No abstract

This paper presents a general theory of stochastic convexity. The notions of stochastic convexity formulated by Shaked and Shanthikumar (1988a, 1988b, 1990) are defined for general partially ordered spaces. All of the closure properties of the one-dimensional real theory are proved to be true in this general framework as well and results concerning the temporal convexity of Markov chains are sharpened. Many proofs are based on new ideas, some of which also provide insightful alternatives for proofs in Shaked and Shanthikumar (1990). The general theory encompasses the (largely one-dimensional) stochastic convexity theory as known from these papers, and at the same time permits treatment of multivariate multiparameter families as well as more general random objects. Among others, it applies to real vector spaces and yields a theory of stochastic convexity for random vectors and stochastic processes. We illustrate this new scope with examples and applications from queueing theory, coverage processes, reliability and branching processes. We show that the virtual waiting time process of an NHPP driven ·/G/1 queue is stochastically convex in the arrival intensity function, which explains the known adverse effect of fluctuating arrival rates; that the expected size of an i.i.d. union of random sets grows concavely; that the expected utility of repairable items under imperfect repair policies is increasing and convex in the probabilities of successful repair.

We define a notion of regularity ordering among stochastic processes called directionally convex (dcx) ordering and give examples of doubly stochastic Poisson and Markov renewal processes where such ordering is prevalent. Furthermore, we show that the class of segmented processes introduced by Chang, Chao, and Pinedo [3] 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 the input processes to two tandem -/M/c, -• / M / c 2 -> • / M / c m queueing systems are dcx ordered, so are the numbers of customers in the systems. The concept of directionally convex functions (Shaked and Shanthikumar [15]) and the notion of multivariate stochastic convexity (Chang, Chao, Pinedo, and Shanthikumar [4]) are employed in our analysis.•Supported in part by NSF grant DDM-9113008.

No abstract

No abstract