# Reliability Analysis of a Two Nonidentical Unit Parallel System with Optional Vacations under Poisson Shocks

Authors: Mohamed S. El-Sherbeny; Zienab M. Hussien
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this article, we study the impact of some system parameters on an industrial system consisting of two nonidentical parallel units with one repairer. The active unit may fail due to essential factors such as aging or deterioration, or due to external phenomena such as Poisson shocks occurring at different time periods. Whenever the value of a shock is greater than the specified threshold of the active unit, the active unit fails. The article assumes that the repairman can make one of two decisions at the beginning of system operation: either he takes a vacation when the two units are operating normally, or he remains in the system to monitor it until the first failure of the system. If a failure occurs in either unit during the repairman’s absence, the failed unit must wait until the repairman is called back to work. We assume that the value of each shock is i.i.d. with a known distribution. The length of the repairman’s vacation, the repair time, and the recall time are arbitrary distributions. Various reliability measures were calculated using the supplementary variable technique and the theory of Markov vector processes. Sensitivity and relative sensitivity analyses were also performed for the system parameters. Finally, numerical calculations and graphical analyses were performed to validate the derived indices.


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## Body

## 1. Introduction

The economic progress of any country depends to a great extent on the development of the industrial and mechanical fields. This progress leads to the emergence of new technical skills that help in solving problems that may arise in these complex industrial systems. However, despite all these kinds of help, consumers want cost-effective and highly efficient industrial systems. Therefore, system designers and researchers in this field face a great challenge to develop many systems and increase their efficiency, reliability, and safety.In the past, many research studies have been done to explain the concept of reliability, to increase the efficiency of many industrial systems, and to analyze the cost of different redundant systems under the effect of some constraints such as the periods of time when the system is on or off, the different types of faults affecting this system, the different types of repair of these units, the different types of leave for repairmen, etc. Reliability measures such as mean time to failure are effective measures when analyzing engineering models. The applications of reliability were introduced in many papers in literature (see, for example, [1]). Lin et al. [2] explained that the maintenance of an offshore wind turbine accounts for more than 30% of the total cost of its life cycle. According to Ren et al. [3], preventive maintenance based on a predetermined period is considered as a more advanced approach. Optimizing the selection of the specified interval can help, in turn, to optimize the preventive maintenance strategies. Preventive maintenance brings various advantages such as avoiding unplanned maintenance, combining maintenance and repairs, and optimizing maintenance tasks.Most research in the past focused on three types of classical random shock models: cumulative shock models, extreme shock models, andδ-shock models. In this context, we present some research papers that deserve to be mentioned. For example, Lam and Zhang [4] studied a shock model for the maintenance problem of a repairable system. Li and Zhao [5] presented complex systems containing n i.i.d. units with the δ-shock model. Recently, El-Sherbeny [6] studied the behavior of the repairable cold standby system under the effect of Poisson shocks. In discussing the above research, we have found that the failed device is repaired immediately, but, in reality, the failed device may not be repaired immediately for various reasons. In discussing the above researches, we have found that the failed device is repaired immediately, but in reality, the failed device may not be repaired immediately because the multiple leaves for the technician or he is busy repairing another device. This happens in medium or small factories because they may not be able to afford to hire a full-time repair technician to take care of their equipment. Instead, a technician may have to take care of many types of equipment, and he may not be able to repair a failed piece of equipment as soon as it fails. For this reason, the reliability theory has been studied with vacation [7–13]. Therefore, the vacation concept was first presented in the model analysis of queuing systems in [14], and the types of vacation policies are defined as follows: single vacation and multiple vacations. For a comprehensive and excellent study of the latest results for a variety of leave models, including some applications [15], Ke and Wang [16] studied a machine repair problem with different vacation policies, both based on a queuing theory viewpoint.In the current research, the deterioration to the system from the first shock can be neglected, and the system fails once the shocks accumulate to a certain level. However, the concept of the policy of optional leave of repairers with the effect of Poisson shocks for any repairable system has not appeared in the previous literature. Therefore, in light of the above observations, we have analyzed here a repairable system with optional vacations for a repairman in starting the system work under the effect of Poisson shocks on active units using the supplementary variables technique. Suppose that the system could be subjected to shocks that follow a Poisson process, and the technician has the choice of either staying in the system to monitor the two units until one fails or taking leave when the two units are operating normally. Note that analyzing the behavior of the system is not a simple task when most of the time the system follows the general distributions. The method of supplementary variable helps to solve the partial differential equations associated with the description of the dynamics of motion between the different states of the system. Using the ergodicity of the investigated process and the method of supplementary variables, we obtain explicit expressions of the reliability metrics such as reliability function, steady-state availabilities, mean time to system failure, the steady-state probability that the repairman is on vacation, the steady-state probability that the system is waiting for repair, and steady-state failure frequency.This article is designed as follows: Section2 introduces more details about the system and assumptions of the system description. In Sections 3 and 4, the state equations of the system are constructed to derive the steady-state availability before first failure and the mean time to first failure using the supplementary variable technique method. The numerical examples are presented in Section 5. In Section 6, sensitivity and relative sensitivity analyses of the reliability characteristics are performed. The conclusions are summarized in the last section.


