Convexity and Complete Monotonicity in Queueing Distributions and Associated Limit Behavior
Congestion theory is often called upon for insight into the behavior of a random system for which governing distribution forms and parameters are unavailable, and the fidelity of any simple mathematical model subject to question. Indeed, most applied problems have this character. To be helpful, the guidance offered should be simple and substantially insensitive to missing structural detail, i.e., should be “robust”. Such robustness may be hoped for when the system behavior of interest is in the limit-theoretic domain. e.g., is associated with the law of large numbers, or the central limit theorem, or the limit theorem for rare events. When limit theorems are of interest, one must know how valid the limit theorem is and how close one is to normality or exponentiality. Inequalities and bounds obtained from underlying structural features of the system may be needed. For such objectives, an awareness of convexity present and exploitation of this convexity is often useful.
Unable to display preview. Download preview PDF.
- 1.Barlow, R. E., and A. W. Marshall. Bounds on interval probabilities for restricted families of distributions. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics & Probability, University of California Press, Berkeley, California, 1970.Google Scholar
- 4.Callaert, H., and J. Keilson. On exponential ergodicity and spectral structure for birth-death processes. Stochastic Processes and Their Applications 1 (1973) Nos. 2 and 3, pages 187216, 217–236.Google Scholar
- 11.Keilson, J., Sojourn time and exit time structure for processes reversible in time; an application to reliability theory. Center for System Science Report 73–06. University of Rochester, Rochester, New York, 1973.Google Scholar
- 18.Keilson, J., and F. W. Steutel. Mixtures of distributions, moment inequalities and measures of exponentiality and normality. Ann. Math. Stat. (to appear).Google Scholar
- 23.Widder, D. V. The Laplace Transform. Princeton University Press, Princeton, New Jersey, 1946.Google Scholar