Expanded Algebraic Solutions for RBDs and FTDs in BlockSim 6.2

Reliability HotWire

Issue 33, November 2003

Hot Topics

Expanded Algebraic Solutions for RBDs and FTDs in BlockSim 6.2

BlockSim 6.2, the latest service release for Standard and FTI editions, is a free upgrade for all current BlockSim 6 users. Click here for free product updates for BlockSim 6.

BlockSim 6 is unique among reliability block diagram (RBD) software because it is the only software package with the analytical power to provide the complete algebraic solution for even the most complex systems. In addition to the complete reliability equation, BlockSim version 6.2 now provides both the complete algebraic pdf and failure rate function for such systems. To illustrate this functionality, consider the following simple series system:

The system equations, namely the reliability equation, R(t), the probability density function, f(t), and the failure rate function, λ(t), are trivial and are given next:

R(t)system=	R(t)A.R(t)B.R(t)C
f(t)system=	f(t)A.R(t)B.R(t)C+f(t)B.R(t)A.R(t)C+f(t)C.R(t)A.R(t)B
λ(t)system=	λ(t)A+λ(t)B+λ(t)C

As the system gets more complex, however, such analysis becomes extremely complex. Consider the following complex system:

Complex system

The system reliability equation is given by:

R(t)system=RA.RE(RN.RD.RB.RM.RC-RN.RD.RB.RM-RN.RM.RC-RD.RB.RC
                        +RN.RM+RD.RB+RC) .{RL(RK.RI.RJ.RO.RF.RH+RK.RJ.RG.RO.RF.RH
                        -RK.RI.RJ.RF.RH-RK.RI.RO.RF.RH-RK.RJ.RG.RO.RF-RK.RJ.RG.RF.RH
                        -RK.RJ.RO.RF.RH+RK.RI.RF.RH+RK.RJ.RF.RH+RK.RJ.RG+RO.RF)}

The system pdf, f_System, as obtained from BlockSim 6.2, is then given by: (where f and R are the pdf, f(t), and the reliability function, R(t), of the identified block and items in bold are tokens identifying sub-expressions)

