Expected Value and MTTF Calculation In the Name of the Most High Expected Value and MTTF Calculation Behzad Akbari Spring 2009 Tarbiat Modares University These slides are based on the slides of Prof. K.S. Trivedi (Duke University) 4/21/2019

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.

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’

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: Mean E[X(t)]:

Binomial Random Variable (cont.) Y(t) is binomial with parameters n,p

Poisson Distribution Probability mass function (pmf) (or discrete density function): Mean E[N(t)] :

Exponential Distribution Distribution Function: Density Function: Reliability: Failure Rate: failure rate is age-independent (constant) MTTF:

Weibull Distribution (cont.) Failure Rate: IFR for DFR for MTTF: Shape parameter  and scale parameter 

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)

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:

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.

MGF Properties If Y=aX+b (translation & scaling), then, Uniqueness property Summation in one domain  convolution in the other domain.

MGF Properties For the LST: For the z-transform case:

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.

Series system (Continued) Other versions of Equation (2)

Series System MTTF (contd.) RV Xi : ith comp’s life time (arbitrary distribution) Case of least common denominator. To prove above

TMR (Continued) Assuming that the reliability of a single component is given by, we get:

TMR (Continued)

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.

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.

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

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()

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 =  .

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()

The WFS Example File Server Computer Network Workstation 1

RBD for the WFS Example Workstation 1 File Server Workstation 2

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

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:

Homework 1: For a 2-component parallel redundant system with EXP( ) behavior, write down expressions for: Rp(t) MTTFp

Solution 1:

Homework 2: For a 2-component parallel redundant system with EXP( ) and EXP( ) behavior, write down expressions for: Rp(t) MTTFp

Solution 2:

Homework 3: specialize the bridge reliability formula to the case where Ri(t) = find Rbridge(t) and MTTF for the bridge

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

Bridge: Rbridge(t) When C3 is working C1 C4 C2 C5 S T

Bridge: Rbridge(t) C1 C5 C2 C4 S T When C3 fails

Bridge: Rbridge(t)

Bridge: MTTF

Homework 4: Derive & compare reliability expressions for Cold, Warm and Hot standby cases.

Cold spare: EXP()

Warm spare: EXP(+ ) EXP()

Hot spare: EXP(2) EXP()

Comparison graph:

TMR and TMR/simplex as hypoexponentials

Conditional Probability and Expectation 4/21/2019

Conditional pmf Conditional probability: Above works if x is a discrete rv. For discrete rv’s X and Y, conditional pmf is, Above relationship also implies, Hence we have another version of the theorem of total probability If x and y are mutually independent, then, p(y|x) = p(y).

Independence, Conditional Distribution Conditional distribution function Using conditional pmf,

Example n total jobs, k are passed on to server A Two servers k jobs p: prob. that the next job goes to server A p A Poisson ( λ) Job stream Bernoulli trial n jobs 1-p B n total jobs, k are passed on to server A pY(k) = [(pk/k!)e-λ \sum_{k-n}^\infty \frac{\lambda^k (\lambda (1-p))^{n-k}}{(n-k)!} = (\lamda p)^k / k! \sum_{n=k}^\infty \frac{(\lambda (1-p))^{n-k}}{n-k!} substituting m= n-k gives the result.

Conditional pdf For continuous rv’s X and Y, conditional pdf is, Also, Independent X, Y  Marginal pdf (cont. version of the TTP), Conditional distribution function

Conditional Reliability Software system after having incurred (i-1) faults, Ri(t) = P(Ti > t) (Ti : inter-failure times) Ti : independent exponentially distributed Exp(λi). λi : Failure rate itself may be random, then Conditional reliability: λi:: random  that with fault occurrence, followed by its removal, failure rate λi changes (in a random manner denoted by the random variable Λi ). If ψi = a1+a2i, then what happens to E[Λi]and R_i(t) ? E[Λi] = 1/ψi = i.e. decreasing with i and there is a likelihood (but no guarantee) that R_i(t) improves with time.

Conditional Moments Conditional Expectation is E[Y|X=x] or E[Y|x] E[Y|x]: a.k.a regression function For the discrete case, In general, then, The mathematical function used to describe the deterministic variation in the response variable is sometimes called the "regression function", the "regression equation", the "smoothing function", or the "smooth". [http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd12.htm]

Conditional MTTF Y: time-to-failure may depend on the temperature, and the conditional MTTF may be: Let Temp be normal, Unconditional MTTF is: