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A NOTE ON THE INCREASING DIRECTIONALLY CONCAVE MONOTONICITY IN QUEUES

Published online by Cambridge University Press:  01 January 2005

Tomasz Rolski
Tomasz Rolski
Affiliation:
Mathematical Institute, University of Wrocław, 50-384 Wrocław, Poland, E-mail: [email protected]
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Abstract

In this article, we study comparison theorems for stochastic functionals like V(∞;C) = sup0≤t {M(t) − C(t)} or V(T;C) = sup0≤t≤T {M(t) − C(t)}, where {M(t)} and {C(t)} are two independent nondecreasing processes with stationary increments. We will tacitly assume that the considered stochastic functionals are proper random variables. We prove that V(T;C′) ≤icxV(T;C) ≤icxV(T;C′′), where and C′′(dt) = c(0) dt, provided dC(t) is absolute continuous with density c(t). Similarly, we show that V(∞;C′) ≤icxV(∞;C) ≤icxV(∞;C′′). For proofs, we develop the theory of the ≤idcv ordering defined by increasing directionally concave functions. We apply the theory to M/G/1 priority queues and M/G/1 queues with positive and negative customers.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 19 , Issue 1 , January 2005 , pp. 33 - 43
Copyright
© 2005 Cambridge University Press

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