## 2. Description of the System and Assumptions

The system which consists of two nonidentical units and a single repairman under Poisson shock is subjected to the following assumptions:(i)
A1: in the initial period, both devices operate at high efficiency, and a repairman has the choice of either staying in the system or taking a vacation.(ii)
A2: the system is exposed to shocks continually. The arrival of shocks is considered as a Poisson processSt,t≥0 with strength λi>0. The magnitude of each shock Y^, independent random variable with distribution function F.(iii)
A3: when a shock occurs, it affects the active units (A, B). The active units will fail when the magnitude of a shock exceeds a threshold. The threshold of units is a non-negative random variableτi with a distribution function ξi.(iv)
A4: when any unit fails with the existence of the repairman in the system, it will be repaired immediately. Once the repairman is done repairing the failed units, the repairman has the choice to either stay at the system or take a vacation, and then, he returns from vacation if at least one unit is failing. The repair rule is ‘‘first-in-first-out’’. If a unit fails while the other is being repaired, the recently failed unit must wait for repair, and the system has to stop working.(v)
A5: shocks are the main reason for units to fail, and the system fails only if both the units fail.Based on the preceding assumptions, we can conclude that the failure probability of units, given the shock valuey^, ξiy^=Pτi≤y^. Since the magnitude of a shock is a random variable Y^, the conditional failure probability of units is a random variable ξiY^i=1,2, and their probability distribution can be written by(1)Piy=PξiY^≤y=PY^≤ξi−1y=Fξi−1y,0≤y≤1,i=1,2.According to the above assumptionsA2andA3, the probability that single shock causes unit (A or B) to fail is(2)ri=PY^>τi=∫0∞Pτi<y^|Y^=y^dPY^≤y^=∫0∞ξiy^dFy^,0≤y≤1,i=1,2.


