Convexity and Complete Monotonicity in Queueing Distributions and Associated Limit Behavior

Abstract

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.

This research was supported in part by the Office of Naval Research under Contract N00014-68-A-0091 and Contract No. N00014-67-A-0398-0014.