Generalizations of Palm's Theorem and Dyna-METRIC's Demand and Pipeline Variability
Palm's theorem is a useful tool in modeling inventory problems in logistics models such as METRIC and Mod-METRIC. However, to fit its limited domain of applicability, time-dependent customer arrival rates have been approximated by the required constant rates, which results in a possible loss of accuracy. This report reviews the extension of Palm's theorem for time-dependent arrival rates and service distributions under Poisson input, and it provides further extensions to compound Poisson input. The extensions make it possible to model processes in which the number either of demands or of customers in service has variance-to-mean ratios greater than or equal to unity. The report introduces the nonhomogeneous Poisson queue with infinite servers, and develops a generalization of Palm's Theorem to nonhomogeneous Poisson input. The author shows that comparable results hold for compound Poisson input. He then relates these results to two-echelon repair systems. Finally, he covers various initial conditions of the queueing system.