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 ℙ(X > x). A general alternating Markov renewal model is proposed and an asymptotic exponential form ℙ(X > x) ~ Ce-γx identified for the case of a deterministic task time L ≡ ℓ. The rate γ is given by equating the spectral radius of a certain matrix to 1, and the asymptotic form of γ = γ (ℓ) as ℓ → ∞ is derived, leading to the asymptotics of ℙ(X > x) for random task times L. A main finding is that X is always heavy-tailed if L has unbounded support. The casewhere 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.
CITATION STYLE
Asmussen, S., Lipsky, L., & Thompson, S. (2015). Markov renewal methods in restart problems in complex systems. In The Fascination of Probability, Statistics and their Applications: In Honour of Ole E. Barndorff-Nielsen (pp. 501–527). Springer International Publishing. https://doi.org/10.1007/978-3-319-25826-3_23
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