Recent issues on stochastic directional convexity, and new results on the analysis of systems for communication, information, time scales and maintenance

Abstract

In an earlier reference, we provided a review of the regular, sample path and strong stochastic concepts of stochastic convexity in both univariate and multivariate settings, jointly with most of their applications, and some other new results for analysing communication systems on the basis of biologically inspired models. This article provides a comprehensive discussion about the regular notion of stochastic increasing and directional convexity, denoted by SI − DCX introduced by Meester and Shanthikumar for a general partially ordered space. We study the connection of the SI − DCX property of X(θ) for θ taking on values over a sublattice and the optimal solution of a maximization problem with objective function given by the expected value of any increasing convex function of the parameterized random variable X(θ), as well as the analysis of the variability ordering of the mixture model with correlated parameters. We illustrate these results with the conditions for the SI − DCX property of the sojourn time at M ∕ δ ∕ 1 processor sharing queues, the arrival time at GI/GI/1‐queues; best monitoring scales for the ageing description; the total score for the best selected items in sequential screening; moments of multivariate distributions; l1 norms; the failure time of systems under opportunistic maintenance; performance measures of the Internet traffic and information systems. These measures are analysed as mixtures with arbitrary mixing distributions, by the increasing convex ordering, and their distributional bounds are derived. Copyright © 2013 John Wiley & Sons, Ltd.