We consider state charts with generally distributed state sojourn times and with parallel regions in composite states. This corresponds to semi-Markov processes (SMPs) with parallel regions consisting again of SMPs. The concept of parallel regions significantly extends the modeling power: it allows for the specification of non-memoryless activities that take place in parallel on many nested hierarchy levels. Parallel regions can be left either by final states or by exit states, corresponding to the maximum and the minimum of the sojourn times in the regions, respectively. Therefore, concurrent activities with synchronization and competition can easily be modeled. An SMP with parallel regions cannot simply be analyzed by flattening the state space. We propose an analysis based on a steady-state analysis of an embedded Markov chain (EMC) at the top level and by a transient analysis at the composite state level with a limited computational effort. An expression for the asymptotic complexity of the analysis is also provided. An example SMP containing all modeling features with parallel regions is illustrated. We carry out experiments on basis of this model and confirm the results by simulations.


Markov regenerative process Semi-Markov Process concurrency 


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Computer Science 7University Erlangen-NurembergErlangenGermany

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