Geometric Brownian Motion and Cox–Ingersoll–Ross Models
This chapter discusses the use of geometric Brownian motion (GBM) and Cox–Ingersoll–Ross (CIR) models in jump diffusion to explain complex queuing system behaviors. It focuses on hypothetical modeling and practical analysis as opposed to tedious empirical polls. The chapter addresses single and networked queueing systems in which the arrival rates or service capacity can be discussed in the form of a jump-diffusion of the GBM/CIR, exhibits transient and diffusion limits and provides numerical techniques to estimate the parameters and performance. Stochastic models such as GBM and CIR integrated into graph-based queueing theory benefit from fuzzy optimization and automated hyperparameter search to match heavy-tailed network behaviors. The chapter highlights the need to incorporate the latest stochastic processes into the analytics of queueing theory to increase the efficiency of performance measurement, risk identification and system design in highly dynamic and chaotic environments.