Convexity of a set of stochastically ordered random variables

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Abstract
It is shown that a set of random variables with increasing and convex (concave) survival functions are stochastically increasing and convex (concave) in the sample path sense. This stochastic convexity (concavity) result is then used to establish convexity (concavity) results for (i) a single-server queueing system with a time-out control policy, (ii) residual life, (iii) stationary renewal excess life and (iv)M/G/1 queues. These results are new and could not be derived without the direct or indirect aid of the above stochastic convexity (concavity) result. Furthermore, we illustrate that the above stochastic convexity (concavity) result can be applied to obtain new bounds for queueing systems. Specifically, letbe the waiting time of thenth customer in aGI/G/1 queue with inter-arrival time survival functionand service time survival function. Using the above convexity result it is shown that ifand for somesuch thatthenfor all increasing convex functionsφ, whenever the expectations exist. A similar result forandis also obtained. Other examples are also included.

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