EE 445 / 850: Final Examination

Transcription

1 EE 445 / 850: Final Examination Date and Time: 3 Dec 0, PM Room: HLTH B6 Exam Duration: 3 hours One formula sheet permitted. - Covers chapters - 5 problems each carrying 0 marks - Must show all calculations involved - Must define any symbols or notations used - Must explain any assumptions made EE 850 project report submission on the same day.

2 Basic Probability Theory P(outcome) = no. of successful outcomes total no. of outcomes Permutations/combinations theory applied to calculate the no. of outcomes. In engineering systems, P(particular event) = lim ( n f ) f = no. of occurrences of the event n = no. of times the experiment is repeated n Rules for combining probabilities: - independent events - mutually exclusive events - conditional events - OR (A or B or both), AND (overlapping) events If event A is dependent on two mutually exclusive events (success Bs and failure Bf) for a component B, then P(A) = P(A Bs). P(Bs) + P(A Bf). P(Bf) Probability density and distribution functions Expected (mean) value, Variance, Standard deviation E(x) = i n = x i p i (discrete) = xf x) dx Variance = mean of square values (mean) ( (continuous)

3 Binomial Distribution (p + q) n = p n + n p n- q n C r p n-r q r q n n! r!(n - r)! Probability of r successes in n trials, P r = n C r p r q (n-r) Coefficients of Binomial Distribution - Pascal s Triangle n= (p+q) = p + q (p+q) = p + pq + q (p+q) 3 = p 3 + 3p q + 3p q + q Consider a system consisting of 4 identical components, each having a failure probability of 0.. Capacity Outage Probability Table System state Individual probability all components working p 4 = (0.9) 4 = working, failed 4p 3 q = 4(0.9) 3 (0.) = 0.96 working, failed 6p q = 6(0.9) (0.) = working, 3 failed 4pq 3 = 4(0.9)( 0.) 3 = all components failed q 4 = (0.) 4 = Σ =

4 Modelling and Evaluation of Simple Systems Series System Parallel System R s = R. R product rule of reliability Q s = Q. Q product rule of unreliability Series Parallel Systems network reduction technique m out of n systems Binomial Expansion

5 Modelling and Evaluation of Complex Systems System success requires continuity from input to output. Evaluation Techniques: - Conditional probability approach P(SS) = P(SS E good).p(e good) + P(SS E bad).p(e bad) Network reduction technique - Cut set method - components (order) in parallel in a minimal cutest - minimal cutsets in series Advantages: - cut sets identify ways in which a system may fail - approximation can be used to simplify evaluation - can be easily programmed on a computer - Tree diagrams - Event Trees - Fault Trees Multi-failure modes - State enumeration method - Conditional Probability Method A B E C D

6 Probability Distributions Failure density function, f(t) Failure distribution function, Q(t) prob. of failure Survivor function, R(t) = Q(t) Hazard rate, λ(t) = f(t)/r(t) Shape of reliability functions bath tub curve -Debugging or infant mortality -Normal operating or useful life (exponential distribution) -Wear-out or fatigue (normal distribution) Probability Distributions - Exponential: (useful life of component) Reliability, R(t) = e λt and Q(t) = - e λt -Normal: (wear-out life of component) Failure density function of variable x expressed in z = Probability (area under curve) calculated from Table x µ σ -Poisson: (useful life of component) Probability of x failures in time t, P x (t) = ( λt) e x! x λt

7 Reliability Evaluation Using Probability Distributions For a component with a hazard rate λ (t), In general, component reliability, R (t) = For exponential distribution, R (t) = e t 0 e λ t λ (t)dt Series System Parallel System product rule of reliability R s (t) = R (t). R (t) R s (t) = e λ et System failure rate, λ e = λ n i= i product rule of unreliability Q s (t) = Q (t). Q (t) n Q s (t) = i= [ - e λ i t ] m out of n (partially redundant) system Binomial Expansion, System MTTF = R s 0 [R(t) + Q(t)] n = ( t) dt n r= 0 nc r R(t) n-r Q(t) r

8 Standby System (with identical components) or Spares Poisson s Distribution: P x (t) = ( λt) e x! x λt Probability of x components failing in time t. Wear-out Region and Reliability R(t) = e λt x Rw (T+ t) R (T) w

9 Markov Model Markov Approach: lack of memory, stationary - discrete (time or space) Discrete Markov Chain - continuous (time) Continuous Markov Process Discrete Markov Chain: - develop Markov model for the component (or system) - evaluate state probability (time dependent or limiting state) using: o Tree diagram: impractical for large systems or a large number of time intervals o Stochastic Transitional Probability Matrix Transient behavior: State probabilities after n intervals is given by, P(n) = P(0).P n Limiting State Probability: αp = α where α = limiting probability vector Absorbing States Expected # of time intervals, N = [ I Q ] - where, Q = truncated matrix created by deleting row(s) and column(s) associated with the absorbing states

10 Continuous Markov Process Develop the system state space diagram Calculate system state probabilities - Time dependent probability: R(t), A(t) o Differential equations method o Matrix multiplication method Mission-oriented system, MTTF = [ I Q ] - - Limiting state probability o Stochastic transitional probability matrix o Frequency balance approach Probability, frequency & duration indices Frequency of encountering State i = P(being in State i) x (rate of departure from State i) Mean duration = probability / frequency Cumulative states Both Up λ µ Up Down λ µ Both Down State Space Diagram Up γ λ Failed µ Repaired but not installed Two Stage Repair and Installation Process

11 Approximate System Reliability Evaluation System Indices using approximate equations: λ s system failure rate (failure frequency) r s system average down (repair) time U s system unavailability (expected annual outage time) For a series system: λ s = Σ λ i U s = Σ λ i r i r s = For a parallel system: λ p = λ (λ r ) + λ (λ r ) r p = rr r + r U λ s s U p = λ p.r p Complex systems: Minimal Cutset method Failure event overlapping maintenance outage: λ p = λ (λ r ) + λ (λ r ) r U p = λ (λ r )( "r r ) + λ (λ r ) ( r "+ r r r p = U p / λ p "r "+ r ) Common mode failures: D U λ µ λ U U µ µ λ U D µ µ λ D D λ