Section: New Results

Stochastic Modeling

Participants : Alain Jean-Marie, Hlib Mykhailenko, Eleni Vatamidou.

Semi-Markov Accumulation Processes

E. Vatamidou and A. Jean-Marie have introduced in [37] a new accumulation process, the Semi-Markov Accumulation Process (SMAP). This class of processes extends the framework of continuous-time Markov Additive Processes (MAPs) by allowing the underlying environmental component to be a semi-Markov process instead of a Markov process. They follow an analytic approach to derive a Master Equation formula of the Renewal type that describes the evolution of SMAPs in time. They show that under exponential holding times, a matrix exponential form analogous to the matrix exponent of a MAP is attained. Finally, they consider an application of these results where closed-form solutions are rather easy to achieve.

The marmoteCore platform

The development of marmoteCore (see Section 6.1) has been pursued. The software library is now mature enough to develop complex models, such as in [33]. Its architecture and main capabilities have been presented in [26]. marmoteCore provides the classes necessary to represent the state space of Markov models, from the elementary bricks that are interval or rectangular domains, simplices, or binary sequences. From there, the user easily programs the construction of probability transition matrices or infinitesimal generators. Structural analysis methods allow to identify recurrent and transient classes, and to compute the period of the model. Numerous methods allow the Monte Carlo simulation of the chain, the computation of transient and stationary distributions, as well as hitting times. marmoteCore is organized in a hierarchy of Markov models, from the simplest ones (Poisson process, two-state chains, ...) to the most general ones, including classes of models with a particular interest, such as QBDs. It is therefore possible to program solution methods specifically optimized and adapted to the level of structure of the model.