Solving Systems of Bilinear Equations for Transition Rate Reconstruction

Compositional models, specified with the help of a Markovian Stochastic Process Algebra (SPA), are widely used in performance and dependability modelling. The paper considers the problem of transition rate reconstruction: Given two SPA components with unknown rates, and given their combined flat model with fixed rates, the task is to reconstruct the rates in the components. This problem occurs frequently during so-called model repair, if a certain subset of transition rates of the flat model needs to be changed in order to satisfy some given requirement. It is important to have a structured approach to decide whether or not the rate reconstruction, satisfying the desired low-level model changes, is possible or not. In order to realize such a reconstruction, every combined model transition is transformed into an equation, resulting – for each action type – in a system of bilinear equations. If the system of equations meets a consistency condition, rate reconstruction is indeed possible. We identify a class of SPA systems for which solving the system of equations is not necessary, since by checking a set of simple conditions we can check the consistency of the system of equations. Furthermore, for general models outside this class, an iterative algorithm for solving the system of equations efficiently is proposed.