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Regularity of Stochastic Processes: A Theory Based on Directional Convexity

Published online by Cambridge University Press:  27 July 2009

Ludolf E. Meester  and
J. George Shanthikumar
Ludolf E. Meester
Affiliation:
Department of Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands
J. George Shanthikumar
Affiliation:
School of Business Administration, University of California, Berkeley, California 94720
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Abstract

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. Further-more, 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/c1→/M/c2→…→/M/cm 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.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 3 , July 1993 , pp. 343 - 360
Copyright
Copyright © Cambridge University Press 1993

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