# Availability Equivalence Analysis for the Simulation of Repairable Bridge Network System

Authors: Jaafar M. Alghazo; Abdelfattah Mustafa; Adel A. El-Faheem
Journal: Complexity
 (2020)

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## Abstract

The performance of a repairable bridge network system is improved by using the availability equivalence factors. All components for the bridge system have constant failure and repair rates. The system is improved through the use of five methods: reduction, increase, hot duplication, warm duplication, and cold duplication methods. The availability of the original and improved systems is derived. Two types of availability equivalence factors of the system are obtained to compare different system designs. Numerical example to interpret how to utilize the obtained results is provided.


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## Body

## 1. Introduction

In reliability analysis, there are two main methods to improve a nonrepairable system design. These two methods are (i) the reduction method which assumes that the system can be improved by reducing the failure rates of a set of components by a factorρ, 0 < ρ < 1, and (ii) the redundancy method which in actuality is divided into more than one redundancy method such as hot, warm, cold, and cold with imperfect switch redundancy [1]. The redundancy and reduction methods can be used to improve the repairable systems as well. In addition, the repairable system can be improved by increasing the rate of repair of some system components by a factor σ, σ > 1 [2].The use of the redundancy method may not be a practical solution for a system in which the minimum size and weight are excessive [3]. Therefore, the concept of reliability/availability equivalence takes place. In such a concept, the design of the improved system according to the reduction or increase method must be equivalent to the design of the improved system in accordance with one of the redundancy methods specified. That is, using this concept, one can say that system performance can be improved through an alternative design [4]. In this case, different system designs must be compared based on performance characteristics such as (i) the reliability function or mean time to failure for nonrepairable systems [5–21] or (ii) the availability in the case of repairable systems [2, 22–24].The reliability systems in life can be divided into the following types:(1)
The failed component is not repaired or replaced, and the operation of the complete system once one or more of its components fail depends on its structure.(2)
The failed component can be replaced by a new component, and the failure rate for the new component may be different from the failed item.(3)
The failed component can be repaired and restored to the system after repair in order for the system to operate again.The nonrepairable systems can sometimes be considered a special case of repairable systems.In this paper, the main objective is to derive the availability equivalence factors of a repairable bridge system with independent and identical components. Recent research has been done on the reliability equivalence of nonrepairable bridge systems. Sarhan [14], Mustafa et al. [17], and Mustafa [18] derived the reliability equivalence factors of the bridge system in the case of no repairs. In [14, 17, 18], the reliability function is computed based on the assumption that the system components are nonrepairable. In addition, the reliability function was computed to obtain the reliability equivalence factors. Four methods are used to improve the reliability of the structure, namely, reduction, hot duplication, perfect switch, and imperfect switch methods. However, this paper assumes that the system components are repairable and could be repaired to function again in the system. Therefore, the goal is to calculate the availability function in order to obtain the availability equivalence factors. In this paper, five different methods were used to improve the system availability, namely, reduction, increase, hot duplication, warm duplication, and cold duplication methods.This paper is organized as follows. In Section2, we introduce the original system and present its availability function. The system availability is improved by five different methods, which are introduced in Section 3. The availability equivalence factors of the system are derived in Section 4. Numerical results and conclusions are discussed in Section 5.


## 2. Materials and Methods

The following notations and abbreviations are used throughout this paper:λ, failure rateμ, repair rateAi, availability for the component iAsys, availability for the systemAR,ρ, availability for the improving system by using the reduction methodAI,σ, availability for the improving system by using the increasing methodABℋ, availability for the improving system by using the hot methodABW, availability for the improving system by using the warm methodABC, availability for the improving system by using the cold methodN, natural numbersρ, reduction factorσ, increasing factorAEF, availability equivalence factorsRAEF, reducing availability equivalence factorIAEF, increasing availability equivalence factorD,ℋ,W,C, duplication, hot, warm, cold


