Transient behaviour in highly dependable Markovian systems: new regimes, many paths.

Transcription

1 Transient behaviour in highly dependable Markovian systems: new regimes, many paths. Daniël Reijsbergen Pieter-Tjerk de Boer Werner Scheinhardt University of Twente RESIM, June 22nd, 2010

2 Outline Problem Setting Path-based Importance Sampling Asymptotic Regimes Many Paths for Fast Repairs What to Remember

3 Problem Setting Outline Problem Setting Path-based Importance Sampling Asymptotic Regimes Many Paths for Fast Repairs What to Remember

4 Problem Setting Model Checking (1) Automated verification of probabilistic systems. System model is specified in high-level language (e.g. Petri net). Model checker then verifies performance properties. E.g. the probability of overflow before time τ is smaller than p.

5 Problem Setting Model Checking (2) Model is translated to continuous time Markov chain with: transition rates q ss, starting state s 0, time bound τ, set of overflow states ( >n components have failed). Our interest: P(s 0, τ) = P(overflow before τ start in s 0 )

6 Problem Setting s 0

7 Problem Setting Model Checking (3) Probability P(s 0, τ) is estimated using simulation. Problem: in reliable systems, P(s 0, τ) is small. Solution: we use importance sampling.

8 Path-based Importance Sampling Outline Problem Setting Path-based Importance Sampling Asymptotic Regimes Many Paths for Fast Repairs What to Remember

9 Path-based Importance Sampling Importance Sampling (1) New measure tries to imitate the zero variance measure. If we are in state s with t time units remaining, then the ideal density of jumping to state s after δ time units is based on ratio P(s, t δ). P(s, t) P(s, t) is not known explicitly, so we use an approximation ˆP(s, t) and normalise.

10 Path-based Importance Sampling Importance Sampling (2) We use an approximation based on probability of the most likely path (or paths) to overflow from a state s. The result is a state-dependent and possibly time-dependent simulation density.

11 Path-based Importance Sampling Most Likely Paths (1) s 0 System with two parallel queues.

12 Path-based Importance Sampling Most Likely Paths (2) s 0 Most likely paths are the straight paths to overflow.

13 Path-based Importance Sampling Most Likely Paths (3) s 0 Most likely paths are the straight paths to overflow.

14 Path-based Importance Sampling Less Likely Paths (1) s 0 Path has an unnecessary arrival in queue 2.

15 Path-based Importance Sampling Less Likely Paths (2) s 0 Path has an unnecessary cycle from the empty state.

16 Path-based Importance Sampling Advantages Advantages of path-based importance sampling: Easy to generalise. Paths are found quickly using high-level model description. Performs well when probability contribution of other paths vanishes quickly.

17 Asymptotic Regimes Outline Problem Setting Path-based Importance Sampling Asymptotic Regimes Many Paths for Fast Repairs What to Remember

18 Asymptotic Regimes Causes of Rarity Probability contribution of non-straight paths may vanish because: 1. the extra jumps occur with vanishing probability 2. the extra sojourns consume an increasingly unlikely amount of time.

19 Asymptotic Regimes Asymptotic Regimes (1) τ 0 0 n τ 0: short mission time. 0: highly dependable system. n : high redundancy. : highly recoverable system.

20 Asymptotic Regimes τ 0 τ 0 0 n Extra sojourns consume too much time. Possible (partial) solution: forcing. Nicola, Nakayama, Heidelberger and Goyal. Fast simulation of highly dependable systems with general failure and repair processes. IEEE Trans on Comp, Nakayama and Shahabuddin. Quick simulation methods for estimating the unreliability of regenerative models of large, highly reliable systems. PEIS, 2004.

21 Asymptotic Regimes τ 0 s 0 Extra sojourns take too much time.

22 Asymptotic Regimes 0 τ 0 0 n Sojourns in empty state consume too much time. Failures before repairs occur with vanishing probability. Possible solution: forcing + (balanced) failure baising. Nicola, Nakayama, Heidelberger and Goyal. Fast simulation of highly dependable systems with general failure and repair processes. IEEE Trans on Comp, de Boer, L Ecuyer, Rubino and Tuffin. Estimating the probability of a rare event over a finite time horizon. Winter Simulation Conference, 2007.

23 Asymptotic Regimes 0 s 0 Extra jumps increasingly unlikely.

24 Asymptotic Regimes n τ 0 0 n Extra sojourns consume too much time. Often considered in combination with and. Possible solutions: based on large deviations.

25 Asymptotic Regimes τ 0 0 n Failures before repairs occur with vanishing probability. So far, only bounds (based on estimates) for large τ. Nakayama and Shahabuddin. Quick simulation methods for estimating the unreliability of regenerative models of large, highly reliable systems. PEIS, 2004.

26 Asymptotic Regimes s 0 If : contribution does not vanish asymptotically!

27 Many Paths for Fast Repairs Outline Problem Setting Path-based Importance Sampling Asymptotic Regimes Many Paths for Fast Repairs What to Remember

28 Many Paths for Fast Repairs Many Paths for Fast Repairs (1) Probability of less likely paths does not vanish when. Solution: use the regenerative behaviour of the model. Probability of the straight paths remains a good guess for π = P( overflow during busy cycle).

29 Many Paths for Fast Repairs Many Paths for Fast Repairs (2) When, duration of busy cycle = sojourn time in s 0. This is exponentially distributed, with rate equal to exit rate of empty state. # busy cycles before overflow geometrically distributed(π). Distribution of time until overflow: exponential.

30 Many Paths for Fast Repairs Many Paths for Fast Repairs (3) ˆP(s 0, t) = 1 e tπ tπ when π small For all states s s 0, we set ˆP(s, t) to be the sum of 1. P(straight paths from s) 2. the probability of the path that falls back to the empty state, multiplied by ˆP(s 0, t) Resulting measure, combined with forcing, leads to measure that works well in regimes 0, τ 0 and.

31 Conclusions Outline Problem Setting Path-based Importance Sampling Asymptotic Regimes Many Paths for Fast Repairs What to Remember

32 Conclusions What to Remember Path-based importance sampling can efficiently be applied in computerised model checking. Challenges due to different forms of rarity. Future work: Quick search for most likely paths. Exploit regenerative behaviour in more general scenarios.