# Download PDF by Attila Csenki: Dependability for Systems with a Partitioned State Space:

By Attila Csenki

ISBN-10: 0387943331

ISBN-13: 9780387943336

ISBN-10: 1461226740

ISBN-13: 9781461226741

Probabilistic versions of technical platforms are studied the following whose finite nation house is partitioned into or extra subsets. The structures thought of are such that every of these subsets of the country house will correspond to a undeniable functionality point of the approach. The crudest technique differentiates among 'working' and 'failed' method states merely. one other, extra refined, procedure will differentiate among many of the degrees of redundancy supplied through the method. The dependability features tested listed here are random variables linked to the nation space's partitioned constitution; a few standard ones are as follows • The series of the lengths of the system's operating sessions; • The sequences of the days spent through the method on the a variety of functionality degrees; • The cumulative time spent by means of the procedure within the set of operating states through the first m operating classes; • the whole cumulative 'up' time of the method until eventually ultimate breakdown; • The variety of fix occasions in the course of a fmite time period; • The variety of fix occasions till ultimate method breakdown; • Any mixture of the above. those dependability features might be mentioned in the Markov and semi-Markov frameworks.

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**Extra info for Dependability for Systems with a Partitioned State Space: Markov and Semi-Markov Theory and Computational Implementation**

**Example text**

Q. Q. ;::: 2. Q. Q. E {I, ... Q. ;::: n + 1. 13. 13. 53) obviously hold. 51) holds. 55) with z = 1. • NOTE. 13 first appeared in [CSE5]. The distribution of the continuoustime counterpart of LAJ,n was considered by Sericola [SER]. Closed form expressions for the distribution of the variable LA\,n seem, however, not to have been considered in the literature prior to [CSE5]. The closest result to ours is in [lOS] pp 162, where a formula is given for the characteristic function of the vector { Vn(s): s E S } with Vn(s) standing for the 'occupation time in state s during the first n steps'.

LW} = 0, QA2{ w} = 0, and OC W = O. The Ai - component of the initial probability vector of Z is OC Ai with OC AIT 1 + OC A/ 1 = 1. For zED = {Z E [ : I Z I :0; 1 }, and S', S" E { At. 63) In the next theorem, the probability generating function of any finite collection of sojourn times in Al and A2 is stated. 16. For mE {I, 2, ... 64) with m-l Ji. :{) k=l m-I Ji. :{) k=l m n (~AjACCj)(Zik)~ ACCjjA/ZC(i)k)) 1. 65) are zero. 16. The proof is by induction on m. 63). 65) readily follows. We first note that if Z starts in some a E Ai, the first sojourn of Z in Ai is of length Ki (= NAj,l) which is defined by Ki = k <=> Zo E Ai, ...

N ::::: 1. • Zn-I }. • by Z during the first n 'time' instances. 45). 46). n was the finite-horizon version of NSAJ,oo, Similarly to LAJ,n. KAI,n is also finite even if Z is not absorbing. , it need not be absorbing or irreducible. 14. The probability mass function o/KAI,I is given by l a A 2T Pr{ KAJ,n=k} = 1 if k = O. oa AI T 1 if k = 1, if k ::::: 2. For n ::::: 2, the probability mass function 0/ KA I,n is given by if k = 0, if k E {I, ... , it is the largest integer not exceeding (n+l)/2. 48) and Hn = (I - QAIAI ® E(n, 2)r l QAIA2 ® E(n, 2) X (I - QA2A2 ® E(n, 2)r 1 QA2AI ® E(n, 2).

### Dependability for Systems with a Partitioned State Space: Markov and Semi-Markov Theory and Computational Implementation by Attila Csenki

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