Some achievements from a statistical queuing theory research

Abstract

It is intended in this paper, that is the corrected and enlarged version of Ferreira (2016) presented at APLIMAT 2016, to give an overview on some achievements, considered the most important, of a research on queues through a long time period. Infinite servers’ queues are mainly considered. A theory of queues overview, single node and in network is presented. And laying on it some important new theoretical results are highlighted. Also practical applications, the most of them uncommon but some very well known, using mainly infinite servers nodes are outlined. In the theoretical results stand out: - An algorithm to compute the sojourn time probability distribution function of a customer in a network which nodes are M/G/? queues, using traffic equations and Laplace transform, - A collection of service times distributions for which the M/G/? queue busy period length probability distribution has a particularly simple structure, very rare in the busy period lengths domain. The practical applications considered in this work, mainly using the M/G/? queue ability for the study of large populations processes, are in: - Logistics, Financial problems, Energy problems, Disease problems, Unemployment situations, Compartment models (Biology, Birth-sickness-death Processes, Hierarchical Systems) and Repairs Shop. Some historical important parameters in queues domain are named and emphasized its role in the applications. Also some problems, still open problems, as the conditions for the existence of product form equilibrium distribution, are touched upon. In the end it is given a considerable list of references on these subjects.