Different Monotonicity Definitions in Stochastic Modelling
In this paper we discuss different monotonicity definitions applied in stochastic modelling. Obviously, the relationships between the monotonicity concepts depends on the relation order that we consider on the state space. In the case of total ordering, the stochastic monotonicity used to build bounding models and the realizable monotonicity used in perfect simulation are equivalent to each other while in the case of partial order there is only implication between them. Indeed, there are cases of partial order, where we can’t move from the stochastic monotonicity to the realizable monotonicity, this is why we will try to find the conditions for which there are equivalences between these two notions. In this study, we will present some examples to give better intuition and explanation of these concepts.
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- 2.Borovkov, A.A., Foss, S.: Two ergodicity criteria for stochastically recursive sequences. Acta Appl. Math. 34 (1994)Google Scholar
- 4.Fill, J.A., Machida, M.: An interruptible algorithm for perfect sampling via markov chains. In: STOC 1997: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, New York, USA, pp. 688–695 (1997)Google Scholar
- 5.Fill, J.A., Machida, M.: Realizable monotonicity and inverse probability transform. Technical report, Department of Mathematical sciences. The Johns Hopkins university (2000)Google Scholar
- 6.Fourneau, J.M., Kadi, I., Pekergin, N., Vienne, J., Vincent, J.M.: Perfect simulation and monotone stochastic bounds. In: ValueTools 2007: Proceedings of the 2nd international conference on Performance 249-263evaluation methodologies and tools, pp. 1–10, ICST (2007)Google Scholar
- 9.Olle Haggstrom. Finite Markov Chains and algorithmic applications, Matematisk Statistik, Chalmers teknisk hogshola och Goteborgs universitet (2001)Google Scholar
- 13.Stenflo, O.: Ergodic theorems fory Iterated Function Systems controlled by stochastic sequences. Doctoral thesis n. 14, Umea university (1998)Google Scholar
- 14.Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. John Wiley & Sons, ChichesterGoogle Scholar
- 15.Vincent, J.-M.: Perfect simulation of queueing networks with blocking and rejection. In: SAINT-W 2005: Proceedings of the 2005 Symposium on Applications and the Internet Workshops, Trento, Italy, pp. 268–271 (2005)Google Scholar