Software Reliability Modeling Steven J Zeil Old Dominion Univ. - PowerPoint PPT Presentation

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Software Reliability Modeling Steven J Zeil Old Dominion Univ. Spring 2012 1

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Software Reliability and System Reliability Recap 1 Historical Perspective and Implementation 2 Exponential Failure Time Models 3 Jelinski-Moranda De-eutrophication Model Goel and Okumoto NHPP Model Schneidewind’s Model Musa’s Basic Execution Time Model Hyperexponential Model Other Finite Failure Models 4 Infinite Failure Models 5 Bayesian Models 6 Pre-Implementation Models 7 2

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Reliability Function Denote the probability that the time to failure T is in some interval ( t , t + ∆ t ) as P ( t ≤ T ≤ t + ∆ t ) � t F ( t ) = P (0 ≤ T ≤ t ) = f ( x ) dx 0 The reliability function is the probability of success at time t (i.e., the prob. that the time to failure exceeds t ) � ∞ R ( t ) = P ( T > t ) = 1 − F ( t ) = f ( x ) dx t 3

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Failure Rate The failure rate is the probability that a failure per unit time occurs in the interval [ t , t + ∆ t ], given that a failure has not occurred before t . P ( t ≤ T < t + ∆ t | T > t ) Failure rate ≡ ∆ t P ( t ≤ T < t + ∆ t ) = (∆ t ) P ( T > t ) F ( t + ∆ t ) − F ( t ) = (∆ t ) R ( t ) Failure rate measurable easier to understand than the prob. density function 4

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Hazard Rate The hazard rate is defined as the limit of the failure rate as the interval ∆ t approaches zero. F ( t + ∆ t ) − F ( t ) = f ( t ) z ( t ) = lim (∆ t ) R ( t ) Rt ∆ t → 0 The hazard rate is an instantaneous rate of failure at time t , given that the system survives up to t . z ( t ) dt represents the probability that a system of age t will fail in the small interval t to t + dt . 5

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Failure Intensity Function � t � � R ( t ) = exp − z ( x ) dx 0 or, differentiating � t � � f ( t ) = z ( t ) exp − z ( x ) dx 0 6

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Relating Reliability to Failure Rate (2) Alternatively, Let t e be time required to execute one test case. t = kt e Assume that there is a finite limit for p / t e as t e becomes vanishingly small p λ = lim t e t e → 0 This is the failure intensity function . 7

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Reliability and the Failure Intensity t e → 0 (1 − p ( t e )) t / t e = exp( − λ t ) R ( t ) = lim which is the exponential distribution 8

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Data Models typically employ failures per time period, or time between failures Conversion between the two is possible, with some caveats. 9

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Faults and Failures In this chapter, no distinction is made between faults and failures. They are assumed to be 1-to-1. Implies that we fix all faults before they have a chance to cause a 2nd failure. Let M ( t ) be a random variable denoting the number of failures (faults) experienced by time t . Let µ ( t ) be the mean value function of M ( t ). 10

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 2 Category: total number of failures over infinite time What is lim t →∞ µ ( t )? 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 2 Category: total number of failures over infinite time What is lim t →∞ µ ( t )? finite 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 2 Category: total number of failures over infinite time What is lim t →∞ µ ( t )? finite infinite 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 2 Category: total number of failures over infinite time What is lim t →∞ µ ( t )? finite infinite 3 Type: distribution of the number of failures experienced by time t 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 2 Category: total number of failures over infinite time What is lim t →∞ µ ( t )? finite infinite 3 Type: distribution of the number of failures experienced by time t Poisson 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 2 Category: total number of failures over infinite time What is lim t →∞ µ ( t )? finite infinite 3 Type: distribution of the number of failures experienced by time t Poisson binomial 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 2 Category: total number of failures over infinite time What is lim t →∞ µ ( t )? finite infinite 3 Type: distribution of the number of failures experienced by time t Poisson binomial 4 Class (finite failure models only): Functional form of the failure intensity as a function of time 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Model Classification Scheme Musa & Okumoto classify models according to 1 Time domain: wall clock versus execution time 2 Category: total number of failures over infinite time What is lim t →∞ µ ( t )? finite infinite 3 Type: distribution of the number of failures experienced by time t Poisson binomial 4 Class (finite failure models only): Functional form of the failure intensity as a function of time 5 Family (infinite failure models only): Functional form of the failure intensity as a function of expected number of failures experienced 11

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Poisson Processes The Poisson distribution describes the probability of a given number of events occurring in a fixed interval given that events occur at a fixed rate and are independent of the time since the last event. f ( k ; λ ) = λ k e − λ k ! Divide time range 0 . . . t into a sequence of observation points t 0 , t 1 , . . . t n with t 0 = 0 and t n = t . Let f i denote the # of failures occurring in t i − 1 . . . t i . If the f i are independent Poisson variables, we have a Poisson process. E [ f i ] = µ ( t i ) − µ ( t i − 1 ) 12

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Reliability of a Poisson Process Let ∆ t be any non-negative value such that t i − 1 + ∆ t < t i . The the prob that the software will run reliably for another ∆ t given that it has not yet failed at t i − 1 is � t +∆ t � � R (∆ t | t ) = P ( f i = 0 | t ) = exp − λ ( x ) dx t 13

Recap Historical Perspective and Implementation Exponential Failure Time Models Other Finite Failure Models Infinite Failure Mo Poisson Process and Single Faults z (∆ t | t i − 1 ) = λ ( t i − 1 + ∆ t ) µ ( t ) = α F a ( t ) where α is the number of faults (as t → ∞ ) and F a ( t ) is the cumulative distribution of the time to failure of a single fault a . λ ( t ) = α f a ( t ) 14