### 2.1. Repairable Bridge System

The bridge system has five components connected as shown in Figure1.Figure 1
A bridge structure.The componenti, i = 1, 2, …, 5, has exponential lifetime distribution with failure rate λ. Also, the repair time is exponential with repair rate μ. The system components are independent and identical with availability Ai,i,i=1,2,…,5. Ai is calculated using the following equation.(1)Ai=μμ+λ=11+η,where η=λ/μ.The availability for the bridge network system can be evaluated by using some techniques such as Tie sets [25]. A tie set is a minimal path of the system. The minimal path sets for the bridge system are 14, 25, 135, and 234.The component for each path must be connected in series. All the tie sets must fail in order for the system to fail; therefore, the tie sets are connected in parallel mode as shown in Figure2.Figure 2
Tie sets for the bridge system.LetA be the availability of the bridge system, then A can be obtained as(2)Asys=Pτ1∪τ2∪τ3∪τ4=A1A4+A2A5+A1A3A5+A2A3A4−∑i=15∏j∈NiAi+2∏j∈NAi,where N=1,2,…,5,Ni=N\i,i∈N. Assuming that the components are identical in equations (1) and (2), we obtain(3)Asys=2η3+8η2+5η+11+η5.


## 2.1. Repairable Bridge System

The bridge system has five components connected as shown in Figure1.Figure 1
A bridge structure.The componenti, i = 1, 2, …, 5, has exponential lifetime distribution with failure rate λ. Also, the repair time is exponential with repair rate μ. The system components are independent and identical with availability Ai,i,i=1,2,…,5. Ai is calculated using the following equation.(1)Ai=μμ+λ=11+η,where η=λ/μ.The availability for the bridge network system can be evaluated by using some techniques such as Tie sets [25]. A tie set is a minimal path of the system. The minimal path sets for the bridge system are 14, 25, 135, and 234.The component for each path must be connected in series. All the tie sets must fail in order for the system to fail; therefore, the tie sets are connected in parallel mode as shown in Figure2.Figure 2
Tie sets for the bridge system.LetA be the availability of the bridge system, then A can be obtained as(2)Asys=Pτ1∪τ2∪τ3∪τ4=A1A4+A2A5+A1A3A5+A2A3A4−∑i=15∏j∈NiAi+2∏j∈NAi,where N=1,2,…,5,Ni=N\i,i∈N. Assuming that the components are identical in equations (1) and (2), we obtain(3)Asys=2η3+8η2+5η+11+η5.


## 3. The Improving Methods

The system availability can be improved by using one of the following methods:(1)
Reduction method(2)
Increase method(3)
Hot duplication method(4)
Warm duplication method(5)
Cold duplication method


### 3.1. The Reduction Method

Assume that the system will improve by reducing the failure rates of the elements in the setR using a factor ρ,0<ρ<1 and R=r,0≤r≤N. Let AR,ρ be the availability function of the improved system in accordance with the reduction method. For a component i∈R, the availability Aρ is given by(4)Aρ=μμ+ρ⋅λ=11+ρ⋅η,where η=λ/μ. The availability AR,ρ can be obtained for some different values of r≤2 as follows:(1)
R∈S1=3:(5)AR,ρ=2η3ρ+4η21+ρ+η4+ρ+11+η41+ηρ.(2)
R∈S2=1,2,4,5:(6)AR,ρ=η31+ρ+η25+3ρ+η4+ρ+11+η41+ηρ.(3)
R∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(7)AR,ρ=η3ρ1+ρ+η22+5ρ+ρ2+η3+2ρ+11+η31+ηρ2.(4)
R∈S4=1,2,4,5:(8)AR,ρ=2η3ρ+η22+6ρ+η3+2ρ+11+η31+ηρ2.(5)
R∈S5=1,4,2,5:(9)AR,ρ=η31+ρ2+η23+4ρ+ρ2+η3+2ρ+11+η31+ηρ2.


