Transient Solution of Markov Chains
Gunter Bolch, Stefan Greiner, Hermann de Meer, Kishor Shridharbhai Trivedi · 2006
The more challenging case of transient analysis of Markov chains is investigated in Chapter 5. The chapter introduces symbolic solutions in simple cases such as small or very regular state spaces. In general, numerical techniques are more suitable and are therefore covered in detail. Uniformization and some variants thereof are introduced as the method of choice for transient analysis in most cases. Particular emphasis is given to stiffness tolerant uniformization that can be of practical relevance in many modeling scenarios where relatively rare and fast events occur concurrently. As an alternative a method for aggregation/disaggregation of stiff Markov chains is introduced for a computation of approximate transient state probabilities. The method is based on a distinction of fast recurrent and fast transient sets of states that can be aggregated with relatively small error. All steps are illustrated by a detailed example model of server breakdown and repair. In addition to numerical results an error analysis is provided as well.