## 3. System Analysis

The states of this systemΩt at time t as a following:(i)
S0: at any time t, unit A is active, unit B is active, and the repairman has the choice to either stay at the system or take a vacation.(ii)
S1: at any time t, unit A is active, unit B is being repaired, the repairman is still in the system, and the system is working.(iii)
S2: at any time t unit A is being repaired, unit B is active, the repairman is still in the system, and the system is working.(iv)
S3: at any time t, unit A is active, unit B is active, the repairman chooses to take a vacation, and the system is working.(v)
S4: at any time t, unit A is still repaired from above S2, unit B is waiting for being repaired, and the system is down.(vi)
S5: at any time t, unit A is active, unit B is waiting for repair, the repairman is on vacation, and the system is working.(vii)
S6: at any time t, unit A is waiting for repair, unit B is active, the repairman is on vacation, and the system is working.(viii)
S7: at any time t, unit A is still waiting for repair from S6, unit B is waiting for repair, the repairman is on vacation, and the system is down.(ix)
S8: at any time t, unit A is waiting for being repaired, unit B is under repair, and the system is down.(x)
S9: at any time t, unit B is still waiting for repair from S5, unit A is waiting for repair, the repairman is on vacation, and the system is down.The state space isE=S0,S1,S2,S3,S4,S5,S6,S7,S8,S9, where the working state is U=S0,S1,S2,S3,S5,S6 and the down state is D=S4,S7,S8,S9 as shown in Figure 1. According to the above assumptions, and Ωt,t≥0 is not a Markov process, we using the supplementary variables as a following:(i)
X1t is time to make a vacation decision. Ωt=S0(ii)
X2t is the elapsed vacation time. Ωt=S5,S6,S7,S9(iii)
Y1t is the elapsed repair time of unit A being repaired at time t. Ωt=S2,S4(iv)
Y2t is the elapsed repair time of unit B being repaired at time t. Ωt=S1,S8Figure 1
Transition diagram of the system.State space is as follows:Ω∗=0,u,1,y,2,x,3,4,x,5,z,6,z,7,z,8,y,9,z where u,z,x, and y are explanatory values of X1t,X2t,Y1t, and Y2t, respectively. Subject to(3)Q0t,u=ρΩt=0,X1t≤u,Qit,z=ρΩt=i,X2t≤z,i=5,6,7,9,Qit,x=ρΩt=i,Y1t≤x,i=2,4,Qit,y=ρΩt=i,Y2t≤y,i=1,8,where ρE is the probability of event E.(4)ρit,w=ddwQit,w,i=0,1,2,4,5,6,7,8,9,ϕit=ρst=i,i=0,1,2,4,5,6,7,8,9.From here, we can formulate the differential equations that represent this system by using the probability arguments and limiting transitions as follows:(5)ρ0t+Δt,u+Δt=ρ0t,u1−λ1r1+λ2r2+φuΔt+oΔt.WhenΔt tend to zero, we get(6)∂∂t+∂∂u+λ1r1+λ2r2+φuρ0t,u=0.In the same style, we get the following:(7)∂∂t+∂∂y+λ1r1+μ2yρ1t,y=0,(8)∂∂t+∂∂x+λ2r2+μ1xρ2t,x=0,(9)ddt+λ1r1+λ2r2ϕ3t=∫0∞ρ0u,tφudu,(10)∂∂t+∂∂x+μ1xρ4t,x=λ2r2ρ2t,x,(11)∂∂t+∂∂z+λ1r1+αzρ5t,z=0,(12)∂∂t+∂∂z+λ2r2+αzρ6t,z=0,(13)∂∂t+∂∂z+αzρ7t,z=λ2r2ρ6t,z,(14)∂∂t+∂∂y+μ2yρ8t,y=λ1r1ρ1t,y,(15)∂∂t+∂∂z+αzρ9t,z=λ1r1ρ5t,z.The boundary conditions are as follows:(16)ρ0t,0=∫0∞ρ1t,yμ2ydy+∫0∞ρ2t,xμ1xdx+εt,(17)ρ1t,0=∫0∞λ2r2ρ0t,udu+∫0∞ρ4t,xμ1xdx+∫0∞ρ5t,zαzdz,(18)ρ2t,0=∫0∞λ1r1ρ0t,udu+∫0∞ρ8t,yμ2ydy+∫0∞ρ6t,zαzdz,(19)ρ6t,0=λ1r1ϕ3t,(20)ρ6t,0=λ1r1ϕ3t,(21)ρ7t,0=ρ9t,0=0,(22)ρ4t,0=∫0∞ρ7t,zαzdz,ρ8t,0=∫0∞ρ9t,zαzdz,(23)∑i=02∫0∞ρit,mdm+ϕ3t+∑i=49∫0∞ρit,mdm=1,m=u,x,y,z.The initial conditions are as follows:(24)∑i=02∫0∞ρit,mdm+ϕ3t+∑i=49∫0∞ρit,mdm=1,m=u,x,y,z.Laplace transform can be defined as follows:(25)k∗s=Lkx=∫0∞kxe−sxdx,s>0.The ergodicity of the investigation process guarantees the presence of the following steady-probability:ϕi=limt⟶∞ϕit,, i=1,…,9. θim=limt⟶∞ρit,m, i=1,2,4,5,6,7,8,9, which follows the following relations: ϕi=∫0∞θimdm,i=1,2,4,5,6,7,8,9.Taking the limitt⟶∞ in the equations (6)–(23), the following equations are obtained:(26)ddu+λ1r1+λ2r2+φuθ0u=0,ddy+λ1r1+μ2yθ1y=0,∂∂x+λ2r2+μ1xθ2x=0,λ1r1+λ2r2ϕ3=∫0∞θ0uφudu,ddx+μ1xθ4x=λ2r2θ2x,ddz+λ1r1+αzθ5z=0,∂∂z+λ2r2+αzθ6z=0,ddz+αzθ7z=λ2r2θ6z,ddy+μ2yθ8y=λ1r1θ1y,ddz+αzθ9z=λ1r1θ5z,θ00=∫0∞θ1yμ2ydy+∫0∞θ2xμ1xdx,θ10=∫0∞λ2r2θ0udu+∫0∞θ4xμ1xdx+∫0∞θ5zαzdz,θ20=∫0∞λ1r1θ0udu+∫0∞θ8yμ2ydy+∫0∞θ6zαzdz,θ50=λ2r2ϕ3,θ60=λ1r1ϕ3,θ70=θ90=0,θ40=∫0∞θ7zαzdz,θ80=∫0∞θ9zαzdz,∑i=02∫0∞θimdm+ϕ3+∑i=49∫0∞θimdm=1,m=u,x,y,z.Steady-state availability is as follows:(27)Av∞=ϕ0+ϕ1+ϕ2+ϕ3+ϕ5+ϕ6,where(28)ϕ0=C0V¯1∗λ1r1+λ2r2,ϕ1=C0a1−λ1r1V¯1∗λ1r1+λ2r2h1∗λ2r2−1+a2H¯2∗λ1r1h1∗λ2r2h2∗λ1r1−1−h2∗λ1r1,ϕ2=−C0λ12r12V¯1∗λ1r1+λ2r2+a3+a4H¯1∗λ2r2λ1r1+λ2r2h1∗λ2r2h2∗λ1r1−1−h2∗λ1r1,ϕ3=C0v1∗λ1r1+λ2r2λ1r1+λ2r2,ϕ5=C0λ2r2V¯2∗λ1r1v1∗λ1r1+λ2r2λ1r1+λ2r2,ϕ6=C0λ1r1V¯2∗λ2r2v1∗λ1r1+λ2r2λ1r1+λ2r2,C0=αλ1r1+λ2r2μ1μ2h1∗λ2r2h2∗λ1r1−1−h2∗λ1r1a5,a1=−λ2r2λ1r1+λ2r2V¯1∗λ1r1+λ2r2+v1∗λ1r1+λ2r21+h1∗λ2r2v2∗λ1r1−1λ1r1+λ2r2,a2=v1∗λ1r1+λ2r21−h1∗λ2r2v2∗λ2r2λ1r1+λ2r2,a3=λ2r2−λ2r2V¯1∗λ1r1+λ2r2h2∗λ1r1−1+v1∗λ1r1+λ2r21−h2∗λ1r1v2∗λ1r1,a4=λ1r1−λ2r2V¯1∗λ1r1+λ2r2h2∗λ1r1−2+v1∗λ1r1+λ2r21+h2∗λ1r1v2∗λ2r2−1,a5=αλ12r12V¯1∗λ1r1+λ2r2−μ2+μ1h1∗λ2r2−1−αλ22r22V¯1∗λ1r1+λ2r2μ1−μ2h2∗λ1r1−1,+αμ2μ1h1∗λ2r2h2∗λ1r1−1−h2∗λ1r1v1∗λ1r1+λ2r2+λ2r2μ1μ2−h1∗λ2r2+h1∗λ2r2−1,h2∗λ1r1αV¯1∗λ1r1+λ2r2+v1∗λ1r1+λ2r2−αv1∗λ1r1+λ2r21+h1∗λ2r2−1h2∗λ1r1,v2∗λ1r1−1+αμ2v1∗λ1r1+λ2r2v2∗λ1r1h2∗λ1r1−1+λ1r1αλ2r2V¯1∗λ1r1+λ2r2,μ1h1∗λ2r2−2+μ2h2∗λ1r1−2−αμ2v1∗λ1r1+λ2r21+h1∗λ2r2h2∗λ1r1−1v2∗λ2r2−1,+μ1μ2h1∗λ2r2−1h2∗λ1r1−h1∗λ2r2v1∗λ1r1+λ2r2+αV¯1∗λ1r1+λ2r2,+αv1∗λ1r1+λ2r2v2∗λ2r2h1∗λ2r2−1.


