Numerical Analysis of Generalized Semi-Markov Processes

Christoph Lindemann

Universität Dortmund


Résumé:

In this talk, we present recent results towards the development of effective numerical methods for stationary and transient analysis of finite-state generalized semi-Markov processes (GSMPs) with exponential and deterministic events. In previous work, we introduced an approach for the analysis of such GSMPs based on a general state space Markov chain (GSSMC) embedded at equidistant time points nD (n=1,2,..) of the continuous-time GSMP. For being practical applicable, this approach requires the algorithmic generation of the transition kernel of this GSSMC from the building blocks of the GSMP. Furthermore, we need efficient numerical solvers for the system of multidimensional Volterra integral equations that constitute the time-dependent and stationary equations of the GSSMC. In this talk, we focus on the algorithmic generation of the transition kernel. The transition kernel of the GSSMC specifies one-step jump probabilities from a given state at instant of time nD to all reachable new states at instant of time (n+1)D. In general, entries of the transition kernel of a GSSMC are functions of clock readings associated with the current state and intervals for clock readings associated with the new state. Key contributions constitutes the derivation of conditions on the building blocks of the GSMP under which kernel entries are constant (i.e., are not functions of clock readings) and under which submatrices of the kernel are separable. Such a submatrix can be expressed as the sum and/or product of a matrix comprising only constant entries, a matrix comprising only functional entries setting new clocks and a matrix comprising only functional entries taking into account old clocks. Due to the exploitation of these properties, the GSSMC approach shows great promise for being effectively applicable for the stationary and transient analysis of GSMPs with large state space and several deterministic events concurrently active.


[Christoph Lindemann]
[Universität Dortmund]