# 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.