### 3.2. The Increase Method

Suppose thatAI,σ refers to the availability function of the improved system through increasing the repair rates of some system components belonging to the set I by a factor σ,σ>1 and I=ℓ,0≤ℓ≤N. For component i∈I, the availability after increasing its repair rate can be given as follows:(10)Ai,σ=σμσμ+λ=σσ+η,where η=λ/μ. The function AI,σ for some different I,ℓ≤2, is obtained as(1)
I∈S1=3:(11)AI,σ=2η3+4η21+σ+η4σ+1+σ1+η4σ+η.(2)
I∈S2=1,2,3,4,5:(12)AI,σ=η31+σ+η23+5σ+η1+4σ+σ1+η4σ+η.(3)
I∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(13)AI,σ=η31+σ+η21+5σ+2σ2+ησ2+3σ+σ21+η3σ+η2.(4)
I∈S4=1,2,4,5:(14)AI,σ=σ2η3+2η23+σ+η2+3σ+σ1+η3σ+η2.(5)
I∈S5=1,4,2,5:(15)AI,σ=η31+σ2+η21+4σ+3σ2+ησ2+3σ+σ21+η3σ+η2.


### 3.3. The Hot Duplication Method

Suppose thatABℋ denotes to the availability of the improved system by hot duplication for the components belonging to the set B and B=m,0≤m≤N. The component i∈B has the availability, say Aiℋ, as follows:(16)Aiℋ=1−1−Ai2=1−η1+η2.Thus, the system availabilityABℋ can be derived as m≤2(1)
B∈S1=3:(17)ABℋ=2η4+12η3+13η2+6η+11+η6.(2)
B∈S2=1,2,4,5:(18)ABℋ=3η4+14η3+14η2+6η+11+η6.(3)
B∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(19)ABℋ=3η5+20η4+29η3+20η2+7η+11+η7.(4)
B∈S4=1,2,4,5:(20)ABℋ=1+2η2η4+10η3+10η2+5η+11+η7.(5)
B∈S5=1,4,2,5:(21)ABℋ=5η5+25η4+33η3+21η2+7η+11+η7.


### 3.4. The Warm Duplication Method

Suppose that, in this method, each component belonging to the setB is connected with a warm standby component (with constant failure rate ν) via a perfect switch. Let AiW be the availability of the component i∈B, then AiW can be obtained as follows [26]:(22)AiW=μ2+λμ+νμμ2+λμ+νμ+1/2λ2+1/2λν=1+η+ξ1+η+ξ+1/2η2+1/2ηξ,where η=λ/μ and ξ=ν/μ. Thus, the availability of the improved system by warm duplication method, ABW, is given by(1)
B∈S1=3:(23)ABW=2η4+2η36+ξ+η217+12ξ+η10+9ξ+21+ξ1+η4η2+η2+ξ+21+ξ.(2)
B∈S2=1,2,4,5:(24)ABW=3η4+3η35+ξ+η219+13ξ+η10+9ξ+21+ξ1+η4η2+η2+ξ+21+ξ.(3)
B∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(25)ABW=11+η3η2+η2+ξ+21+ξ3η5+3η47+2ξ+η342+40ξ+3ξ2+η240+58ξ+19ξ2+4η5+9ξ+4ξ2+41+ξ2.(4)
B∈S4=1,2,4,5:(26)ABW=41+η+ξη4+η35+ξ+η26+5ξ+4η1+ξ+ξ+11+η3η2+η2+ξ+21+ξ2.(5)
B∈S5=1,4,2,5:(27)ABW=11+η3η2+η2+ξ+21+ξ25η5+η429+10ξ+η352+50ξ+5ξ2+η244+64ξ+21ξ2+4η5+9ξ+4ξ2+41+ξ2.


### 3.5. The Cold Duplication Method

The cold duplication method assumes that each component belonging to the setB is connected with an identical component via a perfect switch. The availability of the component i∈B,AiC, can be calculated by [27](28)AiC=μ2+λμμ2+λμ+1/2λ2=1+η1+η+1/2η2,where η=λ/μ. In what follows, we present the availability, ABC, of the improved system in accordance with the cold duplication method for the components belonging to B for some different values of B as(1)
B∈S1=3:(29)ABC=2η4+12η3+17η2+10η+21+η4η2+2η+2.(2)
B∈S2=1,2,4,5:(30)ABC=3η4+15η3+19η2+10η+21+η4η2+2η+2.(3)
B∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(31)ABC=3η4+18η3+24η2+16η+41+η2η2+2η+22.(4)
B∈S4=1,2,4,5:(32)ABC=4η4+5η3+6η2+4η+11+η2η2+2η+22.(5)
B∈S5=1,4,2,5:(33)ABC=5η4+24η3+28η2+16η+41+η2η2+2η+22.