## 4. Mean Time to the First Failure (MTTFF)

In this section, we deduce the mean time to the first failure (MTTFF) of the system. We assumed thatt be the time to the first failure of the system; therefore, the reliability function of this system is calculated as follows: Rt=Pτ>t. To obtain the reliability function, we consider the failure states {4, 7, 8, 9} of the system are absorbing states.Let(29)L0t,u=dduρS˜t=0,X˜1t≤u,Lit,u=ddzρS˜t=i,X˜2t≤z,i=5,6,L1t,y=ddyρS˜t=1,Y˜2t≤y,L3t=ρS˜t=3,L2t,x=ddxρS˜t=2,Y˜1t≤x.In the same manner as previously mentioned in Section3, we conclude the reliability function as follows:(30)∂∂t+∂∂u+λ1r1+λ2r2+φuL0t,u=0,(31)∂∂t+∂∂y+λ1r1+μ2yL1t,y=0,(32)∂∂t+∂∂x+λ2r2+μ1xL2t,x=0,(33)ddt+λ1r1+λ2r2L3t=∫0∞L0u,tφudu,(34)∂∂t+∂∂z+λ1r1+αzL5t,z=0,(35)∂∂t+∂∂z+λ2r2+αzL6t,z=0.The boundary conditions are as follows:(36)L0t,0=∫0∞L1t,yμ2ydy+∫0∞L2t,xμ1xdx+εt,(37)L1t,0=∫0∞λ2r2L0t,udu+∫0∞L5t,zαzdz,(38)L2t,0=∫0∞λ1r1L0t,udu+∫0∞L6t,zαzdz,(39)L5t,0=λ2r2L3t,(40)L6t,0=λ1r1L3t.The initial conditions are as follows:(41)R00,u=εu=1,u=0,0,u≠0.Taking the Laplace transform of the equations (30)–(40), as well as initial conditions, we have(42)ddu+s+λ1r1+λ2r2+φuL0∗s,u=0,ddy+s+λ1r1+μ2yL1∗s,y=0,∂∂x+s+λ2r2+μ1xL2∗s,x=0,s+λ1r1+λ2r2L3∗s=∫0∞L0∗u,sφudu,ddz+s+λ1r1+αzL5∗s,z=0,ddz+s+λ2r2+αzL6∗s,z=0,L0∗s,0=∫0∞L1∗s,yμ2ydy+∫0∞L2∗s,xμ1xdx+εt,L1∗s,0=∫0∞λ2r2L0∗s,udu+∫0∞L5∗s,zαzdz,L2∗s,0=∫0∞λ1r1L0∗s,udu,L5∗s,0=λ2r2L3∗s,L6∗s,0=λ1r1L3∗s.From the previous equations, we defined the reliability function as follows:(43)R∗s=∫0∞L0∗s,udu+∫0∞L1∗s,ydy+∫0∞L2∗s,xdx+∑i=56∫0∞Li∗s,zdz+L3∗s,and the mean time to the first failure of the system (MTTFF) is given by(44)MTTFF=lims⟶0R∗s,where(45)L0∗s=∫0∞L0∗s,udu=L0∗s,0V¯1∗s+r1λ1+r2λ2,L1∗s=∫0∞L1∗s,ydy=L1∗s,0H¯2∗s+r1λ1,L2∗s=∫0∞L2∗s,xdx=L2∗s,0H¯1∗s+r2λ2,L3∗s=L0∗s,0v1∗s+r1λ1+r2λ2s+r1λ1+r2λ2,L5∗s=∫0∞L5∗s,zdz=L5∗s,0V¯2∗s+r1λ1,L6∗s=∫0∞L6∗s,zdz=L6∗s,0V¯2s+r2λ2,L0∗s,0=−ε∗ss+r1λ1+r2λ2−s+r2λ2−1+h2∗s+r1λ1v1∗s+r1λ1+r2λ2v2∗s+r1λ1,+r1λ1−1+h1∗s+r2λ2v1∗s+r1λ1+r2λ2v2∗s+r2λ2+s+r1λ1+r2λ2r1λ1h1∗s+r2λ2,+r2λ2h2∗s+r1λ1V¯1∗s+r1λ1+r2λ2,L1∗s,0=r2λ2L0∗s,0V¯1∗s+r1λ1+r2λ2+v1∗s+r1λ1+r2λ2v2∗s+r2λ2s+r1λ1+r2λ2,L2∗s,0=r1λ1L0∗s,0V¯1∗s+r1λ1+r2λ2+v1∗s+r1λ1+r2λ2v2∗s+r2λ2s+r1λ1+r2λ2,L5∗s,0=r2λ2L0∗s,0v1∗s+r1λ1+r2λ2s+r1λ1+r2λ2,L6∗s,0=r1λ1L0∗s,0v1∗s+r1λ1+r2λ2s+r1λ1+r2λ2.


