Continuous Random Variables Probability and Statistics with Reliability, Queuing and Computer Science Applications: Chapter 3 Continuous Random Variables

Definitions Distribution function: If FX(x) is a continuous function of x, then X is a continuous random variable. FX(x): discrete in x  Discrete rv’s FX(x): piecewise continuous  Mixed rv’s If X is to qualify as a rv, on the space (S,F,P) then we should be able to define prob. measure for X. This in turn implies that P(X <= x) be defined for all x. This would require the existence of the event {s| X<= s} belonging to F. 2. F(x) is actually absolutely continuous i.e. its derivative is well defined except possibly at the end points.

Definitions (Continued) Equivalence: CDF (cumulative distribution function) PDF (probability distribution function) Distribution function FX(x) or FX(t) or F(t)

Probability Density Function (pdf) X : continuous rv, then, pdf properties: Sometimes we may have deal with mixed (discrete+continuous) type of rv’s as well. See Fig. 3.2 and understand it.

Definitions (Continued) Equivalence: pdf probability density function density function density f(t) = For a non-negative random variable

Exponential Distribution Arises commonly in reliability & queuing theory. A non-negative random variable It exhibits memoryless (Markov) property. Related to (the discrete) Poisson distribution Interarrival time between two IP packets (or voice calls) Time to failure, time to repair etc. Mathematically (CDF and pdf, respectively): No. of failure in a given interval may follow Poisson distribution.

CDF of exponentially distributed random variable with  = 0.0001 F(t) 12500 25000 37500 50000 t

Exponential Density Function (pdf) f(t) t

Memoryless property Assume X > t. We have observed that the component has not failed until time t. Let Y = X - t , the remaining (residual) lifetime The distribution of the remaining life, Y, does not depend on how long the component has been operating. Distribution of Y is identical to that of X.

Memoryless property Assume X > t. We have observed that the component has not failed until time t. Let Y = X - t , the remaining (residual) lifetime

Memoryless property (Continued) Thus Gt(y) is independent of t and is identical to the original exponential distribution of X. The distribution of the remaining life does not depend on how long the component has been operating. Its eventual breakdown is the result of some suddenly appearing failure, not of gradual deterioration.

Reliability as a Function of Time Reliability R(t): failure occurs after time ‘t’. Let X be the lifetime of a component subject to failures. Let N0: total no. of components (fixed); Ns(t): surviving ones; Nf(t): failed one by time t. Failure density function: f(t)t is the probability that the component will fail in the interval (t, t ]. Instantaneous Failure rate function h(t): is the conditional probability that the component will fail in the interval (t, t ], given that it survived until time t.

Definitions (Continued) Equivalence: Reliability Complementary distribution function Survivor function R(t) = 1 -F(t)

Failure Rate or Hazard Rate Instantaneous failure rate: h(t) (#failures/10k hrs) Let the rv X be EXP( λ). Then, Using simple calculus the following apples to any rv, R(t): P(X > t). For Exp() distribution, P(-infinity < X <= t) = 1-exp(-λt). Therefore, P(X>t) = 1- (1-exp(-λt)) = exp(-λt)

Hazard Rate and the pdf h(t) t = Conditional Prob. system will fail in (t, t + t) given that it has survived until time t f(t) t = Unconditional Prob. System will fail in (t, t + t) Difference between: probability that someone will die between 90 and 91, given that he lives to 90 probability that someone will die between 90 and 91

Weibull Distribution Frequently used to model fatigue failure, ball bearing failure etc. (very long tails) Reliability: Weibull distribution is capable of modeling DFR (α < 1), CFR (α = 1) and IFR (α >1) behavior. α is called the shape parameter and  is the scale parameter Weibull distribution may include the 3rd parameter sometimes. It is called the location parameter that has the effect of shifting the origin. That is, F(t) = F(t-θ) (RHS is the original Weibull distribution). F(t) = 1 – exp{–λ(t- θ)α}

Failure rate of the weibull distribution with various values of  and  = 1 5.0 1.0 2.0 3.0 4.0

Infant Mortality Effects in System Modeling Bathtub curves Early-life period Steady-state period Wear out period Failure rate models

Bathtub Curve Until now we assumed that failure rate of equipment is time (age) independent. In real-life, variation as per the bathtub shape has been observed Failure Rate l(t) Infant Mortality (Early Life Failures) Steady State Wear out Operating Time

Early-life Period Also called infant mortality phase or reliability growth phase Caused by undetected hardware/software defects that are being fixed resulting in reliability growth Can cause significant prediction errors if steady-state failure rates are used Availability models can be constructed and solved to include this effect Weibull Model can be used

Steady-state Period Failure rate much lower than in early-life period Either constant (age independent) or slowly varying failure rate Failures caused by environmental shocks Arrival process of environmental shocks can be assumed to be a Poisson process Hence time between two shocks has the exponential distribution

Wear out Period Failure rate increases rapidly with age Properly qualified electronic hardware do not exhibit wear out failure during its intended service life (Motorola) Applicable for mechanical and other systems Weibull Failure Model can be used