## 3.1. The Reduction Method

Assume that the system will improve by reducing the failure rates of the elements in the setR using a factor ρ,0<ρ<1 and R=r,0≤r≤N. Let AR,ρ be the availability function of the improved system in accordance with the reduction method. For a component i∈R, the availability Aρ is given by(4)Aρ=μμ+ρ⋅λ=11+ρ⋅η,where η=λ/μ. The availability AR,ρ can be obtained for some different values of r≤2 as follows:(1)
R∈S1=3:(5)AR,ρ=2η3ρ+4η21+ρ+η4+ρ+11+η41+ηρ.(2)
R∈S2=1,2,4,5:(6)AR,ρ=η31+ρ+η25+3ρ+η4+ρ+11+η41+ηρ.(3)
R∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(7)AR,ρ=η3ρ1+ρ+η22+5ρ+ρ2+η3+2ρ+11+η31+ηρ2.(4)
R∈S4=1,2,4,5:(8)AR,ρ=2η3ρ+η22+6ρ+η3+2ρ+11+η31+ηρ2.(5)
R∈S5=1,4,2,5:(9)AR,ρ=η31+ρ2+η23+4ρ+ρ2+η3+2ρ+11+η31+ηρ2.


## 3.2. The Increase Method

Suppose thatAI,σ refers to the availability function of the improved system through increasing the repair rates of some system components belonging to the set I by a factor σ,σ>1 and I=ℓ,0≤ℓ≤N. For component i∈I, the availability after increasing its repair rate can be given as follows:(10)Ai,σ=σμσμ+λ=σσ+η,where η=λ/μ. The function AI,σ for some different I,ℓ≤2, is obtained as(1)
I∈S1=3:(11)AI,σ=2η3+4η21+σ+η4σ+1+σ1+η4σ+η.(2)
I∈S2=1,2,3,4,5:(12)AI,σ=η31+σ+η23+5σ+η1+4σ+σ1+η4σ+η.(3)
I∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(13)AI,σ=η31+σ+η21+5σ+2σ2+ησ2+3σ+σ21+η3σ+η2.(4)
I∈S4=1,2,4,5:(14)AI,σ=σ2η3+2η23+σ+η2+3σ+σ1+η3σ+η2.(5)
I∈S5=1,4,2,5:(15)AI,σ=η31+σ2+η21+4σ+3σ2+ησ2+3σ+σ21+η3σ+η2.


## 3.3. The Hot Duplication Method

Suppose thatABℋ denotes to the availability of the improved system by hot duplication for the components belonging to the set B and B=m,0≤m≤N. The component i∈B has the availability, say Aiℋ, as follows:(16)Aiℋ=1−1−Ai2=1−η1+η2.Thus, the system availabilityABℋ can be derived as m≤2(1)
B∈S1=3:(17)ABℋ=2η4+12η3+13η2+6η+11+η6.(2)
B∈S2=1,2,4,5:(18)ABℋ=3η4+14η3+14η2+6η+11+η6.(3)
B∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(19)ABℋ=3η5+20η4+29η3+20η2+7η+11+η7.(4)
B∈S4=1,2,4,5:(20)ABℋ=1+2η2η4+10η3+10η2+5η+11+η7.(5)
B∈S5=1,4,2,5:(21)ABℋ=5η5+25η4+33η3+21η2+7η+11+η7.