## 5. Numerical Analysis and Discussion

In this section, the usefulness of the proposed system is demonstrated by examining the effects of the number of shocks and other parameters on the system using the following numerical figures, taking into account that(46)V1t=1−e−φt,t>0,0,t≤0,V2t=1−e−αt,t>0,0,t≤0,Hit=1−e−μit,t>0,0,t≤0,∀i=1,2.The study of the behavior of steady-state availability and mean time to system failure for the number of shocks can be clearly seen in Figures2–9, respectively. Figures 2–5 show the effects of both r1 and r2 on steady-state availability versus the effects of various system parameters. Likewise, Figures 6–5 discuss the same effects of both r1 and r2 on mean time to system failure against various system parameters. These results are summarized as follows.Figure 2
Steady-state availability versus rateμ1 when φ=0.2,α=0.1,λ1=0.4,λ2=0.5,andμ2=0.5.Figure 3
Steady-state availability versus rateμ2 when φ=0.2,α=0.1,λ1=0.4,λ2=0.5,andμ1=0.6.Figure 4
Steady-state availability versus rateα when φ=0.2,λ1=0.4,λ2=0.5,μ1=0.6,andμ2=0.5.Figure 5
Steady-state availability versus rateφ when α=0.2,λ1=0.4,λ2=0.5,μ1=0.6,andμ2=0.5.Figure 6
Mean time to system failure versus rateμ1 when φ=0.3,α=0.5,λ1=0.4,λ2=0.5,andμ2=0.5.Figure 7
Mean time to system failure versus rateμ2 when φ=0.3,α=0.5,λ1=0.4,λ2=0.5,andμ1=0.6.Figure 8
Mean time to system failure versus rateα when φ=0.4,λ1=0.4,λ2=0.5,μ1=0.6,andμ2=0.5.Figure 9
Mean time to system failure versus rateφ when α=0.01,λ1=0.4,λ2=0.5,μ1=0.6,andμ2=0.5.In Figures2 and 6, we can clearly observe the effects of the different cases of r1 and r2 on the steady-state availability and mean time to system failure against μ1∈0,1 at φ=0.2,α=0.1,λ1=0.4,λ2=0.5,μ2=0.5. The steady-state availability and the mean time to system failure are increased when r1=r2=0.2, and the curves start to decrease gradually when r1=0.2<r2=0.4 and r1=0.4>r2=0.2, respectively.Figures3 and 7 show the effect of r1 and r2 on steady-state availability and mean time to system failure versus μ2∈0,1 at φ=0.2,α=0.2,λ1=0.4,λ2=0.5,μ1=0.6. The steady-state availability and mean time to system failure increase in the case of r1=r2=0.2, while they decrease regularly when r1=0.2<r2=0.4 and r1=0.4>r2=0.2.In Figures4 and 8, we can discuss the effects of parameters r1 and r2 on steady-state availability and mean time to system failure against parameter α∈0,1 when φ=0.2,λ1=0.4,λ2=0.5,μ1=0.6,μ2=0.5. The steady-state availability and mean time to system failure decrease in the cases r1=0.2<r2=0.4 and r1=0.4>r2=0.2, but they increase when r1=r2=0.2.The effects of parametersr1 and r2 on steady-state availability and mean time to system failure against parameter φ∈0,1 at α=0.2,λ1=0.4,λ2=0.5,μ1=0.6,μ2=0.5 can be seen in Figures 5 and 9. From these curves, it can see that the steady-state availability and mean time to system failure increase when r1=r2=0.2, while they gradually decrease when r1=0.2<r2=0.4 and r1=0.4>r2=0.2.From Figures2–9, we can come to the following conclusion:(1)
In the steady-state availability,Av∞ increase when r1=r2=0.2(2)
In the mean time to system failure,MTTFF increase when r1=r2=0.2


## 6. Sensitivity and Relative Sensitivity Analysis

In this section, we perform sensitivity and relative sensitivity analysis of mean time to system failure (MTTFF) and steady-state availability,Av∞ along with changes in certain values of the system parameters φ,λ1,λ2,μ1,μ2, and α.


### 6.1. Sensitivity and Relative Sensitivity Analysis for MTTFF

We defined the sensitivity analysis for MTTFF via the differentiation of (44) with respect to κ.(47)ηκ=∂MTTFF∂κ.Moreover, relative sensitivity analysis for MTTFF defined as(48)θκ=ηκκMTTFF,where κ=φ,λ1,λ2,μ1,μ2,α.


