Combining Decomposition and Lumping to Evaluate Semi-hierarchical Systems

Determining performance and fault tolerance properties of distributed systems is a challenging task. One common approach to quantify such properties is to construct the state space and a transition model of the distributed system that is to be evaluated. The challenge lies in the state space being exponentially large in the size of the system. One popular approach to tackle this challenge is to combine decomposition and lumping. The system is decomposed, transition models of the subsystems are constructed and minimized by lumping bisimilar states under an equivalence relation, and the intermediate marginal transition systems are composed to construct the minimal aggregate transition model. The approach allows to circumvent the necessity to construct a full transition model while preserving the ability to compute precise measures. The decomposition yet hinges on the structure of the communication within the system. When processes do not influence each other, decomposition is trivial as it is arbitrary. On the contrary, when all processes are influenced by all other processes — known as heterarchical structure — systems cannot be decomposed at all. Between systems of independent and heterarchical processes are i) hierarchically structured systems and ii) systems that are globally hierarchical, but contain locally heterarchical subsystems. The hierarchical type has been addressed elsewhere. This paper targets the second type — referred to as semi-hierarchically structured —, thus expanding the frontier from decomposing purely hierarchically structured systems to decomposing semi-hierarchically structured systems. Furthermore, this paper points out the role of different types of execution semantics regarding the decomposition.