446 0.501 NA 0.342 NA{1} {1, 2} 0.705 NA 0.483 NA 0.407



0.7 {1} Φ 0.219 NA NA NA NAΦ {1} NA 0.202 NA 0.057 NA{1} {1} 0.512 NA 0.291 NA 0.226Φ {1, 2} 0.001 0.384 NA 0.218 NA{1} {1, 2} 0.599 NA 0.392 NA 0.324

0.8 {1} Φ 0.182 NA NA NA NAΦ {1} NA 0.151 NA 0.029 NA{1} {1} 0.455 NA 0.252 NA 0.194Φ {1, 2} NA 0.328 NA NA NA{1} {1, 2} 0.537 NA 0.344 NA 0.283



Table 5: The ρCA,B(δ), when A = A1 ∪A2 and B = B1 ∪B2.


0.1 {1} Φ 0.228 0.077 NA NA NAΦ {1} 0.367 0.282 NA 0.171 NA{1} {1} 0.614 0.575 0.321 0.496 0.279Φ {1, 2} NA 0.436 NA 0.305 NA{1} {1, 2} 0.710 0.656 0.390 0.390 0.340

0.2 {1} Φ 0.2 NA NA NA NAΦ {1} 0.289 0.247 NA 0.126 NA{1} {1} 0.593 0.575 0.293 0.487 0.245Φ {1, 2} NA 0.423 NA 0.272 NA{1} {1, 2} 0.688 0.662 0.374 NA 0.316

0.3 {1} Φ 0.179 NA NA NA NAΦ {1} 0.217 0.22 NA 0.096 NA{1} {1} 0.579 NA 0.272 NA 0.220Φ {1, 2} NA 0.408 NA 0.244 NA{1} {1, 2} 0.668 NA 0.36 NA 0.297






0.9 {1} Φ 0.070 NA NA NA NAΦ {1} NA 0.047 NA 0.003 NA{1} {1} 0.330 NA 0.127 NA 0.090Φ {1, 2} NA NA NA NA NA{1} {1, 2} 0.398 NA 0.193 NA 0.148

According to the results presented in Tables 2–5, at α = 2, it may be observed that:


1. Hot duplication of the set B1 = {1} of system component will increase L(0.1) from3.0751

Λ to 3.4041Λ , see Table 2. The same effect on L(0.1) can occur by reducing the

failure rate of components belong to (i) A1 = {1} from the first subsystem by thefactor ρH = 0.572, (ii) A2 = {1} from the second subsystem by the factor ρH = 0.598,(iii) A1 = {1}, A2 = {1} by the factor ρH = 0.783, (iv) A2 = {1, 2} from thesecond subsystem by the factor ρH = 0.735,(v) A1 = {1}, A2 = {1, 2} by the factorρH = 0.837 see Table 3.

2. Cold duplication with imperfect switch of the set B1 = {1} of system componentwill increase L(0.1) from 3.0751

Λ to 3.5405Λ , see Table 2. The same effect on L(0.1) can

occur by reducing the failure rate of components belong to (i) A1 = {1} from the firstsubsystem by the factor ρI = 421, (ii) A2 = {1} from the second subsystem by thefactor ρI = 0.490, (iii) A1 = {1}, A2 = {1} by the factor ρI = 0.717, (iv) A2 = {1, 2}from the second subsystem by the factor ρI = 0.643,(v) A1 = {1}, A2 = {1, 2} by thefactor ρI = 0.781 see Table 4.

3. Cold duplication of the set B1 = {1} of system component will increase L(0.1) from3.0751

Λ to 3.7360Λ , see Table 2. The same effect on L(0.1) can occur by reducing the

failure rate of components belong to (i) A1 = {1} from the first subsystem by thefactor ρC = 0.228, (ii) A2 = {1} from the second subsystem by the factor ρC = 0.367,(iii) A1 = {1}, A2 = {1} by the factor ρC = 0.614, (iv) A1 = {1}, A2 = {1, 2} by thefactor ρC = 0.710 see Table 5.

4. In the same manner, one can read the rest of results presented in Tables 2–5.

5. The notation NA, means that there is no equivalence between the two improvedsystems: one obtained by reducing the failure rates of the set A of system componentsand the other obtained by improving the set B of the system components accordingto the duplication methods.

7 Conclusions

Reliability equality function of original system is studied in this paper. four improvementmethods, including reduction method, hot and cold duplication method with imperfect andperfect switch, are also applied to improve the reliability of mentioned system. It is shownthat reliability of an improved system is higher than the original system reliability in allcases. Further, it’s shown in this paper that the cold duplication method improves systemreliability much better than the cold method with imperfect switch and the hot duplicationmethod, but it’s not possible to make a general statement for a comparison between reductionmethod and duplication methods. If we put n = 2,m1 = 1,m2 = 2, Ezzati and Rasouli [1],is a special case from our article.

References

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Received: August 12, 2016; Published: October 15, 2016

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Improving The Performance of the Series Parallel System ... · 1038 Abdelfattah Mustafa, Beih S. El-Desouky and Ahmed Taha Keywords: Reliability equivalence, Series-parallel system,