Multimodularity, Convexity and Optimization Properties

Altman, Eitan ; Gaujal, Bruno ; Hordijk, Arie

HAL, inria-00113337 / Harvested from HAL

Résumé

We investigate in this paper the properties of multimodular functions. In doing so we give elementary proofs for properties already established by Hajek, and we generalize some of his results. In particular, we extend the relation bewteen convexity and multimodularity to some convex subsets of the integers numbers. We also obtain general optimizatnio results for average costs related to a sequence of multimodular functions rather than to a single functions. Under this general context, we show that the expected average cost problem is optimized by ising balanced sequences. We finally illustrate the usefulness of this theory in admission control into a D/D/1 queue with fixed bathch arrivals, with no state information. We show that the balanced policy minimizes the average queue length for the case of an intinite queeu, but not for the case of a finite queue. When further adding a constraint on the losses, it si show that a balanced policy is also optimal for the finite queue case.

Détail

Publié le : 2000-07-05
Classification:  multimodularité,  admission control into a queue,  convexité,  contrôle d'admission,  suites équilibrées,  multimodular functions,  convexity,  balanced sequences,  ACM: G.: Mathematics of Computing/G.3: PROBABILITY AND STATISTICS,  ACM: C.: Computer Systems Organization/C.2: COMPUTER-COMMUNICATION NETWORKS,  ACM: C.: Computer Systems Organization/C.4: PERFORMANCE OF SYSTEMS,  [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC],  [MATH.MATH-PR]Mathematics [math]/Probability [math.PR],  [INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI]