Level Crossing Ordering of Stochastic Processes

Stochastic Ordering is an important area of Applied Probability tailored for qualitative comparisons of random variables, random vectors, and stochastic processes. In particular, it may be used to investigate the impact of parameter changes in important performance measures of stochastic systems, avoiding exact computation of those performance measures. In this respect, the great diversity of performance measures used in applied sciences to characterize stochastic systems has inspired the proposal of many types of stochastic orderings.

In this talk we address the level crossing ordering, proposed by A. Irle and J. Gani in 2001, that compares stochastic processes in terms of the times they take to reach high levels (states). After introducing some motivation for the use of the level crossing ordering, we present tailored sufficient conditions for the level crossing ordering of (univariate and multivariate) Markov and semi-Markov processes. These conditions are applied to the comparison of birth-and- death processes with catastrophes, queueing networks, and particle systems.

Our analysis highlights the benefits of properly using the sample path approach, which compares directly trajectories of the compared processes defined on a common probability space. This approach provides, as a by-product, the basis for the construction of algorithms for the simulation of stochastic processes ordered in the level crossing ordering sense. In the case of continuous Markov chains, we resort additionally to the powerful uniformization technique, which uniformizes the rates at which transitions take place in the processes being compared.

Joint work with Fátima Ferreira (CM-UTAD and Universidade de Trás-os-Montes e Alto Douro).