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.