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Lecture Structure

- Fault Forecasting (00:00:30)
- Quantitative methods (00:00:30)
- Qualitative methods (00:00:30)
- conseguences of faults (00:00:30)
- to estimate the gresent number the future incidence and the iikely (00:00:30)
- Boundary _ External (00:01:15)
- Motivation (00:01:42)
- Quantify dependability attributes of components (00:01:42)
- - Use a formalism to mode! System dependability (00:01:42)
- Modelling approaches (00:04:12)
- - Deductive methods: postulate a System faifure, ?nd out what (00:04:12)
- - Inductive methods: reasoningfrom Specific cases t0 a general (00:04:12)
- ModeHing approaches (00:05:00)
- Dependabilitv model tvpes (00:05:36)
- m: Mumm um daqnmRG m nun-Im um9.x „mm; rvmrd um (00:05:36)
- - State-based / Markov models: (00:05:36)
- GSPNiCTMC SRN‘? MRM (00:05:36)
- ° Structural /combinatoria1 models: (00:05:36)
- Boolean algebra basics (00:07:03)
- computing these probabilities? (00:07:03)
- - What assumptions are you making when (00:07:03)
- - What is the probability of exactly one disk (00:07:03)
- two discs are con?gured in a RAID l setug? (00:07:03)
- - What is the overall failure probability ßfthe (00:07:03)
- two discs are configured in a RAID O setug? (00:07:03)
- - What is the overall failure probability if the (00:07:03)
- Boolean algebra basics (ctd.) (00:14:06)
- - Stochastically dependent events need more sophisticated modelling! (00:14:06)
- - Pgx) = 1 — P(X) (00:14:06)
- - P(A AB) = P(A) * P05’) (00:14:06)
- - For stochastically independent events (A, B) (00:14:06)
- [Vlemoryless reliability i: distribution (00:15:30)
- - 1 — cdf(t) — Reliability (00:15:30)
- - CdfOf) — Probability offailure before t (00:15:30)
- - Parameter o)‘ exponential distribution Ä x: f“ 0s (00:15:30)
- 1 0 cumulative distribution function (00:15:30)
- - Time to failure (TTF) modelled as random (00:15:30)
- - History does not affect faiiure time (00:15:30)
- - Assumption: time to faHure is memoryless (00:15:30)
- evolution (00:17:03)
- Reliability Block Diagrams (RBDs) (00:17:18)
- RBD—structure formula (00:20:06)
- um’-_ ‚Vn-G/ (00:20:30)
- RBDs — extensions (00:21:21)
- Fault trees (00:22:45)
- - Quantitative analysis regarding mission time (00:22:45)
- - Deductive modelling in failure space (00:22:45)
- Fault trees - events (00:24:36)
- - lntermediate event: ”a fault event which occurs because of (00:24:36)
- - Undeveioped event: ”an event which is not further (00:24:36)
- External/house event: ”An event which is normally (00:24:36)
- - Basic event: ”a basic initlating fault event that requires no (00:24:36)
- Fault trees — static gates (00:26:03)
- Fault trees — qualitative analysis (00:26:54)
- - Minimal path set: the set cannot be reduced without loosing its Status a5 cut set (00:26:54)
- - Path set: a set of basic events whose non-occurrence (at the same (00:26:54)
- ° Cut set: set of basic events whose occurrence (at the Same time) (00:26:54)
- Method for obtaining cut set (IVIOCUS) (00:30:15)
- Sequence-dependent fault trees (00:31:15)
- - More realistic: considering the sequence (00:31:15)
- - Simplicity ofthe mathematical model v5. (00:31:15)
- Sequence-dependent gates (00:33:12)
- Sequence-dependent fault trees — examples (00:35:27)
- hüpßf/‘?/ipfhgithulzta/marko? (00:36:57)
- Stochastic petri nets (00:36:57)
- Markov chains (00:36:57)
- State transition diagrams (00:36:57)
- State-based modelling (00:36:57)
- State-based models (00:37:12)
- State transition diagrams (00:37:42)
- - Operational equilibrium: number of requests in the System is the Same at the (00:37:42)
- ° Homogeneous workload: requests are indistinguishabie; omy sum counts (00:37:42)
- - Assumptions (00:37:42)
- State transition diagrams: application (00:39:09)
