Regularity of Stochastic Processes: A Theory Based on Directional Convexity

摘要

The authors define a notion of regularity ordering called directionally convex ordering (dcx) among stochastic processes. They give examples of doubly stochastic Poisson and Markov renewal processes where such ordering is prevalent. Furthermore the authors show that the class of segmented processes introduced by Chang, Chao and Pinedo (1991) provides a rich set of stochastic processes where the dcx ordering can be commonly encountered. When the input processes to a large class of queueing systems (single stage as well as networks) are dcx ordered, so are the processes associated with these queueing systems. For example if two input processes to a tandem multiplied by /M/c -> /M/c -> equals as (proportion) -> /M/c queueing system are dcx ordered so are the number of customers in the system. The concept of directionally convex functions (Shaked and Shanthikumar) and the notion of multivariate stochastic convexity (Chang, Chao, Pinedo and Shanthikumar (1991)) are employed in the analysis. (Copyright (c) 1992 by Faculty of Technical Mathematics and Informatics, Delft, The Netherlands.)