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Convexity of a set of stochastically ordered random variables
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- Journal:
- Advances in Applied Probability / Volume 22 / Issue 1 / March 1990
- Published online by Cambridge University Press:
- 01 July 2016, pp. 160-177
- Print publication:
- March 1990
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- Article
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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, let
be the waiting time of the nth customer in a GI/G/1 queue with inter-arrival time survival function 
and service time survival function 
. Using the above convexity result it is shown that if 
and for some 
such that 
then 
for all increasing convex functions φ, whenever the expectations exist. A similar result for 
and 
is also obtained. Other examples are also included.
be the waiting time of the 
and service time survival function 
. Using the above convexity result it is shown that if 
and for some 
such that 
then 
for all increasing convex functions 
and 
is also obtained. Other examples are also included.