Issue 33, November 2003

Expanded Algebraic Solutions for RBDs and FTDs in BlockSim 6.2

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, l(t), are trivial and are given next:

As the system gets more complex, however, such analysis becomes extremely complex. Consider the following 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, l(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)

l(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.