## 3.4. The Warm Duplication Method

Suppose that, in this method, each component belonging to the setB is connected with a warm standby component (with constant failure rate ν) via a perfect switch. Let AiW be the availability of the component i∈B, then AiW can be obtained as follows [26]:(22)AiW=μ2+λμ+νμμ2+λμ+νμ+1/2λ2+1/2λν=1+η+ξ1+η+ξ+1/2η2+1/2ηξ,where η=λ/μ and ξ=ν/μ. Thus, the availability of the improved system by warm duplication method, ABW, is given by(1)
B∈S1=3:(23)ABW=2η4+2η36+ξ+η217+12ξ+η10+9ξ+21+ξ1+η4η2+η2+ξ+21+ξ.(2)
B∈S2=1,2,4,5:(24)ABW=3η4+3η35+ξ+η219+13ξ+η10+9ξ+21+ξ1+η4η2+η2+ξ+21+ξ.(3)
B∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(25)ABW=11+η3η2+η2+ξ+21+ξ3η5+3η47+2ξ+η342+40ξ+3ξ2+η240+58ξ+19ξ2+4η5+9ξ+4ξ2+41+ξ2.(4)
B∈S4=1,2,4,5:(26)ABW=41+η+ξη4+η35+ξ+η26+5ξ+4η1+ξ+ξ+11+η3η2+η2+ξ+21+ξ2.(5)
B∈S5=1,4,2,5:(27)ABW=11+η3η2+η2+ξ+21+ξ25η5+η429+10ξ+η352+50ξ+5ξ2+η244+64ξ+21ξ2+4η5+9ξ+4ξ2+41+ξ2.


## 3.5. The Cold Duplication Method

The cold duplication method assumes that each component belonging to the setB is connected with an identical component via a perfect switch. The availability of the component i∈B,AiC, can be calculated by [27](28)AiC=μ2+λμμ2+λμ+1/2λ2=1+η1+η+1/2η2,where η=λ/μ. In what follows, we present the availability, ABC, of the improved system in accordance with the cold duplication method for the components belonging to B for some different values of B as(1)
B∈S1=3:(29)ABC=2η4+12η3+17η2+10η+21+η4η2+2η+2.(2)
B∈S2=1,2,4,5:(30)ABC=3η4+15η3+19η2+10η+21+η4η2+2η+2.(3)
B∈S3=1,3,1,5,2,3,2,4,3,4,3,5:(31)ABC=3η4+18η3+24η2+16η+41+η2η2+2η+22.(4)
B∈S4=1,2,4,5:(32)ABC=4η4+5η3+6η2+4η+11+η2η2+2η+22.(5)
B∈S5=1,4,2,5:(33)ABC=5η4+24η3+28η2+16η+41+η2η2+2η+22.


## 4. Availability Equivalence Factors

In this section, we will derive two different types of availability equivalence factors (AEF), reducing availability equivalence factor (RAEF) and increasing availability equivalence factor (IAEF).Definition 1.
Availability equivalence factor (AEF) is defined as the factor by which the failure rates (repair rates) of some of the system’s components should be reduced (increased) in order to reach equality of the availability of a better system.


### 4.1. The RAEF

The reducing availability equivalence factor can be obtained by solving the following equation with respect toρ=ρR,BD.(34)AR,ρ=ABD,D=ℋ,W,C.For a specific setR of the system components, we present the different forms of the RAEF of the bridge system that can be derived from equation (34) as follows:(1)
WhenR∈S1:(35)ρR,BD=1+η4ABD−2η+12η2η2+4η+1−1+η4ABD.(2)
WhenR∈S2:(36)ρR,BD=1+η4ABD−η3+5η2+4η+1ηη2+3η+1−1+η4ABD.(3)
WhenR∈S3:(37)ρR,BD=−b1±b12−4a1c12a1,where a1=η2η+11−1+η2ABD,b1=ηη2+5η+2−21+η3ABD, and c1=1+η2η+1−1+ηABD.(4)
WhenR∈S4:(38)ρR,BD=−b2±b22−4a2c22a2,wherea2=1+η3η2ABD, b2=2η1+η3ABD−η2+3η+1, b2=2η1+η3ABD−η2+3η+1, and c2=1+η1+η2ABD−2η+1.(5)
WhenR∈S5:(39)ρR,BD=−b3±b32−4a3c32a3,where a3=η21+η1−1+η2ABD,b3=2η2η+1−1+η3ABD, and c3=η+131−ABD.For the values ofη,ξ and ABD for different B,D, the values for RAEFs, ρR,BD, can be calculated from (35)–(39).


