Efficient Transient Analysis of a Class of Compositional Fluid Stochastic Petri Nets

Fluid Stochastic Petri Nets (FSPNs) which have discrete and continuous places are an established model class to describe and analyze several dependability problems for computer systems, software architectures or critical infrastructures. Unfortunately, their analysis is faced with the curse of dimensionality resulting in very large systems of differential equations for a sufficiently accurate analysis. This contribution introduces a class of FSPNs with a compositional structure and shows how the underlying stochastic process can be described by a set of coupled partial differential equations. Using semi discretization, a set of linear ordinary differential equations is generated which can be described by a (hierarchical) sum of Kronecker products. Based on this compact representation of the transition matrix, a numerical solution approach is applied which also represents transient solution vectors in compact form using the recently developed concept of a Hierarchical Tucker Decomposition. The applicability of the approach is presented in a case study analyzing a degrading software system with rejuvenation, restart, and replication.