Modeling Event Occurrences Using the Borel-Tanner Distribution: Applications and Numerical Analysis

The Borel-Tanner distribution has been one of the powerful means in modelling those phenomena, in which events have taken place according to some structured process. Branching processes and theory of queueing had given birth to this distribution, which quickly found several applications in biology, telecommunications, and industrial reliability. In the present work, we consider the Borel-Tanner distribution for modeling probability, given certain initial input of the number of events, studying in detail the applicability for the representation of complex patterns of dependent events. It is very effective in discrete process modeling where every event depends on the occurrence of the previous one; hence, it finds its application in systems analysis with feedback mechanisms or chains of dependent events.\r\nThe introduction to the Borel-Tanner distribution has been carried out in an orderly, theoretical manner. Key properties and application of the Borel-Tanner distribution have been discussed at length. We use three numerical examples in order to show the practical utility of this distribution in real-world situations regarding biological population growth, telecommunication network congestion, and system failure regarding industrial processes. Each example is constructed for pointing out different aspects of the distribution, such as how it can handle different types of input values and how sensitive it is to changes in parameters. Results indicate that the Borel-Tanner distribution indeed provides an appropriate and illustrative framework in modeling a complex system. We also present discussions on the implications of these results and suggest ways in which these results can be extended to other fields. In general, this paper contributes to the understanding and application of the distribution by Borel-Tanner in different practical situations. \r\n