### 6.2. Sensitivity and Relative Sensitivity Analysis forAv∞

We defined sensitivity analysis forAv∞ via differentiating (27) with respect to κ.(49)σκ=∂Av∞∂κ.In addition, relative sensitivity analysis ofAv∞ can be written in the form(50)∏κ=σkkAv∞.The results of the sensitivity analysis to the MTTFF affected by each system parameter are shown in Tables1–3. From these tables, the order of the sensitivity analysis influenced by each parameter can be derived as follows:(1)
In the caser1=r2, we get λ1>λ2>φ>α>μ2>μ1(2)
In the caser1>r2, we get λ2>λ1>φ>α>μ2>μ1(3)
In the caser1<r2, we get λ1>λ2>φ>α>μ2>μ1Table 1
Sensitivity ofMTTFF when r1=r2.
λ1ηλ1λ2ηλ2μ1ημ1μ2ημ2αηαφηφ0.1−685.960.1−654.430.113.26370.113.6920.131.22020.1−46.4180.2−169.4610.2−161.910.27.553930.28.007390.222.03480.2−17.6510.3−74.16230.3−70.9870.34.870810.35.247720.316.3790.3−9.22420.4−41.02330.4−39.3400.43.399470.43.702910.412.65110.4−5.65550.5−25.82140.5−24.8050.52.506530.52.751930.510.0650.5−3.8186Table 2
Sensitivity ofMTTFF when r1>r2.
λ1ηλ1λ2ηλ2μ1ημ1μ2ημ2αηαφηφ0.1−338.9210.1−587.300.17.591070.17.761910.113.86990.1−24.2430.2−82.04650.2−146.140.24.679820.25.312030.29.679360.2−10.9310.3−35.30590.3−64.5450.33.171010.33.862450.37.16660.3−6.19340.4−19.33990.4−36.0490.42.289620.42.934380.45.52770.4−3.98110.5−12.14290.5−22.9020.51.730470.52.304650.54.396060.5−2.7727Table 3
Sensitivity ofMTTFF when r1<r2.
λ1ηλ1λ2ηλ2μ1ημ1μ2ημ2αηαφηφ0.1−603.2430.1−323.820.17.567290.17.943140.114.14190.1−24.6690.2−149.9520.2−78.6800.25.004910.24.967220.29.933210.2−11.0180.3−66.13840.3−33.9690.33.553720.33.397110.37.368090.3−6.21250.4−36.8860.4−18.6510.42.653070.42.468890.45.684930.4−3.98180.5−23.40.5−11.7270.52.055990.51.874990.54.519990.5−2.7679Moreover, the signs of the sensitivity to MTTFF affected by the parametersλ1,λ2 and φ are negative, which means that increasing the parameter worsens the sensitivity to MTTFF; the signs of the sensitivity to MTTFF affected by the parameters μ1,μ2 and α are positive, which means that increasing the parameter improves the sensitivity to MTTFF.The relative sensitivity is completed. The results of relative sensitivity for MTTFF can be found in Tables4–6. From these tables, we can see that the order of relative sensitivity on MTTFF affected by each system parameter can be ranked as follows:(1)
In the caser1=r2, we get λ1>λ2>φ>α>μ2>μ1(2)
In the caser1>r2, we get λ2>λ1>φ>α>μ2>μ1(3)
In the caser1<r2, we get λ1>λ2>φ>α>μ2>μ1Table 4
Relative sensitivity ofMTTFF when r1=r2.
λ1θλ1λ2θλ2μ1θμ1μ2θμ2αθαφθφ0.1−0.89780.1−0.91070.10.075430.10.077560.10.15530.1−0.19140.2−0.80230.2−0.82490.20.081290.20.085640.20.19390.2−0.16510.3−0.71720.3−0.74660.30.076140.30.08140.30.19950.3−0.13760.4−0.64350.4−0.67730.40.069390.40.074850.40.19410.4−0.11670.5−0.58040.5−0.61680.50.063010.50.068430.50.18510.5−0.1009Table 5
Relative sensitivity ofMTTFF when r1>r2.
λ1θλ1λ2θλ2μ1θμ1μ2θμ2αθαφθφ0.1−0.80220.1−0.95210.10.056850.10.058520.10.092740.1−0.13930.2−0.64350.2−0.90290.20.067090.20.076390.20.120140.2−0.13860.3−0.52670.3−0.85470.30.066370.30.080690.30.12690.3−0.12420.4−0.44220.4−0.80870.40.06270.40.079860.40.12580.4−0.11010.5−0.38040.5−0.76570.50.058450.50.077050.50.121650.5−0.0982Table 6
Relative sensitivity ofMTTFF when r1<r2.
λ1θλ1λ2θλ2μ1θμ1μ2θμ2αθαφθφ0.1−0.94340.1−0.82490.10.06150.10.06350.10.10060.1−0.14920.2−0.8860.2−0.67750.20.07750.20.07560.20.13030.2−0.1480.3−0.83090.3−0.56440.30.079960.30.07510.30.13730.3−0.13260.4−0.77910.4−0.48020.40.07780.40.071340.40.13580.4−0.11750.5−0.73130.5−0.41710.50.07410.50.066690.50.13090.5−0.1046The sensitivity forA∞ can be obtained in Tables 7–9. In these tables, we can see that the order of sensitivity on steady-state availability affected by each system parameter can be ranked as follows:(1)
In the caser1=r2, we get α>μ2>μ1>λ2>λ1>φ(2)
In the caser1>r2, we get α>μ2>μ1>λ1>λ2>φ(3)
In the caser1<r2, we get α>μ2>μ1>λ2>λ1>φTable 7
Sensitivity ofAv∞ when r1=r2.
λ1σλ1λ2σλ2μ1σμ1μ2σμ2ασαφσφ0.1−0.8660.1−0.91430.11.52660.11.54650.12.21480.1−0.78010.2−0.55790.2−0.59310.20.37090.20.39080.20.68290.2−0.3280.3−0.37960.3−0.4060.30.13420.30.14650.30.30470.3−0.17880.4−0.26910.4−0.2890.40.059460.40.067410.40.16510.4−0.11250.5−0.19690.5−0.21230.50.029680.50.035070.50.10090.5−0.0773Table 8
Sensitivity ofAv∞ when r1>r2.
λ1σλ1λ2σλ2μ1σμ1μ2σμ2ασαφσφ0.1−1.1160.1−1.02660.11.6480.11.6590.12.1740.1−0.86940.2−0.53820.2−0.69120.20.41580.20.49170.20.7030.2−0.39660.3−0.29570.3−0.49110.30.14380.30.20220.30.323010.3−0.22610.4−0.17660.4−0.36240.40.057370.40.098570.40.17860.4−0.14590.5−0.11170.5−0.27550.50.023850.50.053330.50.11080.5−0.1018Table 9
Sensitivity ofAv∞ when r1<r2.
λ1σλ1λ2σλ2μ1σμ1μ2σμ2ασαφσφ0.1−0.99780.1−1.18620.11.6530.11.6640.12.2240.1−0.90140.2−0.67010.2−0.57860.20.47520.20.43920.20.73190.2−0.40880.3−0.47380.3−0.31980.30.18790.30.1590.30.3390.3−0.23230.4−0.34820.4−0.19130.40.087890.40.067270.40.18780.4−0.14960.5−0.26370.5−0.12060.50.045370.50.030650.50.11660.5−0.1043Finally, the results of the relative sensitivity on steady-state availability affected by each system parameter are listed in Tables10–12. The order of relative sensitivity on steady-state availability affected by each system parameter can be ranked as follows:(1)
In the caser1=r2, we get α>μ2≃μ1>λ2>φ>λ1(2)
In the caser1>r2, we get α>μ2>μ1>λ1>λ2>φ(3)
In the caser1<r2, we get α>μ1>μ2>λ2>φ>λ1Table 10
Relative sensitivity ofAv∞ when r1=r2.
λ1∏λ1λ2∏λ2μ1∏μ1μ2∏μ2α∏αφ∏φ0.1−0.097580.1−0.10370.10.26180.10.26670.10.31670.1−0.10080.2−0.1370.2−0.14690.20.11230.20.11840.20.16580.2−0.09040.3−0.14760.3−0.16060.30.05890.30.06430.30.10510.3−0.07670.4−0.14560.4−0.15980.40.03440.40.038820.40.073980.4−0.06570.5−0.13750.5−0.15190.50.021280.50.02510.50.055790.5−0.0571Table 11
Relative sensitivity ofAv∞ when r1>r2.
λ1∏λ1λ2∏λ2μ1∏μ1μ2∏μ2α∏αφ∏φ0.1−0.13650.1−0.11810.10.31260.10.32690.10.33280.1−0.1170.2−0.14560.2−0.17620.20.13560.20.16390.20.18010.2−0.11610.3−0.1270.3−0.20260.30.067510.30.095980.30.11730.3−0.10380.4−0.10450.4−0.21170.40.035420.40.061010.40.084030.4−0.09180.5−0.08430.5−0.21080.50.018290.50.040810.50.064050.5−0.0817Table 12
Relative sensitivity ofAv∞ when r1<r2.
λ1∏λ1λ2∏λ2μ1∏μ1μ2∏μ2α∏αφ∏φ0.1−0.1150.1−0.14690.10.33290.10.32610.10.34710.1−0.12310.2−0.16940.2−0.15980.20.16170.20.14680.20.19020.2−0.12170.3−0.19340.3−0.14090.30.091220.30.076180.30.12390.3−0.10880.4−0.20060.4−0.11670.40.055710.40.042290.40.088890.4−0.09610.5−0.19860.5−0.09410.50.035590.50.023910.50.067730.5−0.0855


