Using Rational Approximations for Evaluating the Reliability of Highly Reliable Systems

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Z. Koren, J. Rajagopal, C. M. Krishna, and I. Koren Dept. of Elect. and Comp. Engineering University of Massachusetts Amherst, MA 01003 and W. Wang and J. M. Loman GE Corporate R&D Electrical Systems and Technologies Laboratory Schenectady, NY 12309.

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Z. Koren, J. Rajagopal, C. M. Krishna, and I. Koren

Dept. of Elect. and Comp. Engineering

University of Massachusetts

Amherst, MA 01003

and

W. Wang and J. M. Loman

GE Corporate R&D

Electrical Systems and Technologies Laboratory Schenectady, NY 12309

Using Rational Approximations for Evaluating the Reliability of Highly Reliable Systems

Supported in part by GE

Motivation
• In highly reliable systems failures are rare
• Simulation to obtain reliability takes a very long time
• Current approaches for simulating rare events have drawbacks
• We focus on the Rational Interpolation (RI) approach
• Results are obtained for larger failures rates
• A rational interpolant is then used for the small failure rates
Rational Interpolation
• A rational interpolant has the form

ao+a1x+…+anxn

bo+b1x+…+bmxm

• This is an (n,m) RI
• xis a system parameter
• The constants ai ,bi are determined by the higher-failure-rate simulation results f(x1),...,f(xk )
• The function is then used for predicting the reliability at very low failure rates
• A pre-transformation is sometimes performed on f(x1),...,f(xk )

f(x)=

Reliability Model
• System is a block diagram of modules
• Each module is either up or down
• When a module fails, a repair process begins
• Time to failure and repair time are exponentially distributed
• System reliability at time t – probability that the system has been up during the whole time interval [0,t]
System Model
• For proof of concept – a simple system selected
• System is up if either BEF,ACEF, or ADF are up
• System has an analytic solution

B

A

C

E

F

D

Reliability Analysis
• Rational Interpolation requires one parameter
• Parameter selected - system time t = 0,1,2,...
• System can be described as a Markov chain with 16 states, denoted by 0,...,15
• State 15 – initial state – all modules are up
• State 0 – “system is down” state – absorbing state
• Reliability –R(t)=1 - P15,0(t)
Our RI Approach - Notations
• R(t) – the accurate reliability function
• Rs(t) – the reliability as obtained by simulation
• Řm,n(t) – the estimate obtained using an (m,n) RI
• We select three sets of points on the time-axis
• x1 ,...,xk – input points – the points for which either R(t) or Rs(t) is calculated
• y1 ,...,yl – test points – additional points which assist us in choosing the best RI for our purposes
• z1 ,...,zs – target points – the points for which we are interested in evaluating the reliability
• The average test-error (for a given (m,n) RI)

Ēm,n(Y) = Σ li=1|Řm,n(yi) – R(yi)|

l

Our RI Approach - Procedure
• Select input points x1,...,xk, test points y1,...,yl, target points z1,...,zs
• Calculate R(x1 ),...,R(xk ) and R(y1 ),...,R(yl )
• For a sequence of (m,n)'s, calculate the rational interpolants based on R(x1 ),...,R(xk ) and use them to calculate Řm,n(yi )
• For each function obtained in step 3, find the average error

Ēm,n(Y) = Σ li=1 | Řm,n(yi ) – R(yi ) |

lover the test points

• Select the rational interpolant with the lowest Ēm,n(Y) and use it to predict the reliability for the target points
Failure rates - λi=0.0001(7-i), (i=A,...,F)

Repair rates - μi=0.004, (i=A,...,F)

R(0)=1

Logarithmic transformation used

z1<z2<…<zs<y1<y2<…<yl<x1<x2<…<xk

Numerical Experiments - Parameters

B

A

C

E

F

D

Experiments – Numerical Considerations
• Transition probabilities for the Markov chain calculated using the uniformization method
• For simulation – random number generator selected – Mersenne Twister generator
• Increasing precision of numerical calculations – multi-precision software developed by Bailey et al.
Comparing Exact and RI Reliabilities
• Checking the validity of the RI approach
• We calculated a (5,5)RI based on input points t=13,...,23 and a logarithmic transformation
• Predicted the reliabilities Ř5,5(t) for t=0,...,50
Numerical Experiment I
• Target points z1,...,zs=1,2,3,4,5
• Assessing sensitivity of best RI to input points selection
• Placement and number of input points varied
• For each starting point between 6 and 20, Ēm,n(Z)calculated for each RI between (1,1) and (10,10)
• RIs with Ēm,n(Z)< 10-12 listed
• Results confirm that input points closer to the target points result in higher precision
Numerical Experiment I - Simulation Results
• Same experiment – simulated reliabilities
• Listed RI’ s with Ēm,n(Z) < 10-7
• Effect of points positioning – less prominent
Numerical Experiment II
• Number and placement of test points and target points fixed
• Varied the number and the placement of input points
• Consequently – degree of best RI varies
• purpose – detect a possible correlation between Ēm,n(Y)and Ēm,n(Z)
• Both exact and simulated results used
• Result – correlations around 0.95 for the exact results and 0.85 for the simulated results
Numerical Experiment III
• Assessing the effect of simulation time on prediction accuracy
• Important in case of a restricted simulation budget
• Input points t=13,14,..., target points 1,...,5
• For two RIs: (7,5) and

(8,6), average error of estimate calculated

• Longer simulation time less noisy results  better interpolation
• Not much accuracy

added by simulating longer than 250 million cycles

Discussion
• We reported a case study in the use of Rational Interpolations for calculating very high reliabilities
• We demonstrated the usefulness of a technique for selecting an accurate RI
• We are currently developing a method for selecting a ''least squares" RI for higher accuracy