f_System=fD1.D2+fD2.D1

D2=RL.IL

fD2=fL.IL+fIL.RL

IL=RK.RI.RJ.RO.RF.RH+RK.RJ.RG.RO.RF.RH

-RK.RI.RJ.RF.RH-RK.RI.RO.RF.RH-RK.RJ.RG.RO.RF

-RK.RJ.RG.RF.RH-RK.RJ.RO.RF.RH+RK.RI.RF.RH

+RK.RJ.RF.RH+RK.RJ.RG+RO.RF

fIL=fK.RI.RJ.RO.RF.RH+fI.RK.RJ.RO.RF.RH+fJ.RK.RI.RO.RF.RH

+fO.RK.RI.RJ.RF.RH+fF.RK.RI.RJ.RO.RH+fH.RK.RI.RJ.RO.RF

+fK.RJ.RG.RO.RF.RH+fJ.RK.RG.RO.RF.RH+fG.RK.RJ.RO.RF.RH

+fO.RK.RJ.RG.RF.RH+fF.RK.RJ.RG.RO.RH+fH.RK.RJ.RG.RO.RF

-fK.RI.RJ.RF.RH-fI.RK.RJ.RF.RH-fJ.RK.RI.RF.RH-fF.RK.RI.RJ.RH

-fH.RK.RI.RJ.RF-fK.RI.RO.RF.RH-fI.RK.RO.RF.RH-fO.RK.RI.RF.RH

-fF.RK.RI.RO.RH-fH.RK.RI.RO.RF-fK.RJ.RG.RO.RF-fJ.RK.RG.RO.RF

-fG.RK.RJ.RO.RF-fO.RK.RJ.RG.RF-fF.RK.RJ.RG.RO-fK.RJ.RG.RF.RH

-fJ.RK.RG.RF.RH-fG.RK.RJ.RF.RH-fF.RK.RJ.RG.RH-fH.RK.RJ.RG.RF

-fK.RJ.RO.RF.RH-fJ.RK.RO.RF.RH-fO.RK.RJ.RF.RH-fF.RK.RJ.RO.RH

-fH.RK.RJ.RO.RF+fK.RI.RF.RH+fI.RK.RF.RH+fF.RK.RI.RH+fH.RK.RI.RF

+fK.RJ.RF.RH+fJ.RK.RF.RH+fF.RK.RJ.RH+fH.RK.RJ.RF+fK.RJ.RG+fJ.RK.RG

+fG.RK.RJ+fO.RF+fF.RO

D1=RA.RE.IE

fD1=+fA.RE.IE+fE.RA.IE+fIE.RA.RE

IE=RN.RD.RB.RM.RC-RN.RD.RB.RM-RN.RM.RC

-RD.RB.RC+RN.RM+RD.RB+RC

fIE=fN.RD.RB.RM.RC+fD.RN.RB.RM.RC+fB.RN.RD.RM.RC

+fM.RN.RD.RB.RC+fC.RN.RD.RB.RM-fN.RD.RB.RM-fD.RN.RB.RM

-fB.RN.RD.RM-fM.RN.RD.RB-fN.RM.RC-fM.RN.RC-fC.RN.RM

-fD.RB.RC-fB.RD.RC-fC.RD.RB+fN.RM+fM.RN+fD.RB+fB.RD+fC

The system failure rate equation, λ(t)_System, as obtained from BlockSim version 6.2, then is: (where, again, f and R are the pdf, f(t), and the reliability function, R(t), of the identified block and items in bold are tokens identifying sub-expressions)

λ(t)_System=frD1+frD2

frD2=+frL+frIL

frIL=fIL/IL

IL=RK.RI.RJ.RO.RF.RH+RK.RJ.RG.RO.RF.RH-RK.RI.RJ.RF.RH

-RK.RI.RO.RF.RH-RK.RJ.RG.RO.RF-RK.RJ.RG.RF.RH

-RK.RJ.RO.RF.RH+RK.RI.RF.RH+RK.RJ.RF.RH+RK.RJ.RG

+RO.RF

fIL=fK.RI.RJ.RO.RF.RH+fI.RK.RJ.RO.RF.RH+fJ.RK.RI.RO.RF.RH

+fO.RK.RI.RJ.RF.RH+fF.RK.RI.RJ.RO.RH+fH.RK.RI.RJ.RO.RF

+fK.RJ.RG.RO.RF.RH+fJ.RK.RG.RO.RF.RH+fG.RK.RJ.RO.RF.RH

+fO.RK.RJ.RG.RF.RH+fF.RK.RJ.RG.RO.RH+fH.RK.RJ.RG.RO.RF

-fK.RI.RJ.RF.RH-fI.RK.RJ.RF.RH-fJ.RK.RI.RF.RH-fF.RK.RI.RJ.RH

-fH.RK.RI.RJ.RF-fK.RI.RO.RF.RH-fI.RK.RO.RF.RH-fO.RK.RI.RF.RH

-fF.RK.RI.RO.RH-fH.RK.RI.RO.RF-fK.RJ.RG.RO.RF-fJ.RK.RG.RO.RF

-fG.RK.RJ.RO.RF-fO.RK.RJ.RG.RF-fF.RK.RJ.RG.RO-fK.RJ.RG.RF.RH

-fJ.RK.RG.RF.RH-fG.RK.RJ.RF.RH-fF.RK.RJ.RG.RH-fH.RK.RJ.RG.RF

-fK.RJ.RO.RF.RH-fJ.RK.RO.RF.RH-fO.RK.RJ.RF.RH-fF.RK.RJ.RO.RH

-fH.RK.RJ.RO.RF+fK.RI.RF.RH+fI.RK.RF.RH+fF.RK.RI.RH

+fH.RK.RI.RF+fK.RJ.RF.RH+fJ.RK.RF.RH+fF.RK.RJ.RH

+fH.RK.RJ.RF+fK.RJ.RG+fJ.RK.RG+fG.RK.RJ+fO.RF+fF.RO

frD1=+frA+frE+frIE

frIE=fIE/IE

IE=RN.RD.RB.RM.RC-RN.RD.RB.RM-RN.RM.RC-RD.RB.RC

+RN.RM+RD.RB+RC

fIE=fN.RD.RB.RM.RC+fD.RN.RB.RM.RC+fB.RN.RD.RM.RC

+fM.RN.RD.RB.RC+fC.RN.RD.RB.RM-fN.RD.RB.RM-fD.RN.RB.RM

-fB.RN.RD.RM-fM.RN.RD.RB-fN.RM.RC-fM.RN.RC-fC.RN.RM

-fD.RB.RC-fB.RD.RC-fC.RD.RB+fN.RM+fM.RN+fD.RB+fB.RD+fC

Of course, and given the power of BlockSim, this analysis can be expanded to even more complex systems, such as the one shown next:

[Click to Enlarge]
[View the pdf for this system]

This same concept can also be expanded to fault trees when using BlockSim FTI, BlockSim's fault tree interface edition, since fault trees can be converted to RBDs.

Six years of refinement led ReliaSoft to the new analytical engine used in BlockSim 6.2. In addition to algebraically formulating R(t), f(t) and λ(t), the new engine is capable of solving much larger systems faster. The prior complex example was analyzed in less than 2 seconds using the new engine, whereas, it took more than 20 seconds to analyze the system in BlockSim version 6.1.