Symbolic Solution of Kronecker-Based Structured Markovian Models

Abstract

Abstract-This paper describes a method to obtain symbolic solution of large stochastic models using Gauss-Jordan elimination. Such solution is an efficient alternative to standard simulations and it allows fast and exact solution of very large and complex models that are hard to be dealt even with iterative numerical methods. The proposed method assumes the system described as a structured (modular) Markovian system with discrete states for each system module and transitions among those states ruled by Markovian processes. The mathematical representation of such system is made by a Kronecker (Tensor) formula, i.e., a tensor formulation of small matrices representing each system module transitions and occasional dependencies among modules. Preliminary results of the proposed solution indicate the expected efficiency of the proposed solution.

Stochastic Performance Analysis of Global Software Development Teams

“…After parameterization of the model events, one can obtain the numerical solution, that is, the steady-state probabilities of being in each state. The numerical analysis of SAN models can be performed by dedicated software packages such as Performance Evaluation of Parallel Systems (PEPS) [Brenner et al 2007], SAN Lite-Solver [Sales 2012], or GTAexpress [Czekster et al 2009], which implement numerical solutions using iterative [Czekster et al 2010a[Czekster et al , 2011a, symbolic [Fernandes et al 2013a] and simulation [Czekster et al 2010b] techniques.…”