- State transition diagrams: analysis (00:40:12)
- Markov Chains (00:40:48)
- neous Markov chains: transition probabHities/rates do not change (00:40:48)
- - Time-homogeintime (00:40:48)
- - Recurrent state: probability ofl to retum to this state after unspecified time t (00:40:48)
- - Markov property: next step depends omy on the current step (00:40:48)
- Markov Chains: Time Model (00:43:03)
- DTMC — example (00:44:06)
- =[_4 5 .25] (00:44:06)
- _ 3 .3 .4[3 5 3] .4 .4 .2inltiaßdwstnbution 5 -3 -2 (00:44:06)
- A row sum a ways 1 (00:44:06)
- Dependability Modelling vvith CTMCS (00:45:42)
- - Stationary Distribution: the probability distribution to which the (00:45:42)
- - Transition: assigned with component failure rate (00:45:42)
- - State: represents a particular error state (00:45:42)
- Quiz (00:46:36)
- - In a Markov chain modellingthis System, which are recurring states? (00:46:36)
- - Would (and can) you model this System using (00:46:36)
- Consider a k»out—of»n system with n components. (00:46:36)
- S . I). i.u Zu M (00:49:33)
- - What dependability metric are we computing here? (00:49:33)
- O or1(upto one faiied component)? (00:49:33)
- ° In the stationary distribution, what is the probability of being in state (00:49:33)
- Example: 2/3 System — availability (00:51:27)
- - Stationary distribution: probabilities are in equilibrium (00:51:27)
- We are looking for the steady-state availability ofthe System. (00:51:27)
- Example: 2/3 System F w-EA 2A (00:52:39)
- Equilibrium: Püeavmgs?) : Nenteringsg) („ä : (00:52:39)
- 1. Balance equations (steady-state equihbrium cfterion): (00:52:39)
- l-s). 3). (00:53:42)
- Ö/?g? (00:54:33)
- - Conflict: When two transiüons need the same token, only one can ?re (00:54:33)
- - Transition ?ring: consume tokens from input maces, produce tokens in Output (00:54:33)
- - Places contain tokens (marking) (00:54:33)
- Blpamte graph (00:54:33)
- Petri Nets — semantic mappings (00:55:36)
- stand byx Q1 (00:56:06)
- Stochastic petri nets (00:57:03)
- Stochastic Petri Nets: Transitions (00:57:57)
- ° Reachability set (00:58:36)
- Example: 2-0193 System (00:58:48)
- Stochastic Petri Nets vs Markov Chains (00:59:39)
- Example: K-of-N vvith Standby and Repairmen (01:00:18)
- Example: parallel System vvith input buffer (01:01:24)
- funcüona! fairen pmpage (01:01:42)
- (Ö4; mnzgaxa3mrm A I.‘ : 12mm54m mona! (3x LB (01:01:42)
- fawLA Dmuögate A ** m“Bjuncuona! ajgued »faH a prooagale a (01:01:42)
- Sequence-dependent PAND gate -) GSPN (01:02:42)
- Petri Net Simulation v5 Analysis (01:03:21)
- Rare event Simulation: importance sampling (01:04:09)
- 9 p = R0505’) *W(X)) (01:04:09)
- lmportance Sampling: Compute p : E(q5(X))‚ where (MX) i5 a desireddependability metric, and X a rare random variable (01:04:09)
- Bayesia n Netvvorks (01:05:00)
- Example: Fault Tree 9 Bayesian Netvvork (01:06:03)
- Every anempt t0 use mathematics t0 smdy some real phcnomena must bcgin with building amathemalical model ofthese phcnomena, Of necessily, the model sim- (01:08:16)
- ot the model and I0 compare Lhem with obscrvatinn, (01:08:16)
- possiblc t0 prCdiCI with cenajnty whcther or not a given mathematical model is (01:08:16)
- from Lhe conditioxus of Ihc praclical pmblem considered. Beforehand. it 15 im— (01:08:16)
- hecause the Original assumptions of the mathemalical model diverge essenlially (01:08:16)
- problem may be correct and you may be in vielem con?ict with realities‘, simply (01:08:16)
- ‘In thc dcxclopmcnt of the phenomena studied. The solution ofthe mathemalical (01:08:16)
- The success dcpends on whether or not thc dctails ignoted are rcally unimponant (01:08:16)
- pli?es Ihe matrers l0 a greater or lesser extent and u number ofdetails are ignored. (01:08:16)

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