Dept. of Electrical & Computer engineering Probability and Statistics with Reliability, Queuing and Computer Science Applications: Chapter 4 on Expected Value and Higher Moments Dept. of Electrical & Computer engineering Duke University Email: bbm@ee.duke.edu, kst@ee.duke.edu 1/16/2019

Expected (Mean, Average) Value There are several ways to abstract the information in the CDF into a single number: median, mode, mean. Mean: E(X) may also be computed using distribution function In case, the summation or the integration does is not absolutely convergent, then E(X) does not exist. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Higher Moments RV’s X and Y (=Φ(X)). Then, Φ(X) = Xk, k=1,2,3,.., E[Xk]: kth moment k=1 Mean; k=2: Variance (Measures degree of variability) Example: Exp(λ)  E[X]= 1/ λ; σ2 = 1/λ2 shape of the pdf (or pmf) for small and large variance values. σ is commonly referred to as the ‘standard deviation’ Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Bernoulli Random Variable For a fixed t, X(t) is a random variable. The family of random variables {X(t), t  0} is a stochastic process. Random variable X(t) is the indicator or Bernoulli random variable so that: Probability mass function: Mean E[X(t)]: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Binomial Random Variable (cont.) Y(t) is binomial with parameters n,p Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson Distribution Probability mass function (pmf) (or discrete density function): Mean E[N(t)] : Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Exponential Distribution Distribution Function: Density Function: Reliability: Failure Rate: failure rate is age-independent (constant) MTTF: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Exponential Distribution Distribution Function: Density Function: Reliability: Failure Rate (CFR): Failure rate is age-independent (constant) Mean Time to Failure: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Weibull Distribution (cont.) Failure Rate: IFR for DFR for MTTF: Shape parameter  and scale parameter  Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Using Equations of the underlying Semi-Markov Process (Continued) Time to the next diagnostic is uniformly distributed over (0,T) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Using Equations of the underlying Semi-Markov Process (Continued) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

E[ ] of a function of mutliple RV’s If Z=X+Y, then E[X+Y] = E[X]+E[Y] (X, Y need not be independent) If Z=XY, then E[XY] = E[X]E[Y] (if X, Y are mutually independent) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Variance: function of Mutliple RV’s Var[X+Y]=Var[X]+Var[Y] (If X, Y independent) Cov[X,Y] E{[X-E[X]][Y-E[Y]]} Cov[X,Y] = 0 and (If X, Y independent) Cross Cov[ ] terms may appear if not independent. (Cross) Correlation Co-efficient: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Moment Generating Function (MGF) For dealing with complex function of rv’s. Use transforms (similar z-transform for pmf) If X is a non-negative continuous rv, then, If X is a non-negative discrete rv, then, M[θ] is not guaranteed to exist. But for most distributions of our interest, it does exist. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

MGF (contd.) Complex no. domain characteristics fn. transform is If X is Gaussian N(μ, σ), then, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

MGF Properties If Y=aX+b (translation & scaling), then, Uniqueness property Summation in one domain  convolution in the other domain. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

MGF Properties For the LST: For the z-transform case: For the characteristic function, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

MFG of Common Distributions Read sec. 4.5.1 pp.217-227 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

MTTF Computation R(t) = P(X > t), X: Lifetime of a component Expected life time or MTTF is In general, kth moment is, Series of components, (each has lifetime Exp(λi) Overall lifetime distribution: Exp( ), and MTTF = The last equality follows from by integrating by parts, int_0^∞ t R’(t) = -t R(t)|0 to ∞ + Int_0^∞ R(t) -t R(t) 0 as t ∞ since R(t)  0 faster than t  ∞. Hence, the first term disappears. Note that the MTTF of a series system is much smaller than the MTTF of an individual component. Failure of any component implies failure of the overall system. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Series system (Continued) Other versions of Equation (2) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Series System MTTF (contd.) RV Xi : ith comp’s life time (arbitrary distribution) Case of least common denominator. To prove above Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 2: For a 2-component parallel redundant system with EXP( ) behavior, write down expressions for: Rp(t) MTTFp Further assuming EXP(µ) behavior and independent repair, write down expressions for: Ap(t) Ap downtime Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 3: For a 2-component parallel redundant system with EXP( ) and EXP( ) behavior, write down expressions for: Rp(t) MTTFp Assuming independent repair at rates µ1 and µ2, write down expressions for: Ap(t) Ap downtime Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

