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SUMMARY:Transient behaviour in highly dependable Markovian systems\, new r
 egimes\, many paths - Reijsbergen\, D\, de Boer\, P-T\, Scheinhardt\, W (T
 wente)
DTSTART:20100622T104000Z
DTEND:20100622T110500Z
UID:TALK25321@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:In recent years\, probabilistic analysis of highly dependable 
 Markovian systems has received considerable attention. Such systems typica
 lly consist of several component types\, subject to failures\, with spare 
 components for replacement while repair is taking place. System failure oc
 curs when all (spare) components of one or several types have failed. In t
 his work we try to estimate the probability of system failure before some 
 fixed time bound $	au$ via stochastic simulation. Obviously\, in a highly 
 dependable system\,  system failure is a rare event\, so we apply importan
 ce sampling (IS) techniques\, based on knowledge of the behaviour of the s
 ystem and the way the rare event occurs. \n\nInterestingly\, we can discer
 n quite a few different situations to explain why system failure is rare\,
  each with its own typical way of how the rare event is reached\, namely: 
 (1) low component failure rates\, (2) small value of $	au$\, (3) many spar
 e components and (4) high component repair rates. Each of these can be con
 sidered as a limiting regime in which some model parameter tends to $0$ or
  infinity. Classifying this parameter as the `mph{rarity parameter}'\, we
  can measure the performance of an IS scheme by how well it does in the as
 ymptote involved. We could also combine regimes\, which sometimes leads to
  new cases and sometimes not (e.g. the limit in which both failure and rep
 air rates become small is equivalent to $	au$ becoming small).\n\nFor case
 s (1) and (2)\, a combination of balanced failure biasing and forcing was 
 proven to have bounded relative error in 	te{shahabuddin1994importance}. I
 n 	te{deboer2007estimating} an alternative estimator was proposed\, based 
 on the dominant path to failure\, the idea being that when an event is rar
 e\, deviations from the most likely path to this event become even more ra
 re. However\, in several model checking problems an analysis based on domi
 nant paths fails to identify a well-performing change of measure. The reas
 on is that the contribution of some other paths to the probability of inte
 rest is too large to neglect\, or\, more formally speaking\, that the cont
 ribution of these paths does not vanish asymptotically.\n\nIn our paper\, 
 we first prove that in the asymptote of case (3)\, which is interesting in
  its own right\, the dominant path to failure indeed does determine the en
 tire rare event\, as in cases (1) and (2). Then we demonstrate that this i
 s not true for case (4). We propose a state- and time-dependent change of 
 measure for a simple\, yet nontrivial\, model. Our measure is based on the
  one in 	te{deboer2007estimating} and takes all paths into account that co
 ntribute to the probability of interest.  Finally\, we empirically verify 
 that our estimators have good performance.\n  \n[1] P.T. de Boer\, P. L'ec
 uyer\, G. Rubino\, and B. Tuffin.  Estimating the probability of a rare ev
 ent over a finite time horizon.  In Proceedings of the 2007 Winter Simulat
 ion Conference\, pages 403-411\, 2007.\n[2] P. Shahabuddin.  Importance \n
 \n
LOCATION:Seminar Room 1\, Newton Institute
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