### 4.2. The IAEF

The increasing availability equivalence factor,σ=σI,BD, can be calculated by solving the following equation:(40)AI,σ=ABD,D=ℋ,W,C,with respect to σ.Different forms for IAEF can be calculated from equation (40) for a specific set I as follows:(1)
WhenI∈S1:(41)σI,BD=η1+η4ABD−2η2+4η+12η+12−1+η4ABD.(2)
WhenI∈S2:(42)σI,BD=η1+η4ABD−η2+3η+1η3+5η2+4η+1−1+η4ABD.(3)
WhenI∈S3:(43)σI,BD=−e1±e12−4d1f12d1,whered1=η+12η+1−1+η2ABD, e1=ηη2+5η+2−21+η3ABD, and f1=η21+η1−1+η2ABD.(4)
WhenI∈S4:(44)σI,BD=−e2±e22−4d2f22d2,whered2=η+12η+1−1+η2ABD, e2=2ηη2+3η+1−1+η3ABD, and f2=−η21+η3ABD.(5)
WhenI∈S5:(45)σI,BD=−e3±e32−4d3f32d3,whered3=η+131−ABD, e3=2η2η+1−1+η3ABD, and f3=η2η+11−1+η2ABD.From (41)–(45), the values for IAEFs, σI,BD, can be calculated through specified values of η,ξ and ABD for different B and D=ℋW,C.


## 4.1. The RAEF

The reducing availability equivalence factor can be obtained by solving the following equation with respect toρ=ρR,BD.(34)AR,ρ=ABD,D=ℋ,W,C.For a specific setR of the system components, we present the different forms of the RAEF of the bridge system that can be derived from equation (34) as follows:(1)
WhenR∈S1:(35)ρR,BD=1+η4ABD−2η+12η2η2+4η+1−1+η4ABD.(2)
WhenR∈S2:(36)ρR,BD=1+η4ABD−η3+5η2+4η+1ηη2+3η+1−1+η4ABD.(3)
WhenR∈S3:(37)ρR,BD=−b1±b12−4a1c12a1,where a1=η2η+11−1+η2ABD,b1=ηη2+5η+2−21+η3ABD, and c1=1+η2η+1−1+ηABD.(4)
WhenR∈S4:(38)ρR,BD=−b2±b22−4a2c22a2,wherea2=1+η3η2ABD, b2=2η1+η3ABD−η2+3η+1, b2=2η1+η3ABD−η2+3η+1, and c2=1+η1+η2ABD−2η+1.(5)
WhenR∈S5:(39)ρR,BD=−b3±b32−4a3c32a3,where a3=η21+η1−1+η2ABD,b3=2η2η+1−1+η3ABD, and c3=η+131−ABD.For the values ofη,ξ and ABD for different B,D, the values for RAEFs, ρR,BD, can be calculated from (35)–(39).


## 4.2. The IAEF

The increasing availability equivalence factor,σ=σI,BD, can be calculated by solving the following equation:(40)AI,σ=ABD,D=ℋ,W,C,with respect to σ.Different forms for IAEF can be calculated from equation (40) for a specific set I as follows:(1)
WhenI∈S1:(41)σI,BD=η1+η4ABD−2η2+4η+12η+12−1+η4ABD.(2)
WhenI∈S2:(42)σI,BD=η1+η4ABD−η2+3η+1η3+5η2+4η+1−1+η4ABD.(3)
WhenI∈S3:(43)σI,BD=−e1±e12−4d1f12d1,whered1=η+12η+1−1+η2ABD, e1=ηη2+5η+2−21+η3ABD, and f1=η21+η1−1+η2ABD.(4)
WhenI∈S4:(44)σI,BD=−e2±e22−4d2f22d2,whered2=η+12η+1−1+η2ABD, e2=2ηη2+3η+1−1+η3ABD, and f2=−η21+η3ABD.(5)
WhenI∈S5:(45)σI,BD=−e3±e32−4d3f32d3,whered3=η+131−ABD, e3=2η2η+1−1+η3ABD, and f3=η2η+11−1+η2ABD.From (41)–(45), the values for IAEFs, σI,BD, can be calculated through specified values of η,ξ and ABD for different B and D=ℋW,C.