TMR (Continued) Assuming that the reliability of a single component is given by, we get: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

TMR (Continued) In the following figure, we have plotted RTMR(t) vs t as well as R(t) vs t. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 5: specialize the bridge reliability formula to the case where Ri(t) = find Rbridge(t) and MTTF for the bridge Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

MTTF Computation (contd.) Parallel system: life time of ith component is rv Xi X = max(X1, X2, ..,Xn) If all Xi’s are EXP(λ), then, As n increases, MTTF also increases as does the Var. These are notes. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Standby Redundancy A system with 1 component and (n-1) cold spares. Life time, If all Xi’s same,  Erlang distribution. Read secs. 4.6.4 and 4.6.5 on TMR and k-out of-n. Sec. 4.7 - Inequalities and Limit theorems Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Cold standby Lifetime of Active EXP() Total lifetime 2-Stage Erlang Spare EXP() Total lifetime 2-Stage Erlang EXP() Assumptions: Detection & Switching perfect; spare does not fail Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Warm standby With Warm spare, we have: Active unit time-to-failure: EXP() Spare unit time-to-failure: EXP() 2-stage hypoexponential distribution EXP(+ ) EXP() Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Warm standby derivation First event to occur is that either the active or the spare will fail. Time to this event is min{EXP(),EXP()} which is EXP( + ). Then due to the memoryless property of the exponential, remaining time is still EXP(). Hence system lifetime has a two-stage hypoexponential distribution with parameters 1 =  +  and 2 =  . Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Hot standby With hot spare, we have: Active unit time-to-failure: EXP() Spare unit time-to-failure: EXP() 2-stage hypoexponential EXP(2) EXP() Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

The WFS Example File Server Computer Network Workstation 1 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

RBD for the WFS Example Workstation 1 File Server Workstation 2 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

RBD for the WFS Example (cont.) Rw(t): workstation reliability Rf (t): file-server reliability System reliability R(t) is given by: Note: applies to any time-to-failure distributions Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

RBD for the WFS Example (cont.) Assuming exponentially distributed times to failure: failure rate of workstation failure rate of file-server The system mean time to failure (MTTF) is given by: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Comparison Between Exponential and Weibull Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 2: For a 2-component parallel redundant system with EXP( ) behavior, write down expressions for: Rp(t) MTTFp Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Solution 2: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 3 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 3: For a 2-component parallel redundant system with EXP( ) and EXP( ) behavior, write down expressions for: Rp(t) MTTFp Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Solution 3: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 4: Specialize formula (3) to the case where: Derive expressions for system reliability and system meantime to failure. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 4 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Control channels-Voice channels Example: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 5 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 5: specialize the bridge reliability formula to the case where Ri(t) = find Rbridge(t) and MTTF for the bridge Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Bridge: conditioning Non-series-parallel block diagram C1 C2 C3 fails C3 is working C4 C5 C1 C2 S T Factor (condition) on C3 C4 C5 Non-series-parallel block diagram Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Bridge: Rbridge(t) When C3 is working C1 C4 C2 C5 S T Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Bridge: Rbridge(t) When C3 fails C1 C5 C2 C4 S T Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Bridge: Rbridge(t) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Bridge: MTTF Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 7 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 7: Derive & compare reliability expressions for Cold, Warm and Hot standby cases. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Cold spare: EXP() Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Warm spare: EXP(+ ) EXP() Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Hot spare: EXP(2) EXP() Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Comparison graph: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 8 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Homework 8: For the 2-component system with non-shared repair, use a reliability block diagram to derive the formula for instantaneous and steady-state availability. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Solution 8: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

TMR and TMR/simplex as hypoexponentials Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University