Explicit Solutions, Properties and Applications of DES and RES Markov processes

In this talk I will present work that is part of my PhD research at Leiden University. I will define and study the down entrance state (DES) and the restart entrance state (RES) classes of quasi skip free Markov processes. I will derive explicit solutions and bounds for the steady state probabilities of both processes using successive lumping (SL). To motivate their applicability I will present solutions for queueing and inventory problems. In addition I will present a procedure to decompose Markov processes into separate thinned processes. I will discuss applications when the thinned processes satisfy the DES and RES property and the original process does not. Secondly, I will provide a comparison of SL with various other methods. These methodologies are compared both in terms of applicability limitations, and numerical complexity by analyzing their performance. Finally, under homogeneity and irreducibility assumptions I will show that the stationary distribution of a specific DES process has a product form as a function of the level. In particular I will study some other properties of this stationary distribution and derive monotonicity results.