1. Using Rational Approximations for Evaluating the Reliability of Highly Reliable Systems 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 Supported in part by GE

2. 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

3. Rational Interpolation A rational interpolant has the form a o +a 1 x+…+a n x n b o +b 1 x+…+b m x m This is an (n,m) RI x is a system parameter The constants a i,b i are determined by the higher- failure-rate simulation results f(x 1 ),...,f(x k ) The function is then used for predicting the reliability at very low failure rates A pre-transformation is sometimes performed on f(x 1 ),...,f(x k ) f(x)=

4. 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]

5. 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 AC D B EF

6. 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 - P 15,0 (t)

7. Our RI Approach - Notations R(t) – the accurate reliability function R s (t) – the reliability as obtained by simulation Ř m,n (t) – the estimate obtained using an (m,n) RI We select three sets of points on the time-axis x 1,...,x k – input points – the points for which either R(t) or R s (t) is calculated y 1,...,y l – test points – additional points which assist us in choosing the best RI for our purposes z 1,...,z s – target points – the points for which we are interested in evaluating the reliability The average test-error (for a given ( m,n ) RI) Ē m,n (Y) = Σ l i=1 | Ř m,n (y i ) – R(y i ) | l

8. Our RI Approach - Procedure 1. Select input points x 1,...,x k, test points y 1,...,y l, target points z 1,...,z s 2. Calculate R(x 1 ),...,R(x k ) and R(y 1 ),...,R(y l ) 3. For a sequence of ( m,n )'s, calculate the rational interpolants based on R(x 1 ),...,R(x k ) and use them to calculate Ř m,n (y i ) 4. For each function obtained in step 3, find the average error Ē m,n (Y) = Σ l i=1 | Ř m,n (y i ) – R(y i ) | l over the test points 5. Select the rational interpolant with the lowest Ē m,n (Y) and use it to predict the reliability for the target points

9. Numerical Experiments - Parameters Failure rates - λ i =0.0001(7-i), (i=A,...,F) Repair rates - μ i =0.004, (i=A,...,F) R(0)=1 Logarithmic transformation used z 1 <z 2 <…<z s <y 1 <y 2 <…<y l <x 1 <x 2 <…<x k AC D B EF

10. 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.

11. 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 Ř 5,5 (t) for t=0,...,50

12. Actual vs. RI Reliabilities - Numerical Results

13. Numerical Experiment I Target points z 1,...,z s =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, Ē m,n (Z) calculated for each RI between (1,1) and (10,10) RIs with Ē m,n (Z) < 10 -12 listed Results confirm that input points closer to the target points result in higher precision

14. Impact of Input Points Placement: Exact Values

15. Numerical Experiment I - Simulation Results Same experiment – simulated reliabilities Listed RI’ s with Ē m,n (Z) < 10 -7 Effect of points positioning – less prominent

16. Impact of Input Points Placement: Simulated Values

17. 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 Ē m,n (Y) and Ē m,n (Z) Both exact and simulated results used Result – correlations around 0.95 for the exact results and 0.85 for the simulated results

18. 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

19. 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