## 6.1. Sensitivity and Relative Sensitivity Analysis for MTTFF

We defined the sensitivity analysis for MTTFF via the differentiation of (44) with respect to κ.(47)ηκ=∂MTTFF∂κ.Moreover, relative sensitivity analysis for MTTFF defined as(48)θκ=ηκκMTTFF,where κ=φ,λ1,λ2,μ1,μ2,α.


## 6.2. Sensitivity and Relative Sensitivity Analysis forAv∞

We defined sensitivity analysis forAv∞ via differentiating (27) with respect to κ.(49)σκ=∂Av∞∂κ.In addition, relative sensitivity analysis ofAv∞ can be written in the form(50)∏κ=σkkAv∞.The results of the sensitivity analysis to the MTTFF affected by each system parameter are shown in Tables1–3. From these tables, the order of the sensitivity analysis influenced by each parameter can be derived as follows:(1)
In the caser1=r2, we get λ1>λ2>φ>α>μ2>μ1(2)
In the caser1>r2, we get λ2>λ1>φ>α>μ2>μ1(3)
In the caser1<r2, we get λ1>λ2>φ>α>μ2>μ1Table 1
Sensitivity ofMTTFF when r1=r2.
λ1ηλ1λ2ηλ2μ1ημ1μ2ημ2αηαφηφ0.1−685.960.1−654.430.113.26370.113.6920.131.22020.1−46.4180.2−169.4610.2−161.910.27.553930.28.007390.222.03480.2−17.6510.3−74.16230.3−70.9870.34.870810.35.247720.316.3790.3−9.22420.4−41.02330.4−39.3400.43.399470.43.702910.412.65110.4−5.65550.5−25.82140.5−24.8050.52.506530.52.751930.510.0650.5−3.8186Table 2
Sensitivity ofMTTFF when r1>r2.
λ1ηλ1λ2ηλ2μ1ημ1μ2ημ2αηαφηφ0.1−338.9210.1−587.300.17.591070.17.761910.113.86990.1−24.2430.2−82.04650.2−146.140.24.679820.25.312030.29.679360.2−10.9310.3−35.30590.3−64.5450.33.171010.33.862450.37.16660.3−6.19340.4−19.33990.4−36.0490.42.289620.42.934380.45.52770.4−3.98110.5−12.14290.5−22.9020.51.730470.52.304650.54.396060.5−2.7727Table 3
Sensitivity ofMTTFF when r1<r2.
λ1ηλ1λ2ηλ2μ1ημ1μ2ημ2αηαφηφ0.1−603.2430.1−323.820.17.567290.17.943140.114.14190.1−24.6690.2−149.9520.2−78.6800.25.004910.24.967220.29.933210.2−11.0180.3−66.13840.3−33.9690.33.553720.33.397110.37.368090.3−6.21250.4−36.8860.4−18.6510.42.653070.42.468890.45.684930.4−3.98180.5−23.40.5−11.7270.52.055990.51.874990.54.519990.5−2.7679Moreover, the signs of the sensitivity to MTTFF affected by the parametersλ1,λ2 and φ are negative, which means that increasing the parameter worsens the sensitivity to MTTFF; the signs of the sensitivity to MTTFF affected by the parameters μ1,μ2 and α are positive, which means that increasing the parameter improves the sensitivity to MTTFF.The relative sensitivity is completed. The results of relative sensitivity for MTTFF can be found in Tables4–6. From these tables, we can see that the order of relative sensitivity on MTTFF affected by each system parameter can be ranked as follows:(1)
In the caser1=r2, we get λ1>λ2>φ>α>μ2>μ1(2)
In the caser1>r2, we get λ2>λ1>φ>α>μ2>μ1(3)
In the caser1<r2, we get λ1>λ2>φ>α>μ2>μ1Table 4
Relative sensitivity ofMTTFF when r1=r2.
λ1θλ1λ2θλ2μ1θμ1μ2θμ2αθαφθφ0.1−0.89780.1−0.91070.10.075430.10.077560.10.15530.1−0.19140.2−0.80230.2−0.82490.20.081290.20.085640.20.19390.2−0.16510.3−0.71720.3−0.74660.30.076140.30.08140.30.19950.3−0.13760.4−0.64350.4−0.67730.40.069390.40.074850.40.19410.4−0.11670.5−0.58040.5−0.61680.50.063010.50.068430.50.18510.5−0.1009Table 5
Relative sensitivity ofMTTFF when r1>r2.
λ1θλ1λ2θλ2μ1θμ1μ2θμ2αθαφθφ0.1−0.80220.1−0.95210.10.056850.10.058520.10.092740.1−0.13930.2−0.64350.2−0.90290.20.067090.20.076390.20.120140.2−0.13860.3−0.52670.3−0.85470.30.066370.30.080690.30.12690.3−0.12420.4−0.44220.4−0.80870.40.06270.40.079860.40.12580.4−0.11010.5−0.38040.5−0.76570.50.058450.50.077050.50.121650.5−0.0982Table 6
Relative sensitivity ofMTTFF when r1<r2.
λ1θλ1λ2θλ2μ1θμ1μ2θμ2αθαφθφ0.1−0.94340.1−0.82490.10.06150.10.06350.10.10060.1−0.14920.2−0.8860.2−0.67750.20.07750.20.07560.20.13030.2−0.1480.3−0.83090.3−0.56440.30.079960.30.07510.30.13730.3−0.13260.4−0.77910.4−0.48020.40.07780.40.071340.40.13580.4−0.11750.5−0.73130.5−0.41710.50.07410.50.066690.50.13090.5−0.1046The sensitivity forA∞ can be obtained in Tables 7–9. In these tables, we can see that the order of sensitivity on steady-state availability affected by each system parameter can be ranked as follows:(1)
