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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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References

1.Baccelli, F. & Makowski, A.M. (1986). Stability and bounds for single server queues in random environment. Stochastic Models 2: 281–291.CrossRefGoogle Scholar
2.Bremaud, P. (1981). Point processes and queues. Heidelberg: Springer.CrossRefGoogle Scholar
3.Chang, C.-S., Chao, X.L., & Pinedo, M. (1991). Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture. Advances in Applied Probability 23: 210–228.Google Scholar
4.Chang, C.-S., Chao, X.L., Pinedo, M., & Shanthikumar, J.G. (1991). Stochastic convexity for multidimensional processes and its applications. IEEE Transactions on Automated Control 36: 1347–1355.CrossRefGoogle Scholar
5.Meester, L.E. (1990). Contributions to the theory and applications of stochastic convexity. Ph.D. thesis, Department of Statistics, University of California, Berkeley.Google Scholar
6.Meester, L.E. & Shanthikumar, J.G. (1990). Stochastic convexity on general space. Technical Report, University of California, Berkeley.Google Scholar
7.Miyazawa, M. & Shanthikumar, J.G. (1990). Poisson process is better than geometric branching Poisson process. Technical Report, University of California, Berkeley.Google Scholar
8.Rolski, T. (1981). Queues with non-stationary input streams: Ross's conjecture. Advances in Applied Probability 13: 603–618.CrossRefGoogle Scholar
9.Rolski, T. (1986). Upper bounds for single server queues with doubly stochastic Poisson arrivals. Mathematics of Operations Research 11: 442–450.CrossRefGoogle Scholar
10.Ross, S.M. (1978). Average delay in queues with non-stationary Poisson arrivals. Journal of Applied Probability 15: 602–609.CrossRefGoogle Scholar
11.Ross, S.M. (1983). Stochastic processes. New York: John Wiley.Google Scholar
12.Rüschendorf, L. (1983). Solution of a statistical optimization problem by rearrangement methods. Metrika 30: 55–61.CrossRefGoogle Scholar
13.Shaked, M. & Shanthikumar, J.G. (1988). Stochastic convexity and its applications. Advances in Applied Probability 20: 427–446.CrossRefGoogle Scholar
14.Shaked, M. & Shanthikumar, J.G. (1988). Temporal stochastic convexity and concavity. Stochastic Processes and Their Applications 27: 1–20.CrossRefGoogle Scholar
15.Shaked, M. & Shanthikumar, J.G. (1990). Parametric stochastic convexity and concavity of stochastic processes. Annals of the Institute of Statistical Mathematics 42: 509–531.Google Scholar
16.Shanthikumar, J.G. & Yao, D.D. (1992). Spatiotemporal convexity of stochastic processes and applications. Probability in the Engineering and Informational Sciences 6: 1–16.CrossRefGoogle Scholar
17.Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York: John Wiley.Google Scholar
18.Tchen, A.H. (1980). Inequalities for distributions with given marginals. Annals of Probability 8: 814–827.CrossRefGoogle Scholar