Aggregation and Numerical Techniques for Passage Time Calculations in Large semi-Markov Models


ABSTRACT First-passage time densities and quantiles are important metrics in performance analysis. They are used in the analysis of mobile communication systems, web servers, manufacturing systems as well as for the analysis of the quality of service of hospitals and government organisations. In this report we look at computational techniques for the first-passage time analysis on high-level models that translate to Markov and semi-Markov processes. In particular we study exact first-passage time analysis on semi-Markov processes. Previous studies have shown that it is possible to analytically determine passage times by solving a large set of linear equations in Laplace space. The set of linear equations arises from the state transition graph of the Markov or semi-Markov process, which is usually derived from high-level models such as process algebras or stochastic Petri nets. The difficulty in passage time analysis is that even simple high-level models can produce large state transition graphs with several million states and transitions. These are difficult to analyse on modern hardware, because of limitations in the size of main memory. Whilst for Markov processes there exist several efficient techniques that allow the analysis of large chains with more than 100 million states, in the semi-Markov domain such techniques are still less developed. Consequently parallel passage time analyser tools currently only work on semi-Markov models with fewer than 50 million states. This study extends existing techniques and presents new approaches for state space reduction and faster first-passage time computation on large semi-Markov processes. We show that intelligent state space partitioning methods can reduce the amount of main memory needed for the evaluation of first-passage time distributions in large semi-Markov processes by up to 99% and decrease the runtime by a factor of up to 5 compared to existing semi-Markov passage time analyser tools. Finally we outline a new passage time analysis tool chain that has the potential to solve semi-Markov processes with more than 1 billion states on contemporary computer hardware.

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    ABSTRACT: Semi-Markov modelling paradigms are more expressive than their Markovian counterparts but are equally vulnerable to the state space explosion problem. This paper addresses this issue by presenting an exact state-by-state aggregation algorithm for semi-Markov mod- els. Empirical evidence shows that the computational complexity of our method depends critically on the order in which the states are aggregated. We investigate different state-ordering strategies and find one that aggregates a large proportion (circa 75%) of our exam- ple state spaces at minimal cost. This leaves a significa ntly smaller model which can then be analysed for performance-related quanti- ties such as passage-time quantiles. We demonstrate our technique on several examples, including a state model which is re- duced to states.


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