Bathtub curve DFR phase: Initial design, constant bug fixes CFR phase: Normal operational phase IFR phase: Aging behavior h(t) (burn-in-period) (wear-out-phase) CFR (useful life) DFR IFR t Decreasing failure rate Increasing fail. rate

Failure-Rate Multiplier Failure Rate Models We use a truncated Weibull Model Infant mortality phase modeled by DFR Weibull and the steady-state phase by the exponential 7 6 5 4 3 2 1 Failure-Rate Multiplier 2,190 4,380 6,570 8,760 10,950 13,140 15,330 17,520 Operating Times (hrs)

Failure Rate Models (cont.) This model has the form: where: steady-state failure rate is the Weibull shape parameter Failure rate multiplier =

Failure Rate Models (cont.) There are several ways to incorporate time dependent failure rates in availability models The easiest way is to approximate a continuous function by a decreasing step function 7 6 5 4 3 2 1 Failure-Rate Multiplier 2,190 4,380 6,570 8,760 10,950 13,140 15,330 17,520 Operating Times (hrs)

Failure Rate Models (cont.) Here the discrete failure-rate model is defined by:

Uniform Random Variable All (pseudo) random generators generate random deviates of U(0,1) distribution; that is, if you generate a large number of random variables and plot their empirical distribution function, it will approach this distribution in the limit. U(a,b)  pdf constant over the (a,b) interval and CDF is the ramp function

Uniform density

{ Uniform distribution The distribution function is given by: 0 , x < a, F(x)= , a < x < b 1 , x > b. {

Uniform distribution (Continued)

HypoExponential HypoExp: multiple Exp stages in series. 2-stage HypoExp denoted as HYPO(λ1, λ2). The density, distribution and hazard rate function are: HypoExp results in IFR: 0  min(λ1, λ2) Disk service time may be modeled as a 3-stage Hypoexponential as the overall time is the sum of the seek, the latency and the transfer time

HypoExponential used in software rejuvenation models Preventive maintenance is useful only if failure rate is increasing Robust state A simple and useful model of increasing failure rate: Failure probable state Failed state Time to failure: Hypo-exponential distribution Increasing failure rate aging

Erlang Distribution Special case of HypoExp: All stages have same rate. [X > t] = [Nt < r] (Nt : no. of stresses applied in (0,t]) and Nt is Possion (param λt). This interpretation gives, r=1 case of Erlang distribution reduces to the Exp( ) distribution case.

Is used to approximate the deterministic one Erlang Distribution Is used to approximate the deterministic one since if you keep the same mean but increase the number of stages, the pdf approaches the delta function in the limit Can also be used to approximate the uniform distribution r=1 case of Erlang distribution reduces to the Exp( ) distribution case.

probability density functions (pdf) If we vary r keeping r/ constant, pdf of r-stage Erlang approaches an impulse function at r/ .

cumulative distribution functions (cdf) And the cdf approaches a step function at r/. In other words r-stage Erlang can approximate a deterministic variable.

Comparison of probability density functions (pdf)

Comparison of cumulative distribution functions (cdf)

Gamma Random Variable Gamma density function is, Gamma distribution can capture all three failure modes, viz. DFR, CFR and IFR. α = 1: CFR α <1 : DFR α >1 : IFR Gamma with α = ½ and  = n/2 is known as the chi-square random variable with n degrees of freedom

HyperExponential Distribution Hypo or Erlang  Sequential Exp( ) stages. Alternate Exp( ) stages  HyperExponential. CPU service time may be modeled as HyperExp In workload based software rejuvenation model we found the sojourn times in many workload states have this distribution

Log-logistic Distribution Log-logistic can model DFR, CFR and IFR failure models simultaneously, unlike previous ones. For, κ > 1, the failure rate first increases with t (IFR); after momentarily leveling off (CFR), it decreases (DFR) with time. This is known as the inverse bath tub shape curve Use in modeling software reliability growth

Hazard rate comparison

Defective Distribution If Example: This defect (also known as the mass at infinity) could represent the probability that the program will not terminate (1-c). Continuous part can model completion time of program. There can also be a mass at origin.

Pareto Random Variable Also known as the power law or long-tailed distribution Found to be useful in modeling CPU time consumed by a request Webfile sizes Number of data bytes in FTP bursts Thinking time of a Web browser

Gaussian (Normal) Distribution Bell shaped pdf – intuitively pleasing! Central Limit Theorem: mean of a large number of mutually independent rv’s (having arbitrary distributions) starts following Normal distribution as n  μ: mean, σ: std. deviation, σ2: variance (N(μ, σ2)) μ and σ completely describe the statistics. This is significant in statistical estimation/signal processing/communication theory etc. In these areas, very often, we have to solve optimization problems that call minimizing the variance in estimating a parameter, detecting a signal, testing a hypothesis. By assuming the distributions to be Normal, variance minimization (Least-mean-square-error (LSME) or Min. mean square error (mmse)) results are globally optimal.

Normal Distribution (contd.) N(0,1) is called normalized Guassian. N(0,1) is symmetric i.e. f(x)=f(-x) F(z) = 1-F(z). Failure rate h(t) follows IFR behavior. Hence, N( ) is suitable for modeling long-term wear or aging related failure phenomena.

