Several notions of stochastic convexity and concavity and their properties are described in this survey. The notion of sample path stochastic convexity is a refinement of the well used notion of stochastic ordering, and it can be used to construct, on a common probability space, random variables which have desirable convexity (or concavity) properties with probability one. Three open problems from the literature are described. These problems could not be resolved until the introduction of the stochastic convexity notions which are described in this survey. The solutions of these problems illustrate the strength and the usefulness of these notions. Each notion is accompanied by a description of some of its applications. References for more detailed study of these notions are given. Indications of further work in this area are included.1. Introduction. Regular, Sample Path and Strong Stochastic Convexity notions are very valuable in many areas in probability and statistics such as queueing and reliability theory. Consider, for example, the following three open problems: PROBLEM 1. Consider a single stage queueing system at which customers arrive according to a doubly stochastic Poisson (DSP) process. The stochastic intensity of the DSP is a Markov process on {λχ,λ2} (λ t > 0,i = 1,2). The expected time this Markov process spends in state λ t is 0r t ,i =1,2 for some r t > 0, i = 1,2, and θ > 0. The service times of the customers are independent and identically distributed random variables. Let EW(Θ) be the average work load in this DSP/G/1 queueing system. CONJECTURE 1. (Ross 1978). EW(Θ) is a decreasing function of θ.

A (univariate) random variable is said to be of phase type if it can be represented as the time until absorption in a finite state absorbing Markov chain. Univariate phase type random variables are useful because they arise from processes that are often encountered in applications, they have densities that can be written in a closed form, they possess some useful closure properties, and they can approximate any nonnegative random variable. This paper introduces and discusses several extensions to the multivariate case. It shows that the multivariate random variables possess many of the properties of univariate phase type distributions and derives explicit formulas for various probabilistic quantities of interest. Some examples are included.

The supermodular and the symmetric supermodular stochastic orders have been cursorily studied in previous literature. In this paper we study these orders more thoroughly. First we obtain some basic properties of these orders. We then apply these results in order to obtain comparisons of random vectors with common values, but with different levels of multiplicity. Specifically, we show that if the vectors of the levels of multiplicity are ordered in the majorization order, then the associated random vectors are ordered in the symmetric supermodular stochastic order. In the non-symmetric case we obtain bounds (in the supermodular stochastic order sense) on such random vectors. Finally, we apply the results to problems of optimal assembly of reliability systems, of optimal allocation of minimal repair efforts, and of optimal allocation of reliability items. Academic Press

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Abstract. In this paper, new classes of stochastic order relations are introduced. These can be seen as extensions of the usual convex order and are closely related to the orderings discussed in Lefèvre and Utev (1996), as well as to the stochastic dominances in economics and stop-loss orders in actuarial sciences. These classes are studied in detail, including properties, characterizations, sufficient conditions, and extrema with respect to these orderings in different sets of distribution functions. Some applications illustrate the theory.Mathematics subject classification (1991): 60E15.

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Sample path convexity and concavity, Markov processes, directional convexity and concavity, single stage queues, supermodular and submodular functions, L-superadditive functions, reliability theory, branching processes, shock models, total positivity,