Markov Renewal Methods in Restart Problems in Complex Systems
| Autor: | Stephen Thompson, Søren Asmussen, Lester Lipsky |
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| Přispěvatelé: | Podolskij, Mark, Stelzer, Robert, Thorbjørnsen, Steen, Veraart, D. Almut E. |
| Jazyk: | dánština |
| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | Asmussen, S, Lipsky, L & Thompson, S 2016, Markov Renewal Methods in Restart Problems in Complex Systems . i M Podolskij, R Stelzer, S Thorbjørnsen & D A E Veraart (red), The Fascination of Probability, Statistics and their Applications : In Honour of Ole E. Barndorff-Nielsen . Springer, Cham, s. 501-527 . https://doi.org/10.1007/978-3-319-25826-3_23 The Fascination of Probability, Statistics and their Applications ISBN: 9783319258249 |
| DOI: | 10.1007/978-3-319-25826-3_23 |
| Popis: | A task with ideal execution time L such as the execution of a computer program or the transmission of a file on a data link may fail, and the task then needs to be restarted. The task is handled by a complex system with features similar to the ones in classical reliability: failures may be mitigated by using server redundancy in parallel or k-out-of-n arrangements, standbys may be cold or warm, one or more repairmen may take care of failed components, etc. The total task time X (including restarts and pauses in failed states) is investigated with particular emphasis on the tail \({\mathbb P}(X>x)\). A general alternating Markov renewal model is proposed and an asymptotic exponential form \({\mathbb P}(X>x)\sim C{\mathrm {e}}^{-\gamma x}\) identified for the case of a deterministic task time \(L\equiv \ell \). The rate \(\gamma \) is given by equating the spectral radius of a certain matrix to 1, and the asymptotic form of \(\gamma =\gamma (\ell )\) as \(\ell \rightarrow \infty \) is derived, leading to the asymptotics of \({\mathbb P}(X>x)\) for random task times L. A main finding is that X is always heavy-tailed if L has unbounded support. The case where the Markov renewal model is derived by lumping in a continuous-time finite Markov process with exponential holding times is given special attention, and the study includes analysis of the effect of processing rates that differ with state or time. |
| Databáze: | OpenAIRE |
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