Sur l'Utilisation des Integrales de Contour dans les Problemes de Stocks et de Delais d'Attente

The goal of the article is to show how sometimes the use of contour integrals can help in solving problems of inventory and queueing theories by enabling the target function to be uniquely represented in the analytical form min(0, X). The use of those integrals allows us to rewrite some well-known formulas from queueing theory in a larger setting that enables generalizations. That is our goal in the first part of the paper. In the second part, this idea is applied to a particular problem to determine the inventory that is changing ten months in a year by regular reduction for use in production and by supplies arriving by ships, randomly distributed following the Poisson distribution. Penalties imposed are proportional to the waiting time for a ship to arrive. The mathematical problem that enables to determine the optimum between the costs of waiting for a ship to arrive and the costs of maintaining the inventory is expressed as the residue of a contour integral. An asymptotic formula useful for numerical calculation is derived.