Functions of Random Variables Often, rv’s need to be transformed/operated upon. Y = Φ (X) : so, what is the density of Y ? Example: Y = X2 If X is N(0,1), then, Above Y is also known as the χ2 distribution (with 1-degree of freedom).

Functions of RV’s (contd.) If X is uniformly distributed, then, Y= -λ-1ln(1-X) follows Exp(λ) distribution transformations may be used to generate random variates (or deviates) with desired distributions.

Functions of RV’s (contd.) Given, A monotone differentiable function, Above method suggests a way to get the random variates with desired distribution. Choose Φ to be F. Since, Y=F(X), FY(y) = y and Y is U(0,1). To generate a random variate with X having desired distribution, generate U(0,1) random variable Y, then transform y to x= F-1(y) . This inversion can be done in closed-form, graphically or using a table.

Jointly Distributed RVs Joint Distribution Function: Independent rv’s: iff the following holds:

Joint Distribution Properties

Joint Distribution Properties (contd)

Order statistics: kofn, TMR

Order Statistics: KofN X1 ,X2 ,..., Xn iid (independent and identically distributed) random variables with a common distribution function F(). Let Y1 ,Y2 ,...,Yn be random variables obtained by permuting the set X1 ,X2 ,..., Xn so as to be in increasing order. To be specific: Y1 = min{X1 ,X2 ,..., Xn} and Yn = max{X1 ,X2 ,..., Xn}

Order Statistics: KofN (Continued) The random variable Yk is called the k-th ORDER STATISTIC. If Xi is the lifetime of the i-th component in a system of n components. Then: Y1 will be the overall series system lifetime. Yn will denote the lifetime of a parallel system. Yn-k+1 will be the lifetime of an k-out-of-n system.

Order Statistics: KofN (Continued) To derive the distribution function of Yk, we note that the probability that exactly j of the Xi's lie in (- ,y] and (n-j) lie in (y, ) is:

Applications of order statistics Reliability of a k out of n system Series system: Parallel system: Minimum of n EXP random variables is special case of Y1 = min{X1,…,Xn} where Xi~EXP(i) Y1~EXP( i) This is not true (that is EXP dist.) for the parallel case

Triple Modular Redundancy (TMR) R(t) Voter R(t) R(t) An interesting case of order statistics occurs when we consider the Triple Modular Redundant (TMR) system (n = 3 and k = 2). Y2 then denotes the time until the second component fails. We get:

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

TMR (Continued) In the following figure, we have plotted RTMR(t) vs t as well as R(t) vs t.

TMR (Continued)

Cold standby (dynamic redundancy) X Y Lifetime of Active EXP() Lifetime of Spare EXP() Total lifetime 2-Stage Erlang EXP() Assumptions: Detection & Switching perfect; spare does not fail

Sum of RVs: Standby Redundancy Two independent components, X and Y Series system (Z=min(X,Y)) Parallel System (Z=max(X,Y)) Cold standby: the life time Z=X+Y

Sum of Random Variables Z = Φ(X, Y)  ((X, Y) may not be independent) For the special case, Z = X + Y The resulting pdf is (assuming independence), Convolution integral (modify for the non-negative case) If X and Y are mutually independent, the resulting pdf is just the product of two pdf’s rather than Convolution.

Convolution (non-negative case) Z = X + Y, X & Y are independent random variables (in this case, non-negative) The above integral is often called the convolution of fX and fY. Thus the density of the sum of two non-negative independent, continuous random variables is the convolution of the individual densities.

Cold standby derivation X and Y are both EXP() and independent. Then

Cold standby derivation (Continued) Z is two-stage Erlang Distributed

Convolution: Erlang Distribution The general case of r-stage Erlang Distribution When r sequential phases have independent identical exponential distributions, then the resulting density is known as r-stage (or r-phase) Erlang and is given by:

Convolution: Erlang (Continued) EXP()

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 =  .

Warm standby derivation (Continued) X is EXP(1) and Y is EXP(2) and are independent 1 = 2 Then fZ(t) is

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()

TMR and TMR/simplex as hypoexponentials

Hypoexponential: general case Z = , where X1 ,X2 , … , Xr are mutually independent and Xi is exponentially distributed with parameter i (i = j for i = j). Then Z is a r-stage hypoexponentially distributed random variable. EXP(1) EXP(2) EXP(r)

Hypoexponential: general case

KofN system lifetime as a hypoexponential At least, k out of n units should be operational for the system to be Up. EXP(n) EXP((n-1)) EXP(k) EXP((k-1)) EXP() ... ... Y1 Y2 Yn-k+1 Yn-k+2 Yn

KofN with warm spares At least, k out of n + s units should be operational for the system to be Up. Initially n units are active and s units are warm spares. EXP(n s) EXP(n +(s-1) ) EXP(n + ) EXP(n) EXP(k) ... ...

Sum of Normal Random Variables X1, X2, .., Xk are normal ‘iid’ rv’s, then, the rv Z = (X1+ X2+ ..+Xk) is also normal with, X1, X2, .., Xk are normal. Then, follows Gamma or the χ2 (with n-degrees of freedom) distribution