Markov Modeling of Availability and Unavailability Data
Peter Buchholz and Jan Kriege
Proceedings of the Tenth European Dependable Computing Conference (EDCC 2014), Newcastle upon Tyne, UK, 2014
Abstract
Markov models are often used in performance and dependability analysis and
allow a precise and numerically stable computation of many result measures
including those which result from rare events. It is, however, known that
simple exponential distributions, which are the base of Markov modeling, cannot
adequately describe the duration of availability or unavailability intervals of
components in a distributed system. Commonly used in modeling those durations
are Weibull, log-normal or Pareto distributions that can also capture a
possibly heavy tailed behavior but cannot be analyzed analytically or
numerically. An alternative to applying the mentioned distributions in modeling
availability or unavailability intervals are phase type distributions and Markovian arrival processes that
still result in a Markov model. Based on experiments for a large number of
publically available availability traces, we show that phase type distributions are
a flexible alternative to other commonly known distributions
and even more that Markov models can be easily extended to capture also
correlation in the length of availability or unavailability intervals.
PHDs for failure traces from the FTA
PHDs: A collection of Phase-type distributions (PHDs) fitted to traces from the
Failure Trace Archive
- For fitting each trace was divided into a trace with availability intervals and a trace with unavailability
intervals scaled to hours. The collection contains PHDs of different order fitted with the tool
gfit.
- File names have the format <trace>_avail_<order>.mat for availability times or
<trace>_uavail_<order>.mat for unavailability times.
- The files are ready to use with GNU Octave.
You can load a PHD with
load <filename>,
which sets the two variables pi and D0
to the initial distribution vector and the matrix with transition rates of the PHD, respectively.