## 5. Numerical Results and Conclusions

In this section, a numerical example is given to illustrate the theoretical results obtained in the previous sections. We assumeλ=0.24,μ=1.2, and ν=0.12. In this case, we have η=λ/μ=0.2 and ξ=ν/μ=0.1.Figures3 and 4 show the availability of an improved system through improving different sets of the components in accordance with the reduction (increase) method by the factor ρ, 0<ρ<1, and σ,σ>1, respectively.Figure 3
TheAR,ρ for the reduction method, R∈Si,i=1,2,…,5.Figure 4
TheAI,σ for the increase method, I∈Si,i=1,2,…,5.According to Figures3 and 4, we can conclude that(1)
AR,ρ, decreases with increasing ρ for all possible sets R.(2)
whenσ increases, the AI,σ increases as well for all possible sets I.(3)
A<AS1,ρ<AS2,ρ<AS3,ρ<AS4,ρ<AS5,ρ, for all ρ∈0,1.(4)
A<AS1,σ<AS2,σ<AS3,σ<AS4,σ<AS5,σ, for all σ>1.The availability of the original system isAsys=0.951342 and ABD for the improved system for all possible sets B are presented in Table 1.Table 1
TheABD,D=ℋW,C and B∈Si,i=1,2,…,5.
B
∈
S
i
A
B
ℋ
A
B
W
A
B
C
S
1
0.955415
0.955492
0.955715
S
2
0.971482
0.971862
0.972962
S
3
0.974125
0.974527
0.975687
S
4
0.976518
0.976702
0.977177
S
5
0.992585
0.99338
0.995692From the numerical results in Table1, the following can be concluded:(1)
Asys<ABℋ<ABW<ABC for all B∈Si;(2)
Asys<AS1D<AS2D<AS3D<AS4D<AS5D for all D=ℋW,C.Finally, the AEFs,ρR,BD, and xρR,BD for a different D,R,I, and B are calculated according to the previous theoretical formulas. Tables 2–6 contain the values of AEFs.Table 2
The AEFs for a different setB and R,I∈S1.
B
∈
S
i
ρ
R
,
B
D
σ
I
,
B
D
Hot
Warm
Cold
Hot
Warm
Cold
S
1
0.12963
0.115385
0.0744681
7.71429
8.66667
13.4286
S
2
—
—
—
—
—
—
S
3
—
—
—
—
—
—
S
4
—
—
—
—
—
—
S
5
—
—
—
—
—
—Table 3
The AEFs for a different setB and R,I∈S2.
B
∈
S
i
ρ
R
,
B
D
σ
I
,
B
D
Hot
Warm
Cold
Hot
Warm
Cold
S
1
0.803674
0.800084
0.789690
1.24429
1.24987
1.26631
S
2
0.129630
0.115385
0.074468
7.71429
8.66667
13.4286
S
3
0.031872
0.017290
—
31.3751
57.8362
—
S
4
—
—
—
—
—
—
S
5
—
—
—
—
—
—Table 4
The AEFs for a different setB and R,I∈S3.
B
∈
S
i
ρ
R
,
B
D
σ
I
,
B
D
Hot
Warm
Cold
Hot
Warm
Cold
S
1
0.834270
0.831193
0.822279
1.19865
1.20309
1.21613
S
2
0.224141
0.210474
0.171020
4.46149
4.75119
5.84728
S
3
0.129630
0.115385
0.074468
7.71429
8.66667
13.4286
S
4
0.045323
0.038871
0.022315
22.0639
25.7262
44.813
S
5
—
—
—
—
—
—Table 5
The AEFs for a different setB and R,I∈S4.
B
∈
S
i
ρ
R
,
B
D
σ
I
,
B
D
Hot
Warm
Cold
Hot
Warm
Cold
S
1
0.896350
0.894343
0.888510
1.11564
1.11814
1.12548
S
2
0.393569
0.377786
0.329639
2.54085
2.64700
3.03363
S
3
0.273861
0.253018
0.186415
3.65149
3.95229
5.36438
S
4
0.129630
0.115385
0.074468
7.71429
8.66667
13.4286
S
5
—
—
—
—
—
—Table 6
The AEFs for a different setB and R,I∈S5.
B
∈
S
i
ρ
R
,
B
D
σ
I
,
B
D
Hot
Warm
Cold
Hot
Warm
Cold
S
1
0.900617
0.898776
0.8934460
1.11035
1.11262
1.11926
S
2
0.539715
0.531738
0.5087430
1.85283
1.88062
1.96563
S
3
0.484669
0.476396
0.4526690
2.06326
2.09909
2.20912
S
4
0.435802
0.432071
0.4225060
2.29462
2.31443
2.36683
S
5
0.129630
0.115385
0.0744681
7.71429
8.66667
13.4286


