Abstract
A task with ideal execution time \(\ell \) is handled by a Markovian system with features similar to the ones in classical reliability. The Markov states are of two types, UP and DOWN, such that the task only can be processed in an UP state. Upon entrance to a DOWN state, processing is stopped and must be restarted from the beginning upon the next entrance to an UP state. The total task time \(X=X_\mathrm {r}(\ell )\) (including restarts and pauses in failed states) is investigated with particular emphasis on the expected value \({\pmb {\mathbb {E}}}[X_\mathrm {r}(\ell )]\), for which an explicit formula is derived that applies for all relevant systems. In general, transitions between UP and DOWN states are interdependent, but simplifications are pointed out when the UP to DOWN rate matrix (or the DOWN to UP) has rank one. A number of examples are studied in detail and an asymptotic exponential form \(\exp (\beta _\mathrm {m} \ell )\) is found for the expected total task time \({\pmb {\mathbb {E}}}[X(\ell )]\) as \(\ell \rightarrow \infty \). Also, the asymptotic behavior of the total distribution, \(H_\mathrm {r}(x|\ell )\rightarrow \exp (-x\gamma (\ell ))\), as \(x\rightarrow \infty \) is discussed.
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Notes
- 1.
For clarity, note that in general the boldfaced subscripts identify the entire matrix, not its components. The components are identified by the notation, \((\mathbf{Q_{ud}})_{\mathrm {ij}}\), \(i\in E_\mathrm {u}\), and \(j\in E_\mathrm {d}\).
- 2.
Of course, even if all the Cov coefficients were 0, that would not be sufficient to have independence. Just one nonzero coefficient proves dependence.
- 3.
Strictly speaking, this is only true if \(\sigma \) is unique. If it is a multiple root then the formula is more complicated.
- 4.
spr[\(\mathbf{X}\)] = SPectral Radius := Largest eigenvalue in magnitude of matrix \(\mathbf{X}\).
- 5.
\(\mathbf{D(\ell \beta _\mathrm {m})}\) is a matrix whose components are polynomials of degree less than or equal to the degree of \(p(\ell \beta _\mathrm {m})\). Therefore, \(\mathbf{D}(\ell \beta _\mathrm {m})/p(\ell \beta _\mathrm {m})\sim {O(1)}\).
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Thompson, S., Lipsky, L., Asmussen, S. (2016). Linear Algebraic Methods in RESTART Problems in Markovian Systems. In: Fiondella, L., Puliafito, A. (eds) Principles of Performance and Reliability Modeling and Evaluation. Springer Series in Reliability Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30599-8_17
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DOI: https://doi.org/10.1007/978-3-319-30599-8_17
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