In the caser1=r2, we get α>μ2>μ1>λ2>λ1>φ(2)
In the caser1>r2, we get α>μ2>μ1>λ1>λ2>φ(3)
In the caser1<r2, we get α>μ2>μ1>λ2>λ1>φTable 7
Sensitivity ofAv∞ when r1=r2.
λ1σλ1λ2σλ2μ1σμ1μ2σμ2ασαφσφ0.1−0.8660.1−0.91430.11.52660.11.54650.12.21480.1−0.78010.2−0.55790.2−0.59310.20.37090.20.39080.20.68290.2−0.3280.3−0.37960.3−0.4060.30.13420.30.14650.30.30470.3−0.17880.4−0.26910.4−0.2890.40.059460.40.067410.40.16510.4−0.11250.5−0.19690.5−0.21230.50.029680.50.035070.50.10090.5−0.0773Table 8
Sensitivity ofAv∞ when r1>r2.
λ1σλ1λ2σλ2μ1σμ1μ2σμ2ασαφσφ0.1−1.1160.1−1.02660.11.6480.11.6590.12.1740.1−0.86940.2−0.53820.2−0.69120.20.41580.20.49170.20.7030.2−0.39660.3−0.29570.3−0.49110.30.14380.30.20220.30.323010.3−0.22610.4−0.17660.4−0.36240.40.057370.40.098570.40.17860.4−0.14590.5−0.11170.5−0.27550.50.023850.50.053330.50.11080.5−0.1018Table 9
Sensitivity ofAv∞ when r1<r2.
λ1σλ1λ2σλ2μ1σμ1μ2σμ2ασαφσφ0.1−0.99780.1−1.18620.11.6530.11.6640.12.2240.1−0.90140.2−0.67010.2−0.57860.20.47520.20.43920.20.73190.2−0.40880.3−0.47380.3−0.31980.30.18790.30.1590.30.3390.3−0.23230.4−0.34820.4−0.19130.40.087890.40.067270.40.18780.4−0.14960.5−0.26370.5−0.12060.50.045370.50.030650.50.11660.5−0.1043Finally, the results of the relative sensitivity on steady-state availability affected by each system parameter are listed in Tables10–12. The order of relative sensitivity on steady-state availability affected by each system parameter can be ranked as follows:(1)
In the caser1=r2, we get α>μ2≃μ1>λ2>φ>λ1(2)
In the caser1>r2, we get α>μ2>μ1>λ1>λ2>φ(3)
In the caser1<r2, we get α>μ1>μ2>λ2>φ>λ1Table 10
Relative sensitivity ofAv∞ when r1=r2.
λ1∏λ1λ2∏λ2μ1∏μ1μ2∏μ2α∏αφ∏φ0.1−0.097580.1−0.10370.10.26180.10.26670.10.31670.1−0.10080.2−0.1370.2−0.14690.20.11230.20.11840.20.16580.2−0.09040.3−0.14760.3−0.16060.30.05890.30.06430.30.10510.3−0.07670.4−0.14560.4−0.15980.40.03440.40.038820.40.073980.4−0.06570.5−0.13750.5−0.15190.50.021280.50.02510.50.055790.5−0.0571Table 11
Relative sensitivity ofAv∞ when r1>r2.
λ1∏λ1λ2∏λ2μ1∏μ1μ2∏μ2α∏αφ∏φ0.1−0.13650.1−0.11810.10.31260.10.32690.10.33280.1−0.1170.2−0.14560.2−0.17620.20.13560.20.16390.20.18010.2−0.11610.3−0.1270.3−0.20260.30.067510.30.095980.30.11730.3−0.10380.4−0.10450.4−0.21170.40.035420.40.061010.40.084030.4−0.09180.5−0.08430.5−0.21080.50.018290.50.040810.50.064050.5−0.0817Table 12
Relative sensitivity ofAv∞ when r1<r2.
λ1∏λ1λ2∏λ2μ1∏μ1μ2∏μ2α∏αφ∏φ0.1−0.1150.1−0.14690.10.33290.10.32610.10.34710.1−0.12310.2−0.16940.2−0.15980.20.16170.20.14680.20.19020.2−0.12170.3−0.19340.3−0.14090.30.091220.30.076180.30.12390.3−0.10880.4−0.20060.4−0.11670.40.055710.40.042290.40.088890.4−0.09610.5−0.19860.5−0.09410.50.035590.50.023910.50.067730.5−0.0855


## 7. Conclusion

This article studies the reliability measurements and sensitivity analysis of a system consisting of two nonidentical parallel units and a single repairman. The repairman might take a vacation or not at the beginning of the system operation, and the active units might be attacked by successive shocks. Such a system can be considered as an extension of repairable systems in the engineering industry. It is therefore difficult to analyze because many random variables with general distributions are involved. The expressions for steady-state availability and mean time to system failure with general repair time are derived from Laplace transform equations. Both sensitivity analysis and relative sensitivity analysis were calculated for three different cases of this system; the results are shown in Tables1–12. Finally, the numerical results explain the relationship between the derived reliability measurements and the system parameters. From the numerical results and graphs, we deduce the following:(1)
The steady-state availability increases whenr1=r2(2)
The mean to system failure increases whenr1=r2



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*Source: 2488182-2022-02-23.xml*