## 6. Discussion

According to the numerical results in Tables1–6, the following can be observed:(1)
Improving the system by hot duplication method for the components belonging to the setB∈S1 improves the system availability from 0.951342 to 0.955415 (see Table 1). The system with ABℋ=0.955415 can be obtained by using one of the following:(a)
The failure rate of the components belonging to the set (i)R∈S1 is reduced by the factor ρℋ=0.12963; (ii) R∈S2 is reduced by the factor ρℋ=0.803674; (iii) R∈S3 is reduced by the factor ρℋ=0.834270; (iv) R∈S4 is reduced by the factor ρℋ=0.896350; (v) R∈S5 is reduced by the factor ρℋ=0.900617 (see Tables 2–6).(b)
Increasing the repair rate for the components in the set (i)I∈S1 by the factor σℋ=7.71429; (ii) I∈S2 by the factor σℋ=1.24429; (iii) I∈S3 by the factor σℋ=1.19865; (iv) I∈S4 by the factor σℋ=1.11564; (v) I∈S5 by the factor σℋ=1.11035 (see Tables 2–6).(2)
The system availability increases from 0.951342 to 0.955492 (see Table1), by improving the components belonging to the set B∈S1 using the warm duplication method. Through one of the following, we can obtain the system with availability, ABW=0.955492:(a)
Reducing the failure rate of the components in set (i)R∈S1 by the factor ρW=0.115385; (ii) R∈S2 by the factor ρW=0.800084; (iii) R∈S3 by the factor ρW=0.831193; (iv) R∈S4 by the factor ρW=0.894343; (v) R∈S5 by the factor ρW=0.898776 (see Tables 2–6).(b)
Increasing the repair rate for the components belonging to the set (i)I∈S1 by the factor σW=8.66667; (ii) I∈S2 by the factor σW=1.24987; (iii) I∈S3 by the factor σW=1.20309; (iv) I∈S4 by the factor σW=1.11814; (v) I∈S5 by the factor σW=1.11262 (see Tables 2–6).(3)
Cold duplication of the system components belonging to the setB∈S1 increases the availability from 0.951342 to 0.955715 (see Table 1). The design with ABC=0.955715 can be obtained through one of the following:(a)
Reducing the failure rate of the system components in (i)R∈S1 by the factor ρC=0.0744681; (ii) R∈S2 by the factor ρC=0.789690; (iii) R∈S3 by the factor ρC=0.822279; (iv) R∈S4 by the factor ρC=0.888510; (v) R∈S5 by the factor ρC=0.8934460 (see Tables 2–6).(a)
Increasing the repair rate of the components in (i)I∈S1 by the factor σC=13.4286; (ii) I∈S2 by the factor σC=1.26631; (iii) I∈S3 by the factor σC=1.21613; (iv) I∈S4 by the factor σC=1.12548; (v) I∈S5 by the factor σC=1.11926 (see Tables 2–6).


## 7. Conclusions

In this paper, the equivalence of different designs of a repairable bridge system based on the system availability was derived. The failure rates and repair rates of the system’s components were assumed constant. The system availability was improved using five different methods. The availability of the original and the improved systems and the availability equivalence factors of the systems were derived. Numerical results were presented to illustrate how one can utilize the theoretical results obtained in this work and to compare the different availability factors of the system. Future work includes extending the current paper to include other cases such as systems with nonidentical components, limited repair teams which are available for each component in the system, and failure (repair) rates of the components which are not constant.



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